Fractal-Fractional Co-Infection Model for Hepatitis B and Diphtheria with Mittag-Leffler Kernel: Mathematical Analysis and Optimal Control Strategies

Hepatitis B virus (HBV) and Diphtheria continue to impose substantial co-morbidity burdens in low- and middle-income countries where inadequate vaccination coverage and healthcare access create conditions for simultaneous infection. Understanding their coupled transmission dynamics is essential for designing effective integrated public health responses. This study presents, to our knowledge, the first application of a fractal-fractional derivative operator with Mittag-Leffler kernel to a fifteen-compartment Hepatitis BDiphtheria co-infection model incorporating explicit bacterial load dynamics. The model simultaneously represents sexual, vertical, and environmental HBV transmission routes alongside respiratory diphtheria transmission, extending beyond prior single-disease fractional models. The total human population and bacterial reservoir are partitioned into fifteen compartments (S, V, E D , I D , T D , R D , E HB , I aHB , I cHB , T HB , R HB , E C , I C , T C , R C ) and one bacterial compartment (B). The fractal-fractional derivative operator with Mittag- Leffler kernel (Atangana-Baleanu in Caputo sense) is applied to the governing system. Existence and uniqueness of solutions are established via the Banach and Krasnoselskii fixed-point theorems. The basic reproduction number R 0 is derived via the next-generation matrix method. Local and global asymptotic stability of the disease-free and endemic equilibria are analyzed. Model parameters are estimated by fitting to annual HBV case data from China (2004–2021, CDC) and weekly Diphtheria case data from Nigeria (2023 outbreak, NCDC). Elevated fractal dimension α and fractional order β jointly prolong infection peaks and delay epidemic resolution, with co-infected compartments IC reaching up to 2,500 individuals under high transmission scenarios. Increasing treatment rates and vaccination rates substantially reduce prevalence across all compartments. The basic reproduction numbers R 0H and R 0D (ranging from 0.001 to 0.005 across simulated parameter scenarios) confirm that integrated intervention strategies can drive the system below the epidemic threshold. These results demonstrate that fractal-fractional co-infection modelling, validated against real outbreak data, provides actionable insights for multi-disease intervention design that cannot be obtained from integer-order or single-disease models alone.