Threshold of a stochastic SIQS epidemic model with isolation. (English) Zbl 1504.37097
Summary: 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 \(\widehat{R}\). Precisely, we show that if \(\widehat{R}<1\) 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 \(\widehat{R}>1\), 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.
MSC:
| 37N25 | Dynamical systems in biology |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 92D25 | Population dynamics (general) |
| 92D30 | Epidemiology |
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