Advanced machine learning techniques for mathematical modeling and predicting tuberculosis control in diabetic patients using Physics-Informed Neural Networks

Patients with diabetes mellitus (DM) are at a significantly higher risk of developing tuberculosis (TB) due to impaired immune responses, making the TB–diabetes co-morbidity a major global public health concern. Motivated by this interaction, the present study proposes a mathematical model to describe the transmission dynamics of TB in a diabetic population, explicitly incorporating disease progression and complication stages. The model accounts for immune and environmental influences through an exponential scaling factor, enabling a biologically meaningful representation of susceptibility variations. A comprehensive qualitative analysis is performed, including the derivation of the basic reproduction number, as well as stability and sensitivity analyses to identify key parameters governing disease spread. To further enhance the modeling framework, a Physics-Informed Neural Network (PINN) approach is employed for solving the system and estimating unknown epidemiological parameters from data. The PINN framework integrates the governing differential equations with observational data, enabling accurate recovery of model parameters and reliable prediction of disease dynamics even in data-limited settings. Numerical results demonstrate that the proposed approach effectively captures the interplay between diabetes progression and TB transmission, while sensitivity analysis highlights the most influential factors for disease control. The findings suggest that improving treatment effectiveness, reducing complication rates, and enhancing environmental conditions can significantly mitigate TB prevalence among diabetic individuals. This study provides a unified modeling and computational framework that contributes to a deeper understanding of TB–diabetes interactions and supports the development of effective public health strategies.