Advanced Fractal–Fractional Modeling of HIV-Tuberculosis Co-Infection: Theory, Applications, and Integrated Intervention Strategies

This study presents a novel fractal-fractional order mathematical model for HIV/AIDS and tuberculosis co-infection dynamics, employing fractal-fractional derivatives to capture memory effects and multi-scale heterogeneities inherent in dual pathogen transmission systems. Unlike conventional integer-order models, the fractal-fractional framework integrates fractional calculus, which accounts for non-local dynamics and long-term dependencies in disease progression, with fractal geometry, which represents self-similar patterns in population mixing and co-infection transmission pathways. This dual approach effectively captures the complex, non-linear nature of HIV/TB co-infection spread, where historical infection states and heterogeneous contact patterns significantly influence current epidemiological outcomes. The model stratifies the population into multiple compartments including susceptible individuals, various stages of single infection (HIV-only and TB-only), co-infected states with primary and secondary TB, treatment compartments, and recovered populations. Key model assumptions incorporate differential infectiousness parameters for co-infected individuals, variable progression rates between infection stages, and treatment efficacy for both mono-infections and co-infections.