本文選題：傳染病系統(tǒng) + 參數(shù)辨識(shí)�。� 參考：《北京信息控制研究所》2005年博士論文

【摘要】： 利用動(dòng)力學(xué)的方法建立傳染病傳播的數(shù)學(xué)模型，研究某種傳染病在某一地區(qū)是否會(huì)蔓延下去而成為該地區(qū)的流行病，或者這種傳染病是否會(huì)最終消除，已經(jīng)成為傳染病學(xué)和數(shù)學(xué)相結(jié)合的一個(gè)重要的具有理論和現(xiàn)實(shí)意義的研究課題，它有助于對(duì)傳染病將來(lái)的發(fā)展趨勢(shì)進(jìn)行預(yù)測(cè)，有利于傳染病的預(yù)防與控制。 這篇博士學(xué)位論文就是將傳染病學(xué)的有關(guān)知識(shí)和數(shù)學(xué)理論結(jié)合起來(lái)，對(duì)傳染病的流行規(guī)律及相關(guān)問(wèn)題進(jìn)行探究。 本文第一章為緒論，對(duì)傳染病數(shù)學(xué)模型的國(guó)內(nèi)外研究狀況和最新進(jìn)展作了綜述，并從中引出本文所要研究的問(wèn)題，并簡(jiǎn)要敘述了本文所得到的結(jié)果。 在本文第二章，根據(jù)倉(cāng)室建模的思想建立了由常微分方程組和偏微分方程組描述的SIR傳染病模型，證明了平衡解的存在唯一性及穩(wěn)定性。特別地，我們還研究了系統(tǒng)(2-1-1)的參數(shù)辨識(shí)問(wèn)題，建立了疫情控制區(qū)域。 在本文第三章中，根據(jù)倉(cāng)室建模的思想建立了由常微分方程組描述的SEIR傳染病模型，通過(guò)分析特征方程特征根的分布證明了平衡解的存在性和穩(wěn)定性。在證明該模型疾病不消亡的平衡解的局部穩(wěn)定性時(shí)利用了Routh-Hurwits判別法。第三章中還建立了由偏微分方程組描述的SEIR傳染病模型，利用再生函數(shù)，給出了平衡解的存在性和穩(wěn)定性條件。 考慮有些疾病染病時(shí)間較長(zhǎng)，其流行規(guī)律、傳染能力、治愈效果都依賴染病期，基于這一點(diǎn)，在第四章對(duì)這種含染病期的傳染病模型(P)進(jìn)行了討論，通過(guò)先驗(yàn)估計(jì)的方法，得到了系統(tǒng)(P)正則廣義解的唯一性，應(yīng)用偏微分-積分方程的理論，證明了該模型解的穩(wěn)定性。 在本文第五章，我們假設(shè)某地區(qū)的人口受兩種病癥的困擾，這兩種疾病有排斥性，將總?cè)丝诜譃樗念?lèi)：易感人群，第一類(lèi)染病人群，第二類(lèi)染病人群，康復(fù)人群，根據(jù)倉(cāng)室建模的思想建立了兩種傳染病同時(shí)流行的數(shù)學(xué)模型(5-2-1)，利用小擾動(dòng)的方法討論了系統(tǒng)疾病消亡的平衡解的存在性及全局穩(wěn)定性，用微分-積分方程的知識(shí)證明了系統(tǒng)的傳染病不消亡的平衡解的存在性及局部漸近穩(wěn)定性。 本文的主要?jiǎng)?chuàng)新之處在于： 1、研究了系統(tǒng)(2-4-1)如下兩個(gè)最優(yōu)接種問(wèn)題：(1)在滿足一定效果要求的條件下，追求最低費(fèi)用；(2)在不超過(guò)一定費(fèi)用的前提下，追求接種效果最佳。借助泛函分析的知識(shí)證明了上述兩個(gè)最優(yōu)接種問(wèn)題最優(yōu)接種策略的存在性。 2、對(duì)于給定的目標(biāo)泛函(?)(ψ)，研究了系統(tǒng)(3-4-1)的最優(yōu)接種問(wèn)題，得到了最優(yōu)接種控制滿足的最優(yōu)性組。 3、對(duì)傳染率函數(shù)進(jìn)行了改進(jìn)。雖然各種各樣的年齡結(jié)構(gòu)的流行病模型已被很多作者討論過(guò)，但是大多數(shù)作者討論SIS模型、SIR模型、SEIR模型和MSEIR模型時(shí)，，都假定感染率函數(shù)和染病人群成正比，即假設(shè)感染率函數(shù) λ(t)=∫_0~Aβ(a)I(a,t)da (I(a,t)為染病人群的年齡密度函數(shù)) 但事實(shí)上對(duì)大多數(shù)傳染病而言，感染率函數(shù)應(yīng)該和染病人群與潛伏期人群占總?cè)丝诘谋嚷食烧�。本文將傳染率函�?shù)改進(jìn)為 λ(t)=∫_0~Aβ(a)(E(a,t)+I(a,t)/P_∞(a))da 這里E(a,t)、I(a,t)分別為潛伏期人群和染病人群的年齡密度函數(shù)。 4、研究了含染病期的傳染病模型和兩種傳染病同時(shí)流行的傳染病模型，證明了系統(tǒng)平衡解的存在性和穩(wěn)定性。 本文綜合應(yīng)用非線性泛函分析、微分方程、積分方程以及分布參數(shù)系統(tǒng)控制論等理論和方法，獲得了一批重要的理論成果。這批成果既具有較高的學(xué)術(shù)價(jià)值，也為傳染病系統(tǒng)的實(shí)際研究提供了理論依據(jù)。
[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.

【學(xué)位授予單位】：北京信息控制研究所
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2005
【分類(lèi)號(hào)】：R181.3

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