【作者】 李星；

【导师】 靳祯；

【作者基本信息】 山西大学 ， 应用数学， 2017， 硕士

【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.更多还原

【Abstract】 Epidemics have been along with people’s production and livelihood since ancient times.Some disrupt the regular life of the public and some have threat to human life.Therefore,it has been always focus to know the mechanism of the spread of infectious diseases and make corresponding prevention and controlling measures.As epidemics are not feasible by experiment in the crowd,it is a valid and effective method to research and forecast epidemics by dynamical models.Taking into account the heterogeneity of contacts between individuals,network-based dynamical epidemic models are models which are more in tune with the real life.Pair-approximation model is one of network-based epidemic models,which looks up nodes and edges of networks as variables to explore the laws of the spread of epidemics.The accuracy and rationality of this kind of models depend largely on the pair-approximation methods.In this paper,we compare the accuracy,advantages and disadvantages of different pair-approximation methods by SIS epidemic models on homogeneous,heterogeneous and clustering networks.Chapter 1,firstly,gives the research background 、 significance of epidemic models on network.Then,it introduces several topological network parameters,four traditional network models and the classification of network-based epidemic models.Finally,the progress of pair-approximation models and the current research of approximating methods are depicted,further leading to the focus of our works.In Chapter 2,via SIS pair-approximation models(P-PW,B-PW and ER-PW),the accuracy of three approximation methods in the homogeneous networks is compared.The first method is based on the number of infected neighborhoods of individuals following Poisson distribution.The second is based on the number of infected neighborhoods of individuals following multinomial distribution.The last one is based on the mean-field theory.Then we find the error of the pair-approximation model under Poisson distribution smallest,i.e.the accuracy of approximation methods under Poisson distribution is highest.In chapter 3,we compare the accuracy of heterogeneous approximation method Keeling proposed and super compact pairwise method Simon and Kiss proposed on heterogeneous networks by SIS models;Firstly,the detailed derivation of the super compact pair-approximation formula of Simon and Kiss is given.Then,based on these two approximation methods,K(10)1 dimensions SIS pairwise models(H-PW and HSH-PW)and three dimensions SIS pairwise model(HSL-PW)are given.It is found by calculating the basic reproduction numbers and simulating for the three models that the super compact pair-approximation method can not only reduce the dimension but make the SIS epidemic models incorporate more network topology parameters without losing models accuracy.In chapter 4,for clustering network,we similarly compare the accuracy of approximation methods Keeling presented and super compact pairwise method Sherborne,Blyuss and Kiss presented via SIS pair-approximation model.Utilizing that two methods containing clustering coefficient,we derived three SIS pairwise models with clustering: K(10)1 dimensions SIS pairwise models with clusters(HC-PW and HSHC-PW)and three dimensions SIS pairwise model with clustering(HSLC-PW).Then,simulation verifies the rationality of the model.Finally,it is found via error analysis that super compact pair-approximation method is the most accurate.Chapter 5,gives the conclusions and the prospects.更多还原