Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

Shangzhi Li; Shangjiang Guo

doi:10.3934/dcdsb.2020201

Article Contents

2021, Volume 26, Issue 5: 2693-2719. Doi: 10.3934/dcdsb.2020201

This issue Previous Article Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks Next Article Properties of basins of attraction for planar discrete cooperative maps

Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

Shangzhi Li and
Shangjiang Guo^,

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

^* Corresponding author: Shangjiang Guo
^* Corresponding author: Shangjiang Guo

Received: January 2020

Revised: April 2020

Early access: June 2020

Published: May 2021

The second author is supported by NSFC (Grant No. 11671123)

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Abstract

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold $ \lambda $, it is proved that if $ \lambda>0 $, the disease is permanent and there is a stationary distribution. And when $ \lambda<0 $, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

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Mathematics Subject Classification: Primary: 34K20; Secondary: 92C60.

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Figure 1. Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = b = c = 0.1 $

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Figure 2. Stationary distribution of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = b = c = 0.1 $: (a) the graph of the relative frequency densities of $ S $ and $ I $; (b) the joint density distribution of solution $ (S,I) $

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Figure 3. Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 1 $, $ b = 0.1 $, $ c = 0.1 $

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Figure 4. Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 2 $, $ b = 0.1 $, $ c = 0.1 $

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Figure 5. Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 1 $, $ c = 0.1 $

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Figure 6. Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.31 $

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Figure 7. Numerical simulations of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.31 $: (a) Trajectories of the solution $ S(t) $ of (26) and the solution $ \varphi $ of (10) shows the convergence of $ S $ to a boundary distribution; (b) The dynamics of $ S(t) $ and $ I(t) $ in time average

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Figure 8. Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.1 $

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Figure 9. Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 1 $, $ b = 0.1 $, $ c = 0.1 $

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Figure 10. Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 1 $, $ c = 0.1 $

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Figure 11. Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 1 $

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