Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation

This paper characterizes the qualitative dynamics of a stochastic SIRS epidemic model. The study shows that the dynamics of the model are determined by a certain threshold quantity RSRS expressed in terms of the model parameters and the intensity of the noise. If RS<1RS<1, the disease will be eliminated from the community; whereas an epidemic occurs if RS>1RS>1. Our results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.