一类依靠媒介传染的虫媒传染病模型的数学研究与分析

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英文篇名：Mathematic Study and Analysis for a Kind of Vector-Borne Transmission Disease 作者：张颖 ; 赵静 英文作者：ZHANG Ying;ZHAO Jing;Department of Basic Teaching, Shandong Water Conservancy Vocational College;School of Economics and Management, Chang'an University; 关键词：虫媒传染病模型 ; 基本再生数 ; 无病平衡点 ; 地方病平衡点 ; 数值模拟 英文关键词：vector-borne diseases;;Basic reproduction number;;disease-free equilibrium;;endemic equilibrium;;numerical simulation 中文刊名：SSJS 英文刊名：Mathematics in Practice and Theory 机构：山东水利职业学院基础教学部;长安大学经济与管理学院; 出版日期：2018-10-23 出版单位：数学的实践与认识 年：2018 期：v.48 语种：中文; 页：SSJS201820036 页数：9 CN：20 ISSN：11-2018/O1 分类号：272-280

摘要

针对一类依靠媒介传染的虫媒传染病,建立相应的具有非线性发生率的虫媒传染病模型,定性和定量研究该类虫媒传染病的传播规律.基于此,首先根据微分方程与传染病模型的理论分析与数学推导,推出该模型的基本再生数R_0的代数表达式,并得到无病平衡点和地方病平衡点存在的充分条件;其次,利用Hurwitz判据证明了地方病平衡点的稳定性.最后将具体的结论总结如下:当R_0<1时,模型存在惟一渐进稳定的无病平衡点,此时疾病将随着时间的推移趋于消失;当R_0> 1时,模型不存在无病平衡点,但其存在唯一渐进稳定的地方病平衡点,此时疾病将在人群和媒介中持续传播,即意味着疾病将会在某个地区或国家持续流行下去.
        Aiming at a kind of vector-borne transmission disease, The model of vector-borne transmission diseases with non-linear incidence was established, and the propagation law of the vector-borne transmission diseases was studied qualitatively and quantitatively. Firstly,based on the theoretical analysis and mathematical derivation of infectious disease model and differential equation, the expression of the basic reproduction number R_0 of the model is introduced and the sufficient conditions for the existence of the disease-free equilibrium and the endemic equilibrium are obtained. Secondly, the Hurwitz criterion is used to prove the stability of the equilibrium point of endemic disease. Finally, the concrete conclusion is summarized as follows: When R_0 < 1, the model has the only progressive stable diseasefree equilibrium point, the disease will disappear with the passage of time; when R_0 > 1,the model does not exist disease-free balance, and there is 'only the gradual stability of the endemic disease balance point. At this point the disease will continue to spread in the crowd and the media, which means that the disease will continue to prevail in a region or country.

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