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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (2): 199-228.doi: 10.3969/j.issn.1005-3085.2026.02.001cstr: 32411.14.cjem.CN61-1269/O1.2026.02.001

• •    下一篇

带有扩散的非单调发病率的SIR流行病模型的分歧和斑图形成

袁海龙1,2,  马亚妮1   

  1. 1. 陕西科技大学数学与数据科学学院，西安 710021

    2. 西安交通大学数学与统计学院，西安 710049

  • 收稿日期:2023-06-02 接受日期:2024-06-27 出版日期:2026-04-15 发布日期:2026-06-15
  • 基金资助:
    国家自然科学基金 (11901370)；中国博士后科学基金 (2019M653578)；陕西省自然科学基础研究计划 (2019JQ-516)；陕西省教育厅自然科学基金 (19JK0142).

Bifurcation and Pattern Formation in a SIR Epidemic Model with Diffusion and Nonmonotonic Incidence Rate

YUAN Hailong1,2,   MA Yani1   

  1. 1. School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi'an 710021
    2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2023-06-02 Accepted:2024-06-27 Online:2026-04-15 Published:2026-06-15
  • Supported by:
    The National Natural Science Foundation of China (11901370); the China Postdoctoral Science Foundation (2019M653578); the Natural Science Basic Research Plan in Shannxi Province (2019JQ-516); the Natural Science Foundation of the Education Department of Shaanxi Province (19JK0142).

摘要：

鉴于SIR流行病学模型在传染病研究领域中的重要作用，研究了齐次Neumann边界条件下时空SIR模型的动力学分析。首先，利用线性化分析方法分析常数平衡解的Turing不稳定性，并应用最大值原理给出非常数正解的先验估计。其次，利用能量估计和Leray-Schauder度理论分别证明非常数正解的不存在性和存在性。此外，应用分歧理论，建立单重特征值处的局部和全局分歧，得到确定分歧方向的条件，并采用空间分解和隐函数定理讨论双重特征值处的局部分歧。最后，通过数值模拟对理论分析结果进行验证。

关键词: 分歧, Turing不稳定性, 斑图形成, 数值模拟

Abstract:

Considering the significant role of the SIR epidemic model in infectious disease research field, the dynamical analysis of a spatiotemporal SIR epidemiological model under homogeneous Neumann boundary conditions is investigated. First, the Turing instability of the constant equilibrium solution is analyzed via the linearization method, and a priori estimate of the positive nonconstant solution is derived by applying the maximum principle. Second, the nonexistence and existence of positive nonconstant solutions are proved respectively by using energy estimation and the Leray-Schauder degree theory. Furthermore, local and global bifurcations at simple eigenvalues are established by applying bifurcation theory, and the conditions for determining the bifurcation direction are obtained. The local bifurcation at double eigenvalues is discussed by employing spatial decomposition and the implicit function theorem. Finally, the theoretical analysis results are verified through numerical simulations.

Key words: bifurcation, Turing instability, pattern formation, numerical simulation

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