Analytical Investigation of Hopf Bifurcation and Intrinsic Nonlinear Dynamics in a Fractional-Order HIV/AIDS Transmission Framework

Abstract

We propose and analyze a fractional-order HIV/AIDS transmission model that partitions the popu lation into susceptible men and women, undiagnosed and diagnosed infected individuals, AIDS patients, those undergoing treatment, and cumulative mortality. Incorporating demographic and epidemiological data from South Africa and India, parameter estimates are adjusted to capture regional variations in HIV prevalence and treatment coverage. The model ensures biological plausibility by demonstrating the positivity and bounded ness of all solutions. The disease-free and endemic equilibrium states are derived, with the basic reproduction number R0 formulated through the next-generation matrix approach. Stability analysis both local and global are conducted using RouthUHurwitz criteria, and conditions for Hopf bifurcation are identified to reveal potential ˝ oscillatory infection dynamics. The existence and uniqueness of solutions are assured under a Lipschitz con dition. Overall, this framework offers a rigorous mathematical foundation for understanding HIV transmission mechanisms and informing intervention strategies in heterogeneous populations.