几类脉冲传染病模型的持续生存与周期解
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摘要
微分方程数学模型在描述种群动力学行为中起着非常重要的作用.特别是用脉冲微分方程来描述种群动力学模型能够更合理,更精确的反映各种变化规律,因为现实世界中的许多生命现象和人类的开发行为几乎是脉冲的。本文针对传染病控制的几个问题利用脉冲微分方程的相关理论和方法建立并研究了相应的动力学模型,同时借助计算机模拟讨论了所提模型的各种动力学行为,包括周期解的存在性、系统的持久性与灭绝.本文的主要结果概括如下:
第二章研究了在时滞和脉冲免疫作用下的SIR传染病模型.利用脉冲微分方程比较定理和非线性分析的方法,系统的研究了该模型的动力学性质,给出了无病周期解全局吸引和系统持久的充分条件.
在建立描述传染病模型时,幼年和成年的感染力是不同的,往往不能被忽略.与不含阶段结构的传染病模型相比较,含有阶段结构的传染病模型能更好的刻画疾病的传染情况.在第三章我们建立和研究了一类在时滞和脉冲作用下的阶段结构SIR传染病模型.由于时滞和脉冲及阶段结构的存在,使得模型的研究更为复杂.通过对模型的研究,得到了系统无病周期解全局吸引和系统持久的充分条件.在第四章我们建立和研究了一类具有非单调传染率的时滞脉冲免疫SIRS传染病模型.通过对模型的研究,得到了系统无病周期解全局吸引和系统持久的充分条件.
Mathematical models of differential equations play an important role in describing population dynamic behavior.Especially,impulsive differential equations describe population dynamic models,which is more reasonable and precise on reflecting all kinds of change orderliness,since many life phenomena and human exploitation are almost impulsive in the natural world.In this dissertation,population dynamic models are established to control infection by means of the theory and method of impulsive differential equations.Numerical simulations are used to investigate dynamic behavior including the existence of periodic solution,the permanence and infection free periodic solution.The main results of this dissertation may be summarized as follows:
The differential susceptibility SIR epidemic model with time delay and pulse vaccination are considered in Chapter 3.The dynamics of the epidemic model is globally investigated by using comparison theorem of impulsive differential equation and analytic method.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.
When epidemic models are constructed to describe the transmission of infectious diseases,the immature and mature have different susceptibility,which are not always neglected.Compared with the epidemic models without stage structure,the epidemic models with stage structure can describe the features of the diseases diffusion more well and truly.In Chapter 3,we propose the differential susceptibility SIR epidemic model with stage structure and pulse vaccination.Due to the coexistence of time delays and stage structure,the dynamical behaviors become more complex and are difficult to study,we analyze and study the model.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.In Chapter 4,we propose a delayed SIRS epidemic model with non-monotonic incidence rate and pulse vaccination,we analyze and study the model.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.
第二章研究了在时滞和脉冲免疫作用下的SIR传染病模型.利用脉冲微分方程比较定理和非线性分析的方法,系统的研究了该模型的动力学性质,给出了无病周期解全局吸引和系统持久的充分条件.
在建立描述传染病模型时,幼年和成年的感染力是不同的,往往不能被忽略.与不含阶段结构的传染病模型相比较,含有阶段结构的传染病模型能更好的刻画疾病的传染情况.在第三章我们建立和研究了一类在时滞和脉冲作用下的阶段结构SIR传染病模型.由于时滞和脉冲及阶段结构的存在,使得模型的研究更为复杂.通过对模型的研究,得到了系统无病周期解全局吸引和系统持久的充分条件.在第四章我们建立和研究了一类具有非单调传染率的时滞脉冲免疫SIRS传染病模型.通过对模型的研究,得到了系统无病周期解全局吸引和系统持久的充分条件.
Mathematical models of differential equations play an important role in describing population dynamic behavior.Especially,impulsive differential equations describe population dynamic models,which is more reasonable and precise on reflecting all kinds of change orderliness,since many life phenomena and human exploitation are almost impulsive in the natural world.In this dissertation,population dynamic models are established to control infection by means of the theory and method of impulsive differential equations.Numerical simulations are used to investigate dynamic behavior including the existence of periodic solution,the permanence and infection free periodic solution.The main results of this dissertation may be summarized as follows:
The differential susceptibility SIR epidemic model with time delay and pulse vaccination are considered in Chapter 3.The dynamics of the epidemic model is globally investigated by using comparison theorem of impulsive differential equation and analytic method.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.
When epidemic models are constructed to describe the transmission of infectious diseases,the immature and mature have different susceptibility,which are not always neglected.Compared with the epidemic models without stage structure,the epidemic models with stage structure can describe the features of the diseases diffusion more well and truly.In Chapter 3,we propose the differential susceptibility SIR epidemic model with stage structure and pulse vaccination.Due to the coexistence of time delays and stage structure,the dynamical behaviors become more complex and are difficult to study,we analyze and study the model.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.In Chapter 4,we propose a delayed SIRS epidemic model with non-monotonic incidence rate and pulse vaccination,we analyze and study the model.We obtain the conditions of global attractivity of infection-free periodic solution and permanence.
引文
[1] Lakshmikantham V, Bainov DD, Simeonov PC. Theory of impulsive differential equations. Singapore: World Scientific; 1989.
[2] Beretta E, Takeuchi Y, Global stability of an SIR epidemic model with time delays, J Math Biol 1995;33: 250-260.
[3] Takeuchi Y, Ma W B, Beretta E, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Analysis 2000;42: 931-947.
[4] Wang W, Chen L, A predator-prey system with stage-structure for predator. Comput Math Appl 1997;33: 83-91.
[5] Song X, Chen L, Optimal harvesting and stability for a two species competitive system with stage-structure. Math Biosci 2001;170: 173-186.
[6] Xiao Y, Chen Y, On an SIS epidemic model with stage-structure. J Sys Sci Complexity 2003; 16: 275-288.
[7] Aiello W G, Freedman H I, A time-delay model of single-species growth with stage structure. Math Biosci 1990;101: 139-153.
[8] Aiello W G, Preedman H I, Wu J, Analysis of a model representing stage structured population growth with state-dependent time delay. SIAM J. Appl. Math 1992;52:855-869.
[9] Agur Z, Cojocaru L, Mazor G, Anderson R M, Danon Y L, Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 1993;90:11698-11702.
[10] d'Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math Biosci 2002;179:57-72.
[11] Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model.Bull Math Biol 1998;60:1123-1148.
[12] Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math Comput Modelling 2000;31:207-215.
[13] Kermack W O, Mckendrick A G. Contributions to the mathematical theory of epidemics. Proc Roy Soc A 1927;115:700-721.
[14] Song M, Ma W B, Takeuchi Y, Global Stability of an SIR Epidemic Model with Time Delay, Applied Mathematics Letters 2004;17: 1141-1145.
[15] Zhen J, Ma Z, Han M, Global stability of an SIRS epidemic model with delays, Acta Mathematica Scientia 2006;26: 291-306.
[16] Zhen J, Ma Z, The stability of an SIR epidemic model with time delays, Math 1 Biosci 2006;3: 101-109.
[17] Beretta E, Hara T, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis 2001;47: 4107-4115.
[18] Song M, Ma W B, Takeuchi Y, Permanence of a delayed SIR epidemic model with density dependentbirth rate, Journal of Computational and Applied Mathematics 2007;201: 389-394.
[19] Ma W B, Takeuchi Y, Permanence of an SIR epidemic model with distributed time delays. Tohoku Math J 2002;54: 581-591.
[20] Gao S, Teng Z, Analysis of a delayed SIR epidemic model with pulse vaccination.Chaos, Solution and Fractals(2007), doi:10.1016/j.chaos.2007.08.056.
[21] Li X, Wang W, A discrete epidemic model with stage structure. Chaos Solitions and Fractals 2005;26: 947-958.
[22] Xiao Y, Chen Y, Analysis of a SIS epidemic model with stage structure and a delay.Communications in Nonlinear Science and Numerical Simulation 2001;6: 35-39.
[23] Zeng G, Chen L, Sun L. Complexity of an SIR epidemic dynamics model with impulsive vaccination control. Chaos Solitons and Fractals 2005;26:495-505.
[24] Hui J, Chen L. Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Discrete Cont Dyn Syst (Ser B) 2004;4:595-605.
[25] Korobeinikov A, Wake G C. Lyapunov functions and global stability for SIR, SIRS,and SIS epidemiological models. Appl Math Lett 2002;15:955-960.
[26] Allen L J S, Jones M A, Martin C E. A discrete-time model with vaccination for a measles epidemic. Math Biosci 1991;105:lll-131.
[27] Xiao Y, Chen L. Modelling and analysis of a predator-prey model with disease in the prey. Math Biosci 2001;171:59-82.
[28] Hethcote H, Ma Z, Liao S. Effects of quarantine in six endemic models for infectious diseases. Math Biosci 2002;180:141-160.
[29] Hethcote H W, Van Den Driessche P. Two SIS epidemiological models with delays. J Math Biol 2000;40:2-26.
[30] Huo H F , Existence of Positive Periodic Solutions of a Neutral Delay Lotka-Volterra System. Comp Math Appl 2004;48:1833-1846.
[31] Huo H F, Li W T, Periodic solutions of delay Leslie-Gower predator-prey models.Appl Math Comp 2004; 155: 591-605.
[32] Huo H F, Li W T, Periodic solutions of a delayed ratio-dependent food chain model.Taiwan J Math 2004;8: 211-222.
[33] Pang G, Wang F, Chen L. Extinction and permanence in delayed stage-structure predator-prey system with impulsive effects. Chaos Solutions and Fractals.doi:10.1016/j.chaos.2007.06.071.
[34] Pang G, Chen L. A delayed SIRS epidemic model with pulse vaccination. Chaos Solution and Fractals 2007;34:1629-1635.
[35] Ramsay M, Gay N, Miller E. The epidemiology of measles in England and Wales:Rationale for 1994 nation vaccination campaign. Commun Dis Rep 1994;4(12):141-146.
[36] Anderson R, May R. Infectious disease of humans, dynamical and control. Oxford University Press, Oxford, 1992.
[37] Anderson R, May R. Regulation and stability of host-parasite population interactions Ⅱ: Destabilizing process. J Anim Ecol 1978;47:219-267.
[38] Wang W, Ruan S. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Diffen Equat 2003;188:135-163.
[39] Liu W M, Levin S A, Lwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J Math Biol 1986;23:187-204.
[40] Liu W M, Hethcote H W, Levin S A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol 1987;25:359-380.
[41] Xiao D Ruan S, Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences 2006;208: 419-429.
[42] Hyman J M, Li J, Stanley E A. The differentiated infectivity and staged progression models for the transmission of HIV. Math Biosci 1999;155:77-109.
[43] Lin X, Hethcote H W,Van den Driessche P. An epidemiological model for HIV/AIDS with proportional recruitment. Math. Biosci. 1993;118:181-195.
[44] Thieme H R, Castillo-Chavez C. How may infection-age dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Applied Math 1993;53:1449-1479.
[45] James M. Hyman, Jia Li. Differential susceptibility epidemic models. J Math Biol 2005;50:626-644.
[46] Cull P, Global stability for population models. Bull Math Biol 1981;43:47-58.
[47] Huo H F, Ma Z P. Dynamics of a delayed epidemic model with non-monotonic incidence rate, Communications in Nonlinear Science and Numerical Simulation, in Press.
[48] Kuang Y, Delay Differential Equations with Applications in Population Dynamics,Academic Press, INC., San Diego, CA, 1993.
[2] Beretta E, Takeuchi Y, Global stability of an SIR epidemic model with time delays, J Math Biol 1995;33: 250-260.
[3] Takeuchi Y, Ma W B, Beretta E, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Analysis 2000;42: 931-947.
[4] Wang W, Chen L, A predator-prey system with stage-structure for predator. Comput Math Appl 1997;33: 83-91.
[5] Song X, Chen L, Optimal harvesting and stability for a two species competitive system with stage-structure. Math Biosci 2001;170: 173-186.
[6] Xiao Y, Chen Y, On an SIS epidemic model with stage-structure. J Sys Sci Complexity 2003; 16: 275-288.
[7] Aiello W G, Freedman H I, A time-delay model of single-species growth with stage structure. Math Biosci 1990;101: 139-153.
[8] Aiello W G, Preedman H I, Wu J, Analysis of a model representing stage structured population growth with state-dependent time delay. SIAM J. Appl. Math 1992;52:855-869.
[9] Agur Z, Cojocaru L, Mazor G, Anderson R M, Danon Y L, Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 1993;90:11698-11702.
[10] d'Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math Biosci 2002;179:57-72.
[11] Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model.Bull Math Biol 1998;60:1123-1148.
[12] Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math Comput Modelling 2000;31:207-215.
[13] Kermack W O, Mckendrick A G. Contributions to the mathematical theory of epidemics. Proc Roy Soc A 1927;115:700-721.
[14] Song M, Ma W B, Takeuchi Y, Global Stability of an SIR Epidemic Model with Time Delay, Applied Mathematics Letters 2004;17: 1141-1145.
[15] Zhen J, Ma Z, Han M, Global stability of an SIRS epidemic model with delays, Acta Mathematica Scientia 2006;26: 291-306.
[16] Zhen J, Ma Z, The stability of an SIR epidemic model with time delays, Math 1 Biosci 2006;3: 101-109.
[17] Beretta E, Hara T, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis 2001;47: 4107-4115.
[18] Song M, Ma W B, Takeuchi Y, Permanence of a delayed SIR epidemic model with density dependentbirth rate, Journal of Computational and Applied Mathematics 2007;201: 389-394.
[19] Ma W B, Takeuchi Y, Permanence of an SIR epidemic model with distributed time delays. Tohoku Math J 2002;54: 581-591.
[20] Gao S, Teng Z, Analysis of a delayed SIR epidemic model with pulse vaccination.Chaos, Solution and Fractals(2007), doi:10.1016/j.chaos.2007.08.056.
[21] Li X, Wang W, A discrete epidemic model with stage structure. Chaos Solitions and Fractals 2005;26: 947-958.
[22] Xiao Y, Chen Y, Analysis of a SIS epidemic model with stage structure and a delay.Communications in Nonlinear Science and Numerical Simulation 2001;6: 35-39.
[23] Zeng G, Chen L, Sun L. Complexity of an SIR epidemic dynamics model with impulsive vaccination control. Chaos Solitons and Fractals 2005;26:495-505.
[24] Hui J, Chen L. Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Discrete Cont Dyn Syst (Ser B) 2004;4:595-605.
[25] Korobeinikov A, Wake G C. Lyapunov functions and global stability for SIR, SIRS,and SIS epidemiological models. Appl Math Lett 2002;15:955-960.
[26] Allen L J S, Jones M A, Martin C E. A discrete-time model with vaccination for a measles epidemic. Math Biosci 1991;105:lll-131.
[27] Xiao Y, Chen L. Modelling and analysis of a predator-prey model with disease in the prey. Math Biosci 2001;171:59-82.
[28] Hethcote H, Ma Z, Liao S. Effects of quarantine in six endemic models for infectious diseases. Math Biosci 2002;180:141-160.
[29] Hethcote H W, Van Den Driessche P. Two SIS epidemiological models with delays. J Math Biol 2000;40:2-26.
[30] Huo H F , Existence of Positive Periodic Solutions of a Neutral Delay Lotka-Volterra System. Comp Math Appl 2004;48:1833-1846.
[31] Huo H F, Li W T, Periodic solutions of delay Leslie-Gower predator-prey models.Appl Math Comp 2004; 155: 591-605.
[32] Huo H F, Li W T, Periodic solutions of a delayed ratio-dependent food chain model.Taiwan J Math 2004;8: 211-222.
[33] Pang G, Wang F, Chen L. Extinction and permanence in delayed stage-structure predator-prey system with impulsive effects. Chaos Solutions and Fractals.doi:10.1016/j.chaos.2007.06.071.
[34] Pang G, Chen L. A delayed SIRS epidemic model with pulse vaccination. Chaos Solution and Fractals 2007;34:1629-1635.
[35] Ramsay M, Gay N, Miller E. The epidemiology of measles in England and Wales:Rationale for 1994 nation vaccination campaign. Commun Dis Rep 1994;4(12):141-146.
[36] Anderson R, May R. Infectious disease of humans, dynamical and control. Oxford University Press, Oxford, 1992.
[37] Anderson R, May R. Regulation and stability of host-parasite population interactions Ⅱ: Destabilizing process. J Anim Ecol 1978;47:219-267.
[38] Wang W, Ruan S. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Diffen Equat 2003;188:135-163.
[39] Liu W M, Levin S A, Lwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J Math Biol 1986;23:187-204.
[40] Liu W M, Hethcote H W, Levin S A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol 1987;25:359-380.
[41] Xiao D Ruan S, Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences 2006;208: 419-429.
[42] Hyman J M, Li J, Stanley E A. The differentiated infectivity and staged progression models for the transmission of HIV. Math Biosci 1999;155:77-109.
[43] Lin X, Hethcote H W,Van den Driessche P. An epidemiological model for HIV/AIDS with proportional recruitment. Math. Biosci. 1993;118:181-195.
[44] Thieme H R, Castillo-Chavez C. How may infection-age dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Applied Math 1993;53:1449-1479.
[45] James M. Hyman, Jia Li. Differential susceptibility epidemic models. J Math Biol 2005;50:626-644.
[46] Cull P, Global stability for population models. Bull Math Biol 1981;43:47-58.
[47] Huo H F, Ma Z P. Dynamics of a delayed epidemic model with non-monotonic incidence rate, Communications in Nonlinear Science and Numerical Simulation, in Press.
[48] Kuang Y, Delay Differential Equations with Applications in Population Dynamics,Academic Press, INC., San Diego, CA, 1993.