Dynamic behavior of a stochastic SIR model with nonlinear incidence and recovery rates

Xiangming Zhao; Jianping Shi; Xiangming Zhao; Jianping Shi

doi:10.3934/math.20231278

AIMS Mathematics

2023, Volume 8, Issue 10: 25037-25059. doi: 10.3934/math.20231278

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Research article

Dynamic behavior of a stochastic SIR model with nonlinear incidence and recovery rates

Xiangming Zhao ,
Jianping Shi ^,

Department of System Science and Applied Mathematics, Kunming University of Science and Technology, Kunming 650500, China

Received: 29 June 2023 Revised: 15 August 2023 Accepted: 17 August 2023 Published: 28 August 2023
MSC : 37H05, 37H30, 60H10, 62M15

The spread of infectious diseases are inevitably affected by natural and social factors, and their evolution presents oscillations and other uncertainties. Therefore, it is of practical significance to consider stochastic noise interference in the studies of infectious disease models. In this paper, a stochastic SIR model with nonlinear incidence and recovery rate is studied. First, a unique global positive solution for any initial value of the system is proved. Second, we provide the sufficient conditions for disease extinction or persistence, and the influence of threshold $ \tilde{R_{0}} $ of the stochastic SIR model on disease state transition is analyzed. Additionally, we prove that the system has a stationary distribution under some given parameter conditions by building an appropriate stochastic Lyapunov function as well as using the equivalent condition of the Hasminskii theorem. Finally, the correctness of these theoretical results are validated by numerical simulations.
- stochastic SIR model,
- extinction,
- persistence in the mean,
- stochastic Lyapunov function,
- stationary distribution
Citation: Xiangming Zhao, Jianping Shi. Dynamic behavior of a stochastic SIR model with nonlinear incidence and recovery rates[J]. AIMS Mathematics, 2023, 8(10): 25037-25059. doi: 10.3934/math.20231278

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Abstract

The spread of infectious diseases are inevitably affected by natural and social factors, and their evolution presents oscillations and other uncertainties. Therefore, it is of practical significance to consider stochastic noise interference in the studies of infectious disease models. In this paper, a stochastic SIR model with nonlinear incidence and recovery rate is studied. First, a unique global positive solution for any initial value of the system is proved. Second, we provide the sufficient conditions for disease extinction or persistence, and the influence of threshold $ \tilde{R_{0}} $ of the stochastic SIR model on disease state transition is analyzed. Additionally, we prove that the system has a stationary distribution under some given parameter conditions by building an appropriate stochastic Lyapunov function as well as using the equivalent condition of the Hasminskii theorem. Finally, the correctness of these theoretical results are validated by numerical simulations.

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