| Citation: |
Wenjie Cao, Tao Pan. DYNAMICS OF AN IMPULSIVE STOCHASTIC SIR EPIDEMIC MODEL WITH SATURATED INCIDENCE RATE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1396-1415. doi: 10.11948/20190214
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DYNAMICS OF AN IMPULSIVE STOCHASTIC SIR EPIDEMIC MODEL WITH SATURATED INCIDENCE RATE
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Abstract
In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations. -
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References
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Wenjie Cao, Tao Pan. DYNAMICS OF AN IMPULSIVE STOCHASTIC SIR EPIDEMIC MODEL WITH SATURATED INCIDENCE RATE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1396-1415. doi: 10.11948/20190214
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Figure 1. The pathway simulations of
$ S(t) $ and$ I(t) $ for the deterministic system (1.1) and the stochastic system (1.2) with initial value$ (S(0),I(0)) = (0.7,0.2) $ . (a) Solutions to deterministic system (1.1) with$ R_0 = 1.0738 $ ; (b) Solutions to stochastic system (1.2) with$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.02 $ ,$ R_0^s = 1.0732 $ , ; (c) Solutions to stochastic system (1.2) with$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.3 $ ,$ R_0^s = 0.9515 $ ; (d) Solutions to stochastic system (1.2) with$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.55 $ ,$ R_0^s = 0.7498 $ -
Figure 2. Simulations of the boundary periodic solution and global attraction for the stochastic system (1.2) when disease goes extinct. (a) the positive boundary periodic solution to stochastic system (1.2) with
$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.03 $ and initial value$ (S(0),I(0)) = (0.7,0.2) $ ; (b) the positive boundary periodic solution to stochastic system (1.2) with$ \sigma_1 = 0.015 $ ,$ \sigma_2 = 0.55 $ and initial value$ (S(0),I(0)) = (0.7,0.2) $ ; (c) global attraction for the stochastic system (1.2) with$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.3 $ and different initial values$ (S(0),I(0)) = (0.6,0.2) $ ,$ (S(0),I(0)) = (0.7,0.2) $ ,$ (S(0),I(0)) = (0.8,0.2) $ ; (d) global attraction for the stochastic system (1.2) with$ \sigma_1 = 0.015 $ ,$ \sigma_2 = 0.55 $ and different initial values$ (S(0),I(0)) = (0.6,0.2) $ ,$ (S(0),I(0)) = (0.7,0.2) $ ,$ (S(0),I(0)) = (0.8,0.2) $ -
Figure 3. Simulations of phase trajectories of positive periodic solutions
$ (S(t),I(t)) $ . (a) the deterministic system (1.1); (b) the stochastic system (1.2) with with$ \sigma_1 = 0.02 $ ,$ \sigma_2 = 0.02 $ ; (c) the stochastic system (1.2) with with$ \sigma_1 = 0.01 $ ,$ \sigma_2 = 0.01 $