在线咨询
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 (3): 447-457.doi: 10.3969/j.issn.1005-3085.2024.03.005

• • 上一篇    下一篇

具有季节性反应扩散疟疾模型的全局动力学

张志雯,  白振国   

  1. 西安电子科技大学数学与统计学院，西安 710126
  • 收稿日期:2021-08-31 接受日期:2022-05-16 出版日期:2024-06-15 发布日期:2024-08-15
  • 通讯作者: 白振国 E-mail: zgbai@xidian.edu.cn
  • 基金资助:
    国家自然科学基金 (12371501).

Global Dynamics of a Reaction-diffusion Malaria Model with Seasonality

ZHANG Zhiwen,  BAI Zhenguo   

  1. School of Mathematics and Statistics, Xidian University, Xi'an 710126
  • Received:2021-08-31 Accepted:2022-05-16 Online:2024-06-15 Published:2024-08-15
  • Contact: Z. Bai. E-mail address: zgbai@xidian.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (12371501).

摘要：

疟疾是一种由疟原虫引起的传染病，它是通过成年雌性按蚊叮咬而引发的人与人之间传播。为了探讨空间异质和季节性对疟疾传播的影响，建立了一类周期的反应扩散模型。鉴于蚊子总密度趋于一个正的周期解，故对原系统的研究转而讨论其极限系统。首先定义了模型的基本再生数$\mathcal{R}_0$，然后利用单调次齐性系统理论表明了$\mathcal{R}_0$是决定极限系统全局动力学的一个阈值参数。具体地说，当$\mathcal{R}_0\leq 1$时，无病周期解是全局渐近稳定的；而当$\mathcal{R}_0> 1$时，模型存在唯一正的周期解且它是全局渐近稳定的。最后，利用链传递集理论将极限系统的动力学提升到原系统。

关键词: 疟疾, 季节性, 反应扩散模型, 基本再生数, 阈值动力学

Abstract:

Malaria is an infectious disease caused by the Plasmodium parasite and it is transmitted among humans through bites of adult female Anopheles mosquitoes. To investigate the effects of spatial heterogeneities and seasonality, we develop a periodic reaction-diffusion model. Since the total density of mosquitoes tends to be a positive periodic solution, we are focus on the limiting system associated with the original system. We first introduce the basic reproduction number $\mathcal {R}_0$ and then show that $\mathcal {R}_0$ serves as a threshold parameter in determining the global dynamics of the limiting system by employing the theory of monotone and subhomogeneous systems. More precisely, the disease-free periodic solution is globally asymptotically stable if $\mathcal {R}_0\leq 1$, and the model admits a unique positive periodic solution that is globally asymptotically stable when $\mathcal {R}_0>1$. Finally, the threshold type result for the limiting system is lifted to the original system with the help of the theory of chain transitive sets.

Key words: malaria, seasonality, reaction-diffusion model, basic reproduction number, threshold dynamics

中图分类号: