两个流行病模型的定性分析
摘要
在这篇文章中,我们分别在第二章和第三章中研究了两个不同的流行病模型.
第二章中主要研究了一个具有一般非线性发生率函数的SEIQR流行病模型.该模型始终存在无病平衡点,当且仅当基本再生数R0>1时存在地方病平衡点,且是唯一的.无病平衡点与地方病平衡点的稳定性与基本再生数R0的取值有着密切联系,当R0≤1时,无病平衡点是全局渐进稳定的；当R0>1时,无病平衡点是不稳定的,此时疾病具有持久性.当R0>1且发生率函数满足§2.2条件H,地方病平衡点是全局渐近稳定的.本章最后给出了几个数值模拟来证明结论的正确性,同时通过改变隔离项的系数来说明对感染者进行一定比例的隔离对消灭流行病的重要作用.
第三章我们主要研究了一个具有非单调发生率函数的时滞SIR流行病模型.得到了系统无病平衡点及地方病平衡点的存在与基本再生数之间的关系.证明了当基本再生数R0≤1时无病平衡点是全局渐近稳定的；当R0>1时,疾病是持久的,此时无病平衡点是不稳定的；给出参数满足的一定条件时,地方病平衡点是全局渐近稳定的.
In this paper, we study two epidemic models in chapter two and chapter three, respectively.
In chapter two, we study a SEIQR epidemic model with a class of nonlinear incidence rate. The model always exhibits the disease-free equilibrium, and the unique endemic equilibrium turns up if and only if the basic reproduction number R0>1. It is shown that if R0≤1, the disease-free equilibrium is globally asymptotically stable and if R0>1, the disease-free equilibrium is unstable. Moreover, we show that if R0>1,the disease is uniformly persistent and the unique endemic equilibrium is globally asymptotically stable under certain condition H which we will give in§2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to eliminate the disease.
In chapter three, we consider a SIR epidemic model with non-monotone incidence rate and time delay. Through analysis we get the relative of the existence of the disease free equilibrium and endemic equilibrium between the basic reproduction number R0. We show that the disease-free equilibrium is globally asymptotically stable when R0≤1, and if R0>1, the disease free equilib-rium is unstable but the disease is persistent, and the endemic equilibrium is globally asymptoti-cally stable if the parameters satisfied some conditions.
第二章中主要研究了一个具有一般非线性发生率函数的SEIQR流行病模型.该模型始终存在无病平衡点,当且仅当基本再生数R0>1时存在地方病平衡点,且是唯一的.无病平衡点与地方病平衡点的稳定性与基本再生数R0的取值有着密切联系,当R0≤1时,无病平衡点是全局渐进稳定的；当R0>1时,无病平衡点是不稳定的,此时疾病具有持久性.当R0>1且发生率函数满足§2.2条件H,地方病平衡点是全局渐近稳定的.本章最后给出了几个数值模拟来证明结论的正确性,同时通过改变隔离项的系数来说明对感染者进行一定比例的隔离对消灭流行病的重要作用.
第三章我们主要研究了一个具有非单调发生率函数的时滞SIR流行病模型.得到了系统无病平衡点及地方病平衡点的存在与基本再生数之间的关系.证明了当基本再生数R0≤1时无病平衡点是全局渐近稳定的；当R0>1时,疾病是持久的,此时无病平衡点是不稳定的；给出参数满足的一定条件时,地方病平衡点是全局渐近稳定的.
In this paper, we study two epidemic models in chapter two and chapter three, respectively.
In chapter two, we study a SEIQR epidemic model with a class of nonlinear incidence rate. The model always exhibits the disease-free equilibrium, and the unique endemic equilibrium turns up if and only if the basic reproduction number R0>1. It is shown that if R0≤1, the disease-free equilibrium is globally asymptotically stable and if R0>1, the disease-free equilibrium is unstable. Moreover, we show that if R0>1,the disease is uniformly persistent and the unique endemic equilibrium is globally asymptotically stable under certain condition H which we will give in§2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to eliminate the disease.
In chapter three, we consider a SIR epidemic model with non-monotone incidence rate and time delay. Through analysis we get the relative of the existence of the disease free equilibrium and endemic equilibrium between the basic reproduction number R0. We show that the disease-free equilibrium is globally asymptotically stable when R0≤1, and if R0>1, the disease free equilib-rium is unstable but the disease is persistent, and the endemic equilibrium is globally asymptoti-cally stable if the parameters satisfied some conditions.
引文
[1]马知恩,周义仓等,传染病动力学的数学建模与研究[M],北京：科学出版社,2004.
[2]马知恩,周义仓,常微分方程定性与稳定性方法[M],北京：科学出版社,2007.
[3]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[J],西安交通大学学报,2005,39(2),210-213.
[4]徐文雄,张太雷,徐宗本,非线性高维自治微分系统SEIQR流行病模型全局稳定性[J],工程数学学报,2007,24(1),79-86.
[5]V. Capasso, G. Serio, Ageneration of the Kermack-Mackendrick deterministic epidemic model. Math. Biosci.42(1978)43.
[6]L.-M.Cai, X.-Z.Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model.33(2009)2919-2926.
[7]L.-M. Cai, X.-Z. Li, Mini Ghosh, Global stability of a stage-structured epidemic model with a nonlinear incidence. Appl. Math. Comput.214(2009)73-81.
[8]B. K. Mishra, N. Jha, SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Modelling (2009),doi:10.1016/j.apm.2009.06.011
[9]Dongmei Xiao, Shigui Ruan, Global analysis of an epidemic model with nonmonotone inci-dence rate. Math. Biosci.208(2007)419-429.
[10]X.-Z.Li, L.-L. Zhou, Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate. Chaos, Solitons and Fractals 40(2009)874-884.
[11]Rui Xu, Zhien Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal.:Real Word Appl.10(2009)3175-3189.
[12]Tailei Zhang, Zhidong Teng, Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence. Chaos, Solitons and Fractals 37(2008)1456-1468.
[13]Chi-Wei Lee, Yen-Shuo Tsai, Tai-Wai Wang, Chor-Chiu Lau, A loophole in international quarantine procedures disclosed during the SARS crisis. Traveal Medicine and Infection dis-ease 4(2006),22-28.
[14]Herbert Hethcote, Ma Zhien, Liao Shengbing, Effects of quarantine in six endemic models for infections disease. Math. Biosc.180(2002)141-160.
[15]Meng Fan, M Y Li, Ke Wang. Global stability of an SEIS epidemic model with recruitment and a varying total population size. Math. Biosci.170(2001)199-208.
[16]Guihua Li, Zhen Jin, Global stability of a SEIR epidemic model with infectious in latent, infected and immune period. Chaos, Solitons and Fractals 25(2005)1177-1184.
[17]Chengjun Sun, Yiping Lin, Shoupeng Tang, Global stability fora special SEIR epidemic model with nolinear incidence rates. Chaos, Solitons and Fractals 33(2007)290-297.
[18]P.E.M. Fine, Vectors and vertical transmissions:An epidemiologic perspective. Annals N.Y. Academic Sci.266(1979)173-194. S.N.
[19]Busenberg, K.L. Cooke, Vertically transmitted diseases. Models and dynamics. In:Biomath-ematics Vol.23, Springer-Verlag, Berlin (1993).
[20]M. El-Doma, Analysis of an age-dependent SIS epidemic model with vertical transmission and proportionate mixing assumption. Math. and computer modelling 29(1999)31-43.
[21]Rui Xu, Zhien Ma, Stability of a delayed SIRS epidemic model with nonlinear incidence rate, Chaos, Solitons and Fractals 41(2009)2319-2325.
[22]Rui Xu, Zhien Ma, Zhiping Wang, Global stability of a delayed SIR epidemic model with saturation incidence and temporary immunity. Comput. amd Math. with Application (2010),doi:10.1016/j.camwa.2010.03.009.
[23]K.L.Cooke, Stability analysis for a vector disease model. Rocky Mountain J. Math. 9(1979)31-42.
[24]Naoki Yoshida, Tadayuki Hara, Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J. Comput. and Appl. Math.201(2007)339-347.
[25]Mei Song, Wanbiao Ma, Yasuhiro Takeuchi, Permanence of a delayed SIR with density de-pendent birth rate. J. Comput. and Appl. Math.201(2007)389-394.
[26]Fenfen Zhang, Zhen Jin, Guiquan Sun, Bifurcation analysis of a delayed epidemic model. Appl. Math. and Comput.216(2010)753-767.
[27]Zhang Zhonghua, Peng Jigen, A SIRS epidemic model with infection-age dependence. J. Math. Anal. Appl.331(2007)1396-1414.
[28]Litao Han, Zhien Ma, Tan Shi, An SIRS epidemic model of two competitive species. Math. and computer modelling 37(2003)87-108.
[29]Ram Naresh, Agraj Tripathi, J.M.Tchuenche, Dileep Sharma, Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Computer and Math. with Appl. 58(2009)348-359.
[30]Edoardo Beretta, Yasuhiro Takeuchi, Convergence results in SIR epidemic models with vary-ing population sizes. Nonliear Anal. Theory. Methods and Appl.28(1997)1909-1921.
[31]Tailei Zhang, Zhidong Teng, Permanence and extinction for a nonautonomous SIRS epidemic model with time delayed. Applied Math. Modelling 33(2009)1058-1071.
[32]Suzanne M.O'Regan, Thomas C.Kelly, Andrei Korobeinikov, Michael J.A.O'Callaghan. Lya-punov functions for SIR and SIRS epidemic models. Applied Math. Letters 23(2010)446-448.
[33]C. Connell McCluskey, Global stability for an SIR epidemic model with de-lay and nonlinear incidence. Nonlinear Analysis:Real World Applications (2009),doi:10.1016/j.nonrwa.2009.11.005.
[2]马知恩,周义仓,常微分方程定性与稳定性方法[M],北京：科学出版社,2007.
[3]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[J],西安交通大学学报,2005,39(2),210-213.
[4]徐文雄,张太雷,徐宗本,非线性高维自治微分系统SEIQR流行病模型全局稳定性[J],工程数学学报,2007,24(1),79-86.
[5]V. Capasso, G. Serio, Ageneration of the Kermack-Mackendrick deterministic epidemic model. Math. Biosci.42(1978)43.
[6]L.-M.Cai, X.-Z.Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model.33(2009)2919-2926.
[7]L.-M. Cai, X.-Z. Li, Mini Ghosh, Global stability of a stage-structured epidemic model with a nonlinear incidence. Appl. Math. Comput.214(2009)73-81.
[8]B. K. Mishra, N. Jha, SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Modelling (2009),doi:10.1016/j.apm.2009.06.011
[9]Dongmei Xiao, Shigui Ruan, Global analysis of an epidemic model with nonmonotone inci-dence rate. Math. Biosci.208(2007)419-429.
[10]X.-Z.Li, L.-L. Zhou, Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate. Chaos, Solitons and Fractals 40(2009)874-884.
[11]Rui Xu, Zhien Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal.:Real Word Appl.10(2009)3175-3189.
[12]Tailei Zhang, Zhidong Teng, Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence. Chaos, Solitons and Fractals 37(2008)1456-1468.
[13]Chi-Wei Lee, Yen-Shuo Tsai, Tai-Wai Wang, Chor-Chiu Lau, A loophole in international quarantine procedures disclosed during the SARS crisis. Traveal Medicine and Infection dis-ease 4(2006),22-28.
[14]Herbert Hethcote, Ma Zhien, Liao Shengbing, Effects of quarantine in six endemic models for infections disease. Math. Biosc.180(2002)141-160.
[15]Meng Fan, M Y Li, Ke Wang. Global stability of an SEIS epidemic model with recruitment and a varying total population size. Math. Biosci.170(2001)199-208.
[16]Guihua Li, Zhen Jin, Global stability of a SEIR epidemic model with infectious in latent, infected and immune period. Chaos, Solitons and Fractals 25(2005)1177-1184.
[17]Chengjun Sun, Yiping Lin, Shoupeng Tang, Global stability fora special SEIR epidemic model with nolinear incidence rates. Chaos, Solitons and Fractals 33(2007)290-297.
[18]P.E.M. Fine, Vectors and vertical transmissions:An epidemiologic perspective. Annals N.Y. Academic Sci.266(1979)173-194. S.N.
[19]Busenberg, K.L. Cooke, Vertically transmitted diseases. Models and dynamics. In:Biomath-ematics Vol.23, Springer-Verlag, Berlin (1993).
[20]M. El-Doma, Analysis of an age-dependent SIS epidemic model with vertical transmission and proportionate mixing assumption. Math. and computer modelling 29(1999)31-43.
[21]Rui Xu, Zhien Ma, Stability of a delayed SIRS epidemic model with nonlinear incidence rate, Chaos, Solitons and Fractals 41(2009)2319-2325.
[22]Rui Xu, Zhien Ma, Zhiping Wang, Global stability of a delayed SIR epidemic model with saturation incidence and temporary immunity. Comput. amd Math. with Application (2010),doi:10.1016/j.camwa.2010.03.009.
[23]K.L.Cooke, Stability analysis for a vector disease model. Rocky Mountain J. Math. 9(1979)31-42.
[24]Naoki Yoshida, Tadayuki Hara, Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J. Comput. and Appl. Math.201(2007)339-347.
[25]Mei Song, Wanbiao Ma, Yasuhiro Takeuchi, Permanence of a delayed SIR with density de-pendent birth rate. J. Comput. and Appl. Math.201(2007)389-394.
[26]Fenfen Zhang, Zhen Jin, Guiquan Sun, Bifurcation analysis of a delayed epidemic model. Appl. Math. and Comput.216(2010)753-767.
[27]Zhang Zhonghua, Peng Jigen, A SIRS epidemic model with infection-age dependence. J. Math. Anal. Appl.331(2007)1396-1414.
[28]Litao Han, Zhien Ma, Tan Shi, An SIRS epidemic model of two competitive species. Math. and computer modelling 37(2003)87-108.
[29]Ram Naresh, Agraj Tripathi, J.M.Tchuenche, Dileep Sharma, Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Computer and Math. with Appl. 58(2009)348-359.
[30]Edoardo Beretta, Yasuhiro Takeuchi, Convergence results in SIR epidemic models with vary-ing population sizes. Nonliear Anal. Theory. Methods and Appl.28(1997)1909-1921.
[31]Tailei Zhang, Zhidong Teng, Permanence and extinction for a nonautonomous SIRS epidemic model with time delayed. Applied Math. Modelling 33(2009)1058-1071.
[32]Suzanne M.O'Regan, Thomas C.Kelly, Andrei Korobeinikov, Michael J.A.O'Callaghan. Lya-punov functions for SIR and SIRS epidemic models. Applied Math. Letters 23(2010)446-448.
[33]C. Connell McCluskey, Global stability for an SIR epidemic model with de-lay and nonlinear incidence. Nonlinear Analysis:Real World Applications (2009),doi:10.1016/j.nonrwa.2009.11.005.