【摘要】：傳染病模型是為了方便研究傳染病在個(gè)體之間和地區(qū)之間的發(fā)病機(jī)制及擴(kuò)散規(guī)律,通過運(yùn)用一些合理的假設(shè),建立適當(dāng)?shù)臄?shù)學(xué)模型,并將可決定傳染病擴(kuò)散的各個(gè)因素轉(zhuǎn)化為已建立數(shù)學(xué)模型中的相關(guān)數(shù)學(xué)變量,利用動力學(xué)理論來分析疾病的發(fā)展趨勢,以達(dá)到幫助我們預(yù)測和控制日常生活中疾病的目的.基本再生數(shù)是傳染病模型中的重要參數(shù),而基本再生數(shù)決定了疾病是否蔓延或者消退.但是我們逐漸認(rèn)識到空間擴(kuò)散和環(huán)境的異質(zhì)性不僅是影響疾病消退和蔓延的重要因素,還決定了疾病傳播方式和傳播速度.這樣來說的話通常的基本再生數(shù)不足以描述疾病的傳播,也不能反映所研究區(qū)域的空間特征.從而很有必要研究擴(kuò)散對于疾病在區(qū)域中的傳播和控制所起的作用.伴隨這些要求,樓元老師對一類非均質(zhì)區(qū)域下的SIS傳染病模型的穩(wěn)定性進(jìn)行了分析.先考慮固定區(qū)域上的SIS反應(yīng)擴(kuò)散問題,通過定義具有齊次Neumann邊界條件的反應(yīng)擴(kuò)散問題的基本再生數(shù)R_0~N,討論無病平衡點(diǎn)和染病平衡點(diǎn)的穩(wěn)定性;在此基礎(chǔ)上,引入自由邊界描述傳染病傳播的邊沿,定義具有齊次Dirichlet邊界條件的反應(yīng)擴(kuò)散問題的基本再生數(shù)R_0~D,從而引入具有自由邊界的SIS模型的基本再生數(shù)R_0~F(t),并討論了疾病的消退和蔓延.本文采用了一種新的非齊次的SIRS傳染病模型.基本思路是:先是構(gòu)造本模型在具有Neumann邊界條件下的基本再生數(shù)R_0,同時(shí)討論傳染病者的擴(kuò)散對基本再生數(shù)R_0的影響,即如果R_01,則無病平衡點(diǎn)全局漸近穩(wěn)定,如果R_01,則無病平衡點(diǎn)不穩(wěn)定.再是,在低危險(xiǎn)區(qū)域,我們運(yùn)用分叉理論研究染病平衡點(diǎn)的存在性和穩(wěn)定性.最終結(jié)果顯示,減少染病者的擴(kuò)散并不有利于傳染病的消除,但染病平衡點(diǎn)的不穩(wěn)定性表明傳染病可以得到控制.本文在第一章緒論的第一節(jié)中具體介紹了SIRS傳染病模型的背景來源,第二節(jié)中給出近來研究現(xiàn)狀;第二章第一節(jié)給出了Lyapunov穩(wěn)定性的,第二節(jié)給出Crandall-Rabinowitz分叉理論的知識,第三節(jié)給出局部分叉圖像和穩(wěn)定性變換原則的相關(guān)知識;第三章中討論了所研究的SIRS反應(yīng)擴(kuò)散傳染病模型的基本再生數(shù)的定義和特征、無病平衡點(diǎn)的穩(wěn)定性、染病平衡點(diǎn)的存在性與穩(wěn)定性和局部分叉圖像的方向.第四章對本文的研究做了相關(guān)總結(jié).
[Abstract]:The infectious disease model is to facilitate the study of the pathogenesis and diffusion of infectious diseases between individuals and regions, and to establish appropriate mathematical models by using some reasonable assumptions. The factors that can determine the spread of infectious diseases are transformed into relevant mathematical variables in established mathematical models, and the development trend of diseases is analyzed by using the kinetic theory to help us predict and control diseases in our daily life. The number of basic regeneration is an important parameter in infectious disease model, and the number of basic regeneration determines whether the disease is spreading or fading. However, we have come to realize that spatial diffusion and environmental heterogeneity are not only important factors that affect the extinction and spread of disease, but also determine the mode and speed of disease transmission. In this case, the usual number of basic regeneration is not sufficient to describe the spread of disease, nor can it reflect the spatial characteristics of the region studied. It is therefore necessary to study the role of diffusion in the spread and control of disease in the region. In response to these requirements, the stability of a class of SIS infectious disease models in heterogeneous regions was analyzed. Considering the SIS reaction-diffusion problem in a fixed region, the stability of disease-free equilibrium point and disease-free equilibrium point is discussed by defining the basic regenerative number of the reaction-diffusion problem with homogeneous Neumann boundary condition. The free boundary is introduced to describe the edge of infectious disease propagation, and the basic number of reaction-diffusion problems with homogeneous Dirichlet boundary condition is defined. The basic number of reproduction of SIS model with free boundary is introduced, and the extinction and spread of disease are discussed. In this paper, a new non-homogeneous SIRS infectious disease model is used. The basic ideas are as follows: first, we construct the basic regenerative number R0 with Neumann boundary condition, and at the same time discuss the influence of the diffusion of infectious diseases on the basic regenerative number R0, that is, if R201, the disease-free equilibrium point is globally asymptotically stable. If RW is 01, the disease-free equilibrium is unstable. Then, in the low risk area, we use bifurcation theory to study the existence and stability of the infection equilibrium. The final results show that reducing the spread of infectious diseases is not conducive to the elimination of infectious diseases, but the instability of infection equilibrium points indicates that infectious diseases can be controlled. In the first section of the first chapter, the background of SIRS infectious disease model is introduced in detail. In the second section, the recent research status is given. In the second chapter, the stability of Lyapunov is given, and the knowledge of Crandall-Rabinowitz bifurcation theory is given in the second section. In the third section, we give the knowledge of the local bifurcation image and the principle of stability transformation. In chapter 3, we discuss the definition and characteristics of the basic reproduction number of the SIRS model, and the stability of the disease-free equilibrium. The existence and stability of the equilibrium point and the direction of the local bifurcation image. Chapter four summarizes the research of this paper.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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