耦合网络上的传染病传播动力学研究

发布时间：2018-06-22 09:27

本文选题：耦合网络 + 疾病传播　；参考：《上海大学》2016年博士论文

【摘要】：传染病不仅对人类的身体造成危害,还给社区、国家乃至世界带来灾难。因而研究传染病的传播机制和相应的防控措施有重大意义。复杂网络这一学科的兴起为传染病动力学注入了新的活力。因为很多真实世界中的网络都是互相作用互相依赖的,所以我们在研究这些网络上的疾病传播时,不能孤立地研究单个网络而忽略网络之间的相互联系与作用。能体现这种网络之间相互作用的网络就是耦合网络。本文研究了几类新型耦合网络上传染病传播的动力学性质,主要研究内容如下:1.考虑了包含两个子网的单向耦合网络上的疾病传播。很多人畜共患病具有动物传染人但人不传染动物的特征,针对这一特征我们提出了一种新型的耦合网络——单向耦合网络。通过平均场方法我们建立了疾病传播模型,证明了模型的无病平衡点和地方病平衡点的全局稳定性。通过再生矩阵的方法我们求出了模型基本再生数的准确表达式,发现整个网络的基本再生数R0是两个子网基本再生数的最大值,它与交叉传染率和交叉接触模式无关。这说明在单向耦合网络中,子网之间的交叉传染率不会对疾病的传播阈值产生任何影响。这是一个非常有意义的结论,因为之前有关耦合网络的研究都表明疾病的传播阈值与所有传染率有关。通过数值模拟我们还发现:如果内部接触模式是无标度的,那么R0随着内部传染率的增加而迅速增大;为了在人类中彻底消灭这些人畜共患传染病,我们必须同时在动物中消灭它们;禽对禽的传染率比禽对人的传染率对人类的最终平均感染密度有更大的影响,这一结论与人们的直觉相反,通常人们直觉上认为禽对人的传染对人类影响更大。2.研究了三部图网络上的疾病传播。很多媒介传染病在人类、媒介和动物三个种群中传播,针对这类传染病我们提出三部图网络。通过数学分析,我们发现三部图网络上模型的基本再生数不仅与二阶矩和平均度的比值k2k有关还与平均度k有关。这一结论和二部图网络上基本再生数只与k2k有关的结论是有本质区别的,说明三部图网络比二部图网络更有利于疾病的传播。通过数值模拟,我们还发现:在同样的接触模式下,四个交叉传染率对基本再生数有同样的影响;传染病在三个子网络上同时存在或同时消亡。3.研究了两个典型的三层耦合网络(串状耦合网络和环状耦合网络)上的疾病传播动力学。三层耦合网络可用来表示三个社团的疾病传播。我们研究发现:每个子网内部传染率的增加仅仅能有效影响子网本身和它的邻居;串状耦合网络中,随着网络尺寸或者传染率的增加,中间的子网比两端的子网对整个网络的基本再生数有更大的影响;随着网络尺寸的增加,环状耦合网络的基本再生数比串状耦合网络的基本再生数更大;然而,随着某个传染率的增加,串状耦合网络和环状耦合网络的基本再生数和平均感染密度是几乎相同的。
[Abstract]:Infectious diseases not only cause harm to the human body, but also bring disaster to the community, the country and the world. Therefore, it is of great significance to study the transmission mechanism of infectious diseases and the corresponding prevention and control measures. The rise of the subject of complex networks has injected new vitality into the dynamics of infectious disease dynamics. Because many networks in the real world are interacting and interacting with each other. When we study the spread of the diseases on these networks, we can not study the single network in isolation and ignore the interconnections and functions between the networks. The network which can reflect the interaction between these networks is the coupling network. The following contents are as follows: 1. consider the spread of disease on a one-way coupled network containing two subnets. Many zoonosis has the characteristics of animal infectious but non infectious animals. In view of this, we propose a new type of coupling network, one way coupling network. The disease propagation model is established by means of mean field method. The global stability of the disease-free equilibrium point and the endemic equilibrium point of the model is proved. Through the method of regenerative matrix, we find out the exact expression of the basic regeneration number of the model, and find that the basic regeneration number R0 of the whole network is the maximum value of the basic regeneration number of the two subnet, which is independent of the cross transmission rate and cross contact mode. In a unidirectional coupling network, the cross transmission rate between subnets does not have any effect on the spread threshold of the disease. This is a very meaningful conclusion, because previous studies on the coupling network have shown that the spread threshold of the disease is related to all the rates of transmission. In order to eliminate these zoonotic infectious diseases in humans, we must eliminate them at the same time in animals, and the transmission rate of poultry to birds has a greater impact on the final average density of human infection than the rate of transmission of birds to humans, which is contrary to people's intuition, R0. Generally, people intuitively think that avian infection has a greater impact on human beings..2. studies the spread of diseases on the network of three maps. Many vectors are transmitted among three species of humans, media and animals. We propose three graph networks for this kind of infectious disease. Through mathematical analysis, we find the basic regeneration of the models on the network of three graphs. The ratio k2k of the number not only with the two order moment is also related to the mean degree K. This conclusion and the basic regeneration number on the two graph network are essentially different from the k2k related conclusions, indicating that the three graph network is more beneficial to the spread of the disease than the two graph network. Four cross transmission rates have the same effect on the number of basic regeneration; infectious diseases exist at the same time or at the same time on three subnetworks..3. studies the disease spread dynamics on two typical three layer coupling networks (crosstick coupling networks and ring coupling networks). The three layer coupling network can be used to represent the spread of disease in three societies. It is found that the increase of each subnet's internal infection rate can only effectively affect the subnet itself and its neighbors; in the cascade coupling network, with the increase of the network size or the rate of infection, the subnet of the intermediate subnet has a greater influence on the basic regeneration number of the whole network than the subnet at both ends; with the increase of the network size, the base of the loop coupling network This regenerative number is larger than the basic regeneration number of the cascade coupling network. However, with the increase of a certain rate of infection, the basic regeneration number and the average infection density of the cross coupled and annular coupling networks are almost the same.
【学位授予单位】：上海大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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