Physics-informed deep learning modelling of tuberculosis dynamics with waning immunity in diabetic populations: evidence from Saudi Arabia

Tuberculosis (TB) and diabetes mellitus (DM) pose a growing public health challenge in Saudi Arabia, where diabetes-induced immune dysfunction increases TB susceptibility and post-treatment relapse. This study develops a four-compartment TB-DM mathematical model with a saturating incidence function to capture transmission dynamics under high infectious pressure and waning immunity. A Physics-Informed Neural Network (PINN) framework is proposed for parameter estimation by integrating observational data with the governing nonlinear dynamical system, enabling biologically consistent and data-efficient learning. The basic reproduction number is derived using the next-generation matrix approach, a closed-form endemic equilibrium is obtained, and local stability of the disease-free and endemic equilibria is established. The PINN accurately recovers model parameters and demonstrates strong predictive performance using synthetic and Saudi tuberculosis surveillance data, including a direct, reproducible comparison against a classical least-squares baseline and a standard, non-physics-informed neural network. Sensitivity and intervention analyses identify transmission intensity, treatment-related recovery, early case detection, and integrated diabetes management as the key levers for reducing TB burden, providing a data-driven framework to support evidence-based TB-DM control policy in Saudi Arabia.