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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 ›› 2016, Vol. 33 ›› Issue (2): 175-183.doi: 10.3969/j.issn.1005-3085.2016.02.007

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一类具有非线性发生率的SEIR传染病模型的全局稳定性分析

宋修朝1， 李建全1， 杨亚莉1,2   

  1. 1- 空军工程大学理学院，西安  710051
    2- 陕西师范大学数学与信息科学学院，西安 710062
  • 收稿日期:2014-05-20 接受日期:2014-11-27 出版日期:2016-04-15 发布日期:2016-06-15
  • 基金资助:
    国家自然科学基金 (11371369; 11301320)；陕西省自然科学基金 (2012JQ1019).

Global Stability of an SEIR Epidemic Model with Nonlinear Incidence

SONG Xiu-chao1,  LI Jian-quan1,  YANG Ya-li1,2   

  1. 1- School of Science, Air Force Engineering University, Xi'an 710051
    2- School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062
  • Received:2014-05-20 Accepted:2014-11-27 Online:2016-04-15 Published:2016-06-15
  • Supported by:
    The National Natural Science Foundation of China (11371369; 11301320); the Natural Science Foundation of Shaanxi Province (2012JQ1019).

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摘要： 本文对一类具有非线性发生率的SEIR传染病模型进行了研究．确定了决定疾病灭绝或持续存在的阈值-基本再生数，并分析了模型的平衡点的存在性；通过构造恰当的Lyapunov函数，运用LaSalle不变性原理证明了当基本再生数小于或等于1时，无病平衡点是全局渐进稳定的；利用Lyapunov直接方法证明了当基本再生数大于1时，地方病平衡点是全局渐进稳定的．最后，将发生率具体化用数值模拟验证了所得理论分析结果的正确性．

关键词: 传染病模型, 非线性发生率, 基本再生数, 平衡点, 全局稳定性

Abstract:

In this paper, an SEIR epidemic model with nonlinear incidence is investigated. By applying the next generation matrix, the basic reproduction number determining whether the disease dies is found, and the existence of the equilibria of the model is discussed; according to the suitable Lyapunov function and the LaSalle invariance principle, it is proved that the disease free equilibrium is globally asymptotically stable as the basic reproduction number is less than or equal to unity; by means of the Lyapunov direct method, it is testified that the endemic equilibrium is globally asymptotically stable as the basic reproduction number is greater than unity. Finally, the theoretical results obtained here are verified by numerical simulations for the SEIR model with a specific incidence.

Key words: epidemic model, nonlinear incidence, basic reproduction number, equilibrium, globally stability

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