Threshold dynamics and density function in a stochastic SEIT tuberculosis system with exogenous reinfection

In this paper, we investigate a susceptible–exposed–infected–treated (SEIT) tuberculosis transmission model with exogenous reinfection in the stochastic environment, aiming to deeply analyze the impact of white noise on the transmission dynamics of tuberculosis. The threshold values [Formula: see text], [Formula: see text] and [Formula: see text] are defined and the criteria for the extinction and persistence of the disease are established. That is, when [Formula: see text], the disease persists in the mean and any positive solution has a unique ergodic stationary distribution, indicating that the disease will become endemic, while when [Formula: see text] or [Formula: see text], the disease tends to extinction with probability one. Furthermore, the expression of an approximate normal probability density function near the quasi-positive equilibrium is calculated. A novel calculation method is proposed which differs from the existing results. The numerical examples and simulations are presented not only to illustrate the theoretical results, to compare the difference between the stochastic model and the deterministic model, but also to discuss some open problems.