具有非单调传染率的传染病模型的动力学研究
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摘要
进行数学建模并用动力学方法分析所建模型是研究传染病的一种重要理论方法,通过对模型中一些参数的分析及数学模拟,可以得到影响疾病传播的关键因素,再对这些参数进行生物学意义上的解释,便可向人类提供有效的预防和控制策略,并且可以提供理论基础和数量依据.
本学位论文介绍了传染病模型的背景知识及基本概念,回顾了传染病模型动力学研究的概况.论文中,作者将非单调传染率引入到具有接种的传染病模型中,这是本文的创新点之一.通过分析得到了接种疫苗的有效性强弱,对种群的接种比例大小都能够对模型产生影响,同时定义了决定疾病是否发展成为地方病的阈值,即基本生成数"R_0".进而给出了模型产生平衡点的充分条件,并用Liapunov函数方法,LaSalle不变性原理及特征根方法等,分析了平衡点的存在性与稳定性,并在结论部分给出了一些防控策略.
本论文共分三章,第一章为绪论部分,简单介绍了一些基本知识,第二部分分析了传染率为非线性的传染病模型,并对其平横点的存在性与稳定性做了系统的分析与证明,第三章总结了论文的工作并给出了后继的工作目标.
It is well known that infectious disease are the archenemy of mankind. Over the past decades, human made permanent struggle to all kinds of infectious and have obtained many brilliant achievements, especially after 20th century much more achievements have been obtained. However, it is long way to go for the infectious disease. The reports from the World Health Organization(WHO) show that infectious disease are still the essential factor for human's health and life. So, it is very important to study the infectious disease either from the experiment and from the theoretical analysis.
For the mathematical workers, some mathematical models for the infectious disease have already made. By the theoretical analysis, quantitative analysis and computer simulations to these gotten mathematical models, we have obtained many results which are very close to the real disease. Since from the 1980s, a rapid progress has been made for the mathematical models of many kinds of disease, which makes the theoretical studies more perfect. Liapunov's stability approach, LaSalle invariance principle, limited equation theory, characteristic roots methods, bifurcation theory, K sequence monotone system theory and the center manifold theory are the main methods to study the mathematical models of disease.
Inoculate vaccination for the epidemic individual has become one of effective methods for diseases control. In this paper, we focus on study a SIS epidemic model with vaccination and nonlinear incidence rate (?).We first obtained the equilibria of this model and then defined the basic reproduction number "R_0" determining whether the endemic diseases occur or not. When R_0 < 1,the model has only a global asymptotically stable disease-free equilibrium and the disease will naturally disappear; when R_0 > 1,the model has stable endemic equilibrium and the disease will always exist in the population.
Let N(t) , S(t), I(t) and R(t) be the number of total population, susceptible individuals, infective individuals and vaccinated individuals at time t, respectively. First, we have the following assumption:
1) b is the recruitment rate of the population andδ(0 <δ< 1) is the ratio of vaccination of b;
2) The infectious disease spread by the nonmonotone incidence rate (?),and for the vaccinated individuals, the incidence rate is (?).WhereβI measures the infectious force of the disease in the susceptible individuals,σβI measures the infectious force of the disease in the vaccinated individuals, and the parameterσdescribes the efficiency of the vaccine,the largerσis the worse efficiency of the vaccine,σ= 0 means the efficiency of the vaccine is 100% for the infectious disease control, andσ= 1 means the vaccination is ineffective;
3) d is the natural death rate of the population;
4) The constantμ≥0 is the ratio of vaccination of the susceptible individuals;
5) The constantγ> 0 is the natural recovery rate of the infective individuals;
6)ε> 0 is the rate at which vaccinated individuals lose immunity and return to the susceptible class;
7)α≥0 is the parameter measures the psychological or inhibitory effect. Notice that whenα= 0,the nonmonotone incidence rate (?) becomes the bilinear incidence rateβS I;
8) The recruitment individuals who are not vaccinated are susceptible individuals.
Based on these assumption, we constructed the following mathematical model:
We obtained some results as follow:
CaseⅠ):whenσ= 1,we obtained the basic reproduction number, disease-free equilibrium,epidemic equilibrium are as follows:respectively, where andBy using the Liapunov function, LaSalle invariance principle and characteristic roots methods we have the following results for the stability of the equilibrium:
Theorem 1 If R_0≤1,then there is no endemic equilibrium, and the disease-free equilibrium E_0 is globally asymptotically stable; if R_0 > 1,the disease-free equilibrium E_0 is unstable, the endemic equilibrium E~+ is stable.
Furthermore, we obtained the following schemes that can be used to control the spread of infectious disease:
1) Decrease the effective contact rate small enough, do our most to and prevent the spread way from infectious disease;
2) Reduce the recruitment rate b, decrease the mobility from one area to another;
3) Improve the recovery rate y, speed up the study of drugs.
CaseⅡ): whenσ= 0,we obtained the basic reproduction number, disease-free equilibrium, epidemic equilibrium are as follows:respectively, whereThe stability of the disease-free equilibrium and endemic equilibrium is similar to the- orem 1,and here, we omitted the details. We only give some methods that can be used to control the spread of infectious disease:
1) Methods used as same as the caseⅠ;
2) Increase the probability of vaccination of the recruitment individuals big enough;
3) Increase the vaccination rate of the susceptible individuals, especial in the poor area.
CaseⅢ):when 0 <σ< 1,we obtained the basic reproduction number and disease-freeequilibrium are as follows:respectively.
Then we give a theorem for the global asymptotically stable of the disease-free equilibrium.
Theorem 2 For the system (2.13), if R_0≤1,the disease-free equilibrium E_σ,is global asymptotically stable, and if R_0 > 1,the the disease-free equilibrium E_σis unstable.
It is difficult to obtain the endemic equilibrium in this case. We just give a theorem to illustrate the stability of the endemic equilibrium.
Theorem 3 If R_0 < 1,the system (2.13) nonexist positive equilibrium; if R_0 > 1. the system (2.13)probably exist positive equilibrium, and if the system (2.13) exist positive equilibrium E_σ~+,then it was stable.
本学位论文介绍了传染病模型的背景知识及基本概念,回顾了传染病模型动力学研究的概况.论文中,作者将非单调传染率引入到具有接种的传染病模型中,这是本文的创新点之一.通过分析得到了接种疫苗的有效性强弱,对种群的接种比例大小都能够对模型产生影响,同时定义了决定疾病是否发展成为地方病的阈值,即基本生成数"R_0".进而给出了模型产生平衡点的充分条件,并用Liapunov函数方法,LaSalle不变性原理及特征根方法等,分析了平衡点的存在性与稳定性,并在结论部分给出了一些防控策略.
本论文共分三章,第一章为绪论部分,简单介绍了一些基本知识,第二部分分析了传染率为非线性的传染病模型,并对其平横点的存在性与稳定性做了系统的分析与证明,第三章总结了论文的工作并给出了后继的工作目标.
It is well known that infectious disease are the archenemy of mankind. Over the past decades, human made permanent struggle to all kinds of infectious and have obtained many brilliant achievements, especially after 20th century much more achievements have been obtained. However, it is long way to go for the infectious disease. The reports from the World Health Organization(WHO) show that infectious disease are still the essential factor for human's health and life. So, it is very important to study the infectious disease either from the experiment and from the theoretical analysis.
For the mathematical workers, some mathematical models for the infectious disease have already made. By the theoretical analysis, quantitative analysis and computer simulations to these gotten mathematical models, we have obtained many results which are very close to the real disease. Since from the 1980s, a rapid progress has been made for the mathematical models of many kinds of disease, which makes the theoretical studies more perfect. Liapunov's stability approach, LaSalle invariance principle, limited equation theory, characteristic roots methods, bifurcation theory, K sequence monotone system theory and the center manifold theory are the main methods to study the mathematical models of disease.
Inoculate vaccination for the epidemic individual has become one of effective methods for diseases control. In this paper, we focus on study a SIS epidemic model with vaccination and nonlinear incidence rate (?).We first obtained the equilibria of this model and then defined the basic reproduction number "R_0" determining whether the endemic diseases occur or not. When R_0 < 1,the model has only a global asymptotically stable disease-free equilibrium and the disease will naturally disappear; when R_0 > 1,the model has stable endemic equilibrium and the disease will always exist in the population.
Let N(t) , S(t), I(t) and R(t) be the number of total population, susceptible individuals, infective individuals and vaccinated individuals at time t, respectively. First, we have the following assumption:
1) b is the recruitment rate of the population andδ(0 <δ< 1) is the ratio of vaccination of b;
2) The infectious disease spread by the nonmonotone incidence rate (?),and for the vaccinated individuals, the incidence rate is (?).WhereβI measures the infectious force of the disease in the susceptible individuals,σβI measures the infectious force of the disease in the vaccinated individuals, and the parameterσdescribes the efficiency of the vaccine,the largerσis the worse efficiency of the vaccine,σ= 0 means the efficiency of the vaccine is 100% for the infectious disease control, andσ= 1 means the vaccination is ineffective;
3) d is the natural death rate of the population;
4) The constantμ≥0 is the ratio of vaccination of the susceptible individuals;
5) The constantγ> 0 is the natural recovery rate of the infective individuals;
6)ε> 0 is the rate at which vaccinated individuals lose immunity and return to the susceptible class;
7)α≥0 is the parameter measures the psychological or inhibitory effect. Notice that whenα= 0,the nonmonotone incidence rate (?) becomes the bilinear incidence rateβS I;
8) The recruitment individuals who are not vaccinated are susceptible individuals.
Based on these assumption, we constructed the following mathematical model:
We obtained some results as follow:
CaseⅠ):whenσ= 1,we obtained the basic reproduction number, disease-free equilibrium,epidemic equilibrium are as follows:respectively, where andBy using the Liapunov function, LaSalle invariance principle and characteristic roots methods we have the following results for the stability of the equilibrium:
Theorem 1 If R_0≤1,then there is no endemic equilibrium, and the disease-free equilibrium E_0 is globally asymptotically stable; if R_0 > 1,the disease-free equilibrium E_0 is unstable, the endemic equilibrium E~+ is stable.
Furthermore, we obtained the following schemes that can be used to control the spread of infectious disease:
1) Decrease the effective contact rate small enough, do our most to and prevent the spread way from infectious disease;
2) Reduce the recruitment rate b, decrease the mobility from one area to another;
3) Improve the recovery rate y, speed up the study of drugs.
CaseⅡ): whenσ= 0,we obtained the basic reproduction number, disease-free equilibrium, epidemic equilibrium are as follows:respectively, whereThe stability of the disease-free equilibrium and endemic equilibrium is similar to the- orem 1,and here, we omitted the details. We only give some methods that can be used to control the spread of infectious disease:
1) Methods used as same as the caseⅠ;
2) Increase the probability of vaccination of the recruitment individuals big enough;
3) Increase the vaccination rate of the susceptible individuals, especial in the poor area.
CaseⅢ):when 0 <σ< 1,we obtained the basic reproduction number and disease-freeequilibrium are as follows:respectively.
Then we give a theorem for the global asymptotically stable of the disease-free equilibrium.
Theorem 2 For the system (2.13), if R_0≤1,the disease-free equilibrium E_σ,is global asymptotically stable, and if R_0 > 1,the the disease-free equilibrium E_σis unstable.
It is difficult to obtain the endemic equilibrium in this case. We just give a theorem to illustrate the stability of the endemic equilibrium.
Theorem 3 If R_0 < 1,the system (2.13) nonexist positive equilibrium; if R_0 > 1. the system (2.13)probably exist positive equilibrium, and if the system (2.13) exist positive equilibrium E_σ~+,then it was stable.
引文
[1] 彭文伟.传染病学[M].北京:人民卫生出版社,2001.
[2] WEBB G F. Theory of nonlinear age-dependent population dynamics[M]. New York:Springer-Verlag, 1987.
[3] SONG J, YU J. Population system control[M]. Berlin: Springer-Verlag, 1987.
[4] KERMACK W O, MCKENDRICK A G. Conttributions to the mathematical theory of epidemics[J]. Proceedings Royal Society, 1927, A115, 700-721.
[5] KERMACK W O, MCKENDRICK A G. Conttributions to the mathematical theory of epidemics[J]. Proceedings Royal Society, A138, 55-83.
[6] 陈兰荪,陈健.非线性生物动力系统[M].北京:科学出版社,1983.
[7] BA1LY N T J. The mathematical theory of infectious disease, 2nded[M]. New York:Haner, 1975.
[8] CAPASSA V, SERIO G. A generalization of the Kermack-Mekendrick deterministic epidemic model[J]. Mathematical Biosciences, 1978, 42,43-61.
[9] RUAN S, WANG W. Dynamical behavior of an epidemic model with a nonlinear incidence rate[J]. Journal of Differential Equations, 2003, 188, 135-163.
[10] XIAO D, RUAN S. Global analysis of an epidemic model with nonmonotone incidence rate[J]. Mathematical Biosciences, 2007, 208, 419-429.
[11] LIU W, LEVIN S A, IWASA Y. Influce of nonlinear incidence rates upon the behavior of SIRS epidemiological models[J].Journal of Mathematical Biology, 1986, 23,187-204.
[12] LIU W, HETHCOTE H W, LEVIN S A. Dynamical behavior of epidemiological models with nonlinear incidence rates[J]. Journal of Mathematical Biology, 1987, 25, 359-380.
[13] 刘烁,李建全,王拉娣,马润年.一类带有非线性传染率的SEIR传染病模型的全局分析[J].Mathematics in Practice and Theory,2007,37(23),54-59.
[14] WANG X, JIANG X, SONG X. Global dynamics of a model for HIV infection of CD4~+T cell with cure rate[J]. Mathematica Applicata, 2008, 21(3), 449-456.
[15] 娄洁.艾滋病的动力学模型研究[D].西安:西安交通大学数学系,2004.
[16] FU C, SHEN Y, ZHENG L. Exponential stability of SIS epidemic models with varying total populaition size[J]. Mathematica Applicata, 2007, 20(2), 233-238.
[17] 秦琳.几类传染病模型的动力学研究[D].长春:吉林大学数学研究所,2008.
[18] REBECCA V, CULSHAW, RUAN S, RAYMOND J, SPITERI. Optimal HIV treatment by maximising immune response[J]. Journal of Mathematical Biology, 2004, 48, 545-562.
[19] KATRI P,RUAN S. Dynamics of human T-cell lymphotropic virus I(HTLV-Ⅰ) infection of CD4~+T T-cells[J]. Comptes Rendus Biologies, 2004, 327, 1009-1016.
[20] 李建全.传染病动力学模型研究[D].西安:西安交通大学数学系,2003.
[21] TANG Y, HUANG D, RUAN S, ZHANG W. Coexistence of limit cycles and homoclinic loop in a SIRS model with a nonlinear incidence rate[J]. Society for Industrial and Applied Mathematics, 2008, 69(2), 621-639.
[22] WANG W, RUAN S. Bifurcation in an epidemic model with constant removal rate of the infective[J]. Mathematical Analysis and Applications, 2004, 291, 775-793.
[23] CAI L,LI X, YU J. A two-strain epidemic model with super-infection and vaccination[J]. Mathematica Applicata, 2007, 20(2), 328-335.
[24] 方彬,杨金根,李学志.潜伏期和染病期均具有康复的年龄结构MSEIS传染病模型的稳定性[J].Mathematica Applicata,2009,22(1),90-100.
[25] 刘汉武.有接种的SIS型传染病模型[D].西安:西安交通大学数学系,2001.
[26] LI X, GUPUR G, ZHU G. Analysis for an age-structured SEIR epidemic model with vaccination[J]. International Journal of Differential Equations Applicata, 2003, 7(1), 47-67.
[27] LI X, GUPUR G, ZHU G. Analysis of an age-structured SEIR epidemic model with varying total population size and vaccination[J]. Acta Mathematicae Applicatae Sinica,2004,20(1),25-36.
[28] LI X, GUPUR G, ZHU G. Existence and uniqueness of endemic state for the age-structured MSEIR epidemic model[J]. Acta Mathematicae Applicatae Sinica, 2002,18(3), 441-454.
[29] BUSENERG S, VAN DRIESSCHE P. Analysis of a disease transmission model in a population with varying size[J]. Journal of Mathematical Biology, 1990, 28, 257-270.
[30] MARTCHEVA M, HORST THIEME R. Progression age enhanced backward bifurcation in an epidemic model with super-infection[J]. Journal of Mathematical Biology,2003, 46, 385-424.
[31] BERETTA E, TAKEUCHI Y. Convergence results in SIR epidemic models with varying population size[J]. Nonlinear Analysis Theory Methods & Applications, 1997, 28,1909-1921.
[32] FAN M, LI M, WANG K. Global stability of an SEIS epidemic model with recruitment and a varying total population size[J]. Mathematical Biosciences, 2001, 170, 199-208.
[33] LI J, MA Z. Qualitative analysis of SIS epidemic model with vaccination and varying total population size[J]. Mathematical and Computer Modeling, 2002, 35, 1235-1243.
[34] WANG X, SONG X. Global stability and periodic solution of a model for HIV infection of CD4~+T cells[J]. Applied Mathematics Computation, 2007, 189, 1331-1340.
[35] WANG L, LI M. Mathematical analysis of the global dynamics of a model for HIV infection of CD4~+T cells[J]. Mathematical Bioscience, 2006, 200, 44-57.
[36] HERBERT W, HETHCOTE. The mathematics of infectious diseases[J]. Society for Industrial and Applied Mathematics, 2000,42(4), 599-653.
[37] 伍卓群,李勇.常微分方程[M].北京:高等教育出版社,2004.
[2] WEBB G F. Theory of nonlinear age-dependent population dynamics[M]. New York:Springer-Verlag, 1987.
[3] SONG J, YU J. Population system control[M]. Berlin: Springer-Verlag, 1987.
[4] KERMACK W O, MCKENDRICK A G. Conttributions to the mathematical theory of epidemics[J]. Proceedings Royal Society, 1927, A115, 700-721.
[5] KERMACK W O, MCKENDRICK A G. Conttributions to the mathematical theory of epidemics[J]. Proceedings Royal Society, A138, 55-83.
[6] 陈兰荪,陈健.非线性生物动力系统[M].北京:科学出版社,1983.
[7] BA1LY N T J. The mathematical theory of infectious disease, 2nded[M]. New York:Haner, 1975.
[8] CAPASSA V, SERIO G. A generalization of the Kermack-Mekendrick deterministic epidemic model[J]. Mathematical Biosciences, 1978, 42,43-61.
[9] RUAN S, WANG W. Dynamical behavior of an epidemic model with a nonlinear incidence rate[J]. Journal of Differential Equations, 2003, 188, 135-163.
[10] XIAO D, RUAN S. Global analysis of an epidemic model with nonmonotone incidence rate[J]. Mathematical Biosciences, 2007, 208, 419-429.
[11] LIU W, LEVIN S A, IWASA Y. Influce of nonlinear incidence rates upon the behavior of SIRS epidemiological models[J].Journal of Mathematical Biology, 1986, 23,187-204.
[12] LIU W, HETHCOTE H W, LEVIN S A. Dynamical behavior of epidemiological models with nonlinear incidence rates[J]. Journal of Mathematical Biology, 1987, 25, 359-380.
[13] 刘烁,李建全,王拉娣,马润年.一类带有非线性传染率的SEIR传染病模型的全局分析[J].Mathematics in Practice and Theory,2007,37(23),54-59.
[14] WANG X, JIANG X, SONG X. Global dynamics of a model for HIV infection of CD4~+T cell with cure rate[J]. Mathematica Applicata, 2008, 21(3), 449-456.
[15] 娄洁.艾滋病的动力学模型研究[D].西安:西安交通大学数学系,2004.
[16] FU C, SHEN Y, ZHENG L. Exponential stability of SIS epidemic models with varying total populaition size[J]. Mathematica Applicata, 2007, 20(2), 233-238.
[17] 秦琳.几类传染病模型的动力学研究[D].长春:吉林大学数学研究所,2008.
[18] REBECCA V, CULSHAW, RUAN S, RAYMOND J, SPITERI. Optimal HIV treatment by maximising immune response[J]. Journal of Mathematical Biology, 2004, 48, 545-562.
[19] KATRI P,RUAN S. Dynamics of human T-cell lymphotropic virus I(HTLV-Ⅰ) infection of CD4~+T T-cells[J]. Comptes Rendus Biologies, 2004, 327, 1009-1016.
[20] 李建全.传染病动力学模型研究[D].西安:西安交通大学数学系,2003.
[21] TANG Y, HUANG D, RUAN S, ZHANG W. Coexistence of limit cycles and homoclinic loop in a SIRS model with a nonlinear incidence rate[J]. Society for Industrial and Applied Mathematics, 2008, 69(2), 621-639.
[22] WANG W, RUAN S. Bifurcation in an epidemic model with constant removal rate of the infective[J]. Mathematical Analysis and Applications, 2004, 291, 775-793.
[23] CAI L,LI X, YU J. A two-strain epidemic model with super-infection and vaccination[J]. Mathematica Applicata, 2007, 20(2), 328-335.
[24] 方彬,杨金根,李学志.潜伏期和染病期均具有康复的年龄结构MSEIS传染病模型的稳定性[J].Mathematica Applicata,2009,22(1),90-100.
[25] 刘汉武.有接种的SIS型传染病模型[D].西安:西安交通大学数学系,2001.
[26] LI X, GUPUR G, ZHU G. Analysis for an age-structured SEIR epidemic model with vaccination[J]. International Journal of Differential Equations Applicata, 2003, 7(1), 47-67.
[27] LI X, GUPUR G, ZHU G. Analysis of an age-structured SEIR epidemic model with varying total population size and vaccination[J]. Acta Mathematicae Applicatae Sinica,2004,20(1),25-36.
[28] LI X, GUPUR G, ZHU G. Existence and uniqueness of endemic state for the age-structured MSEIR epidemic model[J]. Acta Mathematicae Applicatae Sinica, 2002,18(3), 441-454.
[29] BUSENERG S, VAN DRIESSCHE P. Analysis of a disease transmission model in a population with varying size[J]. Journal of Mathematical Biology, 1990, 28, 257-270.
[30] MARTCHEVA M, HORST THIEME R. Progression age enhanced backward bifurcation in an epidemic model with super-infection[J]. Journal of Mathematical Biology,2003, 46, 385-424.
[31] BERETTA E, TAKEUCHI Y. Convergence results in SIR epidemic models with varying population size[J]. Nonlinear Analysis Theory Methods & Applications, 1997, 28,1909-1921.
[32] FAN M, LI M, WANG K. Global stability of an SEIS epidemic model with recruitment and a varying total population size[J]. Mathematical Biosciences, 2001, 170, 199-208.
[33] LI J, MA Z. Qualitative analysis of SIS epidemic model with vaccination and varying total population size[J]. Mathematical and Computer Modeling, 2002, 35, 1235-1243.
[34] WANG X, SONG X. Global stability and periodic solution of a model for HIV infection of CD4~+T cells[J]. Applied Mathematics Computation, 2007, 189, 1331-1340.
[35] WANG L, LI M. Mathematical analysis of the global dynamics of a model for HIV infection of CD4~+T cells[J]. Mathematical Bioscience, 2006, 200, 44-57.
[36] HERBERT W, HETHCOTE. The mathematics of infectious diseases[J]. Society for Industrial and Applied Mathematics, 2000,42(4), 599-653.
[37] 伍卓群,李勇.常微分方程[M].北京:高等教育出版社,2004.
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