Analysis on dynamics and tipping points for a stochastic SIM model involving media influence

In this paper, a stochastic SIM epidemic model involving media influences is proposed and the dynamical behavior of its solutions and the problem of tipping points are investigated. The stochastic epidemic model is first established by taking into account the media influence. The existence, extinction and persistence of this stochastic model are then discussed respectively by applying Lyapunov functions method and strong law of large numbers for martingales. Existence of steady-state distribution of this system is studied as well by the method of Lyapunov exponents. Finally, to show the effects of the stochastic noise-induced transient dynamics, the problems of tipping probability, tipping time and the confidence ellipse function of this stochastic model are discussed, and the related numerical simulations are also conducted to illustrate the obtained results.