【摘要】：本文利用穩(wěn)定性理論和中心流形理論研究了幾類離散傳染病模型的穩(wěn)定性與分支問(wèn)題.全文共分為四章.第一章簡(jiǎn)要介紹了問(wèn)題的研究背景和本文的主要工作.第二章研究了具有飽和發(fā)生率βSI1+αI的離散SIR模型的穩(wěn)定性和分支問(wèn)題,利用局部穩(wěn)定性理論和Jury判據(jù),討論了無(wú)病平衡點(diǎn)和地方病平衡點(diǎn)的局部穩(wěn)定性,并證明了當(dāng)基本再生數(shù)R01時(shí),無(wú)病平衡點(diǎn)是全局穩(wěn)定的,最后利用中心流形定理給出了當(dāng)參數(shù)h在小鄰域內(nèi)變化時(shí),地方病平衡點(diǎn)處產(chǎn)生Flip分支和Hopf分支的充分條件.第三章研究了具有飽和發(fā)生率βSI1+αS的離散SIR模型的穩(wěn)定性和分支問(wèn)題,通過(guò)對(duì)模型特征方程的分析,得到了無(wú)病平衡點(diǎn)和地方病平衡點(diǎn)局部穩(wěn)定的充分條件,并證明了當(dāng)基本再生數(shù)R01時(shí),無(wú)病平衡點(diǎn)是全局穩(wěn)定的,最后給出了在地方病平衡點(diǎn)處產(chǎn)生Flip分支和Hopf分支的充分條件.第四章研究了具有l(wèi)ogistic增長(zhǎng)率的離散SI模型首先通過(guò)分析模型的特征方程,得到了無(wú)病平衡點(diǎn)和有病平衡點(diǎn)的局部漸近穩(wěn)定性,其次給出了當(dāng)參數(shù)h在小鄰域內(nèi)變化時(shí),在地方病平衡點(diǎn)處產(chǎn)生Flip分支和Hopf分支的充分條件.
[Abstract]:In this paper, the stability and bifurcation of several discrete infectious disease models are studied by using the stability theory and the center manifold theory. The full text is divided into four chapters. The first chapter briefly introduces the research background and the main work of this paper. In chapter 2, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 偽 I are studied. By using local stability theory and Jury criterion, the local stability of disease-free equilibrium and endemic equilibrium is discussed. It is proved that the disease-free equilibrium point is globally stable when the basic reproducing number R _ (01). Finally, by using the central manifold theorem, the sufficient conditions for the Flip bifurcation and Hopf bifurcation at the endemic equilibrium point are given when the parameter h changes in the small neighborhood. In chapter 3, the stability and bifurcation of discrete SIR model with saturation incidence 尾 SI1 偽 S are studied. By analyzing the characteristic equations of the model, sufficient conditions for local stability of disease-free equilibrium and endemic equilibrium are obtained. It is proved that the disease-free equilibrium is globally stable when the basic reproduction number is R01. At last, the sufficient conditions for producing Flip bifurcation and Hopf bifurcation at endemic equilibrium point are given. In chapter 4, the discrete SI model with logistic growth rate is studied. Firstly, by analyzing the characteristic equations of the model, the local asymptotic stability of disease-free equilibrium and diseased equilibrium is obtained. Secondly, when the parameter h changes in the small neighborhood, the local asymptotic stability of the disease-free equilibrium and the diseased equilibrium is obtained. Sufficient conditions for generating Flip bifurcation and Hopf bifurcation at endemic equilibrium point.
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 李建全;婁潔;婁梅枝;;Some discrete SI and SIS epidemic models[J];Applied Mathematics and Mechanics(English Edition);2008年01期

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本文編號(hào)：2342172