Global stability of a five-dimensional model with immune responses and delay

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January 2012, 17(1): 401-416. doi: 10.3934/dcdsb.2012.17.401

Global stability of a five-dimensional model with immune responses and delay

1.	Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China, China

Received September 2010 Revised December 2010 Published October 2011

Abstract
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In this article, we study a virus model with immune responses and an intracellular delay which is relatively large compared with virus life-cycle and is an indispensable factor in understanding virus infections, such as HIV and HBV infections. By constructing suitable Liapunov functionals, the global dynamics of the model is completely determined by the reproductive numbers for viral infection $R_0$, for CTL immune response $R_1$, for antibody immune response $R_2$, for CTL immune competition $R_3$ and for antibody immune competition $R_4$. The global stability of the model precludes the existence of periodic solution and other complex dynamical behaviors.

Keywords: Complete classification, Lyapuov functional, Reproductive number, Time delay, Invariance principle..

Mathematics Subject Classification: Primary: 92D30; Secondary: 34K2.

Citation: Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401

References:

[1]	A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234. doi: 10.1016/j.physa.2004.04.083.
[2]	R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27. doi: 10.1016/S0025-5564(00)00006-7.
[3]	N. Eshima, M. Tabata, T. Okada and S. Karukaya, Population dynamics of HTLV-I infection: A discrete-time mathematical epidemic model approach,, Math. Med. Biol., 20 (2003), 29. doi: 10.1093/imammb/20.1.29.
[4]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).
[5]	V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247.
[6]	Y. Iwasa, M. Franziska and M. A. Nowak, Virus evolution with patients increases pathogenicity,, J. Theor. Biol., 232 (2005), 17. doi: 10.1016/j.jtbi.2004.07.016.
[7]	M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x.
[8]	J. M. Murray, R. H. Purcell and S. F. Wieland, The half-life of hepatitis B virions,, Hepatology, 44 (2006), 1117. doi: 10.1002/hep.21364.
[9]	P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8.
[10]	H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response,, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796.
[11]	A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A.
[12]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582.
[13]	K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons Fractals, 28 (2006), 90. doi: 10.1016/j.chaos.2005.05.003.
[14]	K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Compute. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020.
[15]	D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses,, J. Gen. Virol., 84 (2003), 1743. doi: 10.1099/vir.0.19118-0.
[16]	H. Zhu and X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.

show all references

References:

[1]	A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234. doi: 10.1016/j.physa.2004.04.083.
[2]	R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27. doi: 10.1016/S0025-5564(00)00006-7.
[3]	N. Eshima, M. Tabata, T. Okada and S. Karukaya, Population dynamics of HTLV-I infection: A discrete-time mathematical epidemic model approach,, Math. Med. Biol., 20 (2003), 29. doi: 10.1093/imammb/20.1.29.
[4]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).
[5]	V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247.
[6]	Y. Iwasa, M. Franziska and M. A. Nowak, Virus evolution with patients increases pathogenicity,, J. Theor. Biol., 232 (2005), 17. doi: 10.1016/j.jtbi.2004.07.016.
[7]	M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x.
[8]	J. M. Murray, R. H. Purcell and S. F. Wieland, The half-life of hepatitis B virions,, Hepatology, 44 (2006), 1117. doi: 10.1002/hep.21364.
[9]	P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8.
[10]	H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response,, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796.
[11]	A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A.
[12]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582.
[13]	K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons Fractals, 28 (2006), 90. doi: 10.1016/j.chaos.2005.05.003.
[14]	K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Compute. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020.
[15]	D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses,, J. Gen. Virol., 84 (2003), 1743. doi: 10.1099/vir.0.19118-0.
[16]	H. Zhu and X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.

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