具有非线性发生率和非定常人口的传染病传播模型分析

发布时间：2018-05-11 14:54

本文选题：传染病传播模型 + 非线性发生率　；参考：《中南大学》2012年硕士论文

【摘要】：传染病历来是人类的大敌,利用动力学方法建立传染病传播的数学模型,并通过模型对传染病进行定性和定量的分析与研究已经取得一些成果,主要集中在判定、预测疾病发展趋势上。本文研究具有非线性发生率和非定常人口的传染病传播网络模型。在本文考察的模型中,非定常人口因素使得传染病传播模型表现为网络大系统,非线性发生率又使得通常的李雅普诺夫分析显得较为复杂。基于这样的观察,本文提出一种将稳定性判定和平衡点类型判定分开处理的方法,内容如下： 1.将协调系统理论与状态近似估计方法相结合,首先运用协调系统理论整体地判定平衡点的全局渐近稳定性,其次使用状态近似估计的方法判定平衡点的类型,即是零平衡点或非零平衡点,得到的判据都用系统参数明确表达,易于验证。采用状态估计的办法还便于得到某些定量估计,如定量估计康复率对疾病的影响； 2.将协调系统理论与Driessche阈值理论相结合,首先运用协调系统理论整体判定平衡点的全局渐近稳定性,其次使用Driessche的阈值理论判定平衡点的类型,得到模型的全局阈值条件。此方法避免了构造李雅普诺夫函数带来的困难。值得强调的是,Driessche阂值理论只能得到零平衡点的局部稳定判据,但是当其与协调系统理论结合时,本文得到的是平衡点的全局稳定判据； 3.本文最后给出上述理论方法的一个实际应用,即将其应用在种群迁移中存在路途感染的两斑块模型中,证明模型的全局渐近稳定性,给出判定平衡点类型的全局阈值条件。本文的结论对于判定疾病的消失或流行及在流行情况下疾病的控制有理论指导意义。
[Abstract]:Infectious diseases have always been the great enemy of human beings. The mathematical model of infectious disease transmission has been established by using dynamic methods, and some achievements have been achieved in qualitative and quantitative analysis and research of infectious diseases through the model, which is mainly focused on the determination of infectious diseases. Predict the trend of disease development. In this paper, the transmission network model of infectious diseases with nonlinear incidence and unsteady population is studied. In the model investigated in this paper, the unsteady population factors make the infectious disease transmission model behave as the network large-scale system, and the nonlinear incidence makes the common Lyapunov analysis more complicated. Based on this observation, this paper proposes a method to deal with the stability judgment and the equilibrium type decision separately. The contents are as follows: 1. By combining the theory of coordinated systems with the method of approximate state estimation, the global asymptotic stability of equilibrium points is determined by the theory of coordinated systems, and the type of equilibrium points is determined by the method of state approximate estimation. That is, the zero equilibrium point or the non zero equilibrium point, the obtained criteria are clearly expressed by the system parameters, which is easy to verify. The use of state estimation also facilitates the availability of certain quantitative estimates, such as quantitative estimates of the impact of rehabilitation rates on disease; 2. By combining the theory of coordinated system with the theory of Driessche threshold, the global asymptotic stability of equilibrium point is determined by the theory of coordinated system theory, and then the type of equilibrium point is determined by Driessche's threshold theory, and the global threshold condition of the model is obtained. This method avoids the difficulty of constructing Lyapunov function. It is worth emphasizing that Driessche's boundary value theory can only obtain the local stability criterion of zero equilibrium point, but when it is combined with the coordinated system theory, the global stability criterion of equilibrium point is obtained in this paper. 3. In the end of this paper, a practical application of the above theoretical method is given, that is to say, it is applied to the two-patch model with path infection in population migration, the global asymptotic stability of the model is proved, and the global threshold condition for judging the type of equilibrium point is given. The conclusion of this paper is of theoretical significance in determining the disappearance or prevalence of diseases and the control of diseases under epidemic conditions.
【学位授予单位】：中南大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O242.1;R311

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