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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 ›› 2024, Vol. 41 ›› Issue (4): 741-756.doi: 10.3969/j.issn.1005-3085.2024.04.010

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一类时间周期SIRS扩散传染病模型的渐近传播速度

王双明1,  李尚芝2,  王  杰3   

  1. 1. 兰州财经大学金融学院 甘肃金融协同创新中心，兰州 730101
    2. 东南大学数学学院，南京 211189
    3. 兰州理工大学理学院，兰州 730050
  • 收稿日期:2023-06-04 接受日期:2023-12-15 出版日期:2024-08-15
  • 基金资助:
    国家自然科学基金 (12161051)；兰州财经大学科研项目 (Lzufe2023C-007)；兰州财经大学高等教育研究项目 (LJY202202)；兰州财经大学科研创新团队支持计划 (202002)；兰州财经大学金融统计科研融合团队 (XKKYRHTD202304).

Asymptotic Spreading Speed of a Time Periodic SIRS Reaction-diffusion Epidemic Model

WANG Shuangming1,  LI Shangzhi2,  WANG Jie3   

  1. 1. Collaborative Innovation Center of Finance in Gansu, School of Finance, Lanzhou University of Finance and Economics, Lanzhou 730101
    2. School of Mathematics, Southeast University, Nanjing 211189
    3. School of Science, Lanzhou University of Technology, Lanzhou 730050
  • Received:2023-06-04 Accepted:2023-12-15 Online:2024-08-15
  • Supported by:
    The National Natural Science Foundation of China (12161051); the Research Program of Lanzhou University of Finance and Economics (Lzufe2023C-007); the Higher Education Research Project of Lanzhou University of Finance and Economics (LJY202202); the Scientific Research and Innovation Team of Lanzhou University of Finance and Economics (202002); the Financial Statistics Research Integration Team of Lanzhou University of Finance and Economics (XKKYRHTD202304).

摘要：

利用渐近传播速度理论，研究了一类模拟时间周期变化传播环境中疾病传播的SIRS反应扩散传染病模型的渐近传播性质。区别于已有针对SI二维系统的传播速度的结果，该模型中R仓室无法解耦于整个系统，为此需要克服高维和非自治性耦合带来的困难，以证明高维系统渐近传播速度的存在性。首先，借助单调系统渐近传播速度的抽象理论和比较原理证明了染病者I仓室在疾病未入侵区域的传播性质，以此结论为基础，利用整体解结合最大值原理进一步验证了恢复者R仓室在未入侵区域具有类似的传播性质。其次，分别对I和R方程运用比较原理结合一致持久思想和最大值原理分析了其在疾病已入侵区域的一致持久性。由此，得到了划分这两个变化区域的渐近传播速度阈值，即证明了整个系统渐近传播速度的存在性。最后，利用数值方法进一步模拟了时间周期传播环境下疾病已入侵区域更具体的传播动态。

关键词: 时间周期, SIRS传染病模型, 渐近传播速度, 比较原理, 一致持久性

Abstract:

The asymptotic spread properties are investigated for an SIRS reaction-diffusion infectious disease model simulating the disease propagation in temporally periodic environment by using the theory of asymptotic speed of spread. Different from two-dimensional SI type systems, the R component in current model cannot be decoupled from the entire system. This means to overcome the difficulties caused by the coupling of high-dimension and non-autonomy to establish the existence of asymptotic speed of spread for current high-dimensional system. Firstly, the asymptotic spread characteristics of I component is obtained in disease-free region by using the abstract theory of asymptotic spreading speed of monotone systems and comparison principle. Further，the asymptotic spread property of R component in disease-free region is verified through the use of the entire solution and the maximum principle. Secondly, the propagation properties of the invaded region are analyzed by applying comparison principle and uniformly persistent idea to I and R equations, respectively. As a result, the value of asymptotic spreading speed by which we divide the two regions is obtained. In addition, numerical experiments are tested for more specific propagation dynamics of the invaded area in temporally periodic environment.

Key words: time-periodic, SIRS reaction-diffusion epidemic model, asymptotic spreading speed, principle of comparison, uniform persistence

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