# On the Optimal Control of HIV-TB Co-Infection and Improvement of Workplace Productivity

Authors: Baba Seidu; Oluwole Daniel Makinde; Ibrahim Yakubu Seini
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

Human immunodeficiency virus (HIV) and tuberculosis (TB) have long been known to have a synergistic relationship. This is a result of each of the diseases impacting negatively on the immune system of the infected persons. The impact of these diseases on workforce productivity is studied in this paper from the viewpoint of dynamical systems. In this paper, we present a nonlinear ordinary differential equation model to study the dynamics of HIV-TB co-infection and its effect on workforce productivity. The main model is first decoupled into two basic submodels of HIV-only and TB-only models, whose qualitative properties are presented before the qualitative properties of the main model are studied. While the HIV-only model is shown to have a globally asymptotically stable disease-free equilibrium whenever its basic reproduction number is less than unity, the TB-only model is shown to exhibit backward bifurcation under some conditions. To investigate the impact of various intervention strategies on the control of the co-infection and improvement of workforce productivity, five time-dependent controls (involving transmission prevention for the two diseases, therapy for the two diseases, and capacity building for improved workforce productivity) are incorporated into the basic model to form an optimal control problem, which is qualitatively analyzed using Pontryagin’s maximum principle and numerically simulated. Incremental cost-effectiveness analysis is conducted with the results of the numerical simulations. It is observed that the most cost-effective strategy for fighting the spread of the co-infection with enhanced productivity is that of combining both preventative and curative measures along with skills training.


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## Body

## 1. Introduction

HIV continues to be among the top diseases that place economic burden on governments and individuals, apart from the many lives that are lost through the disease. The disease has claimed over 39 million lives and continues to claim more even with the stepping up of intervention strategies. Tuberculosis (TB) places second only to HIV as a worldwide killer due to a single infectious agent. TB also continues to place huge burden on the world as unacceptable numbers of people continue to get infected and die from the disease. The low-and-middle-income countries continue to bear the brunt of TB, with over 95% of TB deaths occurring in these countries. An estimated 2 billion dollars is required to fill the resource gap required to fully implement the current intervention strategies [1]. HIV and TB have long since been identified to have a synergistic relationship, with each disease increasing the risk of contracting the other [2, 3]. People living with HIV are said to be about 12–20 times more likely to develop TB than HIV-negative people.Mathematical models have been used to study the dynamics of infectious diseases. These models are especially useful as they help in improving our understanding about the dynamics and the factors that affect the spread of diseases. Most of the mathematical models that deal with HIV and other related diseases generally appear to follow two trends. The first category involves those that deal with modeling the pathogenesis of the diseases [4–8], while the second category involves those that deal with the effects of demographic and other epidemiological factors on the spread of the diseases [9–13]. A lot of research on HIV-TB co-infection has been conducted using mathematical modeling in order to study the dynamics and to identify optimal strategies to curb the spread of the two diseases. The work in [14] appears to be among the very first mathematical models that were proposed to study the spread of HIV-TB co-infection. The goal in [14] was to provide a mathematical understanding of the impact HIV had on the spread of TB. Following [14], a mathematical model was developed in [15] to study the spread of HIV and curable TB co-infection within a variable-sized population. The work in [15] was modified by using density-dependent birth and death rates to study the effect of TB on HIV dynamics, and it was observed that effective treatment of TB may have a positive effect in slowing down the surge of HIV [16]. The work in [17] also modified the model in [15] to include the effect of antiretroviral therapy as well as other transmission dynamics of TB. It was observed that TB treatment is equally effective for persons who were infected with both TB and HIV. They also concluded that administering antiretroviral therapy alongside the treatment of latent and active TB infected persons could slow down progression to the AIDS stage. In [18], an earlier model [19] was extended to study the effect of administering antiretroviral therapy at the various phases of TB infection treatment. They observed that early administration of antiretroviral therapy during TB treatment is very effective in reducing disease-induced deaths. Tanvi et al. [20] proposed a model to study the effect of Holling type II TB treatment. To reduce the gap between actual and reported infection infections, the authors of [21] proposed a model to incorporate treatment and detection of TB infection and studied the optimal strategies in controlling the spread of HIV/TB co-infection.As the world strives to increase productivity to match its increasing population, it is important for the working class to remain healthy. We note however that majority of the people infected with HIV and TB are those in their active working years. This leads to declining workforce productivity, reduced household incomes, and increased poverty. The question of how HIV-TB co-infection impacts workplace productivity is thus pertinent. The research studies discussed so far have provided the basic framework for modeling HIV-TB co-infection. However, not so much research has been done on the impact of infectious diseases on workforce evolution/dynamics. The pioneering work of Okosun et al. [22] is probably one of the first to present a mathematical modeling approach to describe the impact of HIV on workforce productivity. They sought to study the optimal strategies in controlling the spread of HIV and improving productivity using the least cost and observed that the most cost-effective strategy is to use infection preventative measures together with skills improvement measure. Following [22], Seidu and Oluwole [9] proposed an industrial HIV model to determine the optimal control of workforce productivity and HIV spread in the presence of carefree individuals. In a related work [12], a model was proposed to study the impact of HIV-malaria co-infection on workforce productivity. Through cost-effectiveness analysis, they observed that preventative measures are the most cost-effective strategy in the fight against the spread of the co-infection and improved productivity. Also, the work in [23] presented an optimal control problem to study the impact of carefree attitude towards safe sex on the transmission dynamics of HIV and workforce productivity. In this current paper, the goal is to incorporate the concept of workforce productivity developed in [9, 22, 23] to propose an industrial HIV-TB co-infection model which describes the dynamics of the co-infection and its effects on a typical workforce. Thus, this paper presents a deterministic mathematical model that describes the dynamics of HIV and TB infections in a homogeneously mixed workforce.The remainder of the paper is organized as follows. Section2 discusses the formulation of the model in focus. In Section 3, the full model is decomposed into two basic submodels (namely, the HIV-only model and the TB-only submodel) and their qualitative analysis is presented. The qualitative properties of the full model are discussed in Section 4. In Section 5, the main model is extended into an optimal control problem which is qualitatively analyzed using Pontryagin’s maximum principle. In Section 6, numerical experimentation of the resulting optimal control is performed and the results are discussed. The main conclusions of the work are finally presented in Section 7.


## 2. Formulation of the Mathematical Model

Consider a homogeneously mixed workforce of sizeNt consisting of Susceptibles, St, and Infectives, It. The infectives consist of those employees who are either HIV or TB infected or both. Thus, the infectives consist of the following:(1)
HIV-only infected persons, some of whom are productive,Iph, and others are nonproductive, Inh.(2)
Employees without HIV who are exposed to TB,Ep, who are assumed to be productive.(3)
Employees with only active TB infection,Int. These people are assumed to be nonproductive.(4)
Employees in the latent state of TB who have contracted HIV; those who are productive,Ipht and those who are nonproductive, Inht.(5)
Employees, with active TB and HIV infection,Ind, who are assumed to be nonproductive.(6)
Employees who had only TB and recovered,Rp, who are assumed to be productive.(7)
AIDS patients,At, who are assumed to be nonproductive.Thus, the total workforce population is given by(1)N=S+Iph+Inh+E+Int+Rp+Ipht+Inht+Ind+A.In the subscripts, we usep,n,a,t, and d to represent productive, nonproductive, HIV, TB, and dually infected, respectively.HIV/AIDS is spread through contact with an infected person while TB is contracted through inhaling of air containing droplets which are left in the air when an infected person sneezes without covering. Individuals susceptible to HIV (S, E, Int, and Rp) are assumed to get infected through contact with HIV-infected persons (Inh,Inh, Inht, Ipht, Ind) with a force of infection λh=βhchIph+Inh+Ipht+Inht+Ind/N. The parameters βh and ch, respectively, represent the probability of infection per contact and the average number of contacts per infected person. Due to differences in the immune systems of people, a fraction f1 of Susceptibles who are infected with HIV only move to the productive HIV-infective class, Iph, and the remaining fraction, 1−f1, moves to the nonproductive HIV-only infective class, Inh. Thus, the rate of progression of fully susceptible individuals into the Iph and Inh classes is f1λh and 1−f1λh, respectively. Individuals in the Ep and Int classes progress to classes Inht and Ind at rates ξ1λh and ξ2λh, respectively, where ξ1>1 and ξ2>ξ1 are modification parameters that account for increased susceptibility due to the TB infection. Individuals in the Rp class get HIV-only infection to join the Iph and Inh at rates f1λh and 1−f1λh, respectively. Individuals susceptible to TB (S, Iph,Inh, and Rp) get infected with TB through contact with TB-infected persons (Int and Ind) with a force of infection λT=βTcTInt+Ind/N.Individuals who are exposed to TB and are in the latent stage of the infection are assumed incapable of transmitting the disease. Due to differences in the immune systems of people, a fraction,f2, of the susceptible individuals infected with TB only move to the exposed class, Ep, and are assumed to be productive since exposed persons in the latent stage mostly do not show clinical symptoms of TB infection and usually appear and feel healthy, while the remaining fraction moves to the infective class. People in the TB-exposed classes Ep,Ipht, and Inht are assumed to progress to the active TB-infected classes, Int,Ind, and Ind, respectively, at rates θ, η1θ and η2θ, respectively. The parameters η1>1 and η2>1 are modification parameters due to increased possibility of progressing to infectiousness because of the impact of HIV infection. The progression of people in the Inht and Ipht classes to the active stage of TB leads to dual infection. People who recover from TB-only infection, Rp, are assumed to be re-infected with TB at the rate ρλT where ρ<1 is a modification parameter accounting for the increased immunity that leads to reduced susceptibility due to their previous TB infection (individuals who recover from TB are known to often have some increased partial immunity from the disease). HIV-infected people, Iph, Inh, Ipht, Inht, and Ind, are assumed to progress to the AIDS class, A, at rates δ, κ1δ, κ2δ, κ3δ, and κ4δ where 1<κ1<κ2<κ3<κ4 are modification parameters that account for increased chances of failure of HIV treatment due to either lower immunity or TB infection. It is assumed that TB-infected people in classes, Iph and Ind, are treated of TB to join the Rp and Inht classes at rates r and τ2r, respectively, where τ2<1 is a modification parameter accounting for reduced rate of recovery due to HIV infection of people in the Ind class. Administration of HAAR drugs is assumed to lead to reduced viral load which in turn leads to people in the Inh and Inht moving to the Iph class at rates σ and τ3σ where τ3<1 is a modification parameter that accounts for reduced reduction of viral load in people within the Inht class due to the TB infection. Dual infection results when individuals infected with either disease also get infected with the other. Thus, dual infection results from HIV-only infected persons getting infected with TB, TB-only infected persons getting infected with HIV, and HIV-infected persons in the latent stage of TB infection progressing to the active stage of TB infection.Individuals in all classes are assumed to have a natural death rate ofμ, whereas individuals in TB-infected classes Int, Ind suffer TB-induced death rate of μT and individuals in the AIDS class, A, suffer an AIDS-induced death rate of μh. The dynamics described above are represented in Figure 1.Figure 1
Schematic diagram of the industrial HIV-TB model.The mathematical model that describes the dynamics of HIV-TB infection in a workforce describe thus far is given by the following set of differential equations:(2)dSpdt=Q−λh+λT+μSpdIphdt=f1λhSp+Rp+σInh−τ1λT+k1IphdInhdt=1−f1λhSp+Rp−τ1λT+k2InhdEpdt=f2λTSp+ρRp−ξ1λh+k3EpdIphtdt=τ1f2λTIph+σInht−k4IphtdInhtdt=τ11−f2λTInh+ξ1λhEp+τ2rInd−k5InhtdIntdt=1−f2λTSp+ρRp+θEp−ξ2λh+k6IntdRpdt=rInt−ρλT+λh+μRpdInddt=ξ2λhInt+τ1λTf2Inh+1−f2Iph+θη1Inht+η2Ipht−k7InddAdt=δIph+κ1Inh+κ2Ipht+κ3Inht+κ4Ind−k8A,where(3)k1=δ+μ,k2=σ+κ1δ+μ,k3=θ+μ,k4=η2θ+κ2δ+μ,k5=η1θ+κ3δ+σ+μ,k6=r+μ+μT,k7=τ2r+κ4δ+μ+μT,k8=μ+μh.It should be noted that model (2) can be decomposed into two single-disease infection models, namely, the HIV-only model and the TB-only model. In the next few sections, each of the two basic models is discussed before the full model is discussed. The parameters of the model and their values used for numerical simulation purposes only are presented in Table 1.Table 1
Description of model parameters and baseline values.
ParameterDescriptionValueReferenceQRate of recruitment into population100chPer capita contact rate of HIV-infectious persons1cTPer capita contact rate of TB-infectious persons1βhHIV transmission probability per contactVariableβTHIV transmission probability per contactVariableσRate of improvement to productive classes due to training and treatment of HIV-infected peopleVariableδRate of progression of HIV patients to AIDS0.1[24]θRate of progression from latent to active TB0.03[25]rRecovery rate from active TB infectionVariableτ1Modification parameter for increased rate of TB infection due to HIVVariablef1Fraction of newly HIV-infected people who are productiveVariablef2Fraction of newly infected individuals with latent TBρModification parameter for reduced susceptibility to TB due to acquired partial immunityη1,η2Modification parameters for increased progression to active TB forInht and Ipht classesVariableξ1,ξ2Modification parameters for increased rate of HIV infection forEp and Int classesVariableκ1,κ2,κ3,κ4Modification parameters for increased progression to AIDS forInh,Ipht,Inht, and Ind classesVariableμNatural death rate of humans0.02μhAIDS-induced death rate in humans0.1μTTB-induced death rate in humans0.2[8]


## 3. Qualitative Analysis of the Submodels

In this section, we break the main model into two submodels, namely, the HIV-only model and the TB-only model. This is done by setting all parameters and variables related to TB infection to zero in the full model (2) to get the HIV-only model and vice versa.The submodels are thus given by(4)dSpdt=Q−λhh+μSpdIphdt=λhhf1Sp+σInh−k1IphdInhdt=1−f1Spλhh−k2InhdAdt=δIph+κ1Inh−k8A⏞HIV−onlyModelHOMdSpdt=Q−λTT+μSp;dEpdt=f2λTTSp+ρRp−k3Ep;dIntdt=1−f2Sp+ρRpλTT+θEp−k6Int;dRpdt=rInt−ρλTT+μRp,⏞TB−onlyModelTOMwhere λh, λT modify into λhh=βhchIph+Inh/Nh and λTT=βTcTInt/NT, respectively, and the total population N modifies into Nh=Sp+Iph+Inh+A and NT=Sp+Ep+Int+Rp for the HIV-only and TB-only models, respectively.It is easy to show that the feasible regions of the HIV-only model and the TB-only are given byDH and DT, respectively, given as follows:(5)DH=Sp,Iph,Inh,A∈R+4Nh<Qμ,DT=Sp,Ep,Int,Rp∈R+4NT≤Qμ.The following theorem can then be established regarding the positive invariance of the feasible regions.Theorem 1.
(positive invariance of feasible regions).(1)
If the initial conditions of the HIV-only model are inDH, then all its future solutions will remain in DH.(2)
If the initial conditions of the TB-only model are inDT, then all its future solutions will remain in DT.Proof.
Adding all equations of the HIV-only model in equation (2) yields.
dNh/dt=Q−Sp+Iph+Inh+Aμ−Aμh≤Q−μNh.
SincedNh/dt is bounded by Q−μNh, a standard comparison theorem [26] shows that(6)Nt≤N0e−μt+Qμ1−e−μt.
Clearly, wheneverN0≤Q/μ then Nt≤Q/μ, showing that every solution of the HIV-only model in equation (4) starting in DH will remain in DH. This proves that DH is positive and attracting, completing the proof of the first part of the theorem.
Similar arguments can be applied on the TB-only model to prove the second part of the theorem.Theorem1 implies that it is sufficient to consider the dynamics of the submodels in equation (4) in their respective feasible regions, inside which the submodels are epidemiologically and mathematically well-posed [27].


### 3.1. Equilibrium Points of the Submodels

#### 3.1.1. Disease-Free Equilibria and Local Stability

The submodels in equation (4) can be shown to each exhibit two equilibria, namely, the disease-free and endemic equilibrium points.The infection-free equilibrium points of the HIV-only and TB-only submodels are given, respectively, by(7)EH0=Q/μ,0,0,0,ET0=Q/μ,0,0,0.Using the next-generation matrix approach [28], the basic reproduction numbers of the submodels are obtained as given below.(8)RH0=βhchσ+κ1f1+1−f1δ+μδ+μσ+κ1δ+μ⏞BasicReproductionNumberofHOM,RT0=βTcTθ+μ1−f2θ+μr+μ+μT⏞BasicReproductionNumberofTOM.The following result concerns the local stability of the disease-free critical points of the submodels.Lemma 1.
(local stability of disease-free equilibria). The disease-free equilibrium pointEH0 of the HIV-only model and ET0 of the TB-only model are locally asymptotically stable whenever RH0<1 and RT0<1, respectively. The equilibria are unstable otherwise.Proof.
The proof of this lemma is conducted using the second Lyapunov technique which states that a critical point of a given ode model is locally asymptotically stable if all eigenvalues of the Jacobian matrix of the model evaluated at the critical point have negative real part and unstable otherwise. The Jacobian matrices of the submodels at the disease-free critical points are given below:(9)JHEH0=−μ−βaca−βaca00βacaf1−k1βacaf1+σ001−f1βaca1−f1βaca−k200δδκ1−k8⏞HIV−onlymodel,JTET0=−μ0−βTcT00−k3f2βTcT00θ1−f2βTcT−k6000r−μ⏞TB−onlymodel.
Clearly, two of the eigenvalues ofJHEH0 (namely, −μ and −k8) and also two of the eigenvalues of JTET0 (namely, −μ,−μ) are negative. The rest of the eigenvalues are roots of the polynomial equations(10)XH2+k22f1+k1+k2σ+k121−f1+k1k21−RH0k2f1+1−f1σ+k1XH+k1k21−RH0=0,(11)XT2+k3+k6θf2+1−f2k3k61−RT0θf2+1−f2k3XT+k3k61−RT0=0.
Using the Routh–Hurwitz criterion, equations (10) and (11) will have all of their zeros in the left half of the plane if RH0<1 and RT0<1, concluding the proof.


#### 3.1.2. Endemic Equilibria of Submodels

The HIV-only model can be shown to have an endemic equilibrium pointEH∗=Sph∗,Ipah∗,Inah∗,Ah∗ of the HIV-only model (3), where(12)Sp=Qλhh∗+μ;Ipa=Qλhh∗k2f1+1−f1σλhh∗+μk2k1;Ina=1−f1Qλhh∗λhh∗+μk2;A=δ1−f1κ1λhh∗+μk2k8+k2f1+1−f1σλhh∗+μk2k1k8Qλhh∗⏞EndemicEquilibriumPointofHOMSp=QλTT∗+μ;Ep=f2λTT∗ρλT+μk6QλTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;InT=θf2+1−f2k3ρλTT∗+μQλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;Rp=rθf2+1−f2k3QλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗.⏞EndemicEquilibriumPointofTOMSubstituting the expressions in equation (12) into λhh and λTT and simplifying leads to the following equations.(13)K1λhh∗−k1μk2RH0−1k8λhh∗=0,(14)K2λTT∗2+K3λT−μ2k3k6RT0−1λTT∗=0,where(15)K1=k1k2k8−1−f1k1κ1+k2f1+1−f1σδμa>0;K2=ρμr+μTf2+μ+θ;K3=μρk3k61−RT0−θf2+1−f2k3rρ+μT+k3k6.Equation (12) has two solutions, namely, λhh∗=0 (corresponding to the disease-free equilibrium point EH0) and λhh∗=κ1/μk1k2RH0−1k8 (corresponding to the endemic equilibrium point EH∗), which are positive only when RH0>1. The following theorem is thus established.Theorem 2.
(existence of endemic equilibria of the HIV-only model). For the HIV-only model,(i)
Only the disease-free equilibrium exists wheneverRH0≤1.(ii)
The endemic equilibriumEH∗ exists only when RH0>1.It can be observed that wheneverRT0>1, then the last term of equation (13) will be negative and the equation will have a unique positive root. Also, whenever RT0≤1, then equation (13) will have a positive root if K3<0. The following theorem is thus easily established.Theorem 3.
(existence of endemic equilibria of the TB-only model).(i)
The TB-only submodel has a unique endemic equilibrium point whenRT0>1.(ii)
The TB-only submodel may still have an endemic equilibrium even ifRT0≤1.Theorem3 suggests the existence of backward bifurcation in the TB-only submodel. This will be explored later.


### 3.2. Global Stability of Infection-Free Equilibria

In this section, we adopt the first Lyapunov technique to study the global stability of the disease-free equilibria of the HIV-only and TB-only submodels.To do this, we define the Lyapunov functions(16)Lh=k2k8Ipa+k8σ+k1Ina⏞HIV−onlyModel,LT=θEp+k3InT⏞TB−onlyModel.The time derivatives of the above Lyapunov functions are given as follows.(17)dLhdt=k2k8dIpadt+k8σ+k1dInadt=k2k8λhhf1Sp+σInh−k1Iph+k8σ+k11−f1Spλhh−k2Inh,≤k1k2RH0−1Ipa+Ina⏞HIV−onlyModeldLTdt=θdEpdt+k3dInTdt=θf2λTTf1Sp+Rp−k3Ep+k31−f2Sp+RpλTTθEp−k6Int,=k3k6RT0Rp+SpNT−1Int⏞TB−onlyModelTherefore,dLh/dt≤0 if RH0≤1; dLh/dt=0 if and only if Ipa=Ina=0 and the largest compact invariant set in Sp,Ipa,Ina,A∈DHdLh/dt=0 is the singleton set EH0. From LaSalle’s invariance principle [29], we can conclude that all solutions of the HIV-only model with initial conditions in DH will eventually converge to EH0, showing that EH0 is globally asymptotically stable whenever RH0≤1.As for the TB-only model, we observe thatRT0≤1 does not guarantee that dLT/dt≤0. This suggests the presence of backward bifurcation, which we shall explore in the next section. However, we note that when ρ=0, then RT0≤1 will guarantee that dLT/dt≤0. Therefore, the TB-only model has a globally asymptotically stable (GAS) disease-free equilibrium whenever RT0≤1 and there is no re-infection. The following result is thus established.Theorem 4.
(global stability of infection-free equilibria of submodels).(1)
The HIV-only model has a GAS disease-free equilibrium wheneverRH0<1.(2)
The infection-free equilibrium of the TB-only model is GAS wheneverRT0<1 and ρ≤ρ∗ such that ρ∗Rp+Sp≤NT.The essence of the result in Theorem4 is that HIV only can be eradicated by keeping the basic reproduction number below unity, but TB only may be difficult to be eradicated even after keeping the basic reproduction number below unity.


### 3.3. Bifurcation Analysis of Submodels

The center manifold theory described in [25] is used here to study the directions of bifurcation of the submodels. We note that this theory can be used to determine the direction of bifurcation if the Jacobian of the respective models evaluated at the disease-free equilibrium has a simple eigenvalue. The Jacobian of the submodels evaluated at the disease-free equilibria is given by(18)JH=−μ−βhch−βhch00βhchf1−k1βhchf1+σ001−f1βhch1−f1βhch−k200δδκ1−k8⏞HIV−onlyModel,JT=−μ0−βTcT00−k3f2βTcT00θ1−f2βTcT−k6000r−μ⏞TB−onlyModel..It is easy to show that each of the Jacobians has a simple eigenvalue when the associated basic reproduction number equals unity, which then allows us to employ the center manifold theory to study the direction of bifurcation for the submodels.Now, the right and left eigenvector pairs (corresponding to the simple eigenvalues) for the HIV-only and TB-only submodelswH,vH and wT,vT are such that(19)w1H=−k1k2k8δk2f1+1−f1k1κ1+σμ,v1H=0;w2H=k8k2f1+1−f1σδk2f1+1−f1k1κ1+σ,v2H=k2f1+1−f1k1κ1+σδk2k8k22f1+σk1+k2+k121−f1;w3H=k81−f1k1δk2f1+1−f1k1κ1+σ,v3H=σ+k1k2v2H;w4H=1,v4H=0;⏞HIV−onlyModelw1T=−k3k6θf2+1−f2k3r,v1T=0;w2T=f2k6μrθf2+1−f2k3,v2T=rθθf2+1−f2k3θk3+k6f2+1−f2k32μ;w3T=μr,v3T=rθf2+1−f2k3k3θk3+k6f2+1−f2k32μ;w4T=1,v4T=0⏞TB−onlyModelChoosingβh∗=k1k2/chk2f1+1−f1σ+k1 and βT∗=k3k6/θf2+1−f2k3cT as the bifurcation parameters for the HIV-only and TB-only models, the bifurcation co-coefficients of the submodels are obtained as(20)aH=−2chμβh∗Qv2f1+1−f1v3w2+w3+w4w3+w2;bH=v2f1+1−f1v3chw3+w2;⏞HIV−onlyModelaT=2βT∗cTf2v2+1−f2v3μ2rQρ−1−μθf2+1−f2k3+f2k6rθf2+1−f2k3;bT=cTθf2+1−f2k3μθrv2T.⏞TB−onlyModelClearly,aH<0 and bH>0 and hence by Theorem 2 (item (iv)) of [25], the following theorem is established.Theorem 5.
(bifurcation of the HIV-only model). The unique endemic equilibrium pointEH∗ of the HIV-only model is locally asymptotically stable when RH0 is close to 1. Specifically, the HIV-only submodel exhibits forward bifurcation at RH0=1.Obviously,bT>0, and the sign of aT depends on the sign of ρ−1−μθf2+1−f2k3+f2k6/rθf2+1−f2k3.The following theorem is thus obtained.Theorem 6.
(bifurcation of TB-only model). The TB-only model undergoes a backward bifurcation when the following condition holds.(21)ρ>1+μθf2+1−f2k3+f2k6rθf2+1−f2k3.Proof.
We note that wheneverρ>1+μθf2+1−f2k3+f2k6/rθf2+1−f2k3, then aT>0, so that by Theorem 2 of [25], the TB-only model will exhibit backward bifurcation. This concludes the proof.Theorem6 suggests that the control of the spread of TB requires more effort beyond getting the basic reproduction below 1. This phenomenon has been identified in several other studies, especially of diseases in which there is relapse and exogenous re-infection.


### 3.4. Sensitivity Analysis

In this section, the normalized forward sensitivity index is employed to study the effect of marginal changes in model parameters on the spread of the two diseases. This is done to determine the parameters that drive the spread of these infections. Parameters with positive sensitivity indices drive the spread of the contagion and the parameters with negative indices inhibit the spread of the diseases. The sensitivity index of a functionζ that differentiably depends on a variable ϑ is given by(22)SIζϑ=∂ζ∂ϑ×ϑζ.To this end, the sensitivity indices are presented as follows:(23)SIRH0βh=1,SIRH0ch=1,SIRH0κ1=−δκ11−f1σ+δ+μδκ1+μ+σμ+f1κ1+1−f1δ+σ,SIRH0f1=−δ1−κ1f1μ+f1κ1+1−f1δ+σ,SIRH0δ=−δκ1f1κ1+1−f1δ2+μ+σ1−f1κ1+f1μ+2δκ1+σδ+μδκ1+μ+σδf1κ1+1−f1δ+μ+σ,SIRH0σ=−δσ1−κ11−f1δκ1+μ+σμ+f1κ1+1−f1δ+σ,SIRH0μ=−f1κ12+1−f1δ2+f1κ1+1−f12μ+σ+σκ1δ+μ+σ2μδ+μδκ1+μ+σ1+κ1−1f1δ+μ+σ.⏞HIV−onlyModelSIRT0βT=1,SIRT0cT=1,SIRT0f2=−f2μ1−f2μ+θ,SIRT0θ=θf2μθ+μ1−f2μ+θ,SIRT0μT=−μTr+μ+μT,SIRT0r=−rr+μ+μT,SIRT0μ=−μ1−f2μ2+θθ+2μ+f2r+μTθ+μr+μ+μT1−f2μ+θ.⏞TB−onlyModelThe parametersβh, ch, βT, cT, θ have positive sensitivity indices, which implies that an increase (a decrease) in each of these parameters will cause an increase (a decrease) in the corresponding basic reproduction number and subsequently drive (inhibit) the spread of the diseases. Also, the parameters κ1, f1, f2, δ, σ, μ, r, and μT have negative sensitivity indices, which implies that an increase (decrease) in these parameters will cause a decrease (increase) in the respective basic reproduction numbers and subsequently inhibit (drive) the spread of the contagion.


### 3.5. The Impact of HIV on TB

In order to study the impact of HIV on TB progression and vice versa, we relate the basic reproduction numbers of the two submodels. To do this, use the expression ofRT0 to obtain μ2 and substitute it into the expression of RH0 to obtain(24)RH0=βhchδκ1+μ+σf1+1−f1σ+δ+μRT0βTcTθf2+1−f2θ+μ+σ−θμ+δ+μδκ1+σ−θ+μr+μTRT0.WithRH0 now expressed in terms of RT0 and other model parameters, it is then possible to analytically study the impact of the model parameters on RH0 and subsequently the spread of HIV/AIDS.Now, we have the following:(25)∂RH0∂RT0=βTcT1−f2μ+θZ1,∂RH0∂cT=−βT1−f2μ+θZ1RT0,∂RH0∂βT=−cT1−f2μ+θZ1RT0,∂RH0∂θ=r+μ+μTRT0−βTcTZ1RT0,∂RH0∂r=θ+μZ1RT02,∂RH0∂f2=βTcTμZ1RT0,∂RH0∂μT=θ+μZ1RT02HIV−onlymodel∂RT0∂RH0=Z2βh1+κ1−1f1δ+μ+σch∂RT0∂βh=−Z2RH01+κ1−1f1δ+μ+σch∂RT0∂ch=−Z2RH0βh1+κ1−1f1δ+μ+σ;∂RT0∂σ=RH0δ+μRH0−βhchZ2∂RT0∂δ=κ1+1μ+2δκ1+σRH0−f1κ1−f1+1chβhZ2RH0∂RT0∂f1=−δRH0chβhκ1−1Z2∂RT0∂κ1=δRH0δ+μRH0−f1βhchZ2TB−onlyModelwhere(26)Z1=βhchμ+f1κ1+1−f1δ+σβTcTθf2+1−f2θ+μ+σ−θμ+δ+μδκ1+σ−θ+μr+μTRT02>0;Z2=βTcTθ+μ1−f2−δκ1+δ+σμ−δδκ1+σ+θ+r+μTμ+θr+μTRH0+δκ1+μ+σf1+1−f1σ+δ+μchβh2>0.We observe thatRH0 is an increasing function of RT0, r, f2, and μT since its derivatives with respective to these parameters are positive, showing that these parameter drive the spread of HIV/AIDS. Also, RH0 is a decreasing function of βT and cT, indicating that these two parameters inhibit the spread of HIV/AIDS. We observe that since βT and cT drive TB which in turn drives HIV/AIDS, it is expected that these two parameters would transitively drive HIV. Therefore, this result is counter-intuitive.


## 3.1. Equilibrium Points of the Submodels

### 3.1.1. Disease-Free Equilibria and Local Stability

The submodels in equation (4) can be shown to each exhibit two equilibria, namely, the disease-free and endemic equilibrium points.The infection-free equilibrium points of the HIV-only and TB-only submodels are given, respectively, by(7)EH0=Q/μ,0,0,0,ET0=Q/μ,0,0,0.Using the next-generation matrix approach [28], the basic reproduction numbers of the submodels are obtained as given below.(8)RH0=βhchσ+κ1f1+1−f1δ+μδ+μσ+κ1δ+μ⏞BasicReproductionNumberofHOM,RT0=βTcTθ+μ1−f2θ+μr+μ+μT⏞BasicReproductionNumberofTOM.The following result concerns the local stability of the disease-free critical points of the submodels.Lemma 1.
(local stability of disease-free equilibria). The disease-free equilibrium pointEH0 of the HIV-only model and ET0 of the TB-only model are locally asymptotically stable whenever RH0<1 and RT0<1, respectively. The equilibria are unstable otherwise.Proof.
The proof of this lemma is conducted using the second Lyapunov technique which states that a critical point of a given ode model is locally asymptotically stable if all eigenvalues of the Jacobian matrix of the model evaluated at the critical point have negative real part and unstable otherwise. The Jacobian matrices of the submodels at the disease-free critical points are given below:(9)JHEH0=−μ−βaca−βaca00βacaf1−k1βacaf1+σ001−f1βaca1−f1βaca−k200δδκ1−k8⏞HIV−onlymodel,JTET0=−μ0−βTcT00−k3f2βTcT00θ1−f2βTcT−k6000r−μ⏞TB−onlymodel.
Clearly, two of the eigenvalues ofJHEH0 (namely, −μ and −k8) and also two of the eigenvalues of JTET0 (namely, −μ,−μ) are negative. The rest of the eigenvalues are roots of the polynomial equations(10)XH2+k22f1+k1+k2σ+k121−f1+k1k21−RH0k2f1+1−f1σ+k1XH+k1k21−RH0=0,(11)XT2+k3+k6θf2+1−f2k3k61−RT0θf2+1−f2k3XT+k3k61−RT0=0.
Using the Routh–Hurwitz criterion, equations (10) and (11) will have all of their zeros in the left half of the plane if RH0<1 and RT0<1, concluding the proof.


### 3.1.2. Endemic Equilibria of Submodels

The HIV-only model can be shown to have an endemic equilibrium pointEH∗=Sph∗,Ipah∗,Inah∗,Ah∗ of the HIV-only model (3), where(12)Sp=Qλhh∗+μ;Ipa=Qλhh∗k2f1+1−f1σλhh∗+μk2k1;Ina=1−f1Qλhh∗λhh∗+μk2;A=δ1−f1κ1λhh∗+μk2k8+k2f1+1−f1σλhh∗+μk2k1k8Qλhh∗⏞EndemicEquilibriumPointofHOMSp=QλTT∗+μ;Ep=f2λTT∗ρλT+μk6QλTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;InT=θf2+1−f2k3ρλTT∗+μQλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;Rp=rθf2+1−f2k3QλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗.⏞EndemicEquilibriumPointofTOMSubstituting the expressions in equation (12) into λhh and λTT and simplifying leads to the following equations.(13)K1λhh∗−k1μk2RH0−1k8λhh∗=0,(14)K2λTT∗2+K3λT−μ2k3k6RT0−1λTT∗=0,where(15)K1=k1k2k8−1−f1k1κ1+k2f1+1−f1σδμa>0;K2=ρμr+μTf2+μ+θ;K3=μρk3k61−RT0−θf2+1−f2k3rρ+μT+k3k6.Equation (12) has two solutions, namely, λhh∗=0 (corresponding to the disease-free equilibrium point EH0) and λhh∗=κ1/μk1k2RH0−1k8 (corresponding to the endemic equilibrium point EH∗), which are positive only when RH0>1. The following theorem is thus established.Theorem 2.
(existence of endemic equilibria of the HIV-only model). For the HIV-only model,(i)
Only the disease-free equilibrium exists wheneverRH0≤1.(ii)
The endemic equilibriumEH∗ exists only when RH0>1.It can be observed that wheneverRT0>1, then the last term of equation (13) will be negative and the equation will have a unique positive root. Also, whenever RT0≤1, then equation (13) will have a positive root if K3<0. The following theorem is thus easily established.Theorem 3.
(existence of endemic equilibria of the TB-only model).(i)
The TB-only submodel has a unique endemic equilibrium point whenRT0>1.(ii)
The TB-only submodel may still have an endemic equilibrium even ifRT0≤1.Theorem3 suggests the existence of backward bifurcation in the TB-only submodel. This will be explored later.


## 3.1.1. Disease-Free Equilibria and Local Stability

The submodels in equation (4) can be shown to each exhibit two equilibria, namely, the disease-free and endemic equilibrium points.The infection-free equilibrium points of the HIV-only and TB-only submodels are given, respectively, by(7)EH0=Q/μ,0,0,0,ET0=Q/μ,0,0,0.Using the next-generation matrix approach [28], the basic reproduction numbers of the submodels are obtained as given below.(8)RH0=βhchσ+κ1f1+1−f1δ+μδ+μσ+κ1δ+μ⏞BasicReproductionNumberofHOM,RT0=βTcTθ+μ1−f2θ+μr+μ+μT⏞BasicReproductionNumberofTOM.The following result concerns the local stability of the disease-free critical points of the submodels.Lemma 1.
(local stability of disease-free equilibria). The disease-free equilibrium pointEH0 of the HIV-only model and ET0 of the TB-only model are locally asymptotically stable whenever RH0<1 and RT0<1, respectively. The equilibria are unstable otherwise.Proof.
The proof of this lemma is conducted using the second Lyapunov technique which states that a critical point of a given ode model is locally asymptotically stable if all eigenvalues of the Jacobian matrix of the model evaluated at the critical point have negative real part and unstable otherwise. The Jacobian matrices of the submodels at the disease-free critical points are given below:(9)JHEH0=−μ−βaca−βaca00βacaf1−k1βacaf1+σ001−f1βaca1−f1βaca−k200δδκ1−k8⏞HIV−onlymodel,JTET0=−μ0−βTcT00−k3f2βTcT00θ1−f2βTcT−k6000r−μ⏞TB−onlymodel.
Clearly, two of the eigenvalues ofJHEH0 (namely, −μ and −k8) and also two of the eigenvalues of JTET0 (namely, −μ,−μ) are negative. The rest of the eigenvalues are roots of the polynomial equations(10)XH2+k22f1+k1+k2σ+k121−f1+k1k21−RH0k2f1+1−f1σ+k1XH+k1k21−RH0=0,(11)XT2+k3+k6θf2+1−f2k3k61−RT0θf2+1−f2k3XT+k3k61−RT0=0.
Using the Routh–Hurwitz criterion, equations (10) and (11) will have all of their zeros in the left half of the plane if RH0<1 and RT0<1, concluding the proof.


## 3.1.2. Endemic Equilibria of Submodels

The HIV-only model can be shown to have an endemic equilibrium pointEH∗=Sph∗,Ipah∗,Inah∗,Ah∗ of the HIV-only model (3), where(12)Sp=Qλhh∗+μ;Ipa=Qλhh∗k2f1+1−f1σλhh∗+μk2k1;Ina=1−f1Qλhh∗λhh∗+μk2;A=δ1−f1κ1λhh∗+μk2k8+k2f1+1−f1σλhh∗+μk2k1k8Qλhh∗⏞EndemicEquilibriumPointofHOMSp=QλTT∗+μ;Ep=f2λTT∗ρλT+μk6QλTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;InT=θf2+1−f2k3ρλTT∗+μQλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗;Rp=rθf2+1−f2k3QλTT∗λTT∗+μμ+ρλTT∗k3k6−rρθf2+1−f2k3λTT∗.⏞EndemicEquilibriumPointofTOMSubstituting the expressions in equation (12) into λhh and λTT and simplifying leads to the following equations.(13)K1λhh∗−k1μk2RH0−1k8λhh∗=0,(14)K2λTT∗2+K3λT−μ2k3k6RT0−1λTT∗=0,where(15)K1=k1k2k8−1−f1k1κ1+k2f1+1−f1σδμa>0;K2=ρμr+μTf2+μ+θ;K3=μρk3k61−RT0−θf2+1−f2k3rρ+μT+k3k6.Equation (12) has two solutions, namely, λhh∗=0 (corresponding to the disease-free equilibrium point EH0) and λhh∗=κ1/μk1k2RH0−1k8 (corresponding to the endemic equilibrium point EH∗), which are positive only when RH0>1. The following theorem is thus established.Theorem 2.
(existence of endemic equilibria of the HIV-only model). For the HIV-only model,(i)
Only the disease-free equilibrium exists wheneverRH0≤1.(ii)
The endemic equilibriumEH∗ exists only when RH0>1.It can be observed that wheneverRT0>1, then the last term of equation (13) will be negative and the equation will have a unique positive root. Also, whenever RT0≤1, then equation (13) will have a positive root if K3<0. The following theorem is thus easily established.Theorem 3.
(existence of endemic equilibria of the TB-only model).(i)
The TB-only submodel has a unique endemic equilibrium point whenRT0>1.(ii)
The TB-only submodel may still have an endemic equilibrium even ifRT0≤1.Theorem3 suggests the existence of backward bifurcation in the TB-only submodel. This will be explored later.


## 3.2. Global Stability of Infection-Free Equilibria

In this section, we adopt the first Lyapunov technique to study the global stability of the disease-free equilibria of the HIV-only and TB-only submodels.To do this, we define the Lyapunov functions(16)Lh=k2k8Ipa+k8σ+k1Ina⏞HIV−onlyModel,LT=θEp+k3InT⏞TB−onlyModel.The time derivatives of the above Lyapunov functions are given as follows.(17)dLhdt=k2k8dIpadt+k8σ+k1dInadt=k2k8λhhf1Sp+σInh−k1Iph+k8σ+k11−f1Spλhh−k2Inh,≤k1k2RH0−1Ipa+Ina⏞HIV−onlyModeldLTdt=θdEpdt+k3dInTdt=θf2λTTf1Sp+Rp−k3Ep+k31−f2Sp+RpλTTθEp−k6Int,=k3k6RT0Rp+SpNT−1Int⏞TB−onlyModelTherefore,dLh/dt≤0 if RH0≤1; dLh/dt=0 if and only if Ipa=Ina=0 and the largest compact invariant set in Sp,Ipa,Ina,A∈DHdLh/dt=0 is the singleton set EH0. From LaSalle’s invariance principle [29], we can conclude that all solutions of the HIV-only model with initial conditions in DH will eventually converge to EH0, showing that EH0 is globally asymptotically stable whenever RH0≤1.As for the TB-only model, we observe thatRT0≤1 does not guarantee that dLT/dt≤0. This suggests the presence of backward bifurcation, which we shall explore in the next section. However, we note that when ρ=0, then RT0≤1 will guarantee that dLT/dt≤0. Therefore, the TB-only model has a globally asymptotically stable (GAS) disease-free equilibrium whenever RT0≤1 and there is no re-infection. The following result is thus established.Theorem 4.
(global stability of infection-free equilibria of submodels).(1)
The HIV-only model has a GAS disease-free equilibrium wheneverRH0<1.(2)
The infection-free equilibrium of the TB-only model is GAS wheneverRT0<1 and ρ≤ρ∗ such that ρ∗Rp+Sp≤NT.The essence of the result in Theorem4 is that HIV only can be eradicated by keeping the basic reproduction number below unity, but TB only may be difficult to be eradicated even after keeping the basic reproduction number below unity.


## 3.3. Bifurcation Analysis of Submodels

The center manifold theory described in [25] is used here to study the directions of bifurcation of the submodels. We note that this theory can be used to determine the direction of bifurcation if the Jacobian of the respective models evaluated at the disease-free equilibrium has a simple eigenvalue. The Jacobian of the submodels evaluated at the disease-free equilibria is given by(18)JH=−μ−βhch−βhch00βhchf1−k1βhchf1+σ001−f1βhch1−f1βhch−k200δδκ1−k8⏞HIV−onlyModel,JT=−μ0−βTcT00−k3f2βTcT00θ1−f2βTcT−k6000r−μ⏞TB−onlyModel..It is easy to show that each of the Jacobians has a simple eigenvalue when the associated basic reproduction number equals unity, which then allows us to employ the center manifold theory to study the direction of bifurcation for the submodels.Now, the right and left eigenvector pairs (corresponding to the simple eigenvalues) for the HIV-only and TB-only submodelswH,vH and wT,vT are such that(19)w1H=−k1k2k8δk2f1+1−f1k1κ1+σμ,v1H=0;w2H=k8k2f1+1−f1σδk2f1+1−f1k1κ1+σ,v2H=k2f1+1−f1k1κ1+σδk2k8k22f1+σk1+k2+k121−f1;w3H=k81−f1k1δk2f1+1−f1k1κ1+σ,v3H=σ+k1k2v2H;w4H=1,v4H=0;⏞HIV−onlyModelw1T=−k3k6θf2+1−f2k3r,v1T=0;w2T=f2k6μrθf2+1−f2k3,v2T=rθθf2+1−f2k3θk3+k6f2+1−f2k32μ;w3T=μr,v3T=rθf2+1−f2k3k3θk3+k6f2+1−f2k32μ;w4T=1,v4T=0⏞TB−onlyModelChoosingβh∗=k1k2/chk2f1+1−f1σ+k1 and βT∗=k3k6/θf2+1−f2k3cT as the bifurcation parameters for the HIV-only and TB-only models, the bifurcation co-coefficients of the submodels are obtained as(20)aH=−2chμβh∗Qv2f1+1−f1v3w2+w3+w4w3+w2;bH=v2f1+1−f1v3chw3+w2;⏞HIV−onlyModelaT=2βT∗cTf2v2+1−f2v3μ2rQρ−1−μθf2+1−f2k3+f2k6rθf2+1−f2k3;bT=cTθf2+1−f2k3μθrv2T.⏞TB−onlyModelClearly,aH<0 and bH>0 and hence by Theorem 2 (item (iv)) of [25], the following theorem is established.Theorem 5.
(bifurcation of the HIV-only model). The unique endemic equilibrium pointEH∗ of the HIV-only model is locally asymptotically stable when RH0 is close to 1. Specifically, the HIV-only submodel exhibits forward bifurcation at RH0=1.Obviously,bT>0, and the sign of aT depends on the sign of ρ−1−μθf2+1−f2k3+f2k6/rθf2+1−f2k3.The following theorem is thus obtained.Theorem 6.
(bifurcation of TB-only model). The TB-only model undergoes a backward bifurcation when the following condition holds.(21)ρ>1+μθf2+1−f2k3+f2k6rθf2+1−f2k3.Proof.
We note that wheneverρ>1+μθf2+1−f2k3+f2k6/rθf2+1−f2k3, then aT>0, so that by Theorem 2 of [25], the TB-only model will exhibit backward bifurcation. This concludes the proof.Theorem6 suggests that the control of the spread of TB requires more effort beyond getting the basic reproduction below 1. This phenomenon has been identified in several other studies, especially of diseases in which there is relapse and exogenous re-infection.


## 3.4. Sensitivity Analysis

In this section, the normalized forward sensitivity index is employed to study the effect of marginal changes in model parameters on the spread of the two diseases. This is done to determine the parameters that drive the spread of these infections. Parameters with positive sensitivity indices drive the spread of the contagion and the parameters with negative indices inhibit the spread of the diseases. The sensitivity index of a functionζ that differentiably depends on a variable ϑ is given by(22)SIζϑ=∂ζ∂ϑ×ϑζ.To this end, the sensitivity indices are presented as follows:(23)SIRH0βh=1,SIRH0ch=1,SIRH0κ1=−δκ11−f1σ+δ+μδκ1+μ+σμ+f1κ1+1−f1δ+σ,SIRH0f1=−δ1−κ1f1μ+f1κ1+1−f1δ+σ,SIRH0δ=−δκ1f1κ1+1−f1δ2+μ+σ1−f1κ1+f1μ+2δκ1+σδ+μδκ1+μ+σδf1κ1+1−f1δ+μ+σ,SIRH0σ=−δσ1−κ11−f1δκ1+μ+σμ+f1κ1+1−f1δ+σ,SIRH0μ=−f1κ12+1−f1δ2+f1κ1+1−f12μ+σ+σκ1δ+μ+σ2μδ+μδκ1+μ+σ1+κ1−1f1δ+μ+σ.⏞HIV−onlyModelSIRT0βT=1,SIRT0cT=1,SIRT0f2=−f2μ1−f2μ+θ,SIRT0θ=θf2μθ+μ1−f2μ+θ,SIRT0μT=−μTr+μ+μT,SIRT0r=−rr+μ+μT,SIRT0μ=−μ1−f2μ2+θθ+2μ+f2r+μTθ+μr+μ+μT1−f2μ+θ.⏞TB−onlyModelThe parametersβh, ch, βT, cT, θ have positive sensitivity indices, which implies that an increase (a decrease) in each of these parameters will cause an increase (a decrease) in the corresponding basic reproduction number and subsequently drive (inhibit) the spread of the diseases. Also, the parameters κ1, f1, f2, δ, σ, μ, r, and μT have negative sensitivity indices, which implies that an increase (decrease) in these parameters will cause a decrease (increase) in the respective basic reproduction numbers and subsequently inhibit (drive) the spread of the contagion.


## 3.5. The Impact of HIV on TB

In order to study the impact of HIV on TB progression and vice versa, we relate the basic reproduction numbers of the two submodels. To do this, use the expression ofRT0 to obtain μ2 and substitute it into the expression of RH0 to obtain(24)RH0=βhchδκ1+μ+σf1+1−f1σ+δ+μRT0βTcTθf2+1−f2θ+μ+σ−θμ+δ+μδκ1+σ−θ+μr+μTRT0.WithRH0 now expressed in terms of RT0 and other model parameters, it is then possible to analytically study the impact of the model parameters on RH0 and subsequently the spread of HIV/AIDS.Now, we have the following:(25)∂RH0∂RT0=βTcT1−f2μ+θZ1,∂RH0∂cT=−βT1−f2μ+θZ1RT0,∂RH0∂βT=−cT1−f2μ+θZ1RT0,∂RH0∂θ=r+μ+μTRT0−βTcTZ1RT0,∂RH0∂r=θ+μZ1RT02,∂RH0∂f2=βTcTμZ1RT0,∂RH0∂μT=θ+μZ1RT02HIV−onlymodel∂RT0∂RH0=Z2βh1+κ1−1f1δ+μ+σch∂RT0∂βh=−Z2RH01+κ1−1f1δ+μ+σch∂RT0∂ch=−Z2RH0βh1+κ1−1f1δ+μ+σ;∂RT0∂σ=RH0δ+μRH0−βhchZ2∂RT0∂δ=κ1+1μ+2δκ1+σRH0−f1κ1−f1+1chβhZ2RH0∂RT0∂f1=−δRH0chβhκ1−1Z2∂RT0∂κ1=δRH0δ+μRH0−f1βhchZ2TB−onlyModelwhere(26)Z1=βhchμ+f1κ1+1−f1δ+σβTcTθf2+1−f2θ+μ+σ−θμ+δ+μδκ1+σ−θ+μr+μTRT02>0;Z2=βTcTθ+μ1−f2−δκ1+δ+σμ−δδκ1+σ+θ+r+μTμ+θr+μTRH0+δκ1+μ+σf1+1−f1σ+δ+μchβh2>0.We observe thatRH0 is an increasing function of RT0, r, f2, and μT since its derivatives with respective to these parameters are positive, showing that these parameter drive the spread of HIV/AIDS. Also, RH0 is a decreasing function of βT and cT, indicating that these two parameters inhibit the spread of HIV/AIDS. We observe that since βT and cT drive TB which in turn drives HIV/AIDS, it is expected that these two parameters would transitively drive HIV. Therefore, this result is counter-intuitive.


## 4. Qualitative Analysis of the Full Model

It is easy to show that the biologically feasible region of the full model given by(27)D=Sp,Iph,Inh,Ep,Ipht,Inht,Int,Rp,Ind,A∈R+10N≤Qμ,is a positively invariant and attracting set. Therefore, all analyses are carried out inside this set.


### 4.1. Equilibrium Points of the Full Model

The disease-free equilibrium point of the full model is given byE0=Q/μ,0,0,0,0,0,0,0,0,0.Using the next-generation matrix approach [28], the basic reproduction number of the full HIV-TB model (2) is obtained as(28)R0=maxRH0,RT0.The following is obtained.Lemma 2.
The disease-free equilibrium pointE0 of the full HIV-TB model (2) is locally asymptotically stable whenever R0<1 and unstable otherwise.Proof.
The proof of this result is similar to that of Theorem2.Next, we present some result on the global stability of the disease-free equilibrium point of the full model. To do this, refer to [30] and rewrite the model as follows:(29)dUdt=FU,IdIdt=GU,I,where U=Sp,Rp and I=Iph,Inh,Ep,Ipht,Inht,Int,Ind,A denote the uninfected and infected compartments, respectively. Based on [30], the disease-free equilibrium point E0 is globally stable if the following conditions hold.(H1)dU/dt=FU,0 is globally stable.(H2)GU,I=AI−G^U,I,G^U,I≥0∀U,I∈Ω, where A=∂G/∂IU0,0 is an M− matrix.In the full model, we note that(30)FU,I=Q−λh+λT+μSprInt−ρλT+λh+μRp,GU,I=f1λhSp+Rp+σInh−τ1λT+k1Iph1−f1λhSp+Rp−τ1λT+k2Inhf2λTSp+ρRp−ξ1λh+k3Epτ1f2λTIph+σInht−k4Iphtτ11−f2λTInh+ξ1λhEp+τ2rInd−k5Inht1−f2λTSp+ρRp+θEp−ξ2λh+k6Intξ2λhInt+τ1λTf2Inh+1−f2Iph+θη1Inht+η2Ipht−k7IndδIph+κ1Inh+κ2Ipht+κ3Inht+κ4Ind−k8A.We observe thatFU,0=Q−μSp−μRp has E0 as a globally stable critical point, satisfying condition (H1). To test condition (H2), we find G^U,I=AI−GU,I.Now,A=A1,A2 where(31)A1=f1βaca−k1f1βaca+σ0f1βaca1−f1βaca1−f1βaca−k201−f1βaca00−k30000−k4000000θ0000η2θδδκ10δκ2,A2=f1βaca0f1βaca01−f1βaca01−f1βaca00f2βTcTf2βTcT0σ000−k50τ2r001−f2βTcT−k61−f2βTcT0η1θ0−k70δκ30δκ4−k8,G^U,I=−f1N−Sp−Rpλh−Ipaτ1λT−1−f1N−Sp−Rpλh−τ1λTIna−f2N−Sp−ρRpλTT−ξ1λhEpτ1f2λTTIpaInaτ11−f2λTT+ξ1λhEp−1−f2N−Sp−ρRpλTT−ξ2λhInTτ11−f2Ipa+f2InaλTT+ξ2λhInT0.It can be clearly seen thatG^U,I≱0, and hence condition (H2) does not hold. Therefore, the disease-free equilibrium E0 of the full model is not globally asymptotically stable.The endemic equilibrium of the full HIV-TB modelE∗=Sp∗,Iph∗,Inh∗,Ep∗,Ipht∗,Inht∗,Int∗,Rp∗,Ind∗,A∗ is such that we have the following:(32)Sp∗=Qλh∗+λT∗+μIph∗=1τ1λT∗+k1f1+σ1−f1τ1λT∗+k2Sp∗+Rp∗λh∗Inh∗=1−f1τ1λT∗+k2Sp∗+Rp∗λh∗Ep∗=f2ξ1λh∗+k3Sp∗+ρRp∗λT∗Ipht∗=τ1f2k4τ1λT∗+k1f1+σ1−f1τ1λT∗+k2+στ11−f21−f1k4k5τ1λT∗+k2Sp∗+Rp∗λh∗λT∗+σξ1f2k4k5ξ1λh∗+k3Sp∗+ρRp∗λh∗λT∗+στ2rk4k5Ind∗Inht∗=λh∗λT∗τ11−f21−f1k5τ1λT∗+k2Sp∗+Rp∗+ξ1f2k5ξ1λh∗+k3Sp∗+ρRp∗+τ2rk5Ind∗Int∗=1ξ2λh∗+k61−f2+θf2ξ1λh∗+k3Sp∗+ρRp∗λT∗Rp∗=XSp∗λT∗1−ρXλT∗Ind∗=λh∗λTG1ξ2ξ2λh∗+k61−f2+θf2ξ1λh∗+k3+θξ1f2k5ξ1λh∗+k3η1+η2σk4Sp∗+ρRp∗+1τ1λT∗+k2G2+G3τ1λT∗+k1f1τ1λT∗+k2+σ1−f1Sp∗+Rp∗A∗=δk8Iph∗+κ1Inh∗+κ2Ipht∗+κ3Inht∗+κ4Ind∗,,where(33)G1=k7−θτ2rk5η1+η2σk4;G2=τ11−f1θη1+η2σk41−f2k5+f2;G3=τ1η2θf2k4+1−f2;X=r1−f2ξ1λh+k3+θf2ρλT+λh+μξ1λh+k3ξ2λh+k6.Substituting (32) into the infection force functions λh and λT yields the following set of nonlinear equations.(34)λh∗λT∗=F1λh∗,λT∗F2λh∗,λT∗.The equilibrium point of the full HIV-TB model is characterized by the fixed points of equation (34).


### 4.2. Bifurcation of the Full Model

ConsideringR0 as the basic reproduction number of the model, we observe that when RH0=1, the Jacobian of the model possesses a simple eigenvalue, which allows the application of the center manifold theory [25] to study the existence and direction of bifurcation for the full model. The eigenvectors associated with the simple eigenvalues are then given by w=w1,w2,w3,w4,w5,w6,w7,w8,w9,w10T and v=v1,v2,v3,v4,v5,v6,v7,v8,v9,v10 where(35)w1=caσ+k1βaw3f1βaca−k1μ+−βTβacTcaf1+βTcTk1w7f1βaca−k1μw2=−w3f1βaca+σf1βaca−k1;w4=f2βTcTw7k3;w5=w6=w9=0;w8=rw7μw10=−δ−caf1κ1−1βa+k1κ1+σw3f1βaca−k1k8;v1=v8=v10=0;v2=1−f1βacav3−f1βaca+k1;v4=θv7k3;v5=βacak11−f1v3−f1βaca+k1k4+η2θv9k4;v6=βacak11−f1σ+k4v3−f1βaca+k1k4k5+θση2+η1k4v9k4k5;v9=−θ−k3f2+k3cTβTk5k4v7τ2rθση2+η1k4−k7k4k5k3−cak1βav3−1+f1rσ+k4τ2+k4k5τ2rθση2+η1k4−k7k4k5f1βaca−k1;withw7=v7=0,w3≠0,v3≠0whenR0=RH0=1;andw7≠0,v7≠0,w3=v3=0whenR0=RT0=1..The eigenvectors are chosen such thatw•v=1 or equivalently(36)1−f1βacaf1βaca+σf1βaca−k12+1v3w3+θf2βTcT+k32k32w7v7=1.Now, whenβa is taken as bifurcation parameter with R0=RH0=1, we obtain the following bifurcation coefficients:(37)a=aβa∗=−2βaσ+k1f1δca1−κ1βa+δκ1+k8k1+σk8+δca1−f1μk1w32v3k1−f1βaca3k8Q,b=bβa∗=1−f1cak1σ+k1v3w3f1βaca−k12.Further, whenβT is taken as bifurcation parameter with R0=RT0=1, we obtain the following bifurcation coefficients:(38)a=aβT∗=−2βTcTμ+1−ρrk3+f2μcTβT1−f2k3+f2θw72v7k32Qb=bβT∗=1−f2k3+f2θw7v7cTk3.Now, clearly,bβa∗>0,bβT∗>0, and hence the presence and direction of bifurcation in the model depend on the signs of aβa∗ and aβT∗. The following result of bifurcation is thus established.Theorem 7.
(bifurcation of the full model).(i)
IfR0=RH0, then the HIV-TB model exhibits backward bifurcation at R0=1 whenever k1<βacaf1; otherwise, the direction of the bifurcation is forward.(ii)
IfR0=RT0, then the HIV-TB model exhibits backward bifurcation at R0=1 whenever ρ>1+μ/r+f2μcTβT/rk3; otherwise, the direction of the bifurcation is forward.


## 4.1. Equilibrium Points of the Full Model

The disease-free equilibrium point of the full model is given byE0=Q/μ,0,0,0,0,0,0,0,0,0.Using the next-generation matrix approach [28], the basic reproduction number of the full HIV-TB model (2) is obtained as(28)R0=maxRH0,RT0.The following is obtained.Lemma 2.
The disease-free equilibrium pointE0 of the full HIV-TB model (2) is locally asymptotically stable whenever R0<1 and unstable otherwise.Proof.
The proof of this result is similar to that of Theorem2.Next, we present some result on the global stability of the disease-free equilibrium point of the full model. To do this, refer to [30] and rewrite the model as follows:(29)dUdt=FU,IdIdt=GU,I,where U=Sp,Rp and I=Iph,Inh,Ep,Ipht,Inht,Int,Ind,A denote the uninfected and infected compartments, respectively. Based on [30], the disease-free equilibrium point E0 is globally stable if the following conditions hold.(H1)dU/dt=FU,0 is globally stable.(H2)GU,I=AI−G^U,I,G^U,I≥0∀U,I∈Ω, where A=∂G/∂IU0,0 is an M− matrix.In the full model, we note that(30)FU,I=Q−λh+λT+μSprInt−ρλT+λh+μRp,GU,I=f1λhSp+Rp+σInh−τ1λT+k1Iph1−f1λhSp+Rp−τ1λT+k2Inhf2λTSp+ρRp−ξ1λh+k3Epτ1f2λTIph+σInht−k4Iphtτ11−f2λTInh+ξ1λhEp+τ2rInd−k5Inht1−f2λTSp+ρRp+θEp−ξ2λh+k6Intξ2λhInt+τ1λTf2Inh+1−f2Iph+θη1Inht+η2Ipht−k7IndδIph+κ1Inh+κ2Ipht+κ3Inht+κ4Ind−k8A.We observe thatFU,0=Q−μSp−μRp has E0 as a globally stable critical point, satisfying condition (H1). To test condition (H2), we find G^U,I=AI−GU,I.Now,A=A1,A2 where(31)A1=f1βaca−k1f1βaca+σ0f1βaca1−f1βaca1−f1βaca−k201−f1βaca00−k30000−k4000000θ0000η2θδδκ10δκ2,A2=f1βaca0f1βaca01−f1βaca01−f1βaca00f2βTcTf2βTcT0σ000−k50τ2r001−f2βTcT−k61−f2βTcT0η1θ0−k70δκ30δκ4−k8,G^U,I=−f1N−Sp−Rpλh−Ipaτ1λT−1−f1N−Sp−Rpλh−τ1λTIna−f2N−Sp−ρRpλTT−ξ1λhEpτ1f2λTTIpaInaτ11−f2λTT+ξ1λhEp−1−f2N−Sp−ρRpλTT−ξ2λhInTτ11−f2Ipa+f2InaλTT+ξ2λhInT0.It can be clearly seen thatG^U,I≱0, and hence condition (H2) does not hold. Therefore, the disease-free equilibrium E0 of the full model is not globally asymptotically stable.The endemic equilibrium of the full HIV-TB modelE∗=Sp∗,Iph∗,Inh∗,Ep∗,Ipht∗,Inht∗,Int∗,Rp∗,Ind∗,A∗ is such that we have the following:(32)Sp∗=Qλh∗+λT∗+μIph∗=1τ1λT∗+k1f1+σ1−f1τ1λT∗+k2Sp∗+Rp∗λh∗Inh∗=1−f1τ1λT∗+k2Sp∗+Rp∗λh∗Ep∗=f2ξ1λh∗+k3Sp∗+ρRp∗λT∗Ipht∗=τ1f2k4τ1λT∗+k1f1+σ1−f1τ1λT∗+k2+στ11−f21−f1k4k5τ1λT∗+k2Sp∗+Rp∗λh∗λT∗+σξ1f2k4k5ξ1λh∗+k3Sp∗+ρRp∗λh∗λT∗+στ2rk4k5Ind∗Inht∗=λh∗λT∗τ11−f21−f1k5τ1λT∗+k2Sp∗+Rp∗+ξ1f2k5ξ1λh∗+k3Sp∗+ρRp∗+τ2rk5Ind∗Int∗=1ξ2λh∗+k61−f2+θf2ξ1λh∗+k3Sp∗+ρRp∗λT∗Rp∗=XSp∗λT∗1−ρXλT∗Ind∗=λh∗λTG1ξ2ξ2λh∗+k61−f2+θf2ξ1λh∗+k3+θξ1f2k5ξ1λh∗+k3η1+η2σk4Sp∗+ρRp∗+1τ1λT∗+k2G2+G3τ1λT∗+k1f1τ1λT∗+k2+σ1−f1Sp∗+Rp∗A∗=δk8Iph∗+κ1Inh∗+κ2Ipht∗+κ3Inht∗+κ4Ind∗,,where(33)G1=k7−θτ2rk5η1+η2σk4;G2=τ11−f1θη1+η2σk41−f2k5+f2;G3=τ1η2θf2k4+1−f2;X=r1−f2ξ1λh+k3+θf2ρλT+λh+μξ1λh+k3ξ2λh+k6.Substituting (32) into the infection force functions λh and λT yields the following set of nonlinear equations.(34)λh∗λT∗=F1λh∗,λT∗F2λh∗,λT∗.The equilibrium point of the full HIV-TB model is characterized by the fixed points of equation (34).


## 4.2. Bifurcation of the Full Model

ConsideringR0 as the basic reproduction number of the model, we observe that when RH0=1, the Jacobian of the model possesses a simple eigenvalue, which allows the application of the center manifold theory [25] to study the existence and direction of bifurcation for the full model. The eigenvectors associated with the simple eigenvalues are then given by w=w1,w2,w3,w4,w5,w6,w7,w8,w9,w10T and v=v1,v2,v3,v4,v5,v6,v7,v8,v9,v10 where(35)w1=caσ+k1βaw3f1βaca−k1μ+−βTβacTcaf1+βTcTk1w7f1βaca−k1μw2=−w3f1βaca+σf1βaca−k1;w4=f2βTcTw7k3;w5=w6=w9=0;w8=rw7μw10=−δ−caf1κ1−1βa+k1κ1+σw3f1βaca−k1k8;v1=v8=v10=0;v2=1−f1βacav3−f1βaca+k1;v4=θv7k3;v5=βacak11−f1v3−f1βaca+k1k4+η2θv9k4;v6=βacak11−f1σ+k4v3−f1βaca+k1k4k5+θση2+η1k4v9k4k5;v9=−θ−k3f2+k3cTβTk5k4v7τ2rθση2+η1k4−k7k4k5k3−cak1βav3−1+f1rσ+k4τ2+k4k5τ2rθση2+η1k4−k7k4k5f1βaca−k1;withw7=v7=0,w3≠0,v3≠0whenR0=RH0=1;andw7≠0,v7≠0,w3=v3=0whenR0=RT0=1..The eigenvectors are chosen such thatw•v=1 or equivalently(36)1−f1βacaf1βaca+σf1βaca−k12+1v3w3+θf2βTcT+k32k32w7v7=1.Now, whenβa is taken as bifurcation parameter with R0=RH0=1, we obtain the following bifurcation coefficients:(37)a=aβa∗=−2βaσ+k1f1δca1−κ1βa+δκ1+k8k1+σk8+δca1−f1μk1w32v3k1−f1βaca3k8Q,b=bβa∗=1−f1cak1σ+k1v3w3f1βaca−k12.Further, whenβT is taken as bifurcation parameter with R0=RT0=1, we obtain the following bifurcation coefficients:(38)a=aβT∗=−2βTcTμ+1−ρrk3+f2μcTβT1−f2k3+f2θw72v7k32Qb=bβT∗=1−f2k3+f2θw7v7cTk3.Now, clearly,bβa∗>0,bβT∗>0, and hence the presence and direction of bifurcation in the model depend on the signs of aβa∗ and aβT∗. The following result of bifurcation is thus established.Theorem 7.
(bifurcation of the full model).(i)
IfR0=RH0, then the HIV-TB model exhibits backward bifurcation at R0=1 whenever k1<βacaf1; otherwise, the direction of the bifurcation is forward.(ii)
IfR0=RT0, then the HIV-TB model exhibits backward bifurcation at R0=1 whenever ρ>1+μ/r+f2μcTβT/rk3; otherwise, the direction of the bifurcation is forward.


## 5. Optimal Control of the Industrial HIV-TB

In this section, the basic HIV-TB co-infection model (2) is extended incorporating time-dependent controls to study the impact of those controls on the dynamics of the diseases. The goal here is study how public health interventions could be employed on a dynamical basis to eradicate or to drive the spread of the diseases to acceptable levels. One very important strategy for controlling the spread of diseases is to adopt strategies aimed at reducing the contact and/or transmission rate of the disease. To incorporate this strategy into our basic model, we introduce controls u1t and u2t such that as these controls increase, the transmission rates of HIV and TB respectively are reduced. We define u1t and u2t as the respective preventative strategies for reduction of HIV and TB transmission. These include abstinence, use of condoms in the case of HIV, and use of nose masks and isolation in the case of TB. Another very important strategy that is useful in curtailing the spread of HIV is administration of antiretroviral therapy, which helps in reducing the viral load in infected persons. We introduce this strategy as the control u3t such that as u3t increases, the viral load reduces and subsequently helps in reducing the progressing of infected persons into the AIDS stage. Since treatment is available for TB-infected persons, we introduce the control u4t such that a timely application of the control will increase the recovery rate of the TB-infected persons. Finally, in order to improve productivity, human resource departments will often implement in-service training to increase the capacity of the employees. We introduce a control u5t to measure the capacity building strategy adopted to increase productivity. The extended model with controls is thus given by(39)dSpdt=Q−1−u1λa+1−u2λT+μSpdIphdt=f1λh1−u1Sp+Rp+σu5Inh−τ11−u2λT+δ1−u3+μIphdInhdt=1−f1Sp+Rp1−u1λh−τ11−u2λT+σu5+κ1δ1−u3+μInhdEpdt=f21−u2λTSp+ρRp−ξ11−u1λh+θ+μEpdIphtdt=τ1f21−u2λTIph+σu5Inht−η2θ+κ2δ1−u3+μIphtdInhtdt=τ11−f21−u2λTInh+ξ11−u1λhEp+τ2ru4Ind−η1θ+κ3δ1−u3+σu5+μInhtdIntdt=1−f2Sp+ρRp1−u2λT+θEp−ξ21−u1λh+ru4+μ+μTIntdRpdt=ru4Int−ρ1−u2λT+1−u1λh+μRpdInddt=ξ21−u1λhInt+τ1f21−u2λTInh+τ11−f21−u2λTIph+η1θInht+η2θIpht−τ2ru4+κ4δ1−u3+μ+μTInddAdt=δ1−u3Iph+κ1Inh+κ2Ipht+κ3Inht+κ4Ind−μ+μaA.The main goal of incorporating the time-depended controls into the basic model is to help minimize the number of infections and improve productivity at minimal cost. Thus, an objective functional defined by(40)J=minui∫0Tb1Inh+b2Inht+b3Int+b4Ind+∑j=15ωjuj2dt,can be considered as a proxy of the main goal. The constants bis are positive weights associated with the corresponding populations and ωiui2s are the costs associated with implementing the controls uis. Thus, ω1u12 is the cost of implementing measures to prevent transmission of HIV. The objective function J is chosen in line with previous studies concerning optimal control of infectious diseases [9]. The main aim in setting up the optimal control problem is to minimize the number of nonproductive infectives in subclasses Inh,Inht, and Ind and the costs of implementing the controls. Thus, an optimal tuple u1∗,u2∗,u3∗,u4∗,u5∗ is sought such that(41)Ju1∗,u2∗,u3∗,u4∗,u5∗=MinJu1,u2,u3,u4,u5ui∈U,where U is the set of admissible controls with 0≤ui≤1 for t ∈ [0, tf]. Pontryagin’s maximum principle [31] provides the conditions necessary for existence of an optimal tuple. With this principle, the problem of solving equations (39) and (40) is converted to one of a point-wise minimization with respect to the controls of a Hamiltonian, H, function given by(42)H=b1Inh+b2Inht+b3Int+b4Ind+∑j=15ωjuj2+∑i=110λidXidt,where X=Sp,Iph,Inh,Ep,Ipht,Inht,Int,Rp,Ind,A. λis are the co-state or adjoint variables associated with the state variables. The following theorem follows from application of Pontryagin’s maximum principle [31] and the existence result for optimal control from [32].Theorem 8.
Letx∗=Sp∗,Iph∗,Inh∗,Ep∗,Ipht∗,Inht∗,Int∗,Rp∗,Ind∗,A∗be the solution of the state equation (39) and u∗=u1∗,u2∗,u3∗,u4∗,u5∗ be the corresponding optimal control tuple. Then, there exist adjoint variables li,i∈1,10 satisfying the following set of differential equations.(43)dldt=Fx,l,t,with transversality conditions litf=0,∀i=1,2…,10, F=F1,F2,…,F10T, where Fi=−∂H/∂xi.
The optimal controls are also shown to be given by(44)u1∗=λh2ω1l2−l3f1+l3−l1Sp+l2−l3f1+l3−l8Rp+l6−l4ξ1Ep+l9−l7ξ2InT;u2∗=λTT2ω2l6−l9f2+l3−l6Ina+l9−l5f2+l2−l9Ipaτ1−l4−l7f2+l7ρRp+Sp+ρRpl8+Spl1;u3∗=−δ2ω3l2−l10Ipa+κ1l3−l10Ina+κ2l5−l10Ipa1+κ3l6−l10Ina1+κ4l9−l10Ind;u4∗=r2ω4τ2l9−l6Ind+l7−l8InT;u5∗=σ2ω5l6−l5Ina1+Inal3−l2..
The control is bounded as follows:(45)u˜1=0ifu1∗≤0u1∗if0<u1∗<1,1ifu1∗≥1u˜2=0ifu2∗≤0u2∗if0<u2∗<1,1ifu2∗≥1u˜3=0ifu3∗≤0u3∗if0<u3∗<1,1ifu3∗≥1u˜4=0ifu4∗≤0u4∗if0<u4∗<11ifu4∗≥1,u˜5=0ifu5∗≤0u5∗if0<u5∗<1.1ifu5∗≥1Proof.
Due to the convexity of the function under the integral inJ with respect to ui,∀i=1…5, a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables, the existence of the optimal control tuple is established using Corollary 4.1 of Fleming and Rishel [32]. The adjoint system (43) is obtained by differentiating the Hamiltonian H with respect to the state variables. The characterizations in equation (44) are obtained by solving the stationarity conditions ∂H/∂ui=0 for ui. This completes the proof.


## 6. Numerical Experimentation

This section presents a numerical analysis of the optimal control problem (39) resulting from incorporation of the control efforts into model equation (2). The solution process involves the solution of the optimality system consisting of the state system (39) and the adjoint system (43). Following [33], the following iterative scheme is employed:(i)
Make a guess on the control profilesui, usually ui=0(ii)
Using the guess in step (i) above, solve the state system forward using a fourth-order Runge–Kutta method.(iii)
Using the guesseduis and the values of the state variables in step (ii) above, solve the adjoint system (43) backward using a fourth-order Runge–Kutta scheme.(iv)
With the current state and adjoint variables values, update the control using the characterizations (44) and bounds.(v)
With the current adjoint variables values and updated controls, repeat step (ii) and continue the process until the current iteration’s values are closer to the previous iteration’s values.To determine the most efficient strategy that can be employed to fight the spread of the diseases and also improve productivity, we seek to compare the following possible strategies:(1)
Fighting only HIV and skills training (i.e.,u1≠0,u3≠0,u5≠0,u2=u4=0).(2)
Fighting only TB and skills training (i.e.,u2≠0,u4≠0,u5≠0,u1=u3=0).(3)
Fighting both diseases and skills training (i.e.,u1≠0,u2≠0,u3≠0,u4≠u5≠0).(4)
Implementing only preventive controls and skills training (i.e.,u1≠0,u2≠0,u3=u4=0,u5≠0).(5)
Implementing only curative controls and skills training (i.e.,u1=u2=0,u3≠0,u4≠0,u5≠0).The comparison of the above strategies is achieved using the numerical simulation results, which are computed using the same initial state values and the model parameter values in Table1. The number of infections averted is used as a proxy for the benefits of interventions. The total number of infections averted is calculated as the difference in the total number of infectives at final time between the controlled and uncontrolled problems. Table 2 presents the health benefit of various intervention strategies.Table 2
Costs and health benefits of various intervention strategies.
StrategiesTotal infections avertedCost ($)1488263.88710440.182256.03350015.7283488263.92510433.5734488263.88710440.151556.24249950.050To compare the various strategies, use is made of the incremental cost-effectiveness ratio (ICER) which is one of three ratios used to measure effectiveness of different interventions. The ICER is used to compare the differences between the cost and health benefits of two alternative intervention strategies that compete for same resources. The ICER is defined as the extra cost incurred for an extra benefit. The ICER between two intervention strategies is the ratio of differences in cost to differences in health outcomes. Thus, for our case, the ICER between strategies 1 and 2 is given by(46)ICER1=ostofstrategy1−ostofstrategy2otalinfectionssavedbystrategy1−otalinfectionssavedbystrategy2.Thus, we carry out the comparisons as follows. Comparing strategies 1 and 2, we have the following (see Table3).Table 3
Comparing cost and benefit of strategies 1 and 2.
StrategiesTotal infections avertedTotal cost ($)ICER256.0325950015.72772892.618561488263.8872910440.18163−0.08106The cost-effectiveness ratio of strategy 1 over 2 is calculated as follows:(47)ICER1=10440.18163−50015.72772488263.88729−56.03259=−0.08106.Thus, a comparison between strategies 1 and 2 reveals that a cost saving of $0.08106 is made by preferring strategy 1 over strategy 2. Hence, we conclude that strategy 1 should be preferred to strategy 2.Next, we compare strategies 1 and 3 as follows (see Table4).Table 4
Comparing cost and benefit of strategies 1 and 3.
StrategiesTotal infections avertedTotal cost ($)ICER1488263.8872910440.181630.021383488263.9247610433.57277−176.36800A comparison between strategies 1 and 3 reveals that a cost saving of $176.36800 is made by preferring strategy 3 over strategy 1. Hence, we conclude that strategy 3 should be preferred to strategies 1 and 2.Next, we compare strategies 3 and 4 as follows (see Table5).Table 5
Comparing cost and benefit of strategies 3 and 4.
StrategiesTotal infections avertedTotal cost ($)ICER4488263.8873810440.151490.021383488263.9247610433.57277−175.98549A comparison between strategies 3 and 4 reveals that a cost saving of $175.98549 is made by preferring strategy 3 over strategy 4. Hence, we conclude that strategy 3 should be preferred to strategies 1, 2, and 4.Next, we compare strategies 3 and 5 as follows (see Table6).Table 6
Comparing cost and benefit of strategies 3 and 5.
StrategiesTotal infections avertedTotal cost ($)ICER556.2420049950.04995888.1271823488263.9247610433.57277−0.08094A comparison between strategies 3 and 5 reveals that a cost saving of $0.08094 is made by preferring strategy 3 over strategy 5. Hence, we conclude that strategy 3 should be preferred to strategies 1, 2, 4, and 5.These results imply that strategy 3, which is the implementation of all control strategies, strongly dominates over all other combinations of controls. Thus, the most cost-effective combination of our chosen controls is the one that involves all the controls, namely, prevention, treatment, and skills training. This goes to imply that multiple strategies in the combat against the spread of the two diseases should be adopted if cost-effectiveness is desired. The simulation results of the best strategy are presented in Figures2 and 3.Figure 2
Simulation results for the implementation of both preventative and curative measures along with skills training in combating the spread of HIV-TB co-infection.
(a)(b)(c)(d)(e)(f)Figure 3
Plots of adjoint variables and control profiles for the best strategy in combating spread of HIV-TB co-infection.
(a)(b)


## 7. Conclusion

In this paper, a deterministic mathematical model that describes the spread and effects of HIV-TB co-infection on workforce evolution has been proposed and analyzed. The basic model is split into two submodels, namely, the HIV-only and TB-only models, which are qualitatively analyzed for boundedness and stability of disease-free equilibrium states. The local stability and global stability of the disease-free equilibrium points of the submodels and the full model together with other qualitative properties are discussed. The full model is modified by incorporating time-dependent control variables into it, and an optimal control problem is formed by considering a minimization of a quadratic objective functional, which is chosen in line with extant literature on epidemic models due to the nonlinearity of the cost associated with implementation of controls, subject to the modified model. The optimal control problem is qualitatively analyzed using Pontryagin’s maximum principle, and numerical simulations are carried out considering various implementations of the interventions aimed at fighting the spread of the co-infection and improving productivity. Through the cost-effectiveness analysis, it was observed that combining preventive and curative measures together with skills training/improvement is the most cost-effective strategy that should be adopted in the fight against the spread of HIV-TB co-infection within the workforce.



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