# Modeling TB-HIV Syndemic and Treatment

Authors: Cristiana J. Silva; Delfim F. M. Torres
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Tuberculosis (TB) and human immunodeficiency virus (HIV) can be considered a deadly
human syndemic. In this paper, we formulate a model for TB and HIV transmission dynamics. The model considers both TB and acquired immune deficiency syndrome (AIDS)
treatment for individuals with only one of the two infectious diseases or both. The basic reproduction number and equilibrium points are determined and stability is analyzed. Through
simulations, we show that TB treatment for individuals with only TB infection reduces the
number of individuals that become coinfected with TB and HIV/AIDS and reduces the
diseases (TB and AIDS) induced deaths. Analogously, the treatment of individuals with
only AIDS also reduces the number of coinfected individuals. Further, TB treatment for
coinfected individuals in the active and latent stage of TB disease implies a decrease of the
number of individuals that passes from HIV-positive to AIDS.


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## Body

## 1. Introduction

Tuberculosis (TB) and human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) are the leading causes of death from an infectious disease worldwide [1]. Individuals infected with HIV are more likely to develop TB disease because of their immunodeficiency, and HIV infection is the most powerful risk factor for progression from TB infection to disease [2]. This interaction justifies the fact that HIV and TB can be considered a deadly humansyndemic, where syndemic refers to the convergence of two or more diseases that act synergistically to magnify the burden of disease [3].Following UNAIDS global report on AIDS epidemic 2013 [4], globally, an estimated 35.3 million people were living with HIV in 2012, an increase from previous years as more people are receiving the life-saving antiretroviral therapy (ART). There were approximately 2.3 million new HIV infections globally, showing a 33% decline in the number of new infections with respect to 2001. At the same time, the number of AIDS deaths is also declining with around 1.6 million AIDS deaths in 2012, down from about 2.3 million in 2005. In 2012, 1.1 million of 8.6 million people who developed TB worldwide were HIV-positive. The number of people dying from HIV-associated TB has been falling since 2003. However, there were still 320 000 deaths from HIV-associated TB in 2012 and further efforts are needed to reduce this burden [1]. ART is a critical intervention for reducing the risk of TB morbidity and mortality among people living with HIV and, when combined with isoniazid preventive therapy, it can have a significant impact on TB prevention [1].Collaborative TB/HIV activities (including HIV testing, ART therapy, and TB preventive measures) are crucial for the reduction of TB-HIV coinfected individuals. The World Health Organization (WHO) estimates that these collaborative activities prevented 1.3 million people from dying, from 2005 to 2012. However, significant challenges remain: the reduction of tuberculosis related deaths among people living with HIV has slowed in recent years; the ART therapy is not being delivered to TB-HIV coinfected patients in the majority of the countries with the largest number of TB/HIV patients; the pace of treatment scale-up for TB/HIV patients has slowed; less than half of notified TB patients were tested for HIV in 2012; and only a small fraction of TB/HIV-infected individuals received TB preventive therapy [4].The study of the joint dynamics of TB and HIV presents formidable mathematical challenges due to the fact that the models of transmission are quite distinct [5]. Few mathematical models have been proposed for TB-HIV coinfection (see, e.g., [5–9]). Kirschner [7] developed a cellular model for HIV-1 and TB coinfection inside a host. Roeger et al. [5] proposed a population model for TB-HIV/AIDS coinfection transmission dynamics, assuming that TB-infected individuals in the active stage of the disease are too ill to remain sexually active and therefore they are unable to transmit HIV. In this work, we assume that active TB-infected individuals are susceptible to HIV infection. Naresh and Tripathi [8] proposed a model for TB-HIV coinfection in a variable size population with only TB treatment. Here we consider TB and HIV treatment in different stages of the disease. Bhunu et al. [6] studied a TB-HIV coinfection model with both TB and HIV treatment. The authors did not take into account that an individual coinfected with TB and HIV can effectively recover from TB infection. We assume that TB can be cured, even in HIV-positive individuals [1]. Sharomi et al. [9] also considered these assumptions, subdividing the total population into 15 classes. It is our aim in this work to develop a model that balances two goals: simplicity and useful information.The paper is organized as follows. Section2 describes our model for TB-HIV syndemic with TB and HIV treatment. In Section 3, the positivity and boundedness of solutions of the model are proved and in Section 4 equilibrium points and respective stability are analyzed. Section 5 is devoted to numerical simulations and discussion of results.


## 2. TB-HIV/AIDS Model

The model subdivides the human population into 10 mutually exclusive compartments, namely, susceptible individuals (S), TB-latently infected individuals, who have no symptoms of TB disease and are not infectious (LT), TB-infected individuals, who have active TB disease and are infectious (IT), TB-recovered individuals (RT), HIV-infected individuals with no clinical symptoms of AIDS (IH), HIV-infected individuals with AIDS clinical symptoms (A), TB-latent individuals coinfected with HIV (pre-AIDS) (LTH), HIV-infected individuals (pre-AIDS) coinfected with active TB disease (ITH), TB-recovered individuals with HIV infection without AIDS symptoms (RTH), and HIV-infected individuals with AIDS symptoms coinfected with TB (AT). The total population at time t, denoted by N(t), is given by
(1)N(t)=S(t)+LT(t)+IT(t)+RT(t)+IH(t)+A(t)+ITH(t)+LTH(t)+RTH(t)+AT(t).
The susceptible population is increased by the recruitment of individuals (assumed susceptible) into the population, at a rate Λ. All individuals suffer from natural death, at a constant rate μ. Susceptible individuals acquire TB infection from individuals with active TB at a rate λT, given by
(2)λT=β1N(IT+ITH+AT),
where β1 is the effective contact rate for TB infection. Similarly, susceptible individuals acquire HIV infection, following effective contact with people infected with HIV at a rate λH, given by
(3)λH=β2N[IH+ITH+LTH+RTH+η(A+AT)],
where β2 is the effective contact rate for HIV transmission and the modification parameter η⩾1 accounts for the relative infectiousness of individuals with AIDS symptoms, in comparison to those infected with HIV with no AIDS symptoms. Individuals with AIDS symptoms are more infectious than HIV-infected individuals (pre-AIDS) because they have a higher viral load and there is a positive correlation between viral load and infectiousness [14].Individuals leave the latent TB classLT by becoming infectious, at a rate k1, or recovered, with a treatment rate τ1. The treatment rate for active TB-infected individuals is τ2. We assume that TB-recovered individuals RT acquire partial immunity and the transmission rate for this class is given by β1′λT with β1′⩽1. Individuals with active TB disease suffer induced death at a rate dT. We assume that individuals in the class RT are susceptible to HIV infection at a rate λH. On the other hand, TB-active infected individuals IT are susceptible to HIV infection, at a rate δλH, where the modification parameter δ⩾1 accounts for higher probability of individuals in class IT to become HIV-positive.HIV-infected individuals (with no AIDS symptoms) progress to the AIDS classA, at a rate ρ1. HIV-infected individuals with AIDS symptoms are treated for HIV at the rate α1 and suffer induced death at a rate dA. Individuals in the class IH are susceptible to TB infection at a rate ψλT, where ψ⩾1 is a modification parameter traducing the fact that HIV infection is a driver of TB epidemic [3].HIV-infected individuals (pre-AIDS) coinfected with TB-disease, in the active stageITH, are treated for TB at the rate τ3 and progress to the AIDS-TB coinfection class AT at a rate ρ2. Individuals in the class ITH suffer TB induced death at a rate dT. The anti-TB drugs can prevent or decrease the likelihood of TB infection progression to active TB disease in individuals in the class LTH [13]. The treatment rate for individuals in this class is given by τ4. However, individuals in the class LTH are more likely to progress to active TB disease than individuals infected only with latent TB. In our model, this progression rate is given by k2. Similarly, HIV infection makes individuals more susceptible to TB reinfection when compared with non-HIV-positive patients. The modification parameter associated with the TB reinfection rate, for individuals in the class RTH, is given by β2′, where β2′⩾1. Individuals in this class progress to class AT, at a rate ρ3.HIV-infected individuals (with AIDS symptoms), coinfected with TB, are treated for HIV, at a rateα2. Individuals in the class AT suffer from AIDS-TB coinfection induced death rate, at a rate dTA.The aforementioned assumptions result in the following system of differential equations that describes the transmission dynamics of TB and HIV disease:(4)S˙(t)=Λ-λTS(t)-λHS(t)-μS(t),L˙T(t)=λTS(t)+β1′λTRT(t)-(k1+τ1+μ)LT(t),I˙T(t)=k1LT(t)-(τ2+dT+μ+δλH)IT(t),R˙T(t)=τ1LT(t)+τ2IT(t)-(β1′λT+λH+μ)RT(t),I˙H(t)=λHS(t)-(ρ1+ψλT+μ)IH(t)+α1A(t)+λHRT(t),A˙(t)=ρ1IH(t)-α1A(t)-(μ+dA)A(t),L˙TH(t)=β2′λTRTH(t)-(k2+τ4+μ)LTH(t),I˙TH(t)=δλHIT(t)+ψλTIH(t)+α2AT(t)+k2LTH(t)-(τ3+ρ2+μ+dT)ITH(t),R˙TH(t)=τ3ITH(t)+τ4LTH(t)-(β2′λT+ρ3+μ)RTH,A˙T(t)=ρ2ITH(t)+ρ3RTH-(α2+μ+dTA)AT(t).
The model flow is described in Figure 1. The initial conditions of model (4) satisfy
(5)S(0)=S0⩾0,LT(0)=LT0⩾0,IT(0)=IT0⩾0,RT(0)=RT0⩾0,IH(0)=IH0⩾0,A(0)=A0⩾0,LTH(0)=LTH0⩾0,ITH(0)=ITH0⩾0,RTH(0)=RTH0⩾0,AT(0)=AT0⩾0.
Note that if we consider the submodel of (4) with no HIV/AIDS disease, that is, IH=A=LTH=ITH=RTH=AT=0, then we obtain the TB model from [12]. On the other hand, if we consider the submodel with no TB, that is, LT=IT=RT=LTH=ITH=RTH=AT=0, then we obtain an HIV/AIDS model based on the models proposed in [6, 15].Figure 1
Model for TB-HIV/AIDS transmission with treatment.


## 3. Positivity and Boundedness of Solutions

Let(S,LT,IT,RT,IH,A,LTH,ITH,RTH,AT)∈R+10 be any solution of (4) with initial conditions (5). Consider the biologically feasible region given by
(6)Ω={Λμ(S,LT,IT,RT,IH,A,LTH,ITH,RTH,AT)∈R+10:0⩽N(t)⩽Λμ}.
For the model system (4) to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time t>0. Suppose, for example, that at some t¯>0 the variable LT becomes zero, that is, LT(t¯)=0, while all other variables are positive. Then, from the LT equation we have dLT(t¯)/dt>0. Thus, LT(t)⩾0 for all t>0. Analogously, we can prove that all variables remain nonnegative for all time t>0.Adding all equations in model (4) gives
(7)dNdt(t)=Λ-μN(t)-dTIT(t)-dAA(t)-dTITH(t)-dTAAT(t).
Since N(t)⩾IT(t)+A(t)+ITH(t)+AT(t), then
(8)Λ-(μ+dT+dA+dTA)N(t)⩽dNdt(t)⩽Λ-μN(t).
Therefore, we conclude that N(t) is bounded for all t>0 and every solution of system (4) with initial condition in Ω remains in Ω. This result is summarized below.Lemma 1.
The regionΩ is positively invariant for model (4) with nonnegative initial conditions in R+10.


## 4. Stability Analysis

Model (4) has four nonnegative equilibria, namely,(i)
the disease-free equilibrium (no disease):(9)Σ0=(S0,LT0,IT0,RT0,IH0,A0,LTH0,ITH0,RTH0,AT0)Σ0=(Λμ,0,0,0,0,0,0,0,0,0),(ii)
the HIV-AIDS free equilibrium:(10)ΣT=(S⋄,LT⋄,IT⋄,RT⋄,IH⋄,A⋄,LTH⋄,ITH⋄,RTH⋄,AT⋄)
with IT⋄>0 and IH⋄=A⋄=LTH⋄=ITH⋄=RTH⋄=AT⋄=0 for R1>1, where R1 is the basic reproduction number of model (4) with IH=A=LTH=ITH=RTH=AT=0 (only TB model) that is given by
(11)R1=ΛNμ(β1dT+μ+τ2)(k1k1+τ1+μ)
(see [12]),(iii)
the TB-free equilibrium:(12)ΣH=(S⋆,LT⋆,IT⋆,RT⋆,IH⋆,A⋆,LTH⋆,ITH⋆,RTH⋆,AT⋆)
with LT⋆=IT⋆=RT⋆=LTH⋆=ITH⋆=RTH⋆=AT⋆=0 and
(13)S⋆=ΛμR2,(14)IH⋆=(R2-1)μNH(α1+dA+μ)β2(α1+dA+μ+ηρ1),(15)A⋆=(R2-1)ρ1μNHβ2(α1+dA+μ+ηρ1),
for R2>1, where R2 is the basic reproduction number of model (4) with LT=IT=RT=LTH=ITH=RTH=AT=0 (only HIV-AIDS model); that is,
(16)R2=ΛNμβ2(μ+α1+dA+ηρ1μα1+(μ+ρ1)(μ+dA)),(iv)
the syndemic equilibrium:(17)Σ*=(S*,LT*,IT*,RT*,IH*,A*,LTH*,ITH*,RTH*,AT*)
with IT*>0, IH*>0, A*>0, LTH*>0, ITH*>0, RTH*>0, and AT*>0, for R0>1, where R0 is the basic reproduction number of model (4); that is,
(18)R0=max⁡⁡{R1,R2}.The details of the computation of the basic reproduction number R0 are given in Appendix A.The following theorem states the stability of the equilibrium points.Theorem 2.
The disease-free equilibriumΣ0 is locally asymptotically stable if R0<1 and unstable if either Ri>1 with i=1,2. The HIV-AIDS free equilibrium ΣT is locally asymptotically stable if R1>1, and the TB-free equilibrium ΣH is locally asymptotically stable for R2 near 1.Details of the proof of Theorem2 are given in Appendix B.Explicit expressions for the coinfection endemic equilibriumΣ* are very difficult to compute analytically. In Section 5, we consider an example, with R0>1, for which there exists a syndemic equilibrium, and analyze, numerically, the local asymptotical stability of the syndemic equilibrium Σ*.


## 5. Numerical Analysis and Discussion

For numerical simulations, we consider the following initial conditions for system (4):
(19)(S(0),LT(0),IT(0),RT(0),IH(0),A(0),LTH(0),ITH(0),RTH(0),AT(0))=(60N100,14N100,3N100,0,4N100,N100,12N100,5N100,0,N100)
with N=50000. The parameters of model (4) take the values of Table 1.Table 1
Parameters of the TB-HIV/AIDS model (4).
Symbol
Value
References
Symbol
Value
References
Λ
714
τ
4
1 yr−1
μ
1
/
70 yr−1
ρ
1
0.1 yr−1
[10, 11]
β
1
Variable
ρ
2
0.25 yr−1
β
2
Variable
ρ
3
0.125 yr−1
β
1
′
0.9
α
1
0.33 yr−1
[6]
β
2
′
1.1
α
2
0.33 yr−1
k
1
1
[12]
ψ
1.07
k
2
1.3
k
1
[13]
d
T
1
/
8 yr−1
τ
1
1 yr−1
[12]
d
A
0.3 yr−1
τ
2
2 yr−1
[12]
d
T
A
0.33 yr−1
τ
3
2 yr−1
η
1.02
δ
1.03


### 5.1. Equilibrium Points and Stability Analysis

In Table2 we show the effect of the transmission coefficient β1 on the state IT⋄ of the HIV-free equilibrium ΣT and on the basic reproduction number R1. Table 3 shows the effect of the transmission coefficient β2 on the states IH⋆ and A⋆ of the TB-free equilibrium ΣH and on the basic reproduction number R2. We conclude that the equilibrium states IT⋄ and (IH⋆,A⋆) increase with the transmission coefficients β1 and β2, respectively.Table 2
Effect ofβ1 on IT⋄ and R1.
β
1
4.3
6
10
15
50
R
1
0.99788
1.39239
2.32065
3.48097
11.60326
I
T
⋄
0.00397
903.93492
2206.57268
2870.72755
3804.50589Table 3
Effect ofβ2 on IH⋆, A⋆, and R2.
β
2
0.051
0.055
0.07
0.09
0.99
R
2
0.93669
1.01016
1.28566
1.65299
1.81829
I
H
⋆
0.01708
135.73817
2516.54721
4472.84980
4930.48696
A
⋆
0.00266
21.07182
390.59491
694.23361
765.26396In Figure2 we considered different initial conditions in a neighborhood of the initial conditions given by (19) and R0<1 (R1<1 and R2<1) to illustrate the stability of the disease-free equilibrium Σ0 given by (9). In these numerical simulations we considered β1=2.7 and β2=0.03, corresponding to R1=0.62632 and R2=0.55077, while the rest of the parameters take the values in Table 1.Stability of the disease-free equilibrium (9).Figure3 shows that, for R0>1, the syndemic equilibrium Σ* exists. We considered different initial conditions for the state variables of system (4) in a neighborhood of (19), β1=6 and β2=0.1, corresponding to R1=1.39239 and R2=1.83593, and the rest of the parameters take the values in Table 1. We observe that the state variables converge to Σ* when t→∞. In this case, Σ* is given by
(20)Σ*=(S*,LT*,IT*,RT*,IH*,A*,LTH*,ITH*,RTH*,AT*)=(4766.84,2019.66,943.06,28621.89,362.66,56.29,31.39,55.15,495.68,112.33).Figure 3
Stability of the syndemic equilibriumΣ*.


### 5.2. Treatment Impact on TB-HIV/AIDS Coinfection

Considerβ1=13 and β2=0.06, while the rest of the parameters take the values of Table 1. Figure 4 shows the impact of treating the individuals with active and latent TB on the number of individuals coinfected with TB-HIV/AIDS. The treatment of individuals with only TB, IT and LT, has a positive impact on the reduction of the number of individuals coinfected with TB-HIV/AIDS. Moreover, the number of individuals that suffered from disease (TB and AIDS) induced death is higher when individuals with TB-single infection are not treated. In this case, the total population at the end of 20 years is around 10509 and, in the case where individuals with only TB are treated, the total population at the end of 20 years is around 29758 individuals. In Figure 5, we assume that there are no disease induced deaths; that is, dT=dA=dTA=0. The impact of treating individuals with only TB on the reduction of the number of coinfected individuals is more evident.Impact of TB treatment on single-infected individuals with disease induced death.Impact of TB treatment on single-infected individuals with no disease induced death.Figure6 illustrates the case where we compare the number of individuals coinfected with TB-HIV/AIDS when individuals with only AIDS symptoms AT are or are not treated. We observe that treating this class of individuals is important for the reduction of the number of individuals that become coinfected, with special attention to the individuals that have AIDS symptoms and TB infection. In Figure 7, we considered that there are no disease induced deaths (dT=dA=dTA=0). It is crucial that TB-infected individuals (in the latent and active stage), which are also HIV-positive, take anti-TB drugs, since they can recover from TB. We analyze the impact of treating TB-HIV/AIDS coinfected individuals LTH, ITH, and AT on the reduction of the number of individuals coinfection. If anti-TB drugs are supplied, then latent and active TB individuals with HIV can recover and pass to the class RTH (the number of individuals in the class RTH tends to zero when TB is not treated). In Figure 8, we observe that, after 7 years, the number of individuals infected with active TB and HIV, in the case without treatment, becomes lower than in the case with treatment. This is due to the fact that coinfection precipitates AIDS symptoms.Impact of AIDS treatment on single-infected individuals with disease induced death.Impact of AIDS treatment on single-infected individuals with no disease induced death.Impact of TB and AIDS treatment on coinfected individuals with no disease induced death.


## 5.1. Equilibrium Points and Stability Analysis

In Table2 we show the effect of the transmission coefficient β1 on the state IT⋄ of the HIV-free equilibrium ΣT and on the basic reproduction number R1. Table 3 shows the effect of the transmission coefficient β2 on the states IH⋆ and A⋆ of the TB-free equilibrium ΣH and on the basic reproduction number R2. We conclude that the equilibrium states IT⋄ and (IH⋆,A⋆) increase with the transmission coefficients β1 and β2, respectively.Table 2
Effect ofβ1 on IT⋄ and R1.
β
1
4.3
6
10
15
50
R
1
0.99788
1.39239
2.32065
3.48097
11.60326
I
T
⋄
0.00397
903.93492
2206.57268
2870.72755
3804.50589Table 3
Effect ofβ2 on IH⋆, A⋆, and R2.
β
2
0.051
0.055
0.07
0.09
0.99
R
2
0.93669
1.01016
1.28566
1.65299
1.81829
I
H
⋆
0.01708
135.73817
2516.54721
4472.84980
4930.48696
A
⋆
0.00266
21.07182
390.59491
694.23361
765.26396In Figure2 we considered different initial conditions in a neighborhood of the initial conditions given by (19) and R0<1 (R1<1 and R2<1) to illustrate the stability of the disease-free equilibrium Σ0 given by (9). In these numerical simulations we considered β1=2.7 and β2=0.03, corresponding to R1=0.62632 and R2=0.55077, while the rest of the parameters take the values in Table 1.Stability of the disease-free equilibrium (9).Figure3 shows that, for R0>1, the syndemic equilibrium Σ* exists. We considered different initial conditions for the state variables of system (4) in a neighborhood of (19), β1=6 and β2=0.1, corresponding to R1=1.39239 and R2=1.83593, and the rest of the parameters take the values in Table 1. We observe that the state variables converge to Σ* when t→∞. In this case, Σ* is given by
(20)Σ*=(S*,LT*,IT*,RT*,IH*,A*,LTH*,ITH*,RTH*,AT*)=(4766.84,2019.66,943.06,28621.89,362.66,56.29,31.39,55.15,495.68,112.33).Figure 3
Stability of the syndemic equilibriumΣ*.


## 5.2. Treatment Impact on TB-HIV/AIDS Coinfection

Considerβ1=13 and β2=0.06, while the rest of the parameters take the values of Table 1. Figure 4 shows the impact of treating the individuals with active and latent TB on the number of individuals coinfected with TB-HIV/AIDS. The treatment of individuals with only TB, IT and LT, has a positive impact on the reduction of the number of individuals coinfected with TB-HIV/AIDS. Moreover, the number of individuals that suffered from disease (TB and AIDS) induced death is higher when individuals with TB-single infection are not treated. In this case, the total population at the end of 20 years is around 10509 and, in the case where individuals with only TB are treated, the total population at the end of 20 years is around 29758 individuals. In Figure 5, we assume that there are no disease induced deaths; that is, dT=dA=dTA=0. The impact of treating individuals with only TB on the reduction of the number of coinfected individuals is more evident.Impact of TB treatment on single-infected individuals with disease induced death.Impact of TB treatment on single-infected individuals with no disease induced death.Figure6 illustrates the case where we compare the number of individuals coinfected with TB-HIV/AIDS when individuals with only AIDS symptoms AT are or are not treated. We observe that treating this class of individuals is important for the reduction of the number of individuals that become coinfected, with special attention to the individuals that have AIDS symptoms and TB infection. In Figure 7, we considered that there are no disease induced deaths (dT=dA=dTA=0). It is crucial that TB-infected individuals (in the latent and active stage), which are also HIV-positive, take anti-TB drugs, since they can recover from TB. We analyze the impact of treating TB-HIV/AIDS coinfected individuals LTH, ITH, and AT on the reduction of the number of individuals coinfection. If anti-TB drugs are supplied, then latent and active TB individuals with HIV can recover and pass to the class RTH (the number of individuals in the class RTH tends to zero when TB is not treated). In Figure 8, we observe that, after 7 years, the number of individuals infected with active TB and HIV, in the case without treatment, becomes lower than in the case with treatment. This is due to the fact that coinfection precipitates AIDS symptoms.Impact of AIDS treatment on single-infected individuals with disease induced death.Impact of AIDS treatment on single-infected individuals with no disease induced death.Impact of TB and AIDS treatment on coinfected individuals with no disease induced death.



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*Source: 248407-2014-07-08.xml*