【摘要】：傳染病動力學(xué)是對傳染病進行理論性定量研究的一種重要方法.它是根據(jù)種群生長的特性,建立模型來分析疾病的發(fā)生及在種群內(nèi)的傳播、流行規(guī)律,并通過分析和數(shù)值模擬,來揭示疾病的流行規(guī)律,預(yù)測其變化發(fā)展趨勢,尋找對其預(yù)防和控制的策略.本文用常微分方程組描述了兩類傳染病動力學(xué)模型,同時討論了所提出模型的一些動力學(xué)行為,其中包括解的正不變區(qū)域,平衡點的存在性和穩(wěn)定性,系統(tǒng)的持續(xù)性與滅絕性等動力學(xué)行為,并討論其生物學(xué)意義.主要有以下兩個方面的內(nèi)容: 第二章研究具有接種效應(yīng)的霍亂模型.提出了一類自治的五維SVIR-B模型,給出了決定霍亂消亡與否的控制再生數(shù)Rv,當Rv1時,利用Routh-Hurwitz條件證明了無病平衡點是局部漸近穩(wěn)定的,又得到了無病平衡點是全局漸近穩(wěn)定的,此時霍亂會消亡;當Rv1時,存在唯一一個地方病平衡點,利用Routh-Hurwitz條件證明了地方病平衡點是局部漸近穩(wěn)定的,接著證明了系統(tǒng)的持續(xù)性,并使用復(fù)合矩陣理論討論了地方病平衡點的全局漸近穩(wěn)定性,得到了地方病平衡點是全局漸近穩(wěn)定的充分條件,霍亂將持續(xù)存在.最后,對Rv的參數(shù)靈敏性分析并進行數(shù)值模擬,提出需要同時增加接種率和減小免疫失去率達到某個臨界值后,霍亂才得以控制. 第三章研究了具有媒體效應(yīng)的傳染病模型.建立了一類具有一般性接觸率SIRS模型,得到了基本再生數(shù)的表達式R0,當R01時,利用Routh-Hurwitz條件證明了無病平衡點是局部漸近穩(wěn)定的,此時，傳染病會消亡;當R01時,利用函數(shù)的零點定理證明了地方病平衡點的唯一存在性,使用Routh-Hurwitz條件證明了地方病平衡點是局部漸近穩(wěn)定的,并使用Bendixson判據(jù)討論了地方病平衡點的全局漸近穩(wěn)定性,得到了地方病平衡點全局漸近穩(wěn)定的充分條件,此時，傳染病將一直存在.最后,進行了數(shù)值模擬,顯示了媒體報道對傳染病傳播和控制的影響.
[Abstract]:Infectious disease dynamics is an important method for theoretical and quantitative study of infectious diseases. According to the characteristics of population growth, it establishes a model to analyze the occurrence, spread and prevalence of disease within the population, and by means of analysis and numerical simulation, it reveals the epidemic law of disease and predicts its changing trend. Look for strategies to prevent and control them. In this paper, two kinds of infectious disease dynamics models are described by ordinary differential equations, and some dynamic behaviors of the proposed model are discussed, including the positive invariant region of solution, the existence and stability of equilibrium point. The dynamics of the system such as persistence and extinction are discussed and its biological significance is discussed. The main contents are as follows: chapter 2 studies the cholera model with inoculation effect. In this paper, an autonomous five-dimensional SVIR-B model is proposed, and the control reproducing number Rv, is given to determine the extinction of cholera. When Rv1 is used, it is proved that the disease-free equilibrium is locally asymptotically stable and that the disease-free equilibrium is globally asymptotically stable. At this point, cholera will die out; when Rv1, there is only one endemic equilibrium, and the Routh-Hurwitz condition is used to prove that the endemic equilibrium is locally asymptotically stable, and then the persistence of the system is proved. The global asymptotic stability of endemic equilibrium is discussed by using the compound matrix theory. A sufficient condition is obtained that the endemic equilibrium is globally asymptotically stable, and cholera will continue to exist. Finally, the sensitivity of the parameters of Rv is analyzed and simulated. It is suggested that cholera can only be controlled if the immunization coverage rate and the immune loss rate reach a certain critical value at the same time. Chapter three studies the infectious disease model with media effect. In this paper, a class of SIRS model with general contact rate is established, and the expression R0 is obtained. When R01, we prove that the disease-free equilibrium is locally asymptotically stable by using the Routh-Hurwitz condition, and the infectious disease will die out when R01. The unique existence of endemic equilibrium is proved by using the zero point theorem of function, the local asymptotic stability of endemic equilibrium is proved by using Routh-Hurwitz condition, and the global asymptotic stability of endemic equilibrium is discussed by using Bendixson criterion. A sufficient condition for global asymptotic stability of endemic equilibrium is obtained. Finally, numerical simulation is carried out to show the influence of media reports on the transmission and control of infectious diseases.
【學(xué)位授予單位】：北京建筑大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2014
【分類號】：R181

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相關(guān)期刊論文 前1條

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本文編號：2216348