Asymptotic behavior and numerical simulation of a stochastic multi-group SEIR epidemic model with infinite distributed delays

Die Sun; Yingjia Guo; Die Sun; Yingjia Guo

doi:10.3934/math.2026553

AIMS Mathematics

2026, Volume 11, Issue 5: 13412-13448. doi: 10.3934/math.2026553

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Research article

Asymptotic behavior and numerical simulation of a stochastic multi-group SEIR epidemic model with infinite distributed delays

Die Sun ,
Yingjia Guo ^,

School of Mathematics and Statistics, Beihua University, Jilin, Jilin 132000, China

Received: 17 February 2026 Revised: 25 April 2026 Accepted: 08 May 2026 Published: 14 May 2026
MSC : 34K20, 60G51, 60H10, 92D30

In this paper, we studied the asymptotic behavior of a stochastic multi-group Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model with infinite distributed delays and Lévy jumps. First, by the methods of Lyapunov functions, Itô's formula, and the theory of stopping times, we proved the existence and uniqueness of the global positive solution to the stochastic delayed system. Furthermore, by using appropriate Lyapunov functions, graph theory and stochastic analysis, we established the asymptotic dynamical behaviors around the disease-free equilibrium $ P_{0} $ and the endemic equilibrium $ P^{*} $ of the deterministic system, respectively. It was shown that if the threshold $ R_{0} < 1 $, the solution of the stochastic delayed system oscillates around the disease-free equilibrium $ P_{0} $; while if $ R_{0} > 1 $, the solution fluctuates around the endemic equilibrium $ P^{*} $. Finally, numerical simulations were performed to intuitively analyze the impact of Lévy noise on the dynamical behavior of the stochastic delayed system.
- stochastic multi-group epidemic model,
- infinite distributed delays,
- Lévy jumps,
- asymptotic behavior,
- numerical simulation
Citation: Die Sun, Yingjia Guo. Asymptotic behavior and numerical simulation of a stochastic multi-group SEIR epidemic model with infinite distributed delays[J]. AIMS Mathematics, 2026, 11(5): 13412-13448. doi: 10.3934/math.2026553

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Abstract

In this paper, we studied the asymptotic behavior of a stochastic multi-group Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model with infinite distributed delays and Lévy jumps. First, by the methods of Lyapunov functions, Itô's formula, and the theory of stopping times, we proved the existence and uniqueness of the global positive solution to the stochastic delayed system. Furthermore, by using appropriate Lyapunov functions, graph theory and stochastic analysis, we established the asymptotic dynamical behaviors around the disease-free equilibrium $ P_{0} $ and the endemic equilibrium $ P^{*} $ of the deterministic system, respectively. It was shown that if the threshold $ R_{0} < 1 $, the solution of the stochastic delayed system oscillates around the disease-free equilibrium $ P_{0} $; while if $ R_{0} > 1 $, the solution fluctuates around the endemic equilibrium $ P^{*} $. Finally, numerical simulations were performed to intuitively analyze the impact of Lévy noise on the dynamical behavior of the stochastic delayed system.

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