research article mathematical model of mdr-tb and xdr-tb...
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Research ArticleMathematical Model of MDR-TB and XDR-TB with Isolationand Lost to Follow-Up
F. B. Agusto, J. Cook, P. D. Shelton, and M. G. Wickers
Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA
Correspondence should be addressed to F. B. Agusto; [email protected]
Received 24 February 2015; Accepted 31 May 2015
Academic Editor: Juan-Carlos CorteΜs
Copyright Β© 2015 F. B. Agusto et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a deterministic model with isolation and lost to follow-up for the transmission dynamics of three strains ofMycobacterium tuberculosis (TB), namely, the drug sensitive, multi-drug-resistant (MDR), and extensively-drug-resistant (XDR)TB strains. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoreticaland epidemiological findings indicate that the model has locally asymptotically stable (LAS) disease-free equilibrium when theassociated reproduction number is less than unity. Furthermore, the model undergoes in the presence of disease reinfection thephenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemicequilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection. The result of the global sensitivityanalysis indicates that the dominant parameters are the disease progression rate, the recovery rate, the infectivity parameter, theisolation rate, the rate of lost to follow-up, and fraction of fast progression rates. Our results also show that increase in isolation rateleads to a decrease in the total number of individuals who are lost to follow-up.
1. Introduction
Mycobacterium tuberculosis (TB) is caused by bacteria thatare transmitted from person to person through the air by aninfected personβs coughing, sneezing, speaking, or singing [1].TB usually affects the lungs, but it can also affect other parts ofthe body, such as the brain, the kidneys, or the spine [1]. TheTB bacteria can stay in the air for several hours, depending onthe environment. In 2013, 9 million people were ill with TB,and 1.5millionmortalities occurred from the disease [2]; over95% of deaths occurred in low- andmiddle-income countries[2]. About one-third of the worldβs population has latent TB[2]. TB is also among the top three causes of death in womenaged 15 to 44 [2]. Tuberculosis is second only to HIV/AIDSas the greatest killer worldwide due to a single infectiousagent [2]. Those who have a compromised immune system,like those who are living with HIV, malnutrition, or diabetes,or people who use tobacco products, have a much higherrisk of falling ill. Individuals who develop TB are providedwith a six-month course of four antimicrobial drugs alongwith supervision and support by a health worker. Improper
treatment compliance or use of poor quality medicines canall lead to the development of drug-resistant tuberculosis [2].
Multi-drug-resistant (MDR) TB is a form of TB causedby bacteria that do not respond to, at least, isoniazid andrifampicin, which are the two most powerful, standard anti-tuberculosis drugs.MDR-TB is treatable and curable by usingsecond-line drugs [3]. However, these treatment options arelimited and recommendedmedicines are not always available[3]. In some cases, more severe drug resistance can develop.Extensively-drug-resistant (XDR)TB is a formofmulti-drug-resistant TB that responds to even fewer available medicines,including the second-line drugs [3]. In 2013, there were about480,000 cases of MDR-TB present in the world [4]; it wasestimated that about 9%-10% of these cases were XDR-TB[2, 4β6].
The increase in drug-resistant TB strains has called for anincreased urgency for isolating individuals infected with suchstrains of TB [7]. High priority is being placed on identifyingand curing these individuals. With proper identification andtreatment, about 40% of XDR-TB cases could potentially becured [6, 8]. However, only 10% of MDR-TB cases are ever
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 828461, 21 pageshttp://dx.doi.org/10.1155/2015/828461
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2 Abstract and Applied Analysis
identified, leaving the potential development of XDR-TB tobecome more prevalent worldwide [2]. The need for a mod-ern and effective approach to curtail the rise of drug-resistantTB strains is being sought after, one of which is constructedisolation, whether it is an at-home isolation or isolation ata medical facility [6]. An event that portrays the need forcarefully constructed isolation was the identification of anindividual infected with MDR-TB in Atlanta InternationalAirport in 2007 [9]. The individual had flown to Atlantaafter visiting Paris, Greece, Italy, CzechRepublic, andCanada.The individual unknowingly was infected withMDR-TB and,after twelve days of travel, the individual was involuntarilyisolated by the CDC in Atlanta under the Public HealthService Act [9]. The CDC held the individual in isolationfor one week and then moved him to a hospital in Denver,Colorado [9]. In this case, there are no reported infectionsresulting from the travel of the individual, perhaps due tothe slightly lower infection rate of MDR-TB as comparedto drug sensitive TB strains [9]. It should be noted herethat the isolation of this individual, albeit brief, potentiallyhelped prevent a rise in drug-resistant TB in the UnitedStates, which is currently at admittedly low levels [10]. Anoccurrence such as this one clearly demonstrates the need forcarefully executed isolation procedures for individuals withdrug-resistant TB strains.
Several reports have shown the effectiveness of isola-tion in reducing the number of people with TB [6, 7, 26,27]. Weis et al. [27] showed the effectiveness of isolation,reporting lower occurrences of primary and acquired drugresistance among individuals with TB. Historically, to treatthe infection, individuals were isolated in a sanatoriumwherethey would receive proper nutrition and a constant supplyof fresh air [28]. However, while this method is successfulin certain situations, this treatment methodology is difficultto implement without proper infrastructure in place [29].Locations without proper facilities such as South Africa havepoor treatment and success rates [29]; effective isolation isessential in areas such as these which have high treatmentfailure rates. Sutton et al. [26] studied three hospitals inCalifornia; they found that implementing the CDC isolationguidelines for hospitals was feasible; however, since not everyhospital could afford the necessary equipment, the isolationwas not the same for each hospital. Even though the resultsvaried, it is noticeably more efficient to isolate patients whoare infected with TB.
According to the National Committee of Fight againstTuberculosis of Cameroon [30], about 10% of infectious indi-viduals who start the recommended WHO DOTS treatmenttherapy in the hospital do not return to the hospital for therest of sputumexaminations and check-up and are thus lost tofollow-up.This can be attributed to the long duration of treat-ment regimen, negligence, or lack of information about TB[30], a brief relief from the long term treatment [22], poverty,and so forth. As such, health-care personnel do not knowtheir epidemiological status, that is, if they died, recovered,or are still infectious and this lack of epidemiological status ofthese individuals can affect the spread of TB in a population[30]. A number of mathematical models for tuberculosisdeveloped account for this population by the inclusion of
either the lost sight class [23, 30β32] or lost to follow-up class[22].
The aim of this study is to develop a new deterministictransmission model for TB to gain qualitative insight intothe effects of isolation in the presence of individuals who arelost to follow-up on TB transmission dynamics. A notablefeature of themodel is the incorporation of isolated and lost tofollow-up classes for the three TB strains. The paper is orga-nized as follows.Themodel formulation and analysis is givenin Section 2. Sensitivity analysis of the model is considered inSection 3. Analysis of the reproduction number is carried outin Section 4.The effect of isolation is numerically investigatedin Section 5. The key theoretical and epidemiological resultsfrom this study are summarized in Section 6.
2. Model Formulation
Themodel is formulated as follows: the population is dividedinto susceptible (π), latently infected (πΈ
π), symptomatically
infectious with drug sensitive strain (π), MDR strain (π),XDR strain (π), symptomatically infectious individuals whoare lost to follow-up (πΏ
π), isolated (π½), and recovered (π ),
where π = π,π,π. Thus, the total population is π(π‘) =π(π‘) + πΈ
π(π‘) + π(π‘) + πΈ
π(π‘) + π(π‘) + πΈ
π(π‘) + π(π‘) +
πΏπ(π‘) + πΏ
π(π‘) + πΏ
π(π‘) + π½(π‘) + π (π‘).
As the disease evolves individuals move from one class tothe other with respect to their disease status. The populationof susceptible (π) is generated by new recruits (either via birthor immigration) who enter the population at a rate π. Theparameter π denotes the recruitment rate. It is assumed thatthere is no vertical transmission or immigration of infectious;thus, these new inflow does not enter the infectious classes.All individuals, whatever their status, are subject to naturaldeath, which occurs at a rate π. The susceptible population isreduced by infection following effective contact with infectedindividuals with drug sensitive, MDR-, and XDR-TB strainsat the rates π
π, ππ, and π
π, where
ππ=π½π(π + π
ππΏπ)
π,
ππ=π½π(π + π
ππΏπ)
π,
ππ=π½π(π + π
ππΏπ)
π.
(1)
The parameters π½π, π½π, and π½
πare the effective transmission
probability per contact; we assume that π½π
< π½π
< π½π
[9] and the parameters ππ
> 1, ππ
> 1, and ππ
> 1are the modification parameter that indicates the increasedinfectivity of individuals who are lost to follow-up.
A fraction ππ1 of the newly infected individuals with drug
sensitive strainmove into the latently infected class (πΈπ), with
ππ2 fractionmoving into the symptomatic-infectious class (π)and the other fraction [1 β (π
π1 + ππ2)] moving into the lostto follow-up class (πΏ
π) with π
π1 + ππ2 < 1. The latentlyinfected individuals become actively infectious as a result ofendogenous reactivation of the latent bacilli at the rate π
π.
Similarly a fraction ππ1 of the newly infected individuals
with MDR stain move into the latently infected class with
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Abstract and Applied Analysis 3
MDR (πΈπ), with π
π2 fractions moving into the symptomat-ically infectious class (π) and the other fraction [1 β (π
π1 +ππ2)] moving into the symptomatically infectious lost tofollow-up class (πΏ
π) with π
π1 + ππ2 < 1.The latently infectedindividuals with MDR strain become actively infectious as aresult of endogenous reactivation of the latent bacilli at therate π
π.
Lastly, a fraction ππ1 of the newly infected individualswith
XDR stain moves into the latently infected class with XDR(πΈπ), with π
π2 fractions moving into the symptomaticallyinfectious class (π) and the other fraction [1 β (π
π1 + ππ2)]moving into the symptomatically infectious lost to follow-upclass (πΏ
π) with π
π1 + ππ2 < 1.The latently infected individualswith XDR strain become actively infectious as a result ofendogenous reactivation at the rate π
π.
Members of the symptomatically infectious class withthe drug sensitive strain (π) are lost to follow-up at therate π
πand move to the class of lost to follow-up with
drug sensitive strain (πΏπ). They are isolated into the isolated
class (π½) at the rate πΌπ. They undergo the WHO recom-
mended DOTS treatment (Directly Observed Treatment,Short Course (DOTS)); however due to treatment failure(from treatment noncompliance) they move into the latentlyinfected class at the rate π
π1πΎπ or the latently infected classwith MDR at the rate π
π2πΎπ. The remaining fraction moveinto the recovered class (π ) following effective treatment atthe rate (1 β π
π1 β ππ2)πΎπ (where ππ1 + ππ2 < 1). Or they candie from the infection at the rate πΏ
π.
Similarlymembers of the symptomatically infectiouswithMDR strain (π) are lost to follow-up at the rate π
πand
move to the class of lost to follow-up with MDR strain (πΏπ).
They are isolated at the rate πΌπ. And a fraction of them
move into the population of the latently infected with XDRstrain as a result of treatment failure of the symptomaticallyinfectious individuals with MDR strain at the rate π
π1πΎπ.The remaining fraction move into the recovered class at therate (1 β π
π1)πΎπ (where ππ1 < 1). Or they can die from theinfection at the rate πΏ
π.
Lastly, members of the symptomatically infectious withXDR strain (π) are lost to follow-up at the rate π
πand move
to the class of lost to follow-up with XDR strain (πΏπ). Or they
move into recovered class at the rate πΎπ. Or they can die from
the infection at the rate πΏπ. We assume that πΎ
π< πΎπ< πΎπ.
The individuals who are lost to follow-up (πΏπ) with drug
sensitive strain return at the rateππandmove into the class of
individuals with drug sensitive strain. Or they die at the rateπΏπΏπ. Similarly, the individuals who are lost to follow-up (πΏ
π)
withMDR strain return at the rateππandmove into the class
of individuals with MDR strain. Or they die at the rate πΏπΏπ
.Lastly, the individuals who are lost to follow-up (πΏ
π) with
XDR strain return at the rate ππand move into the class of
individuals with XDR strain. Or they die at the rate πΏπΏπ.
Recovered individuals (π ) are reinfected with drug sen-sitive strain at the rate ππ
π, with π
π1πππ fraction movinginto the latently infected class with drug sensitive strain,ππ2πππ fraction moving into the symptomatically infectiousclass with drug sensitive strain, and the other [1 β (π
π1 +ππ2)]πππ moving into the symptomatically infectious lostto follow-up class with drug sensitive strain. Also, these
individuals experience reinfection with MDR and XDRstrains and fractions of these move into the latently infectedand symptomatically infectious classes, respectively.
It follows, from the above descriptions and assump-tions, that the model for the transmission dynamics of thetuberculosis with isolation and lost to follow-up is given bythe following deterministic system of nonlinear differentialequations (the variables and parameters of the model aredescribed in Table 1; a schematic diagram of the model isdepicted in Figure 1):
ππ
ππ‘= πβ[
π½π(π + π
ππΏπ)
π+π½π(π + π
ππΏπ)
π
+π½π(π + π
ππΏπ)
π] πβππ,
ππΈπ
ππ‘=ππ1π½π (π + πππΏπ) (π + ππ )
π+ππ1πΎππβ (ππ
+π) πΈπ,
ππ
ππ‘=ππ2π½π (π + πππΏπ) (π + ππ )
π+πππΈπ+πππΏπ
β (ππ+πΌπ+ πΎπ+π+ πΏ
π) π,
ππΈπ
ππ‘=ππ1π½π (π + πππΏπ) (π + ππ )
π+ππ2πΎππβ (ππ
+π) πΈπ,
ππ
ππ‘=ππ2π½π (π + πππΏπ) (π + ππ )
π+πππΈπ
+πππΏπβ (ππ+πΌπ+ πΎπ+π+ πΏ
π)π,
ππΈπ
ππ‘=ππ1π½π (π + πππΏπ) (π + ππ )
π+ππ1πΎππβ(ππ
+π) πΈπ,
ππ
ππ‘=ππ2π½π (π + πππΏπ) (π + ππ )
π+πππΈπ+πππΏπ
β (ππ+πΌπ+ πΎπ+π+ πΏ
π)π,
ππΏπ
ππ‘=(1 β ππ1 β ππ2) π½π (π + πππΏπ) (π + ππ )
π+πππ
β (ππ+π+ πΏ
πΏπ) πΏπ,
ππΏπ
ππ‘=(1 β ππ1 β ππ2) π½π (π + πππΏπ) (π + ππ )
π
+πππβ(π
π+π+ πΏ
πΏπ) πΏπ,
ππΏπ
ππ‘=(1 β ππ1 β ππ2) π½π (π + πππΏπ) (π + ππ )
π+πππ
β (ππ+π+ πΏ
πΏπ) πΏπ,
ππ½
ππ‘= πΌππ+πΌππ+πΌ
ππβ (πΎ
π½+π+ πΏ
π½) π½,
ππ
ππ‘= (1βπ
π1 βππ2) πΎππ+ (1βππ1) πΎππ+πΎππ
+πΎπ½π½ β ππ β π [
π½π(π + π
ππΏπ)
π+π½π(π + π
ππΏπ)
π
+π½π(π + π
ππΏπ)
π]π .
(2)
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4 Abstract and Applied Analysis
Table 1: Variables and parameters description of model (2).
Variable Descriptionπ(π‘) Susceptible individualsπΈπ(π‘), πΈπ(π‘), πΈπ(π‘) Latently infected individuals with drug sensitive, MDR, and XDR strains
π(π‘),π(π‘), π(π‘) Symptomatically infectious individuals with drug sensitive, MDR, and XDR strainsπΏπ(π‘), πΏπ(π‘), πΏπ(π‘) Symptomatically infectious individuals who are lost to sight with drug sensitive, MDR, and XDR strains
π½(π‘) Isolated individualsπ (π‘) Recovered individualsParameter Descriptionπ Recruitment rateπ Natural death rateπ TB reinfection rateπ½π, π½π, π½π
Transmission probabilityππ, ππ, ππ
Infectivity modification parameterππ1, ππ2, ππ1, ππ2, ππ1, ππ2 Fraction of fast disease progressionππ1, ππ2, ππ1 Fraction that failed treatmentππ, ππ, ππ
Disease progression rateπΎπ, πΎπ, πΎπ, πΎπ½
Recovery rateπΌπ, πΌπ, πΌπ
Isolation rateπΏπ, πΏπ, πΏπ, πΏπ½
Disease-induced death rateππ, ππ, ππ
Lost to follow-up rateππ, ππ, ππ
Return rate from lost to follow-upπΏπΏπ, πΏπΏπ, πΏπΏπ
Disease-induced death rate in individuals who are lost to follow-up
lT1πT
lM1πT lM2πT
lX1πX lX2πX
πlM1πM πlM2πM
pT1πΎT
pT2πΎT
ππ
ππ π
π
π π
π
ππ
π
ET
EM
EX
S
T
M
X
LT
LM
LX
J R
πTπM
πX
π
πX
πT
πM
πΏM
πΏX
πΏJ
πΎJ
πT
πM
πX
πΏLM
lT2πT(1 β lT1 β lT2)πT π(1 β lT1 β lT2)πT
(1 β pT1 β pT2)πΎT
(1 β pM1)πΎM
(1 β lM1 β lM2)πT
π(1 β lM1 β lM2)πM
π(1 β lX1 β lX2)πX
(1 β lX1 β lX2)πX
pM1πΎM
πlX1πXπlX2πX
πΏLT
πΏLX
πlT2πT
πΌM
πΌT
πΌX
πlT1πT
πΏT
πX
πT
πM
Figure 1: Systematic flow diagram of the tuberculosis model (2).
2.1. Basic Properties
2.1.1. Positivity and Boundedness of Solutions. For TB model(2) to be epidemiologically meaningful, it can be shown(using the method in Appendix A of [33]) that all itsstate variables are nonnegative for all time. In other words,
solutions of the model system (2) with nonnegative initialdata will remain nonnegative for all time π‘ > 0.
Lemma 1. Let the initial data π(0) β₯ 0, πΈπ(0) β₯ 0, π(0) β₯
0, πΈπ(0) β₯ 0, π(0) β₯ 0, πΈ
π(0) β₯ 0, π(0) β₯ 0, πΏ
π(0) β₯
0, πΏπ(0) β₯ 0, πΏ
π(0) β₯ 0, π½(0) β₯ 0, π (0) β₯ 0.
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Abstract and Applied Analysis 5
Then the solutions (π, πΈπ, π, πΈπ,π, πΈ
π, π, πΏπ, πΏπ, πΏπ, π½, π )
of the tuberculosis model (2) are nonnegative for all π‘ > 0.Furthermore,
lim supπ‘ββ
π(π‘) β€π
π, (3)
with
π = π+πΈπ+π+πΈ
π+π+πΈ
π+π+πΏ
π+πΏπ
+πΏπ+ π½ +π .
(4)
2.1.2. Invariant Regions. Since model (2) monitors humanpopulations, all variables and parameters of the model arenonnegative. Model (2) will be analyzed in a biologicallyfeasible region as follows. Consider the feasible region
Ξ¦ β R12+
(5)
with
Ξ¦ = {(π, πΈπ, π, πΈπ,π, πΈ
π, π, πΏπ, πΏπ, πΏπ, π½, π )
βR12+: π (π‘) β€
π
π} .
(6)
The following steps are followed to establish the positiveinvariance ofΞ¦ (i.e., solutions inΞ¦ remain inΞ¦ for all π‘ > 0).The rate of change of the population is obtained by adding theequations of model (2) and this gives
ππ (π‘)
ππ‘= πβππ (π‘) β πΏππ (π‘) β πΏππ(π‘) β πΏππ(π‘)
β πΏπΏππΏπ (π‘) β πΏπΏππΏπ (π‘) β πΏπΏππΏπ (π‘) .
(7)
And it follows that
ππ (π‘)
ππ‘β€ πβππ (π‘) . (8)
A standard comparison theorem [34] can then be used toshow that
π(π‘) β€ π (0) πβππ‘ + ππ(1β πβππ‘) . (9)
In particular, π(π‘) β€ π/π, if π(0) β€ π/π. Thus, region Ξ¦is positively invariant. Hence, it is sufficient to consider thedynamics of the flow generated by (2) in Ξ¦. In this region,the model can be considered as being epidemiologicallyand mathematically well-posed [35]. Thus, every solution ofmodel (2) with initial conditions in Ξ¦ remains in Ξ¦ for allπ‘ > 0. Therefore, the π-limit sets of system (2) are containedin Ξ¦. This result is summarized below.
Lemma 2. The region Ξ¦ β R12+Γ is positively invariant for
model (2) with nonnegative initial conditions in R12+.
2.2. Stability of the Disease-Free Equilibrium (DFE). Tubercu-losis model (2) has a DFE, obtained by setting the right-handsides of the equations in the model to zero, given by
E0
= (πβ, πΈβ
π, πβ, πΈβ
π,πβ, πΈβ
π, πβ, πΏβ
π, πΏβ
π, πΏβ
π, π½β, π β)
= (π
π, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) .
(10)
The linear stability of E0 can be established using thenext generation operator method on system (2). Using thenotations in [36], the matrices πΉ andπ, for the new infectionterms and the remaining transfer terms, are, respectively,given by
πΉ = [πΉ1 | πΉ2] , (11)
where
πΉ1 =
((((((((((((((((
(
0 ππ1π½π 0 0 0
0 ππ2π½π 0 0 0
0 0 0 ππ1π½π 0
0 0 0 ππ2π½π 0
0 0 0 0 00 0 0 0 00 (1 β π
π1 β ππ2) π½π 0 0 00 0 0 (1 β π
π1 β ππ2) π½π 00 0 0 0 00 0 0 0 0
))))))))))))))))
)
,
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6 Abstract and Applied Analysis
πΉ2 =
((((((((((((((((
(
0 ππ1π½πππ 0 0 0
0 ππ2π½πππ 0 0 0
0 0 ππ1π½πππ 0 0
0 0 ππ2π½πππ 0 0
ππ1π½π 0 0 ππ1π½πππ 0ππ2π½π 0 0 ππ2π½πππ 00 (1 β π
π1 β ππ2) π½πππ 0 0 00 0 (1 β π
π1 β ππ2) π½πππ 0 0(1 β ππ1 β ππ2) π½π 0 0 (1 β ππ1 β ππ2) π½πππ 0
0 0 0 0 0
))))))))))))))))
)
,
π =
((((((((((((((((
(
π1 βππ1πΎπ 0 0 0 0 0 0 0 0βππ
π2 0 0 0 0 βππ 0 0 00 βπ
π2πΎπ π3 0 0 0 0 0 0 00 0 βπ
ππ4 0 0 0 βππ 0 0
0 0 0 βππ1πΎπ π5 0 0 0 0 0
0 0 0 0 βππ
π6 0 0 βππ 00 βπ
π0 0 0 0 π7 0 0 0
0 0 0 βππ
0 0 0 π8 0 00 0 0 0 0 βπ
π0 0 π9 0
0 βπΌπ
0 βπΌπ
0 βπΌπ
0 0 0 π10
))))))))))))))))
)
,
(12)
whereπ1 = ππ+π, π2 = ππ+πΎπ+πΌπ+ππ+π, π3 = ππ+π, π4 =ππ+ πΎπ+ πΌπ+ ππ+ π, π5 = ππ + π, π6 = ππ + πΎπ +
πΌπ+ ππ+ π, π7 = ππ + π + πΏπΏπ, π8 = ππ + π + πΏπΏπ, π9 =
ππ+ π + πΏ
πΏπ, π10 = πΎπ½ + π + πΏπ½.
It follows that the basic reproduction number of tubercu-losis model (2), denoted byR0, is given by
R0 = π (πΉπβ1) = max (R
π,Rπ,Rπ) , (13)
where
Rπ=π½π{ππ(π7 + ππππ) ππ1 + (π7 + ππππ) π1ππ2 + [(π2ππ + ππ) π1 β ππππππ1πΎπ] (1 β ππ1 β ππ2)}
[π1 (π7π2 β ππππ) β ππππππ1πΎπ],
Rπ=π½π[ππ(π8 + ππππ) ππ1 + (π8 + ππππ) π3ππ2 + (π4ππ + ππ) π3 (1 β ππ1 β ππ2)]
π3 (π8π4 β ππππ),
Rπ=π½π[ππ(π9 + ππππ) ππ1 + (π9 + ππππ) π5ππ2 + (π6ππ + ππ) π5 (1 β ππ1 β ππ2)]
π5 (π9π6 β ππππ).
(14)
Quantity Rπrepresents the reproduction number of TB
drug sensitive-only population. Similarly, quantity Rπ
isthe reproduction number for MDR-only population and thequantity R
πrepresents the reproduction number for XDR-
only TB population.Further, using Theorem 2 in [36], the following result is
established.
Lemma 3. TheDFE of the tuberculosis model (2), given byE0,is locally asymptotically stable (LAS) ifR0 < 1, and unstable ifR0 > 1.
The threshold quantity (R0, i.e., the basic reproductionnumber) measures the average number of new infectionsgenerated by a single infected individual in a completely
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Abstract and Applied Analysis 7
πXπMπTlX2lX1lM2lM1lT2lT1
πΏJπΏXπΏMπΏTπXπMπTπXπMπTπ
pM1pT2pT1πΌTπΌXπΌMπXπMπTπ½Xπ½Mπ½TπΎJπΎXπΎMπΎTπ
0 0.5 1β0.5
π
πΏLXπΏLMπΏLT
(a)
πXπMπTlX2lX1lM2lM1lT2lT1
πΏJπΏXπΏMπΏTπXπMπTπXπMπTπ
pM1pT2pT1πΌTπΌXπΌMπXπMπTπ½Xπ½Mπ½TπΎJπΎXπΎMπΎTπ
0 0.2 0.4 0.6 0.8 1β0.8 β0.6 β0.4 β0.2
π
πΏLXπΏLMπΏLT
(b)πXπMπTlX2lX1lM2lM1lT2lT1
πΏJπΏXπΏMπΏTπXπMπTπXπMπTπ
pM1pT2pT1πΌTπΌXπΌMπXπMπTπ½Xπ½Mπ½TπΎJπΎXπΎMπΎTπ
0 0.2 0.4 0.6 0.8 1β0.8 β0.6 β0.4 β0.2
π
πΏLXπΏLMπΏLT
(c)
Figure 2: PRCC values for model (2), using as the response function (a) the basic reproduction number (Rπ), (b) the basic reproduction
number (Rπ), and (c) the basic reproduction number (R
π). Parameter values (baseline) and ranges used are as given in Table 2.
susceptible population [35β38]. Thus, Lemma 3 implies thattuberculosis can be eliminated from the population (whenR0 < 1) if the initial sizes of the subpopulations of model(2) are in the basin of attraction of the DFE, E0.
3. Sensitivity Analysis
A global sensitivity analysis [39β42] is carried out, on theparameters of model (2), to determine which of the param-eters have the most significant impact on the outcome ofthe numerical simulations of the model. Figure 2(a) depictsthe partial rank correlation coefficient (PRCC) values foreach parameter of the models, using the ranges and baselinevalues tabulated in Table 2 (with the basic reproductionnumbers, R
π, as the response function), from which it
follows that the parameters that have the most influence ondrug sensitive TB transmission dynamics are the fraction offast progression rate (π
π2) into the drug sensitive TB class,the infectivity modification parameter (π
π), the recovery rate
(πΎπ) from drug sensitive TB, disease progression rate (π
π),
rate of lost to follow-up (ππ) of those with drug sensitive
TB, fraction of latently infected with drug sensitive TB thatfailed treatment (π
π1), the fraction of fast progression rate(ππ1) into the latently infected with drug sensitive TB class,
and the isolation rate (πΌπ) from drug sensitive TB class.
The identification of these key parameters is vital to theformulation of effective control strategies for combating thespread of the disease, as this study identifies the most impor-tant parameters that drive the transmissionmechanism of thedisease. In other words, the results of this sensitivity analysissuggest that, to effectively control the spread of drug sensitiveTB in the community, the effective strategy will be to reducethe disease progression rate (reduceπ
π), increase the recovery
rate (increase πΎπ) from drug sensitive TB, reduce the disease
modification parameter (reduce ππ), increase the isolation
rate (increase πΌπ) from drug sensitive TB class, and reduce
the fraction of fast progression rates (reduce ππ1 and ππ2) into
the drug sensitive TB class and lost to follow-up class andreduce the rate of lost to follow-up (reduce π
π) of those with
drug sensitive TB.The result of this analysis suggests that thefraction of latently infected with drug sensitive TB that failedtreatment (π
π1) be increased, since this has a negative impactof the basic reproduction number, R
π. This of course is
counter intuitive; however increasing this rate lowers the rateof development of drug-resistant TB due to treatment failure.
The sensitivity analysis was also carried out using model(2) with the basic reproduction number, R
π, for drug-
resistant TB, as the response function (see Figure 2(b)). Thedominant parameters in this case are πΌ
π, ππ, ππ, πΎπ, ππ,
ππ, π, ππ1, and ππ2. Similarly, when using as response func-
tion the basic reproduction number, Rπ, for extended drug
-
8 Abstract and Applied Analysis
resistance TB (see Figure 2(c)), the dominant parameters inthis case are πΌ
π, ππ, ππ, πΎπ, ππ, ππ, π, ππ1, and ππ2.
The results from these analyses using as response func-tions the basic reproduction numbers,R
πandR
π, suggest
that the natural death rate (π) be increased, since it has anegative impact on the basic reproduction numbers,R
πand
Rπ. However increasing this rate is not epidemiologically
relevant, as it implies reducing the population size thatwe wish to preserve by other means aside from death bytuberculosis and should therefore be ignored in any controlmeasures.
4. Analysis of the Reproduction Number
Following the result obtained in Section 3, we investigate inthis section whether or not treatment-only, isolation-onlyof individuals with tuberculosis or a combination of bothcan lead to tuberculosis elimination in the population. Theanalysis will be carried out using the reproduction numberfor drug sensitive TB (R
π) since R0 is the maximum of the
reproduction number of drug sensitive TB (Rπ), MDR-TB
(Rπ), and XDR-TB (R
π); similar result can be obtained for
the MDR- and XDR-TB.In the absence of isolation (πΌ
π= 0), the reproduction
number (Rπ) reduces to
RππΌ
=π½π[(πππ7 + ππππππ) ππ1 + (πππ1ππ + π1π7) ππ2 + (π1ππ + πππ1 (ππ + πΎπ + π + πΏπ) β ππππππ‘1πΎπ) (1 β ππ1 β ππ2)]
π1 (ππ + πΎπ + π + πΏπ) π7 β π1ππππ β ππππ1πΎππ7.
(15)
The reproduction number (Rπ) can be written as
Rπ= π΄πΌRππΌ
, (16)
where
π΄πΌ
=(π1π7 (ππ + πΎπ + π + πΏπ) β π1ππππ β ππππ1πΎππ7) [(πππ7 + ππππππ) ππ1 + (πππ1ππ + π1π7) ππ2 + (π1ππ + πππ1π2 β ππππππ1πΎπ) (1 β ππ1 β ππ2)]{(π1π2π7 β π1ππππ β ππππ1πΎππ7) [ππ (π7 + ππππ) ππ1 + (ππππ + π7) π1ππ2 + (πππ1 (ππ + πΎπ + π + πΏπ) + π1ππ β ππππππ1πΎπ) (1 β ππ1 β ππ2)]}
.
(17)
The difference between Rπand R
ππΌ
is in the isolation rate(πΌπ); as such the factor π΄
πΌcompares a population with
and without isolation; however, this is in the presence oftreatment and individuals who are lost to follow-up andindividuals returning from lost to follow-up. If R
ππΌ
< 1,then drug sensitive TB cannot develop into an epidemicin the community. However, if R
ππΌ
> 1, it is imperative
to investigate the effect of isolation on the transmission ofdrug sensitive TB among the populace and determine thenecessary condition for slowing down its development in thecommunity. Following [43] we have
ΞπΌ= RππΌ
βRπ= (1βπ΄
πΌ)RππΌ
, (18)
where
ΞπΌ=[π½ππ1 (π7 + ππππ) (ππ β π2 + πΎπ + π + πΏπ) (πππ7ππ1 + π1π7ππ2 + π1ππ (1 β ππ1 β ππ2))]
{(π1π2π7 β π1ππππ β ππππ1πΎππ7) [π1π7 (ππ + πΎπ + π + πΏπ) β π1ππππ β ππππ1πΎππ7]}. (19)
To slow down the spread of drug sensitive tuberculosis inthe population via effective isolation, proper treatment, andidentification of individuals who return from lost to follow-up and in the presence of individuals who are lost to follow-up, we expect that Ξ
πΌ> 0, and this is satisfied if π΄
πΌ< 1
in (18). Now, setting Rπ= 1 and solving for π΄
πΌgives the
threshold effectiveness of isolation and taking into account
treatment and identification of individuals who return fromlost to follow-up and who are also lost to follow-up:
π΄β
πΌ=
1RππΌ
. (20)
Hence, drug sensitive tuberculosis can be eradicated inthe community in the presence of isolation taking into
-
Abstract and Applied Analysis 9
consideration proper treatment and identification of individ-uals who return from lost to follow-up and are lost to follow-up if π΄
πΌ< π΄β
πΌis attained. Note that π΄β
πΌis a decreasing
function ofRππΌ
, thus indicating that higher values forπ΄βπΌwill
result in smaller values forRππΌ
, a desired outcome. But a large
value for RππΌ
results in a small value for π΄βπΌ, an indication
that eradication may not be attainable.The following limits of π΄
πΌprovide a further insight into
possible ways of reducing the burden of drug sensitive TB inthe community:
limπ΄πΌπΌπββ
=ππ2ππ [(ππ + π + πΏπ + πΎπ) π1π7 β πππππ1 β ππππ1πΎππ7]
π7 {ππ (π7 + ππππ) ππ1 + (ππππ + π7) π1ππ2 + [πππ1 + ππ (ππ + πΎπ + π + πΏπ) π1 β ππππππ1πΎπ] (1 β ππ1 β ππ2)},
(21)
limπ΄πΌππββ
= 1,
limπ΄πΌππββ
= 1,
limπ΄πΌπΎπββ
= 1.
(22)
The limits in (21) will be less than unity. In practice, πΌπβ β
implies high rate of isolating individuals with drug sensitiveTB, π
πβ β implies high rate of individuals who are lost
to follow-up, ππ
β β implies a high rate of individualswho return from lost to follow-up, and πΎ
πβ β implies
high treatment rate. Ideally, the results obtained from theselimits can be pursued for the reduction of the burden of
drug sensitive TB in the community, provided of course thatit is feasible and practicable economically. Therefore, with alook at factor π΄
πΌ, one observes that an effective isolation will
lead to a reduction in the burden of drug sensitive TB in thepopulation.
Next, we consider the case when the treatment rate is setto zero (πΎ
π= 0). The reproduction number is given as
RππΎ
=π½π[(πππ7 + ππππππ) ππ1 + (πππ1ππ + π1π7) ππ2 + [π1πππππ1 (ππ + πΌπ + π + πΏπ)] (1 β ππ1 β ππ2)]
π1 (ππ + πΌπ + π + πΏπ) π7 β π1ππππ. (23)
The reproduction numberRπcan be expressed as
Rπ= π΄πΎRππΎ
, (24)
where
π΄πΎ
={[(πππ7 + ππππππ) ππ1 + (πππ1ππ + π1π7) ππ2 + (π1ππ + πππ1π2 β ππππππ1πΎπ) (1 β ππ1 β ππ2)] π1 [(ππ + πΌπ + π + πΏπ) π7 β ππππ]}
{(π1π2π7 β π1ππππ β ππππ‘1πΎππ7) [(πππ7 + ππππππ) ππ1 + (πππ1ππ + π1π7) ππ2 + (π1ππ + (ππ + πΌπ + π + πΏπ) π1ππ) (1 β ππ1 β ππ2)]}.
(25)
The difference between RππΎ
and Rπis in the treatment
rate (πΎπ); thus π΄
πΎcompares a population with and without
treatment in the presence of isolation of infected individualswith drug sensitive TB, individuals who are lost to follow-up and who return from of lost to follow-up. If R
ππΎ
< 1,then drug sensitive TB cannot develop into an epidemic in
the community and no control strategy will be required forits control. Take the difference betweenR
πandR
ππΎ
; that is,
ΞπΎ= RπβRππΎ
= (1βπ΄πΎ)RππΎ
, (26)
where
ΞπΎ={π½π(π7 + ππππ) [π1 (ππ + π + πΏπ + πΌπ β π2) + πΎπππ1ππ] [π1π7ππ2 + πππ7ππ1 + π1ππ (1 β ππ1 β ππ2)]}
{π1 (π1π2π7 β π1ππππ β ππππ1πΎππ7) [π7 (ππ + πΌπ + π + πΏπ) β ππππ]}. (27)
-
10 Abstract and Applied Analysis
To slow down the spread of drug sensitive tuberculosis inthe population using proper treatment, effective isolationand identification of individuals who return from lost tofollow-up and in the presence of those who are of lost to
follow-up, we expect that ΞπΎ> 0, and this is satisfied if
π΄πΎ< 1 in (18).Take the following limits of π΄
πΎ:
limπ΄πΎπΎπββ
=[(ππ+ πΌπ+ π + πΏ
π) π7 β ππππ] πππ1 (1 β ππ1 β ππ2)
π7 {ππ (π7 + ππππ) ππ1 + (ππππ + π7) π1ππ2 + [πππ1 + ππ (ππ + πΎπ + π + πΏπ) π1 β ππππππ1πΎπ] (1 β ππ1 β ππ2)},
limπ΄πΎππββ
= 1,
limπ΄πΎππββ
= 1,
limπ΄πΎπΌπββ
= 1.
(28)
From the limit ofπ΄πΎ, one observes that an effective treatment
will lead to a reduction in the burden of drug sensitive TB inthe population.
Comparing the quantities π΄πΌand π΄
πΎshows that π΄
πΌ<
π΄πΎ; that is, size
π΄πΌβπ΄πΎ= β {[(π7 + ππππ) + (π7 + ππππ) ππ2π1 + (πππ1 + πππ1π2 β ππππππ1πΎπ) (1β ππ1 β ππ2)] [π1 (πΌπ β πΎπ)
+ ππππ1πΎπ] (π7 + ππππ) [πππ7ππ1 +π1π7ππ2 +π1ππ (1β ππ1 β ππ2)]} ({(π1π2π7 βπ1ππππ βππππ1πΎππ7)
β [ππ(π7ππππ) ππ1 + (π7 + ππππ) π1ππ2 + (πππ1 + πππ1 (ππ + πΌπ + π + πΏπ)) (1 β ππ1 β ππ2)]
β [ππ(π7 + ππππ) ππ1 + (π7 + ππππ) π1ππ2 + (πππ1 + πππ1 (ππ + πΎπ + π + πΏπ) β ππππππ1πΎπ) (1 β ππ1 β ππ2)]})
β1< 0.
(29)
This implies that π΄πΎwill provide better results in slowing
down drug sensitive tuberculosis spread, using reduction inthe prevalence, than usingπ΄
πΌ. On the other hand ifπ΄
πΌβπ΄πΎ>
0, this means that π΄πΌwill give better outcome in slowing
down drug sensitive tuberculosis spread compared to usingπ΄πΎ.Thus, from the above discussions, to slow down the
spread of the disease and reduce the number of secondaryinfections in the population, we require control strategieswith parameter values that would make π΄
πΌ< 1 or π΄
πΎ<
1. Hence, the necessary condition for slowing down thedevelopment of drug sensitive TB at the population level isthat Ξ
πΌ> 0 or Ξ
πΎ> 0. However, Ξ
πΌgives a better result
in terms of reduction in the prevalence of the disease overΞπΎprovided Ξ
πΎ> ΞπΌ; otherwise if Ξ
πΌ> ΞπΎ, then Ξ
πΎ
gives a better result over ΞπΌ. Using parameters in Table 1, we
have that π΄πΌ= 0.8524, π΄
πΎ= 0.8802 and Ξ
πΌ= 0.1061,
ΞπΎ= 0.0833; thus π΄
πΌβ π΄πΎ= β0.02789. It follows that, the
isolation-only strategy provides more effective control mea-sures in curtailing the disease transmission in the community.
Using the threshold quantity, Rπ, we determine how
isolation and treatment rates could lead to tuberculosiselimination in the population. Thus
limRππΌπββ
=π½π(1 β ππ1 β ππ2) ππ
(ππ+ π + πΏ
π)
> 0,
limRππΎπββ
=π½π(1 β ππ1 β ππ2) ππ
(ππ+ π + πΏ
π)
> 0.
(30)
Thus a sufficient effective TB control program that isolates (ortreats) the identified cases at a high rate πΌ
πβ β (or πΎ
πβ
β) can lead to effective disease control if it results in makingthe respective right-hand side of (30) less than unity.
Differentiating partially the reproduction number of πRπ
with respect to the key parameters (πΌπand πΎπ), this gives
πRπ
ππΌπ
=βπ½ππ1 (π7 + ππππ) [ππ1πππ7 + ππ2π1π7 + (1 β ππ1 β ππ2) π1ππ]
[π1π7 (ππ + πΌπ + πΎπ + π + πΏπ) β π1ππππ β ππππ1πΎππ7]2 , (31)
πRπ
ππΎπ
=βπ½π(π1 β ππππ‘1) (π7 + ππππ) [ππ1πππ7 + ππ2π1π7 + (1 β ππ1 β ππ2) π1ππ][π1π7 (ππ + πΌπ + πΎπ + π1π7π + πΏπ) β π1ππππ β ππππ1πΎππ7]
2 . (32)
-
Abstract and Applied Analysis 11
Table 2: Values of the parameters of the model (2).
Parameter Baselinevalues Range Reference
π π Γ 105 (0.0143 Γ105, 0.04 Γ 105) [11]
π 0.0159 (0.0143, 0.04) [12]π 0.06 (0.01, 0.5) [13]π½π 9.75 (4.5, 15.0) [14]π½π 1.5 (1.5, 3.5) [13, 15]π½π 0.0000085 (0.01, 0.1) [13]ππ, ππ, ππ 0.5 (0, 1) Assumed
ππ1, ππ2, ππ1, ππ2, ππ1, ππ2 0.14 (0.02, 0.3) [12, 16, 17]ππ1 0.3 (0.1, 0.5) [14, 18]ππ2 0.03 (0.01, 0.05) [14, 18]ππ1 0.03 (0.01, 0.1) [14, 18]
ππ, ππ, ππ
0.05, 0.0018,0.013 (0.005, 0.05) [12, 19, 20]
πΎπ, πΎπ½ 1.5 (1.5, 2.5) [11, 16, 21]
πΎπ, πΎπ 0.75 (0.5, 1.0) [13]
πΌπ, πΌπ, πΌπ 0.6 (0.2, 1.0) Assumed
ππ, ππ, ππ 0.2511 (0.0022, 0.5) [22]
ππ, ππ, ππ 0.1 (0.5, 1.0) [23]
πΏπ, πΏπ½ 0.365 (0.22, 0.39) [20, 24, 25]
πΏπ, πΏπ 0.028 (0.01, 0.03) [13]
πΏπΏπ, πΏπΏπ
, πΏπΏπ 0.02 (0.01, 0.039) [22, 23]
Thus, it follows from (31) that πRπ/ππΌπ
< 0, henceshowing further the effectiveness of the control measures.Thus, isolation (πΌ
π) of drug sensitive tuberculosis will have
a positive impact in reducing the drug sensitive TB burdenin the community, regardless of the values of the otherparameters. This result is stated in the following lemma.
Lemma 4. The use of isolation (πΌπ) will have a positive
impact on the reduction of the drug sensitive TB burden in acommunity regardless of the values of other parameters in thebasic reproduction number under isolation.
Similarly, from (32), we have that πRπ/ππΎπ
< 0.Thus, effective treatment (πΎ
π) of drug sensitive tuberculosis
will have a positive impact in reducing the drug sensitivetuberculosis burden in the community, irrespective of thevalues of the other parameters. This result is summarizedbelow.
Lemma 5. The use of effective treatment (πΎπ) will have a
positive impact on the reduction of the drug sensitive TB burdenin a community irrespective of the values of other parameters inthe basic reproduction number under treatment.
A contour plot of the reproduction number Rπ, as a
function of the effective treatment rate (πΎπ) and isolation
rate (πΌπ), is depicted in Figure 3(a). As expected, the plot
shows a decrease in Rπvalues with increasing values of
the treatment and isolation rates. For instance, if the useof effective treatment result in πΎ
π= 0.8 and πΌ
π= 0.8,
drug sensitive TB burden will be reduced considerably in thepopulation. Similarly in Figure 3(b) the plot shows a decreasein Rπvalues with decreasing values of the return rate from
lost to follow-up (ππ) and lost to follow-up rate (π
π).
A contour plot of the reproduction number Rπ, as a
function of return rate from lost to follow-up (ππ) and the
effective treatment rate (πΎπ), is depicted in Figure 4(a). The
plot shows a decrease inRπvalues with increasing values of
the treatment rate. Similarly the plot in Figure 4(b) shows adecrease inR
πvalues with increasing values of the isolation
rate (πΌπ).
4.1. Backward Bifurcation Analysis. Model (2) is now investi-gated for the possibility of the existence of the phenomenonof backward bifurcation (where a stable DFE coexists with astable endemic equilibrium when the reproduction number,R0, is less than unity) [44β53]. The epidemiological impli-cation of backward bifurcation is that the elimination (oreffective control) of the TB (and various strains) in the systemis no longer guaranteed when the reproduction number isless than unity but is dependent on the initial sizes of thesubpopulations. The possibility of backward bifurcation inmodel (2) is explored using the centre manifold theory [47],as described in [54] (Theorem 4.1).
Theorem 6. Model (2) undergoes a backward bifurcation atRπ= 1whenever inequality (A.9), given inAppendix A, holds.
Theproof ofTheorem 6 is given in Appendix A (the proofcan be similarly given for the case whenR
π= 1 orR
π= 1).
The backward bifurcation property of model (2) is illustratedby simulating themodel using a set of parameter values givenin Table 2 (such that the bifurcation parameters, π and π,given in Appendix A, take the values π = 558.61 > 0 and π =1.59 > 0, resp.). The backward bifurcation phenomenon ofmodel (2) makes the effective control of the TB strains in thepopulation difficult, since, in this case, disease control whenR0 < 1 is dependent on the initial sizes of the subpopulationsof model (2). This phenomenon is illustrated numerically inFigures 5 and 6 for individuals with drug sensitive, MDR-,and XDR-TB, as well as individuals who are lost to follow-upwith drug sensitive, MDR-, and XDR-TB, respectively.
It is worth mentioning that when the reinfection param-eter of model (2), for the recovered individuals, is set to zero(i.e., π = 0), the bifurcation parameter, π, becomes negative(see Appendix A). This rules out backward bifurcation (inline with Item (iv) of Theorem 4.1 of [54]) in this case. Thus,this study shows that the reinfection of recovered individualscauses backward bifurcation in the transmission dynamicsof TB in the system. To further confirm the absence of thebackward bifurcation phenomenon in model (2) for thiscase, the global asymptotic stability of the DFE of the modelis established below for the case when no reinfection ofrecovered individuals occurs.
-
12 Abstract and Applied Analysis
4
4
6
6
6
6
8
8
8
10
10
1214
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1πΎT
πΌT
(a)
2.5
33
33
3.53.5
3.5
4
4
4.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
πT
πT
(b)
Figure 3: Contour plot of the reproduction number (Rπ) of model (2) as: (a) a function of isolation rate (πΌ
π) and treatment rate (πΎ
π); (b) a
function of return rate from lost to follow-up (ππ) and lost to follow-up rate (π
π). Parameter values used are as given in Table 2.
4.5 4.5 4.55
5 5 55.5 5.5 5.56
6 6 66.5 6.5 6.57
7 7 77.5 7.5 7.58 8 8 88.5 8.5 8.590 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
πΎT
πT
(a)
2.6
2.8 2.82.8
33 3 3
3.2 3.23.2
3.4 3.43.4
3.6 3.63.6
3.8 3.83.8
4 40 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1πΌT
πT
(b)
Figure 4: Contour plot of the reproduction number (Rπ) of model (2) as: (a) a function of return rate from lost to follow-up (π
π) and
treatment rate (πΎπ); (b) a function of return rate from lost to follow-up (π
π) and treatment rate (πΌ
π). Parameter values used are as given in
Table 2.
4.2. Global Stability of the DFE: Special Case. Consider thespecial case of model (2) where the reinfection parametersare set to zero (i.e., π = 0). It is convenient to define thereproduction threshold RΜ0 = R0|π=0.
Theorem 7. The DFE of model (2), with π = 0, is GAS in Ξ¦whenever RΜ0 < 1.
The proof of Theorem 7 is given in Appendix B.The epidemiological significance ofTheorem 7 is that, for
the special case of model (2) with π = 0, TB will be eliminatedfrom the community if the reproduction number (RΜ0) can bebrought to (and maintained at) a value less than unity.
5. The Effects of Isolation
Following the result obtained from the sensitivity analysis, weinvestigate the impact of the isolation parameters πΌ
π, πΌπ, and
πΌπ, which are one of the dominant parameters of model (2).
We start by individually varying these parameters for (say)πΌπ= 0.2, 0.4, 0.6, 0.8, 1.0, with the other parameters given
in Table 2 kept constant. We observed that (see Figure 7) asthe isolation rate, πΌ
π, for drug sensitive TB increases, the
total number of individuals (with drug sensitive, MDR, andXDR) isolatedwith each strain of TB increases, while the totalnumber of individuals who are lost to follow-up decreases.We observed similar result for the isolation rate πΌ
π(see
Figure 8). However, for isolation rate πΌπ, negligible change
was observed and the plots are not shown.
6. Conclusion
In this paper, we have developed and analyzed a system ofordinary differential equations for the transmission dynam-ics of drug-resistant tuberculosis with isolation. From ouranalysis, we have the following results which are summarizedbelow:
-
Abstract and Applied Analysis 13
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
800
700
500
300
100
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M
0
20
40
60
80
100
120
140
160
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
0
5
10
15
20
25
(c)
Figure 5: Backward bifurcation plot of model (2) as a function of time. (a) Individuals with drug sensitive TB (π). (b) Individuals with MDR(π). (c) Individuals with XDR (π). Parameter values used are as given in Table 2.
(i) Themodel is locally asymptotically stable (LAS)R0 <1 and unstable whenR0 > 1.
(ii) The model exhibits in the presence of disease reinfec-tion the phenomenon of backward bifurcation, wherethe stable disease-free coexists with a stable endemicequilibrium, when the associated reproduction num-ber is less than unity.
(iii) As the isolation rate for each strain of TB increases,the total number of individuals infected with theparticular strain of TB decreases.
(iv) Model (2) in the absence of disease reinfection isglobally asymptotically stable (GAS)R0 < 1.
(v) The sensitivity analysis of the model shows that thedominant parameters for the drug sensitive TB are
the disease progression rate (ππ), the recovery rate
(πΎπ) from drug sensitive TB, the infectivity parameter
(ππ), the isolation rate (πΌ
π) from drug sensitive TB
class, fraction of fast progression rates and (ππ1 and
ππ2) into the drug sensitive TB class and lost to follow-up class, and the rate of lost to follow-up (π
π). Similar
parameters and return rates from lost to follow-up(ππ
and ππ) are dominant for MDR- and XDR-
TB. The natural death rate (π), although dominantin MDR- and XDR-TB, is however epidemiologicallyirrelevant.
(vi) Increase in isolation rate leads to increase in totalnumber of individuals isolated with each TB strainresulting in decreases in the total number of individ-uals who are lost to follow-up.
-
14 Abstract and Applied Analysis
Appendices
A. Proof of Theorem 6
Proof. Theproof is based on using the centremanifold theory[47], as described in [54]. It is convenient to make thefollowing simplification and change of variables.
Let π = π₯1, πΈπ = π₯2, π = π₯3, πΈπ = π₯4, π = π₯5, πΈπ =π₯6, π = π₯7, πΏπ = π₯8, πΏπ = π₯9, πΏπ = π₯10, π½ = π₯11and π = π₯12, so that π = π₯1 + π₯2 + π₯3 + π₯4 + π₯5 +π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12. Using the vectornotation x = (π₯1, π₯2, π₯3, π₯4, π₯5, π₯6, π₯7, π₯8, π₯9, π₯10, π₯11, π₯12)
π,model (2) can be written in the form πx/ππ‘ = m(x), wherem = (π1, π2, π3, π4, π5, π6, π7, π8, π9, π10, π11, π12)
π, asfollows:
ππ₯1ππ‘
= π1 = πβ[π½π(π₯3 + πππ₯8) + π½π (π₯5 + πππ₯9) + π½ππ₯7 (π₯7 + πππ₯10)] π₯1
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)β ππ₯1,
ππ₯2ππ‘
= π2 =(1 β ππ1 β ππ2) π½π (π₯3 + πππ₯8) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)β π1π₯2,
ππ₯3ππ‘
= π3 =ππ1π½π (π₯3 + πππ₯8) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯2 +πππ₯8 βπ2π₯3,
ππ₯4ππ‘
= π4 =(1 β ππ1 β ππ2) π½π (π₯5 + πππ₯9) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)β π3π₯4,
ππ₯5ππ‘
= π5 =ππ1π½π (π₯5 + πππ₯9) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯4 +πππ₯3 +πππ₯9 βπ4π₯5,
ππ₯6ππ‘
= π6 =(1 β ππ1 β ππ2) π½π (π₯7 + πππ₯10) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)β π5π₯6,
ππ₯7ππ‘
= π7 =ππ1π½π (π₯7 + πππ₯10) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯6 +πππ₯5 +πππ₯10 βπ6π₯7,
ππ₯8ππ‘
= π8 =ππ2π½π (π₯3 + πππ₯8) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯3 βπ7π₯8,
ππ₯9ππ‘
= π9 =ππ2π½π (π₯5 + πππ₯6) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯5 βπ8π₯9,
ππ₯10ππ‘
= π10 =ππ2π½π (π₯7 + πππ₯10) (π₯1 + ππ₯12)
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12)+ πππ₯7 βπ9π₯10,
ππ₯11ππ‘
= π11 = πΌππ₯3 +πΌππ₯5 +πΌππ₯7 βπ10π₯11,
ππ₯12ππ‘
= π12 = πΎππ₯3 + πΎππ₯5 + πΎππ₯7 βπ11π₯12 βπ [π½π(π₯3 + πππ₯8) + π½π (π₯5 + πππ₯9) + π½π (π₯7 + πππ₯10)] π₯12
(π₯1 + π₯2 + π₯3 + π₯4 + π₯5 + π₯6 + π₯7 + π₯8 + π₯9 + π₯10 + π₯11 + π₯12).
(A.1)
The Jacobian of the transformed system (A.1), at the disease-free equilibriumE1, is given by
π½ (E1) = (π½1 | π½2) , (A.2)
-
Abstract and Applied Analysis 15
where
π½1 =
((((((((((((((((((((((
(
βπ 0 βπ½π
0 βπ½π
00 βπ1 ππ1π½π + ππ1πΎπ 0 0 00 ππ
ππ2π½π β π2 0 0 0
0 0 0 βπ3 ππ1π½π 00 0 π
π2πΎπ ππ ππ2π½π β π4 00 0 0 0 π
π1πΎπ βπ5
0 0 0 0 0 ππ
0 0 (1 β ππ1 β ππ2) π½π + ππ 0 0 0
0 0 0 0 (1 β ππ1 β ππ2) π½π + ππ 0
0 0 0 0 0 00 0 πΌ
π0 πΌ
π0
0 0 (1 β ππ1 β ππ2) πΎπ 0 (1 β ππ1) πΎπ 0
))))))))))))))))))))))
)
,
π½2
=
((((((((((((((((((((((
(
βπ½π
βπ½πππ
βπ½πππ
βπ½πππ
0 00 π
π1π½πππ 0 0 0 00 π
π2π½πππ + ππ 0 0 0 00 0 π
π1π½πππ 0 0 00 0 π
π2π½πππ + ππ 0 0 0ππ1π½π 0 0 ππ1π½πππ 0 0
ππ2π½π β π6 0 0 ππ2π½πππ + ππ 0 0
0 (1 β ππ1 β ππ2) π½πππ β π7 0 0 0 0
0 0 (1 β ππ1 β ππ2) π½πππ β π8 0 0 0
(1 β ππ1 β ππ2) π½π + ππ 0 0 (1 β ππ1 β ππ2) π½πππ β π9 0 0
πΌπ
0 0 0 βπ10 0πΎπ
0 0 0 0 βπ11
))))))))))))))))))))))
)
.
(A.3)
Consider the case when R0 = 1. Suppose, further, that π½πis chosen as a bifurcation parameter. Solving (2) for π½
πfrom
R0 = 1 gives π½π = π½β
π. The transformed system (A.1) at
the DFE evaluated at π½π= π½β
πhas a simple zero eigenvalue
(and all other eigenvalues having negative real parts). Hence,the centre manifold theory [47] can be used to analyze thedynamics of (A.1) near π½
π= π½β
π. In particular, the theorem
in [54] (see also [36, 47, 48]) is used (it is reproduced inthe Appendix for convenience). To apply the theorem, thefollowing computations are necessary (it should be noted thatwe are using π½
πinstead of π for the bifurcation parameter).
Eigenvectors of π½(E1)|π½π=π½β
π
. The Jacobian of (A.1) at π½π= π½β
π,
denoted by π½(E1)|π½π=π½β
π
, has a right eigenvector (associatedwith the zero eigenvalue) given by
w = (π€1, π€2, π€3, π€4, π€5, π€6, π€7, π€8, π€9, π€10, π€11,
π€12)π,
(A.4)
where
π€1 = β1π[π½ππ€3 +π½ππ€5 +π½ππ€7 +π½ππππ€8
+π½ππππ€9 +π½ππππ€10] ,
π€2 =1π1
{[ππ1πΎπ + ππ1π½π] π€3 + ππ1π½ππππ€8} ,
π€3 > 0,
π€5 > 0,
π€4 =1π3
[ππ2πΎππ€3 + ππ1π½ππ€5 + ππ1π½ππππ€9] ,
π€6 =1π5
[ππ1πΎππ€5 + ππ1π½ππ€7 + ππ1π½ππππ€10] ,
π€7 > 0,
π€8 =[(1 β π
π1 β ππ2) π½π + ππ] π€3
[π7 β (1 β ππ1 β ππ2) π½πππ],
-
16 Abstract and Applied Analysis
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LT
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1000
800
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LM
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
80
70
60
50
40
30
20
10
0
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LX
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
15
10
5
0
(c)
Figure 6: Backward bifurcation plot of model (2) as a function of time. (a) Individuals who are lost to follow-up with drug sensitive TB (π).(b) Individuals who are lost to follow-up with MDR (π). (c) Individuals who are lost to follow-up with XDR (π). Parameter values used areas given in Table 2.
π€9 =[(1 β π
π1 β ππ2) π½π + ππ] π€5
[π8 β (1 β ππ1 β ππ2) π½πππ],
π€10 =[(1 β π
π1 β ππ2) π½π + ππ] π€7
(π9 β (1 β ππ1 β ππ2) π½πππ),
π€11 =(πΌππ€5 + πΌππ€7)
π10,
π€12 =1π11
[(1βππ1 βππ2) πΎππ€3 + (1βππ1) πΎππ€5
+ πΎππ€7] .
(A.5)
Also, π½(E1)|π½π=π½β
π
has a left eigenvector k = (V1, V2, V3,V4, V5, V6, V7, V8, V9, V10, V11, V12) (associated with the zeroeigenvalue), where
V1 = 0,
V2 =ππV3π1
,
V4 =ππV4
π3,
V6 =ππV7
π5,
V3 > 0,
-
Abstract and Applied Analysis 17
πΌT = 0.20
πΌT = 0.40
πΌT = 0.60
πΌT = 0.80
πΌT = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
1200
1400
1600To
tal n
umbe
r of i
sola
ted
indi
vidu
als
(a)
πΌT = 0.20
πΌT = 0.40
πΌT = 0.60
πΌT = 0.80
πΌT = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
2500
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 7: Simulation of model (2) as a function of time varying πΌπfor (a) total number of isolated individuals and (b) total number of
individuals who are lost to follow-up. Parameter values used are as given in Table 2.
πΌM = 0.20
πΌM = 0.40
πΌM = 0.60
πΌM = 0.80
πΌM = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
Tota
l num
ber o
f iso
late
d in
divi
dual
s
(a)
πΌM = 0.20
πΌM = 0.40
πΌM = 0.60
πΌM = 0.80
πΌM = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 8: Simulation of model (2) as a function of time varying πΌπ
for (a) total number of isolated individuals and (b) total number ofindividuals who are lost to follow-up. Parameter values used are as given in Table 2.
V7 > 0,
V5 =1
(π4 β ππ2π½π)[ππ1π½πV4 +ππ1πΎπV6
+ [(1β ππ1 β ππ2) π½π +ππ] V9] ,
V8 =1
[π7 β (1 β ππ1 β ππ2) π½πππ][ππ1π½πππV2
+ (ππ2π½πππ +ππ) V3] ,
V9 =1
(π8 β (1 β ππ1 β ππ2) π½πππ)[ππ1π½πππV4
+ (ππ2π½πππ +ππ) V5] ,
V10 =1
(π9 β (1 β ππ1 β ππ2) π½πππ)[ππ1π½πππV6
+ (ππ2π½πππ +ππ) V7] ,
V11 = 0,
V12 = 0.(A.6)
Computations of Bifurcation Coefficients π and π. The appli-cation of the theorem (given in the Appendix) entails the
-
18 Abstract and Applied Analysis
computation of two bifurcation coefficients π and π. It can beshown, after some algebraic manipulations, that
π = V212β
π,π=1π€ππ€π
π2π2
ππ₯πππ₯π
+ V312β
π,π=1π€ππ€π
π2π3
ππ₯πππ₯π
+ V412β
π,π=1π€ππ€π
π2π4
ππ₯πππ₯π
+ V512β
π,π=1π€ππ€π
π2π5
ππ₯πππ₯π
+ V612β
π,π=1π€ππ€π
π2π6
ππ₯πππ₯π
+ V712β
π,π=1π€ππ€π
π2π7
ππ₯πππ₯π
=2π₯1
(βπ€2 βπ€3 βπ€4 βπ€5 βπ€6 βπ€7 βπ€8 βπ€9
βπ€10 βπ€11 βπ€12 +π€12π) {π½π (π€3 +π€8ππ)
β [ππ1V2 + ππ2V3 + (1β ππ1 β ππ2) V8]
+ π½π(π€5 +π€9ππ)
β [ππ1V4 + ππ2V5 + (1β ππ1 β ππ2) V9]
+ π½π(π€7 +π€10ππ)
β [ππ1V6 + ππ2V7 + (1β ππ1 β ππ2) V10]} .
(A.7)
Furthermore,
π = V212β
π=1π€π
π2π2
ππ₯πππ½βπ
+ V312β
π=1π€π
π2π3
ππ₯πππ½βπ
= (π€3 +π€8ππ) [ππ1V2 + ππ2V3 + (1β ππ1 β ππ2) V8]
> 0.
(A.8)
Hence, it follows from Theorem 4.1 of [54] that the trans-formed model (A.1) (or, equivalently, (2)) undergoes back-ward bifurcation atR0 = 1whenever the following inequalityholds:
π > 0. (A.9)
It is worth noting that if π = 0 (i.e., reinfectionof recovered individuals does not occur), the bifurcationcoefficient, π, given in (A.7), reduces to
π = β2π₯1
(π€2 +π€3 +π€4 +π€5 +π€6 +π€7 +π€8 +π€9
+π€10 +π€11) {π½π (π€3 +π€8ππ)
β [ππ1V2 + ππ2V3 + (1β ππ1 β ππ2) V8]
+ π½π(π€5 +π€9ππ)
β [ππ1V4 + ππ2V5 + (1β ππ1 β ππ2) V9]
+ π½π(π€7 +π€10ππ)
β [ππ1V6 + ππ2V7 + (1β ππ1 β ππ2) V10]} .
(A.10)
It follows from (A.10) that the bifurcation coefficient π < 0(ruling out backward bifurcation in this case, in line withTheorem 4.1 in [54]). Thus, this study shows that the back-ward bifurcation phenomenon ofmodel (A.1) is caused by thereinfection of the recovered individuals in the population.
B. Proof of Theorem 7
Proof. The proof is based on using a comparison theorem.The equations for the infected components of model (2), withπ = 0, can be rewritten as
((((((((((((((((((((((((((((((((((((
(
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππ½ (π‘)
ππ‘
))))))))))))))))))))))))))))))))))))
)
= (πΉβπ)
(((((((((((((((((((
(
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΏπ (π‘)
πΏπ (π‘)
πΏπ (π‘)
π½ (π‘)
)))))))))))))))))))
)
βππ
(((((((((((((((((((
(
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΏπ (π‘)
πΏπ (π‘)
πΏπ (π‘)
π½ (π‘)
)))))))))))))))))))
)
,
(B.1)
-
Abstract and Applied Analysis 19
where π = 1 β π, the matrices πΉ and π are as given inSection 2.2, andπ is nonnegativematrices given, respectively,by
π = [π1 | π2] , (B.2)
where
π1 =
(((((((((((((
(
0 ππ1π½π 0 0 0
0 ππ2π½π 0 0 0
0 0 0 ππ1π½π 0
0 0 0 ππ2π½π 0
0 0 0 0 00 0 0 0 00 (1 β π
π1 β ππ2) π½π 0 0 00 0 0 (1 β π
π1 β ππ2) π½π 00 0 0 0 00 0 0 0 0
)))))))))))))
)
,
π2 =
(((((((((((((
(
0 ππ1π½πππ 0 0 0
0 ππ2π½πππ 0 0 0
0 0 ππ1π½πππ 0 0
0 0 ππ2π½πππ 0 0
ππ1π½π 0 0 ππ1π½πππ 0ππ2π½π 0 0 ππ2π½πππ 00 (1 β π
π1 β ππ2) π½πππ 0 0 00 0 (1 β π
π1 β ππ2) π½πππ 0 0(1 β ππ1 β ππ2) π½π 0 0 (1 β ππ1 β ππ2) π½πππ 0
0 0 0 0 0
)))))))))))))
)
.
(B.3)
Thus, since π(π‘) β€ π(π‘) in Ξ¦ for all π‘ β₯ 0, it follows from(B.1) that
((((((((((((((((((((((((((((((((
(
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΈπ (π‘)
ππ‘
ππ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππΏπ (π‘)
ππ‘
ππ½ (π‘)
ππ‘
))))))))))))))))))))))))))))))))
)
β€ (πΉβπ)
(((((((((((((((((((
(
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΈπ (π‘)
π (π‘)
πΏπ (π‘)
πΏπ (π‘)
πΏπ (π‘)
π½ (π‘)
)))))))))))))))))))
)
. (B.4)
Using the fact that the eigenvalues of the matrix πΉ β π allhave negative real parts (see the local stability result givenin Lemma 3, where π(πΉπβ1) < 1 if R0 < 1, which isequivalent to πΉ β π having eigenvalues with negative realparts when R0 < 1 [36]), it follows that the linearizeddifferential inequality system (B.4) is stable whenever R0 <1. Consequently, by comparison of theorem [34] (Theorem1.5.2, p. 31),
(πΈπ (π‘) , π (π‘) , πΏπ (π‘) , πΈπ (π‘) ,π (π‘) , πΏπ (π‘) , πΈπ (π‘) ,
π (π‘) , πΏπ (π‘) , π½ (π‘)) β (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) ,
as π‘ β β.
(B.5)
Substituting πΈπ= π = πΏ
π= πΈπ= π = πΏ
π= πΈπ= π =
πΏπ= π½ = 0 into the equations of π and π in model (2), and
noting that π = 0, gives π(π‘) β πβ, π (π‘) β 0 as π‘ β β.Thus, in summary,
(π (π‘) , πΈπ (π‘) , π (π‘) , πΏπ (π‘) , πΈπ (π‘) ,π (π‘) , πΏπ (π‘) ,
πΈπ (π‘) , π (π‘) , πΏπ (π‘) , π½ (π‘) , π (π‘)) β (π
β, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0) ,
(B.6)
-
20 Abstract and Applied Analysis
as π‘ β β. Hence, the DFE (E0) of model (2), with π = 0, isGAS in Ξ¦ if RΜ0 < 1.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
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