| 两种传染病模型的动力过程分析 |
| Alternative Title | The Dynamical Process Analysis of Two Epidemic Models
|
| 杨允海 |
| Thesis Advisor | 李自珍
|
| 2008-05-23
|
| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
|
| Keyword | 传染病模型
非线性接触率
平衡点
全局稳定
传染率
时滞
|
| Abstract | 据世界卫生组织报告，传染病仍是人类的第一大杀手，人类正面临着各种传染病的长期威胁。由于对传染病的研究不能以试验形式，因此，对各类传染病发病机理、流行规律、预测预报就更需要理论性分析、定量分析、模拟仿真来进行，上述离不开建立数学模型。因次通过建立传染病模型，来研究传染病的传播过程和预测传染病发展行势，是传染病研究的一个重要方法，也是数学知识应用的一个重要领域。传染病动力学就是针对传染病的流行规律进行理论性定量研究
的一种方法。
第一章着重介绍了传染病的背景，传染病数学模型中的几个重要概念，用图框简单介绍了传染病模型的一些重要形式，使大家对传染病数学模型有一个大致了解。
第二章是本文的预备知识，主要把本文用到的一些重要概念、定理做了一下简要描述。
第三章对一类具有非线性接触率SIRS的传染病模型进行重点分析。在传染病动力学中，传染率是不可缺少的项。在经典的传染病模型中，大量使用双线性和标准型传染率，这两种传染率只是两种极端的情形，不能合理解释传染病传播过程中的一些复杂现象。本文提出较双线性和标准传染率更一般的非线性传染率，并将之引入到SIRS模中, 通过构造Liapunov等办法，得到模型无病平衡点和地方病平衡点存在的阈值以及全局稳定性的完整结果，并对结果进行了计算机模拟。依据所得结果，提出减少种群的外界输入，提高染病者治疗率或采取捕杀染病者以加大染病死亡率来控制此传染病的策略。
第四章本章建立了一类具有时滞的SIR传染病模型。
本章得到控制此类传染病的阈值R0。证明了当R0 < 1时，无病平衡点是局部渐进稳定且是全局吸引的；当R0 > 1时，无病平衡点不稳定，此时存在唯一的地方病平衡点，且它是全局稳定的。由以上结论提出防控措施为：加大染病者因染病的死亡率和治愈率。若患者禽畜，可采取捕杀的办法；若是人类，可加快研究治疗药物以提高治愈率。 |
| Other Abstract | In recent years, there is a great development after the pioneering work of Lotka and
Volterra focusing on population dynamics, and Kermack and McKendrick majoring in
epidemic dynamics, those help us to understand the importance in exploring the resource
and reducing disease transmission. In real word, species does not exist alone, and they
spread the disease, so we should merge these two areas of research, but little attention has
been paid so far. In this paper, we studied the disease transmission in the predator-prey
system, the mathematical models were established, through mathematical analysis and
numerical simulation , we obtain the following main results:
In chapter one ,we mainly introduced the background of epidemics and some important
concepts of mathematical epidemic models. To understand the models best ,we use
some figures to show the form of epidemic models simply.
In chapter two, some pretest knowledge is introduced,such as important concepts
theories which are used in our paper.
In chapter three ，the paper deals with a kind of SIRS epidemic model with constant
input and nonlinear incidence rate. The incidence rate is a very important item in
the epidemic models.In the classical epidemic disease models,bilinear incidence rate and
standard incidence rate have been frequently used.Actually ,they are two extreme types
and can not explain better the complex phenomena of disease transmission.In this paper
,bilinear incidence rate and standard incidence rate are improved into nonlinear type as
¯SI
1+aI and was introduced into the SIRS models.By constructing Liapunov function the
threshold of existence of endemic equilibrium and disease-free equilibrium are obtained
. According the globally stable results obtained of the four models, the controlling epidemic
disease strategy that including the immigration and increasing the recovery rate
and death rate is put forward.
In chapter four ,we mainly derive and study one type of time-delay SIR epidemic
model.The basic reproduction number (R0) is obtained in this chapter .It is proved that
the disease-free equilibrium is unstable,and endemic equilibrium is globally asymptotically
stable when (R0) is not greater than one; the disease-free equilibrium and endemic
equilibrium is globally asymptotically stable when (R0) is greater than one.According the
results above,the suggestion following are put forward:Raising the death rate and recover rate of the infective individuals.If the infective
individu... |
| URL | 查看原文
|
| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224877
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
杨允海. 两种传染病模型的动力过程分析[D]. 兰州. 兰州大学,2008.
|
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