几类传染病模型的稳定性分析

发布时间：2018-10-25 19:56

【摘要】：传染病模型是生物数学研究的主要内容,运用传染病动力学知识,建立传染病数学模型,并进行数值模拟,得到传染病的传播规律,分析传染病爆发和流行的主要原因,从而找到预防传染病的最好方法。本文的主要研究内容如下:首先,建立了一类具有CTL免疫的乙肝病毒模型,研究分析了该模型的平衡点的动态稳定性,利用谱半径的方法求出基本再生数R_0。当R_0≤1时,通过构造Lyapunov函数,利用Lassalle不变性原理验证了系统无病平衡点的局部稳定性;当R_0＞1时,研究分析了系统地方病平衡点的局部渐近稳定性。再通过选取恰当的参数进行数值模拟验证了理论结果。其次,研究了一类具有时滞和饱和发生率的乙肝病毒模型,考虑到感染细胞的恢复率,分析确定了疾病是否流行的阈值R_0,通过构造Lyapunov函数,利用Lassalle不变集原理证明了当R_01时,对于任意时滞,系统在无病平衡点处是全局渐近稳定的;当R_01时,分析了地方病平衡点的局部渐近稳定性。再通过选取恰当的参数进行数值模拟验证了理论结果。最后,研究了一类带有接种的非线性发生率的传染病模型,分析了该模型的平衡点的动态稳定性,得到了疾病流行与否的阈值R_0。假设所有输入者都是易感者,当R_01时,通过构造Lyapunov函数,验证了无病平衡点的全局渐近稳定性;当R_01时,利用Huwitz判据证明了地方病平衡点的局部渐近稳定性。再通过选取恰当的参数进行数值模拟验证了理论结果。
[Abstract]:The infectious disease model is the main content of the biological mathematics research. By using the knowledge of infectious disease dynamics, the mathematical model of the infectious disease is established, and the numerical simulation is carried out to obtain the law of the spread of the infectious disease, and the main reasons for the outbreak and epidemic of the infectious disease are analyzed. To find the best way to prevent infectious diseases. The main contents of this paper are as follows: firstly, a hepatitis B virus model with CTL immunity is established, and the dynamic stability of the equilibrium point of the model is analyzed. The basic regenerative number R _ S _ 0 is obtained by the method of spectral radius. The local stability of the disease-free equilibrium point of the system is verified by constructing the Lyapunov function when R _ S _ 0 鈮，

本文编号：2294677