Bifurcation and Pattern Formation of a Susceptible-Infected-Recovered Model with Diffusion

YUAN Hailong; CHONG Xiaoyan; DING Siyu

doi:10.1051/wujns/2026315481

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Bifurcation and Pattern Formation of a Susceptible-Infected-Recovered Model with Diffusion

Mathematics | 更新时间：2026-10-29

- Bifurcation and Pattern Formation of a Susceptible-Infected-Recovered Model with Diffusion
- Wuhan University Journal of Natural Sciences Vol. 31, Issue 5, Pages: 481-499(2026)
- 作者机构：
  
  陕西科技大学数学与数据科学学院，陕西西安 710021
- 作者简介：
  
  YUAN Hailong, male, Ph.D., Lecturer, research direction: biomathematics. E-mail: yuanhailong@sust.edu.cn
- 基金信息：
- DOI：10.1051/wujns/2026315481
  CLC： O175.26
- Received：29 September 2025，
  
  Published：24 October 2026
- 稿件说明：
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袁海龙,种晓艳,丁思雨.一类具有扩散项的SIR传染病模型的分歧与斑图动力学分析[J].武汉大学学报自然科学英文版,2026,31(05):481-499.

YUAN Hailong, CHONG Xiaoyan, DING Siyu. Bifurcation and Pattern Formation of a Susceptible-Infected-Recovered Model with Diffusion[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 481-499.
袁海龙,种晓艳,丁思雨.一类具有扩散项的SIR传染病模型的分歧与斑图动力学分析[J].武汉大学学报自然科学英文版,2026,31(05):481-499. DOI： 10.1051/wujns/2026315481.

YUAN Hailong, CHONG Xiaoyan, DING Siyu. Bifurcation and Pattern Formation of a Susceptible-Infected-Recovered Model with Diffusion[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 481-499. DOI： 10.1051/wujns/2026315481.

摘要

本文研究一类带扩散的易感-感染-康复（SIR）传染病模型。该模型引入心理影响系数

和

，采用饱和发生率刻画疾病带来的心理效应。首先，推导正解的先验估计，利用能量方法证明非

常数正解的不存在性。其次，分析正平衡点的图灵失稳，并借助Leray-Schauder度理论证明非常数正解的存在性。随后，利用分支理论刻画稳态分支的局部结构与全局结构。通过空间分解和隐函数定理，构造二重特征值处稳态分支的局部结构并推导出判定分支方向的判别准则。最后，数值模拟对理论结果进行补充与验证。

Abstract

This paper investigates a susceptible-infected-recovered (SIR) epidemic model with diffusion. The model incorporates psychological impact coefficients

and

and employs a saturated incidence rate to characterize the psychological effects induced by the disease. First

a priori estimates for positive solutions are derived

and the nonexistence of nonconstant positive solutions is proved by the energy method. Second

the Turing instability of the positive equilibrium is analyzed

and the existence of nonconstant positive solutions is established by means of Leray‑Schauder degree theory. Subsequently

bifurcation theory is utilized to characterize the local and global structures of steady state bifurcations. Through spatial decomposition and the implicit function theorem

the local structure of steady state bifurcations at a double eigenvalue is constructed

and a criterion for judging the direction of bifurcation is derived. Finally

numerical simulations are performed to supplement and verify the theoretical results.

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Keywords

references

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