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