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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 ›› 2023, Vol. 40 ›› Issue (3): 413-424.doi: 10.3969/j.issn.1005-3085.2023.03.006

• • 上一篇    下一篇

具有年龄结构的流行病模型的全局稳定性

王  飞,   付丽婷   

  1. 新疆农业大学数理学院，乌鲁木齐 830052
  • 收稿日期:2021-02-18 接受日期:2022-12-15 出版日期:2023-06-15 发布日期:2023-08-15
  • 基金资助:
    新疆维吾尔自治区高校科研计划项目 (XJEDU2018Y021)；新疆农业大学大学生创新创业训练计划项目 (dxscx2023492).

Global Stability of an Epidemic Model with Age-structure

WANG Fei,  FU Liting   

  1. College of Mathematics and Physics, Xinjiang Agricultural University, Urumqi 830052
  • Received:2021-02-18 Accepted:2022-12-15 Online:2023-06-15 Published:2023-08-15
  • Supported by:
    The College Scientific Research Project of Xinjiang Uygur Autonomous Region (XJEDU2018Y021); the College Student Innovation and Entrepreneurship Training Program (dxscx2023492).

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193

摘要：

基于重新感染情形，建立了一个具有接种、潜伏和染病年龄结构的流行病模型，目的在于讨论疫苗接种年龄、潜伏年龄和感染年龄对模型全局动力学的影响，得到了模型的全局动力学由基本再生数决定。首先，利用偏微分方程沿特征线积分理论，给出了模型解的存在唯一性、连续有界性和渐近光滑性；其次，利用微分方程解的理论，得到模型的平衡点和基本再生数。再次，结合引入的基本再生数和构造的Lyapunov函数，应用LaSalle不变性原理得到结论：若基本再生数小于1，则无病平衡点全局渐近稳定；若基本再生数大于1，则无病平衡点不稳定。最后，数值模拟验证了所讨论模型的解收敛于无病平衡点。

关键词: 年龄结构, 感染年龄, 重新感染, 稳定性, 渐近光滑性

Abstract:

Based on reinfection, an epidemic model with vaccination, incubation and infection ages is presented to understand the impact of age of vaccination, age of latency and age of infection on global dynamics of the model. It is shown that the global dynamics of the model is determined by the basic regeneration number. First, the boundedness, existence, nonnegativity and asymptotic smoothness of the model are established by the theory of integrating partial differential equations along characteristic lines. Then, the steady states and basic reproduction number of the model are obtained by the theory of solutions to differential equations. By constructing a suitable Lyapunov functional and using the LaSalle invariance principle, it is verified that the disease-free steady state is globally asymptotically stable if the basic regeneration number is less than 1, and unstable if the basic regeneration number is larger than 1. Finally, by numerical simulation it is verified that the solution converges to the disease-free equilibrium point.

Key words: age-structured, infection age, reinfection, stability, asymptotic smoothness

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