A novel hybrid fractional approach to nonlinear dynamics of HIV transmission among men who have sex with men in Taiwan

In Taiwan, human immunodeficiency virus (HIV) transmission occurs primarily among men who have sex with men (MSM). High-risk behaviors within this population contribute significantly to the variability and spread of the virus. This study introduces a fractional-order mathematical framework to better understand the dynamics of HIV transmission among MSM. We employ the modified Atangana-Baleanu derivative in the Caputo sense (ABC) to model the system. Theoretical analysis focuses on the existence, positivity, uniqueness, and sensitivity of solutions, utilizing the Leray-Schauder nonlinear alternative theorem and Banach's fixed-point theorem. Successive iterative sequences are constructed to verify the existence of solutions. Furthermore, we demonstrate the stability and uniqueness of the model solution within the Hyers-Ulam framework using tools from functional analysis. A chaos control method based on linear response regulation is applied to control the system dynamics and stabilize equilibrium points. Numerical simulations are conducted, and a graphical comparison between classical and fractional-order derivatives is presented using Lagrange interpolation within the modified ABC-fractional framework. Our findings highlight the significant impact of HIV transmission dynamics among MSM on public health systems. The modified ABC operator proves effective in capturing the long-term memory effects of the disease, enabling more precise and adaptive intervention strategies. This research provides policymakers and epidemic managers with a novel and robust modeling tool to develop more flexible and sustainable strategies for managing HIV spread, especially in light of evolving public health challenges.