Modeling Transmission Dynamics of Tuberculosis-HIV Co-Infection in South Africa

Abstract

South Africa has the highest number of people living with the human immunodeficiency virus (HIV) in the world, accounting for nearly one in five people living with HIV globally. As of 2021, 8 million people in South Africa were infected with HIV, which is 13% of the country's total population. Approximately 450,000 people in the country develop tuberculosis (TB) disease every year, and 270,000 of those are HIV positive. This suggests that being HIV positive significantly increases one's susceptibility to TB, accelerating the spread of the epidemic. To better understand the disease burden at the population level, a Susceptible-Infected-Recovered-Dead (SIRD) TB-HIV co-infection epidemic model is presented. Parameter values are estimated using the method of moments. The disease-free equilibrium and basic reproduction number of the model are also obtained. Finally, numeric simulations are carried out for a 30-year period to give insights into the transmission dynamics of the co-infection.

Keywords: HIV; South Africa; co-infection dynamics; mathematical modeling; tuberculosis.

Figure 1

Deterministic SIRD for TB–HIV co-infection model. $S$ represents susceptible individuals, $I_T$ represents TB-infected individuals, $I_H$ represents HIV-infected individuals, $C$ represents TB–HIV-co-infected individuals, $R_T$ represents TB-recovered individuals, $R_C$ represents TB-recovered individuals with HIV infection, $D_T$ represents TB death, $D_C$ represents TB–HIV co-infection death, and $D_H$ represents HIV death.

Figure 2

(a) Susceptible population trajectory; (b) TB-infected population trajectories for HIV-positive individuals (blue) and HIV-negative individuals (red).

Figure 3

(a) HIV-infected population trajectory; (b) TB recovery trajectory for HIV-positive individuals (blue) and HIV-negative individuals (red).

Figure 4

(a) HIV death trajectory; (b) TB death trajectories for HIV-positive individuals (blue) and HIV-negative individuals (red).

Figure 5

TB dynamics trajectory. Class C represents TB–HIV-co-infected individuals, $D_C$ represents TB–HIV co-infection death, $D_T$ represents TB death, $I_T$ represents TB-infected individuals, $R_C$ represents TB-recovered individuals with HIV infection, and $R_T$ represents TB-recovered individuals.

Figure 6

(a) Co-infected population trajectory (observed (red)) vs. simulated (blue)); (b) HIV-infected population trajectory (observed (red) vs. simulated (blue)).