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Research ArticleTransmission Dynamics of Hepatitis C with Control Strategies

Adnan Khan Sultan Sial and Mudassar Imran

Department of Mathematics Lahore University of Management Sciences Lahore 54792 Pakistan

Correspondence should be addressed to Mudassar Imran mudassarimranlumsedupk

Received 29 October 2013 Revised 6 December 2013 Accepted 17 December 2013 Published 13 February 2014

Academic Editor Darryl D DrsquoLima

Copyright copy 2014 Adnan Khan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a rigorousmathematical analysis of a deterministicmodel for the transmission dynamics of hepatitis C using a standardincidence function The infected population is divided into three distinct compartments featuring two distinct infection stages(acute and chronic) along with an isolation compartment It is shown that for basic reproduction number 119877

0le 1 the disease-free

equilibrium is locally and globally asymptotically stable The model also has an endemic equilibrium for 1198770gt 1 Uncertainty and

sensitivity analyses are carried out to identify and study the impact of critical parameters on 1198770 In addition we have presented the

numerical simulations to investigate the influence of different important parameters on 1198770 Since we have a locally stable endemic

equilibrium optimal control is applied to the deterministic model to reduce the total infected population Two different optimalcontrol strategies (vaccination and isolation) are designed to control the disease and reduce the infected population PontryaginrsquosMaximum Principle is used to characterize the optimal controls in terms of an optimality system which is solved numericallyNumerical results for the optimal controls are compared against the constant controls and their effectiveness is discussed

1 Introduction

Hepatitis C (HCV) is an important public health problem asit is the common cause of liver diseases throughout the world[1] The disease was first recognized in 1975 and its causativeagent was identified in 1989 Hepatitis C is characterized byan acute (often asymptotic) stage which in most cases isfollowed by a chronic stage that can result in cirrhosis andliver cancer The hepatitis C virus (causative agent) is anenveloped RNA virus which is further characterized to be apositive-sense single stranded virus belonging to the familyFlaviviridae and is considerably small in size Replicationof the RNA-based virus involves the use of the enzymeRNA-dependent RNA polymerase (RdRP) which has a higherror rate while going through this process World HealthOrganizationrsquos report suggests that around 3 of the worldpopulation has been infected with HCV The populationinfected with chronicHCV who are at risk of developing livercancer or cirrhosis is estimated to be around 170 millionFurthermore nearly 350000 people die annually throughoutthe globe as a result of HCV-related liver diseases [2]

Hepatitis C can be characterized by two distinct stagesan acute stage and a chronic stage Initially infection by HCV

causes an acuteHCVwhich is usually asymptotic Only about15of the cases showmild symptoms like decreased appetitenausea fatigue joint ormuscle pains and weight loss In 20of the cases the infection may resolve spontaneously Andthe remaining 80 of the people exposed to HCV progressto the chronic stage of the infection by developing a chronicinfection which can last for decades During the startingyears of infection most people experience minimal or nosymptoms at all However HCV becomes the main cause ofliver cancer and cirrhosis after several years of living withit About 1ndash5 of chronic HCV patients die from livercancer or cirrhosis and nearly 5ndash20develop cirrhosis over30 years Patients with cirrhosis are 20 times more likely todevelop hepatocellular carcinoma at the rate of 13 eachyear Moreover 27 of cirrhosis cases and 25 of hepatocel-lular carcinoma cases worldwide are estimated to be causedby HCV [3ndash5]

Depending on the genotype of the HCV the standardtreatment of infected patients includes a combination ofpegylated interferon (peg IFN-120572) and the robust antiviraldrug Ribavirin for a period of 24 or 48 weeks [6] The reac-tion to treatment also differs by genotype and lies between70 and 80 for genotypes 2 and 3 while it is almost non

Hindawi Publishing CorporationJournal of Computational MedicineVolume 2014 Article ID 654050 18 pageshttpdxdoiorg1011552014654050

2 Journal of Computational Medicine

existent for genotype 6 Recently there have been promisingtreatment advances of genotype 1 using directly acting antivi-ral agents (DAAs) However to prevent the infection there isstill a need to create effective vaccine strategyVaccines whichare used in preclinical and clinical trials involve DNA-basedproteins recombinant proteins synthetic peptide vaccinesand so forth The future design of vaccines along withthe use and success of previously mentioned vaccines hasbeen discussed here [7] However no long-term immunityis granted in recovering from hepatitis C infection So thisabsence of acquired immunity must be shown in any modelfor hepatitis CThis is done by allowing recovered patients tobecome susceptible again

Several studies have been carried out [8ndash13] and theyare pertinent to our work Most of these papers classifyindividuals in the population into different states and thenformulate a system of ordinary differential equations (ODE)to analyze the time-evolution of each of these populationstates Reade et al [12] present an ODE model of infectionswith acute and chronic stages Similarly by Luo and Xiang[10] a four state system was analyzed with exposed acute andchronic states Suna et al [13] present a study on a SEIRSmodel where it was assumed that recovered individuals losetheir infection-acquired immunity Martcheva and Castillo-Chavez in [11] have formulated amodel for hepatitis C lackingan exposed class and have discussed the stability of theequilibrium states

We will formulate a five-state deterministic model withindividuals in the population being classified as susceptibleacute chronic isolated and recovered Individuals sufferingfrom acute and chronic stages of the infections are repre-sented by acute and chronic states respectively The isolatedstate represents the chronically infected individuals gettingisolated The isolation of those with disease symptoms isprobably the first infection control measure in recordedhuman history [14] Over the decades quarantine and isola-tion have been used to reduce the transmission of numerousemerging and reemerging human diseases such as leprosyplague cholera typhus yellow fever smallpox diphtheriatuberculosis measles Ebola pandemic influenza and morerecently SARS [15ndash20]

Previous models of HCV particularly the models cal-culating the threshold quantity basic reproduction number1198770 have included the treatment andor vaccination and have

discussed the control of the disease by looking at the roleof disease transmission parameters in the reduction of 119877

0

and the prevalence of the disease However these modelsdid not account for time-dependent control strategies sincetheir discussions are based on prevalence of the disease atequilibria Optimal control theory has been employed tomake decisions involving epidemic and biological modelsThe desired results and performance of the control functionsdepend on the different situations Lenhartrsquos HIVmodels [2122] used optimal control to design the treatment strategiesJung et al [23] provide a very good example of deciding howto divide the efforts between two treatment strategies (caseholding and case finding) of the two-strain TB model Yanet al [24] used an optimal isolation campaign to fight theSARS epidemic Study control strategies produce valuable

theoretical results which can be used to suggest or designepidemic control programs Depending on a chosen goal (orgoals) various objective criteria may be adopted

Our model extends previous work done on modeling thespread of hepatitis C in several key ways First we introducean isolation class and qualitatively assess the effects of thisisolation class on the transmission dynamics Quarantine ofindividuals suspected of being exposed to a disease and theisolation of those with disease symptoms constitute whatprobably is the first infection control measure since thebeginning of recorded human history [14] However almostno analysis of the effects of an isolation class on diseases witha chronic stage has been done and therefore our paper willbe one of the first attempts to study the effect of isolationon the spread of a disease with a chronic stage In additionwe will model the force of infection by a proportionatemixing with the possibility of secondary infections due tocontact with individuals who belong to the acute chronicor isolation class Furthermore we consider the disease-induced death rates for HCV in our model and will alsotake into account the possibility of recovery at every stageof the disease These features add to the complexity of ourmodel and make it considerably more insightful from anepidemiological perspective than previous models [11 25]

In the paper Section 2 presents a rigorous mathematicalanalysis of the deterministic model It is shown that for basicreproduction number 119877

0le 1 the disease-free equilibrium

is locally and globally asymptotically stable Further themodel has an endemic equilibrium which exists if 119877

0gt 1

and persists in this case The effect of using isolation onpopulation is discussed using a threshold quantity Sensitivityand uncertainty analyses are carried out to study the impactof crucial parameters on 119877

0 The existence of a locally stable

endemic equilibrium in case of 1198770gt 1 encourages us to

use a time-dependent optimal control strategy to prevent andcontrol HCV In Section 3 we designed two different optimalcontrol strategies (vaccination and isolation) and performednumerical simulations to illustrate the effects of an optimalcontrol strategy Conclusion is presented in the last section

2 Model Formulation andSteady State Analysis

21Model Formulation We formulate a five-statemodel withindividuals classified as susceptible acute chronic isolatedand recovered Hep C has an extremely slow progressionthat makes it difficult to characterize the natural history ofthe disease [3] The following assumptions will therefore bemade

(1) All infected individuals develop the acute formofHepC first

(2) Individuals with either the acute or chronic form ofHep C are capable of transmitting the disease

(3) Individuals with the acute form of the disease eitherprogress to the chronic form or recover naturallySince the acute form of the disease is largely asymp-tomatic there is little chance of treatment at this stage

Journal of Computational Medicine 3

(4) There is no permanent immunity against HCV afterrecovery thus the recovered individuals move backto the susceptible

In addition the model will assume that the susceptiblepopulation 119878 has a constant recruitment rate Π and naturaldeath rate 120583 Susceptible individuals who get infected sufferfrom the acute form of hepatitis C and move to the compart-ment 119860 with the force of infection given by 120582 Individuals in119860 in addition to the natural death rate 120583 die at a disease-induced death rate 120575

119886 They also have a natural recovery rate

of 120581 Individuals with the acute form of the infection progressto the chronic form of the disease at a rate 120585 in which casethe individual is shifted to compartment 119862 Individuals in 119862in addition to the natural death rate 120583 also die at a disease-induced death rate 120575

119888 Furthermore these individuals recover

at a rate 120595 and thus move to the recovered compartment119878 Also the individuals in compartment 119862 are isolated andmoved to compartment 119876 at a rate 120572 Individuals in 119876 inaddition to the natural death rate 120583 also die at a disease-induced death rate 120575

119902 Isolated individuals can either recover

at a rate 120574119891 or become acutely infected with HCV at a rate120574(1 minus 119891) The individuals in 119877 are prone to the natural deathrate 120583 or they become susceptible again and enter the 119878compartment at a rate of 120596 Recovery from HCV does notresult in immunity

Mathematically the model is as follows

119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595 + 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(1)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (2)

The descriptions of variables and parameters of model (1) areas follows

Variable

119873(119905) Total population119878(119905) Population of susceptible individuals119860(119905) Population of individuals with acute HepC119862(119905) Population of individuals with chronicHep C119876(119905) Population of isolated individuals119877(119905) Population of recovered individuals

Parameter

Π Recruitment rate120583 Natural death rate120575119886 Disease-induced death rate for individuals

with acute Hep C120575119888 Disease-induced death rate for individuals

with chronic Hep C120575119902 Disease-induced death rate for isolated indi-

viduals120574 Recovery rate of isolations119891 Fraction of isolated that becomes susceptibles120585 Progression rate from acute to chronic120572 Isolation rate of chronic individuals120581 Natural recovery rate for acute individuals120595 Recovery rate for chronic individuals120596 Progression rate of recovered individuals tosusceptible individuals120573 Effective contact rate120578 Modification parameter for reduction ininfectiousness of acute individuals120577 Modification parameter for reduction ininfectiousness of quarantined individuals

Since (1) is a model for human populations all the associ-ated parameters are nonnegative Furthermore the followingresult holds and can be proved easily

Lemma 1 The variables of model (1) are nonnegative for alltime 119905 gt 0 In other words solutions of the system (1) withpositive initial data will exist and remain positive for all 119905 gt 0Moreover the closed set

119863 = (119878 119860 119862 119876 119877) isin 1198775

+ 119878 + 119860 + 119862 + 119876 + 119877 le

Π

120583 (3)

is positively invariant and global attractor

Since the region 119863 is positively invariant and globalattractor it is sufficient to consider the dynamics of the flowgenerated by model (1) (Figure 1) in region 119863 where themodel is considered to be epidemiologically and mathemati-cally well posed

22 Disease-Free Equilibrium (DFE) In this section we dis-cuss the existence and stability of the disease-free equilibrium(DFE)

221 Local Stability Model (1) has a disease-free equilibriumDFE obtained by setting the right-hand sides of the equationsin (1) to zero given by

alefsym0= (119878lowast 119860lowast 119862lowast 119876lowast 119877lowast) = (

Π

120583 0 0 0 0) (4)

The local stability property of alefsym0will be determined using

the next generation operator method described in [26]


120583

120583

120583

120583120583

S

120582

120596 120585

120581C

A

120595 120572

R Qf120574

120575a

120575q

120575c

(1 minus f)120574

prod

Figure 1 Schematic diagram of model (1)

The nonnegativematrix119865 of the new infection terms and the119872-matrix 119881 of the transition terms associated with model(1) are given by

119865 = (

120573120578 120573 120573120577

0 0 0

0 0 0

)

119881 = (

120585 + 120581 + 120583 + 120575119886

0 minus120574 (1 minus 119891)

minus120585 120572 + 120595 + 120583 + 120575119888

0

0 minus120572 120574 + 120583 + 120575119902

)

(5)

The eigenvalues of matrix 119865119881minus1 are

0 0120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

(6)

It follows that the basic reproduction number 1198770

=

120588(119865119881minus1) is given by

1198770=120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

gt 0 (7)

where

1198961= (120585 + 120581 + 120583 + 120575

119886)

1198962= (120572 + 120595 + 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902)

1198964= 120574 (1 minus 119891)

(8)

The basic reproduction number is interpreted as the averagenumber of new infections that one infectious individualcan produce if introduced into a population composedof susceptible individuals Susceptible individuals acquireinfection following contact with either an acute (119860) chronic(119862) or isolated (119876) individual The number of infectionsproduced by an acutely infected individual (near the DFE)is 120573120578119896

1given by the product of the infection rate of

an acute individual (120573120578) and the average duration in theacute class (1119896

1) Furthermore the number of infections

produced by a chronically infected individual (near the DFE)is 120573120585119896

11198962given by the product of the infection rate of a

chronic individual (120573) the average duration in the chronicC class (1119896

2) and the probability that an acute individual

survives and progresses to the chronic stage 1205851198961 Similarly

the number of infections produced by an isolated individual(near the DFE) is 120573120577120585120572119896

111989621198963given by the product of the

infection rate of an isolated individual (120573120577) the averageduration in the isolated class (1119896

3) and the probability that

an acute individual survives and progresses to the isolatedstage 120585120572119896

11198962 Finally we observe that a fraction 120585120572120574(1 minus

119891)119896111989621198963of newly infected individuals will reenter the acute

class Thus the average number of new infections generatedby a single infectious individual is given by

(120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)

infin

sum

119899=0

[120585120572120574(1 minus 119891)

119896111989621198963

]

119899

= (120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)[1

1 minus (120585120572120574 (1 minus 119891) 119896111989621198963)]

= 1198770

(9)

The local stability of the DFE holds due toTheorem 2 of [26]

Lemma 2 The DFE alefsym0 of model (1) is locally asymptotically

stable if 1198770lt 1 and unstable if 119877

0gt 1

Lemma 22 implies that with 1198770lt 1 a small influx of

infectious individuals will not lead to a large outbreak of thedisease To ensure that disease elimination is independent ofthe initial sizes of subpopulations it is necessary to show thatthe DFE is globally asymptotically stable if 119877

0lt 1 This is

explored below

222 Global Stability

Theorem 3 The DFE of model (1) given by (4) is globallyasymptotically stable whenever 119877

0le 1

Proof Consider the following Lyapunov function

119871 = 119886119860 + 119888119862 + 119902119876 (10)

where

119886 =11989621198963

119896111989621198963minus 120572120585119896

4

119888 =1205731198963+ 120572 [120573120577 + 119896

4]

119896111989621198963minus 120572120585119896

4

119902 =1198962[120573120577 + 119896

4]

119896111989621198963minus 1205721205851198964

(11)


Clearly 119871 is positive definite We have

= 119886 + 119888 + 119902

= 119886120582119878 + 1198861198964119876 minus 119886119896

1119860 + 119888120585119860 minus 119902119896

3119876 + 119902120572119862 minus 119888119896

2119862

= 119886120573[120578119860 + 119862 + 120577119876]

119873119878 + [119886119896

4minus 1199021198963] 119876

+ [119888120585 minus 1198861198961] 119860 + [119902120572 minus 119888119896

2] 119862

le [119886120573120578 + 119888120585 minus 1198861198961] 119860 + [119886120573 + 119902120572 minus 119888119896

2] 119862

+ [119886120573120577 + 1198861198964minus 1199021198963] 119876

=[12057312057811989621198963+ 120585120573119896

3+ 120585120572120573120577 + 120585120572120573119896

4minus 119896111989621198963]

119896111989621198963minus 120572120585119896

4

119860

+[11989621198963120573 + 120572119896

2120573120577 + 120572119896

21198964minus 12057311989631198962minus 1205721198962120573120577 minus 120572119896

21198964]

119896111989621198963minus 120572120585119896

4

119862

+[11989621198963120573120577 + 119896

211989631198964minus 11989621198963120573120577 minus 119896

211989631198964]

119896111989621198963minus 120572120585119896

4

119876

= (1198770minus 1)119860

(12)

Thus

le (1198770minus 1)119860 le 0 for 119877

0lt 1 (13)

It follows that le 0 for 1198770lt 1 with = 0 if and only if

119860 = 119862 = 119876 = 0 or 1198770= 1 Hence 119871 is a Lyapunov function

on119863The largest invariant set in (119878 119860 119862 119876 119877) isin 119863 | = 0

is the singleton alefsym0 According to the LaSalle Invariance

Principle alefsym0is globally asymptotically stable in 119863 if 119877

0lt 1

This means that with 1198770lt 1 every solution to the system (1)

with initial conditions in119863 approaches alefsym0as 119905 rarr infin

(119878 119860 119862 119876 119877) 997888rarr alefsym0= (

Π

120583 0 0 0 0) as 119905 997888rarr infin

(14)

The epidemiological implication of the above result isthat the disease can be eliminated from the population ifthe basic reproduction number 119877

0can be brought down to

and maintained at a value less than unity (ie the condition1198770lt 1 is sufficient and necessary for disease elimination)

Figure 2 depicts numerical results by simulating model (1)using various initial conditions with119877

0lt 1 It is evident from

the simulation that all initial solutions converged to DFEalefsym0

in-line withTheorem 3

23 Endemic Equilibrium In this section the existence andstability of endemic equilibrium of model (1) will be dis-cussed We define endemic equilibrium to be those fixedpoints of the system (1) in which at least one of the infectedcompartments of the model is nonzero

0 50 100 150 200 2500

100

200

300

400

500

600

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Disease-free equilibrium

Figure 2 Disease-free equilibrium 1198770= 07994 Simulation shows

the total infected population with different initial infected popula-tion sizes The parameter values are given in the appendix

Let alefsym1= (119878lowastlowast 119860lowastlowast 119862lowastlowast 119876lowastlowast 119877lowastlowast) denote an arbitrary

endemic equilibrium of model (1) so that119873lowastlowast = 119878lowastlowast +119860lowastlowast +119862lowastlowast+119876lowastlowast+119877lowastlowast Solving the equations of model (1) at steady

state gives

119860lowastlowast=1198962

120585119862lowastlowast


1198963

119862lowastlowast

119878lowastlowast=

1

120582lowastlowast(119896111989621198963minus 120572120585119896

4

1205851198963

)119862lowastlowast


1198965

(119891120574120572

1198963

+1205811198962

120585+ 120595)119862

lowastlowast

(15)

where

120582lowastlowast= 120573

[120578119860lowastlowast+ 119862lowastlowast+ 120577119876lowastlowast]

119873lowastlowast 1198965= (120596 + 120583) (16)

Consider 119878lowastlowast Then using 120582lowastlowast and (15) from above wehave the following endemic states


120585[1198770minus 1

119884] 119878lowastlowast

119862lowastlowast= [

1198770minus 1



1198963

[1198770minus 1



1198965

[11989621198963+ 1198963120581120585 + 120574119891120572119896

2

11989621198963

] [1198770minus 1


(17)

where

119884 = [1198962

120585+ 1 +

120572

1198963

] (18)

Hence we have the following result


Lemma 4 Model (1) has endemic equilibria given byalefsym1(15)

whenever 1198770gt 1

Now we address the question of uniform persistence ofthe infected population

Theorem 5 If 1198770gt 1 then the disease is uniformly persistent

there exists an 120598 gt 0 such that

lim119905rarrinfin

inf 119860 (119905) gt 120598 lim119905rarrinfin

inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)

for all solutions (119878 119860 119862 119876 119877) of (1) with 119860(0) gt 0 119862(0) gt 0and 119876(0) gt 0

Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus

119883 is the set of all disease-free states of (1) and it can be easilyverified that 119883 is positively invariant Let119872 = 119863 cap 119883 Sinceboth 119863 and 119883 are positively invariant 119872 is also positivelyinvariant Also note that alefsym

0isin 119872 and alefsym

0attracts all the

solutions in 119883 So Ω(119872) = alefsym0 The equations for the

infected components of (1) can be written as

1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It

is clear that 119884(alefsym0) = 119865 minus 119881 Also it is easy to check that

119884(alefsym0) is irreducibleWewill apply LemmaA4 in [27] to show

that119872 is a uniform weak repeller Since alefsym0is a steady state

solution we can consider it to be a periodic orbit of period119879 = 1 119875(119905 119909) the fundamental matrix of the solutions for(20) is 119890119905119884 Since the spectral radius of 119884(alefsym

0) = 119877

0gt 1 the

spectral radius of 119890119884(alefsym0) gt 1 So condition 2 of Lemma A4 issatisfied Taking 119909 = alefsym

0 we get 119875(119879alefsym

0) = 119890119884(alefsym0) which is a

primitive matrix because 119884(alefsym0) is irreducible as mentioned

in Theorem A12(i) [28] This satisfies the condition 1 ofLemmaA4Thus119872 is a uniformweak repeller and disease isweakly persistent119872 is trivially closed and bounded relativeto 119863 and hence compact Therefore by Theorem 13 [29]we have that 119872 is a uniform strong repeller and disease isuniformly persistent

The epidemiological implication of Theorem 5 is thatthe disease will persist in the population whenever 119877

0gt

1 Numerical simulation results depicted in Figure 3 usingdifferent initial conditions shows convergence of solutions tothe alefsym1in-line withTheorem 5

24 Sensitivity Analysis The asymptotic dynamics of themodel are completely determined by the threshold quantity1198770 which determines the prevalence of the disease Since we

have a deterministic model the only uncertainty is generatedby the input variation and parameters Therefore we presentparameter-related global uncertainty and sensitivity analyseson1198770 Parameter estimates can be uncertain because of many

reasons including natural variation error in measurementsor a lack of measuring techniques Uncertainty analysisqualitatively decides which parameters aremost influential in

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium

Figure 3 Endemic equilibrium 1198770= 15545 Simulation shows the

total infected population with different initial infected populationsizes The parameter values are given in the appendix

the model output (1198770in our case) and quantifies the degree

of confidence in the existing data and parameter estimationOn the other hand the sensitivity analysis identifies criticalmodel parameters and quantifies the impact of each inputparameter on the value of an output in the presence of otherinput parameters

Ideally uncertainty and sensitivity analyses should beperformed simultaneously Here we use the Latin-hypercubesampling based method to quantify uncertainty and sensi-tivity of 119877

0as a function of 12 model parameters (120583 120574 120585

120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575

119902) For the sensitivity analysis

partial rank correlation coefficient (PRCC) measures theimpact of the parameters on the output variable PRCCprovides a measure of monotonicity after the removal of thelinear effects of all but one variable PRCC method uses therank transformation of the data (ie replacing the valueswith their ranks) to reduce the effects of nonlinearity TheRank Correlation Coefficient (RCC) indicates the degree ofmonotonicity between the input and output variables Theresultant data are considered partially in some sense that ispartial rank correlation coefficients (PRCC) are computedthat take into account the correlations among other inputvariables The basic reproduction number 119877

0is the output

measure in the sensitivity and uncertainty analysesThe assumed distributions of the model parameters used

in the two analyses are mentioned in the appendix Ourestimate of119877

0forHepC fromuncertainty analysis is 133 with

95 CI (11 195) as shown in Figure 4 The probability that1198770gt 1 is 90 This suggests that Hep C will get endemic

under the preset conditions However the time taken to reachthat state could be large

The sensitivity analysis suggests that the most significant(PRCC values above 05 or below minus05 in Figure 5) sensitivityparameters to 119877

0are 120572 120581 120573 and 120577 This suggests that

these parameters need to be estimated with precision tocapture the transmission dynamics of theHepCThe analysesfurther suggest that the isolation strategy aimed to reduce theinfected population yields the desired result


05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3

Mean = 0004843 Std = 00088012120575a Mean = 0006078 Std = 000805459120575c Mean = 0003296 Std = 00097452120575q

Mean = 1333360 Std = 0274529R0

Figure 4 Uncertainty analysis the probability that 1198770gt 1 is 90 with 95 confidence interval (11 195) This suggests that hepatitis C will

get endemic under the present conditions However the time taken to reach that state could be large 10000 values were generated for eachparameter according to their distributions and mean values Values of parameters given in Appendix were used to calculate 119877

0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)

Sensitivity of R0 with respect to model parameter

120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115

Figure 5 Sensitivity analysis the proportion of chronically infectedbeing quarantined 120572 proportion of acute infections recoveringspontaneously 120581

1 effective contact rate 120573 and modification param-

eter for infectiousness of quarantined 120577 are the most significantparameters This means that even a small error in the estimation ofthese parameters can greatly affect the value of 119877

0and hence the

analysis of our model Partial rank correlation coefficients (PRCC)are calculated with respect to119877

0 Parameters withmodulus of PRCC

values in excess of 05 are declared sensitive to 1198770

Since we are interested in the influence of critical modelparameters on the basic reproductive number and hencethe prevalence of chronic Hep C we conduct numericalsimulation to investigate it In order to qualitatively measurethe effect of isolation on the transmission dynamics of HepC a threshold analysis of the parameter associated with theisolation of chronically infected individuals is discussed (120572)We computed the partial derivative of 119877

0with respect to this

parameterFor the case of the isolation of chronically infected

individuals it is easy to see that1205971198770

120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1

(21)which simplifies to

1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion

Without quarantineWith quarantine

(a) Positive effect of isolation measures on the infected population with1198961015840

3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion


(b) Negative effect of isolation measures on the infected population with1198961015840

3= 00803 gt 1198963 = 0517

Figure 6 Effect of isolation on the infected population

where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)

Thus the isolation of chronically infected individuals willreduce 119877

0and therefore reduce disease burden (new infec-

tions mortality etc) if 11989610158403does not exceed the threshold 119896

3

(which is the rate of transfer of individuals out of the isolationstate) This case is presented in Figure 6(a)

On the other hand if 11989610158403gt 1198963 then the use of isolation

(of chronically infected individuals) will increase 1198770 and

consequently increase disease burden (hence the use ofisolation is detrimental to the community in this case) Thiscase is presented in Figure 6(b) This important result issummarized below

Lemma 6 The use of isolation of the chronically infected indi-viduals will have positive (negative) population-level impact if1198961015840

3lt (gt)119896

3

Nowwe present the simulations of the critical parameters(as identified by sensitivity analysis) and119877

0 Figure 7 presents

the dependence of the basic reproductive number on theparameters 120572 and 120585 where 120572 denotes the isolation rate ofchronic and 120585 denotes the progression rate to chronic fromacute From the contour plot we see that if 120585 is larger then 119877

0

is always greater than one which implies that it is importantto control the acute Hep C Figure 7(b) shows that the basicreproductive number may be less than one if 120572 and 120585 can berestricted to a range leading to the potential extinction of thedisease

The dependence of basic reproductive number 1198770on

the recovery rate 120581 isolation rate 120572 and effective contactrate is explored in Figure 8 From Figure 8(a) it is clear thathigh isolation rate with low effective contact will result insmaller value of 119877

0 Furthermore 119877

0is very sensitive to 120573

and basic reproductive number increases sharply when 120573 isslightly increased Therefore keeping the effective contactrate lowwill result in disease extinction In Figure 8(b) largerrecovery rate of chronic individuals 120581 results in smaller valuesof 1198770 However still the 119877

0increases as 120573 increases but

smoothly and not sharply as seen in Figure 8(a)

3 Optimal Control Strategies

Pontryagin and Boltyanskii [30] formulated the optimalcontrol theory for models with underlying dynamics definedby a system of ordinary differential equations The theoryapplication areas and the numerical methods have pro-gressed considerably PontryaginrsquosMaximumPrinciple allowsus to adjust the control in a model to achieve the desiredresults The control parameters are mostly functions of timeappearing as coefficients in the model

Optimal control theory has been employed to make deci-sions involving epidemic and biological models The desiredresults and performance of the control functions depend onthe different situations Lenhartrsquos HIV models [21 22] usedoptimal control to design the treatment strategies Jung et al


01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)

Figure 7 Plots of the basic reproductive number 1198770in terms of the parameters 120572 and 120585 which show the estimated effects of 120572 and 120585 on 119877

0

(a) A contour plot of the surface 1198770for the values of 119877

0= 1 2 3 (b) Two surfaces 119877

0and the constant 1 are plotted to show the curve on

which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)

Figure 8 Plots of the basic reproductive number 1198770in terms of the parameters 120573 120572 and 120581 which show the effects of 120573 120572 and 120581 on 119877

0 (a)

Effect of isolation rate 120572 and effective contact rate 120573 on the 1198770 (b) Effect of recovery rate of chronic 120581 and effective contact rate 120573 on the 119877

0

[23] provide a very good example of deciding how to dividethe efforts between two treatment strategies (case holdingand case finding) of the two-strain TB model Yan et al[24] used an optimal isolation strategy to fight the SARSepidemic In [31] Joshi formulated two control functionsas coefficients of the ODE system representing treatmenteffects in a two-drug regime in an HIV immunology modelThe goal was to maximize the concentration of T cells whileminimizing the toxic effects of the drug The analytic andnumerical results illustrated the level of two drugs to be usedover the chosen time interval The required balancing effectbetween two competing goals was well predicted by optimalcontrol theory Behncke [32] studied SIR models includingvaccination isolation and health promotion campaign andobtained analytical results for optimal control The optimal

control intervention policies for stochastic epidemic modelswere treated by Clancy [33]

Pontryaginrsquos Maximum Principle appends an adjointsystem of differential equations with terminal boundaryconditions to the original model (state system) of differentialequations in the attempt to characterize an optimal con-trol The optimality system which characterizes the optimalcontrols consists of the differential equations of the originalmodel (state system) along with the adjoint differentialequations (adjoint system) The adjoint system has the samenumber of equations as in the state system The adjointfunctions behave very similar to the Lagrange multipliers(appending constraints to the function of several variables tobemaximized orminimized)The adjoint variablesmaximizeor minimize the state variables with respect to the desired


objective functional The details of the necessary conditionsfor the adjoint and optimal controls are presented here [3034 35] For the application of these results see [21]

Optimal control techniques are used to optimize themodels given by system of differential equations Whileformulating an optimal control problem deciding how andwhere to introduce the control (through vaccination drugtreatment etc) in the system of differential equation is veryimportant The formulation of the optimal control problemmust be a reasonable and practical representation of thesituation to be considered The form of the optimal controldepends heavily on the system being analyzed and the objec-tive functional to be optimized We will consider a quadraticdependence on the control in the objective functional

The questions of the existence and uniqueness of theoptimal control in an optimality system can be dealt withusing the Lipschitz properties of the differential equations[30 34] For a detailed example see the work of Fister etal [21] and Fleming and Rishel [34] After establishing theexistence and the uniqueness results we can confidentlycontinue to numerically solve the optimality system to get thedesired optimal control

In this section we consider two different time-dependentcontrol strategies to prevent and control the spread of Hep Cin the population The first strategy is to introduce vaccina-tion to our population and see the effects on the prevalenceof the Hep C We introduce a vaccination control which isa function of time in our model of differential equations tominimize the population of infected individual while keepingthe cost of the vaccination to minimum through objectivefunctional Next strategy introduces control function for theisolation rate of chronically infected individuals to minimizethe total infected population The objective functional wasdesigned carefully to minimize the infected population andthe cost of the isolation facility

31 Vaccination Control The use of vaccination is an impor-tant disease control and prevention measure Optimal vac-cination control strategy for an SIR model has been devisedusing dynamic programming technique [36] Also the opti-mal control strategies have been investigated for TB control[23 37]We have analyzed the dynamics of themodel withoutvaccination in the last section which had a locally stableequilibrium In this section we will explore the effects of avaccination campaign on our deterministic model of HepC by including vaccination After including vaccination themodel is given as follows

119889119878

119889119905= Π + 120596119877 minus 120582 (1 minus V) 119878 minus (1 minus 120590) 120582V119878 minus 120583119878 minus 120590V119878

119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)

In addition to the dynamics of the original model (1) now thesusceptible population is being vaccinated at per capita rate V(vaccine coverage rate) The vaccination rate V includes boththe medical inoculation and the educational campaigns toprevent Hep CWe also have to consider the partial efficiencyof the vaccine due to which only 120590 (0 le 120590 le 1) fraction of thevaccinated susceptible individuals go to the recovered class (120590is the vaccine efficacy) The remaining (1 minus 120590) fraction of thevaccinated susceptible individuals goes to the acute class asa result of their contact with the infected individuals When120590 = 0 it means that the vaccine has no effect at all and when120590 = 1 the vaccine is perfectly effective

It iswell understood that in order to eradicate an epidemicwe have to vaccinate a large fraction of the susceptiblepopulation Take the example of smallpox its eradication wasachieved after an intensive worldwide vaccination campaignand a very high vaccination rate [38] However in someinfectious diseases such as measles and TB the diseasepersists despite the extensive usage of vaccinemainly becauseof low vaccine efficacy [39] and vaccination campaigns thatcould not reach everyone Different vaccination policies havebeen implemented in different parts of the world Practicallythe cost of vaccine is a very important factor influencingthe policy Hence we need to find a balance between therate of vaccinating susceptible individuals and the cost of thevaccination Now we design an optimal vaccination strategyto minimize an objective functional that takes into accountboth the cost and the number of infectious individuals Nowlet the vaccination rate be given as function of time V(119905) inmodel (26) The control set V is

V = V (119905) 0 le V (119905) le 119887 0 le 119905

le 119879 V (119905) is Lebesgue measurable (28)

The goal is to minimize the cost function defined as

119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)

This performance specification involves the numbers ofindividuals of acute chronic and isolated respectively aswell as the cost of applying vaccination control (V) This costalso includes the cost for organization and management ofvaccine and so forth Hence the cost function should benonlinear In this paper a quadratic function is implementedfor measuring the control cost [21ndash24] The coefficients 119875

1

1198752 1198753 and 119882 are balancing cost factors due to scales and


importance of the four parts of the objective function Weneed to find an optimal control Vlowast(119905) such that

119869 [Vlowast] = minVisin119881

119869 [V] (30)

The existence of a solution to the optimal control problem canbe obtained by verifying sufficient conditions We refer to theconditions inTheorem III41 and its corresponding corollaryin [34] The boundedness of solutions to the system (26) forthe finite time interval is needed to establish these conditionsPontryaginrsquos Maximum Principle [30] provides the necessaryconditions to be satisfied by the optimal vaccination V(119905)This principle reduces (26) (28) and (29) into a problem ofminimizing pointwise a Hamiltonian119867 with respect to V

119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)

where 119896119894represents the right hand side of modelrsquos (26) 119894th

differential equation Using Pontryaginrsquos Maximum principle[30] and the optimal control existence result from [34] wehave the following

Theorem 7 There exists a unique optimal Vlowast(119905) which min-imizes 119869 over V Also there exists adjoint system of 120582

119894rsquos such

that

1198891205821

119889119905= (120582 (1 minus Vlowast) + (1 minus 120590) Vlowast120582 + 120590Vlowast + 120583) 120582

1

minus ((1 minus Vlowast) 120582 + (1 minus 120590) Vlowast120582) 1205822

1198891205822

119889119905= (

120573120578 (1 minus Vlowast + (1 minus 120590) Vlowast) 119878119873

)1205821

+ (1198961minus120573120578 (1 minus Vlowast + (1 minus 120590) Vlowast) 119878

119873)1205822

minus 12058512058231205821minus 1205811205825minus 1198752

1198891205823

119889119905= (


)1205821

minus120573 (1 minus Vlowast + (1 minus 120590) Vlowast) 119878

1198731205822

+ 11989621205823minus 1205721205824minus 1205951205825minus 1198751

1198891205824

119889119905= (


)1205821

minus (1198964+120573120577 (1 minus Vlowast + (1 minus 120590) Vlowast) 119878

119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)

The transversality condition gives

120582119894(119879) = 0 (34)

The vaccination control is characterized as

Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822

minus1205821205821minus 1205821205822(1 minus 120590)) )]

(35)

Proof Clearly the integrand of 119869 is convexwith respect to V(119905)Also the solutions of (26) are bounded as119873(119905) le Π120583 for alltime Also it is easily verifiable that the state system (26) hasthe Lipschitz propertywith respect to the state variablesWiththese properties and using Corollary 41 of [34] we have theexistence of the optimal control

Since we have the existence of the optimal vaccinationcontrol using Pontryaginrsquos Maximum Principle we obtain

1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)

evaluated at the optimal control which results in the statedadjoint system (32) The optimality condition is

120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)

Therefore on the set 119905 0 lt Vlowast(119905) lt 119887 we obtain

Vlowast =119878

119882(1205821205821 (1 minus 120590) + 1205901205821 + 1205821205822 minus 1205821205821 minus 1205821205822 (1 minus 120590))

(38)

Considering the bounds on Vlowast we have the characterizationsof the optimal control as in (35) Clearly the state and theadjoint functions are bounded Also it is easily verifiablethat state system and adjoint system have Lipschitz structurewith respect to the corresponding variables we obtain theuniqueness of the optimal control for sufficiently small time119879 [30 34] The uniqueness of the optimal control pairfollows from the uniqueness of the optimality system whichconsists of (26) and (32) with characterizations (35) Thereis a restriction on the length of the time interval in order


0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol

(a) Optimal vaccination control V(119905)

0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion

Constant controlNo controlOptimal control

(b) Total infected population with different control strategies

Figure 9 Simulations show the optimal vaccination control and its effectiveness The left simulation presents the vaccination strategy to befollowed to prevent the epidemic and disease spread The right simulation presents comparison of the total chronically infected individualsunder optimal and constant control Clearly optimal strategy prevents the epidemic and retains the infected population to a minimum

to guarantee the uniqueness of the optimality system Thisrestriction on the length of the time interval is due to theopposite time orientations of (26) and (32) the state problemhas initial values and the adjoint problem has final values Forexample see [21] This restriction is very common in controlproblems [22]

The following optimality system consisting of 10 equa-tions characterizes the optimal vaccination control asdefined in above theorem It consists of five equations of(26) with initial condition 119878(0) = 119878

0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877

0 and five equations of (32) with

transversality condition 1205821(119879) = 120582

2(119879) = 120582

3(119879) = 120582

4(119879) =

1205825(119879) = 0Next we discuss the numerical solutions of the optimality

system and the corresponding optimal control pairs and theparameters The optimal vaccination strategy is obtained bysolving the optimality system (26) and (32) consisting of10 ODEs from the state and adjoint equations An iterativemethod is used for solving the optimality system We startto solve the state equations with a guess for the controlV(119905) over the simulated time using a forward fourth-orderRunge-Kutta scheme The adjoint functions have final timeconditions Because of this transversality condition on theadjoint functions (32) the adjoint equations are solvedby a backward fourth-order Runge-Kutta scheme usingthe current iteration solution of the state equations Thenthe controls are updated by using a convex combination ofthe previous control and the value from the characterizationsin (35)This process is repeated and iteration is stopped if thevalues of unknowns at the previous iteration are very close tothe ones at the present iteration

Numerical solutions to the optimal system (26) and (32)are carried out using MATLAB and are presented here Theparameter values and the initial conditions are given in theappendix The parameter values used have 119877

0gt 1 when the

model without control is considered Thus the disease is notexpected to die out without intervention strategies

Figure 9(a) represents the control strategy to be employedfor the optimal results This control strategy minimizes boththe cost and the infected population (119860 + 119862 + 119876) It is wellunderstood that in order to eradicate an epidemic a largefraction of susceptible population needs to be vaccinatedTherefore an upper bound of 119887 = 07was chosen for vaccina-tion control V(119905) The optimal control V is at its upper boundduring the first 60 days and then V is steadily decreasing to0 In fact at the beginning of simulated time the optimalcontrol is staying at its upper bound in order to vaccinateas many susceptible individuals as possible to prevent theindividuals from getting infectedThe steady decrease of the Vis determined by the balance between the cost of the infectedindividuals and the cost of the controls Figure 9(b) shows thetotal infected population for the optimal vaccination controlconstant vaccination andwithout control It is easy to see thatthe optimal control is much more effective for reducing thenumber of infected individuals and decreasing the time-spanof the epidemic As normally expected in the early phase ofthe epidemic breakouts keeping the vaccination controls attheir upper bound will directly lead to the decrease of thenumber of the infected people

Figure 10 shows the cost associated with the optimaland constant control strategy It is clear that the cost ofoptimal strategy is much less than the cost of constantstrategy In fact the costs differ by order of magnitude often Figure 11 represents the population sizes of the distinct


0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J

Optimal controlConstant control

Figure 10 Simulation of accumulated cost of different controlstrategies

infected states (acute chronic and isolated) andwith optimalcontrol strategy population sizes go to minimum in shortperiod of time

32 Isolation Control Several approaches to Hep C vaccinedevelopment have now been studied and include syntheticpeptides DNA [40] recombinant E1 and E2 proteins [41]and prime-boost strategies [42] Success of these approacheshas been limited for a number of reasons including thedelivery of a limited number of protective viral epitopesthe inclusion of incorrectly folded recombinant proteinsthe limited humoral and cell mediated responses that areassociated with DNA vaccines and the use of adjuvants withrelatively poor potency It is also now apparent that vaccineinducing strong T cell responses alone may not be sufficientto prevent hepatitis C infection [40] An effective preventivevaccine against Hep C will therefore need to induce strongneutralizing and cellular immune responses [7]

Since an effective vaccine is not available against all thegenotypes of Hep C we have to look for alternate strategiesto control the spread of Hep C The quarantine and isolationof those individuals with disease symptoms constitute whatis probably the first infection control measure since thebeginning of recorded human history [14] In our modelof Hep C the isolation compartment was introduced toinvestigate the effect on the infected population size andresults were discussed in the last section Nowwe can attemptto control the isolation rate of the chronically infected indi-viduals in order to control theHepCThis sectionwill explorethe effects of isolation control rate of chronically infectedindividuals on the total size of the infected population Themodel including the required control is as follows

119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)

In addition to the dynamics of the original model (1)now we have a control parameter for the isolation rate of thechronically infected individuals labeled as 119906

1 The control 119906

1

represents the fraction of the chronically infected individualsthat are being isolated in order to decrease the rate of thespread of infection

Now we design an optimal control strategy to minimizean objective functional that takes into account both thecost and the number of infectious individuals Now let theisolation rate be 119906

1(119905) for model (39) The control set U is

U = 1199061 (119905) 0 le 1199061 (119905) le 119887

0 le 119905 le 119879 1199061(119905) are Lebesgue measurable

(41)

The objective functional is defined as

119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)

where we want to minimize the infectious individuals whilealso keeping the cost of the isolation facilities low119882

1 1198751 1198752

and 1198753are the weight parameters We need to find an optimal

control 119906lowast1(119905) such that

119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)

Pontryaginrsquos Maximum Principle [30] provides the necessaryconditions to be satisfied by the optimal vaccination V(119905)This principle reduces (39) (41) and (42) into a problem ofminimizing pointwise a Hamiltonian 119867 with respect to 119906

1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)


differential equation Using Pontryaginrsquos Maximum Principle[30] and the optimal control existence result from [34] wehave the following


0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion

Optimal controlWithout controlConstant control

(a) Simulation of acute (Hep C) population with different controlstrategies


0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n

(b) Simulation of chronic (Hep C) population with different controlstrategies


0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n

(c) Simulation of isolated population with different control strategies

Figure 11 Population sizes of different infected compartments under vaccination strategy This simulation presents comparison of the totalchronically infected isolated and acute individuals under optimal and constant control Clearly optimal strategy prevents the epidemic andretains the infected population to a minimum

Theorem 8 There exists a unique optimal control 119906lowast1(119905)which

minimizes 119869 overU Also there exists adjoint system of 120582119894rsquos such

that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575

119888) and all other 119896rsquos are given above

The transversality condition is

120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l

(a) Optimal isolation control V(119905)

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion

Optimal controlConstant controlNo control


Figure 12 Simulations show the optimal isolation control and its effectivenessThe left simulation presents the isolation strategy to be followedto prevent the epidemic and disease spread The right simulation presents comparison of the total chronically infected individuals underoptimal and constant control Clearly optimal strategy prevents the epidemic and retains the infected population to a minimum

0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J


Figure 13 Accumulated cost of different control strategies simula-tion presents comparison of the cost incurred to implement optimaland different constant control strategies to control hepatitis C

The optimal treatment control pair is characterised as

119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)

The proof is identical to the proof of Theorem 7In this case the system of 10 differential equations as

stated above characterizes the optimal treatment control pairconsisting of five equations of system (39) and five equationsof system (45)

Now we discuss the numerical solutions of the optimalitysystem and the corresponding optimal control pair and the

parameters The optimal treatment strategy is obtained bysolving the optimality system (39) and (45) consisting of10 ODEs from the state and adjoint system The method isdiscussed in the last section

Figure 12(a) represents the optimal isolation strategy to beemployed to minimize the cost and the infected populationConsidering the practical constraints an upper bound of119887 = 07 was chosen for the optimal isolation control 119906

1(119905)

The optimal control 1199061is at its upper bound during the

first 70 days and then 1199061is steadily decreasing to 0 In

fact at the beginning of simulated time the optimal controlis staying at its upper bound in order to isolate as manychronically infected individuals as possible to prevent theinfected population from increasing The steady decrease of1199061is determined by the balance between the cost of the

infected individuals and the cost of the controls Figure 12(b)shows the total infected population for the optimal controlconstant control and without control It is clear that withthe use of an optimal control strategy the disease can beeradicated in a short period of time Figure 13 shows the costassociated with the optimal and constant control strategy Itis clear that the cost of optimal strategy is much less thanthe cost of constant strategy and in fact differs by order ofmagnitude of tens Figure 14 represents the population sizesof the distinct infected states (acute chronics and isolated)and with optimal control strategy population sizes go tominimum in short period of time

4 Conclusions

This paper has discussed the transmission dynamics of theHep C and eventually formulated an optimal control strategyto prevent disease spread At first a deterministic epidemic


0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n

Optimal controlConstant controlWithout control

(a) Simulation of chronic (Hep C) population with different controlstrategies

0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n


(b) Simulation of isolated population with different control strategies

Figure 14 Population Sizes of different infected compartments under optimal Isolation strategy Simulation presents comparison of the totalchronically infected and isolated individuals under optimal and constant control Clearly optimal strategy prevents the epidemic and retainsthe infected population to a minimum

model for the spread of Hep C which incorporates thepossibility of an isolation state is formulated Global analysisof the equilibrium solution is performed The existenceof a disease-free equilibrium and an endemic equilibriumis shown It is further demonstrated that the disease-freeequilibrium is globally asymptotically stable for values ofthe basic reproduction number (119877

0lt 1) The disease

uniformly persists when 1198770gt 1 In addition it is shown

that the endemic equilibrium is locally asymptotically stablefor values of the basic reproduction number (119877

0gt 1)

assuming constant total population It is also shown thatnumerical simulations of the deterministic model agree withthe theoretical results Finally the sensitivity and uncertaintyanalyses were performed alongwith numerical simulations tostudy the influence of vital parameters on the disease spread

The model is most sensitive to the control variables120572 (proportion of chronic population being isolated) and120573 (effective contact rate) given the nature of the diseasecontrolling the isolation parameter (ie devising an effectiveisolation strategy) seems to be the most workable solution

Since we have a stable endemic equilibrium we formu-lated (time-dependent) optimal control strategies to fightHep C We considered the two most effective strategiesknown to prevent disease spread vaccination and isolationseparately Absence of an effective vaccine against all knowngenotypes of Hep C made us consider vaccination andisolation strategy separatelyWe assumed that the vaccinationnot only includes medical vaccination but also the educationand awareness schemes about Hep C

The numerical results show that the proportion thatis isolated optimally with respect to time has a higherfavorable impact (as compared to implementing a high but

constant isolation rate) on keeping the cost of disease controllow However it should be pointed out that the ideal timevarying optimal strategy might not be applied easily Stillit does provide a basis on which can be designed practicalquasioptimal control strategies

Next we consider the possibility of an effective vaccinebecoming available for Hep C We formulate a possibleoptimal time varying vaccination strategy our analysis showsthat it is not required to vaccinate a constant proportion ofthe susceptible population over time but rather the optimalpolicy (being cost effective) is to start with vaccinating a con-stant proportion of the susceptibles and then progressivelyreducing the fraction of the susceptibles vaccinated

Although the early diagnosis and treatment of Hep Cvirus infection are desirable to prevent spread of infectionand to reduce the risk of progression of disease the majorityof acute individuals are asymptomatic and most infectedpersons are unaware of their exposure to the virus Morepublic awareness of Hep C may also increase the recoveryfrom acute stage in the model The goal of therapy is toprevent complications and death from the infectionTherapyalong with an effective isolation strategy can greatly reducethe prevalence of Hep C

Appendix

Parameter Values

Sensitivity analysis is shown in Table 1Disease-free equilibrium Π = 012 120574 = 02 119891 = 08 120581 =

02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018


Table 1 Sensitivity analysis

Parameter Mean standard (distribution)120583 1119864

minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)

120574 015 001 (119873)120585 08 005 (119873)120572 02 005 (119880)120581 02 minus005 (119873)120595 005 5119864minus3 (119873)120573 03 005 (119866)120578 04 005 (119866)120577 01 005 (119866)The 119873 119880 and 119866 stand for normal uniform and gamma distributionrespectively

Endemic equilibrium 120573 = 035 all other values are thesame

Optimal vaccination control Π = 12 120574 = 02 119891 =

08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15

Optimal isolation control 1198751= 01 119875

2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A M Di Bisceglie ldquoHep CrdquoThe Lancet vol 350 pp 1209ndash12111998

[2] N Jiwani and R Gul ldquoA Silent Storm Hep C in PakistanrdquoJournal of Pakistan Medical Students vol 1 pp 89ndash91 2011

[3] A M Di Bisceglie S E Order J L Klein et al ldquoThe role ofchronic viral hepatitis in hepatocellular carcinoma in theUnitedStatesrdquo American Journal of Gastroenterology vol 86 no 3 pp335ndash338 1991

[4] G Fattovich G Giustina F Degos et al ldquoMorbidity andmortality in compensated cirrhosis type C a retrospectivefollow-up study of 384 patientsrdquo Gastroenterology vol 112 no2 pp 463ndash472 1997

[5] YHutinM E Kitler G J Dore et al ldquoGlobal Burden ofDisease(GBD) forHepCrdquoThe Journal of Clinical Pharmacology vol 44pp 20ndash29 2004

[6] A Jawaid and A K Khuwaja ldquoTreatment and vaccination forHep C present and futurerdquo Journal of Ayub Medical CollegeAbbottabad vol 20 pp 129ndash133 2008

[7] J Torresi D Johnson and H Wedemeyer ldquoProgress in thedevelopment of preventive and therapeutic vaccines for Hep Cvirusrdquo Journal of Hepatology vol 54 pp 1273ndash1285 2011

[8] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[9] I K Dontwi N K Frempong D E Bentil I Adetundeand E Owusu-Ansah ldquoMathematical modeling of Hep CVirus transmission among injecting drug users and the impactof vaccinationrdquo American Journal of Scientific and IndustrialResearch vol 1 pp 41ndash46 2010

[10] F Luo and Z Xiang ldquoGlobal analysis of an endemic model withacute and chronic stagesrdquo International Mathematical Forumvol 7 pp 75ndash81 2012

[11] M Martcheva and C Castillo-Chavez ldquoDiseases with chronicstage in a population with varying sizerdquo Mathematical Bio-sciences vol 182 no 1 pp 1ndash25 2003

[12] B Reade R G Bowers M Begon and R Gaskell ldquoA model ofdisease and vaccination for infections with acute and chronicphasesrdquo Journal of Theoretical Biology vol 190 pp 355ndash3671998

[13] S Suna C Guob andC Lia ldquoGlobal analysis of an SEIRSmodelwith saturating contact raterdquo Applied Mathematical Sciencesvol 6 pp 3991ndash4003 2012

[14] H W Hethcote ldquoMathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[15] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004

[16] M Lipsitch ldquoTransmission dynamics and control of severeacute respiratory syndromerdquo Science vol 300 pp 1966ndash19702003

[17] J O Lloyd-Smith A P Galvani and W M Getz ldquoCurtailingtransmission of severe acute respiratory syndrome within acommunity and its hospitalrdquo Proceedings of the Royal Society Bvol 170 pp 1979ndash1989 2003

[18] R G McLeod J F Brewster A B Gumel and D A SlonowskyldquoSensitivity and uncertainty analyses for a sars model withtime-varying inputs and outputsrdquoMathematical Biosciences andEngineering vol 3 pp 527ndash544 2006

[19] X Yan and Y Zou ldquoOptimal and sub-optimal quarantineand isolation control in SARS epidemicsrdquo Mathematical andComputer Modelling vol 47 pp 235ndash245 2008

[20] X Yan and Y Zou ldquoControl of Epidemics by quarantine andisolation strategies in highly mobile populationsrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp271ndash286 2009

[21] K R Fister S Lenhart and J S McNally ldquoOptimizingchemotherapy in an HIV modelrdquo Electronic Journal of Differ-ential Equations vol 1998 no 32 pp 1ndash12 1998

[22] D Kirschner S Lenhart and S Serbin ldquoOptimal control of thechemotherapy of HIVrdquo Journal of Mathematical Biology vol 35no 7 pp 775ndash792 1997

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] X Yan Y Zou and J Li ldquoOptimal quarantine and isolationstrategies in epidemics controlrdquoWorld Journal of Modelling andSimulation vol 33 pp 202ndash211 2007

[25] I K Dontwi N K Frempong D E Bentil I Adetundeand E Owusu-Ansah ldquoMathematical modeling of Hepatitis CVirus transmission among injecting drug users and the impact


of vaccinationrdquo American Journal of Scientific and IndustrialResearch vol 1 pp 41ndash46 2010

[26] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[27] A S Ackleha B Maa and P L Salceanua ldquoPersistence andglobal stability in a selectionmutation size-structured modelrdquoJournal of Biological Dynamics vol 5 no 5 pp 436ndash453 2011

[28] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press Cambridge UK 1995

[29] H R Thieme ldquoPersistence under relaxed point-dissipativityrdquoSIAM Journal on Mathematical Analysis pp 407ndash435 1993

[30] L S Pontryagin andV G BoltyanskiiTheMathematicalTheoryof Optimal Processes Golden and Breach Science 1986

[31] H R Joshi ldquoOptimal control of an HIV immunology modelrdquoOptimal Control Applications and Methods vol 23 no 4 pp199ndash213 2002

[32] H Behncke ldquoOptimal control of deterministic epidemicsrdquoOptimal Control Applications and Methods vol 21 no 6 pp269ndash285 2000

[33] D Clancy ldquoOptimal intervention for epidemic models withgeneral infection and removal rate functionsrdquo Journal of Math-ematical Biology vol 39 no 4 pp 309ndash331 1999

[34] W H Fleming and R W Rishel Deterministic and StochasticOptimal Control Springer 1975

[35] M L Kamien and N L SchwartzDynamic Optmisation NorthHolland Amsterdam The Netherlands 1991

[36] H W Hethcote and P Waltman ldquoOptimal vaccination sched-ules in deterministic epidemic modelrdquo Mathematical Bio-sciences vol 18 pp 365ndash381 1973

[37] F B Agusto ldquoOptimal chemoprophylaxis and treatment controlstrategies of a tuberculosis transmission modelrdquo World Journalof Modelling and Simulation vol 5 no 3 pp 163ndash173 2009

[38] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[39] A A Saylers and D D Whitt Bacterial Paathogenesis A ASMPress Washington DC USA 2001

[40] Q-L Choo G Kuo R Ralston et al ldquoVaccination of chim-panzees against infection by the hepatitis C virusrdquoProceedings ofthe National Academy of Sciences of the United States of Americavol 91 no 4 pp 1294ndash1298 1994

[41] W Hackbusch ldquoA numerical method for solving parabolicequations with opposite orientationsrdquo Computing vol 20 no3 pp 229ndash240 1978

[42] M Puig K Mihalik J C Tilton et al ldquoCD4+ immune escapeand subsequent t-cell failure following chimpanzee immuniza-tion against hepatitis C virusrdquo Hepatology vol 44 no 3 pp736ndash745 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


existent for genotype 6 Recently there have been promisingtreatment advances of genotype 1 using directly acting antivi-ral agents (DAAs) However to prevent the infection there isstill a need to create effective vaccine strategyVaccines whichare used in preclinical and clinical trials involve DNA-basedproteins recombinant proteins synthetic peptide vaccinesand so forth The future design of vaccines along withthe use and success of previously mentioned vaccines hasbeen discussed here [7] However no long-term immunityis granted in recovering from hepatitis C infection So thisabsence of acquired immunity must be shown in any modelfor hepatitis CThis is done by allowing recovered patients tobecome susceptible again

Several studies have been carried out [8ndash13] and theyare pertinent to our work Most of these papers classifyindividuals in the population into different states and thenformulate a system of ordinary differential equations (ODE)to analyze the time-evolution of each of these populationstates Reade et al [12] present an ODE model of infectionswith acute and chronic stages Similarly by Luo and Xiang[10] a four state system was analyzed with exposed acute andchronic states Suna et al [13] present a study on a SEIRSmodel where it was assumed that recovered individuals losetheir infection-acquired immunity Martcheva and Castillo-Chavez in [11] have formulated amodel for hepatitis C lackingan exposed class and have discussed the stability of theequilibrium states

We will formulate a five-state deterministic model withindividuals in the population being classified as susceptibleacute chronic isolated and recovered Individuals sufferingfrom acute and chronic stages of the infections are repre-sented by acute and chronic states respectively The isolatedstate represents the chronically infected individuals gettingisolated The isolation of those with disease symptoms isprobably the first infection control measure in recordedhuman history [14] Over the decades quarantine and isola-tion have been used to reduce the transmission of numerousemerging and reemerging human diseases such as leprosyplague cholera typhus yellow fever smallpox diphtheriatuberculosis measles Ebola pandemic influenza and morerecently SARS [15ndash20]

Previous models of HCV particularly the models cal-culating the threshold quantity basic reproduction number1198770 have included the treatment andor vaccination and have

discussed the control of the disease by looking at the roleof disease transmission parameters in the reduction of 119877

0

and the prevalence of the disease However these modelsdid not account for time-dependent control strategies sincetheir discussions are based on prevalence of the disease atequilibria Optimal control theory has been employed tomake decisions involving epidemic and biological modelsThe desired results and performance of the control functionsdepend on the different situations Lenhartrsquos HIVmodels [2122] used optimal control to design the treatment strategiesJung et al [23] provide a very good example of deciding howto divide the efforts between two treatment strategies (caseholding and case finding) of the two-strain TB model Yanet al [24] used an optimal isolation campaign to fight theSARS epidemic Study control strategies produce valuable

theoretical results which can be used to suggest or designepidemic control programs Depending on a chosen goal (orgoals) various objective criteria may be adopted

Our model extends previous work done on modeling thespread of hepatitis C in several key ways First we introducean isolation class and qualitatively assess the effects of thisisolation class on the transmission dynamics Quarantine ofindividuals suspected of being exposed to a disease and theisolation of those with disease symptoms constitute whatprobably is the first infection control measure since thebeginning of recorded human history [14] However almostno analysis of the effects of an isolation class on diseases witha chronic stage has been done and therefore our paper willbe one of the first attempts to study the effect of isolationon the spread of a disease with a chronic stage In additionwe will model the force of infection by a proportionatemixing with the possibility of secondary infections due tocontact with individuals who belong to the acute chronicor isolation class Furthermore we consider the disease-induced death rates for HCV in our model and will alsotake into account the possibility of recovery at every stageof the disease These features add to the complexity of ourmodel and make it considerably more insightful from anepidemiological perspective than previous models [11 25]

In the paper Section 2 presents a rigorous mathematicalanalysis of the deterministic model It is shown that for basicreproduction number 119877

0le 1 the disease-free equilibrium

is locally and globally asymptotically stable Further themodel has an endemic equilibrium which exists if 119877

0gt 1

and persists in this case The effect of using isolation onpopulation is discussed using a threshold quantity Sensitivityand uncertainty analyses are carried out to study the impactof crucial parameters on 119877

0 The existence of a locally stable

endemic equilibrium in case of 1198770gt 1 encourages us to

use a time-dependent optimal control strategy to prevent andcontrol HCV In Section 3 we designed two different optimalcontrol strategies (vaccination and isolation) and performednumerical simulations to illustrate the effects of an optimalcontrol strategy Conclusion is presented in the last section

2 Model Formulation andSteady State Analysis

21Model Formulation We formulate a five-statemodel withindividuals classified as susceptible acute chronic isolatedand recovered Hep C has an extremely slow progressionthat makes it difficult to characterize the natural history ofthe disease [3] The following assumptions will therefore bemade

(1) All infected individuals develop the acute formofHepC first

(2) Individuals with either the acute or chronic form ofHep C are capable of transmitting the disease

(3) Individuals with the acute form of the disease eitherprogress to the chronic form or recover naturallySince the acute form of the disease is largely asymp-tomatic there is little chance of treatment at this stage











119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595 + 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(1)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (2)


Variable


Parameter







119863 = (119878 119860 119862 119876 119877) isin 1198775

+ 119878 + 119860 + 119862 + 119876 + 119877 le

Π

120583 (3)






Π

120583 0 0 0 0) (4)




120583

120583

120583

120583120583

S

120582

120596 120585

120581C

A

120595 120572

R Qf120574

120575a

120575q

120575c

(1 minus f)120574

prod



119865 = (

120573120578 120573 120573120577

0 0 0

0 0 0

)

119881 = (

120585 + 120581 + 120583 + 120575119886

0 minus120574 (1 minus 119891)

minus120585 120572 + 120595 + 120583 + 120575119888

0

0 minus120572 120574 + 120583 + 120575119902

)

(5)


0 0120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

(6)


=

120588(119865119881minus1) is given by

1198770=120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

gt 0 (7)

where

1198961= (120585 + 120581 + 120583 + 120575

119886)

1198962= (120572 + 120595 + 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902)

1198964= 120574 (1 minus 119891)

(8)


















(120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)

infin

sum

119899=0

[120585120572120574(1 minus 119891)

119896111989621198963

]

119899

= (120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)[1

1 minus (120585120572120574 (1 minus 119891) 119896111989621198963)]

= 1198770

(9)




0gt 1



0lt 1 This is

explored below



0le 1


119871 = 119886119860 + 119888119862 + 119902119876 (10)

where

119886 =11989621198963

119896111989621198963minus 120572120585119896

4

119888 =1205731198963+ 120572 [120573120577 + 119896

4]

119896111989621198963minus 120572120585119896

4

119902 =1198962[120573120577 + 119896

4]

119896111989621198963minus 1205721205851198964

(11)



= 119886 + 119888 + 119902

= 119886120582119878 + 1198861198964119876 minus 119886119896

1119860 + 119888120585119860 minus 119902119896

3119876 + 119902120572119862 minus 119888119896

2119862

= 119886120573[120578119860 + 119862 + 120577119876]

119873119878 + [119886119896

4minus 1199021198963] 119876

+ [119888120585 minus 1198861198961] 119860 + [119902120572 minus 119888119896

2] 119862

le [119886120573120578 + 119888120585 minus 1198861198961] 119860 + [119886120573 + 119902120572 minus 119888119896

2] 119862

+ [119886120573120577 + 1198861198964minus 1199021198963] 119876

=[12057312057811989621198963+ 120585120573119896

3+ 120585120572120573120577 + 120585120572120573119896

4minus 119896111989621198963]

119896111989621198963minus 120572120585119896

4

119860

+[11989621198963120573 + 120572119896

2120573120577 + 120572119896


21198964]

119896111989621198963minus 120572120585119896

4

119862

+[11989621198963120573120577 + 119896

211989631198964minus 11989621198963120573120577 minus 119896

211989631198964]

119896111989621198963minus 120572120585119896

4

119876

= (1198770minus 1)119860

(12)

Thus

le (1198770minus 1)119860 le 0 for 119877

0lt 1 (13)






0lt 1



(119878 119860 119862 119876 119877) 997888rarr alefsym0= (

Π

120583 0 0 0 0) as 119905 997888rarr infin

(14)









0 50 100 150 200 2500

100

200

300

400

500

600

Time (days)

Tota

l in

fect

ed p

opul

atio

n






state gives




1198963

119862lowastlowast

119878lowastlowast=

1


4

1205851198963

)119862lowastlowast


1198965

(119891120574120572

1198963

+1205811198962

120585+ 120595)119862

lowastlowast

(15)

where



119873lowastlowast 1198965= (120596 + 120583) (16)



120585[1198770minus 1



1198770minus 1



1198963

[1198770minus 1



1198965

[11989621198963+ 1198963120581120585 + 120574119891120572119896

2

11989621198963

] [1198770minus 1


(17)

where

119884 = [1198962

120585+ 1 +

120572

1198963

] (18)








lim119905rarrinfin


inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)


Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus



0attracts all the



1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It





0) = 119877

0gt 1 the







0gt





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium







120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575



0is the output










05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


























119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595 + 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(1)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (2)


Variable


Parameter







119863 = (119878 119860 119862 119876 119877) isin 1198775

+ 119878 + 119860 + 119862 + 119876 + 119877 le

Π

120583 (3)






Π

120583 0 0 0 0) (4)




120583

120583

120583

120583120583

S

120582

120596 120585

120581C

A

120595 120572

R Qf120574

120575a

120575q

120575c

(1 minus f)120574

prod



119865 = (

120573120578 120573 120573120577

0 0 0

0 0 0

)

119881 = (

120585 + 120581 + 120583 + 120575119886

0 minus120574 (1 minus 119891)

minus120585 120572 + 120595 + 120583 + 120575119888

0

0 minus120572 120574 + 120583 + 120575119902

)

(5)


0 0120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

(6)


=

120588(119865119881minus1) is given by

1198770=120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

gt 0 (7)

where

1198961= (120585 + 120581 + 120583 + 120575

119886)

1198962= (120572 + 120595 + 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902)

1198964= 120574 (1 minus 119891)

(8)


















(120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)

infin

sum

119899=0

[120585120572120574(1 minus 119891)

119896111989621198963

]

119899

= (120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)[1

1 minus (120585120572120574 (1 minus 119891) 119896111989621198963)]

= 1198770

(9)




0gt 1



0lt 1 This is

explored below



0le 1


119871 = 119886119860 + 119888119862 + 119902119876 (10)

where

119886 =11989621198963

119896111989621198963minus 120572120585119896

4

119888 =1205731198963+ 120572 [120573120577 + 119896

4]

119896111989621198963minus 120572120585119896

4

119902 =1198962[120573120577 + 119896

4]

119896111989621198963minus 1205721205851198964

(11)



= 119886 + 119888 + 119902

= 119886120582119878 + 1198861198964119876 minus 119886119896

1119860 + 119888120585119860 minus 119902119896

3119876 + 119902120572119862 minus 119888119896

2119862

= 119886120573[120578119860 + 119862 + 120577119876]

119873119878 + [119886119896

4minus 1199021198963] 119876

+ [119888120585 minus 1198861198961] 119860 + [119902120572 minus 119888119896

2] 119862

le [119886120573120578 + 119888120585 minus 1198861198961] 119860 + [119886120573 + 119902120572 minus 119888119896

2] 119862

+ [119886120573120577 + 1198861198964minus 1199021198963] 119876

=[12057312057811989621198963+ 120585120573119896

3+ 120585120572120573120577 + 120585120572120573119896

4minus 119896111989621198963]

119896111989621198963minus 120572120585119896

4

119860

+[11989621198963120573 + 120572119896

2120573120577 + 120572119896


21198964]

119896111989621198963minus 120572120585119896

4

119862

+[11989621198963120573120577 + 119896

211989631198964minus 11989621198963120573120577 minus 119896

211989631198964]

119896111989621198963minus 120572120585119896

4

119876

= (1198770minus 1)119860

(12)

Thus

le (1198770minus 1)119860 le 0 for 119877

0lt 1 (13)






0lt 1



(119878 119860 119862 119876 119877) 997888rarr alefsym0= (

Π

120583 0 0 0 0) as 119905 997888rarr infin

(14)









0 50 100 150 200 2500

100

200

300

400

500

600

Time (days)

Tota

l in

fect

ed p

opul

atio

n






state gives




1198963

119862lowastlowast

119878lowastlowast=

1


4

1205851198963

)119862lowastlowast


1198965

(119891120574120572

1198963

+1205811198962

120585+ 120595)119862

lowastlowast

(15)

where



119873lowastlowast 1198965= (120596 + 120583) (16)



120585[1198770minus 1



1198770minus 1



1198963

[1198770minus 1



1198965

[11989621198963+ 1198963120581120585 + 120574119891120572119896

2

11989621198963

] [1198770minus 1


(17)

where

119884 = [1198962

120585+ 1 +

120572

1198963

] (18)








lim119905rarrinfin


inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)


Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus



0attracts all the



1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It





0) = 119877

0gt 1 the







0gt





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium







120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575



0is the output










05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















120583

120583

120583

120583120583

S

120582

120596 120585

120581C

A

120595 120572

R Qf120574

120575a

120575q

120575c

(1 minus f)120574

prod



119865 = (

120573120578 120573 120573120577

0 0 0

0 0 0

)

119881 = (

120585 + 120581 + 120583 + 120575119886

0 minus120574 (1 minus 119891)

minus120585 120572 + 120595 + 120583 + 120575119888

0

0 minus120572 120574 + 120583 + 120575119902

)

(5)


0 0120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

(6)


=

120588(119865119881minus1) is given by

1198770=120573 [12057811989621198963+ 1205851198963+ 120577120572120585]

119896111989621198963minus 120572120585119896

4

gt 0 (7)

where

1198961= (120585 + 120581 + 120583 + 120575

119886)

1198962= (120572 + 120595 + 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902)

1198964= 120574 (1 minus 119891)

(8)


















(120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)

infin

sum

119899=0

[120585120572120574(1 minus 119891)

119896111989621198963

]

119899

= (120573120578

1198961

+120573120585

11989611198962

+120573120577120585120572

119896111989621198963

)[1

1 minus (120585120572120574 (1 minus 119891) 119896111989621198963)]

= 1198770

(9)




0gt 1



0lt 1 This is

explored below



0le 1


119871 = 119886119860 + 119888119862 + 119902119876 (10)

where

119886 =11989621198963

119896111989621198963minus 120572120585119896

4

119888 =1205731198963+ 120572 [120573120577 + 119896

4]

119896111989621198963minus 120572120585119896

4

119902 =1198962[120573120577 + 119896

4]

119896111989621198963minus 1205721205851198964

(11)



= 119886 + 119888 + 119902

= 119886120582119878 + 1198861198964119876 minus 119886119896

1119860 + 119888120585119860 minus 119902119896

3119876 + 119902120572119862 minus 119888119896

2119862

= 119886120573[120578119860 + 119862 + 120577119876]

119873119878 + [119886119896

4minus 1199021198963] 119876

+ [119888120585 minus 1198861198961] 119860 + [119902120572 minus 119888119896

2] 119862

le [119886120573120578 + 119888120585 minus 1198861198961] 119860 + [119886120573 + 119902120572 minus 119888119896

2] 119862

+ [119886120573120577 + 1198861198964minus 1199021198963] 119876

=[12057312057811989621198963+ 120585120573119896

3+ 120585120572120573120577 + 120585120572120573119896

4minus 119896111989621198963]

119896111989621198963minus 120572120585119896

4

119860

+[11989621198963120573 + 120572119896

2120573120577 + 120572119896


21198964]

119896111989621198963minus 120572120585119896

4

119862

+[11989621198963120573120577 + 119896

211989631198964minus 11989621198963120573120577 minus 119896

211989631198964]

119896111989621198963minus 120572120585119896

4

119876

= (1198770minus 1)119860

(12)

Thus

le (1198770minus 1)119860 le 0 for 119877

0lt 1 (13)






0lt 1



(119878 119860 119862 119876 119877) 997888rarr alefsym0= (

Π

120583 0 0 0 0) as 119905 997888rarr infin

(14)









0 50 100 150 200 2500

100

200

300

400

500

600

Time (days)

Tota

l in

fect

ed p

opul

atio

n






state gives




1198963

119862lowastlowast

119878lowastlowast=

1


4

1205851198963

)119862lowastlowast


1198965

(119891120574120572

1198963

+1205811198962

120585+ 120595)119862

lowastlowast

(15)

where



119873lowastlowast 1198965= (120596 + 120583) (16)



120585[1198770minus 1



1198770minus 1



1198963

[1198770minus 1



1198965

[11989621198963+ 1198963120581120585 + 120574119891120572119896

2

11989621198963

] [1198770minus 1


(17)

where

119884 = [1198962

120585+ 1 +

120572

1198963

] (18)








lim119905rarrinfin


inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)


Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus



0attracts all the



1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It





0) = 119877

0gt 1 the







0gt





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium







120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575



0is the output










05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















= 119886 + 119888 + 119902

= 119886120582119878 + 1198861198964119876 minus 119886119896

1119860 + 119888120585119860 minus 119902119896

3119876 + 119902120572119862 minus 119888119896

2119862

= 119886120573[120578119860 + 119862 + 120577119876]

119873119878 + [119886119896

4minus 1199021198963] 119876

+ [119888120585 minus 1198861198961] 119860 + [119902120572 minus 119888119896

2] 119862

le [119886120573120578 + 119888120585 minus 1198861198961] 119860 + [119886120573 + 119902120572 minus 119888119896

2] 119862

+ [119886120573120577 + 1198861198964minus 1199021198963] 119876

=[12057312057811989621198963+ 120585120573119896

3+ 120585120572120573120577 + 120585120572120573119896

4minus 119896111989621198963]

119896111989621198963minus 120572120585119896

4

119860

+[11989621198963120573 + 120572119896

2120573120577 + 120572119896


21198964]

119896111989621198963minus 120572120585119896

4

119862

+[11989621198963120573120577 + 119896

211989631198964minus 11989621198963120573120577 minus 119896

211989631198964]

119896111989621198963minus 120572120585119896

4

119876

= (1198770minus 1)119860

(12)

Thus

le (1198770minus 1)119860 le 0 for 119877

0lt 1 (13)






0lt 1



(119878 119860 119862 119876 119877) 997888rarr alefsym0= (

Π

120583 0 0 0 0) as 119905 997888rarr infin

(14)









0 50 100 150 200 2500

100

200

300

400

500

600

Time (days)

Tota

l in

fect

ed p

opul

atio

n






state gives




1198963

119862lowastlowast

119878lowastlowast=

1


4

1205851198963

)119862lowastlowast


1198965

(119891120574120572

1198963

+1205811198962

120585+ 120595)119862

lowastlowast

(15)

where



119873lowastlowast 1198965= (120596 + 120583) (16)



120585[1198770minus 1



1198770minus 1



1198963

[1198770minus 1



1198965

[11989621198963+ 1198963120581120585 + 120574119891120572119896

2

11989621198963

] [1198770minus 1


(17)

where

119884 = [1198962

120585+ 1 +

120572

1198963

] (18)








lim119905rarrinfin


inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)


Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus



0attracts all the



1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It





0) = 119877

0gt 1 the







0gt





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium







120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575



0is the output










05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















lim119905rarrinfin


inf 119862 (119905) gt 120598

lim119905rarrinfin

inf 119876 (119905) gt 120598(19)


Proof Let 119883 = (119878 119860 119862 119876 119877) isin 1198775

+ 119860 = 119862 = 119876 = 0 Thus



0attracts all the



1199091015840(119905) = 119884 (119909) 119909 (119905) (20)

where 119909(119905) = (119860(119905) 119862(119905) 119876(119905))119879 119884(119909) = [(119878119873)119865 minus 119881] It





0) = 119877

0gt 1 the







0gt





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

700

Time (days)

Tota

l in

fect

ed p

opul

atio

n

Endemic equilibrium







120572 120581 120595 120573 120578 120577 120575119886 120575119888and 120575



0is the output










05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















05 1 15 2 250

20004000

120583 Mean = 0001399 Std = 0000197696

01 012 014 016 018 020

20004000

120574 Mean = 0150031 Std = 00100622

07 08 09 1 110

20004000

120585 Mean = 0793539 Std = 00497404

016 018 02 022 0240

10002000

120572 Mean = 0199959 Std = 00291144

0 01 02 03 040

20004000

120581 Mean = 0201005 Std = 00500279

002 003 004 005 006 007 0080

20004000

120595 Mean = 0050104 Std = 00050286

01 02 03 04 05 060

20004000

120573 Mean = 0298526 Std = 00497006

0 02 04 06 080

20004000

120578 Mean = 0400407 Std = 00706492

0 01 02 03 04 050

20004000

120577 Mean = 0099884 Std = 00497535

0 002 004 006 008 01 0120

500010000

0 002 004 006 0080

500010000

0 005 01 015 020

500010000

05 1 15 2 25 30

20004000

times10minus3


Mean = 1333360 Std = 0274529R0



0

1 2 3 4 5 6 7 8 9 10 11 12minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Valu

es o

f sen

sitiv

ity in

dexe

s (PR

CC)


120583 minus0135

120574 minus0105

120585 0153

120572 minus0848

120581 minus0717

120595 minus0353

120573 0955

120578 0333

120585 0689

120575a minus0147

120575c minus0425

120575q minus0115




0and hence the








120597120572= ((120573120578119896

1+ 120573120577120585) (119896

111989621198963minus 120572120585119896

4)

minus120573 (12057811989611198962+ 1205851198963+ 120577120572120585) (119896

11198963minus 1205851198964))

times ((119896111989621198963minus 120572120585119896

4)2)minus1


1205971198770

120597120572= 120573120585119896

11198963[120577 (1198962minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

minus 1198963]

(22)


0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 5 10 15 20 25 30 35 40 45 5060

80

100

120

140

160

180

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00414 lt 1198963 = 2017

0 50 100 150120

125

130

135

140

145

150

155

160

165

170

Time (days)

Tota

l inf

ecte

d po

pulat

ion



3= 00803 gt 1198963 = 0517


where

1198962minus 120572 = 120595 + 120583 + 120575

119888gt 0 (23)

Now we have

1205971198770

120597120572lt 0 (gt 0) iff 119896

1015840

3lt 1198963(gt 1198963) (24)

where

1198961015840

3= 120577 (119896

2minus 120572) +

1205781198964(1198962minus 120572)

1198963

+1205851198964

1198961

gt 0 (25)




3






3lt (gt)119896

3


0 Figure 7 presents


0













01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















01 015 02 025 03 035 04 045 05 05501

02

03

04

05

06

07

08

09

120572

120585

R0 = 3

R0 = 2

R0 = 1

(a)

05

1

15

2

25

3

35

4

R0

002

0406

081

12058501

0203

0405

06

120572

0608

05

(b)


0


0= 1 2 3 (b) Two surfaces 119877


which 1198770= 1

R0

001

0203

0405

06

12057201 02

03 0405 06

07 0809

120573

0123456789

(a)

R0

00501

01502

02503

12058102 03 04 05 06 07 08 09 1

120573

0

05

1

15

2

25

3

35

(b)


0 (a)


0










119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















119889119878


119889119860

119889119905= 120582 (1 minus V) 119878 + (1 minus 120590) 120582V119878

+ 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575119886) 119860

119889119862

119889119905= 120585119860 minus (120572 + 120595

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 120572119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120595119862 + 120581119860 minus (120596 + 120583) 119877

(26)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (27)



V = V (119905) 0 le V (119905) le 119887 0 le 119905



119869 [V] = int119879

0

1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) (29)


1





119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119869 [V] (30)


119867 = 1198751119862 + 1198752119860 + 119875

3119876 +

1

2119882V2 (119905) +

119894=5

sum

119894=1

120582119894119896119894 (31)




119894rsquos such

that

1198891205821


1


1198891205822

119889119905= (


)1205821


119873)1205822


1198891205823

119889119905= (


)1205821


1198731205822


1198891205824

119889119905= (


)1205821


119873)1205822

+ 11989631205824minus 120574119891120582

5minus 1198753

(32)

1198891205825

119889119905= 1205961205821+ 11989651205825

1198961= (120585 + 120581 + 120583 + 120575

119886) 119896

2= (120572 + 120595

1+ 120583 + 120575

119888)

1198963= (120574 + 120583 + 120575

119902) 119896

4= 120574 (1 minus 119891)

1198965= (120596 + 120583)

(33)


120582119894(119879) = 0 (34)


Vlowast (119905)

= min [119887max (0 119878119882

(1205821205821(1 minus 120590) + 120590120582

1+ 1205821205822


(35)



1198891205821

119889119905= minus

120597119867

120597119878 120582

1(119879) = 0

1198891205825

119889119905= minus

120597119867

120597119877 120582

5(119879) = 0

(36)


120597119867

120597V

10038161003816100381610038161003816100381610038161003816Vlowast= 0 (37)


Vlowast =119878


(38)



0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 10 20 30 40 50 60 70 800

01

02

03

04

05

06

07

Time (days)

Opt

imal

vac

cina

tion

cont

rol


0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

Time (days)

Tota

l inf

ecte

d po

pulat

ion






0 119860(0) = 119860

0 119862(0) =

1198620 119876(0) = 119876

0 119877(0) = 119877



2(119879) = 120582

3(119879) = 120582

4(119879) =




0gt 1 when the





0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

1400

1600

1800

Time (days)

Accu

mul

ated

cost J






119889119878

119889119905= Π + 120596119877 minus 120582119878 minus 120583119878

119889119860

119889119905= 120582119878 + 120574 (1 minus 119891)119876 minus (120585 + 120581 + 120583 + 120575

119886) 119860

119889119862

119889119905= 120585119860 minus (120595 + 119906

1+ 120583 + 120575

119888) 119862

119889119876

119889119905= 1199061119862 minus (120574 + 120583 + 120575

119902)119876

119889119877

119889119905= 120574119891119876 + 120581119860 + 120595119862 minus (120596 + 120583) 119877

(39)

where

120582 =120573 (120578119860 + 119862 + 120577119876)

119873 (40)



1




U = 1199061 (119905) 0 le 1199061 (119905) le 119887


(41)


119869 [1199061] = int

119879

0

1198751119862 + 1198752119876 +

1

21198821199062

1(119905) (42)


1 1198751 1198752



119869 [119906lowast

1] = min1199061isin119880

119869 [1199061] (43)


1

defined as

119867 = 1198751119862 + 1198752119876 +

1

21198821199062

1(119905) +

119894=4

sum

119894=1

120582119894119896119894 (44)




0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

Time (days)

Acut

e pop

ulat

ion




0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

Time (days)

Qua

rant

ine p

opul

atio

n





that

1198891205821

119889119905= (120582 + 120583) 120582

1minus 1205821205822

1198891205822

119889119905= (

120573120578119878

119873)1205821+ (1198961minus120573120578119878

119873)1205822minus 1205851205823minus 1205811205825

1198891205823

119889119905= (

120573120578119878

119873)1205821minus120573119878

1198731205822+ 11989621205823minus 119906lowast

11205824minus 1205951205825minus 1198751

1198891205824

119889119905= (

120573120577119878

119873)1205821minus (1198964+120573120577119878

119873)1205822+ 11989631205824minus 120574119891120582

5minus 1198752

1198891205825

119889119905= 1205961205821+ 11989651205825

(45)

where 1198962= (1199061+ 120595 + 120583 + 120575



120582119894(119879) = 0 (46)


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

Time (days)

Opt

imal

qua

rant

ine c

ontro

l


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

Time (days)

Tota

l inf

ecte

d po

pulat

ion




0 5 10 15 20 25 30 35 40 45 500

5000

10000

15000

Time (days)

Accu

mul

ated

cost J




119906lowast

1(119905) = min[119887max(0

119862 (1205823minus 1205824

119882)] (47)






1(119905)





4 Conclusions



0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 20 40 60 80 100 120 1400

200

400

600

800

1000

1200

Time (days)

Chro

nic p

opul

atio

n



0 20 40 60 80 100 120 1400

50

100

150

200

250

300

350

Time (days)

Qua

rant

ine p

opul

atio

n





0lt 1) The disease



0gt 1)








Appendix

Parameter Values


02 120596 = 095 120583 = 00004566 120585 = 08 120575119886= 0000233 120575

119888=

000233 120575119902= 0001667 120578 = 04 120577 = 01 120573 = 018




minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















minus3 2119864minus4 (119873)120575119886

63119864minus3 119864minus4 (119866)

120575119888

93119864minus3 119864minus4 (119866)

120575119902

33119864minus3 119864minus4 (119866)




08 120581 = 02 120596 = 095 120583 = 0004566 120585 = 08 120572 = 02 120595 =

005 120575119886= 0000233 120575

119888= 000233 120575

119902= 0001667 120578 =

04 120577 = 01 120573 = 035 1198751= 1 119875

2= 005 119875

3= 005119882 =

1 120590 = 09 1198861= 0 1198871= 09 V = 15


2= 001 119875

3=

001 1198821= 2 119886

1= 0 119887

1= 07



References


















































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Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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