具有非线性传染率的传染病模型研究
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摘要
本文研究了具有非线性传染率的四类传染病模型:
首先,研究了一类易感者、潜伏者和染病者均有常数输入,且传染率是非线性传染率βf(S)I的SEIR传染病模型.研究表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性,进一步利用Li和Muldowney所发展的几何方法证明了地方病平衡点的全局稳定性.
其次,研究了两类SIQR传染病模型,第一类为各仓室均有常数输入(除了隔离仓室),且传染率为一般形式非线性饱和传染率的SIQR模型,第二类为具有强非线性传染率的SIQR模型.对第一个模型,当不考虑隔离者的因病死亡时,引入变量代换将四维模型转化为二维渐近自治系统,而后利用Dulac函数和极限方程理论证明了地方病平衡点的全局稳定性.对第二个模型,运用Hurwitz判别法分析了各平衡点的局部稳定性,发现了在一定的条件下,该模型会发生Hopf分支产生周期解,进一步我们应用Dulac函数和极限方程理论证明了当0＜p≤1时地方病平衡点的全局稳定性.
最后,研究了一类易感者和染病者均有常数输入,疾病具有垂直传染,且传染率是一般形式非线性饱和传染率的SIRI传染病模型.结果表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性.当传染率为双线性传染率和标准传染率时,利用广义BendixsonDulac定理排除了三维系统的周期解,从而证明了地方病平衡点的全局稳定性.
In this paper,we study four kinds of epidemic models with nonlinear incidence rates.
First, we consider an SEIR epidemic model with nonlinear incidence rate and constant immigration, which includes susceptibles、exposeds and infectives. It has been shown that this model has only unique endemic equilibrium. The local asymptotical stable results of the epidemic equilibrium was proved by using the Hurwitz criterion and the global asymptotical stable results of its by means of the geometric approach developed by Li and Muldowney .
Secondly, we study two kinds of SIQR epidemic models. The first one is the SIQR model with general form nonlinear saturated infectivity and constant inflows, which includes susceptibles、infectives and recovered, and the second one is the SIQR model with nonlinear incidence ratesβI~pS~q. For the first model, we reduce the four-dimensional model to a two-dimensional asymptotical autonomous system by means of a transformation of variables. Furthermore, we prove the global asymptotical stability of the epidemic equilibrium by means of Dulac's function and the theory of limit systems. For the second model, we analyse the stability of the equilibria by using the Hurwitz criterion, and obtained the existence of periodic solutions by Hopf bifurcation for some parameter values. Furthermore,using Dulac's function and the theory of limit systems,we prove the global asymptotical stability of the epidemic equilibrium when 0 < p≤1.
Finally, we formulate a kind of SIRI model with the vertical transmission,general form nonlinear saturated infectivity and constant immigration which includes new susceptibles and infectives. It is also found that the system exists only one equilibrium. Using the Routh-Hurwitz criterion, we prove the local asymptotical stability of the epidemic equilibrium. For the important cases of mass action incidence and standard incidence,applying Bendixson-Dulac theorem, the existence of the periodic solutions of the three-dimensional system is excluded, thereby the global stability of the endemic equilibrium is proved provided the endemic equilibrium exists.
首先,研究了一类易感者、潜伏者和染病者均有常数输入,且传染率是非线性传染率βf(S)I的SEIR传染病模型.研究表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性,进一步利用Li和Muldowney所发展的几何方法证明了地方病平衡点的全局稳定性.
其次,研究了两类SIQR传染病模型,第一类为各仓室均有常数输入(除了隔离仓室),且传染率为一般形式非线性饱和传染率的SIQR模型,第二类为具有强非线性传染率的SIQR模型.对第一个模型,当不考虑隔离者的因病死亡时,引入变量代换将四维模型转化为二维渐近自治系统,而后利用Dulac函数和极限方程理论证明了地方病平衡点的全局稳定性.对第二个模型,运用Hurwitz判别法分析了各平衡点的局部稳定性,发现了在一定的条件下,该模型会发生Hopf分支产生周期解,进一步我们应用Dulac函数和极限方程理论证明了当0＜p≤1时地方病平衡点的全局稳定性.
最后,研究了一类易感者和染病者均有常数输入,疾病具有垂直传染,且传染率是一般形式非线性饱和传染率的SIRI传染病模型.结果表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性.当传染率为双线性传染率和标准传染率时,利用广义BendixsonDulac定理排除了三维系统的周期解,从而证明了地方病平衡点的全局稳定性.
In this paper,we study four kinds of epidemic models with nonlinear incidence rates.
First, we consider an SEIR epidemic model with nonlinear incidence rate and constant immigration, which includes susceptibles、exposeds and infectives. It has been shown that this model has only unique endemic equilibrium. The local asymptotical stable results of the epidemic equilibrium was proved by using the Hurwitz criterion and the global asymptotical stable results of its by means of the geometric approach developed by Li and Muldowney .
Secondly, we study two kinds of SIQR epidemic models. The first one is the SIQR model with general form nonlinear saturated infectivity and constant inflows, which includes susceptibles、infectives and recovered, and the second one is the SIQR model with nonlinear incidence ratesβI~pS~q. For the first model, we reduce the four-dimensional model to a two-dimensional asymptotical autonomous system by means of a transformation of variables. Furthermore, we prove the global asymptotical stability of the epidemic equilibrium by means of Dulac's function and the theory of limit systems. For the second model, we analyse the stability of the equilibria by using the Hurwitz criterion, and obtained the existence of periodic solutions by Hopf bifurcation for some parameter values. Furthermore,using Dulac's function and the theory of limit systems,we prove the global asymptotical stability of the epidemic equilibrium when 0 < p≤1.
Finally, we formulate a kind of SIRI model with the vertical transmission,general form nonlinear saturated infectivity and constant immigration which includes new susceptibles and infectives. It is also found that the system exists only one equilibrium. Using the Routh-Hurwitz criterion, we prove the local asymptotical stability of the epidemic equilibrium. For the important cases of mass action incidence and standard incidence,applying Bendixson-Dulac theorem, the existence of the periodic solutions of the three-dimensional system is excluded, thereby the global stability of the endemic equilibrium is proved provided the endemic equilibrium exists.
引文
[1]Li M Y,Muldowney J S.A geometric approach to global stability problems.SIAM J Math Anal,1996,27(4):1070-1083.
[2]Helmar nunes moreira,Yuquan Wang.Global stability in an S → I → R → I model .SIAM REV,1997,Vol.39,No.3,pp.496-502.
[3]Michael Y.Li,Liancheng Wang.Global stability in some SEIR epidemic models.Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part Ⅱ:Models,Methods and Theory(Castillo-Chavez et al.,eds.),IMA Volumes in Mathematics and its Applications 126,Springer-Verlag,2002,pp.295.
[4]Chengjun Sun,Yiping Lin,Shoupeng Tang.Global stability for an special SEIR epidemic model with nonlinear incidence rates.Chaos,Solitons and Fractals,2007,33:290-297.
[5]Horst R.Thieme and Carlos Castillo-Chavez.How may infection age-dependent infectivity affect the dynamics of HIV/AIDs.SIAM J.Appl.Math,1993,53:1447-1479.
[6]Herbert W.Hethcote,Ma Zhien,Liao Shengbing.Effects of quarantine in six endemic models for infectious diseases.Mathematical Biosciences,180(2002):141-160.
[7]余洋,胡志兴.一类具有常数移民且带隔离项的传染病模型的分析.石家庄学院学报,2006,8(6):43-48.
[8]陈军杰.一类具有常数迁入且总人口在变化的SIRI传染病模型的稳定性.生物数学学报,2004,19(3):310-316.
[9]彭文伟.传染病学,北京:人民卫生出版社,2001.
[10]World health organization report,1995.
[11]Anderson R M,May R M.Population biology of infectious diseases I.Nature,1979,180:361-367.
[12]Liu W M,Simon A,Yoh I.Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models.J Math Biol,1986(23):187-204.
[13]H.W.Hethcote.Qualitative analysis of communicable disease models.Math Biosci,1976,28:335-356.
[14]H.W.Hethcote and P.Van Den Drieesche.Some epidemiological models with nonlinear incidence.J.Math.Biol,29(1991),pp.271-287.
[15]J.A.Jacquez,C.P.Simon,and J.S.Koopman.The reproduction number in deterministic models of contagious diseases.Comments Theor.Biol,2(1991),pp.159-209.
[16]Fred Brauer,P.van den Driessche.Models for transmission of disease with immigration of infectives.Math.Biosci,2001,171:143-154.
[17]H.W.Hethcote.A thousand and one epidemic models,in:S.A.Levin(Ed.),Frontiers in Mathematical Biology,Lecture Notes in Biomathematics 100.Berlin,Heideiberg,New York:Springer-Verlag,1994,504-515.
[18]H.W.Hethcote.The mathematics of infectious disease.SIAM Review,2000,42:599-653.
[19]杨建雅,张凤琴.染病者有常数输入的传染病模型.数学的实践和认识,2006,36(12):14-18.
[20]Fiedler M.Additive compound matrices and inequality for eigenvalues of stochastic matrices.Czerch Math J,1974,99:392-410.
[21]Muldowney J S.Compound matrices and ordinary differential equations.Rocky Mountain J Math,1990,20:857-872.
[22]Coppel WA.Stability and asymptotic behavior of differential equations.Bosten:Heath;1965.
[23]J.S.Muldowney.On the dimension of the zero or infinity tending set of linear differential equations.Proc.Amer.Math.Soc,1981,280:705-709.
[24]Li Jianquan,Ma Zhien.Qualitative analyses of SIS epidemic model with vaccination and varying total population size.Mathematical and Computer modelling,2002,35:1235-1243.
[25]R.H.Martin.Jr.Logarithmic norms and projections applied to linear differential systems.J.Math.Anal.Appl,1974,45:432-454.
[26]马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究,北京:科学出版社,2004:1-3,58-59.
[27]Michael Y.Li,Liancheng Wang,Janos Karsai.Global dynamics of a SEIR model with varying total population size.Mathematical Biosciences,1999,160:191-213.
[28]陈军杰.若干具有非线性传染力的传染病模型的稳定性分析.生物数学学报,2005,20(3):286-x296.
[29]Zhang Juan,Li Jianquan,Ma Zhien.Global dynamics of an SEIR epidemic model with immigration of different compartments.Acta Mathematica Scientia,2006,26(B)(3):551-567.
[30]J.Hofbaue and J.W.H.So.Uniform persistence and repellors for maps.Proc.Amer.Math.Soc,1989,107:1137-1142.
[31]Butler G,Waltman P.Persistence in dynamical systems.Proc Amer Math Soc,1986,96:425-430.
[32]Feng,Thieme H R.Endemic models with arbitrarily distributed periods of infection,Ⅰ:General Heory.SIAM.J.Appl.Math,2000,61:803.
[33]Feng,Thieme H R.Endemic models with arbitrarily distributed periods of infection,Ⅱ:Fast disease dynamics and permanent recovery.SIAM.J.Appl.Math,2000,61:983.
[34]Wu L,Feng Z.Homoclinic bifurcation in an SIQR model for childhood disease.J.Diff.Eqns,2000,168:150-167.
[35]徐光辉,邵嘉裕.树的稳定子集和稳定指标.应用数学学报,2003,26(2):253-x263.
[36]张锦炎,冯贝叶.常微分方程几何理论与分支问题.北京:北京大学出版社,1981:88-89,128-129.
[37]Tudor D.A deterministic model for herpes infections in human and animal populations.SIAM Rev,1990,32:136-139.
[38]徐文雄,张仲华,徐宗本.具有一般形式饱和接触率SEIS模型渐近分析.生物数学学报,2005,20(3):297-302.
[39]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.
[40]张彤,楼敏.一类具一般形式饱和接触率SEIS模型的定性分析.浙江工业大学学报,2007,Vol.35,No.1:110-112.
[41]邓东锐,闻良珍,凌霞珍.母儿间单纯疱疹病毒感染的研究.中国医师杂志,2004,10:1339-1340.
[42]张蓉,赵欣,陈孝琴,刘兰女,敖黎明.单纯性疱疹病毒Ⅱ型母婴垂直传播初步研究.生殖与避孕,2002,Vol.22,No.2:121-123.
[43]杨天娇.新生儿单纯疱疹病毒感染的研究进展.国外医学.儿科学分册,2004,31:128-x129.
[44]Ruan S G,Wang W D.Dynamical behavior of an epidemic model with a nonlinear incidence rate.J Diff Equs,2003,188:135-163.
[45]Liu W M,Herbert W,Simon A.Dynamical behavior of epidemiological models with nonlinear incidence rates.J Math Biol.1986,25:359-380.
[46] Hale JK. Ordinary differential equations. New York:Wiley-Interscience,1969:296-297.
[47] Wang La-di,Li Jian-quan. Qualitative analysis of an SEIS epidemic model with nonlinear incidence rate. Applied Mathematics and Mechanics,2006,27:591-596.
[48] Li Jianquan,Zhang Juan,Ma Zhien. Global analysis of some epidemic models with general contact rate and constant immigration. Applied Mathematics and Mechanics,2004,25:396-404.
[49] Juan Zhang,Zhien Ma. Global dynamics of an SEIR epidemic model with saturating contact rate. Mathematical Biosciences,2003,185:15-32.
[50] Xuezhi Li,Linlin Zhou. Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate. Chaos,Solitions and Fractals.
[51] 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.
[52] Zhien Ma,Jianpin Liu,Jia Li. Stability analysis for differential infectivity epidemic models. Nonlinear Analysis,2003,4:841-856.
[53] Ladi Wang,Jianquan Li. Global stability of an epidemic model with nonlinear incidence rate and differential infectivity. Applied Mathematics and Computation, 2005,161:769-778.
[54] Guihua Li,Zhen jin. Global stability of a SEIR epidemic model with infectious force in latent,infected and immune period. Chaos,Solitions and Practals,2005,25:1177-1184.
[55] Guihua Li,Zhen jin. Global stability of an SEIR epidemic model with constant immigration. Chaos,Solitions and Fractals,2006,30:1012-1019.
[56] Dongmei Xiao,Shigui Ruan. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences,2007,208:419-429.
[57] Yugui Zhou,Dongmei Xiao,Yilong Li. Bifurcations of an epidemic model with nonmonotone incidence rate of saturated mass action. Chaos,Solitions and Fractals,2007,32:1903-1915.
[58] W.R.Derrick,P.van den Driessche. A disease transmission model in a nonconstant population. J.Math.Biol,1993,31:495-512.
[59] Junli Yuan,Zuodong Yang. Global dynamics of an SEI model with acute and chronic stages. Journal of computational and Applied Mathematics.
[60] Yu Jin,Wengdi Wang,Shiwu Xiao. An SIRS model with a nonlinear incidence rate. Chaos,Solitions and Practals,2007,34:1482-1497.
[61] Yuliya N.Kyrychko,Konstantin B.Blyuss. Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Analysis,2005,6:495-507.
[62] James M.Hyman,Jia Li. Differential susceptibility and infectivity epidemic models.Mathematical Biosciences and engineering,2006,3:89-100.
[63] Andrei Korobeinikov,Philip K.Maini. Nonlinear incidence and stability of infectious disease models. Mathematical Medicine and Biology,2005,22:113-128.
[2]Helmar nunes moreira,Yuquan Wang.Global stability in an S → I → R → I model .SIAM REV,1997,Vol.39,No.3,pp.496-502.
[3]Michael Y.Li,Liancheng Wang.Global stability in some SEIR epidemic models.Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part Ⅱ:Models,Methods and Theory(Castillo-Chavez et al.,eds.),IMA Volumes in Mathematics and its Applications 126,Springer-Verlag,2002,pp.295.
[4]Chengjun Sun,Yiping Lin,Shoupeng Tang.Global stability for an special SEIR epidemic model with nonlinear incidence rates.Chaos,Solitons and Fractals,2007,33:290-297.
[5]Horst R.Thieme and Carlos Castillo-Chavez.How may infection age-dependent infectivity affect the dynamics of HIV/AIDs.SIAM J.Appl.Math,1993,53:1447-1479.
[6]Herbert W.Hethcote,Ma Zhien,Liao Shengbing.Effects of quarantine in six endemic models for infectious diseases.Mathematical Biosciences,180(2002):141-160.
[7]余洋,胡志兴.一类具有常数移民且带隔离项的传染病模型的分析.石家庄学院学报,2006,8(6):43-48.
[8]陈军杰.一类具有常数迁入且总人口在变化的SIRI传染病模型的稳定性.生物数学学报,2004,19(3):310-316.
[9]彭文伟.传染病学,北京:人民卫生出版社,2001.
[10]World health organization report,1995.
[11]Anderson R M,May R M.Population biology of infectious diseases I.Nature,1979,180:361-367.
[12]Liu W M,Simon A,Yoh I.Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models.J Math Biol,1986(23):187-204.
[13]H.W.Hethcote.Qualitative analysis of communicable disease models.Math Biosci,1976,28:335-356.
[14]H.W.Hethcote and P.Van Den Drieesche.Some epidemiological models with nonlinear incidence.J.Math.Biol,29(1991),pp.271-287.
[15]J.A.Jacquez,C.P.Simon,and J.S.Koopman.The reproduction number in deterministic models of contagious diseases.Comments Theor.Biol,2(1991),pp.159-209.
[16]Fred Brauer,P.van den Driessche.Models for transmission of disease with immigration of infectives.Math.Biosci,2001,171:143-154.
[17]H.W.Hethcote.A thousand and one epidemic models,in:S.A.Levin(Ed.),Frontiers in Mathematical Biology,Lecture Notes in Biomathematics 100.Berlin,Heideiberg,New York:Springer-Verlag,1994,504-515.
[18]H.W.Hethcote.The mathematics of infectious disease.SIAM Review,2000,42:599-653.
[19]杨建雅,张凤琴.染病者有常数输入的传染病模型.数学的实践和认识,2006,36(12):14-18.
[20]Fiedler M.Additive compound matrices and inequality for eigenvalues of stochastic matrices.Czerch Math J,1974,99:392-410.
[21]Muldowney J S.Compound matrices and ordinary differential equations.Rocky Mountain J Math,1990,20:857-872.
[22]Coppel WA.Stability and asymptotic behavior of differential equations.Bosten:Heath;1965.
[23]J.S.Muldowney.On the dimension of the zero or infinity tending set of linear differential equations.Proc.Amer.Math.Soc,1981,280:705-709.
[24]Li Jianquan,Ma Zhien.Qualitative analyses of SIS epidemic model with vaccination and varying total population size.Mathematical and Computer modelling,2002,35:1235-1243.
[25]R.H.Martin.Jr.Logarithmic norms and projections applied to linear differential systems.J.Math.Anal.Appl,1974,45:432-454.
[26]马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究,北京:科学出版社,2004:1-3,58-59.
[27]Michael Y.Li,Liancheng Wang,Janos Karsai.Global dynamics of a SEIR model with varying total population size.Mathematical Biosciences,1999,160:191-213.
[28]陈军杰.若干具有非线性传染力的传染病模型的稳定性分析.生物数学学报,2005,20(3):286-x296.
[29]Zhang Juan,Li Jianquan,Ma Zhien.Global dynamics of an SEIR epidemic model with immigration of different compartments.Acta Mathematica Scientia,2006,26(B)(3):551-567.
[30]J.Hofbaue and J.W.H.So.Uniform persistence and repellors for maps.Proc.Amer.Math.Soc,1989,107:1137-1142.
[31]Butler G,Waltman P.Persistence in dynamical systems.Proc Amer Math Soc,1986,96:425-430.
[32]Feng,Thieme H R.Endemic models with arbitrarily distributed periods of infection,Ⅰ:General Heory.SIAM.J.Appl.Math,2000,61:803.
[33]Feng,Thieme H R.Endemic models with arbitrarily distributed periods of infection,Ⅱ:Fast disease dynamics and permanent recovery.SIAM.J.Appl.Math,2000,61:983.
[34]Wu L,Feng Z.Homoclinic bifurcation in an SIQR model for childhood disease.J.Diff.Eqns,2000,168:150-167.
[35]徐光辉,邵嘉裕.树的稳定子集和稳定指标.应用数学学报,2003,26(2):253-x263.
[36]张锦炎,冯贝叶.常微分方程几何理论与分支问题.北京:北京大学出版社,1981:88-89,128-129.
[37]Tudor D.A deterministic model for herpes infections in human and animal populations.SIAM Rev,1990,32:136-139.
[38]徐文雄,张仲华,徐宗本.具有一般形式饱和接触率SEIS模型渐近分析.生物数学学报,2005,20(3):297-302.
[39]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.
[40]张彤,楼敏.一类具一般形式饱和接触率SEIS模型的定性分析.浙江工业大学学报,2007,Vol.35,No.1:110-112.
[41]邓东锐,闻良珍,凌霞珍.母儿间单纯疱疹病毒感染的研究.中国医师杂志,2004,10:1339-1340.
[42]张蓉,赵欣,陈孝琴,刘兰女,敖黎明.单纯性疱疹病毒Ⅱ型母婴垂直传播初步研究.生殖与避孕,2002,Vol.22,No.2:121-123.
[43]杨天娇.新生儿单纯疱疹病毒感染的研究进展.国外医学.儿科学分册,2004,31:128-x129.
[44]Ruan S G,Wang W D.Dynamical behavior of an epidemic model with a nonlinear incidence rate.J Diff Equs,2003,188:135-163.
[45]Liu W M,Herbert W,Simon A.Dynamical behavior of epidemiological models with nonlinear incidence rates.J Math Biol.1986,25:359-380.
[46] Hale JK. Ordinary differential equations. New York:Wiley-Interscience,1969:296-297.
[47] Wang La-di,Li Jian-quan. Qualitative analysis of an SEIS epidemic model with nonlinear incidence rate. Applied Mathematics and Mechanics,2006,27:591-596.
[48] Li Jianquan,Zhang Juan,Ma Zhien. Global analysis of some epidemic models with general contact rate and constant immigration. Applied Mathematics and Mechanics,2004,25:396-404.
[49] Juan Zhang,Zhien Ma. Global dynamics of an SEIR epidemic model with saturating contact rate. Mathematical Biosciences,2003,185:15-32.
[50] Xuezhi Li,Linlin Zhou. Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate. Chaos,Solitions and Fractals.
[51] 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.
[52] Zhien Ma,Jianpin Liu,Jia Li. Stability analysis for differential infectivity epidemic models. Nonlinear Analysis,2003,4:841-856.
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