具有年龄结构的麻疹传染病模型的稳定性分析

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1950-1968.

具有年龄结构的麻疹传染病模型的稳定性分析

孙丹丹^1,*(),李盈科¹(),滕志东²,张太雷³

¹ 新疆农业大学数理学院乌鲁木齐 830052
² 新疆大学数学与系统科学学院乌鲁木齐 830046
³ 长安大学理学院西安 710064

收稿日期:2020-07-03 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 孙丹丹 E-mail:dandan_1990@126.com;307129154@qq.com
作者简介:李盈科, E-mail: 307129154@qq.com
基金资助:
中国博士后科学基金资助项目(2020M683714XB);新疆维吾尔自治区自然科学计划（自然科学基金）面上项目(2021D01A98);陕西省自然科学基础研究计划项目(2021JM-445)

Analysis of the Stability for Measles Epidemic Model with Age-Structured

Dandan Sun^1,*(),Yingke Li¹(),Zhidong Teng²,Tailei Zhang³

¹ School of Mathematics and Physics, Xinjiang Agriculture University, Urumqi 830052
² School of Mathematics and System Science, Xinjiang University, Urumqi 830046
³ School of Sciences, Changan University, Xi an 710064

Received:2020-07-03 Online:2021-12-26 Published:2021-12-02
Contact: Dandan Sun E-mail:dandan_1990@126.com;307129154@qq.com
Supported by:
the China Postdoctoral Science Foundation(2020M683714XB);the NSF of Xinjiang(2021D01A98);the Natural Science Basic Research Plan in Shaanxi Province(2021JM-445)

摘要/Abstract

摘要：

该文研究一类具有年龄结构的SVEIR麻疹模型.首先将模型化为所谓的Volterra型积分方程，得到了模型解的适定性，包括非负性、有界性、渐近光滑性等.其次得到了模型的平衡点和基本再生数${\cal R}_0$，并证明了当基本再生数${\mathcal R}_{0}>1$时疾病的一致持续性.进一步通过分析特征方程和构造适当的Lyapunov函数得到了：若${\cal R}_{0}<1$，则模型仅存在全局渐近稳定的无病平衡点；若${\cal R}_{0}>1$，则无病平衡点不稳定，地方病平衡点存在且全局渐近稳定.这些理论结果应用在关于全国麻疹传染病数据的趋势分析方面.

关键词: 年龄结构麻疹模型, 解的适定性, 基本再生数, 一致持续性, 平衡点的稳定性

Abstract:

In this paper, a kind of SVEIR measles epidemic model with age structure is established. Firstly, the model is transformed into Volterra integral equation and the well-possdness of solutions of the model is obtained, including non-negativity, boundedness, asymptotic smoothness, etc. Then the equilibria and the basic reproduction number ${{\cal R}}_{0}$ of the model is derived, and it is proved that the epidemic is uniformly persistent when ${{\cal R}}_{0}>1$. Further by analyzing the characteristic equations and selecting suitable Lyapunov functions, we get the model only has the disease-free equilibrium that is globally asymptotically stable if ${{\cal R}}_{0}<1$; if ${{\cal R}}_{0}>1$, the disease-free equilibrium is unstable, the endemic disease equilibrium exist and is globally asymptotically stable. These main theoretical results are applied in the analysis of the trend in data on measles infectious diseases across the country.

Key words: Age-structured measles model, Well-posedness of solutions, Basic reproduction number, Uniform persistence, Stability of equilibrium

中图分类号:

O175.1

孙丹丹,李盈科,滕志东,张太雷. 具有年龄结构的麻疹传染病模型的稳定性分析[J]. 数学物理学报, 2021, 41(6): 1950-1968.

Dandan Sun,Yingke Li,Zhidong Teng,Tailei Zhang. Analysis of the Stability for Measles Epidemic Model with Age-Structured[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1950-1968.

图/表 4

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