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In this paper, the stochastic SIS epidemic model with vaccination under regime switching is further investigated. A new threshold R s 0 which is different from that given in [22] is established. A new technique to deal with the nonlinear incidence and vaccination for stochastic epidemic model under regime switching is proposed. When R s 0 > 0, the existence of a unique stationary distribution and the ergodic property are obtained by constructing a new stochastic Lyapunov function with Markov switching. The corresponding result which is acquired in [22] is improved and extended.

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In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

<abstract><p>In this paper, a class of multi-group SEIQR models with random perturbation in computer network is investigated. The existence and uniqueness of global positive solution with any positive initial value are obtained. The sufficient conditions on the asymptotic behavior of solutions around the disease-free equilibrium and endemic equilibrium of the corresponding deterministic model are established. Furthermore, the existence and uniqueness of stationary distribution are also obtained. Lastly, the analytical results are illustrated by the numerical simulations.</p></abstract>