#### 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.

#### 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 human* syndemic*, 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. Section 2 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 (), TB-latently infected individuals, who have no symptoms of TB disease and are not infectious (), TB-infected individuals, who have active TB disease and are infectious (), TB-recovered individuals (), HIV-infected individuals with no clinical symptoms of AIDS (), HIV-infected individuals with AIDS clinical symptoms (), TB-latent individuals coinfected with HIV (pre-AIDS) (), HIV-infected individuals (pre-AIDS) coinfected with active TB disease (), TB-recovered individuals with HIV infection without AIDS symptoms (), and HIV-infected individuals with AIDS symptoms coinfected with TB (). The total population at time , denoted by , is given by The susceptible population is increased by the recruitment of individuals (assumed susceptible) into the population, at a rate . All individuals suffer from natural death, at a constant rate . Susceptible individuals acquire TB infection from individuals with active TB at a rate , given by where 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 , given by where is the effective contact rate for HIV transmission and the modification parameter 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 class by becoming infectious, at a rate , or recovered, with a treatment rate . The treatment rate for active TB-infected individuals is . We assume that TB-recovered individuals acquire partial immunity and the transmission rate for this class is given by with . Individuals with active TB disease suffer induced death at a rate . We assume that individuals in the class are susceptible to HIV infection at a rate . On the other hand, TB-active infected individuals are susceptible to HIV infection, at a rate , where the modification parameter accounts for higher probability of individuals in class to become HIV-positive.

HIV-infected individuals (with no AIDS symptoms) progress to the AIDS class , at a rate . HIV-infected individuals with AIDS symptoms are treated for HIV at the rate and suffer induced death at a rate . Individuals in the class are susceptible to TB infection at a rate , where 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 stage , are treated for TB at the rate and progress to the AIDS-TB coinfection class at a rate . Individuals in the class suffer TB induced death at a rate . The anti-TB drugs can prevent or decrease the likelihood of TB infection progression to active TB disease in individuals in the class [13]. The treatment rate for individuals in this class is given by . However, individuals in the class 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 . 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 , is given by , where . Individuals in this class progress to class , at a rate .

HIV-infected individuals (with AIDS symptoms), coinfected with TB, are treated for HIV, at a rate . Individuals in the class suffer from AIDS-TB coinfection induced death rate, at a rate .

The aforementioned assumptions result in the following system of differential equations that describes the transmission dynamics of TB and HIV disease: The model flow is described in Figure 1. The initial conditions of model (4) satisfy Note that if we consider the submodel of (4) with no HIV/AIDS disease, that is, , then we obtain the TB model from [12]. On the other hand, if we consider the submodel with no TB, that is, , then we obtain an HIV/AIDS model based on the models proposed in [6, 15].

#### 3. Positivity and Boundedness of Solutions

Let be any solution of (4) with initial conditions (5). Consider the biologically feasible region given by For the model system (4) to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time . Suppose, for example, that at some the variable becomes zero, that is, , while all other variables are positive. Then, from the equation we have . Thus, for all . Analogously, we can prove that all variables remain nonnegative for all time .

Adding all equations in model (4) gives Since , then Therefore, we conclude that is bounded for all and every solution of system (4) with initial condition in remains in . This result is summarized below.

Lemma 1. *The region is positively invariant for model (4) with nonnegative initial conditions in .*

#### 4. Stability Analysis

Model (4) has four nonnegative equilibria, namely,(i)the disease-free equilibrium (no disease): (ii)the HIV-AIDS free equilibrium: with and for , where is the basic reproduction number of model (4) with (only TB model) that is given by (see [12]),(iii)the TB-free equilibrium: with and for , where is the basic reproduction number of model (4) with (only HIV-AIDS model); that is, (iv)the syndemic equilibrium: with , , , , , , and , for , where is the basic reproduction number of model (4); that is, The details of the computation of the basic reproduction number are given in Appendix A.

The following theorem states the stability of the equilibrium points.

Theorem 2. *The disease-free equilibrium is locally asymptotically stable if and unstable if either with . The HIV-AIDS free equilibrium is locally asymptotically stable if , and the TB-free equilibrium is locally asymptotically stable for near 1.*

Details of the proof of Theorem 2 are given in Appendix B.

Explicit expressions for the coinfection endemic equilibrium are very difficult to compute analytically. In Section 5, we consider an example, with , for which there exists a syndemic equilibrium, and analyze, numerically, the local asymptotical stability of the syndemic equilibrium .

#### 5. Numerical Analysis and Discussion

For numerical simulations, we consider the following initial conditions for system (4): with . The parameters of model (4) take the values of Table 1.

##### 5.1. Equilibrium Points and Stability Analysis

In Table 2 we show the effect of the transmission coefficient on the state of the HIV-free equilibrium and on the basic reproduction number . Table 3 shows the effect of the transmission coefficient on the states and of the TB-free equilibrium and on the basic reproduction number . We conclude that the equilibrium states and increase with the transmission coefficients and , respectively.

In Figure 2 we considered different initial conditions in a neighborhood of the initial conditions given by (19) and ( and ) to illustrate the stability of the disease-free equilibrium given by (9). In these numerical simulations we considered and , corresponding to and , while the rest of the parameters take the values in Table 1.

Figure 3 shows that, for , the syndemic equilibrium exists. We considered different initial conditions for the state variables of system (4) in a neighborhood of (19), and , corresponding to and , and the rest of the parameters take the values in Table 1. We observe that the state variables converge to when . In this case, is given by

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

Consider and , 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, and , 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, . The impact of treating individuals with only TB on the reduction of the number of coinfected individuals is more evident.

Figure 6 illustrates the case where we compare the number of individuals coinfected with TB-HIV/AIDS when individuals with only AIDS symptoms 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 (). 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 , , and 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 (the number of individuals in the class 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.

#### Appendices

#### A. Computation of

The basic reproduction number represents the expected average number of new infections produced by a single infectious individual when in contact with a completely susceptible population [16]. Following [16], the basic reproduction number is obtained as the spectral radius of the matrix at the disease-free equilibrium , given by (9), with , and withThe dominant eigenvalues of the matrix are Thus, the basic reproduction number of model (4) is given by Note that is the basic reproduction number of model (4) with (only TB model), and is the basic reproduction number of model (4) with (only HIV-AIDS model).

#### B. Proof of Theorem 2

In this Appendix, we provide details of the proof of Theorem 2.

*Local Asymptotical Stability of the Disease-Free Equilibrium *. Following Theorem 2 of [16], the disease-free equilibrium, , is locally asymptotically stable if all the eigenvalues of the Jacobian matrix of the system (4), here denoted by , computed at the disease-free equilibrium , given by (9), have negative real parts.

The Jacobian matrix of the system (4) at disease-free equilibrium is given by withwhere ; ; ; ; ; ; ; . One has for We have just proved that the disease-free equilibrium of model (4) is locally asymptotically stable if and unstable if either , .

*Global Asymptotical Stability of the Disease-Free Equilibrium *. For convenience, let us rewrite the model system (4) as
where and , with denoting (its components) the number of uninfected individuals and denoting (its components) the number of infected individuals including the latent and infectious.

The disease-free equilibrium is denoted by

Following [6], if(H1) is globally asymptotically stable for ,(H2) for , where , is a Metzler matrix, and is given by (6),then the fixed point is a globally asymptotically stable equilibrium of system (B.5). We haveand . Thus, withFrom (B.9) the condition (H2) is not satisfied, since is not true. Therefore, the disease-free equilibrium may not be globally asymptotically stable. Following [17], the backward bifurcation occurs at and the double endemic equilibria can be supported for , where is a positive constant.

*Existence and Stability of HIV-AIDS Free Equilibrium *. The expressions for , , , and are obtained if we consider a submodel of (4) for which and the total population is given by . The basic reproduction number of this submodel is given by (11). The existence, uniqueness, and local asymptotic stability of are proven in [12, Theorem 1].

*Existence and Stability of TB-Free Equilibrium *. To prove the existence of , consider the submodel of (4) for which and the total population is given by . The equations of this submodel are
where . Setting the right-hand sides of submodel (B.10) to zero, we obtain the endemic equilibrium given by
where and , whenever .

In what follows we prove the local asymptotic stability of the endemic equilibrium , using the center manifold theory [18], as described in [19, Theorem 4.1] (see also [16]), considering ART treatment. The basic reproduction number of this submodel is given by (16). Choose bifurcation parameter, , by solving for from : Submodel (B.10) has a disease-free equilibrium given by .

The Jacobian of the system (B.10), evaluated at and with , is given by The eigenvalues of the linearized system (B.13) are We observe that there is a simple eigenvalue with zero real part and the other two eigenvalues have negative real part. Thus, the system (B.10), with , has a hyperbolic equilibrium point and the center manifold theory [18] can be used to analyze the dynamics of submodel (B.10) near .

The Jacobian at has a right eigenvector (associated with the zero eigenvalue) given by , where Further, for has a left eigenvector (associated with the zero eigenvalue), where To apply Theorem 4.1 in [19] it is convenient to let represent the right-hand side of the th equation of the system (B.10) and let be the state variable whose derivative is given by the th equation for . The local stability near the bifurcation point is then determined by the signs of two associated constants, denoted by and , defined (respectively) by with .

For the system (B.10), the associated partial derivatives at the disease-free equilibrium are given by It follows from the above expressions that with .

For the sign of , it can be shown that the associated nonvanishing partial derivatives are It also follows from the above expressions that Thus, and . Using Theorem 4.1 of [19], the endemic equilibrium is locally asymptotically stable for near 1.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA, University of Aveiro) and* The Portuguese Foundation for Science and Technology* (FCT—Fundação para a Ciência e a Tecnologia), within Project PEst-OE/MAT/UI4106/2014. Silva was also supported by FCT through the Postdoc Fellowship SFRH/BPD/72061/2010 and Torres by the FCT Project PTDC/EEI-AUT/1450/2012, cofinanced by FEDER under POFC-QREN with COMPETE Reference FCOMP-01-0124-FEDER-028894.