A mathematical reaction–diffusion model describing the connection between smokers and tuberculosis patients

In this paper, we introduce a new mathematical model that incorporates both temporal and spatial factors to characterize the relationship between the smoking population and tuberculosis (TB) cases. We validate the model’s relevance by demonstrating the existence and positivity of the solutions. Furthermore, we propose three control measures aimed at reducing TB and smoking transmission, focusing on limiting contact between smokers and non-smokers, improving awareness of the health risks associated with smoking, and ensuring timely and effective medical treatment for TB patients. Mathematical analysis is employed to demonstrate the existence and characteristics of optimal controls for these strategies. The theoretical results are verified using numerical simulations, which also confirm the effectiveness of the suggested control strategies. Our results underline the importance of considering spatiotemporal dynamics in disease modeling and control, and they offer crucial insights for public health initiatives aimed at reducing the prevalence of smoking and TB.