A stochastic differential equation SIS epidemic model with regime switching

Siyang Cai; Yongmei Cai; Xuerong Mao

doi:10.3934/dcdsb.2020317

Article Contents

2021, Volume 26, Issue 9: 4887-4905. Doi: 10.3934/dcdsb.2020317

This issue Previous Article Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model Next Article A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

A stochastic differential equation SIS epidemic model with regime switching

1.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
2.
School of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, 315100, China

^* Corresponding author: yongmei.cai@nottingham.edu.cn
^* Corresponding author: yongmei.cai@nottingham.edu.cn

Received: February 2020

Revised: September 2020

Early access: November 2020

Published: September 2021

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Abstract

In this paper, we combined the previous model in [2] with Gray et al.'s work in 2012 [8] to add telegraph noise by using Markovian switching to generate a stochastic SIS epidemic model with regime switching. Similarly, threshold value for extinction and persistence are then given and proved, followed by explanation on the stationary distribution, where the $ M $-matrix theory elaborated in [20] is fully applied. Computer simulations are clearly illustrated with different sets of parameters, which support our theoretical results. Compared to our previous work in 2019 [2, 3], our threshold value are given based on the overall behaviour of the solution but not separately specified in every state of the Markov chain.

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Mathematics Subject Classification: Primary: 60H30, 92D25; Secondary: 37H05.

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Figure 1. Extinction with $ I(0) = 90 $

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Figure 2. Extinction with $ I(0) = 10 $

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Figure 3. Persistence Case 1 with $ I(0) = 90 $

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Figure 4. Persistence Case 1 with $ I(0) = 10 $

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Figure 5. Persistence Case 2 with $ I(0) = 90 $

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Figure 6. Persistence Case 2 with $ I(0) = 10 $

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Figure 7. Stationary Distribution Case 1 with $ I(0) = 90 $

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Figure 8. Stationary Distribution Case 1 with $ I(0) = 10 $

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Figure 9. Stationary Distribution Case 2 with $ I(0) = 90 $

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Figure 10. Stationary Distribution Case 2 with $ I(0) = 10 $

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