A Mathematical Model of HIV and TB Co-infection
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Keywords:
Basic Reproduction Number, Co-infection, Epidemiology, HIV/AIDS, Tuberculosis, StabilityAbstract
The Human Deficiency Virus (HIV) is the major cause of mortality among individuals also having TB infection. The two diseases interact concurringly in their epidemiological characteristics. A non linear mathematical model with six compartments of Susceptible, Infectious TB, Treated TB, Infectious HIV, Infectious with both HIV and TB and AIDS classes is constructed to examine the interaction between TB and HIV epidemic. The Basic Reproduction Number (BRR) for TB ($R^{T}_{0}$) and HIV (${R}^{H}_{0}$) and the overall BRR ${R}_{0} = \max \{{R}^{T}_{0},{R}^{H}_{0}\}$ is calculated. The model shows three equilibria, viz. a disease-free equilibrium, TB only (HIV-free) equilibrium and HIV only (TB-free) equilibrium. Stability criterion for the three equilibria points is determined. The disease-free equilibrium is found to be globally asymptotically stable if $R_{0} \le 1$. The TB only equilibrium is locally stable if $R_{0}^{T} > 1$ along with some other conditions. The HIV only equilibrium is stable if $R_{0}^{H} >1$ along with some other conditions. Numerical simulations presented in the paper examine the role of some essential epidemiological parameters in disease spread.
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