带有非线性发生率和时滞的HIV-1和HBV模型的动力学研究

发布时间：2018-04-12 18:23

本文选题：HIV-1感染模型 + HBV感染模型　；参考：《云南师范大学》2017年硕士论文

【摘要】：艾滋病和乙型肝炎作为目前全世界两种极具危害的传染病,己经给人类的健康造成了很大程度的威胁.本文则从它们的源头-病毒入手,通过分析相应微分方程平衡点的稳定性,探讨HIV-1和HBV感染系统的动力学行为.主要内容如下:第一章简单叙述了传染病的研究背景和研究意义、HIV-1和HBV感染系统的国内外研究状况、本文的主要研究内容和用到的定理、定义等.第二章,我们对一个带有CTL免疫反应、饱和发生率和三个时滞的五维HIV-1感染系统进行了动力学研究.首先,我们给出了模型的基本性质,其中包括模型的适定性、基本再生数以及平衡点的存在性.然后通过分析平衡点的对应特征方程,我们确定了每个可行性平衡点的局部稳定性和出现Hopf分岔的条件.接下来运用波动引理和构造适当的李雅普诺夫泛函,我们验证了在局部稳定的条件下前两个不动点仍然是全局稳定的.最后,我们对系统进行数值计算来检验理论结果.第三章,我们研究一个新的包含空间扩散,一般发生率和三个时滞的慢性HBV感染模型的动力学.首先我们分析了在有界区域内模型初始值问题的适定性,然后我们定义了一个被称为基本再生数的阀值参数,并且表明了我们的模型存在两个可能的平衡点.接下来通过构造两个适当的李雅普诺夫泛函,说明了平衡点的全局动力学行为完全由系统阈值决定.最终我们给出数值计算来验证之前所得结论的正确性.
[Abstract]:AIDS and hepatitis B are two harmful infectious diseases all over the world, which have posed a great threat to human health.In this paper, the dynamic behavior of HIV-1 and HBV infection systems is discussed by analyzing the stability of the equilibrium point of the corresponding differential equations.The main contents are as follows: in the first chapter, the research background and significance of infectious diseases are briefly described. The main research contents, theorems and definitions of HIV-1 and HBV infection systems are also discussed.In chapter 2, we study the dynamics of a five-dimensional HIV-1 infection system with CTL immune response, saturation incidence and three delays.First, we give the basic properties of the model, including the fitness of the model, the number of basic reproducing and the existence of equilibrium point.Then, by analyzing the corresponding characteristic equations of the equilibrium point, we determine the local stability of each feasible equilibrium point and the conditions for the occurrence of Hopf bifurcation.Then by using wave Lemma and constructing appropriate Lyapunov Functionals we prove that the first two fixed points are globally stable under locally stable conditions.Finally, we verify the theoretical results by numerical calculation of the system.In chapter 3, we study the dynamics of a new chronic HBV infection model with spatial diffusion, general incidence and three delays.First, we analyze the fitness of the initial value problem of the model in a bounded region, then we define a threshold parameter called the basic reproducing number, and show that there are two possible equilibrium points in our model.Then, by constructing two proper Lyapunov Functionals, it is shown that the global dynamical behavior of the equilibrium point is completely determined by the system threshold.Finally, we give a numerical calculation to verify the correctness of the previous conclusions.
【学位授予单位】：云南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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