具有非线性发生率的传染病模型的研究
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摘要
研究传染病的传播和预测传染病的发展趋势,是研究传染病的一个重要方面,它是政府部门和卫生医疗机构制定相应措施的基础。本文仍然采用建立数学模型的方法来研究传染病的传播机理。
全文共分五章:第一章介绍了传染病的研究概况和一些关于研究流行病模型的基本方法;第二章介绍了一些理论知识;第三章建立并分析了一类非线性高维自治微分系统SEIQR流行病传播数学模型。该模型采用的非线性发生率包括了传统的双线性发生率。通过对模型的全局性态的分析,我们对疾病的传播规律有一个透彻的了解,疾病的灭绝与否由基本再生数R_0决定。这在理论上表明,增大隔离强度和提高疾病的治愈率可以控制疾病的蔓延;第四章主要研究了具有常数输入和非线性发生率的SEIS传染病模型,利用LaSalle不变集原理、Liapunov函数、Hurwith判据证明了无病平衡点的全局渐近稳定性。同时利用第二加性复合矩阵原理证明了惟一地方病平衡点的全局渐近稳定性。本章结果表明,给予病人及时的治疗,提高治疗的有效性是控制疾病的有效方法;第五章主要研究了一类潜伏期和染病期均具有传染力的SEIS传染病模型。该模型的全局性态的分析显的更加复杂,通过理论分析和数值模拟我们发现给予病人及时有效的治疗和减少染病系数是控制疾病蔓延的有效方法。
Exploring epidemic spreading and predicting its development trend are important aspects for epidemic study,and they are the basis of control policy adopted by the government and medical department.The aim of this work is to construct several mathematical models in epidemiology and to analyze the asymptotic behavior of these models.
The full text is divided into five chapters:The first chapter introduced the study of infectious diseases and some methods about epidemiological models.The second chapter introduced some theoretical knowledge.The third chapter not only establish but also analyze a nonlinear high dimensional autonomous differential system SEIQR model in epidemiology.That the nonlinear incidence rate is used in the model,has included the traditional bilinear incidence rate.By analyzing global behavior of the model,we get a incisive comprehension about the disease propagation law and the disease extinction is determined by the basic reproduction number R_0.It showed that incresing isolation and improving the cure rate of the disease can control the spread of diaease;An SEIS epidemic model with constant input and nonlinear incidence rate is maily studied in the fourth chapter.The disease-free equilibrium is globally asymptotical stable by the means of Lasalle's invariant set theorem,Liapunov function and Hurwitze criterion.Using second additive compound matrix,it is proved that the unique endemic equilibrium is globally asymptotical stable.The results in this chapter suggest that giving the patients timely treatment and improving the cure rate are an effective way to control the disease.A kind of SEIS model infections in both latent and infected period is studied in the fifth chapter. The global analysis of this model becomes more complex.Through mathematical studied and simulation we find that giving the patients timely treatment,improving the cure efficiency and decreasing the infective coefficient are effective methods for the control of the disease.
全文共分五章:第一章介绍了传染病的研究概况和一些关于研究流行病模型的基本方法;第二章介绍了一些理论知识;第三章建立并分析了一类非线性高维自治微分系统SEIQR流行病传播数学模型。该模型采用的非线性发生率包括了传统的双线性发生率。通过对模型的全局性态的分析,我们对疾病的传播规律有一个透彻的了解,疾病的灭绝与否由基本再生数R_0决定。这在理论上表明,增大隔离强度和提高疾病的治愈率可以控制疾病的蔓延;第四章主要研究了具有常数输入和非线性发生率的SEIS传染病模型,利用LaSalle不变集原理、Liapunov函数、Hurwith判据证明了无病平衡点的全局渐近稳定性。同时利用第二加性复合矩阵原理证明了惟一地方病平衡点的全局渐近稳定性。本章结果表明,给予病人及时的治疗,提高治疗的有效性是控制疾病的有效方法;第五章主要研究了一类潜伏期和染病期均具有传染力的SEIS传染病模型。该模型的全局性态的分析显的更加复杂,通过理论分析和数值模拟我们发现给予病人及时有效的治疗和减少染病系数是控制疾病蔓延的有效方法。
Exploring epidemic spreading and predicting its development trend are important aspects for epidemic study,and they are the basis of control policy adopted by the government and medical department.The aim of this work is to construct several mathematical models in epidemiology and to analyze the asymptotic behavior of these models.
The full text is divided into five chapters:The first chapter introduced the study of infectious diseases and some methods about epidemiological models.The second chapter introduced some theoretical knowledge.The third chapter not only establish but also analyze a nonlinear high dimensional autonomous differential system SEIQR model in epidemiology.That the nonlinear incidence rate is used in the model,has included the traditional bilinear incidence rate.By analyzing global behavior of the model,we get a incisive comprehension about the disease propagation law and the disease extinction is determined by the basic reproduction number R_0.It showed that incresing isolation and improving the cure rate of the disease can control the spread of diaease;An SEIS epidemic model with constant input and nonlinear incidence rate is maily studied in the fourth chapter.The disease-free equilibrium is globally asymptotical stable by the means of Lasalle's invariant set theorem,Liapunov function and Hurwitze criterion.Using second additive compound matrix,it is proved that the unique endemic equilibrium is globally asymptotical stable.The results in this chapter suggest that giving the patients timely treatment and improving the cure rate are an effective way to control the disease.A kind of SEIS model infections in both latent and infected period is studied in the fifth chapter. The global analysis of this model becomes more complex.Through mathematical studied and simulation we find that giving the patients timely treatment,improving the cure efficiency and decreasing the infective coefficient are effective methods for the control of the disease.
引文
[1]彭文伟.传染病学.北京:人民教育出版社,2001.
[2]马知恩,周义仓,王稳地等.传染病动力学的数学建模与研究.北京:科学出版社,2004.
[3]Kermack W O,McKendrick A C.Contributions to the mathematical theory of epidemics.Proc.Roy.Soc.,1927;(A115):700-721.
[4]Bailey N T J.The mathematical theory of infectious disease:2nd ed.,Hafner,New York,1975.
[5]Zhonghua Lu,Xianning Liu,Lansun Chen.Hopf bifurcation of nonlinear incidence rates SIR epidemiological models with stage structure.Communications in nonlinear science and nemerical simulation,2001,6(4):205-209.
[6]Shigui Ruan,Weidi Wang.Dynamical behavior of an epidemic model with a nonlinear incidence rate.Differential Equations,2003,188:135-163.
[7]陆征一,周义仓.数学生物学进展.北京:科学出版社,2006.
[8]K Dietz.Overall population patterns in the transmission cycle of infectious disease agents.In Population Biology of Infectious Disease,Springer,1982.
[9]Heesterbeek J A P,Metz J A J.The saturating contact rate in marriage and epidemic models.J.Math.Biol.,1993,31:529-539.
[10]Weimin Liu,S A Levin.Y Lawasa.Influence on nonlinear incidence rates upon the behavior of SIRS epidemiological models.J.Math.Biol.,1986,23:187-204.
[11]Weimin Liu.H W Hethcote,S A Levin.Dynamical behavior of epidemiological models with nonlinear incidence rates.J.Math.Bioi.,1987,25:359-380.
[12]Meng Fan,Michael Y.Li.Ke Wang.Global stability of an SEIS epidemic model with recruitment and a varying total population size.Mathematical Biosciences,2001,170:199-208.
[13]Jianquan Li,Juan Zhang.Zhisn Ma.Global analysis of some epidemic models with general contact rate and constant immigration.Applied Mathematics and Mechanics,2004,25(4):396-404.
[14]Guihua Li,Zhen Jin.Global stability of an SEI epidemic model with general contact rate.Chaos,Solitons and Fractals,2005,23:997-1004.
[15]Guihua Li,Zhen Jin.Global stability of a SEIR epidemic model with infectious force in latent,infected and immune period.Chaos,Solitons and Fractals,2005,25:1177-1184.
[16]Guihua Li,Zhen Jin.Global stability of an SEIR epidemic model with constant immigration.Chaos,Solitons and Fractals,2006,30:1012-1019.
[17]李健全,马知恩.一类带有接种的流行病模型的全局稳定性.数学物理学报,2006,26(1:21-30.
[18]Weidi Wang.Backward bifurcation of an epidemic model with treatment.Mathematical Biosciences,2006,201:58-71.
[19]Tang Yilei,Li Weiguo.Global analysis of an epidemic model with a constant removal rate.Mathematical and Computer Modelling,2007,45:834-843.
[20]樊爱军,王开发.一类具有非线性接触率的种群动力学流行病模型分析.四川师范大学学报.2002,25(3):261-263.
[21]张娟,马知恩.具有饱和接触率的SEIS模型的动力学性质.西安交通大学学报,2002,36(2):204-207.
[22]张彤.一类具潜伏期和非线性传染率的流行病模型.浙江师范大学学报.2004,27(1):16-19.
[23]徐文雄,张太雷.一类非线性SEIRS流行病传播数学模型.西北大学学报,2004,34(6):627-630.
[24]徐文雄,张太雷.具有隔离仓室流行病传播数学模型的全局稳定性.西安交通大学学报,2005,39(2):210-213.
[25]王拉娣.具有非线性传染率的两类传染病模型的全局分析.工程数学学报 2005,22(4):640-644.
[26]Xinan Zhang,Lansun Chen.The periodic solution of a class of epidemic models.Computers and Mathematics with Applications,1999,38:61-71.
[27]H Hethcote.Ma Zhien,Liao Shengbing. Effects of quarantine in six endemic models for infectious diseases.Math Biol. 2002.180:141-160
[28]M.E.Alexander. S,M.Moghadas. Periodicity in an epidemic model with a generalized nonlinear incidence. Mathematical Biosciences,2004,189:75-96.
[29]Ladi Wang, Jianquan Li. Global stability of an epidemic model with nonlinear incidence and differential infectivity.Appiled mathematics and computation,2005,161:769-778.
[30]Juan Zhang, Jianquan Li,Zhien Ma. Global dynamics of an SEIR epidemic model with immigration of different compartment.Acta Mathematica scientina, 2006, 26B(3):551-567.
[31]Chengjun Sun,Yinping Lin, Shoupeng Tang. Global stability for an special SEIR epidemic model with nonlinear incidence rates.Chaos Solitions and Fractals, 2007,33: 290-297.
[32]Dongmei Xiao, Shigui Ruan. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences,2007,15(1):73-93.
[33]Junli Liu,Tailei Zhang . Bifurcation analysis of an SIS epidemic model with nonlinear birth rate.Chaos Solitions and Fractals, 2007.
[34]566Junli Liu,Yicang Zhou. Global stability of an SIRS epidemic model with trandport-related infection.Chaos Solitions and Fractals, 2007.
[35]Yu Jin, Wendi Wang, Shiwu Xiao.An SIRS model with a nonlinear incidence rate.Chaos Solitions and Fractals,2007.34:1482-1497.
[36] Tailei Zhang, Junli Liu, Zhidong Teng . Bifurcation analysis of a delayed model with stage structure. Chaos Solitions and Fractals,2007.
[37]Liming Cai.Xuezhi Li. Analysis of a SEIV epidemic model with a nonlinear incidence rate. Applied mathematical modellmg,2009.33:2919-2926.
[38]B.Mukhopadhyay. R.Bhattacharyya. Analysis of a spatially extend nonlinear SEIS epidemic model with distinct incidence for exposed and infectives.Nonlinear analysis :Real World Applications.2008,9: 585-598.
[39]张芷芬,丁同仁,黄火灶等.微分方程定性理论,北京:科学出版社,1985.
[40]马知恩,周义仓.常微分方程定性与稳定性理论,北京:科学出版社,2001.
[41]王高雄,周之铭等.常微分方程,北京:高等教育出版社,2003.
[42]M Fiedler.Additive compound matrices and inequality for eigenvalues of stochastic matrices.Czech.Math.J.,1974,99:392-402.
[43]J S Muidowney.Compound matrices and ordinary differential equations.Rockymont.J.Math.,1990,20:857-872.
[44]Michael Y.Li,Graef J.R.,Wang L.C.,Karsai J.Global dynamics of a SEIR model with a varying total population size.Math.Biosci.,1999,160:191-213.
[45]W.A.Coppel.Stability and asymptotic behavior of differential equations.Health Boston,1996.
[46]Michael Y.Li,Muldowney J.S.A geometric approach to global-stability problems.SIAM J Math Anal,1996,27:1070-1083.
[2]马知恩,周义仓,王稳地等.传染病动力学的数学建模与研究.北京:科学出版社,2004.
[3]Kermack W O,McKendrick A C.Contributions to the mathematical theory of epidemics.Proc.Roy.Soc.,1927;(A115):700-721.
[4]Bailey N T J.The mathematical theory of infectious disease:2nd ed.,Hafner,New York,1975.
[5]Zhonghua Lu,Xianning Liu,Lansun Chen.Hopf bifurcation of nonlinear incidence rates SIR epidemiological models with stage structure.Communications in nonlinear science and nemerical simulation,2001,6(4):205-209.
[6]Shigui Ruan,Weidi Wang.Dynamical behavior of an epidemic model with a nonlinear incidence rate.Differential Equations,2003,188:135-163.
[7]陆征一,周义仓.数学生物学进展.北京:科学出版社,2006.
[8]K Dietz.Overall population patterns in the transmission cycle of infectious disease agents.In Population Biology of Infectious Disease,Springer,1982.
[9]Heesterbeek J A P,Metz J A J.The saturating contact rate in marriage and epidemic models.J.Math.Biol.,1993,31:529-539.
[10]Weimin Liu,S A Levin.Y Lawasa.Influence on nonlinear incidence rates upon the behavior of SIRS epidemiological models.J.Math.Biol.,1986,23:187-204.
[11]Weimin Liu.H W Hethcote,S A Levin.Dynamical behavior of epidemiological models with nonlinear incidence rates.J.Math.Bioi.,1987,25:359-380.
[12]Meng Fan,Michael Y.Li.Ke Wang.Global stability of an SEIS epidemic model with recruitment and a varying total population size.Mathematical Biosciences,2001,170:199-208.
[13]Jianquan Li,Juan Zhang.Zhisn Ma.Global analysis of some epidemic models with general contact rate and constant immigration.Applied Mathematics and Mechanics,2004,25(4):396-404.
[14]Guihua Li,Zhen Jin.Global stability of an SEI epidemic model with general contact rate.Chaos,Solitons and Fractals,2005,23:997-1004.
[15]Guihua Li,Zhen Jin.Global stability of a SEIR epidemic model with infectious force in latent,infected and immune period.Chaos,Solitons and Fractals,2005,25:1177-1184.
[16]Guihua Li,Zhen Jin.Global stability of an SEIR epidemic model with constant immigration.Chaos,Solitons and Fractals,2006,30:1012-1019.
[17]李健全,马知恩.一类带有接种的流行病模型的全局稳定性.数学物理学报,2006,26(1:21-30.
[18]Weidi Wang.Backward bifurcation of an epidemic model with treatment.Mathematical Biosciences,2006,201:58-71.
[19]Tang Yilei,Li Weiguo.Global analysis of an epidemic model with a constant removal rate.Mathematical and Computer Modelling,2007,45:834-843.
[20]樊爱军,王开发.一类具有非线性接触率的种群动力学流行病模型分析.四川师范大学学报.2002,25(3):261-263.
[21]张娟,马知恩.具有饱和接触率的SEIS模型的动力学性质.西安交通大学学报,2002,36(2):204-207.
[22]张彤.一类具潜伏期和非线性传染率的流行病模型.浙江师范大学学报.2004,27(1):16-19.
[23]徐文雄,张太雷.一类非线性SEIRS流行病传播数学模型.西北大学学报,2004,34(6):627-630.
[24]徐文雄,张太雷.具有隔离仓室流行病传播数学模型的全局稳定性.西安交通大学学报,2005,39(2):210-213.
[25]王拉娣.具有非线性传染率的两类传染病模型的全局分析.工程数学学报 2005,22(4):640-644.
[26]Xinan Zhang,Lansun Chen.The periodic solution of a class of epidemic models.Computers and Mathematics with Applications,1999,38:61-71.
[27]H Hethcote.Ma Zhien,Liao Shengbing. Effects of quarantine in six endemic models for infectious diseases.Math Biol. 2002.180:141-160
[28]M.E.Alexander. S,M.Moghadas. Periodicity in an epidemic model with a generalized nonlinear incidence. Mathematical Biosciences,2004,189:75-96.
[29]Ladi Wang, Jianquan Li. Global stability of an epidemic model with nonlinear incidence and differential infectivity.Appiled mathematics and computation,2005,161:769-778.
[30]Juan Zhang, Jianquan Li,Zhien Ma. Global dynamics of an SEIR epidemic model with immigration of different compartment.Acta Mathematica scientina, 2006, 26B(3):551-567.
[31]Chengjun Sun,Yinping Lin, Shoupeng Tang. Global stability for an special SEIR epidemic model with nonlinear incidence rates.Chaos Solitions and Fractals, 2007,33: 290-297.
[32]Dongmei Xiao, Shigui Ruan. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences,2007,15(1):73-93.
[33]Junli Liu,Tailei Zhang . Bifurcation analysis of an SIS epidemic model with nonlinear birth rate.Chaos Solitions and Fractals, 2007.
[34]566Junli Liu,Yicang Zhou. Global stability of an SIRS epidemic model with trandport-related infection.Chaos Solitions and Fractals, 2007.
[35]Yu Jin, Wendi Wang, Shiwu Xiao.An SIRS model with a nonlinear incidence rate.Chaos Solitions and Fractals,2007.34:1482-1497.
[36] Tailei Zhang, Junli Liu, Zhidong Teng . Bifurcation analysis of a delayed model with stage structure. Chaos Solitions and Fractals,2007.
[37]Liming Cai.Xuezhi Li. Analysis of a SEIV epidemic model with a nonlinear incidence rate. Applied mathematical modellmg,2009.33:2919-2926.
[38]B.Mukhopadhyay. R.Bhattacharyya. Analysis of a spatially extend nonlinear SEIS epidemic model with distinct incidence for exposed and infectives.Nonlinear analysis :Real World Applications.2008,9: 585-598.
[39]张芷芬,丁同仁,黄火灶等.微分方程定性理论,北京:科学出版社,1985.
[40]马知恩,周义仓.常微分方程定性与稳定性理论,北京:科学出版社,2001.
[41]王高雄,周之铭等.常微分方程,北京:高等教育出版社,2003.
[42]M Fiedler.Additive compound matrices and inequality for eigenvalues of stochastic matrices.Czech.Math.J.,1974,99:392-402.
[43]J S Muidowney.Compound matrices and ordinary differential equations.Rockymont.J.Math.,1990,20:857-872.
[44]Michael Y.Li,Graef J.R.,Wang L.C.,Karsai J.Global dynamics of a SEIR model with a varying total population size.Math.Biosci.,1999,160:191-213.
[45]W.A.Coppel.Stability and asymptotic behavior of differential equations.Health Boston,1996.
[46]Michael Y.Li,Muldowney J.S.A geometric approach to global-stability problems.SIAM J Math Anal,1996,27:1070-1083.