Sharp Threshold for the Dynamics of a SIRS Epidemic Model with General Awareness- Induced Incidence and Four Independent Brownian Motions

Abstract

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R{0}^{S} which is a sharp threshold for the dynamics: If R_{0}^{S} < 1 , the disease will die out; if R_{0}^{S} > 1 , the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.