【摘要】：傳染病動(dòng)力學(xué)是理論性定量,定性研究傳染病流行規(guī)律的一種方法.年齡結(jié)構(gòu)傳染病模型及其動(dòng)力學(xué)性質(zhì)的研究又是傳染病動(dòng)力學(xué)研究的一個(gè)重要課題.本論文主要分析傳染病系統(tǒng)的性質(zhì)以及建立相應(yīng)的Lyapunov函數(shù)來分析年齡結(jié)構(gòu)模型的全局性態(tài)問題,本課題的研究對于傳染病發(fā)展趨勢的估計(jì)是重要的且是具有實(shí)際意義的工作.本文通過研究兩類具有類年齡結(jié)構(gòu)的傳染病模型:一類為HIV病毒傳染模型,考慮了發(fā)生率函數(shù)為非線性時(shí),平衡點(diǎn)依賴于閾值(基本再生數(shù))的動(dòng)力學(xué)行為;另一類模型是流行病模型,本文考慮了SVIR以及SVEIR兩種流行病模型,研究加入接種疫苗倉室后,平衡點(diǎn)依賴于閾值(基本再生數(shù))的動(dòng)力學(xué)行為.重點(diǎn)考慮模型的適定性以及構(gòu)造合適的Lyapunov函數(shù)去解決平衡點(diǎn)全局穩(wěn)定性的問題.另外本文構(gòu)造了一類Lyapunov函數(shù)(泛函)方法來解決非線性以及年齡結(jié)構(gòu)模型的穩(wěn)定性問題,提供了一些簡單有效的判定定理,對實(shí)際的傳染病預(yù)防控制工作可以起到?jīng)Q策依據(jù)以及理論參考.
[Abstract]:Infectious disease dynamics is a theoretical quantitative, qualitative study of epidemic law of an infectious disease. The study of age-structured infectious disease model and its dynamic properties is also an important subject of infectious disease dynamics research. This paper mainly analyzes the properties of infectious disease system and establishes the corresponding Lyapunov function to analyze the global problem of age structure model. The research in this paper is important and meaningful for the estimation of infectious disease development trend. In this paper, two kinds of age-like infectious disease models are studied: one is HIV virus infection model, and the equilibrium point is dependent on threshold (basic reproduction number) when the incidence function is nonlinear; The other one is epidemic model. In this paper, SVIR and SVEIR epidemic models are considered, and the dynamic behavior of equilibrium point dependent on threshold (basic regeneration number) is studied. The suitability of the model and the construction of appropriate Lyapunov functions are considered to solve the problem of global stability of the equilibrium point. In addition, a class of Lyapunov function (functional) method is constructed to solve the problem of nonlinearity and the stability of age structure model, and some simple and effective decision theorems are provided. To the actual infectious disease prevention and control work can play the decision-making basis and the theory reference.
【學(xué)位授予單位】：黑龍江大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175
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本文編號(hào)：2151129