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Research ArticleOptimal (Control of) Intervention Strategies for MalariaEpidemic in Karonga District Malawi
Peter M Mwamtobe12 Shirley Abelman1 J Michel Tchuenche3 and Ansley Kasambara2
1 School of Computational and Applied Mathematics University of Witwatersrand Private Bag 3 WitsJohannesburg 2050 South Africa
2Department of Mathematics and Statistics University of Malawi The Malawi Polytechnic Private Bag 303 ChichiriBlantyre 3 Malawi
3 3253 Flowers Road South Atlanta GA 30341 USA
Correspondence should be addressed to Peter M Mwamtobe pmwamtobegmailcom and J Michel Tchuenchejmtchuenchegmailcom
Received 18 February 2014 Revised 22 April 2014 Accepted 29 April 2014 Published 19 June 2014
Academic Editor Mariano Torrisi
Copyright copy 2014 Peter M Mwamtobe et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Malaria is a public health problem for more than 2 billion people globally About 219 million cases of malaria occur worldwideand 660000 people die mostly (91) in the African Region despite decades of efforts to control the disease Although the diseaseis preventable it is life-threatening and parasitically transmitted by the bite of the female Anopheles mosquito A deterministicmathematical model with intervention strategies is developed in order to investigate the effectiveness and optimal control strategiesof indoor residual spraying (IRS) insecticide treated nets (ITNs) and treatment on the transmission dynamics ofmalaria inKarongaDistrict Malawi The effective reproduction number is analytically computed and the existence and stability conditions of theequilibria are explored The model does not exhibit backward bifurcation Pontryaginrsquos Maximum Principle which uses both theLagrangian and Hamiltonian principles with respect to a time dependent constant is used to derive the necessary conditions forthe optimal control of the disease Numerical simulations indicate that the prevention strategies lead to the reduction of both themosquito population and infected human individuals Effective treatment consolidates the prevention strategiesThus malaria canbe eradicated in Karonga District by concurrently applying vector control via ITNs and IRS complemented with timely treatmentof infected people
1 Introduction
Malaria is a vector-borne infectious disease found mainlyin tropical regions (Sub-Saharan Africa Central and SouthAmerica the Indian subcontinent Southeast Asia and thePacific islands) [1] It is a life-threatening disease transmittedthrough the bites of infected mosquitoes [2] There are fourdifferent types of Plasmodium parasites Plasmodium falci-parum (the only parasite which causes malignant malaria)Plasmodium vivax (causes benign malaria with less severesymptoms the vector can remain in the liver for up to threeyears and can lead to a relapse) Plasmodium malariae (alsocauses benignmalaria and is relatively rare) and Plasmodiumovale (causes benign malaria and can remain in the blood
and liver for many years without causing symptoms) Thisstudy focuses mainly on malignant malaria Severe malariacan affect the patientrsquos brain and central nervous system andcan be fatal [2 3] FurthermoreMedicinenet [4] has reportedanother relatively new species Plasmodium knowlesi whichhas been causing malaria in Malaysia and areas of SoutheastAsia It is also a dangerous species that is typically foundonly in long-tailed and pigtail macaque monkeys Like Pfalciparum P knowlesi may be deadly to anyone infected[5]
Beyond the human toll malaria wreaks significant eco-nomic havoc in endemic regions decreasing gross domesticproduct (GDP) by as much as 13 in countries with highlevels of transmissionmdashthe disease accounts for up to 40
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 594256 20 pageshttpdxdoiorg1011552014594256
2 Abstract and Applied Analysis
of public health expenditures 30ndash50 of in-patient hospitaladmissions and up to 60 of out-patient health clinic visits[2] The Government of Malawi has put in place severalcontrol strategies through the National Malaria ControlProgramme to reduce and possibly (ultimately) eliminatemalaria The main strategic areas that have been identifiedfor scaling-up of malaria control activities include malariacase management intermittent preventive treatment (IPT) ofpregnantwomenusing sulfadoxine-pyrimethamine (SP) andmalaria prevention with special emphasis on the use of ITNsas well as IRS [8]
Various compartmental models for the spread of malariahave been proposed [10ndash12] Extensive review of mathemat-ical models of malaria dynamics using 119878119864119868119877119878-type modelsand their variants can be found in Chiyaka et al [13 14]For information on the unexpected stability of malaria elimi-nation and eradication and additional literature on malariadynamics see Smith et al [15] who also showed that ifmosquito birth rate is increased by 25 then the minimaleffective spraying period is reduced by half and if doubledthe period is reduced by three-quarters Our goal is to assessthe role of optimal control of preventive measures namelyITNs and IRS as well as treatment of the transmission dynam-ics of malaria in Karonga District Malawi without actuallytargeting a certain high-risk group Despite a plethora ofstudies on the dynamics of malaria and its control (seeChiyaka et al [14] and Smith et al [15]) to the best of ourknowledge the proposed model with exposed immigrants isseemingly new the use of optimal values of a combinationof three intervention strategies namely indoor residualspraying (IRS) insecticide treated nets (ITNs) and treatmentto investigate the effectiveness and optimal control strategiesin the transmission dynamics of malaria in a locality in aresource constrained setting
2 Model Formulation and Analysis
We formulate an optimal control model for malaria with thepopulation under study being subdivided into compartmentsaccording to individualsrsquo disease status We consider thetotal population sizes denoted by 119873ℎ(119905) and 119873V(119905) for thehuman hosts and Anopheles female mosquitoes respectivelyWe employ the 119878119864119868119877119878 framework to describe a disease withtemporary immunity on recovery from infection The 119878119864119868119877119878model indicates that the passage of individuals is from thesusceptible class 119878ℎ to the exposed class 119864ℎ then to theinfectious class 119868ℎ and finally to the recovery class 119877ℎ 119878ℎ(119905)represents the number of individuals not yet infectedwith themalaria parasite at time 119905 The latent or exposed class 119864ℎ(119905)represents individuals who are infected but not yet infectiousIndividuals in the 119868ℎ(119905) class are infected with malariaand are capable of transmitting the disease to susceptiblemosquitoes 119877ℎ(119905) represents the class of individuals whohave temporarily recovered from the disease The susceptiblehuman population is increased by recruitment (birth) ata constant rate Λ ℎ while others are generated throughmigration by (1 minus 1205811)120579119873ℎ where 1205811 is the proportion ofinfected immigrants into the exposed class and 120579 is the rate
at which people migrate into Karonga District All the re-cruited individuals are assumed to be naive when joining thecommunity
There is some finite probability 120573Vℎ that the parasites(in the form of sporozoites) will be passed onto the humanswhen an infectious female Anopheles mosquito bites a sus-ceptible human The parasite then moves to the liver whereit develops into its next life stage merozoites Using theapproach adopted in [6] susceptible individuals acquiremalaria through contact with infectious mosquitoes at therate 120599 The infected person moves to the exposed class atthe rate (1 minus 1199061)120582ℎ119868V119878ℎ The preventive variable 1199061(119905) isin
[0 1] represents the use of ITNs as a means of minimizingor eliminating mosquito-human contacts After a certainperiod of time the parasite (in the form of merozoites)enters the blood stream usually signaling the clinical onsetof malaria Then the exposed individuals become infectiousand progress to the infected state at a constant rate 120572ℎ Theindividuals who have experienced infectionmay recover withtemporary immunity at a constant rate 120588 and move to therecovery class while some infectious humans after recoverywithout immunity become immediately susceptible again atthe rate (1minus120588) Infectious individuals recover due to treatmentat a rate 1205781199062 with 1199062(119905) isin [0 1] representing the controleffort on treatment and 120601 being the proportion of indi-viduals who recover spontaneously Recovered individualslose immunity at a rate 120595 The natural and disease induceddeath rates are 120583ℎ and 120575ℎ respectively The disease induceddeath rate is very small in comparison with the recoveryrate
The mosquito population 119873V is divided into three com-partments susceptible 119878V(119905) exposed 119864V(119905) and infectious119868V(119905) FemaleAnophelesmosquitoes enter the susceptible classthrough birth at a rate Λ V The parasites in the form ofgametocytes enter the mosquito population with probability120573ℎV This happens when the mosquito bites an infectioushuman and the mosquito moves from the susceptible tothe exposed class Mosquitoes are assumed to suffer deathdue to natural causes at a rate 120583V The exposed mosquitoesprogress to the class of symptomatic mosquitoes 119868V at a rate120572V It is assumed that the disease does not induce deathto the mosquito population Finally the mortality rate ofthe mosquito population increases at a rate proportional to1205911199063(119905) where 120591 gt 0 is a rate constant when using IRS and1199063(119905) isin [0 1] is a constant rate of IRS Figure 1 represents aschematic flow diagram of the proposed model
The model state variables are represented in Table 1Table 2 represents prevention and control strategies practisedin the district Table 3 shows parameters of the model
The state variables in Table 1 the prevention and controlparameters in Table 2 and the model parameters in Table 3for the malaria model satisfy (1) It is assumed that allstate variables and parameters of the model which moni-tors human and mosquito populations are positive for all119905 ge 0 We will therefore analyse the model in a suitableregion
The assumptions lead to the following deterministicsystem of nonlinear ordinary differential equations which
Abstract and Applied Analysis 3
Sh
S
E
Rh
Ih
I
Eh
120583hEh
120572hEh
Λh
120595Rh120583hSh
120583hIh 120575hIh
120583hRh
Λ
120583I
120583E
120583S
120572E 120582S
120591u3I
120591u3E
120591u3S
(1 minus u1)120582hSh
1( (120601 + 120578u2) minus 120588)Ih
(120601 + 120578u2)120588Ih
1205811120579Nh
(1 minus k1)120579Nh
Figure 1 The malaria model with interventions flowchart
describe the evolutionary dynamics of a malaria model witha combination of interventions
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ119878ℎ + 1205811120579119873ℎ minus 120572ℎ119864ℎ minus 120583ℎ119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062) (1 minus 120588) 119868ℎ minus (120601 + 1205781199062) 120588119868ℎ
minus (120583ℎ + 120575ℎ) 119868ℎ
119889119877ℎ
119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ
119889119878V
119889119905= Λ V minus 120582V119878V minus (120583V + 1205911199063) 119878V
119889119864V
119889119905= 120582V119878V minus 120572V119864V minus (120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(1)
where 120582ℎ = 120573Vℎ120599119868V119873ℎ 120582119881 = 120573ℎV120599119868ℎ119873ℎThe term 120573Vℎ120599119878ℎ119868V119873ℎ denotes the rate at which the
human hosts 119878ℎ become infected by infectious mosquitoes 119868Vand 120573ℎV120599119878V119868ℎ119873ℎ refers to the rate at which the susceptiblemosquitoes 119878V are infected by the infectious human hosts 119868ℎ
Adding the first six equations of model (1) and assumingthat there is no disease induced death that is 120575ℎ = 0 gives119889119873ℎ119889119905 = Λ ℎ minus 120583ℎ119873ℎ so that 119873ℎ(119905) rarr Λ ℎ120583ℎ as 119905 rarr infin
[16] Thus Λ ℎ120583ℎ is an upper bound of 119873ℎ(119905) provided that119873ℎ(0) le Λ120583ℎ Further if 119873ℎ(0) gt Λ ℎ120583ℎ then 119873ℎ(119905)
will decrease to this level Λ ℎ120583ℎ Similar calculation for thevector equations shows that119873V rarr Λ V120583V as 119905 rarr infin
Table 1 State variables of the malaria model
Symbol Description119878ℎ(119905) Number of susceptible individuals at time t119864ℎ(119905) Number of exposed individuals at time t119868ℎ(119905) Number of infectious humans at time 119905119877ℎ(119905) Number of recovered humans at time 119905119878V(119905) Number of susceptible mosquitoes at time 119905119864V(119905) Number of infected mosquitoes at time 119905119868V(119905) Number of infectious mosquitoes at time 119905119873ℎ(119905) Total number of individuals at time 119905119873V(119905) Total mosquito population at time 119905
Table 2 Prevention and control variables in the model
Symbol Description
1199061(119905)Preventive measure using insecticide treated bed nets(ITNs)
1199062(119905)The control effort on treatment of infectiousindividuals
1199063(119905)Preventing measure using indoor residual spraying(IRS)
120591 Rate constant due to use of indoor residual spraying120578 Rate constant due to use of treatment effort
Table 3 Parameters variables of the malaria model
Symbol DescriptionΛ ℎ Recruitment rate of individuals by birth1205811
Proportion of exposed immigrants into exposed class120579 Proportion of people migrating into Karonga District120583ℎ Per capita natural death rate of humans120583V Per capita natural death rate of mosquitoes120575ℎ
Per capita disease induced death rate for humans
120572ℎ
Progression rate of humans from exposed state to theinfectious state
120573VℎProbability that a bite results in transmission ofinfection to human
120573ℎV
Probability that a bite results in transmission of theparasite from an infectious human to the susceptiblemosquitoes
120599 Biting rate of mosquito
120582ℎForce of infection for susceptible humans to exposedindividuals
120582VForce of infection for susceptible mosquitoes toexposed mosquito class
120588 Per capita rate of recovery with temporary immunity120601 Proportion of spontaneous individual recovery120595 Per capita rate of loss immunity
120572VProgression of exposed mosquitoes into infectedmosquitoes
4 Abstract and Applied Analysis
The feasible region
Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7
+119873ℎ le
Λ ℎ
120583ℎ
= 119873lowast
ℎ
119873V leΛ V
120583V= 119873lowast
V
(2)
is positive-invariant and attracting
Lemma 1 The region Φ isin R7+is positively invariant for the
model system (1) with initial conditions in R7+
Proof Let 119905 = sup119905 gt 0 119878ℎ gt 0 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt0 119864V gt 0 119868V gt 0 isin [0 119905] give 119905 gt 0 The first equation ofmodel (1) gives
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
ge Λ ℎ minus ((1 minus 1199061) 120582ℎ + 120583ℎ) 119878ℎ
(3)
which can be rewritten as
119889
119889119905[119878ℎ (119905) 119890
int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905]
ge Λ ℎ119890int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905
(4)
Therefore
119878ℎ (119905) 119890int119905
0(1minus119906
1)120582ℎ(119904)119889119904+120583
ℎ119905minus 119878ℎ (0)
ge int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+119906
ℎ119906119889119906
(5)
so that
119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
+ 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
times int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+120583
ℎ119906119889119906 gt 0
(6)
Similarly it can be shown that 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt 0119864V gt 0 and 119868V gt 0 for all 119905 gt 0This completes the proof
21 Existence and Stability of Equilibrium Points We analysesystem (1) to obtain the equilibrium points of the system andtheir stability Let (119878lowast
ℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) be the equilib-rium points of system (1) At an equilibrium point we have
1198781015840
ℎ(119905) = 119864
1015840
ℎ(119905) = 119868
1015840
ℎ(119905) = 119877
1015840
ℎ(119905) = 119878
1015840
V (119905) = 1198641015840
V (119905) = 1198681015840
V (119905) = 0
(7)
211 Disease-Free Equilibrium 1198640 In the absence of malariathat is 119864lowast
ℎ= 119868lowast
ℎ= 119877lowast
ℎ= 119864lowast
V = 119868lowast
V = 0 model system (1) hasan equilibrium point called the disease-free equilibrium 1198640and is given by
1198640 = (Λ ℎ
120583ℎ
0 0 0Λ ℎ
120583V 0 0) (8)
To establish the linear stability of 1198640 we employ van denDriessche and Watmoughrsquos next generation matrix approach[17] A reproduction number obtained this way determinesthe local stability of the disease-free equilibrium point for119877119890 lt 1 and instability for119877119890 gt 1 Following van denDriesscheand Watmough [17] the associated next generation matrices119865 and 119881 of system (1) can be determined from F119894 and V119894respectively where
F119894 =
[[[[[[[[[
[
(1 minus 1199061) 120573Vℎ120599119868V119878ℎ
119873ℎ
+ 1205811120579119873ℎ
0
120573ℎV120599119868ℎ119878V
119873ℎ
0
]]]]]]]]]
]
V119894 =[[[[
[
(120572ℎ + 120583ℎ) 119864ℎ
(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ
(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V
]]]]
]
(9)
The Jacobian matrix ofF119894 andV119894 is given by
119865 =[[[
[
0 0 0 1199031
0 0 0 0
0 119895 0 0
0 0 0 0
]]]
]
119881 =[[[
[
119898 0 0 0
minus120572ℎ 119899 0 0
0 0 119886 0
0 0 minus120572V b
]]]
]
(10)
where
1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ
Λ ℎ120583V
119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ
119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063
(11)
Algebraic manipulation of the matrices leads to the effectivereproduction number
R119890 = (120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061) Λ V120583ℎ
times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)
times(120572V + 120583V + 1205911199063)(120583V + 1205911199063)120583VΛ ℎ)minus1
)
12
(12)
where
R119890V =120573ℎV120572V120599Λ V
120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)
Abstract and Applied Analysis 5
is the contribution of themosquito populationwhen it infectsthe humans and
R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)
Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)
is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number
R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms
(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector
(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host
The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is
R119890119905 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(15)
Also if ITNs are the only intervention strategy then
R119890119873 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(16)
Similarly if IRS is the only means of protection then
R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V
times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)
times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12
(17)
The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)
R0 = radic120573ℎV120573119906ℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(18)
In general it is easy to prove that
R119890 le R0 (19)
for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds
Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1
212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system
119889119904
119889119905=
1
119873ℎ
[119889119878ℎ
119889119905minus 119904
119889119873ℎ
119889119905]
=1
119873ℎ
[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]
(20)
Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then
119889119904
119889119905=
1
119873ℎ
[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]
minus119904
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
=Λ ℎ
119873ℎ
minus [Λ ℎ
119873ℎ
+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579
+ (120601 + 1205781199062) (1 minus 120588) 119894
minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903
119889119890
119889119905=
1
119873ℎ
[119889119864
119889119905minus 119890
119889119873ℎ
119889119905]
=1
119873ℎ
[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]
minus119890
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
of public health expenditures 30ndash50 of in-patient hospitaladmissions and up to 60 of out-patient health clinic visits[2] The Government of Malawi has put in place severalcontrol strategies through the National Malaria ControlProgramme to reduce and possibly (ultimately) eliminatemalaria The main strategic areas that have been identifiedfor scaling-up of malaria control activities include malariacase management intermittent preventive treatment (IPT) ofpregnantwomenusing sulfadoxine-pyrimethamine (SP) andmalaria prevention with special emphasis on the use of ITNsas well as IRS [8]
Various compartmental models for the spread of malariahave been proposed [10ndash12] Extensive review of mathemat-ical models of malaria dynamics using 119878119864119868119877119878-type modelsand their variants can be found in Chiyaka et al [13 14]For information on the unexpected stability of malaria elimi-nation and eradication and additional literature on malariadynamics see Smith et al [15] who also showed that ifmosquito birth rate is increased by 25 then the minimaleffective spraying period is reduced by half and if doubledthe period is reduced by three-quarters Our goal is to assessthe role of optimal control of preventive measures namelyITNs and IRS as well as treatment of the transmission dynam-ics of malaria in Karonga District Malawi without actuallytargeting a certain high-risk group Despite a plethora ofstudies on the dynamics of malaria and its control (seeChiyaka et al [14] and Smith et al [15]) to the best of ourknowledge the proposed model with exposed immigrants isseemingly new the use of optimal values of a combinationof three intervention strategies namely indoor residualspraying (IRS) insecticide treated nets (ITNs) and treatmentto investigate the effectiveness and optimal control strategiesin the transmission dynamics of malaria in a locality in aresource constrained setting
2 Model Formulation and Analysis
We formulate an optimal control model for malaria with thepopulation under study being subdivided into compartmentsaccording to individualsrsquo disease status We consider thetotal population sizes denoted by 119873ℎ(119905) and 119873V(119905) for thehuman hosts and Anopheles female mosquitoes respectivelyWe employ the 119878119864119868119877119878 framework to describe a disease withtemporary immunity on recovery from infection The 119878119864119868119877119878model indicates that the passage of individuals is from thesusceptible class 119878ℎ to the exposed class 119864ℎ then to theinfectious class 119868ℎ and finally to the recovery class 119877ℎ 119878ℎ(119905)represents the number of individuals not yet infectedwith themalaria parasite at time 119905 The latent or exposed class 119864ℎ(119905)represents individuals who are infected but not yet infectiousIndividuals in the 119868ℎ(119905) class are infected with malariaand are capable of transmitting the disease to susceptiblemosquitoes 119877ℎ(119905) represents the class of individuals whohave temporarily recovered from the disease The susceptiblehuman population is increased by recruitment (birth) ata constant rate Λ ℎ while others are generated throughmigration by (1 minus 1205811)120579119873ℎ where 1205811 is the proportion ofinfected immigrants into the exposed class and 120579 is the rate
at which people migrate into Karonga District All the re-cruited individuals are assumed to be naive when joining thecommunity
There is some finite probability 120573Vℎ that the parasites(in the form of sporozoites) will be passed onto the humanswhen an infectious female Anopheles mosquito bites a sus-ceptible human The parasite then moves to the liver whereit develops into its next life stage merozoites Using theapproach adopted in [6] susceptible individuals acquiremalaria through contact with infectious mosquitoes at therate 120599 The infected person moves to the exposed class atthe rate (1 minus 1199061)120582ℎ119868V119878ℎ The preventive variable 1199061(119905) isin
[0 1] represents the use of ITNs as a means of minimizingor eliminating mosquito-human contacts After a certainperiod of time the parasite (in the form of merozoites)enters the blood stream usually signaling the clinical onsetof malaria Then the exposed individuals become infectiousand progress to the infected state at a constant rate 120572ℎ Theindividuals who have experienced infectionmay recover withtemporary immunity at a constant rate 120588 and move to therecovery class while some infectious humans after recoverywithout immunity become immediately susceptible again atthe rate (1minus120588) Infectious individuals recover due to treatmentat a rate 1205781199062 with 1199062(119905) isin [0 1] representing the controleffort on treatment and 120601 being the proportion of indi-viduals who recover spontaneously Recovered individualslose immunity at a rate 120595 The natural and disease induceddeath rates are 120583ℎ and 120575ℎ respectively The disease induceddeath rate is very small in comparison with the recoveryrate
The mosquito population 119873V is divided into three com-partments susceptible 119878V(119905) exposed 119864V(119905) and infectious119868V(119905) FemaleAnophelesmosquitoes enter the susceptible classthrough birth at a rate Λ V The parasites in the form ofgametocytes enter the mosquito population with probability120573ℎV This happens when the mosquito bites an infectioushuman and the mosquito moves from the susceptible tothe exposed class Mosquitoes are assumed to suffer deathdue to natural causes at a rate 120583V The exposed mosquitoesprogress to the class of symptomatic mosquitoes 119868V at a rate120572V It is assumed that the disease does not induce deathto the mosquito population Finally the mortality rate ofthe mosquito population increases at a rate proportional to1205911199063(119905) where 120591 gt 0 is a rate constant when using IRS and1199063(119905) isin [0 1] is a constant rate of IRS Figure 1 represents aschematic flow diagram of the proposed model
The model state variables are represented in Table 1Table 2 represents prevention and control strategies practisedin the district Table 3 shows parameters of the model
The state variables in Table 1 the prevention and controlparameters in Table 2 and the model parameters in Table 3for the malaria model satisfy (1) It is assumed that allstate variables and parameters of the model which moni-tors human and mosquito populations are positive for all119905 ge 0 We will therefore analyse the model in a suitableregion
The assumptions lead to the following deterministicsystem of nonlinear ordinary differential equations which
Abstract and Applied Analysis 3
Sh
S
E
Rh
Ih
I
Eh
120583hEh
120572hEh
Λh
120595Rh120583hSh
120583hIh 120575hIh
120583hRh
Λ
120583I
120583E
120583S
120572E 120582S
120591u3I
120591u3E
120591u3S
(1 minus u1)120582hSh
1( (120601 + 120578u2) minus 120588)Ih
(120601 + 120578u2)120588Ih
1205811120579Nh
(1 minus k1)120579Nh
Figure 1 The malaria model with interventions flowchart
describe the evolutionary dynamics of a malaria model witha combination of interventions
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ119878ℎ + 1205811120579119873ℎ minus 120572ℎ119864ℎ minus 120583ℎ119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062) (1 minus 120588) 119868ℎ minus (120601 + 1205781199062) 120588119868ℎ
minus (120583ℎ + 120575ℎ) 119868ℎ
119889119877ℎ
119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ
119889119878V
119889119905= Λ V minus 120582V119878V minus (120583V + 1205911199063) 119878V
119889119864V
119889119905= 120582V119878V minus 120572V119864V minus (120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(1)
where 120582ℎ = 120573Vℎ120599119868V119873ℎ 120582119881 = 120573ℎV120599119868ℎ119873ℎThe term 120573Vℎ120599119878ℎ119868V119873ℎ denotes the rate at which the
human hosts 119878ℎ become infected by infectious mosquitoes 119868Vand 120573ℎV120599119878V119868ℎ119873ℎ refers to the rate at which the susceptiblemosquitoes 119878V are infected by the infectious human hosts 119868ℎ
Adding the first six equations of model (1) and assumingthat there is no disease induced death that is 120575ℎ = 0 gives119889119873ℎ119889119905 = Λ ℎ minus 120583ℎ119873ℎ so that 119873ℎ(119905) rarr Λ ℎ120583ℎ as 119905 rarr infin
[16] Thus Λ ℎ120583ℎ is an upper bound of 119873ℎ(119905) provided that119873ℎ(0) le Λ120583ℎ Further if 119873ℎ(0) gt Λ ℎ120583ℎ then 119873ℎ(119905)
will decrease to this level Λ ℎ120583ℎ Similar calculation for thevector equations shows that119873V rarr Λ V120583V as 119905 rarr infin
Table 1 State variables of the malaria model
Symbol Description119878ℎ(119905) Number of susceptible individuals at time t119864ℎ(119905) Number of exposed individuals at time t119868ℎ(119905) Number of infectious humans at time 119905119877ℎ(119905) Number of recovered humans at time 119905119878V(119905) Number of susceptible mosquitoes at time 119905119864V(119905) Number of infected mosquitoes at time 119905119868V(119905) Number of infectious mosquitoes at time 119905119873ℎ(119905) Total number of individuals at time 119905119873V(119905) Total mosquito population at time 119905
Table 2 Prevention and control variables in the model
Symbol Description
1199061(119905)Preventive measure using insecticide treated bed nets(ITNs)
1199062(119905)The control effort on treatment of infectiousindividuals
1199063(119905)Preventing measure using indoor residual spraying(IRS)
120591 Rate constant due to use of indoor residual spraying120578 Rate constant due to use of treatment effort
Table 3 Parameters variables of the malaria model
Symbol DescriptionΛ ℎ Recruitment rate of individuals by birth1205811
Proportion of exposed immigrants into exposed class120579 Proportion of people migrating into Karonga District120583ℎ Per capita natural death rate of humans120583V Per capita natural death rate of mosquitoes120575ℎ
Per capita disease induced death rate for humans
120572ℎ
Progression rate of humans from exposed state to theinfectious state
120573VℎProbability that a bite results in transmission ofinfection to human
120573ℎV
Probability that a bite results in transmission of theparasite from an infectious human to the susceptiblemosquitoes
120599 Biting rate of mosquito
120582ℎForce of infection for susceptible humans to exposedindividuals
120582VForce of infection for susceptible mosquitoes toexposed mosquito class
120588 Per capita rate of recovery with temporary immunity120601 Proportion of spontaneous individual recovery120595 Per capita rate of loss immunity
120572VProgression of exposed mosquitoes into infectedmosquitoes
4 Abstract and Applied Analysis
The feasible region
Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7
+119873ℎ le
Λ ℎ
120583ℎ
= 119873lowast
ℎ
119873V leΛ V
120583V= 119873lowast
V
(2)
is positive-invariant and attracting
Lemma 1 The region Φ isin R7+is positively invariant for the
model system (1) with initial conditions in R7+
Proof Let 119905 = sup119905 gt 0 119878ℎ gt 0 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt0 119864V gt 0 119868V gt 0 isin [0 119905] give 119905 gt 0 The first equation ofmodel (1) gives
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
ge Λ ℎ minus ((1 minus 1199061) 120582ℎ + 120583ℎ) 119878ℎ
(3)
which can be rewritten as
119889
119889119905[119878ℎ (119905) 119890
int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905]
ge Λ ℎ119890int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905
(4)
Therefore
119878ℎ (119905) 119890int119905
0(1minus119906
1)120582ℎ(119904)119889119904+120583
ℎ119905minus 119878ℎ (0)
ge int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+119906
ℎ119906119889119906
(5)
so that
119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
+ 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
times int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+120583
ℎ119906119889119906 gt 0
(6)
Similarly it can be shown that 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt 0119864V gt 0 and 119868V gt 0 for all 119905 gt 0This completes the proof
21 Existence and Stability of Equilibrium Points We analysesystem (1) to obtain the equilibrium points of the system andtheir stability Let (119878lowast
ℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) be the equilib-rium points of system (1) At an equilibrium point we have
1198781015840
ℎ(119905) = 119864
1015840
ℎ(119905) = 119868
1015840
ℎ(119905) = 119877
1015840
ℎ(119905) = 119878
1015840
V (119905) = 1198641015840
V (119905) = 1198681015840
V (119905) = 0
(7)
211 Disease-Free Equilibrium 1198640 In the absence of malariathat is 119864lowast
ℎ= 119868lowast
ℎ= 119877lowast
ℎ= 119864lowast
V = 119868lowast
V = 0 model system (1) hasan equilibrium point called the disease-free equilibrium 1198640and is given by
1198640 = (Λ ℎ
120583ℎ
0 0 0Λ ℎ
120583V 0 0) (8)
To establish the linear stability of 1198640 we employ van denDriessche and Watmoughrsquos next generation matrix approach[17] A reproduction number obtained this way determinesthe local stability of the disease-free equilibrium point for119877119890 lt 1 and instability for119877119890 gt 1 Following van denDriesscheand Watmough [17] the associated next generation matrices119865 and 119881 of system (1) can be determined from F119894 and V119894respectively where
F119894 =
[[[[[[[[[
[
(1 minus 1199061) 120573Vℎ120599119868V119878ℎ
119873ℎ
+ 1205811120579119873ℎ
0
120573ℎV120599119868ℎ119878V
119873ℎ
0
]]]]]]]]]
]
V119894 =[[[[
[
(120572ℎ + 120583ℎ) 119864ℎ
(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ
(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V
]]]]
]
(9)
The Jacobian matrix ofF119894 andV119894 is given by
119865 =[[[
[
0 0 0 1199031
0 0 0 0
0 119895 0 0
0 0 0 0
]]]
]
119881 =[[[
[
119898 0 0 0
minus120572ℎ 119899 0 0
0 0 119886 0
0 0 minus120572V b
]]]
]
(10)
where
1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ
Λ ℎ120583V
119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ
119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063
(11)
Algebraic manipulation of the matrices leads to the effectivereproduction number
R119890 = (120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061) Λ V120583ℎ
times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)
times(120572V + 120583V + 1205911199063)(120583V + 1205911199063)120583VΛ ℎ)minus1
)
12
(12)
where
R119890V =120573ℎV120572V120599Λ V
120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)
Abstract and Applied Analysis 5
is the contribution of themosquito populationwhen it infectsthe humans and
R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)
Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)
is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number
R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms
(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector
(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host
The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is
R119890119905 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(15)
Also if ITNs are the only intervention strategy then
R119890119873 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(16)
Similarly if IRS is the only means of protection then
R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V
times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)
times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12
(17)
The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)
R0 = radic120573ℎV120573119906ℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(18)
In general it is easy to prove that
R119890 le R0 (19)
for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds
Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1
212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system
119889119904
119889119905=
1
119873ℎ
[119889119878ℎ
119889119905minus 119904
119889119873ℎ
119889119905]
=1
119873ℎ
[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]
(20)
Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then
119889119904
119889119905=
1
119873ℎ
[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]
minus119904
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
=Λ ℎ
119873ℎ
minus [Λ ℎ
119873ℎ
+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579
+ (120601 + 1205781199062) (1 minus 120588) 119894
minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903
119889119890
119889119905=
1
119873ℎ
[119889119864
119889119905minus 119890
119889119873ℎ
119889119905]
=1
119873ℎ
[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]
minus119890
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Sh
S
E
Rh
Ih
I
Eh
120583hEh
120572hEh
Λh
120595Rh120583hSh
120583hIh 120575hIh
120583hRh
Λ
120583I
120583E
120583S
120572E 120582S
120591u3I
120591u3E
120591u3S
(1 minus u1)120582hSh
1( (120601 + 120578u2) minus 120588)Ih
(120601 + 120578u2)120588Ih
1205811120579Nh
(1 minus k1)120579Nh
Figure 1 The malaria model with interventions flowchart
describe the evolutionary dynamics of a malaria model witha combination of interventions
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ119878ℎ + 1205811120579119873ℎ minus 120572ℎ119864ℎ minus 120583ℎ119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062) (1 minus 120588) 119868ℎ minus (120601 + 1205781199062) 120588119868ℎ
minus (120583ℎ + 120575ℎ) 119868ℎ
119889119877ℎ
119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ
119889119878V
119889119905= Λ V minus 120582V119878V minus (120583V + 1205911199063) 119878V
119889119864V
119889119905= 120582V119878V minus 120572V119864V minus (120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(1)
where 120582ℎ = 120573Vℎ120599119868V119873ℎ 120582119881 = 120573ℎV120599119868ℎ119873ℎThe term 120573Vℎ120599119878ℎ119868V119873ℎ denotes the rate at which the
human hosts 119878ℎ become infected by infectious mosquitoes 119868Vand 120573ℎV120599119878V119868ℎ119873ℎ refers to the rate at which the susceptiblemosquitoes 119878V are infected by the infectious human hosts 119868ℎ
Adding the first six equations of model (1) and assumingthat there is no disease induced death that is 120575ℎ = 0 gives119889119873ℎ119889119905 = Λ ℎ minus 120583ℎ119873ℎ so that 119873ℎ(119905) rarr Λ ℎ120583ℎ as 119905 rarr infin
[16] Thus Λ ℎ120583ℎ is an upper bound of 119873ℎ(119905) provided that119873ℎ(0) le Λ120583ℎ Further if 119873ℎ(0) gt Λ ℎ120583ℎ then 119873ℎ(119905)
will decrease to this level Λ ℎ120583ℎ Similar calculation for thevector equations shows that119873V rarr Λ V120583V as 119905 rarr infin
Table 1 State variables of the malaria model
Symbol Description119878ℎ(119905) Number of susceptible individuals at time t119864ℎ(119905) Number of exposed individuals at time t119868ℎ(119905) Number of infectious humans at time 119905119877ℎ(119905) Number of recovered humans at time 119905119878V(119905) Number of susceptible mosquitoes at time 119905119864V(119905) Number of infected mosquitoes at time 119905119868V(119905) Number of infectious mosquitoes at time 119905119873ℎ(119905) Total number of individuals at time 119905119873V(119905) Total mosquito population at time 119905
Table 2 Prevention and control variables in the model
Symbol Description
1199061(119905)Preventive measure using insecticide treated bed nets(ITNs)
1199062(119905)The control effort on treatment of infectiousindividuals
1199063(119905)Preventing measure using indoor residual spraying(IRS)
120591 Rate constant due to use of indoor residual spraying120578 Rate constant due to use of treatment effort
Table 3 Parameters variables of the malaria model
Symbol DescriptionΛ ℎ Recruitment rate of individuals by birth1205811
Proportion of exposed immigrants into exposed class120579 Proportion of people migrating into Karonga District120583ℎ Per capita natural death rate of humans120583V Per capita natural death rate of mosquitoes120575ℎ
Per capita disease induced death rate for humans
120572ℎ
Progression rate of humans from exposed state to theinfectious state
120573VℎProbability that a bite results in transmission ofinfection to human
120573ℎV
Probability that a bite results in transmission of theparasite from an infectious human to the susceptiblemosquitoes
120599 Biting rate of mosquito
120582ℎForce of infection for susceptible humans to exposedindividuals
120582VForce of infection for susceptible mosquitoes toexposed mosquito class
120588 Per capita rate of recovery with temporary immunity120601 Proportion of spontaneous individual recovery120595 Per capita rate of loss immunity
120572VProgression of exposed mosquitoes into infectedmosquitoes
4 Abstract and Applied Analysis
The feasible region
Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7
+119873ℎ le
Λ ℎ
120583ℎ
= 119873lowast
ℎ
119873V leΛ V
120583V= 119873lowast
V
(2)
is positive-invariant and attracting
Lemma 1 The region Φ isin R7+is positively invariant for the
model system (1) with initial conditions in R7+
Proof Let 119905 = sup119905 gt 0 119878ℎ gt 0 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt0 119864V gt 0 119868V gt 0 isin [0 119905] give 119905 gt 0 The first equation ofmodel (1) gives
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
ge Λ ℎ minus ((1 minus 1199061) 120582ℎ + 120583ℎ) 119878ℎ
(3)
which can be rewritten as
119889
119889119905[119878ℎ (119905) 119890
int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905]
ge Λ ℎ119890int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905
(4)
Therefore
119878ℎ (119905) 119890int119905
0(1minus119906
1)120582ℎ(119904)119889119904+120583
ℎ119905minus 119878ℎ (0)
ge int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+119906
ℎ119906119889119906
(5)
so that
119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
+ 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
times int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+120583
ℎ119906119889119906 gt 0
(6)
Similarly it can be shown that 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt 0119864V gt 0 and 119868V gt 0 for all 119905 gt 0This completes the proof
21 Existence and Stability of Equilibrium Points We analysesystem (1) to obtain the equilibrium points of the system andtheir stability Let (119878lowast
ℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) be the equilib-rium points of system (1) At an equilibrium point we have
1198781015840
ℎ(119905) = 119864
1015840
ℎ(119905) = 119868
1015840
ℎ(119905) = 119877
1015840
ℎ(119905) = 119878
1015840
V (119905) = 1198641015840
V (119905) = 1198681015840
V (119905) = 0
(7)
211 Disease-Free Equilibrium 1198640 In the absence of malariathat is 119864lowast
ℎ= 119868lowast
ℎ= 119877lowast
ℎ= 119864lowast
V = 119868lowast
V = 0 model system (1) hasan equilibrium point called the disease-free equilibrium 1198640and is given by
1198640 = (Λ ℎ
120583ℎ
0 0 0Λ ℎ
120583V 0 0) (8)
To establish the linear stability of 1198640 we employ van denDriessche and Watmoughrsquos next generation matrix approach[17] A reproduction number obtained this way determinesthe local stability of the disease-free equilibrium point for119877119890 lt 1 and instability for119877119890 gt 1 Following van denDriesscheand Watmough [17] the associated next generation matrices119865 and 119881 of system (1) can be determined from F119894 and V119894respectively where
F119894 =
[[[[[[[[[
[
(1 minus 1199061) 120573Vℎ120599119868V119878ℎ
119873ℎ
+ 1205811120579119873ℎ
0
120573ℎV120599119868ℎ119878V
119873ℎ
0
]]]]]]]]]
]
V119894 =[[[[
[
(120572ℎ + 120583ℎ) 119864ℎ
(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ
(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V
]]]]
]
(9)
The Jacobian matrix ofF119894 andV119894 is given by
119865 =[[[
[
0 0 0 1199031
0 0 0 0
0 119895 0 0
0 0 0 0
]]]
]
119881 =[[[
[
119898 0 0 0
minus120572ℎ 119899 0 0
0 0 119886 0
0 0 minus120572V b
]]]
]
(10)
where
1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ
Λ ℎ120583V
119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ
119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063
(11)
Algebraic manipulation of the matrices leads to the effectivereproduction number
R119890 = (120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061) Λ V120583ℎ
times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)
times(120572V + 120583V + 1205911199063)(120583V + 1205911199063)120583VΛ ℎ)minus1
)
12
(12)
where
R119890V =120573ℎV120572V120599Λ V
120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)
Abstract and Applied Analysis 5
is the contribution of themosquito populationwhen it infectsthe humans and
R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)
Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)
is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number
R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms
(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector
(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host
The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is
R119890119905 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(15)
Also if ITNs are the only intervention strategy then
R119890119873 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(16)
Similarly if IRS is the only means of protection then
R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V
times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)
times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12
(17)
The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)
R0 = radic120573ℎV120573119906ℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(18)
In general it is easy to prove that
R119890 le R0 (19)
for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds
Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1
212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system
119889119904
119889119905=
1
119873ℎ
[119889119878ℎ
119889119905minus 119904
119889119873ℎ
119889119905]
=1
119873ℎ
[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]
(20)
Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then
119889119904
119889119905=
1
119873ℎ
[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]
minus119904
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
=Λ ℎ
119873ℎ
minus [Λ ℎ
119873ℎ
+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579
+ (120601 + 1205781199062) (1 minus 120588) 119894
minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903
119889119890
119889119905=
1
119873ℎ
[119889119864
119889119905minus 119890
119889119873ℎ
119889119905]
=1
119873ℎ
[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]
minus119890
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
The feasible region
Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7
+119873ℎ le
Λ ℎ
120583ℎ
= 119873lowast
ℎ
119873V leΛ V
120583V= 119873lowast
V
(2)
is positive-invariant and attracting
Lemma 1 The region Φ isin R7+is positively invariant for the
model system (1) with initial conditions in R7+
Proof Let 119905 = sup119905 gt 0 119878ℎ gt 0 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt0 119864V gt 0 119868V gt 0 isin [0 119905] give 119905 gt 0 The first equation ofmodel (1) gives
119889119878ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ
ge Λ ℎ minus ((1 minus 1199061) 120582ℎ + 120583ℎ) 119878ℎ
(3)
which can be rewritten as
119889
119889119905[119878ℎ (119905) 119890
int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905]
ge Λ ℎ119890int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905
(4)
Therefore
119878ℎ (119905) 119890int119905
0(1minus119906
1)120582ℎ(119904)119889119904+120583
ℎ119905minus 119878ℎ (0)
ge int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+119906
ℎ119906119889119906
(5)
so that
119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
+ 119890minus(int119905
0(1minus1199061)120582ℎ(119904)119889119904+120583
ℎ119905)
times int
119905
0
Λ ℎ119890int119906
0(1minus1199061)120582ℎ(119908)119889119908+120583
ℎ119906119889119906 gt 0
(6)
Similarly it can be shown that 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt 0119864V gt 0 and 119868V gt 0 for all 119905 gt 0This completes the proof
21 Existence and Stability of Equilibrium Points We analysesystem (1) to obtain the equilibrium points of the system andtheir stability Let (119878lowast
ℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) be the equilib-rium points of system (1) At an equilibrium point we have
1198781015840
ℎ(119905) = 119864
1015840
ℎ(119905) = 119868
1015840
ℎ(119905) = 119877
1015840
ℎ(119905) = 119878
1015840
V (119905) = 1198641015840
V (119905) = 1198681015840
V (119905) = 0
(7)
211 Disease-Free Equilibrium 1198640 In the absence of malariathat is 119864lowast
ℎ= 119868lowast
ℎ= 119877lowast
ℎ= 119864lowast
V = 119868lowast
V = 0 model system (1) hasan equilibrium point called the disease-free equilibrium 1198640and is given by
1198640 = (Λ ℎ
120583ℎ
0 0 0Λ ℎ
120583V 0 0) (8)
To establish the linear stability of 1198640 we employ van denDriessche and Watmoughrsquos next generation matrix approach[17] A reproduction number obtained this way determinesthe local stability of the disease-free equilibrium point for119877119890 lt 1 and instability for119877119890 gt 1 Following van denDriesscheand Watmough [17] the associated next generation matrices119865 and 119881 of system (1) can be determined from F119894 and V119894respectively where
F119894 =
[[[[[[[[[
[
(1 minus 1199061) 120573Vℎ120599119868V119878ℎ
119873ℎ
+ 1205811120579119873ℎ
0
120573ℎV120599119868ℎ119878V
119873ℎ
0
]]]]]]]]]
]
V119894 =[[[[
[
(120572ℎ + 120583ℎ) 119864ℎ
(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ
(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V
]]]]
]
(9)
The Jacobian matrix ofF119894 andV119894 is given by
119865 =[[[
[
0 0 0 1199031
0 0 0 0
0 119895 0 0
0 0 0 0
]]]
]
119881 =[[[
[
119898 0 0 0
minus120572ℎ 119899 0 0
0 0 119886 0
0 0 minus120572V b
]]]
]
(10)
where
1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ
Λ ℎ120583V
119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ
119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063
(11)
Algebraic manipulation of the matrices leads to the effectivereproduction number
R119890 = (120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061) Λ V120583ℎ
times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)
times(120572V + 120583V + 1205911199063)(120583V + 1205911199063)120583VΛ ℎ)minus1
)
12
(12)
where
R119890V =120573ℎV120572V120599Λ V
120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)
Abstract and Applied Analysis 5
is the contribution of themosquito populationwhen it infectsthe humans and
R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)
Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)
is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number
R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms
(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector
(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host
The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is
R119890119905 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(15)
Also if ITNs are the only intervention strategy then
R119890119873 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(16)
Similarly if IRS is the only means of protection then
R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V
times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)
times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12
(17)
The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)
R0 = radic120573ℎV120573119906ℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(18)
In general it is easy to prove that
R119890 le R0 (19)
for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds
Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1
212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system
119889119904
119889119905=
1
119873ℎ
[119889119878ℎ
119889119905minus 119904
119889119873ℎ
119889119905]
=1
119873ℎ
[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]
(20)
Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then
119889119904
119889119905=
1
119873ℎ
[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]
minus119904
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
=Λ ℎ
119873ℎ
minus [Λ ℎ
119873ℎ
+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579
+ (120601 + 1205781199062) (1 minus 120588) 119894
minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903
119889119890
119889119905=
1
119873ℎ
[119889119864
119889119905minus 119890
119889119873ℎ
119889119905]
=1
119873ℎ
[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]
minus119890
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
is the contribution of themosquito populationwhen it infectsthe humans and
R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)
Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)
is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number
R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms
(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector
(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host
The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is
R119890119905 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(15)
Also if ITNs are the only intervention strategy then
R119890119873 = radic120573ℎV120573Vℎ120599
2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(16)
Similarly if IRS is the only means of protection then
R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V
times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)
times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12
(17)
The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)
R0 = radic120573ℎV120573119906ℎ120599
2120572ℎ120572VΛ V120583ℎ
(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ
(18)
In general it is easy to prove that
R119890 le R0 (19)
for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds
Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1
212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system
119889119904
119889119905=
1
119873ℎ
[119889119878ℎ
119889119905minus 119904
119889119873ℎ
119889119905]
=1
119873ℎ
[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]
(20)
Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then
119889119904
119889119905=
1
119873ℎ
[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]
minus119904
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
=Λ ℎ
119873ℎ
minus [Λ ℎ
119873ℎ
+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579
+ (120601 + 1205781199062) (1 minus 120588) 119894
minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903
119889119890
119889119905=
1
119873ℎ
[119889119864
119889119905minus 119890
119889119873ℎ
119889119905]
=1
119873ℎ
[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]
minus119890
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
119889119894
119889119905=
1
119873ℎ
[119889119868
119889119905minus 119894119889119873ℎ
119889119905]
=1
119873ℎ
[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ
minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]
minus119894
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= 120572ℎ119890 minus [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905=
1
119873ℎ
[119889119877
119889119905minus 119903
119889119873ℎ
119889119905]
=1
119873ℎ
[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]
minus119903
119873ℎ
[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]
= (120601 + 1205781199062) 120588119894 minus [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
119889119909
119889119905=
1
119873V[119889119878V
119889119905minus 119909
119889119873V
119889119905]
=1
119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]
minus119909
119873V[Λ V minus 120583V119873V]
= [Λ V
119873Vminus 119909
Λ V
119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909
119889119910
119889119905=
1
119873V[119889119864V
119889119905minus 119910
119889119873V
119889119905]
=1
119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]
minus119910
119873V[Λ V minus 120583V119873V]
= 120573ℎV120599119894119909 minus [Λ V
119873V+ 120572V + 1205911199063]119910
119889119911
119889119905=
1
119873V[119889119868V
119889119905minus 119885
119889119873V
119889119905]
=1
119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]
minus119911
119873V[Λ V minus 120583V119873V]
= 120572V119910 minus [Λ V
119873V+ 1205911199063] 119911
119889119873ℎ
119889119905= [
Λ ℎ
119873ℎ
minus 120583ℎ minus 120575ℎ119894]119873ℎ
119889119873V
119889119905= [
Λ V
119873Vminus 120583V]119873V
(21)
The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region
Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7
+| 119890 ge 0 119894 ge 0 119903 ge 0
119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ
120583ℎ
119910 ge 0 119911 ge 0 119910 + 119911 le 1
119873V leΛ V
120583V
(22)
can be shown to be positively invariant R7+denotes the
nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations
119889119890
119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
minus [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
119889119894
119889119905= 120572ℎ119890 minus [
Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
119889119903
119889119905= (120601 + 1205781199062) 120588119894 minus [
Λ ℎ
119873ℎ
+ 120595 + 120579 + minus120575ℎ119894] 119903
119889119873ℎ
119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ
119889119910
119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [
Λ V
119873V+ 120572V + 1205911199063] 119910
119889119911
119889119905= 120572V minus [
Λ V
119873V+ 1205911199063] 119911
119889119873V
119889119905= Λ V minus 120583V119873V
(23)
We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective
Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
after simplification the following algebraic equations areobtained
(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ
119873ℎ
minus 120575ℎ119894 + 120572ℎ + 120579] 119890
120572ℎ119890 = [Λ ℎ
119873ℎ
+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894
(120601 + 1205781199062) 120588119894 = [Λ ℎ
119873ℎ
+ 120595 + 120579 minus 120575ℎ119894] 119903
Λ ℎ
119873ℎ
= 120575ℎ119894 + 120583ℎ
(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V
119873V+ 120572V + 1205911199063]119910
120572V119910 = [Λ V
119873V+ 1205781199063] 119911
Λ V
119873V= 120583V
(24)
To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)
(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890
(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890
= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579
120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(25)
where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(26)
Equating these two equations for 119890 we obtain
(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894
120572ℎ
(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894
= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911
(27)
Therefore 119894 with 119903 and 119911 becomes
119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ
+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)
times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894
+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1
(28)
Now solve for 119903 from the third equation
(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903
119903 =(120601 + 1205781199062) 119894
120583ℎ + 120595
(29)
Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain
119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =
1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911
119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911
1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911
(30)
Using equation five of the system (24) we obtain
(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894
119910 =(1 minus 119911) 120573ℎV120599119894
120583V + 120572V + 1205911199063 + 120573ℎV120599119894
(31)
Solving for 119911 gives
120572V119910 = (120583V + 1205911199063) 119911
119911 =120572V119910
120583V + 1205911199063
(32)
Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives
119911 =(1 minus 119911) 120572V120573ℎV120599119894
(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)
=1198982119894
1198992 (1199011 + 1199021119894)minus
1198982119894119911
1198992 (1199011 + 1199021119894)
=1198982119894
11989921199011 + 11989921199021119894 + 1198982119894
(33)
Substituting 119911 in the 119894 equation gives
119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))
(34)
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain
1198601198942+ 119861119894 + 119862 = 0 (35)
where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982
= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992
+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992
+ (1 minus 1199061) (120601 + 1205781199062
120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599
2
gt 0
119862 = minus11988911989921199011 minus 11989921199011
= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]
lt 0
119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)
minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)
= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599
+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2
119890)
(36)
ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider
119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599
+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599
+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)
times (120583V + 120572V + 1205911199063) (1 minusR2
119890)
lt 0
(37)
Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic
persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem
Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641
The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1
Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death
213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting
Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ
= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0
(38)
then the following can be claimed
Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2
Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system
119889119864ℎ
119889119905= (1 minus 1199061) 120582ℎ (
Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ
120583ℎ
minus (120572ℎ + 120583ℎ) 119864ℎ
119889119868ℎ
119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ
119889119864V
119889119905= 120582V (
Λ V
120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V
119889119868V
119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V
(39)
Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives
120597
120597119864ℎ
[(1 minus 1199061) 120573Vℎ120599
119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ
120583ℎ
minus 119864ℎ minus 119868ℎ minus 119877ℎ)
+1205811120579Λ ℎ
119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)
119868ℎ119864V119868V]
+120597
120597119868ℎ
[120572ℎ
119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)
119864ℎ119864V119868V]
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
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MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
+120597
120597119864V[
120573ℎV120599
119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V
120583Vminus 119864V minus 119868V)
minus(120572V + 120583V + 1205911199063)
119864ℎ119868ℎ119864V]
+120597
120597119868V[
120572V
119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)
119864ℎ119868ℎ119864V]
= minus120573Vℎ120599
1198642
ℎ119868ℎ119864V
minus120573Vℎ120599120583ℎ
1198642
ℎ119864V
(1 minus119877ℎ
Λ ℎ120583ℎ
) minus1205811120579Λ ℎ
1198642
ℎ119868ℎ119864V119868V120583ℎ
minus120572ℎ
1198682
ℎ119864V119868V
minus120573ℎV120599
119864ℎ1198642V119868V
[1 minus120583ℎ
Λ ℎ
] minus120572V
119864ℎ119868ℎ1198682V
lt 0
(40)
since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits
in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1
The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1
In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi
3 Analysis of Optimal Control ofthe Malaria Model
The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)
with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as
Γ (1199061 1199062 1199063) = int
119879119891
0
(1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)) 119889119905
(41)
subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]
The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906
2
1 treatment of infected
individuals 11986021199062
2 and spraying of houses 1198603119906
2
3 A lin-
ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906
2
1 1198602119906
2
2 and 1198603119906
2
3 such that the
terms (12)11986011199062
1 (12)1198602119906
2
2 and (12)1198603119906
2
3describe the cost
associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062
119894(119905) (119894 = 1 2 3) and the constants
119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast
1(119905) 119906lowast
2(119905) and 119906
lowast
3(119905) such
that
Γ (119906lowast
1 119906lowast
2 119906lowast
3) = min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) (42)
where
Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable
0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3
(43)
is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]
31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by
119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
(44)
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as
119867 = 119871 + 1205821
119889119878ℎ
119889119905+ 1205822
119889119864ℎ
119889119905+ 1205823
119889119868ℎ
119889119905+ 1205824
119889119877ℎ
119889119905
+ 1205825
119889119878V
119889119905+ 1205826
119889119864V
119889119905+ 1205827
119889119868V
119889119905
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1
2(11986011199062
1+ 1198602119906
2
2+ 1198603119906
2
3)
+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ
minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]
+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]
+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]
+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]
+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]
+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]
+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]
(45)
Next we prove the existence of an optimal control for themodel system (1)
Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast
1 119906lowast
2 119906lowast
3) isin Φ2 such that
min(119906111990621199063)isinΦ2
Γ (1199061 1199062 1199063) = Γ (119906lowast
1 119906lowast
2 119906lowast
3) (46)
Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that
Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816
2+100381610038161003816100381611990621003816100381610038161003816
2100381610038161003816100381611990631003816100381610038161003816
2)1205982
minus 1205852(47)
This concludes the existence of an optimal control becausethe state variables are bounded
32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control
problem then there exists a nontrivial vector function 120582 =
(1205821 1205822 1205823 120582119899) satisfying the following equations
0 =120597119867 (119905 120594 119906 120582)
120597119906
1205821015840=120597119867 (119905 120594 119906 120582)
120597120594
119889120594
119889119905= minus
120597119867 (119905 120594 119906 120582)
120597120582
(48)
Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)
Theorem 6 For the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) with
their optimal state solutions (119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119878lowast
V 119864lowast
V 119868lowast
V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying
minus1205821015840
1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1205821015840
2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1205821015840
3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]
+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823
+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1205821015840
4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1205821015840
5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1205821015840
6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1205821015840
7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623
(49)
with transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(50)
Additionally the optimal control triple (119906lowast1 119906lowast
2 119906lowast
3) that mini-
mizes Γ over Φ2 satisfies the optimality condition
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(51)
Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby
minus1198891205821
119889119905=120597119867
120597119878ℎ
= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V
minus 120583ℎ1205821 + 12057912058111205822
minus1198891205822
119889119905=120597119867
120597119864ℎ
= (1 minus 1205811) 1205791205821 + 12057912058111205822
+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621
minus1198891205823
119889119905=120597119867
120597119868ℎ
= (1 minus 1205811) 1205791205821
+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822
minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622
minus1198891205824
119889119905=120597119867
120597119877ℎ
= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824
minus1198891205825
119889119905=120597119867
120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623
minus1198891205826
119889119905=120597119867
120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623
minus1198891205827
119889119905=120597119867
120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ
minus (120583V + 1205911199063) 1205827 + 1198623
(52)
with the transversality conditions
1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)
= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0
(53)
Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878
lowast
ℎ
119864ℎ = 119864lowast
ℎ 119868ℎ = 119868
lowast
ℎ 119877ℎ = 119877
lowast
ℎ 119878V = 119878
lowast
V 119864V = 119864lowast
V and 119868V = 119868lowast
Vyields
120597119867
1205971199061
= 119860119906lowast
1+ 120573Vℎ1205991205821119868
lowast
V 119878lowast
ℎminus 120573Vℎ1205991205822119868
lowast
V 119878lowast
ℎ= 0
120597119867
1205971199062
= 11986021199062 + (1 minus 120588) 1205781205821119868lowast
ℎminus 1205781205823119868
lowast
ℎ+ 1205781205881205824119868
lowast
ℎ= 0
120597119867
1205971199063
= 1198603119906lowast
3minus 1205911205825119878
lowast
V minus 1205911205826119864lowast
V minus 1205911205827119868lowast
V = 0
(54)
for which
119906lowast
1=120573Vℎ120599 (1205822 minus 1205821) 119868
lowast
V 119878lowast
ℎ
1198601
119906lowast
2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868
lowast
ℎ
1198602
1199063 =120591 (1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
(55)
Then
119906lowast
1= max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
)
119906lowast
2= max0min(1
120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
1198602
)
119906lowast
3= max0min(1
120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
)
(56)
We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)
The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast
1 119906lowast2 and 119906lowast
3
Substituting 119906lowast1 119906lowast2 and 119906lowast
3in the system (1) we obtain
119889119878lowast
ℎ
119889119905= Λ ℎ + (1 minus 1205811) 120579119873
lowast
ℎ
+ (120601 + 1205781199062) (1 minus 120588) 119868lowast
ℎminus 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
minus 120583ℎ119878lowast
ℎ+ 120595119877lowast
ℎ
119889119864lowast
ℎ
119889119905= 120573Vℎ120599119868
lowast
V 119878lowast
ℎ
times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868
lowast
V 119878lowast
ℎ
1198601
))
+ 1205811120579119873lowast
ℎminus 120572ℎ119864
lowast
ℎminus 120583ℎ119864
lowast
ℎ
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
119889119868lowast
ℎ
119889119905
= 120572ℎ119864lowast
ℎ
minus (120601 + 120578
times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868
lowast
ℎ
1198602
))
+120583ℎ + 120575ℎ) 119868lowast
ℎ
119889119877lowast
ℎ
119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)
times 119868lowast
ℎtimes (1198602)
minus1))) 120588119868ℎ
minus (120583ℎ + 120595)119877ℎ
119889119878lowast
V
119889119905
= Λ V minus (120583V + 120591
times (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119878
lowast
V minus 120573ℎV120599119868lowast
ℎ119878lowast
V
119889119864lowast
V
119889119905= 120573ℎV120599119868
lowast
ℎ119878lowast
V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
times(1198603)minus1))) 119864
lowast
V minus 120572V119864lowast
V
119889119868lowast
V
119889119905= 120572V119864V
minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast
V + 1205826119864lowast
V +1205827119868lowast
V )
times(1198603)minus1))) 119868
lowast
V
(57)
with119867lowast at (119905 119878lowastℎ 119864lowast
ℎ 119868lowast
ℎ 119877lowast
ℎ 119906lowast
1 119906lowast
2 119906lowast
3 1205821 1205822 1205827)
119867lowast
= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V
+1
2(1198601(max0min(1
120573Vℎ120599 (1205822 minus 1205821) 119868lowast
V 119878lowast
ℎ
1198601
))
2
+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast
ℎ
times(1198602)minus1))2
+1198603(max0min(1120591(1205825119878
lowast
V + 1205826119864lowast
V + 1205827119868lowast
V )
1198603
))
2
)
+ 1205821
119889119878lowast
ℎ
119889119905+ 1205822
119889119864lowast
ℎ
119889119905+ 1205823
119889119868lowast
ℎ
119889119905+ 1205824
119889119877lowast
ℎ
119889119905+ 1205825
119889119878lowast
V
119889119905
+ 1205826
119889119864lowast
V
119889119905+ 1205827
119889119868lowast
V
119889119905
(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state
33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11
Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites
Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))
In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 13
u1 ne 0 u3 ne 0 u2 = 0
u1 ne 0 u2 ne ne0 u3 0
0
10
20
30
40
50
60
70
times102In
fect
ed h
uman
pop
ulat
ion
0 10 20 30 40 50 60 70 80 90 100
Time (years)
(a)
Constant control u1 ne 0 u2 ne ne0 u3 0
=Constant control u1 ne 0 u2 ne 0 u3 0In
fect
ed h
uman
pop
ulat
ion
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90 100
times102
Time (days)
(b)
u2 ne 0 u3 ne 0 u1 = 0
u1 ne 0 u2 ne ne0 u3 0
0 10 20 30 40 50 600
5
10
15
20
25
30
35
40
times102
Infe
cted
hum
an p
opul
atio
n
Time (years)
(c)
u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0
=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0
0
05
1
15
2
25
3
times104
0 10 20 30 40 50 60 70
Infe
cted
hum
an p
opul
atio
n
Time (years)
(d)
Figure 2 Simulations of model (1) showing the effects of intervention measures
some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control
The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Abstract and Applied Analysis
Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)
Malaria transmission-bite from an infected mosquito TotalNo Yes
Major diseases in the community-malariaNo Count 11 15 26
of total 22 30 53
Yes Count 73 393 466 of total 148 799 94
Total Count 84 408 492 of total 171 829 1000
4 Analysis of Malaria Data fromKaronga District
A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents
We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies
Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito
According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)
409
93 53 47 7 4484
400440 446
486449
9 9 9 9 9 90
100
200
300
400
500
600
Bite
from
anin
fect
edm
osqu
ito
Fem
ale
mos
quito
Stag
nant
wat
er
Envi
ronm
ent
man
agem
ent
Oth
er
YesNo
Malaria transmission
wat
erso
aked
in ra
inRa
ined
on
Anopheles
Do not know
Figure 3 Knowledge of malaria transmission
Seek treatment
Methods of preventing malaria
47
418
493
462
409
454
83
8
38
92
0 100 200 300 400 500 600
Other
YesNo
IPT for pregnantwomen
House sprayed(IRS)
Use of ITNor LLN
Figure 4 Methods of preventing malaria
A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 15
Table 5 Malaria occurrence after spraying and without using net
Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000
Table 6 Malaria occurrence after spraying and using net
Frequency PercentNo 195 388Yes 178 355No response 129 257Total 502 1000
hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying
Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)
5 Numerical Results
The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]
51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease
52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place
53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Abstract and Applied Analysis
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
Susc
eptib
le in
divi
dual
s
Susceptible individualstimes102
(a)
R0 = 18894
0 05 1 15 2 25 3 35 4
Time (years)
0
20
40
60
80
100
120
140
times102
Expo
sed
indi
vidu
als
Exposed individuals
(b)
R0 = 18894
0 05 1 15 2 25 3 35 49
10
11
12
13
14
15
16
17
Time (years)
Infe
cted
indi
vidu
als
Infected individuals without any intervention strategiestimes102
(c)
0 100 200 300 400 500 600 7000
1
2
3
4
5
6
7
8
Time (days)
Reco
vere
d in
divi
dual
s
Recovered individuals without any intervention strategies
R0 = 18894
times102
(d)
Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals
the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and
at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals
A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 17
Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)
Use of net and malariaTotalOccurrence
No Yes
Method of preventing use of ITN or LLITNNo Count 17 27 44
of total 36 57 92
Yes Count 214 219 433 of total 449 459 908
Total Count 231 246 477 of total 484 516 1000
Time (days)
R0 = 18894
Prev
alen
ce
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
Figure 6 Exposed humans without any intervention strategies
Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs
6 Conclusions
An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential
Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence
Value df Asymp sig (2-sided)Pearson chi-square 1861
119886 1 0173Continuity correctionb 1454 1 0228
impact of these strategies on the transmission dynamics ofthe disease
Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease
Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Abstract and Applied Analysis
Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)
Malaria occurrence after spraying and using net TotalNo Yes
Method of preventing house spread (IRS)No Count 158 143 301
of total 425 384 809
Yes Count 36 35 71 of total 97 94 191
Total Count 194 178 372 of total 522 478 1000
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
times104
Time (years)0 05 1 15 2 25 3 35 4
0
05
1
15
2
25
(a)
times105
Hum
an p
opul
atio
n
Recovered individualsExposed individualsInfected individuals
Time (years)0 10 20 30 40 50 60
0
02
04
06
08
1
12
14
16
18
2
(b)
Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922
of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment
Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease
Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)
To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 19
Table 10 House sprayed IRS malaria occurrence after spraying andusing net
Value df Asymp sig (2-sided)Pearson chi-square 0074
119886 1 0786Continuity correctionb 0019 1 0889
Table 11 Table for parameter values of the optimal malaria model
Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ
117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ
003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed
transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community
The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgments
P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper
References
[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561
[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten
[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease
[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm
[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria
[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011
[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009
[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008
[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010
[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008
[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000
[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007
[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010
[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009
[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013
[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982
[17] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Abstract and Applied Analysis
[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990
[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009
[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000
[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012
[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011
[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010
[24] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962
[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011
[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf
[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml
[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012
[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012
[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of