Fractional-order modeling of tuberculosis and diabetes mellitus co-existence dynamics

This paper proposes a fractional-order model using the Atangana-Baleanu-Caputo derivative to study the co-dynamics of tuberculosis and diabetes mellitus among susceptible (S), TB-infected (I), DM-infected (D), and co-existence (C) populations. The model's well-posedness is established via the Banach fixed-point theorem, ensuring the uniqueness and positivity of solutions. Basic reproduction numbers (R₀^TB,R₀^DM,R₀) are derived, with values exceeding unity indicating the instability of the disease-free equilibrium and progression toward endemicity. Sensitivity analysis highlights key parameters (β₁,β₂,δ₁,δ₃,δ₅) affecting co-existence dynamics. Numerical simulation is conducted over T=365 days (1 year) with a unit step h=1 day, using the Adams-Bashforth method to reveal that lower fractional orders α∈(0,0.8] slow disease decay. The model is validated against real data over 90 days at α=0.5 using logistic growth for C(t). Results underscore the effectiveness of fractional calculus in modeling chronic co-existence and guiding control strategies.