Dynamics and approximation of positive solution of the stochastic SIS model affected by air pollutants

Qi Zhou; Huaimin Yuan; Qimin Zhang; Qi Zhou; Huaimin Yuan; Qimin Zhang

doi:10.3934/mbe.2022207

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4481-4505. doi: 10.3934/mbe.2022207

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Research article

Dynamics and approximation of positive solution of the stochastic SIS model affected by air pollutants

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
School of Information Engineering, Ningxia University, Yinchuan 750021, China

Academic Editor: Hao Wang

Received: 09 December 2021 Revised: 07 January 2022 Accepted: 10 February 2022 Published: 03 March 2022

In this paper, we develop a stochastic susceptible-infective-susceptible (SIS) model, in which the transmission coefficient is a function of air quality index (AQI). By using Markov semigroup theory, the existence of kernel operator is obtained. Then, the sufficient conditions that guarantee the stationary distribution and extinction are given by Foguel alternative, Khasminsk$\check{\rm l}$ function and Itô formula. Next, a positivity-preserving numerical method is used to approximate the stochastic SIS model, meanwhile for all $ p > 0 $, we show that the algorithm has the $ p $th-moment convergence rate. Finally, numerical simulations are carried out to illustrate the corresponding theoretical results.
- stochastic SIS model,
- stationary distribution,
- numerical approximation,
- Markov semigroup,
- positive solution
Citation: Qi Zhou, Huaimin Yuan, Qimin Zhang. Dynamics and approximation of positive solution of the stochastic SIS model affected by air pollutants[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4481-4505. doi: 10.3934/mbe.2022207

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Abstract

In this paper, we develop a stochastic susceptible-infective-susceptible (SIS) model, in which the transmission coefficient is a function of air quality index (AQI). By using Markov semigroup theory, the existence of kernel operator is obtained. Then, the sufficient conditions that guarantee the stationary distribution and extinction are given by Foguel alternative, Khasminsk$\check{\rm l}$ function and Itô formula. Next, a positivity-preserving numerical method is used to approximate the stochastic SIS model, meanwhile for all $ p > 0 $, we show that the algorithm has the $ p $th-moment convergence rate. Finally, numerical simulations are carried out to illustrate the corresponding theoretical results.

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