Mathematical modeling and analysis of HIV and tuberculosis co-infection dynamics with optimal control

This study develops and analyzes a comprehensive mathematical model for human immunodeficiency virus (HIV)–Mycobacterium tuberculosis (TB) co-infection dynamics, incorporating both direct human-to-human transmission and a healthcare-associated contamination pathway for HIV infection. This model is formulated as a system of nonlinear ordinary differential equations with multiple epidemiological compartments, including a contamination variable M(t) representing the level of infectious contamination arising from unsafe healthcare practices such as improperly sterilized medical instruments, unsafe injections, and exposure to contaminated biological materials. The basic reproduction number is derived using the next-generation matrix approach and is expressed as the maximum of two threshold quantities corresponding to the tuberculosis transmission pathway and the coupled HIV–contamination transmission pathway. Rigorous qualitative analysis establishes positivity and boundedness of solutions and investigates the local and global asymptotic stability of the disease-free equilibrium when R0 < 1 as well as the local asymptotic stability of the endemic equilibrium when R0 > 1. Furthermore, bifurcation analysis establishes the occurrence of a forward (supercritical) transcritical bifurcation at R0 = 1. Sensitivity analysis reveals that transmission rates are the primary drivers of disease spread, whereas tuberculosis treatment and contamination-removal mechanisms contribute significantly to reducing the reproduction threshold. An optimal control framework based on Pontryagin’s maximum principle is developed to assess intervention strategies and their implementation costs. Numerical simulations demonstrate that integrated HIV and TB control measures produce a synergistic effect, substantially outperforming individual interventions in reducing disease prevalence. Although the integrated strategy incurs higher initial costs, it yields considerable long-term epidemiological and economic benefits. Cost-effectiveness analysis further identifies the most economically efficient intervention policies.