Computational aspects and dynamical analysis of a novel HIV/AIDS transmission model with fractional temporal evolution

The profound global health impact of HIV underscores its devastating consequences, particularly evident in the high mortality rates associated with its advanced stage, AIDS. In this article, a novel mathematical model is introduced to analyze the dynamics of HIV/AIDS transmission, employing the Caputo fractional derivative. The solution’s positivity and boundedness, as well as the criteria for a unique solution, are provided. For the model under consideration, the endemic and disease-free equilibrium points are determined. The next-generation matrix technique is used to calculate the basic reproduction number (R0). The Routh–Hurwitz criterion is employed to establish the local asymptotic stability of the disease-free point when R0<1. The global asymptotic stability of the endemic steady state is established when R0>1, using the graph theoretic method together with Lyapunov functions and LaSalle’s invariance assumption. The important parameters of the model are estimated using real data on HIV infections in Taiwan from 2000 to 2023, under the influence of a fractional-order framework that incorporates the memory effect through the Mittag-Leffler function. The considered model shows the best fit with the infected cases when the fractional order α=0.95. The impact of the fractional order α on the HIV/AIDS dynamics is examined using the Adams–Bashforth predictor–corrector approach across various values of α. Furthermore, an analysis is conducted to determine how the AIDS progression rate (η) and the HIV transmission rate (η1) affect the model equations, which are then graphically shown. Increasing the values of these parameters will result in a higher number of infected patients. Our findings indicate that a decrease in transmission rates is required for the disease to be successfully managed.