Fractional modeling of the HIV/AIDS epidemic using the Caputo derivative and Haar wavelet collocation method

Human immunodeficiency virus (HIV) remains a major global health concern, with South Africa experiencing particularly high prevalence. Despite this, few studies have applied fractional-order mathematical models to real HIV data from South Africa. In this work, we develop a fractional-order HIV/AIDS epidemic model using the Caputo derivative to account for memory effects and the non-local nature of disease dynamics. The model’s well-posedness is established by proving the existence, uniqueness, positivity and boundedness of solutions. We analyze both the disease-free and endemic equilibria, derive the basic reproduction number [Formula: see text] using the next-generation matrix approach and investigate global stability through Lyapunov functions and LaSalle’s invariance principle. Model parameters are estimated from reported HIV cases in South Africa, and numerical simulations are carried out in MATLAB using the Haar wavelet collocation method. The results show good agreement between the model and real data, and demonstrate how fractional order and key parameters influence transmission dynamics. These findings provide deeper insight into HIV spread and highlight the potential of fractional-order models in guiding effective prevention and control strategies.