【作者】 王毅；

【导师】 曹进德；

【作者基本信息】 东南大学 ， 数学， 2016， 博士

【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。更多还原

【Abstract】 Although classic disease spread dynamic models make some modest success in predicting some particular disease behaviors, they are usually overly simplistic, and neglect certain im-portant aspects, which include, for example, multiple stages/groups, the number of contacts and other disease states. In this paper, we investigate some virus and epidemic spread models under complex network framework. We establish the global stability for multi-stage/group epi-demic models and the global dynamics for several spread models on complex networks. We also consider modeling problems for SIR disease spread on degree correlated networks. The paper is divided into five main chapters. The second chapter deals with multi-stage/group epidemic models on coupled networks. The third chapter contains several spread models on complex networks. The fourth chapter is concerned with edge-based network SIR disease modeling.In Chapter 2, first, we consider the existence and uniqueness and global stability of equilibria for a multi-stage waterborne disease model. Furthermore, we propose a general class of multi-stage cholera spread model. Under biologically motivated assumptions, the basic reproduction number is derived, and the global stability of equilibria is obtained by using global Lyapunov functions, Kirchhoff’s matrix tree theorem and LaSalle Invariance Principle. Next, we study the global dynamics of multi-group SEI animal disease models with indirect transmission. Under biologically motivated assumptions, the basic reproduction number is derived, and the global stability of the disease-free equilibrium is proved. On the other hand, since the weight matrix for weighted digraphs may be reducible, a new combinatorial identity, combined with global Lyapunov functions and Kirchhoff’s matrix tree theorem, is employed to show the global stability of the endemic equilibrium.In Chapter 3, by using the comparison principle for Ordinary Differential Equations and Kirchhoff’s matrix tree theorem in Digraph, we investigate the global dynamics of several spread models on complex networks. First, we obtain the global dynamics of a network water-borne diseases spread model with birth and death, and discuss the effect of various immuniza-tion strategies on disease spread. Second, we establish the global stability of a heterogeneous network disease model that incorporates balanced birth and death event and has infectious force both in the infected state and carrier state. When individual birth and death is ignored, the final epidemic size formula is obtained. The effect of various immunization strategies is also investigated and compared by numerical simulations. In the end, by utilizing Lyapunov functions and Kirchhoff’s matrix tree theorem, the global stability of the virose equilibrium for a network-based computer viruses spread model is discussed. Applying comparison principle, a new brief proof for the global stability of the virus-free equilibrium is provided.In Chapter 4, first, an edge-based SIR disease model on configuration networks and two node-based SIR disease models on degree correlated networks are revisited, and then two growing network models that will lead to the appearance of degree correlations are introduced. By employing continuous-time stochastic simulation algorithms, the predicted behavior of an edge-based and two node-based SIR models on correlated networks is compared with the ensemble average of 100 runs of stochastic simulations. The simulation results show that, on degree correlated networks, the edge-based SIR model that only utilizes the degree distribution agrees much better than the two node-based SIR models that also utilizes the degree correlation information with stochastic SIR simulations in many respects, including the initial exponential growth rate, the epidemic peak and the arrival time of the peak, which suggests that the Miller model developed for configuration model or uncorrelated networks may have wider applicability than it should have been.更多还原