Stochastic modeling and asymptotic analysis of tuberculosis with environmental noise

Tuberculosis remains a significant public health challenge, with transmission occurring both directly between individuals and indirectly via environmental reservoirs. To capture the impact of environmental and demographic fluctuations, we propose a stochastic TB model incorporating white noise into both human and environmental compartments. We establish the global existence and positivity of solutions, and derive sufficient conditions for stochastic persistence and extinction of the disease. When the persistence condition is satisfied, the system admits a unique ergodic stationary distribution; otherwise, all infected compartments tend to zero almost surely. Numerical simulations based on the Euler-Maruyama method support our analytical results. This study highlights the role of stochasticity in tuberculosis dynamics and extends classical deterministic thresholds, offering a more realistic framework for analyzing long-term disease behavior under uncertainty.