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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 ›› 2019, Vol. 36 ›› Issue (4): 451-460.doi: 10.3969/j.issn.1005-3085.2019.04.008

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一类具有Beverton-Holt出生函数的阶段结构传染病模型的全局分析

王玉萍,  蔺小林,  李建全   

  1. 陕西科技大学文理学院，西安  710021
  • 收稿日期:2018-03-16 接受日期:2018-09-19 出版日期:2019-08-15 发布日期:2019-10-15
  • 通讯作者: 李建全 E-mail: jianq_li@263.net
  • 基金资助:
    国家自然科学基金(11771259; 11371369)；陕西省教育厅专项科研计划项目(14JK1081).

Global Analysis for an Epidemic Model with the Beverton-Holt Birth Function and Stage Structure

WANG Yu-ping,  LIN Xiao-lin,  LI Jian-quan   

  1. School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021
  • Received:2018-03-16 Accepted:2018-09-19 Online:2019-08-15 Published:2019-10-15
  • Contact: J. Li. E-mail address: jianq_li@263.net
  • Supported by:
    The National Natural Science Foundation of China (11771259; 11371369); the Special Scientific Research Project of Shaanxi Education Department (14JK1081).

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摘要： 针对一些疾病仅在成年个体间传播和成年个体的成长受到密度制约等因素，建立了一类具有幼年和成年两个阶段且疾病仅在成年个体间传播的传染病模型，其中以具有饱和性质的Beverton-Holt函数作为幼年出生函数．通过构造恰当的Lyapunov函数和定性分析，得到了模型的全局动力学性态，并确定了决定模型动力学性态的种群存活的基本再生数和疾病传播的基本再生数．所得结果表明：当种群的基本再生数不大于 1 时，种群灭绝；当种群的基本再生数大于 1 而疾病传播的基本再生数不大于 1 时，种群持续生存而疾病灭绝；当疾病传播的基本再生数大于 1 时，种群持续存活且疾病会发展成地方病．

关键词: 阶段结构, 传染病模型, 平衡点, 全局稳定性, 基本再生数

Abstract: Based on the facts that some diseases spread only among adults, and that the growth of adult individuals is density-dependent an adult epidemic model with stage structure is proposed in this paper, where it is assumed that individuals in the population consist of juveniles and adults, that the birth function of juveniles is of the Beverton-Holt type. The global stability of the model is completely investigated by constructing the appropriate Lyapunov functions and qualitative analysis, and the basic reproduction numbers of the population growth and the disease transmission, determining the dynamics of the model, is found. The obtained results suggest that, when the basic reproduction number of the population growth is not greater than unity, the population eventually dies out; when the basic reproduction number of the population growth is greater than unity, and the number of the disease transmission is less than or equal to unity, the population persists and the disease dies out; when the basic reproduction number of the disease transmission is greater than unity, the disease persists and becomes endemic as the population survives.

Key words: stage structure, epidemic model, equilibrium, global stability, basic reproduction number

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