Journal Article

A novel fractal fractional mathematical model for HIV/AIDS transmission stability and sensitivity with numerical analysis

Khan, Nadeem, Khan, Mukhtiar, Ullah, Ibad, Shah, Kamal, Abdalla, Bahaaeldin, Abdeljawad, Thabet

2025

Overview

Understanding the complex dynamics of HIV/AIDS transmission requires models that capture real-world progression and intervention impacts. This study introduces an innovative mathematical framework using fractal-fractional calculus to analyze HIV/AIDS dynamics, emphasizing memory effects and nonlocal interactions critical to disease spread. By dividing populations into four distinct compartments-susceptible individuals, infected individuals, those undergoing treatment, and individuals in advanced AIDS stages-the model reflects key phases of infection and therapeutic interventions. Unlike conventional approaches, the proposed nonlinear transmission function, , accounts for varying infectivity levels across stages (where is the total population and denotes the effective contact rate), offering a nuanced view of how treatment efficacy ( ) and progression to AIDS ( ) shape transmission. The analytical framework combines rigorous mathematical exploration with practical insights. We derive the basic reproduction number to assess outbreak potential and employ Lyapunov theory to establish global stability conditions. Using the Schauder fixed-point theorem, we prove the existence and uniqueness of solutions, while bifurcation analysis via center manifold theory reveals critical thresholds for disease persistence or elimination. We use a computational scheme that combines the Adams-Bashforth method with an interpolation-based correction technique to ensure numerical precision and confirm theoretical results. Sensitivity analysis highlights medication accessibility and delaying the spread of AIDS as a vital control strategy by identifying ( ) and ( ) as critical parameters. The numerical simulations illustrate the predictive ability of the model, which shows how fractal-fractional order affects outbreak trajectories and long-term disease burden. The framework outperforms conventional integer order models and produces more accurate epidemiological predictions by integrating memory-dependent transmission with fractional order flexibility. These findings demonstrate the model’s value in developing targeted public health initiatives, particularly in environments with limited resources where disease monitoring and balancing treatment allocation is essential. In the end, our work provides a tool to better predict and manage the evolving challenges of HIV/AIDS by bridging the gap between theoretical mathematics and actual disease control.

• Publisher: Nature Publishing Group UK,Nature Publishing Group,Nature Portfolio
• Subject: 631/114 / 639/925 / Acquired immune deficiency syndrome / Acquired Immunodeficiency Syndrome - epidemiology / Acquired Immunodeficiency Syndrome - transmission / AIDS / Basic Reproduction Number / Disease control / Disease spread / Disease transmission / Epidemiology / Fixed Point Theorem / Fractal-Fractional Derivatives / Fractals / Health promotion / HIV / HIV Infections - epidemiology / HIV Infections - transmission / Human immunodeficiency virus / Humanities and Social Sciences / Humans / Infectivity / Lyapunov Stability / Mathematical models / Memory Dependent Dynamics / Models, Biological / Models, Theoretical / multidisciplinary / Nonlinear Transmission / Numerical analysis / Outbreaks / Public health / Science / Science (multidisciplinary) / Sensitivity analysis / Therapeutic applications