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In this paper we study the Kernack - MacKendrick model under telegraph noise.The telegraph noise switches at random between two SIRS models. We give outconditions for the persistence of the disease and the stability of a diseasefree equilibrium. We show that the asymptotic behavior highly depends on thevalue of a threshold $\lambda$ which is calculated from the intensities ofswitching between environmental states, the total size of the population aswell as the parameters of both SIRS systems. According to the value of$\lambda$, the system can globally tend towards an endemic case or a diseasefree case. The aim of this work is also to describe completely the omega-limitset of all positive solutions to the model. Moreover, the attraction of theomega-limit set and the stationary distribution of solutions will be pointedout.

Dynamical Behavior of a stochastic SIRS epidemic model