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# On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times

• As it is well known, the dynamics of the stochastic SIRS epidemic model with mass action is governed by a threshold $\mathcal{R_S}$. If $\mathcal{R_S}<1$ the disease dies out from the population, while if $\mathcal{R_S}> 1$ the disease persists. However, when $\mathcal{R_S} = 1$, classical techniques used to study the asymptotic behaviour do not work any more. In this paper, we give answer to this open problem by using a new approach involving some adequate stopping times. Our results show that if $\mathcal{R_S} = 1$ then, small noises promote extinction while the large one promote persistence. So, it is exactly the opposite role of the noises in case when $\mathcal{R_S}\neq 1$.

Mathematics Subject Classification: 92B05, 60G51, 60H30, 60G57.

 Citation: • • Figure 1.  simulation results of $S(t),I(t),R(t)$ respectively for the SDE model (1.1) with different initial conditions $(S_{0},I_{0},R_{0})$ and the parameters: $\mu = 0.1$, $\beta = 0.62$, $\lambda = 0.5$, $\gamma = 0.5$, $\sigma = 0.2$. Here $\mathcal{R_S} = 1$ and $\sigma^{2}<\beta$. For the different values of initial conditions the infection-free equilibrium $E_0(1,0,0)$ is asymptotically stable in probability

Figure 2.  simulation results of $S(t),I(t),R(t)$ respectively for the SDE model (1.1) with different initial conditions $(S_{0},I_{0},R_{0})$ and the parameters: $\mu = 0.1$, $\beta = 0.7802$, $\lambda = 0.2$, $\gamma = 0.2$, $\sigma = 0.98$. Here $\mathcal{R_S} = 1$ and $\beta<\sigma^{2}$. For the different values of initial conditions, respectively, $S(t)$, $I(t)$ and $R(t)$ rises to or above the level $\xi_s = 0.6247$, $\xi_i = 0.2252$ and $\xi_r = 0.1501$ infinitely often with probability one

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