American Institute of Mathematical Sciences

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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

Yincui Yan 1, and  Wendi Wang 1,

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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