本文選題：時滯 + Hopf分支　； 參考：《湖北師范大學》2016年碩士論文

【摘要】：本文主要研究了一類具有不同時滯和不同傳染率的SVIR傳染病模型,運用時滯微分方程的穩(wěn)定性理論得到正平衡點局部穩(wěn)定和Hopf分支存在的充分條件,運用標準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.揭示了時滯和傳染率對正平衡點穩(wěn)定性的影響.第一章,介紹了傳染病數(shù)學模型的研究背景,以及其理論成果對于控制和預防傳染病傳播有著重要現(xiàn)實意義.通過分析國內(nèi)外研究現(xiàn)狀,從而引出SVIR傳染病模型的形成過程.第二章,研究了一類具有潛伏期時滯和雙線性傳染率的SVIR傳染病模型.當基本再生數(shù)10R?時模型存在正平衡點,得到正平衡點局部穩(wěn)定的條件,當時滯經(jīng)過某些特定臨界值時模型出現(xiàn)Hopf分支,導出分支周期解的屬性公式,用Matlab等軟件計算一組數(shù)值模擬驗證了所得的理論結果.第三章,研究了一類具有免疫期時滯和非線性傳染率的SVIR傳染病模型.其存在正平衡點,且是局部漸近穩(wěn)定的.同時,研究了時滯對正平衡點穩(wěn)定性的影響,證明了當時滯經(jīng)過某些特定臨界值時模型出現(xiàn)Hopf分支.第四章,總結全文,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.
【學位授予單位】：湖北師范大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O175

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本文編號：2107246