Abstract: We considered a class of jump-diffusion SEIR epidemic models and determined the threshold for the disease to occur or not. Firstly, the threshold A was theoretically determined through a simplified limiting system. Secondly, by using comparison principle, we proved that when A<0, the disease became exponentially extinct. Finally, by constructing an appropriate Lyapunov function, we proved that when A>0, the transition probability function of the solution of a stochastic system converged to the unique invariant measure, with the convergence speed of arbitary order polynomial convergence.

Key words: threshold, ergodicity, extinction, invariant measure