Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein–Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results.

This paper is concerned with a stochastic Mycobacterium tuberculosis model, which is perturbed by both white noise and colored noise. First, we prove that the stochastic model has a unique global positive solution. Second, we derive an important condition $R_0^{\ast }$ depending on environmental noise for this stochastic model. We construct an appropriate Lyapunov function, and show that the model possesses a unique ergodic stationary distribution when $R_0^{\ast }<0$, in other words, it indicates the long-term persistence of the disease. Finally, we investigate the related conditions of extinction.