对于SIS传染病模型几种不同逼近的比较
发布时间：2018-10-19 18:24
【摘要】：自古以来,传染病都伴随着人们的生产和生活,有的扰乱人们的健康生活,有的则威胁着人的生命.因此,了解传染病的传播机理,制定相应的预防控制措施始终是人们关注的焦点.由于传染病在人群中实验的不可行性,通过动力学模型来了解疾病在人群中的传播规律进而做出预测是一种行之有效的手段和方法.考虑到人与人之间接触的异质性,网络传染病动力学模型是更加切合实际的传染病模型,而对逼近或对近似模型是这类传染病模型的一种,以网络中的单节点和节点之间的边作为变量来研究疾病的传播规律,其精确合理的关键是逼近方法的选择.因此,本文的主要内容是借助SIS对逼近传染病模型比较均匀、异质和聚类网络上不同逼近方法的精度和优劣性.第一章,首先给出传染病模型在网络上的研究背景、研究意义;然后介绍网络的几个拓扑参数和四个传统模型,以及传染病模型在网络中的分类;最后,介绍对逼近或对近似模型进展以及现有的几种对逼近或对近似方法进而引出本文的研究重点.第二章,借助SIS对逼近(Pair-Approximation)或对近似传染病模型(分别是Poisson分布下的对近似模型P-PW、Multinomial分布下的对近似模型B-PW、以平均域(场)思想为基础的对近似模型MF-PW),在均匀网络上比较节点的染病态邻居个数服从Poisson分布、节点的染病态邻居个数服从Multinomial分布、以平均域思想为基础的三种近似方法的精确度和优劣性.经理论分析和模拟得出泊松分布下模型的基本再生数大于其他两种近似下的基本再生数,并且节点相邻的染病节点个数服从Poisson分布时的对近似方法精确度最高.第三章,借助SIS对逼近(Pair-Approximation)或对近似传染病模型,比较异质网络上由Keeling提出的异质对近似近方法和Simon和Kiss提出的超紧对近似近方法的精度和优劣性.首先给出了Simon和Kiss提出的超紧对近似公式的详细推导过程,之后利用这两种近似逼近方法得到三种模型:K(10)1维异质SIS对逼近模型H-PW、K(10)1维异质SIS超紧对逼近模型HSH-PW、三维异质SIS超紧对逼近模型HSL-PW.通过理论分析和模拟发现两种近似下所得模型的基本再生数相同,两种近似方法的精度相差不大,但在超紧对近似下的模型维数低易分析,且包含更多的网络的拓扑参数.第四章,针对聚类网络上的Keeling提出的异质对近似方法与Sherborne、Blyuss和Kiss提出的异质超紧对近似方法,给出近似后的三种SIS对近似传染病模型:K(10)1维含有聚类的异质对逼近模型HC-PW、K(10)1维含有聚类的异质超紧对逼近模型HSHC-PW、三维含有聚类的异质SIS超紧对逼近模型HSLC-PW.进而通过数值和随机模拟验证模型的合理性,并通过误差分析得出含聚类的超紧对逼近公式更精确.第五章,总结本文,给出展望.
[Abstract]:Since ancient times, infectious diseases have been accompanied by people's production and life, some disturbing people's healthy life, some threatening people's lives. Therefore, understanding the transmission mechanism of infectious diseases and formulating corresponding prevention and control measures have always been the focus of attention. Because of the infeasibility of the experiment of infectious diseases in the population, it is an effective means and method to understand the spread law of the disease in the population by using the kinetic model and to make the prediction. Considering the heterogeneity of human contact, the dynamic model of an infectious disease is a more realistic one, and the approximation or pair model is one of the models of such an infectious disease, Taking the edges between single node and node as variables to study the law of disease transmission, the key to the accuracy and reasonableness is the choice of approximation method. Therefore, the main content of this paper is to use SIS to approximate the infectious disease model homogeneously, heterogeneity and clustering network of different approximation methods accuracy and advantages and disadvantages. In the first chapter, the research background and significance of infectious disease model in the network are given. Then, several topological parameters and four traditional models of the network are introduced, and the classification of the infectious disease model in the network is also introduced. This paper introduces the progress of approximation or pair approximation models and the existing methods of pair approximation or pair approximation, which leads to the emphasis of this paper. Chapter 2 By means of SIS pair approximation (Pair-Approximation) or pair approximate infectious disease model (P-PWM Multinomial distribution under Poisson distribution respectively, pair approximate model MF-PW based on the mean domain (field) idea), the nodes are compared on the uniform network. The number of diseased neighbors is distributed from Poisson. The number of diseased neighbors of nodes is based on the Multinomial distribution and the accuracy and superiority of the three approximate methods based on the mean domain theory. The theoretical analysis and simulation show that the number of basic regeneration in Poisson distribution is larger than that in the other two approximations, and the accuracy of the approximate method is the highest when the number of infected nodes adjacent to the nodes is obtained from the Poisson distribution. In chapter 3, by means of SIS pair approximation (Pair-Approximation) or pair approximate infectious disease model, we compare the accuracy and merits of the approaches proposed by Keeling and Simon and Kiss. Firstly, the derivation process of the supercompact pair approximation formula proposed by Simon and Kiss is given in detail. Then, using these two approximate approximation methods, three models: K (10) 1-dimensional heterogeneous SIS pair approximation model H-PWK (10) 1-dimensional heterogeneous SIS hypercompact pair approximation model HSH-PW, 3D heterogeneous SIS hypercompact pair approximation model HSL-PW. are obtained. Through theoretical analysis and simulation, it is found that the basic reproduction number of the two approximate models is the same, and the accuracy of the two approximation methods is not different, but the model dimension under the supercompact approximation is easy to analyze and contains more topological parameters of the network. In chapter 4, the hetero-pair approximation method proposed by Keeling and the hetero-compact pair approximation method proposed by Sherborne,Blyuss and Kiss are discussed. In this paper, three approximate SIS pair approximation models of infectious diseases,: K (10) 1-D hetero-pair approximation model HC-PW,K (10) 1-dimensional heterocompact-pair approximation model with clustering, HSHC-PW, model HSHC-PW, 3-dimensional hetero-SIS hypercompact pair approximation model HSLC-PW. with clustering are given. Furthermore, the rationality of the model is verified by numerical and stochastic simulation, and the formula of super-compact pair approximation with clustering is obtained by error analysis. The fifth chapter, summarizes this article, gives the prospect.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2281976
[Abstract]:Since ancient times, infectious diseases have been accompanied by people's production and life, some disturbing people's healthy life, some threatening people's lives. Therefore, understanding the transmission mechanism of infectious diseases and formulating corresponding prevention and control measures have always been the focus of attention. Because of the infeasibility of the experiment of infectious diseases in the population, it is an effective means and method to understand the spread law of the disease in the population by using the kinetic model and to make the prediction. Considering the heterogeneity of human contact, the dynamic model of an infectious disease is a more realistic one, and the approximation or pair model is one of the models of such an infectious disease, Taking the edges between single node and node as variables to study the law of disease transmission, the key to the accuracy and reasonableness is the choice of approximation method. Therefore, the main content of this paper is to use SIS to approximate the infectious disease model homogeneously, heterogeneity and clustering network of different approximation methods accuracy and advantages and disadvantages. In the first chapter, the research background and significance of infectious disease model in the network are given. Then, several topological parameters and four traditional models of the network are introduced, and the classification of the infectious disease model in the network is also introduced. This paper introduces the progress of approximation or pair approximation models and the existing methods of pair approximation or pair approximation, which leads to the emphasis of this paper. Chapter 2 By means of SIS pair approximation (Pair-Approximation) or pair approximate infectious disease model (P-PWM Multinomial distribution under Poisson distribution respectively, pair approximate model MF-PW based on the mean domain (field) idea), the nodes are compared on the uniform network. The number of diseased neighbors is distributed from Poisson. The number of diseased neighbors of nodes is based on the Multinomial distribution and the accuracy and superiority of the three approximate methods based on the mean domain theory. The theoretical analysis and simulation show that the number of basic regeneration in Poisson distribution is larger than that in the other two approximations, and the accuracy of the approximate method is the highest when the number of infected nodes adjacent to the nodes is obtained from the Poisson distribution. In chapter 3, by means of SIS pair approximation (Pair-Approximation) or pair approximate infectious disease model, we compare the accuracy and merits of the approaches proposed by Keeling and Simon and Kiss. Firstly, the derivation process of the supercompact pair approximation formula proposed by Simon and Kiss is given in detail. Then, using these two approximate approximation methods, three models: K (10) 1-dimensional heterogeneous SIS pair approximation model H-PWK (10) 1-dimensional heterogeneous SIS hypercompact pair approximation model HSH-PW, 3D heterogeneous SIS hypercompact pair approximation model HSL-PW. are obtained. Through theoretical analysis and simulation, it is found that the basic reproduction number of the two approximate models is the same, and the accuracy of the two approximation methods is not different, but the model dimension under the supercompact approximation is easy to analyze and contains more topological parameters of the network. In chapter 4, the hetero-pair approximation method proposed by Keeling and the hetero-compact pair approximation method proposed by Sherborne,Blyuss and Kiss are discussed. In this paper, three approximate SIS pair approximation models of infectious diseases,: K (10) 1-D hetero-pair approximation model HC-PW,K (10) 1-dimensional heterocompact-pair approximation model with clustering, HSHC-PW, model HSHC-PW, 3-dimensional hetero-SIS hypercompact pair approximation model HSLC-PW. with clustering are given. Furthermore, the rationality of the model is verified by numerical and stochastic simulation, and the formula of super-compact pair approximation with clustering is obtained by error analysis. The fifth chapter, summarizes this article, gives the prospect.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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