Ergodic stationary distribution and extinction of stochastic pertussis model with immune and Markov switching

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô's lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value ${\mathcal{R}}_0^e$ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when ${\mathcal{R}}_0^e>1$, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.