| Citation: |
Yanan Zhao, Xiaoying Zhang, Donal O'Regan. THRESHOLD DYNAMICS IN A STOCHASTIC SIRS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2096-2110. doi: 10.11948/20180041
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THRESHOLD DYNAMICS IN A STOCHASTIC SIRS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE
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Abstract
We discuss the dynamic of a stochastic Susceptible-Infectious-Recovered-Susceptible (SIRS) epidemic model with nonlinear incidence rate.The crucial threshold $\tilde{R}_0$ is identified and this will determine the extinction and persistence of the epidemic when the noise is small. We also discuss the asymptotic behavior of the stochastic model around the endemic equilibrium of the corresponding deterministic system. When the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. Finally, these results are illustrated by computer simulations.-
Keywords:
- SIRS epidemic model /
- nonlinear incidence rate /
- extinction /
- persistence /
- threshold
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References
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Yanan Zhao, Xiaoying Zhang, Donal O'Regan. THRESHOLD DYNAMICS IN A STOCHASTIC SIRS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2096-2110. doi: 10.11948/20180041
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Figure 1. Computer simulation of the path
$ S(t) $ ,$ I(t) $ ,$ R(t) $ for model (1.3) and its corresponding deterministic model (1.1) for parameter values$ h = 0.5, \; \Lambda = 0.2, \; \beta = 0.4, \; \mu = 0.1, \; \alpha = 0.2, \; \gamma = 0.3, \; \varepsilon = 0.2, \; \delta = 0.1 $ and$ \sigma = 0.4 $ , such that$ \tilde{R}_0<1 $ , but$ R_0>1 $ . -
Figure 2. Computer simulation of the path
$ S(t) $ ,$ I(t) $ ,$ R(t) $ for the SDE SIRS model (1.3) and its corresponding deterministic model (1.1) for parameter values$ \Lambda = 0.2, \; \beta = 0.6, \; \mu = 0.1, \; \alpha = 0.2, \; \gamma = 0.3, \; \varepsilon = 0.2, \; \delta = 0.1, \; h = 0.5 $ and$ \sigma = 0.1 $ (left plot), and differing values of$ \sigma = 0.3 $ (middle plot) and$ \sigma = 0.7 $ (right plot).