Theoretical and computational analysis of a novel fractional-order mathematical model for HIV transmission dynamics

We develop a five-class fractional mathematical model to investigate HIV transmission using the Caputo–Fabrizio (CF) derivative. The population is divided into susceptible, acute, chronic, treated and AIDS classes to capture the key stages of HIV progression and the effects of antiretroviral therapy. Existence and uniqueness of solutions are established using the fixed point theorem, a bounded invariant region is identified, and the basic reproduction threshold $R_0$ is derived using the next-generation matrix. Analytical results demonstrate the local and global stability of both the infection-free equilibrium and the endemic equilibrium. Lower fractional orders are shown to slow convergence to equilibrium, reflecting the memory effects inherent in fractional dynamics. The CF operator therefore provides a flexible and realistic framework for understanding HIV transmission and assessing the influence of treatment, disease progression, and memory effects on long-term epidemic outcomes. The Ulam–Hyers (UH) stability of the considered HIV model under CF derivative is proved. In addition, numerical solutions are obtained by using the Adams–Bashforth method corresponding to the CF derivative. Simulations show that early initiation of antiretroviral therapy, reduced treatment dropout, and improved viral suppression substantially decrease HIV prevalence.