Lyapunov Stability and Optimization of a Fractional‐Order HIV/AIDS Model Coupling Vertical Transmission With Saturated Treatment

This research introduces a fractional‐order nonlinear model for the dynamics of human immunodeficiency virus (HIV) and acquired immune deficiency syndrome (AIDS) using Caputo‐type derivatives of noninteger order. Solution properties of the model are investigated by analyzing positivity and boundedness characteristics via the generalized mean value theorem. Model solutions are demonstrated to satisfy existence and uniqueness criteria by employing Banach fixed point theorem techniques. Global dynamics are explored using suitable Lyapunov functionals, and model trajectories are quantitatively illustrated. The model is subsequently enhanced to incorporate three time‐varying control mechanisms: strategies for preventing vertical transmission from mother to child, barrier contraceptive interventions, and therapeutic interventions through highly active antiretroviral therapy (HAART). Analysis of the controlled system dynamics is conducted through optimal control theoretical methods, building upon Pontryagin’s maximum principle framework. Computational simulations are conducted to visually represent theoretical outcomes and analyze the role of memory effects in system performance. Specifically, the study demonstrates that implementing all three optimal control strategies simultaneously, combined with memory effects at fractional order values ( α < 1), represents the most effective approach for reducing HIV/AIDS transmission. The study’s results emphasize the critical role of memory‐based optimal control strategies in addressing both vertical and horizontal HIV/AIDS transmission, particularly in scenarios involving treatment delays within endemic populations.