【摘要】：自古以來(lái),傳染病都伴隨著人們的生產(chǎn)和生活,有的擾亂人們的健康生活,有的則威脅著人的生命.因此,了解傳染病的傳播機(jī)理,制定相應(yīng)的預(yù)防控制措施始終是人們關(guān)注的焦點(diǎn).由于傳染病在人群中實(shí)驗(yàn)的不可行性,通過(guò)動(dòng)力學(xué)模型來(lái)了解疾病在人群中的傳播規(guī)律進(jìn)而做出預(yù)測(cè)是一種行之有效的手段和方法.考慮到人與人之間接觸的異質(zhì)性,網(wǎng)絡(luò)傳染病動(dòng)力學(xué)模型是更加切合實(shí)際的傳染病模型,而對(duì)逼近或?qū)颇Ｐ褪沁@類(lèi)傳染病模型的一種,以網(wǎng)絡(luò)中的單節(jié)點(diǎn)和節(jié)點(diǎn)之間的邊作為變量來(lái)研究疾病的傳播規(guī)律,其精確合理的關(guān)鍵是逼近方法的選擇.因此,本文的主要內(nèi)容是借助SIS對(duì)逼近傳染病模型比較均勻、異質(zhì)和聚類(lèi)網(wǎng)絡(luò)上不同逼近方法的精度和優(yōu)劣性.第一章,首先給出傳染病模型在網(wǎng)絡(luò)上的研究背景、研究意義;然后介紹網(wǎng)絡(luò)的幾個(gè)拓?fù)鋮?shù)和四個(gè)傳統(tǒng)模型,以及傳染病模型在網(wǎng)絡(luò)中的分類(lèi);最后,介紹對(duì)逼近或?qū)颇Ｐ瓦M(jìn)展以及現(xiàn)有的幾種對(duì)逼近或?qū)品椒ㄟM(jìn)而引出本文的研究重點(diǎn).第二章,借助SIS對(duì)逼近(Pair-Approximation)或?qū)苽魅静∧Ｐ?分別是Poisson分布下的對(duì)近似模型P-PW、Multinomial分布下的對(duì)近似模型B-PW、以平均域(場(chǎng))思想為基礎(chǔ)的對(duì)近似模型MF-PW),在均勻網(wǎng)絡(luò)上比較節(jié)點(diǎn)的染病態(tài)鄰居個(gè)數(shù)服從Poisson分布、節(jié)點(diǎn)的染病態(tài)鄰居個(gè)數(shù)服從Multinomial分布、以平均域思想為基礎(chǔ)的三種近似方法的精確度和優(yōu)劣性.經(jīng)理論分析和模擬得出泊松分布下模型的基本再生數(shù)大于其他兩種近似下的基本再生數(shù),并且節(jié)點(diǎn)相鄰的染病節(jié)點(diǎn)個(gè)數(shù)服從Poisson分布時(shí)的對(duì)近似方法精確度最高.第三章,借助SIS對(duì)逼近(Pair-Approximation)或?qū)苽魅静∧Ｐ?比較異質(zhì)網(wǎng)絡(luò)上由Keeling提出的異質(zhì)對(duì)近似近方法和Simon和Kiss提出的超緊對(duì)近似近方法的精度和優(yōu)劣性.首先給出了Simon和Kiss提出的超緊對(duì)近似公式的詳細(xì)推導(dǎo)過(guò)程,之后利用這兩種近似逼近方法得到三種模型:K(10)1維異質(zhì)SIS對(duì)逼近模型H-PW、K(10)1維異質(zhì)SIS超緊對(duì)逼近模型HSH-PW、三維異質(zhì)SIS超緊對(duì)逼近模型HSL-PW.通過(guò)理論分析和模擬發(fā)現(xiàn)兩種近似下所得模型的基本再生數(shù)相同,兩種近似方法的精度相差不大,但在超緊對(duì)近似下的模型維數(shù)低易分析,且包含更多的網(wǎng)絡(luò)的拓?fù)鋮?shù).第四章,針對(duì)聚類(lèi)網(wǎng)絡(luò)上的Keeling提出的異質(zhì)對(duì)近似方法與Sherborne、Blyuss和Kiss提出的異質(zhì)超緊對(duì)近似方法,給出近似后的三種SIS對(duì)近似傳染病模型:K(10)1維含有聚類(lèi)的異質(zhì)對(duì)逼近模型HC-PW、K(10)1維含有聚類(lèi)的異質(zhì)超緊對(duì)逼近模型HSHC-PW、三維含有聚類(lèi)的異質(zhì)SIS超緊對(duì)逼近模型HSLC-PW.進(jìn)而通過(guò)數(shù)值和隨機(jī)模擬驗(yàn)證模型的合理性,并通過(guò)誤差分析得出含聚類(lèi)的超緊對(duì)逼近公式更精確.第五章,總結(jié)本文,給出展望.
[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.
【學(xué)位授予單位】：山西大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175

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本文編號(hào)：2281976