Fractional-order approaches to optimal control in HIV-AIDS dynamics

HIV-AIDS, caused by the human immunodeficiency virus (HIV), remains a pressing global health issue, particularly in regions such as South Africa and India. The virus weakens the immune system, making the body vulnerable to infections and diseases. To better understand and control its spread, we propose a nonlinear mathematical model of HIV-AIDS using fractional order derivatives (Caputo) of order $ \tau $ ($ 0 < \tau \leq 1 $). This model focuses on the interaction between two distinct populations: men and women.Our approach employs fractional-order differential equations to capture the dynamics of disease progression. The study investigates critical aspects of the model, including the existence and uniqueness of solutions, the basic reproduction number, equilibrium points, and stability. We further apply Pontryagins maximum principle to design optimal control strategies and conduct sensitivity analysis to assess the impact of key parameters on disease transmission.The fractional-order framework enhances our ability to analyze disease behavior and better tools for controlling potential outbreaks. This study demonstrates the importance of fractional calculus in improving the understanding and management of infectious diseases.