Asymptotic analysis on a new stochastic epidemic model involving isolation mechanism

In this paper, a new stochastic epidemic model is established and the dynamical behavior of its solutions is studied for this model. A deterministic epidemic model (ordinary differential equation) is first proposed by considering the isolation mechanism, and the transmission probability function is determined by a Wells-Riley model method to analyze the transmission in the quarantine. For this deterministic model, the basic reproduction number R0 is computed and it is used to determine the existence of disease-free and positive equilibria. The linearized stability of the equilibria is also discussed by analyzing the distribution of eigenvalues of the linear system. Following that, a corresponding stochastic epidemic model is further established by introducing stochastic disturbance. Then, the extinction result of the model is derived also with the help of the basic reproduction number R0s. Furthermore, by applying the theory of Markov semigroups, it is proved that the densities of the distributions of the solutions can converge to an invariant density or sweeping under certain conditions. At last, some numerical simulations are provided and discussed to illustrate the practicability of the model and the obtained theoretical results.