Fractional-Order Analysis of HIV–TB Co-Infection Model Using Caputo, Atangana–Baleanu–Caputo, and Caputo–Fabrizio Operators with TB Treatment

The global burden of HIV/AIDS-tuberculosis co-infection presents significant public health challenges, particularly in regions with varying levels of treatment accessibility. This study develops and analyzes a novel mathematical model examining the dynamics of HIV/AIDS-tuberculosis co-infection transmission across treated and untreated human populations, while incorporating disease progression through multiple compartments. We investigate the system through three distinct arbitrary-order derivative operators: the Caputo derivative with power law, the Caputo-Fabrizio derivative with non-singular kernel, and the Atangana-Baleanu derivative incorporating the Mittag-Leffler function. The model explicitly considered the impact of treatment accessibility on disease transmission rates, recovery patterns, and intervention effectiveness. Through numerical simulations, we demonstrate that dually infected populations experience significantly higher infection peaks (approximately 8.2 million cases) compared to singly infected populations (approximately 5.8 million cases). Our analysis reveals how varying fractional orders ( Ф = 0.95, 0.85, and 0.75) influence the temporal memory effects and overall disease dynamics. The model parameters, estimated from current epidemiological data and literature, provide insights into the critical role of treatment accessibility in disease mitigation. Surface plots analyzing the basic reproduction number R₀ against various parameters demonstrate the sensitivity of disease spread to contact rates, treatment rates, and accessibility status. These findings emphasize the importance of integrating treatment accessibility measures into public health interventions for effective HIV/AIDS-tuberculosis co-infection control, particularly in vulnerable populations.