Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching

Combining a deterministic SEQIHRS model proposed by Sahu et al. (2015) and the present mathematical modelling of media coverage effect (2020), a hybrid stochastic SEQIHR model, perturbed by both nonlinear white noise and colored noise, is formulated and studied for the transmission dynamics of an infectious disease with media coverage, quarantine strategies and pre-existing immunity in a community. First, we prove that the stochastic model possesses a unique global positive solution. Second, by means of the basic reproduction number R0 of the corresponding deterministic model, two relevant critical values which include R¯0 and R0C are derived. Next, we obtain the disease extinction under R0<1 and R¯0<1. Moreover, we further establish the sufficient condition R0C>1 for the existence and uniqueness of an ergodic stationary distribution of the stochastic model, which means the infectious disease will be prevailing and persistent in a community. Finally, several numerical simulations are performed to validate the above theoretical results. Besides, the impact of media coverage and nonlinear hybrid noises on the dynamical behavior of the stochastic model are studied at the end of this paper.