考虑免疫作用和细胞内时滞的病毒动力学性态
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摘要
本文主要从数学上研究具有免疫作用和细胞内时滞的细胞动力学行为.通过构造Lyapunov函数和Lyapunov泛函将模型的动力学性态进行完整的分析.
首先,研究了两个考虑CTL免疫反应的细胞动力学模型的全局性态.两个模型都可得出当基本再生数R0≤1时,无病毒感染的平衡点是全局渐近稳定的.当R0>1时,平凡平衡点失去稳定性,第一个模型的感染平衡点是全局渐近稳定的,第二个模型是一致持续的.
其次,我们考虑含细胞内时滞和体液免疫反应的细胞动力学模型的全局性态.得出模型的动力学行为完全由基本再生数R0和体液免疫基本再生数R1决定,且R1 最后,我们研究了含细胞内时滞和免疫反应(CTL免疫反应和体液免疫反应)的细胞动力学模型的全局性态.得出模型的动力学行为完全由病毒感染基本再生数R0,CTL免疫再生数R1,抗体免疫再生数R2,CTL免疫竞争再生数R3和抗体免疫再生数R4决定.当R0≤1时,无病毒感染平衡点是全局渐近稳定的；当R0>1,R1≤1和R2≤1时,无免疫感染平衡点是全局渐近稳定的；当R1>1和R4≤1时,CTL免疫介导的感染平衡点是全局渐近稳定的；当R2>1和R3≤1时,抗体免疫介导的感染平衡点是全局渐近稳定的；当R3>1和R4>1时,CTL免疫反应和抗体免疫反应共同介导的感染平衡点是全局渐近稳定的.
In this paper, the virus dynamics with immune responses and intracellular delay are mainly studied in mathematics. We analyse the global dynamics of these models completely by con-structing Lyapunov function and Lyapunov functional.
First, the properties of two virus dynamic models with CTL immune responses are studied. The two models imply that when the basic reproductive number R0< 1, the infection-free steady state is globally asymptotically stable. when R0>1, the trivial equilibriums lose stability, for the first model the infection steady state is globally asymptotically stable, and the second model is uniformly persistent.
Second, the global properties of a virus dynamic model with intracellular delay and hu-moral immune responses are studied. We obtain that the dynamics of this model is completely determined by the basic reproductive number R0 and the humoral immune basic reproductive number R1, with R1 In the end, the global properties of a virus dynamic model with intracellular delay and im-mune responses (CTL and humoral immune responses) are studied. We obtain that the global dynamics of this model is completely determined by the reproductive number for viral infection R0, for CTL immune response R1, for antibody immune response R2, for CTL immune competi-tion R3 and for antibody immune competition R4. when R0≤1, the infection-free steady state is globally asymptotically stable; when R0>1, R1≤1 and R2≤1, the immune-free infected equi-librium is globally asymptotically stable; when R1>1 and R4≤1, the CTL immune infected equilibrium is globally asymptotically stable; when R2>1 and R3≤1, the antibody immune infected equilibrium is globally asymptotically stable; when R3>1 and R4>1, the CTL and the antibody immune infected equilibrium is globally asymptotically stable.
首先,研究了两个考虑CTL免疫反应的细胞动力学模型的全局性态.两个模型都可得出当基本再生数R0≤1时,无病毒感染的平衡点是全局渐近稳定的.当R0>1时,平凡平衡点失去稳定性,第一个模型的感染平衡点是全局渐近稳定的,第二个模型是一致持续的.
其次,我们考虑含细胞内时滞和体液免疫反应的细胞动力学模型的全局性态.得出模型的动力学行为完全由基本再生数R0和体液免疫基本再生数R1决定,且R1
In this paper, the virus dynamics with immune responses and intracellular delay are mainly studied in mathematics. We analyse the global dynamics of these models completely by con-structing Lyapunov function and Lyapunov functional.
First, the properties of two virus dynamic models with CTL immune responses are studied. The two models imply that when the basic reproductive number R0< 1, the infection-free steady state is globally asymptotically stable. when R0>1, the trivial equilibriums lose stability, for the first model the infection steady state is globally asymptotically stable, and the second model is uniformly persistent.
Second, the global properties of a virus dynamic model with intracellular delay and hu-moral immune responses are studied. We obtain that the dynamics of this model is completely determined by the basic reproductive number R0 and the humoral immune basic reproductive number R1, with R1
引文
[1]舒红兵,抗病毒天然免疫[M].北京：科学出版社,2009,13-248.
[2]陆征一,王稳地,王开发,数学生物学进展[M].北京：科学出版社,2006,58-59.
[3]A. Korobeinikov, Global Properties of Basic Virus Dynamics Models [J]. Bulletin of Mathematical Biology,2004,66:879-883.
[4]K. Wang, W. Wang, X. Liu, Viral infection model with periodic lytic immune response [J]. Chaos, Solitons Fractals,2006.28:90-99.
[5]K. Wang, W. Wang, X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses [J], Compute. Math. Appl..2006,51:1593-1610.
[6]R.V. Culshaw, S.G. Ruan, A delay-differential equation model of HIV infection of CD4+T-cells [J]. Math. Biosci..2000.165:27-39.
|7] N. Eshima, M. Tabata, T. Okada, S. Karukaya, Population dynamics of HTLV-I infection:A discrete-time mathematical epidemic model approach [J]. Math. Med. Biol.,2003,20:29-45.
[8]P.W. Nelson, A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection [J]. Math. Biosci.,2002,179:73-94.
[91 M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses [J]. Science,1996,272:74-79.
[10]Haiyan Pang, Wendi Wang, Kaifa Wang, Deterministic and Stochastic Analisis of a Cell-to-cell Virus Dynamics Model with Immune Impairment [J]西南师范大学学报(自然科学版),2006.31(5):26-30.
[11]Zhiping Wang, Xianning Liu, Dynamics of a Chronic Viral Infection Model with Humoral Immunity and Immune Impairment [J]西南大学学报(自然科学版),2007,29(12)：1-5.
[12]D. Wodarz, Hepatitis C virus dynamics and pathology:the role of CTL and antibody responses [J]. J. Gen. Virol.,2003,84:1743-1750.
[13]陈兰荪.宋新宁,陆征一,数学生态学模型与研究方法[M].成都：四川科学技术出版社,2003.
[14]A.S. Perelson, D.E. Kirschner. Dynamics of HIV infection of CD4+T cells [J]. Math.Biosci.,1993, 114:81-125.
[15]A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo [J]. SI AM Review, 1999,41:3-44.
[16]O. Krakovska, L.M. Wahl, Costs versus benefits:best possible and best practical treatment regimens for HIV [J]. J. Math. Biol.,2007,54:385-406.
[17]A.S. Perelson, D.E. Kirschner, R. De Boer, Dynamics of HIV-I infection of CD4 T cells [J]. Math. Biosci.,1993,114:81-125.
[18]M.A. Nowak, S. Bonhoeffer, A.H. Hill, et al, Viral dynamics in hepatetis B virus [J]. PNAS.1996. 93:4398-4402.
[191 A. A. Canabarro. I.M. Gleria and M.L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response [J]. Physica A,2004,342:234-241.
[20]Y. Iwasa, M. Franziska, M.A. Nowak, Virus evolution with patients increases pathogenicity [J]. J. Theor. Biol.,2005,232:17-26.
[21]T.斯科恩著；朱俊萍,李相辉,安静译,病毒学精要概览[M].北京：科学出版社,2010,1-2.
[22]M.A. Nowak, R.M. May, Virus Dynamics:Mathematical Principles of Immunology and Virol ogy [M]. New York:Oxford University Press.2000:16-66.
[23]M.A. Nowak, Dynamies of hcpatitis B virus infection. Edited by I.M. Arias, J.L. Boyer, F.V. Chis-ari, N. Fausto, D.Schehter and D.A. Shafritz. The liver:biology and Pathobiology, Fourth Edition. Lippincott Williams and Wilkins, PhiladePhia, Chapter 66,2001,995-999.
[24]S. Bonhoeffcr, R.M. May, C.M. Shaw, M.A. Nowak, Virus dynamics and drug therapy [J]. Proc. Natl. Acad. Sci. USA,1997,94:6971-6976.
[25]R.M. Ribeiro, A. Lo, A.S. Perelson, Dynamics of hepatitis B virus infection [J], Microbes and Infection,2002,4:52-535.
[26]Tsuyoshi Kajiwara, Toru Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics [J]. Discrete and Continuous Dynamical Systems-Series,2004, B4:615-622.
[27]廖晓昕,稳定性的数学理论及应用(第二版)[M].武汉：华中师范大学出版社,2001.
[28]H. I. Freedman, Shigui Ruan, Moxun Tang, Uniform persistence and flows near a closed positively invariant set [J]. Journal of Dynamics and Equations,1994.6(4):583-600.
129] Yang Kuang, DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN POPULATION DYNAMICS [M]. San Diego:Academic Press,1993,274-275.
[30]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,2001,65-72.
[31]张锦炎,冯贝叶,常微分方程的几何理论与分支问题(第三版)[M].北京：北京大学出版社,2000.
[32]Haiyan Pang, Wendi Wang, Kaifa Wang, Global properties of virus dynamics with CTL immune response [J]西南师范大学学报(自然科学版),2005,30(5)：796-799.
[33]Wendi Wang, Lansun Chen, A Predator-Prey System with Stage-Structure for Predator. Computres Math [J], Applic,1997,33(8):83-91.
[34]H. Smith, Monotone semiflows generated by functional differential equations [J]. J. Diff. Equa., 1987,16:420-442.
[35]V. Herz, S. Bonhoeffer, R. Anderson, R. May, M. Nowak, Viral dynamics in vivo:limitations on estimations on intracellular delay and virus decay [J]. Proc. Nat. Acad. Sci.,1996,93:7247-7251.
[36]A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time [J]. Science,1996,271: 1582-1586.
[37]J.M. Murray, R.H. Purcell, S.F. Wieland, The half-life of hepatitis B virions [J]. Hepatology,2006, 44:1117-1121.
[38]M.Y. Li, H. Shu, Global dynamics of an in-host viral model with intracellular delay [J]. Bull. Math. Biol.,2010,72:1492-505.
[39]庞海燕,考虑免疫反应的病毒动力学的全局性态：[西南大学学位论文].重庆,西南大学,2006.10-44
[40]H. Zhu, X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay [J]. Discrete Contin. Dyn. Syst. Ser. B.,2009,12:511-524.
[2]陆征一,王稳地,王开发,数学生物学进展[M].北京：科学出版社,2006,58-59.
[3]A. Korobeinikov, Global Properties of Basic Virus Dynamics Models [J]. Bulletin of Mathematical Biology,2004,66:879-883.
[4]K. Wang, W. Wang, X. Liu, Viral infection model with periodic lytic immune response [J]. Chaos, Solitons Fractals,2006.28:90-99.
[5]K. Wang, W. Wang, X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses [J], Compute. Math. Appl..2006,51:1593-1610.
[6]R.V. Culshaw, S.G. Ruan, A delay-differential equation model of HIV infection of CD4+T-cells [J]. Math. Biosci..2000.165:27-39.
|7] N. Eshima, M. Tabata, T. Okada, S. Karukaya, Population dynamics of HTLV-I infection:A discrete-time mathematical epidemic model approach [J]. Math. Med. Biol.,2003,20:29-45.
[8]P.W. Nelson, A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection [J]. Math. Biosci.,2002,179:73-94.
[91 M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses [J]. Science,1996,272:74-79.
[10]Haiyan Pang, Wendi Wang, Kaifa Wang, Deterministic and Stochastic Analisis of a Cell-to-cell Virus Dynamics Model with Immune Impairment [J]西南师范大学学报(自然科学版),2006.31(5):26-30.
[11]Zhiping Wang, Xianning Liu, Dynamics of a Chronic Viral Infection Model with Humoral Immunity and Immune Impairment [J]西南大学学报(自然科学版),2007,29(12)：1-5.
[12]D. Wodarz, Hepatitis C virus dynamics and pathology:the role of CTL and antibody responses [J]. J. Gen. Virol.,2003,84:1743-1750.
[13]陈兰荪.宋新宁,陆征一,数学生态学模型与研究方法[M].成都：四川科学技术出版社,2003.
[14]A.S. Perelson, D.E. Kirschner. Dynamics of HIV infection of CD4+T cells [J]. Math.Biosci.,1993, 114:81-125.
[15]A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo [J]. SI AM Review, 1999,41:3-44.
[16]O. Krakovska, L.M. Wahl, Costs versus benefits:best possible and best practical treatment regimens for HIV [J]. J. Math. Biol.,2007,54:385-406.
[17]A.S. Perelson, D.E. Kirschner, R. De Boer, Dynamics of HIV-I infection of CD4 T cells [J]. Math. Biosci.,1993,114:81-125.
[18]M.A. Nowak, S. Bonhoeffer, A.H. Hill, et al, Viral dynamics in hepatetis B virus [J]. PNAS.1996. 93:4398-4402.
[191 A. A. Canabarro. I.M. Gleria and M.L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response [J]. Physica A,2004,342:234-241.
[20]Y. Iwasa, M. Franziska, M.A. Nowak, Virus evolution with patients increases pathogenicity [J]. J. Theor. Biol.,2005,232:17-26.
[21]T.斯科恩著；朱俊萍,李相辉,安静译,病毒学精要概览[M].北京：科学出版社,2010,1-2.
[22]M.A. Nowak, R.M. May, Virus Dynamics:Mathematical Principles of Immunology and Virol ogy [M]. New York:Oxford University Press.2000:16-66.
[23]M.A. Nowak, Dynamies of hcpatitis B virus infection. Edited by I.M. Arias, J.L. Boyer, F.V. Chis-ari, N. Fausto, D.Schehter and D.A. Shafritz. The liver:biology and Pathobiology, Fourth Edition. Lippincott Williams and Wilkins, PhiladePhia, Chapter 66,2001,995-999.
[24]S. Bonhoeffcr, R.M. May, C.M. Shaw, M.A. Nowak, Virus dynamics and drug therapy [J]. Proc. Natl. Acad. Sci. USA,1997,94:6971-6976.
[25]R.M. Ribeiro, A. Lo, A.S. Perelson, Dynamics of hepatitis B virus infection [J], Microbes and Infection,2002,4:52-535.
[26]Tsuyoshi Kajiwara, Toru Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics [J]. Discrete and Continuous Dynamical Systems-Series,2004, B4:615-622.
[27]廖晓昕,稳定性的数学理论及应用(第二版)[M].武汉：华中师范大学出版社,2001.
[28]H. I. Freedman, Shigui Ruan, Moxun Tang, Uniform persistence and flows near a closed positively invariant set [J]. Journal of Dynamics and Equations,1994.6(4):583-600.
129] Yang Kuang, DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN POPULATION DYNAMICS [M]. San Diego:Academic Press,1993,274-275.
[30]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,2001,65-72.
[31]张锦炎,冯贝叶,常微分方程的几何理论与分支问题(第三版)[M].北京：北京大学出版社,2000.
[32]Haiyan Pang, Wendi Wang, Kaifa Wang, Global properties of virus dynamics with CTL immune response [J]西南师范大学学报(自然科学版),2005,30(5)：796-799.
[33]Wendi Wang, Lansun Chen, A Predator-Prey System with Stage-Structure for Predator. Computres Math [J], Applic,1997,33(8):83-91.
[34]H. Smith, Monotone semiflows generated by functional differential equations [J]. J. Diff. Equa., 1987,16:420-442.
[35]V. Herz, S. Bonhoeffer, R. Anderson, R. May, M. Nowak, Viral dynamics in vivo:limitations on estimations on intracellular delay and virus decay [J]. Proc. Nat. Acad. Sci.,1996,93:7247-7251.
[36]A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time [J]. Science,1996,271: 1582-1586.
[37]J.M. Murray, R.H. Purcell, S.F. Wieland, The half-life of hepatitis B virions [J]. Hepatology,2006, 44:1117-1121.
[38]M.Y. Li, H. Shu, Global dynamics of an in-host viral model with intracellular delay [J]. Bull. Math. Biol.,2010,72:1492-505.
[39]庞海燕,考虑免疫反应的病毒动力学的全局性态：[西南大学学位论文].重庆,西南大学,2006.10-44
[40]H. Zhu, X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay [J]. Discrete Contin. Dyn. Syst. Ser. B.,2009,12:511-524.