Bifurcation and Optimal Control Analysis of an HIV/AIDS Model with Saturated Incidence Rate

HIV/AIDS is a chronic disease that weakens the immune system if untreated. This study develops and analyzes an optimal control model of HIV/AIDS transmission with a saturated incidence rate to assess the effectiveness of time-dependent interventions. The model is described by seven nonlinear differential equations. We first establish that the model solutions are nonnegative and bounded. The existence and stability of equilibrium points are explored, and the effective reproduction number is derived using the next-generation matrix method. The Routh–Hurwitz criterion is applied to determine the local stability of the disease-free equilibrium, while center manifold theory is used to demonstrate the occurrence of a forward bifurcation and to analyze the local stability of the endemic equilibrium. Global stability is proven using the Castillo–Chavez method for the disease-free equilibrium and a Lyapunov function for the endemic equilibrium. Sensitivity analysis highlights the impact of parameter variations on disease dynamics, identifying the effective contact rate as the most influential parameter. The model is extended into an optimal control framework by introducing educational campaigns, screening, and antiretroviral treatment as control measures. The existence of optimal control is shown, and the Pontryagin Maximum Principle is used to derive optimal control. Numerical simulations support the theoretical findings, showing that combining multiple controls significantly reduces infection levels. A cost-effectiveness analysis identifies the combination of educational campaigns and antiretroviral treatment (Strategy C) as the most efficient intervention. Overall, the results yield important implications for formulating effective and economically sustainable HIV/AIDS intervention policies.