离散的阶段结构传染病模型

发布时间：2018-04-28 03:26

本文选题：流行病模型 + 阶段结构　；参考：《西南师范大学》2005年硕士论文

【摘要】：本文主要讨论了离散的阶段结构传染病模型。第一章建立并研究了离散的单种群阶段结构模型,讨论了平衡点的存在性和稳定性,证明了种群的持续生存。对于Beverton-Holt形式的出生率,证明了其正平衡点的渐近稳定性,计算机模拟表明当不考虑时滞时,正平衡点是全局渐近稳定的;对于Ricker形式的出生率,作出了其在几种参数条件下的分支图,倍周期分支会出现,然后产生混沌,和一般的Logistic模型的分支图相比,分支图变得更为复杂,时滞和阶段结构会延缓混沌的出现。本文第二章建立并研究了离散的阶段结构成年病模型。得到了疾病的基本再生数R_0,其计算过程比一般的微分方程模型更为复杂,这是因为此时无病空间上的吸引子可以是平衡点,也可以是周期解或者奇怪吸引子。对于Beverton-Holt形式出生率的离散阶段结构成年病模型,当恢复率为零时证明了无病平衡点的全局稳定性。我们还可以发现基本再生数R_0关于系统内所有的参数单调。对于Ricker形式的出生率,我们发现随着自然增长率的变化疾病的存在和消除可能会交替出现。此种计算基本再生数的方法可以被其它的一些离散模型所用。本文第三章研究了离散的阶段结构时滞传染病模型。运用第二章寻求基本再生数的方法对两种出生函数分别进行了研究,得出了基本再生数,可以看到,时滞模型比非时滞模型得性态更为复杂.发现随着自然增长率的变化疾病的存在和消除仍然可能会交替出现。
[Abstract]:In this paper, the discrete structural infectious disease model is discussed. In chapter 1, the discrete single-population structure model is established and studied. The existence and stability of the equilibrium point are discussed, and the persistence of the population is proved. For the birth rate of Beverton-Holt form, the asymptotic stability of the positive equilibrium point is proved. The computer simulation shows that the positive equilibrium point is globally asymptotically stable when time delay is not considered, and for the birth rate in Ricker form, the positive equilibrium point is globally asymptotically stable. In this paper, the bifurcation graph under several parameter conditions is given, the double periodic bifurcation will appear, and then chaos will occur. Compared with the general branching graph of Logistic model, the branching graph becomes more complex, and the delay and phase structure will delay the emergence of chaos. In chapter 2, discrete structural adult disease models are established and studied. The basic regenerative number R _ S _ 0 of the disease is obtained, and the calculation process is more complicated than the ordinary differential equation model. This is because the attractor in the disease-free space can be an equilibrium point, a periodic solution or a strange attractor. The global stability of disease-free equilibrium point is proved when the recovery rate is 00:00 for the discrete stage structured adult disease model of Beverton-Holt form birth rate. We can also find that the basic reproduction number R _ S _ 0 is monotone for all the parameters in the system. For the birth rate in the form of Ricker, we find that the existence and elimination of diseases may alternate with the change of natural growth rate. This method can be used in other discrete models. In the third chapter, we study the discrete stage structural time-delay infectious disease model. In the second chapter, we study the two kinds of birth function by using the method of searching for the basic reproducing number, and get the basic reproducing number. It can be seen that the delay model is more complex than the non-delay model. It is found that the existence and elimination of diseases may alternate with the change of natural growth rate.
【学位授予单位】：西南师范大学
【学位级别】：硕士
【学位授予年份】：2005
【分类号】：R181.3

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