某些传染病系统的建模、分析与控制研究

发布时间：2018-05-01 22:05

本文选题：传染病系统 + 参数辨识　；参考：《北京信息控制研究所》2005年博士论文

【摘要】： 利用动力学的方法建立传染病传播的数学模型，研究某种传染病在某一地区是否会蔓延下去而成为该地区的流行病，或者这种传染病是否会最终消除，已经成为传染病学和数学相结合的一个重要的具有理论和现实意义的研究课题，它有助于对传染病将来的发展趋势进行预测，有利于传染病的预防与控制。这篇博士学位论文就是将传染病学的有关知识和数学理论结合起来，对传染病的流行规律及相关问题进行探究。本文第一章为绪论，对传染病数学模型的国内外研究状况和最新进展作了综述，并从中引出本文所要研究的问题，并简要叙述了本文所得到的结果。在本文第二章，根据仓室建模的思想建立了由常微分方程组和偏微分方程组描述的SIR传染病模型，证明了平衡解的存在唯一性及稳定性。特别地，我们还研究了系统(2-1-1)的参数辨识问题，建立了疫情控制区域。在本文第三章中，根据仓室建模的思想建立了由常微分方程组描述的SEIR传染病模型，通过分析特征方程特征根的分布证明了平衡解的存在性和稳定性。在证明该模型疾病不消亡的平衡解的局部稳定性时利用了Routh-Hurwits判别法。第三章中还建立了由偏微分方程组描述的SEIR传染病模型，利用再生函数，给出了平衡解的存在性和稳定性条件。考虑有些疾病染病时间较长，其流行规律、传染能力、治愈效果都依赖染病期，基于这一点，在第四章对这种含染病期的传染病模型(P)进行了讨论，通过先验估计的方法，得到了系统(P)正则广义解的唯一性，应用偏微分-积分方程的理论，证明了该模型解的稳定性。在本文第五章，我们假设某地区的人口受两种病症的困扰，这两种疾病有排斥性，将总人口分为四类：易感人群，第一类染病人群，第二类染病人群，康复人群，根据仓室建模的思想建立了两种传染病同时流行的数学模型(5-2-1)，利用小扰动的方法讨论了系统疾病消亡的平衡解的存在性及全局稳定性，用微分-积分方程的知识证明了系统的传染病不消亡的平衡解的存在性及局部渐近稳定性。本文的主要创新之处在于： 1、研究了系统(2-4-1)如下两个最优接种问题：(1)在满足一定效果要求的条件下，追求最低费用；(2)在不超过一定费用的前提下，追求接种效果最佳。借助泛函分析的知识证明了上述两个最优接种问题最优接种策略的存在性。 2、对于给定的目标泛函(?)(ψ)，研究了系统(3-4-1)的最优接种问题，得到了最优接种控制满足的最优性组。 3、对传染率函数进行了改进。虽然各种各样的年龄结构的流行病模型已被很多作者讨论过，但是大多数作者讨论SIS模型、SIR模型、SEIR模型和MSEIR模型时，，都假定感染率函数和染病人群成正比，即假设感染率函数 λ(t)=∫_0~Aβ(a)I(a,t)da (I(a,t)为染病人群的年龄密度函数) 但事实上对大多数传染病而言，感染率函数应该和染病人群与潜伏期人群占总人口的比率成正比。本文将传染率函数改进为 λ(t)=∫_0~Aβ(a)(E(a,t)+I(a,t)/P_∞(a))da 这里E(a,t)、I(a,t)分别为潜伏期人群和染病人群的年龄密度函数。 4、研究了含染病期的传染病模型和两种传染病同时流行的传染病模型，证明了系统平衡解的存在性和稳定性。本文综合应用非线性泛函分析、微分方程、积分方程以及分布参数系统控制论等理论和方法，获得了一批重要的理论成果。这批成果既具有较高的学术价值，也为传染病系统的实际研究提供了理论依据。
[Abstract]:The mathematical model of the spread of infectious diseases is established by using the method of dynamics to study whether a certain infectious disease will spread in a certain area and become an epidemic in this area, or whether the infectious disease will eventually be eliminated or not, has become an important theoretical and practical research subject of the combination of infectious diseases and mathematics. It helps to predict the future development trend of infectious diseases and is conducive to the prevention and control of infectious diseases.
This doctoral dissertation combines the knowledge of infectious diseases with mathematical theory, and explores the epidemic rules and related problems of infectious diseases.
In the first chapter, the first chapter is an introduction to the research status and recent progress of the mathematical model of infectious diseases at home and abroad, and the problems to be studied in this paper are drawn out, and the results obtained in this paper are briefly described.
In the second chapter, the SIR infectious disease model, which is described by the ordinary differential equations and partial differential equations, is established according to the idea of the chamber modeling. The existence and uniqueness and stability of the equilibrium solution are proved. In particular, we also study the parameter identification of the system (2-1-1) and establish the epidemic control area.
In the third chapter, the SEIR infectious disease model described by the ordinary differential equations is established according to the idea of the chamber modeling, and the existence and stability of the equilibrium solution are proved by the analysis of the distribution of characteristic roots of the characteristic equations. The Routh-Hurwits discrimination method is used to prove the local stability of the equilibrium solution of the model disease without extinction. Third The SEIR epidemic model described by partial differential equations is also established in this chapter. The existence and stability conditions of the equilibrium solution are given by using the reproducing function.
Considering that some diseases have been infected for a long time, their epidemic law, infectious ability and cure effect depend on the infected period. Based on this, the fourth chapter is discussed in the fourth chapter of the infectious disease model of the infected period, and the uniqueness of the generalized solution of the system (P) is obtained by means of prior estimation, and the theory of partial differential integral equation is applied to prove that the theory of partial differential integral equation is applied. The stability of the solution of the model is clear.
In the fifth chapter of this article, we assume that the population of a certain area is plagued by two diseases, the two diseases are excluded, and the total population is divided into four categories: susceptible population, the first type of infected people, second kinds of infected people, and the rehabilitation crowd, and the mathematical model of two infectious diseases (5-2-1) is established according to the idea of room modeling, and the use of small disturbance is used. The dynamic method is used to discuss the existence and global stability of the equilibrium solution of the disappearance of the system disease. The existence and local asymptotic stability of the equilibrium solution of the system's contagious disease is proved by the knowledge of differential integral equation.
The main innovations of this paper are as follows:
1, we studied the following two optimal inoculation problems of the system (2-4-1) as follows: (1) pursuing the minimum cost under the conditions of satisfying the requirements of a certain effect; (2) the pursuit of the best inoculation effect on the premise of not exceeding a certain cost. The existence of the optimal inoculation strategy for the above two optimal inoculation problems was proved by the knowledge of functional analysis.
2, for the given objective functional (()), we study the optimal vaccination problem of the system (3-4-1) and obtain the optimal set of optimal vaccination control.
3, the infection rate function is improved. Although a variety of age structure epidemic models have been discussed by many authors, most authors discuss the SIS model, the SIR model, the SEIR model and the MSEIR model, all assume that the infection rate function is proportional to the infected population, that is, the infection rate function is assumed.
T (= _0~A) (a) I (a, t) DA (I (a, t) is the age density function of the infected population).
But in fact, for most infectious diseases, the infection rate function should be proportional to the ratio of the infected people to the population in the latent period. This paper improves the infection rate function as a function of the rate of infection.
Lambda (T) = Da _0~A beta (a) (E (a, t) +I (a, t) /P_ infinity (a)) Da
Here, E (a, t), I (a, t) are age density functions of incubation period and infected population respectively.
4, we studied infectious disease models with infectious diseases and two epidemic models with epidemic diseases simultaneously, and proved the existence and stability of the equilibrium solutions.
In this paper, the theory and methods of nonlinear functional analysis, differential equation, integral equation and distributed parameter system control theory are used in this paper. A lot of important theoretical results have been obtained. The results not only have high academic value, but also provide a theoretical basis for the practical research of infectious diseases system.

【学位授予单位】：北京信息控制研究所
【学位级别】：博士
【学位授予年份】：2005
【分类号】：R181.3

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