复杂网络上疾病传播的建模及其动力学

发布时间：2018-03-23 18:26

  本文选题：仓室模型　切入点：基本再生数　出处：《东南大学》2016年博士论文

【摘要】：尽管经典的疾病传播动力学模型在预测某些具体疾病方面取得了一定的成功,但是它们往往过于简单且忽视了一些重要的方面,如多阶段／多群体、接触人数和其它的疾病状态等。本文考虑了在复杂网络框架下的病毒和流行病传播模型,讨论了多阶段／多群体模型的全局稳定性,复杂网络上几类模型的全局动力学以及度相关网络上SIR疾病传播的建模问题。全文共五章。第二章讨论耦合网络上的多阶段／多群体传染病模型。第三章讨论了复杂网络上几类传播模型。第四章为基于网络连边的SIR疾病传播建模问题。在第二章,首先,研究了一个多阶段水传播疾病模型平衡点的存在唯一性及全局稳定性,并在此基础上,进一步提出了一类具有普适性的多阶段霍乱传播模型,在合理的生物学假设下,推导了基本再生数,利用全局Lyapunov函数、Kirchhoff矩阵树定理和LaSalle不变性原理研究了平衡点的全局稳定性。其次,研究了具有间接传播途径多群体SEI动物疾病模型的全局动力学。在合理的生物学假设下,推导了模型的基本再生数并证明了无病平衡点的全局稳定性；另一方面,由于加权有向图的权重矩阵是可约的,故结合全局Lyapunov函数和Kirchhoff矩阵树定理,利用了一个新的组合等式来讨论地方病平衡点的全局稳定性。在第三章,利用比较原理和有向图中的Kirchhoff矩阵树定理研究了复杂网络上几类传播模型的全局动力学。首先,讨论了一个带有出生与死亡网络水传播疾病模型的全局动力学及各种免疫策略对传播的影响。其次,研究了一个考虑平衡出生与死亡事件的异质网络中染病期和携带期都具有传染力疾病模型的全局稳定性,且当不考虑个体出生与死亡时,得到了疾病的最终规模表达式,并利用数值模拟比较了不同免疫策略对疾病传播的影响。最后,基于Lyapunov函数和Kirchhc off黾阵树定理,讨论了一个基于网络的计算机病毒模型有毒平衡点的全局稳定性问题,并利用比较原理,给出了无毒平衡点全局渐近稳定的一个更简洁证明。在第四章,首先回顾了配置网络上基于连边的SIR疾病传播模型和度相关网络上两个基于节点的SIR疾病传播模型,随后介绍了两个会导致度相关性出现的增长网络模型。利用连续时间随机模拟算法,对比了度相关网络上基于连边及基于节点SIR模型的预测结果和100次随机模拟SIR结果的均值。仿真结果表明,在度相关网络上,仅利用度分布信息的基于连边的SIR模型预测结果在许多方面如疾病初始指数增长率、峰值和峰值到达的时间,要优于利用度相关信息的基于节点的SIR模型预测结果,这说明配置网络上基于连边的SIR模型可能具有更广阔的适用范围。
[Abstract]:Although the epidemic model of classic and has achieved some success in predicting some specific diseases, but they are often too simple and ignore some important aspects, such as multi stage / multi group, contact number and other disease states. The virus and epidemic spreading model in complex networks under the framework of the discussion the global stability of multi stage / multi population model, the global dynamics of several kinds of models on complex networks and SIR network modeling problem of the spread of the disease. This paper consists of five chapters. The second chapter discusses the multi stage / multi group coupling network epidemic model. The third chapter discusses the propagation models of several complex the fourth chapter is the SIR network. The spread of the disease based on the network modeling problem of edges. In the second chapter, firstly, study the existence and uniqueness of a multi stage model of waterborne disease equilibrium And global stability, and on this basis, put forward a kind of multi stage model of cholera spread universality, in biology reasonable assumption, derived the basic reproduction number, using the global Lyapunov function, global stability theorem and LaSalle invariance principle of equilibrium matrix Kirchhoff tree. Secondly, the global dynamics the study has indirect transmission SEI group of animal disease model in biology. Reasonable assumption, the basic reproduction number derived model and prove the global stability of the disease-free equilibrium; on the other hand, the weighted directed graph weight matrix is reducible, so with the global Lyapunov function and Kirchhoff matrix the tree theorem, using a new combined equation to discuss the global stability of the endemic equilibrium. In the third chapter, using the comparison principle and theorem to the Kirchhoff matrix in the tree graph Several classes of complex network propagation model of global dynamics is studied. Firstly, the influence of birth and death with a network of waterborne disease model of global dynamics and various immunization strategies on propagation are discussed. Secondly, the study has global stability of infectious disease model with a consideration of the balance of birth and death events in different diseases quality of network and carry, and when not considering the individual birth and death, the ultimate expression of the disease scale, and the effect of different immunization strategies for the spread of the disease were compared by numerical simulation. Finally, based on the Lyapunov function and Kirchhc off array Strider tree theorem, we discuss a global stability problem of computer virus model of network based on toxic balance, and using the comparison principle, given no global asymptotic stability of the equilibrium point of a more concise proof. In the fourth chapter, first review The configuration of the network based on the SIR model and the spread of disease related to network even on the edge of the two SIR based models of disease transmission node, then introduces two growth network model leads to correlation appears. Using the continuous time stochastic simulation algorithm, compares the mean edges and prediction results based on SIR model and 100 nodes random simulation of SIR based on the results of correlation network. Simulation results show that the degree of correlation in the network, using only the degree distribution information based on SIR model predictions of edges in many aspects such as the initial disease index growth rate, peak value and peak arrival time is superior to the utilization of information related to the prediction results of SIR model based on node, the network configuration of SIR model on the edges may have a wider applicable range based on.

【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5
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本文编号：1654649