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

Adel Settati; Aadil Lahrouz; Mustapha El Jarroudi; Mohamed El Fatini; Kai Wang

doi:10.3934/dcdsb.2020012

Article Contents

2020, Volume 25, Issue 5: 1985-1997. Doi: 10.3934/dcdsb.2020012

This issue Previous Article Traveling waves for nonlocal Lotka-Volterra competition systems Next Article Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment

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

1.
Department of Mathematics, Faculty of Sciences and Technics, Abdelmalek Essaâdi University, BP 416, Tangier, Morocco
2.
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133 Kénitra, Morocco
3.
Department of Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China

^*Correponding author (melfatini@gmail.com)
^*Correponding author (melfatini@gmail.com)

Received: April 2019

Revised: July 2019

Early access: December 2019

Published: May 2020

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Abstract

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 $.

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

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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

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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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