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

In this paper, we develop an HIV/AIDS epidemic model that incorporates a saturated incidence rate to reflect the limited transmission capacity and the impact of behavioral saturation in contact patterns. The model is formulated as a system of seven non-linear ordinary differential equations representing key population compartments. In addition to model formulation, we introduce an optimal control problem involving three control measures: educational campaigns, screening of unaware infected individuals, and antiretroviral treatment for aware infected individuals. We begin by establishing the positivity and boundedness of the model solutions under constant control inputs. The existence and local and global stability of both the disease-free and endemic equilibrium points are analyzed, depending on the effective reproduction number (R_e). Bifurcation analysis reveals that the model undergoes a forward bifurcation at R_e = 1. A local sensitivity analysis of R_e identifies the disease transmission rate as the most sensitive parameter. The optimal control problem is then formulated by incorporating the dynamics of infected subpopulations, control costs, and time-dependent controls. The existence of optimal control solutions is proven, and the necessary conditions for optimality are derived using Pontryagin’s Maximum Principle. Numerical simulations support the theoretical analysis and confirm the stability of the equilibrium points. The optimal control strategies, evaluated using the Incremental Cost-Effectiveness Ratio (ICER), indicate that implementing both screening and treatment (Strategy D) is the most cost-effective intervention. These results provide important insights for designing effective and economically sustainable HIV/AIDS intervention policies.