# HIV/AIDS Model with Early Detection and Treatment

Authors: Augustine S. Mbitila; Jean M. Tchuenche
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

A classical epidemiological framework is used to qualitatively assess the impact of early detection and treatment on the dynamics of HIV/AIDS. Within this theoretical framework, two classes of infected populations: those infected but unaware of their serological status and those who are aware of their disease status, are considered. In this context, we formulate and analyze a deterministic model for the transmission dynamics of HIV/AIDS and assess the potential population-level impact of early detection in curtailing the epidemic. A critical threshold parameter for which case detection will have a positive impact is derived. Model parameters sensitivity analysis indicates that the number of partners is the most sensitive (in increasing the average number of secondary transmission) parameter. However, the case detection coverage is the main drivers in reducing the initial disease transmission. Numerical simulations of the model are provided to support the analytical results. Early detection and treatment alone are insufficient to eliminate the disease, and other control strategies are to be explored.


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## Body

## 1. Introduction

HIV/AIDS has killed more than 25 million people globally since its emergence in 1981, making it one of the most destructive epidemics in recorded history. The disease continues to inflict a significant morbidity, mortality, and social-economic and public health burden. For the estimated 33.3 million people living with HIV after nearly 30 years into a very complex epidemic, the gains are real but still fragile, even as the number of annual AIDS-related deaths worldwide has steadily decreased from the peak of 2.1 million in 2004 to an estimated 1.8 million in 2009 [1].Various preventative and therapeutic measures have been embarked upon, aiming at combating one of the greatest pandemics in modern times [2, 3]. In sub-Saharan Africa, many infected individuals are unaware of their disease status. Recent randomized control trials have found that treating HIV-positive individuals with antiretroviral drugs reduces the risk of them transmitting the disease to their heterosexual partners by more than 90% [4]. As the current treatment therapy has been proven beyond reasonable doubt to reduce transmission, it is imperative to identify those who are infected and put them on treatment when eligible. It is therefore desirable to encourage voluntary testing that will increase case detection, thereby reducing the number of secondary infections of individuals receiving treatment. There is no explicit mathematical account of the potential population level impact of case detection when treatment is available known to us. Thus, a dynamical system model is formulated in order to assess the trade-off/population-level impact between treatment and early detection of HIV positive. The results are sensitive to parameter values, and for this reason, a deterministic sensitivity analysis is carried out.The rest of this work is organized as follows. The basic model formulation and its analysis are provided in Section2. The extended model incorporating case detection and treatment is described and analyzed in Section 3. The model simulation using heuristic parameter values for the purpose of illustration follows in Section 4.


## 2. Model Formulation and Analysis

We begin by formulating a deterministic sex-structured basic HIV/AIDS model (i.e., without interventions). Individuals are identified as male and female only in connection with features peculiar to their sex. The male to female infectivity rate is greater than the female to male [5]. We also assume that the mixing between individuals is homogeneous; individuals may become HIV-infected only through sexual contacts with HIV infected individuals. Those in the final disease stage are considered too ill to remain sexually active. We ignore important HIV transmission path such as intravenous drug injections, vertical transmission, breast feeding, blood transfusion, and needle sharing. It is also assumed that there is no recruitment of HIV positive. The total heterosexual population is divided into male and female subpopulations with the following epidemiological subgroups (the classification is based on individuals disease status): susceptible male and female (Sm,Sf), infected male and female (Im,If), and symptomatic individuals in the final disease (AIDS) stage (Am,Af).New recruits enter the heterosexually active population at constant ratesΛm and Λf for male and female, respectively (all recruits into the population are assumed susceptible). Male and female susceptible acquire infection at time-dependent rates λm and λf and become infectious. Infectious individuals exhibit AIDS clinical defining symptoms at rates h and z for male and female, respectively. In the absence of the disease, individuals in the population die of natural death at the rate μ. The disease-induced mortality rate is d for both individuals in the infectious and AIDS classes. A full description of the model variables and parameters used in the model is described in Tables 2 and 3, respectively.Based on our model description and assumptions, we establish the following equations. We note that the red and dash arrows, respectively, in Figures1 and 2 are the feedback branches which indicate how the male and female subpopulations are coupled (via the force of infection)(2.1)MaledSmdt=Λm-(μ+λm)Sm,dImdt=λmSm-(μ+d+h)Im,dAmdt=hIm-(μ+d)Am,FemaledSfdt=Λf-(μ+λf)Sf,dIfdt=λfSf-(μ+d+z)If,dAfdt=zIf-(μ+d)Af.
The forces of infection for male and female are, respectively, given by(2.2)λm=ηfβmIfNf,λf=ηmβfImNm,
where βm, βf and ηm, ηf are, respectively, the probabilities of acquiring HIV and the average number of male and female sexual partners, respectively. The basic model (2.1) with nonnegative initial conditions is epidemiologically meaningful and mathematically well posed. Thus, system (2.1) is dissipative (i.e., all feasible solutions are uniformly bounded [6, 7]). System (2.1) has a disease-free equilibrium (DFE) given by(2.3)E0=(S0m,I0m,A0m,S0f,I0f,A0f)=(Λmμ,0,0,Λfμ,0,0).
Using the next-generation operator method [8], the basic model reproduction number R0 of model (2.1), defined as the number of secondary infections caused by a typical infected individual introduced into the entire susceptible population during his entire period of infectiousness [9], is given by(2.4)R0=βfηm(μ+d+h)βmηf(μ+d+z).
The expression of R0 is a geometric mean of the average number of secondary male infections produced by one female, and the average number of secondary female infections produced by one male. From Theorem 2 of van den Driessche and Watmough [8], the following result holds.Figure 1
The basic HIV/AIDS model flowchart.Figure 2
Flow diagram of the HIV/AIDS model with interventions.Lemma 2.1.
The DFEE0 of system (2.1) is locally asymptotically stable if R0<1, and unstable if R0>1.IfR0<1, then on average an infected individual produces less than one new infection over its infectious period, and the epidemic cannot grow. That is, a small influx of infected individuals would not generate large outbreaks if R0<1. Conversely, if R0>1, then each infected individual produces on average more than one new infection, and the disease can invade the population. However, in order for disease elimination to be independent of the initial sizes of the subpopulations of the model when R0<1, global stability of E0 is required.Lemma 2.2.
TheE0 is globally asymptotically stable if R0<1, and unstable otherwise.Proof.
The proof is based on a comparison theorem [10]. The rate of change of the variables representing the infected components of the system (2.1) can be written as
(2.5)(dImdtdAmdtdIfdtdAfdt)=(F-V)(ImAmIfAf)-(λmSm0λfSf0),
where the matrices F and V are given, respectively, by
(2.6)F=(00βmηfΛmμNf00000βfηmΛfμNm0000000),V=(μ+d+h000-hμ+d0000μ+d+z000-zμ+d).
Thus,
(2.7)(dImdtdAmdtdIfdtdAfdt)≤(F-V)(ImAmIfAf).
Using the fact that the eigenvalues of the matrix F-V all have negative real parts, it follows that the linearized differential inequality above is stable whenever R0<1. Consequently, (Im,Am,If,Af)→(0,0,0,0) as t→∞. By a comparison Theorem [10], (Im,Am,If,Af)→(0,0,0,0) as t→∞. Thus, (Sm,Im,Am,Sf,If,Af)→(Λm/μ,0,0,Λf/μ,0,0) as t→∞ for R0<1, and hence, the DFE is globally asymptotically stable if R0<1.The above result indicates that HIV could be eliminated from the community if the threshold quantityR0 can be brought to (and maintained at) a value less than unity. The endemic equilibrium (EE) of (2.1) is given by(2.8)E*=(Sm*,Im*,Am*,Sf*,If*,Af*),
where(2.9)Sm*=Λm(λm*+μ),Im*=Λmλm*(λm*+μ)(μ+d+h),Am*=hΛmλm*(λm*+μ)(μ+d+h)(μ+d),Sf*=Λf(λf*+μ),If*=Λfλf*(λf*+μ)(μ+d+z),Af*=zΛfλf*(λf*+μ)(μ+d+z)(μ+d),
where λm*=ηfβm(If*/Nf*) and λf*=ηmβf(Im*/Nm*). Solving for λm* using the values of Im*, If* in (2.9) and the value of λf*, after some lengthy algebraic manipulations, the endemic equilibrium of HIV/AIDS basic model satisfies the following linear equation:(2.10)Aλm*-B=0,
where(2.11)A=μNmNf(μ+d+h)(μ+d+z)+NfηmβfΛm(μ+d+z),B=μ2NmNf(μ+d+h)(μ+d+z)[1-ηfβmΛfηmβfΛmμ2NmNf(μ+d+h)(μ+d+z)]=μ2NmNf(μ+d+h)(μ+d+z)[1-R02],A>0 while B<0 provided R0>1, and consequently, the linear system Aλm*-B=0 has a unique positive solution λm*=B/A, whenever R0>1. The components of the endemic equilibrium E* are then determined by substituting λm*=B/A into (2.9). Noting that R0<1 implies B>0; thus, for R0<1, the force of infection λm* at steady state is negative (and biologically meaningless). Hence, the model has no endemic equilibrium in this case. Thus, we have established the following result.Lemma 2.3.
The HIV/AIDS model (2.1) has a unique positive EE E* whenever R0>1 and none otherwise.The uniqueness ofE* and the global stability of the DFE imply that the model does not exhibit the phenomenon of backward or subcritical bifurcation where a locally stable EE coexists with a stable DFE when the reproduction number is less than unity. Because the model parameters are taken from different sources, a deterministic sensitivity analysis is carried out using the approach in [11]. Sensitivity indices (of the reproduction number) which measure initial disease transmission allow us to estimate the relative change in a state variable when a parameter changes. The sensitivity indices of R0 to the parameters for the HIV/AIDS model are given in Table 1. The negative parameters simply means that an increase in that parameter leads to a decrease in the reproductive number. For instance, a 10% increase of the number of sexual partners will lead to a 5% increase of the value of R0 (initial disease transmission threshold).Table 1
Numerical values of sensitivity indices ofR0.
Parameter symbolSensitivity indexd−0.6935Λm, Λf, ηm, ηf, βm, βf+0.5000h−0.1302z−0.1302Table 2
Model variables and their description.
SymbolDescriptionSm(t)Number of susceptible males at timetIm(t)Number of infectious males at timetAm(t)Number of fully symptomatic AIDS males at timetSf(t)Number of susceptible females at timetIf(t)Number of infectious females at timetAf(t)Number of fully symptomatic AIDS females at timetDm(t)The number of detected (screened) males at timetTm(t)The number of treated males at timetDf(t)The number of detected (screened) females at timetTf(t)The number of treated females at timetTable 3
Parameter definitions and their values.
ParameterDefinitionValue(yr)-1SourceμNatural mortality rate0.0222[1]dDisease-induced death rate0.333[12]βmMale effective contact rate0.025AssumedβfFemale effective contact rate0.015AssumedηmAverage number of male partners per female3BMCηfAverage number of female partners per male5BMCτmTreatment rate of detected male0<τm<1AssumedτfTreatment rate of detected female0<τf<1AssumedhProgression rate of infected females to AIDS0.125[13]zProgression rate of infected males to AIDS0.125[13]αmProgression rate of males under treatment to AIDS0.004AssumedαfProgression rate of females under treatment to AIDS0.004AssumedpmReduction in transmission of HIV from treated males0.02BMCpfReduction in transmission of HIV from treated females0.02BMCrmReduction in transmission of HIV from detected male0.004AssumedrfReduction in transmission of HIV from detected female0.0025AssumedσmRate at which infected males are detected0.512BMCσfRate at which infected females are detected0.675BMCΛmNet flow rate into the susceptible male class100,000STHΛfNet flow rate into the susceptible female class100,000STH


## 3. Analysis of the Model with Interventions

The basic model is extended to include screening (detected classes,Dm, Df) and treatment classes (Tm, Tf). It is assumed that the number of contacts made by susceptible individuals under treatment is less than or equal to the number of contacts made with an untreated infective due to behavioral change. This is captured via the parameters rm and rf which are both less than or equal to unity (0≤rm, rf≤1). Also, treatment reduces infectivity, accounted herein by the parameters pm, pf<1. The model compartments and flow are depicted in Figure 2, while the additional variables and parameters for the extended model are described in Tables 2 and 3, respectively.With the above assumptions and terminology, the model is given by the following system of nonlinear equations:(3.1)MaledSmdt=Λm-(λm+μ)Sm,dImdt=λmSm-(μ+d+h+σm)Im,dDmdt=σmIm-(μ+d+τm)Dm,dTmdt=τmDm-(μ+d+αm)Tm,dAmdt=αmTm+hIm-(μ+d)Am,FemaledSfdt=Λf-(λf+μ)Sf,dIfdt=λfSf-(μ+d+z+σf)If,dDfdt=σfIf-(μ+d+τf)Df,dTfdt=τfDf-(μ+d+αf)Tf,dAfdt=αfTf+zIf-(μ+d)Af.
The force of infections of male and female is, respectively, given by (3.2)λm=ηfβmIf+rfDf+pfTfNf,λf=ηmβfIm+rmDm+pmTmNm.
The DFE of model system (3.1) denoted by E00 is given by (3.3)E00=(Λmμ,0,0,0,0,0,Λfμ,0,0,0,0,0).
Using the next-generation matrix operator [8], the reproduction number of (3.1) is (3.4)RT=RmTRfT,
where (3.5)RmT=βfηmμ+d+h+σm(1+rmσmμ+d+τm+pmτmσm(μ+d+τm)(μ+d+αm)),RfT=βmηfμ+d+z+σf(1+rfσfμ+d+τf+pfτfσf(μ+d+τf)(μ+d+αf)).
Thus, using Theorem 2 of van den Driessche and Watmough [8], the following result holds.Lemma 3.1.
The DFEE00 of model system (3.1) is locally asymptotically stable if RT<1, and unstable if RT>1.RmT and RfT are the reproduction numbers for males and females, respectively (i.e., RmT represents the number of females infected by a single male during his entire period of infectiousness in a population where treatment is available). If the interventions are dropped, that is, σm=0, σf=0, and τm=0, τf=0, then the effective reproduction number RT reduces to the basic reproduction number R0. The endemic equilibrium of (3.1) denoted by E** is (3.6)E**=(Sm**,Im**,Dm**,Tm**,Am**,Sf**,If**,Df**,Tf**,Af**),
(3.7)Sm**=Λmμ+AIf**,Sf**=Λfμ+BIm**,Dm**=σmμ+d+τmIm**,Tm**=σmτm(μ+d+τm)(μ+d+αm)Im**,Tf**=σfτf(μ+d+τf)(μ+d+αf)If**,Im**=AΛf[ABΛmΛf-μ2(μ+d+z+σf)(μ+d+h+σm)](μ+d+h+σm)(μABΛm+A2BΛmΛf),If**=ABΛmΛf-μ2(μ+d+z+σf)(μ+d+h+σm)μ(μ+d+z+σf)(μ+d+h+σm)A+ABΛm,Df**=σfμ+d+τfIf**,Am**=αmσmτm+h(μ+d+τm)(μ+d+αm)(μ+d)(μ+d+τm)(μ+d+αm)Im**,Af**=αfσfτf+z(μ+d+τf)(μ+d+αf)(μ+d)(μ+d+τf)(μ+d+αf)If**,
where (3.8)A=βmηfNf(1+rfσfμ+d+τf+pfτfσf(μ+d+τf)(μ+d+αf)),B=βfηmNm(1+rmσmμ+d+τm+pmτmσm(μ+d+τm)(μ+d+αm)).
The endemic equilibrium exists provided Im**>0 and If**>0. Consider If** given in (3.7), then (3.9)If**=ABΛmΛf-μ2(μ+d+z+σf)(μ+d+h+σm)μ(μ+d+z+σf)(μ+d+h+σm)A+ABΛm=ABΛmΛf-μ2(μ+d+z+σf)(μ+d+h+σm)=[ABΛmΛfμ2(μ+d+z+σf)(μ+d+h+σm)-1].
Substituting A and B in (3.9), after some rearrangement, we obtain (3.10)If**=RT2-1.
It is therefore evident that If**>0 provided RT>1. A similar expression can be derived for Im**. Thus, we have established the following result.Lemma 3.2.
The model system (3.1) has a unique positive EE whenever RT>1 and none otherwise.We analytically investigate the impact of case detection on HIV/AIDS dynamics. By partially differentiatingRmT with respect to the case detection rate σm, we obtain (3.11)∂RmT∂σm=βfηm(μ+d+h+σm)2(Δ-1),
where Δ=((μ+d+h)/(μ+d+h+τm))(rm+pmτm/(μ+d+αm)). For Δ<1, (3.11) is negative and consequently, early detection will always have a positive impact on the dynamics of HIV/AIDS. If Δ=1, then case detection has no impact (this case will only arise if treatment of those eligible does not follow). From epidemiological and demographical standpoint, the threshold parameter Δ≤1 is to be expected.The sensitivity indices ofRT are given in Table 4. Next, we numerically investigate the impact of the number of partners, case detection, and treatment on the disease dynamics. Tanzania started HIV/AIDS care and treatment in October 2004, and the target for the first year was to cover 44,000 patients. About 96 care and treatment providing facilities were selected to initiate the services, which included four referral hospitals; Muhimbili, Kilimanjaro Christian Medical Centre (KCMC), Bugando Medical Centre (BMC), and Mbeya Medical Centre. Some of the parameter values (see Table 3) are provided courtesy of the regional medical officer estimated based on data from the Bugando Medical Centre (BMC) and the Sekou Toure Hospital (STH) both located in Mwanza City in northern Tanzania. Others are taken from the literature, and the remaining ones are assumed within realistic range for the purpose of illustration.Table 4
Sensitivity indices ofRT.
ParameterSensitivity indexΛm, Λf, ηm, ηf, βm, βf+0.5000σf−0.3206σm−0.2808h−0.0194z−0.0072rf+0.0023rm+0.0015τm−0.0003τf−0.0002pm+0.0002pf+0.0001Multiple partnerships increase the risk factor of acquiring HIV. When the number of partners is small over a long time period, the rate of infection is minimal. In this case, the disease tends to die down (Figure3(a)). The disease will persist when multiple and concurrent partnerships are frequent in the community, in which case infections are on the increase over time (Figure 3(b)).Effect of the number of partners on the dynamics of HIV-positive (a) low sexual activity:ηm=2 and ηf=3, (b) high sexual activity: ηm=4 and ηf=6.
(a)(b)Since it is assumed that all detected HIV-positive individuals are treated if eligible, the shapes of the time trends of detected (Figure4(a)) and treated individuals (Figure 4(b)) are similar. The slight difference is due to the rate of developing full blown AIDS from the treated class.Time series of susceptible, detected, and treated individuals.
(a)(b)Figure5 depicts the effect of increasing early detection and treatment on the reproduction numbers R0 and RT.Graphical representation ofR0 as detection and treatment rates are varied.
(a)(b)


## 4. Conclusion

The dynamic and determinants of the HIV epidemics are multiple and are shaped by the sexual patterns which are related to social, cultural, and economic factors: for example, promiscuity, low and inconsistent condom use, intergenerational sex, concurrent sexual partners, and various opportunistic infections. Screening is a barometer for achieving success in the fight against the epidemic. It is against this background that a simple deterministic HIV/AIDS model which accounts for case detection and antiretroviral therapy was formulated and analyzed. Conditions for the global stability which rules out any possibility for the model to exhibit the phenomenon of backward bifurcation were provided. Therefore, the classical requirement of the reproduction number to be less than unity might be sufficient for disease elimination. However, HIV/AIDS is inherently a multifaceted disease with poverty-drug use-behavioral/attitudinal change and gender inequality are some of the social factors that need to be addressed. Sensitivity results point to the case detection rate as a driving factor in stemming the tide of the epidemic. Thus, increasing voluntary and or mass screening will always have a positive impact on HIV/AIDS control in reducing the disease burden.Numerical simulations clearly show that early detection and treatment alone are insufficient to eliminate the disease (Figures4 and 5). Other control strategies such as condom and microbicides used are to be explored. The study is not exhaustive and can be extended in various ways by incorporating a potential (imperfect) vaccine, withdrawal from sexual activity of a fraction of individuals in the AIDS defining stage, and development of drug resistance and superinfection (with different virus strains).



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