Fractional-order model for the transmission dynamics and control of HIV/AIDS

Human immunodeficiency virus, along with acquired immunodeficiency syndrome (HIV/AIDS), is a serious global health concern, with AIDS representing the advanced stage of HIV infection. To better understand HIV transmission dynamics, we developed a fractional-order (FO) mathematical model, building on conventional integer-order models. The model is first formulated using nonlinear integer-order differential equations and then extended with fractional-order derivatives and power laws to capture more realistic disease dynamics. Analytical results include conditions for the existence and uniqueness of solutions and stability analysis of the endemic equilibrium using the Lyapunov function. A sensitivity analysis of the basic reproduction number identified key factors affecting disease spread. By fitting the model to real-world HIV data, important parameters are estimated. Numerical simulations, performed using the fractional Adams–Bashforth–Moulton method, reveal how model parameters and FO values influence HIV dynamics. Results showed that improved treatment strategies and adherence to safe practices reduce HIV prevalence, while increased contact rates and lower treatment efficacy exacerbate it.