一类流行病数学模型的研究及应用
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摘要
近几年，一些新型的传染病如SARS，禽流感出现了，在很短的时间里，迅速的在我国和全球一些国家爆发流行，极大的威胁到人类的身体健康和生命安全，直接影响到社会稳定和经济发展。对于这种新的突发传染病，人类对它的防治还处于初步摸索阶段。如何有效地从宏观上了解和掌握这些流行病的传播规律，控制传染病的蔓延就显得越来越重要。数学模型作为研究流行病动态规律和机理的有效手段，近些年以来，已经在控制流行病的蔓延方面显现出越来越重要的作用。
本文主要研究了固定人口下SIR传染病流行的数学模型及其在SARS中的应用。首先介绍了流行病学数学模型的建立、应用、发展的回顾及展望。第二章则介绍了传染病流行的随机模型的一般形式及结果。第三章讨论了SIR模型的解及其参数的贝叶斯估计。接着第四章和第五章分别用SIR模型和神经网络模型对北京市SARS疫情流行规律的拟合及预测进行了探讨。最后一章给出了SIR模型对流行病探讨的作用进行了分析。
本文充分利用了贝叶斯统计的理论，使用了矩阵的分块方法给出了SIR模型参数的贝叶斯估计。例于表明，该方法有很强的实用性。
In recent years, some new infectious diseases like SARS and the Bird Flu appeared. They erupted rapidly and has been widespread in our country and some other countries in short time. They threat the people's health and safety greatly and affect social stability and the economical development directly. Humanity is in the first stage in finding out how to prevent and control the new-emerged infectious diseases. It is more and more important to master these epidemic diseases' rule effectively and prevent the epidemic diseases to spread widely.The mathematical model has the important part in controlling the epidemics' prevailing situation as the effective method of researching the dynamic rule and the mechanism of these epidemics in the recent years .
This article mainly studied the SIR mathematical model of infectious diseases under the fixed population and its application in SAKS. Firstly it introduced its establishment, the application, the review and the forecast of its development .The second chapter introduced the general form and the result of the general stochastic model of the infectious diseases. The third chapter discussed the SIR model's solution and its parameters' Baycsion estimations. Then the fourth and the fifth chapter used the SIR model and the nerve network model separately fitting and forecasting the prevailing situation of SARS in Beijing . Lastly it analyzed the effect of the SIR model with which we discussed the epidemic diseases.
This article fully used the Bayes Theory and the method of matrix patition to produce the Baycsion estimations of the SIR model's parameters. The example indicated that, this method is very effective in practice.
本文主要研究了固定人口下SIR传染病流行的数学模型及其在SARS中的应用。首先介绍了流行病学数学模型的建立、应用、发展的回顾及展望。第二章则介绍了传染病流行的随机模型的一般形式及结果。第三章讨论了SIR模型的解及其参数的贝叶斯估计。接着第四章和第五章分别用SIR模型和神经网络模型对北京市SARS疫情流行规律的拟合及预测进行了探讨。最后一章给出了SIR模型对流行病探讨的作用进行了分析。
本文充分利用了贝叶斯统计的理论，使用了矩阵的分块方法给出了SIR模型参数的贝叶斯估计。例于表明，该方法有很强的实用性。
In recent years, some new infectious diseases like SARS and the Bird Flu appeared. They erupted rapidly and has been widespread in our country and some other countries in short time. They threat the people's health and safety greatly and affect social stability and the economical development directly. Humanity is in the first stage in finding out how to prevent and control the new-emerged infectious diseases. It is more and more important to master these epidemic diseases' rule effectively and prevent the epidemic diseases to spread widely.The mathematical model has the important part in controlling the epidemics' prevailing situation as the effective method of researching the dynamic rule and the mechanism of these epidemics in the recent years .
This article mainly studied the SIR mathematical model of infectious diseases under the fixed population and its application in SAKS. Firstly it introduced its establishment, the application, the review and the forecast of its development .The second chapter introduced the general form and the result of the general stochastic model of the infectious diseases. The third chapter discussed the SIR model's solution and its parameters' Baycsion estimations. Then the fourth and the fifth chapter used the SIR model and the nerve network model separately fitting and forecasting the prevailing situation of SARS in Beijing . Lastly it analyzed the effect of the SIR model with which we discussed the epidemic diseases.
This article fully used the Bayes Theory and the method of matrix patition to produce the Baycsion estimations of the SIR model's parameters. The example indicated that, this method is very effective in practice.
引文
[1]A.T Bharucha-Rcid马尔可夫过程初步及其应用[M]上海科学挂术出版社，1979
[2]Severo,N.C Two theorems on solutions of differential-difference equations and applications to epidemic ihcory.[J]J.Appl.Prob.1967,4,271-280
[3]Severo,N.C A recursion theorem on solving differential-difference equations and applications to some stochastic processes,[J]J,Appl.Prob.1969,6,673-681
[4]L,Billard,Factorial moments and probabilities for the general stochastic epidemic.[J]
[5]James.O.Bergcr Statistical Decision Theory and Beyasian Analysis [M]Springer—Vcrlag New Yorklnc 1985
[6]Ingemar Nasell.Stochastic models of some endemic infeclions [J].Math.Biosci,2002,179:1-19
[7]Linda J.S.Allen,Amy M.Burgin.Comparison of deterministic and stochastic SIS and SIR models in discrete time [J].Math.Biosci,2000,163:1-33
[8]Frank Ball ,Peter Donnelly.Strong approximations for epidemic models [J].Stochastic Proc.Appl,1995,55:1-21
[9]耿贯一．流行病学．[M]．第一卷．第2版．北京：人民卫生出版社,1997．279-297.
[10]Lilienfeld AM．俞焕文译．流行病学基础．[M]．上海:上海科技出版社1981．227-238．
[11]Herbert W,hethcotc.The Mathematics of Infectious Diseases.IJ],SLAM REVIEW,2000,4; 599-653
[12]闻新，周露等．MATLAB神经网络应用设计[M]．北京：科学出版社,2000.
[13]阎平凡，张长水．人工神经网络与模拟进化计算[M]．北京：清华大学出版社,2000．
[14]崔恒建．李仲米．杨华,等．SARS疫情预测预报中的分段非线性回归方法．[J].遥感学报,2003，7；245-250．
[15]李仲来，崔恒建，杨华,等．SARS预测的sI模型和分段sI模型．[J]．遥感学报，2003，7：345-349．
[16]杨华,李小文，施宏,婷．SARS沿交通线的“飞点”传播模型．[J]．遥感学报,2003．7:251-255．
[17]吴开琛．吴开录，陈文江，等．SARS传播数学模型与流行趋势预测研究．[J]．中国热带医学,2003．3：421-426．
[18]Me Kendrick,A.G:Applications of mathematics to medical problems,Proc.Edinburgh Maih.Soc.1926,44,98-130.
[19]Bailey,N.T.J,The mathematical theory of epidemics,Hafncr Publishing Company,New York 1957.
[20]飞思科技产品研发中心．Matlab 6．5辆助神经网格分析与设计[M]，北京电子工业出版社．2003年．
[21]周开利,康耀红，神经网络模型及其MAFLAB仿真程序设计．[M]．北京．清化大学出版社．2003年．
[22]高隽，人工神经网络原理及仿真实例．[M]．北京．机械工业出版社,2003年．
[23]V.SISKIND.A Solution of the general stochastic epidemic [J].Biometrika,1965,52,613.
[24]E.Y.RODIN.Continuous model of an epidemic.[J].Mathl.Comput.Modelling ,1989,12,247-252.
[25]Paul.SF.Yip,Qizhi chen,Statistical infenence for a multitype epidemic model.[J].Jornnal of statistical Planning and Inference,1998,71,229-244.
[26]Akc Svensson.On the Simultaneous distribution of size and costs of an Epidemic in a closed multigroup population.[J].Math,Biosci,1995.127,167-180.
[27]Ingemar Nasell.On the quasi-stationary distribution of the stochastic logistic epidemic.[J]Math.Biosci,1999,156,21-40.
[28]Jianquan Li,zhien Ma,Qualitative analyses of SIS epidemic model with vaccination and varying total population size.[J].Mathematical and Computer Modlling ,2002,35,1235-1243.
[29]Norman T.J.Bailey.The simple stochastic epidemic:a complete solution in terms of known functions,[J]-Biometrika,1963.50,235-240.
[30]J.Gami,D.Jerwood.The cost of a general stochastic epidemic.[J].J.Appl.Prob,1972,9,257-269.
[31]D.A.Griffiths.Multivariate birth-and-death processes as approximations to epidemic processes[J].J.Appl.Prob.1973,10,15-26.
[32]R.H.Norden.On the distribution of the time to extinction in the stochastic logistic population model.[J].Adv.Appl.Prob.1982,14,687-708.
[33]李琼，叶鹰等，一类SIR流行病模型及参数的贝叶斯估计．[J].湖北民族学院学报，2005，4．
[34]李琼．叶鹰等．一粪SIR传染病模型的解．[J]．应用数学,2005．增
[35]陈奇志，随机模型在非典型肺炎预测及疫情分析中的应用.[J]．北京大学学报(医学版)．2003，增
[36]韩卫国等，SARS传播时间过程的参数反演和趋势预测．[J]．地球科学进展,2004，12
[37]姚玉华等．SARS流行病传染动力学模型．[J]．数学的实践与认识，2004，3．
[38]石耀霖，SARS传染扩散的动力学模型．[J]．科学通报．2003，7
[39]赵锡英等．SARS疫情分析模型．[J]．兰州工业高等专科学校学报，2004，9
[40]李建逵等，SARS病毒传染的数学模型．[J]．科技情报开发与经济，2004．2
[41]耻建平等，流行病学中催化模型的研究与应用．[J]．解放军预防医学杂志，1999，2.
[42]黄德生婷，Logistic回归模型拟台SARS发病及流行特征．[J]．中国公共卫生．2003，6
[43]王建锋，SARS流行预测分析．[J]．中国工程科学，2003，8
[44]王正行，北京SARS疫情走势的模型分析与预测.[J].物理学与社会，2003，5
[45]赵晓飞等,SARS疫情的实证分析和预测．[J].北京大学学报(医学版)，2003，增
[46]陈庆华,“SARS”病死率数学模型实力分析.[J]2003,8
[47]半仲来等,SARS预测的数学模型及其研究进展.[J]．2004，6
[2]Severo,N.C Two theorems on solutions of differential-difference equations and applications to epidemic ihcory.[J]J.Appl.Prob.1967,4,271-280
[3]Severo,N.C A recursion theorem on solving differential-difference equations and applications to some stochastic processes,[J]J,Appl.Prob.1969,6,673-681
[4]L,Billard,Factorial moments and probabilities for the general stochastic epidemic.[J]
[5]James.O.Bergcr Statistical Decision Theory and Beyasian Analysis [M]Springer—Vcrlag New Yorklnc 1985
[6]Ingemar Nasell.Stochastic models of some endemic infeclions [J].Math.Biosci,2002,179:1-19
[7]Linda J.S.Allen,Amy M.Burgin.Comparison of deterministic and stochastic SIS and SIR models in discrete time [J].Math.Biosci,2000,163:1-33
[8]Frank Ball ,Peter Donnelly.Strong approximations for epidemic models [J].Stochastic Proc.Appl,1995,55:1-21
[9]耿贯一．流行病学．[M]．第一卷．第2版．北京：人民卫生出版社,1997．279-297.
[10]Lilienfeld AM．俞焕文译．流行病学基础．[M]．上海:上海科技出版社1981．227-238．
[11]Herbert W,hethcotc.The Mathematics of Infectious Diseases.IJ],SLAM REVIEW,2000,4; 599-653
[12]闻新，周露等．MATLAB神经网络应用设计[M]．北京：科学出版社,2000.
[13]阎平凡，张长水．人工神经网络与模拟进化计算[M]．北京：清华大学出版社,2000．
[14]崔恒建．李仲米．杨华,等．SARS疫情预测预报中的分段非线性回归方法．[J].遥感学报,2003，7；245-250．
[15]李仲来，崔恒建，杨华,等．SARS预测的sI模型和分段sI模型．[J]．遥感学报，2003，7：345-349．
[16]杨华,李小文，施宏,婷．SARS沿交通线的“飞点”传播模型．[J]．遥感学报,2003．7:251-255．
[17]吴开琛．吴开录，陈文江，等．SARS传播数学模型与流行趋势预测研究．[J]．中国热带医学,2003．3：421-426．
[18]Me Kendrick,A.G:Applications of mathematics to medical problems,Proc.Edinburgh Maih.Soc.1926,44,98-130.
[19]Bailey,N.T.J,The mathematical theory of epidemics,Hafncr Publishing Company,New York 1957.
[20]飞思科技产品研发中心．Matlab 6．5辆助神经网格分析与设计[M]，北京电子工业出版社．2003年．
[21]周开利,康耀红，神经网络模型及其MAFLAB仿真程序设计．[M]．北京．清化大学出版社．2003年．
[22]高隽，人工神经网络原理及仿真实例．[M]．北京．机械工业出版社,2003年．
[23]V.SISKIND.A Solution of the general stochastic epidemic [J].Biometrika,1965,52,613.
[24]E.Y.RODIN.Continuous model of an epidemic.[J].Mathl.Comput.Modelling ,1989,12,247-252.
[25]Paul.SF.Yip,Qizhi chen,Statistical infenence for a multitype epidemic model.[J].Jornnal of statistical Planning and Inference,1998,71,229-244.
[26]Akc Svensson.On the Simultaneous distribution of size and costs of an Epidemic in a closed multigroup population.[J].Math,Biosci,1995.127,167-180.
[27]Ingemar Nasell.On the quasi-stationary distribution of the stochastic logistic epidemic.[J]Math.Biosci,1999,156,21-40.
[28]Jianquan Li,zhien Ma,Qualitative analyses of SIS epidemic model with vaccination and varying total population size.[J].Mathematical and Computer Modlling ,2002,35,1235-1243.
[29]Norman T.J.Bailey.The simple stochastic epidemic:a complete solution in terms of known functions,[J]-Biometrika,1963.50,235-240.
[30]J.Gami,D.Jerwood.The cost of a general stochastic epidemic.[J].J.Appl.Prob,1972,9,257-269.
[31]D.A.Griffiths.Multivariate birth-and-death processes as approximations to epidemic processes[J].J.Appl.Prob.1973,10,15-26.
[32]R.H.Norden.On the distribution of the time to extinction in the stochastic logistic population model.[J].Adv.Appl.Prob.1982,14,687-708.
[33]李琼，叶鹰等，一类SIR流行病模型及参数的贝叶斯估计．[J].湖北民族学院学报，2005，4．
[34]李琼．叶鹰等．一粪SIR传染病模型的解．[J]．应用数学,2005．增
[35]陈奇志，随机模型在非典型肺炎预测及疫情分析中的应用.[J]．北京大学学报(医学版)．2003，增
[36]韩卫国等，SARS传播时间过程的参数反演和趋势预测．[J]．地球科学进展,2004，12
[37]姚玉华等．SARS流行病传染动力学模型．[J]．数学的实践与认识，2004，3．
[38]石耀霖，SARS传染扩散的动力学模型．[J]．科学通报．2003，7
[39]赵锡英等．SARS疫情分析模型．[J]．兰州工业高等专科学校学报，2004，9
[40]李建逵等，SARS病毒传染的数学模型．[J]．科技情报开发与经济，2004．2
[41]耻建平等，流行病学中催化模型的研究与应用．[J]．解放军预防医学杂志，1999，2.
[42]黄德生婷，Logistic回归模型拟台SARS发病及流行特征．[J]．中国公共卫生．2003，6
[43]王建锋，SARS流行预测分析．[J]．中国工程科学，2003，8
[44]王正行，北京SARS疫情走势的模型分析与预测.[J].物理学与社会，2003，5
[45]赵晓飞等,SARS疫情的实证分析和预测．[J].北京大学学报(医学版)，2003，增
[46]陈庆华,“SARS”病死率数学模型实力分析.[J]2003,8
[47]半仲来等,SARS预测的数学模型及其研究进展.[J]．2004，6
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