Dynamics of stochastic SIQS model based on nonlinear incidence: disease extinction and stationary distribution under degenerate diffusion

Shuantu He; Zhongyi Xiang; Shuantu He; Zhongyi Xiang

doi:10.3934/era.2025193

Electronic Research Archive

2025, Volume 33, Issue 7: 4259-4283. doi: 10.3934/era.2025193

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Research article

Dynamics of stochastic SIQS model based on nonlinear incidence: disease extinction and stationary distribution under degenerate diffusion

Shuantu He ,
Zhongyi Xiang ^,

School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China

Received: 29 April 2025 Revised: 24 June 2025 Accepted: 08 July 2025 Published: 21 July 2025

This paper investigated the dynamical behaviors of the SIQS (susceptible, infected, isolated, and again susceptible) infectious disease model with nonlinear incidence rate and degenerate diffusion in a stochastic environment. By introducing nonlinear contagion rate, the model was able to more realistically reflect the complexity of real-world disease transmission, including the effects of social behavior, medical resource constraints, and public health interventions. It was proved that the infectious disease will be extinct when $ R_0^s < 1 $. Furthermore, by utilizing Markov semigroup theory, we obtained that there existed stationary distribution for the system when $ R_0^s > 1 $. Numerical simulations were conducted by introducing three different forms of nonlinear incidence rates (standard incidence, non-monotonic incidence, Beddington-DeAngelis incidence) to verify our results.
- stochastic SIQS model,
- nonlinear incidence rate,
- degenerate diffusion,
- extinction,
- stationary distribution
Citation: Shuantu He, Zhongyi Xiang. Dynamics of stochastic SIQS model based on nonlinear incidence: disease extinction and stationary distribution under degenerate diffusion[J]. Electronic Research Archive, 2025, 33(7): 4259-4283. doi: 10.3934/era.2025193

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Abstract

This paper investigated the dynamical behaviors of the SIQS (susceptible, infected, isolated, and again susceptible) infectious disease model with nonlinear incidence rate and degenerate diffusion in a stochastic environment. By introducing nonlinear contagion rate, the model was able to more realistically reflect the complexity of real-world disease transmission, including the effects of social behavior, medical resource constraints, and public health interventions. It was proved that the infectious disease will be extinct when $ R_0^s < 1 $. Furthermore, by utilizing Markov semigroup theory, we obtained that there existed stationary distribution for the system when $ R_0^s > 1 $. Numerical simulations were conducted by introducing three different forms of nonlinear incidence rates (standard incidence, non-monotonic incidence, Beddington-DeAngelis incidence) to verify our results.

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