Dynamic analysis of stochastic SE2I2T model with saturation incidence rate

Tuberculosis (TB) remains one of the world’s deadliest infectious diseases, responsible for numerous deaths annually despite being preventable and treatable. In this paper, we propose and analyze a novel stochastic SE2I2T epidemiological model that accounts for two strains of tuberculosis (TB): drug-sensitive and drug-resistant under a saturation incidence rate. This model reflects realistic transmission dynamics by incorporating treatment inefficacy and environmental fluctuations modeled via Gaussian white noise. By creating an appropriate Lyapunov function, we first establish the existence of a unique global positive solution to the underlying problem. Sufficient conditions for extinction and persistence of the disease are investigated. Conditions for the existence of a stationary distribution is also found. The theoretical findings are then validated through numerical simulations using the Milstein scheme. Notably, the results indicate that higher noise intensities and reduced contact between treated individuals and drug-sensitive strains significantly increase the likelihood of disease eradication. Our threshold analysis reveals that increased environmental noise can drive the system toward extinction even in the presence of high transmission, while elevated resistance probabilities can undermine treatment success, leading to the persistence of the resistant strain.