Threshold of a stochastic siqs epidemic model with isolation

The aim of this paper is to give sufficient conditions, very close to the necessary one, to classify the stochastic permanence of SIQS epidemic model with isolation via a threshold value (R) over cap. Precisely, we show that if (R) over cap < 1 then the stochastic SIQS system goes to the disease free case in sense the density of infected I-z (t) and quarantined Q(z) (t) classes extincts to 0 at exponential rate and the density of susceptible class S-z (t) converges almost surely at exponential rate to the solution of boundary equation. In the case <(R)over cap> > 1, the model is permanent. We show the existence of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to this invariant measure. Some numerical examples are also provided to illustrate our findings.