Threshold dynamics of a stochastic SIRS epidemic model with transfer from infected individuals to susceptible individuals and log-normal Ornstein-Uhlenbeck process

Miaomiao Gao; Yanhui Jiang; Daqing Jiang; Miaomiao Gao; Yanhui Jiang; Daqing Jiang

doi:10.3934/era.2025133

Electronic Research Archive

2025, Volume 33, Issue 5: 3037-3064. doi: 10.3934/era.2025133

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Research article

Threshold dynamics of a stochastic SIRS epidemic model with transfer from infected individuals to susceptible individuals and log-normal Ornstein-Uhlenbeck process

1.
Business college, Qingdao University, Qingdao 266071, China
2.
School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
3.
College of Science, China University of Petroleum (East China), Qingdao 266580, China

Received: 31 January 2025 Revised: 26 April 2025 Accepted: 28 April 2025 Published: 19 May 2025

This paper focused on a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with standard incidence and transfer from infected individuals to susceptible individuals. We assumed that the incidence rate satisfied the log-normal Ornstein-Uhlenbeck process. First, by using stochastic Lyapunov analysis method, the sufficient condition for the existence of stationary distribution was obtained. After that, we established the sufficient criteria for the extinction of the infectious disease. It was worth noting that the dynamical behavior of the considered model was governed by a threshold. In addition, we derived the exact expression of probability density function near the positive equilibrium point of the corresponding deterministic system. Finally, some numerical simulations were carried out to confirm theoretical results.
- SIRS model,
- log-normal Ornstein-Uhlenbeck process,
- stationary distribution,
- density function,
- extinction
Citation: Miaomiao Gao, Yanhui Jiang, Daqing Jiang. Threshold dynamics of a stochastic SIRS epidemic model with transfer from infected individuals to susceptible individuals and log-normal Ornstein-Uhlenbeck process[J]. Electronic Research Archive, 2025, 33(5): 3037-3064. doi: 10.3934/era.2025133

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Abstract

This paper focused on a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with standard incidence and transfer from infected individuals to susceptible individuals. We assumed that the incidence rate satisfied the log-normal Ornstein-Uhlenbeck process. First, by using stochastic Lyapunov analysis method, the sufficient condition for the existence of stationary distribution was obtained. After that, we established the sufficient criteria for the extinction of the infectious disease. It was worth noting that the dynamical behavior of the considered model was governed by a threshold. In addition, we derived the exact expression of probability density function near the positive equilibrium point of the corresponding deterministic system. Finally, some numerical simulations were carried out to confirm theoretical results.

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