本文選題：Hopf分支 + 數(shù)值模擬��； 參考：《蘭州大學(xué)》2008年碩士論文

【摘要】： 傳染病的傳播模型可追述到1760年Daniel Bernoulli對天花的分析。傳染病動力學(xué)的建模與研究于二十世紀(jì)中葉開始蓬勃發(fā)展,作為標(biāo)志性的著作是Bailey于1957年出版的專著《數(shù)理流行病學(xué)》。近年來,國際上傳染病動力學(xué)的研究進(jìn)展迅速,大量的數(shù)學(xué)模型被用于分析各種各樣的傳染病問題,也有部分是針對諸如麻疹、流感等諸多具體疾病的模型。從模型的角度來劃分,主要可以分為四類:一類是研究以常微分方程描述的流行病模型;一類是研究以偏微分方程描述的流行病模型;第三類是研究同時(shí)含有常微分方程和偏微分方程的流行病模型,這主要包括類年齡結(jié)構(gòu)的流行病模型;另外一類模型為隨機(jī)模型,可以在相應(yīng)常微分方程的基礎(chǔ)上增加隨機(jī)考慮或利用Markov鏈進(jìn)行Monte Carlo模擬。本文是針對艾滋病,狂犬病而建立了常微分方程模型。 艾滋病是由于人體感染人類免疫缺陷病毒而引起的一種病死率極高的惡性傳染病。狂犬病是由狂犬病毒引起的人獸共患烈性傳染病,一旦發(fā)病病死率幾乎100%。本文基于其相關(guān)的病理知識,分別對應(yīng)建立了數(shù)學(xué)模型,并用動力學(xué)的知識對其進(jìn)行分析,通過數(shù)值模擬,得到以下結(jié)論: 1.在病毒對健康細(xì)胞的感染率是非線性時(shí),找出了決定系統(tǒng)周期解的穩(wěn)定性,周期解的周期的因素。 2.通過對目前狂犬病流行現(xiàn)狀的研究,在考慮中國實(shí)際情況的基礎(chǔ)上,建立了新的數(shù)學(xué)模型。本模型不僅涉及到狂犬病在犬中傳播,還考慮了暴露和患病犬以及攜帶病毒的”健康犬”使人感染狂犬病的現(xiàn)象。通過尋找”基本再生數(shù)”對比了捕殺、免疫、捕殺和免疫相結(jié)合三種不同策略在控制狂犬病傳播中的有效性。分析和模擬結(jié)果表明三種控制狂犬病的方法中捕殺的效果最好,免疫的效果次之,捕殺和免疫相結(jié)合的效果最差。同時(shí)根據(jù)中國目前城市和農(nóng)村發(fā)展不平衡的現(xiàn)狀,提出了在城市以免疫為主,在農(nóng)村采用捕殺和免疫相結(jié)合的控制狂犬病的措施,從而為中國目前控制狂犬病的流行提供了理論依據(jù)。
[Abstract]:The transmission model of infectious diseases can be traced back to Daniel Bernoulli's analysis of smallpox in 1760. The modeling and research of infectious disease dynamics began to flourish in the middle of the 20th century. As a landmark work Bailey published his monograph "Mathematical Epidemiology" in 1957. In recent years, the dynamics of infectious diseases has made rapid progress in the world. A large number of mathematical models have been used to analyze various infectious disease problems, and some models are aimed at many specific diseases such as measles, influenza and so on. From the point of view of model, it can be divided into four categories: one is to study epidemic model described by ordinary differential equation, the other is to study epidemic model described by partial differential equation. The third is to study epidemic models with both ordinary differential equations and partial differential equations, which mainly include age-like epidemic models and stochastic models. It is possible to add random consideration to the corresponding ordinary differential equations or to use Markov chains for Monte Carlo simulation. In this paper, an ordinary differential equation model is established for AIDS and rabies. AIDS is a malignant infectious disease caused by human immunodeficiency virus infection. Rabies is a zoonotic infectious disease caused by rabies virus. Based on the relevant pathological knowledge, the mathematical models are established and analyzed with the knowledge of dynamics. By numerical simulation, the following conclusions are obtained: 1. When the infection rate of virus to healthy cells is nonlinear, the factors that determine the stability of the periodic solution and the period of the periodic solution of the system are found. 2. A new mathematical model was established based on the study of the current situation of rabies in China. This model not only deals with the spread of rabies in dogs, but also takes into account the fact that exposed and diseased dogs and "healthy dogs" carrying the virus cause rabies infection in humans. By looking for "basic regeneration number", this paper compares the effectiveness of three different strategies of killing, immunization, killing and immunization in controlling rabies transmission. The results of analysis and simulation show that among the three methods of rabies control, the killing effect is the best, the immune effect is the second, and the combination of killing and immunization is the worst. At the same time, according to the current situation of unbalanced development between urban and rural areas in China, the measures of controlling rabies are put forward, which are mainly immunization in cities and combination of killing and immunization in rural areas. It provides a theoretical basis for controlling the prevalence of rabies in China.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2008
【分類號】：R311

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