Mathematical Analysis and Numerical Simulation of the Integrated HIV/TB Co-Infection Model

Abstract

In this study, we examined an integrated model of HIV and TB co-infection. Specifically, we proposed a system of nonlinear differential equations to describe the co-dynamics of both diseases and conducted a comprehensive stability analysis of its critical points alongside numerical simulations. The stability analysis of the equilibrium points revealed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number, R_0, is less than one (R_0<1). This result indicates that, under these conditions, both diseases can be eradicated from the population over time. A local sensitivity analysis was performed to identify parameters with the greatest influence on disease dynamics. The analysis demonstrated that the parameter ω_1 is a key driver for tuberculosis (TB) transmission, as evidenced by its high sensitivity index in the TB-specific reproduction number, R_T. From a biological perspective, ω_1 represents a control measure related to reducing aerosol transmission, highlighting the critical importance of respiratory hygiene, such as covering one's mouth when coughing or sneezing. A similar analysis for HIV indicated that the parameter ω_2 exerts the most significant influence on HIV transmission, as shown by its impact on the HIV-specific reproduction number, R_H. Biologically, ω_2 encapsulates behavioral controls, emphasizing the necessity of promoting safe sexual practices and preventing contact with infected bodily fluids to curb HIV spread. Numerical simulations supported our analytical findings. They demonstrated that in a co-infection setting, the disease with the higher basic reproduction number tends to dominate the epidemiological landscape. Furthermore, the simulations illustrated that targeted increases in the control measures ω_1 and ω_2 lead to a substantial reduction in the number of co-infection cases within the community, underscoring the effectiveness of combined, disease-specific intervention strategies.