Threshold behavior in a stochastic SIRS epidemic model with a logarithmic Ornstein-Uhlenbeck process

Yanan Zhao; Chang Liu; Yanan Zhao; Chang Liu

doi:10.3934/era.2026191

Electronic Research Archive

2026, Volume 34, Issue 6: 4290-4324. doi: 10.3934/era.2026191

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Threshold behavior in a stochastic SIRS epidemic model with a logarithmic Ornstein-Uhlenbeck process

Yanan Zhao ^,,
Chang Liu

School of Mathematics and Statistics, Changchun University, Changchun 130021, China

Received: 23 February 2026 Revised: 26 April 2026 Accepted: 06 May 2026 Published: 20 May 2026

This study investigates the threshold behavior of a population-varying stochastic susceptible-infectious-recovered-susceptible (SIRS) model driven by a logarithmic Ornstein-Uhlenbeck process. Introducing the logarithmic Ornstein-Uhlenbeck process to account for random environmental fluctuations enhances the biological significance of the model. By applying the Itô stochastic integral, we construct a suitable Lyapunov function, proving the existence and uniqueness of the global positive solution of the model, thereby ensuring biological feasibility. Then, a critical threshold parameter $ R_0^s $ is derived: If $ R_0^s > 1 $, the system admits a unique invariant probability measure; if $ R_0^s < 1 $, the infection dies out almost surely around the disease-free equilibrium. Furthermore, near the quasi-equilibrium point, the invariant probability density admits a local Gaussian approximation and converges weakly to a normal distribution as the environmental noise tends to zero. Numerical simulations in MATLAB illustrate the theoretical results, further reveal the sensitivity of the stochastic threshold to the key parameters, and confirm that population variation influences the threshold structure and long-term infection level.
- threshold,
- SIRS model,
- Markov semigroup,
- stationary distribution,
- Ornstein-Uhlenbeck process
Citation: Yanan Zhao, Chang Liu. Threshold behavior in a stochastic SIRS epidemic model with a logarithmic Ornstein-Uhlenbeck process[J]. Electronic Research Archive, 2026, 34(6): 4290-4324. doi: 10.3934/era.2026191

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Abstract

This study investigates the threshold behavior of a population-varying stochastic susceptible-infectious-recovered-susceptible (SIRS) model driven by a logarithmic Ornstein-Uhlenbeck process. Introducing the logarithmic Ornstein-Uhlenbeck process to account for random environmental fluctuations enhances the biological significance of the model. By applying the Itô stochastic integral, we construct a suitable Lyapunov function, proving the existence and uniqueness of the global positive solution of the model, thereby ensuring biological feasibility. Then, a critical threshold parameter $ R_0^s $ is derived: If $ R_0^s > 1 $, the system admits a unique invariant probability measure; if $ R_0^s < 1 $, the infection dies out almost surely around the disease-free equilibrium. Furthermore, near the quasi-equilibrium point, the invariant probability density admits a local Gaussian approximation and converges weakly to a normal distribution as the environmental noise tends to zero. Numerical simulations in MATLAB illustrate the theoretical results, further reveal the sensitivity of the stochastic threshold to the key parameters, and confirm that population variation influences the threshold structure and long-term infection level.

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