發(fā)布時(shí)間：2018-07-08 11:12

  本文選題：時(shí)滯 + Hopf分支��； 參考：《湖北師范大學(xué)》2016年碩士論文

【摘要】：本文主要研究了一類(lèi)具有不同時(shí)滯和不同傳染率的SVIR傳染病模型,運(yùn)用時(shí)滯微分方程的穩(wěn)定性理論得到正平衡點(diǎn)局部穩(wěn)定和Hopf分支存在的充分條件,運(yùn)用標(biāo)準(zhǔn)型理論和中心流形定理,分析了分支周期解的方向和穩(wěn)定性等性質(zhì).揭示了時(shí)滯和傳染率對(duì)正平衡點(diǎn)穩(wěn)定性的影響.第一章,介紹了傳染病數(shù)學(xué)模型的研究背景,以及其理論成果對(duì)于控制和預(yù)防傳染病傳播有著重要現(xiàn)實(shí)意義.通過(guò)分析國(guó)內(nèi)外研究現(xiàn)狀,從而引出SVIR傳染病模型的形成過(guò)程.第二章,研究了一類(lèi)具有潛伏期時(shí)滯和雙線(xiàn)性傳染率的SVIR傳染病模型.當(dāng)基本再生數(shù)10R?時(shí)模型存在正平衡點(diǎn),得到正平衡點(diǎn)局部穩(wěn)定的條件,當(dāng)時(shí)滯經(jīng)過(guò)某些特定臨界值時(shí)模型出現(xiàn)Hopf分支,導(dǎo)出分支周期解的屬性公式,用Matlab等軟件計(jì)算一組數(shù)值模擬驗(yàn)證了所得的理論結(jié)果.第三章,研究了一類(lèi)具有免疫期時(shí)滯和非線(xiàn)性傳染率的SVIR傳染病模型.其存在正平衡點(diǎn),且是局部漸近穩(wěn)定的.同時(shí),研究了時(shí)滯對(duì)正平衡點(diǎn)穩(wěn)定性的影響,證明了當(dāng)時(shí)滯經(jīng)過(guò)某些特定臨界值時(shí)模型出現(xiàn)Hopf分支.第四章,總結(jié)全文,展望SVIR傳染病模型的研究前景.
[Abstract]:In this paper, we study a class of SVIR infectious disease models with different time delays and different infection rates. By using the stability theory of delay differential equations, we obtain the sufficient conditions for the existence of local stability and Hopf bifurcation of positive equilibrium points. By using the theory of canonical form and the theorem of center manifold, the direction and stability of the bifurcation periodic solution are analyzed. The effects of time delay and infection rate on the stability of positive equilibrium point are revealed. The first chapter introduces the research background of the mathematical model of infectious diseases and its theoretical results have important practical significance to control and prevent the spread of infectious diseases. By analyzing the current research situation at home and abroad, the formation process of SVIR infectious disease model is introduced. In chapter 2, we study a class of SVIR infectious disease models with latency delay and bilinear infection rate. When the basic regeneration number is 10R? The local stability condition of the positive equilibrium point is obtained. The Hopf bifurcation appears when the model passes through certain critical values at that time, and the attribute formula of the periodic solution of the bifurcation is derived. The theoretical results are verified by a set of numerical simulations calculated by Matlab and other software. In chapter 3, we study a class of SVIR infectious disease models with immune delay and nonlinear infection rate. There exists a positive equilibrium and it is locally asymptotically stable. At the same time, the influence of time delay on the stability of positive equilibrium point is studied, and it is proved that Hopf bifurcation occurs when the time delay passes through certain critical values. Chapter 4 summarizes the full text and looks forward to the future of SVIR infectious disease model.
【學(xué)位授予單位】：湖北師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O175

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本文編號(hào)：2107246