Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching

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.