对逼近网络传染病动力学建模及全局分析

发布时间：2018-09-01 13:53

【摘要】：传染病在人群中的传播可以看作是疾病沿网络连边按某种规律传播的行为，以网络边为基础的传染病模型刻画了不同的网络拓扑结构与疾病传播的相互影响，比传统的均匀混合模型更加实际。基于对逼近建立的传染病模型是网络传染病模型的一种，它是以网络中不同属性的边作为变量并研究其在网络中的动态演化。针对对逼近模型的研究已有很多，但关于这些模型的动力学分析却没有，，因此文章针对规则网、随机网上的SIS对逼近模型做了全局动力学分析，并做了相应的数值模拟，补充并丰富了网络上对逼近传染病模型的研究。此外，随着对传染病研究的不断深入，随机性在传染病建模中的角色越来越重要，随机性模型的优点在于考虑了确定性模型没有考虑的不可或缺随机因素。为此文章应用随机过程中的马尔科夫过程构建了有出生和死亡的SIS随机网络对逼近传染病模型并得到了相应的确定性模型，最后对模型做了动力学分析和数值模拟，结果不仅研究了动态网络中不同属性边的变化情况而且丰富了基于对逼近的网络传染病建模方法。第一章，首先介绍研究网络传染病模型的意义、反应网络拓扑结构的统计学特征、以及经典的四种网络。然后介绍网络对逼近传染病模型的发展概况，进而介绍随机过程中的马尔科夫过程及相关基础知识，最后介绍随机对逼近传染病模型的发展概况。第二章，针对规则和随机网上的SIS对逼近模型，根据两种网络的拓扑特性对模型降维封闭，然后得到疾病传播的基本再生数并应用Lyapunov函数、Dulac函数等动力学理论知识分析且证明了模型的全局动力学性态，最后通过数值仿真验证了理论的正确性，为对逼近模型的研究提供了理论基础。第三章，根据随机过程中的马尔科夫过程，应用转移概率，Q矩阵，Kolmogorov方程及矩生成函数等数学工具推导出有出生和死亡的SIS网络传染病对逼近确定性模型，然后得到疾病的基本再生数并用数值模拟加以验证。
[Abstract]:The transmission of infectious diseases in the population can be regarded as the behavior of disease spreading along the network according to a certain law. The infectious disease model based on the network edge depicts the interaction between different network topology and disease transmission. It is more practical than the traditional uniform mixing model. The infectious disease model based on pair approximation is a kind of network infectious disease model. It takes the edges of different attributes in the network as the variable and studies its dynamic evolution in the network. There have been many researches on approximation models, but there is no dynamic analysis on these models. Therefore, in this paper, the global dynamics analysis and numerical simulation of approximation models are made for SIS on random networks. It complements and enriches the research of approaching infectious disease model on the network. In addition, with the deepening of infectious disease research, randomness plays an increasingly important role in infectious disease modeling. The advantage of stochastic model is that it takes into account the indispensable random factors that are not taken into account by deterministic models. In this paper, the SIS stochastic network pair with birth and death is used to approximate the infectious disease model, and the corresponding deterministic model is obtained. Finally, the dynamic analysis and numerical simulation of the model are done. Results not only the variation of different attribute edges in dynamic network is studied, but also the modeling method of network infectious disease based on pair approximation is enriched. In the first chapter, we introduce the significance of studying the network infectious disease model, the statistical characteristics of the topological structure of the response network, and the classical four kinds of networks. Then it introduces the development of network pair approach infectious disease model, then introduces the Markov process and related basic knowledge in random process, and finally introduces the development of stochastic pair approaching infectious disease model. In the second chapter, according to the topological characteristics of the two networks, the model is closed to the SIS pair approximation model based on the rules and stochastic networks. Then the basic reproduction number of disease transmission is obtained and the global dynamic behavior of the model is proved by using the knowledge of dynamics such as Lyapunov function and Dulac function. Finally, the correctness of the theory is verified by numerical simulation. It provides a theoretical basis for the study of approximation model. In chapter 3, according to Markov process in stochastic process, the deterministic model of infectious disease pair approximation in SIS network with birth and death is derived by using the transfer probability Q matrix Kolmogorov equation and moment generating function. Then the basic regeneration number of the disease is obtained and verified by numerical simulation.
【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

【共引文献】