
Abstract
Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB ($\R_1$) and HIV ($\R_2$) and the overall reproductive number for the system, $\R =\max \{\R_1, \R_2\}$. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if $\R <1$ then the diseasefree equilibrium is locally asymptotically stable.
The TBonly equilibrium $E_T$ is locally asymptotically
stable if $\R_1>1$ and $\R_2<1$. However, the symmetric condition, $\R_1<1$ and $\R_2>1$,
does not necessarily guarantee the stability of the HIVonly equilibrium $E_H$,
and it is possible that TB can coexist with HIV when $\R_2>1$. In other words, in the case when $\R_1<1$ and $\R_2>1$ (or when $\R_1>1$ and $\R_2>1$), we are able to find a stable HIV/TB coexistence equilibrium.
Moreover, we show that the prevalence level of TB increases with $\R_2>1$ under
certain conditions. Through simulations, we find that
i) the increased progression rate from latent to active TB in coinfected individuals may play a significant role in the rising prevalence of TB; and ii)
the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 34C60.
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