Stability Analysis of HIV Control System in African Regions Using Caputo Fractional Dynamic Equations on Time Scales

Human Immunodeficiency Virus (HIV) remains a critical public health challenge in Africa with heterogeneous epidemic dynamics. This study models HIV control in Nigeria, South Africa, Sudan, and Uganda using Caputo fractional dynamic equations on time scales to capture non-local infection and treatment dynamics. Stability is analysed via eigenvalue and Lyapunov methods, with the basic reproduction number \((R_0)\) quantifying control. South Africa exhibits robust stability \((R_0=0.540)\), Uganda maintains steady control \((R_0=0.684)\), while Sudan and Nigeria show borderline and unstable dynamics due to insufficient diagnosis and treatment coverage. Scaling effective interventions is essential for sustainable epidemic control.