具有“健康带菌者”传染模型的稳定性分析

发布时间：2018-06-07 04:04

  本文选题：病毒 + 地方病平衡点　； 参考：《新疆大学》2015年硕士论文

【摘要】：传染病的防制是关系到人类健康的国计民生的重大问题,对疾病流行规律的定量研究是防制工作的重要依据,长期以来,人类与各种传染病进行了不屈不挠的斗争,使得天花被消灭,麻风病,脊髓灰质炎被彻底消灭也指日可待,多种抗生素的问世,如1922年英格兰细菌学家亚历山大.弗莱明(Alexander.Fleming,1881-1955)发现了青霉素拯救了成千上万的人生命,然而,世界卫生组织(WHO)发表的世界卫生报告表明传染病依然是人类的第一杀手.本文的第一部分是引言,介绍了一些具有健康带菌者的传染病模型及研究背景,目的和意义.第二部分是预备知识,包括一些基本定理和论文要用的定理,引理,准则等,第三部分通过不同的三类数学模型来说明健康带菌者对传染病模型的稳定性影响,模型的主要内容概述如下首先研究具有Bedding-De Angelis发生率的健康带菌者模型,引入第二加性复合矩阵,利用求谱半径的方法得到系统的基本再生数,证明了解的正性和地方平衡点的存在,并说明了若基本再生数小于1,无病平衡点全局稳定的;若基本再生数大于1,则疾病是不稳定的,我们利用Bendixson判据方法分析持续带毒平衡点的全局稳定性,其次考虑带一个时滞模拟传染病,我们分离特征方程的实部与虚部,利用反证法来论断特征根实部的符号,在讨论平衡点的全局稳定性时构造了一个无限大正定的Liapunov泛函,得到相应平衡点在所讨论的区域内全局渐近稳定的,建立合适的阈值R0,得到当R01时,系统的无病平衡点是全局渐近稳定的,当R01时,得出地方病平衡点是全局渐近稳定的,最后考虑带两个时滞模拟传染病的潜伏期,患者对疾病的感染期,首先我们分离特征方程的实部与虚部,利用反证法来论断特征根实部的符号,在讨论平衡点的全局稳定性时构造了一个无限大正定的Liapunov泛函,得到相应平衡点在所讨论的区域内全局渐近稳定的,最后对两个时滞对模型的影响作了数值模拟,验证了结论的正确性.
[Abstract]:The prevention and control of infectious diseases is an important issue related to the national economy and the people's livelihood of human health. The quantitative study on the law of disease prevalence is an important basis for the prevention and control work. For a long time, human beings have been engaged in an indomitable struggle against various infectious diseases. The eradication of smallpox, leprosy and poliomyelitis is imminent, and many antibiotics, such as the English bacteriologist Alexander, came out in 1922. Alexander.Flemingn (1881-1955) found penicillin has saved thousands of lives. However, the World Health report published by the World Health Organization (WHO) shows that infectious diseases are still the number one killer of human beings. The first part of this paper is an introduction, which introduces some infectious disease models with healthy carriers and their research background, purpose and significance. The second part is the preparatory knowledge, including some basic theorems and the theorems, Lemma and criteria to be used in the paper. The third part explains the influence of healthy carriers on the stability of infectious disease models through three different mathematical models. The main contents of the model are summarized as follows: firstly, the healthy carrier model with the incidence of Bedding-De Angelis is studied, and the second additive compound matrix is introduced, and the basic regeneration number of the system is obtained by the method of calculating the spectral radius. The existence of positive solutions and local equilibrium points is proved, and it is shown that the disease-free equilibrium points are globally stable if the number of fundamental reproductions is less than 1, and if the number of basic regenerations is greater than 1, the disease is unstable. We use the Bendixson criterion method to analyze the global stability of the persistent poison equilibrium point, and then consider the simulated infectious disease with a delay. We separate the real part from the imaginary part of the characteristic equation, and use the counter-proof method to judge the symbol of the real part of the characteristic root. In this paper, we construct an infinite positive definite Liapunov functional when we discuss the global stability of the equilibrium point. We obtain the globally asymptotically stable equilibrium point in the region under discussion. The disease-free equilibrium of the system is globally asymptotically stable. When R01, the endemic equilibrium is globally asymptotically stable. Finally, the incubation period of the simulated infectious disease with two delays and the infection period of the patient to the disease are considered. First of all, we separate the real part and the imaginary part of the characteristic equation, use the counter-proof method to judge the sign of the real part of the characteristic root, and construct an infinite positive definite Liapunov functional when we discuss the global stability of the equilibrium point. The globally asymptotically stable equilibrium points are obtained. Finally, the effects of two delays on the model are numerically simulated, and the correctness of the conclusions is verified.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
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本文编号：1989683