An investigation on the dynamics of a stochastic SEIR epidemic model with psychological effects and Ornstein-Uhlenbeck perturbations

Yongle Ma; Hongxia Liu; Wencai Zhao; Yongle Ma; Hongxia Liu; Wencai Zhao

doi:10.3934/era.2026258

Electronic Research Archive

2026, Volume 34, Issue 8: 5813-5841. doi: 10.3934/era.2026258

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An investigation on the dynamics of a stochastic SEIR epidemic model with psychological effects and Ornstein-Uhlenbeck perturbations

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 08 May 2026 Revised: 26 June 2026 Accepted: 30 June 2026 Published: 08 July 2026

This work proposed a stochastic SEIR epidemic model incorporating psychological effects coupled with an Ornstein-Uhlenbeck (OU) process. The model focuses on the fact that the incidence rate of diseases is disturbed by environmental noise, and it is characterized by a mean-reverting OU process. Unlike traditional white noise, OU noise is consistent with the continuity and stability characteristics of real-world environmental disturbances. First, the existence and stability of equilibria for the corresponding deterministic model were discussed, laying the foundation for the subsequent dynamical analysis. Second, utilizing Itô's formula, the existence and uniqueness of a globally nonnegative solution for the stochastic model were proven, and the suffcient condition for disease extinction was discussed. Subsequently, by constructing appropriate Lyapunov functions and employing ergodic theory, the sufficient condition for the existence of a stationary distribution was established, namely, $ R_0^s > 1 $, which characterizes the stochastic persistence of the disease. Furthermore, by linearizing the stochastic system around the endemic equilibrium and solving the resulting covariance matrix equation, we derived an explicit expression for the probability density function characterizing the stationary distribution in the vicinity of the endemic equilibrium. Finally, we employed numerical simulations to validate the theoretical findings and to investigate the influence of fluctuation intensity on the system's dynamics.
- SEIR model,
- Ornstein-Uhlenbeck process,
- extinction,
- stationary distribution,
- probability density function
Citation: Yongle Ma, Hongxia Liu, Wencai Zhao. An investigation on the dynamics of a stochastic SEIR epidemic model with psychological effects and Ornstein-Uhlenbeck perturbations[J]. Electronic Research Archive, 2026, 34(8): 5813-5841. doi: 10.3934/era.2026258

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Abstract

This work proposed a stochastic SEIR epidemic model incorporating psychological effects coupled with an Ornstein-Uhlenbeck (OU) process. The model focuses on the fact that the incidence rate of diseases is disturbed by environmental noise, and it is characterized by a mean-reverting OU process. Unlike traditional white noise, OU noise is consistent with the continuity and stability characteristics of real-world environmental disturbances. First, the existence and stability of equilibria for the corresponding deterministic model were discussed, laying the foundation for the subsequent dynamical analysis. Second, utilizing Itô's formula, the existence and uniqueness of a globally nonnegative solution for the stochastic model were proven, and the suffcient condition for disease extinction was discussed. Subsequently, by constructing appropriate Lyapunov functions and employing ergodic theory, the sufficient condition for the existence of a stationary distribution was established, namely, $ R_0^s > 1 $, which characterizes the stochastic persistence of the disease. Furthermore, by linearizing the stochastic system around the endemic equilibrium and solving the resulting covariance matrix equation, we derived an explicit expression for the probability density function characterizing the stationary distribution in the vicinity of the endemic equilibrium. Finally, we employed numerical simulations to validate the theoretical findings and to investigate the influence of fluctuation intensity on the system's dynamics.

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