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Stability and Hopf bifurcations for a delayed diffusion system in population dynamics
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 |
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), 234-241.
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), 27-39.
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), 29-45.
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, Springer-Verlag, New York, 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), 7247-7251.
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), 17-26.
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), 1492-1505.
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), 1117-1121.
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), 73-94.
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), 796-799. |
[11] |
A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells, Math. Biosci., 114 (1993), 81-125.
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), 1582-1586.
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), 90-99.
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), 1593-1610.
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), 1743-1750.
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), 511-524. |
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), 234-241.
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), 27-39.
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), 29-45.
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, Springer-Verlag, New York, 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), 7247-7251.
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), 17-26.
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), 1492-1505.
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), 1117-1121.
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), 73-94.
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), 796-799. |
[11] |
A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells, Math. Biosci., 114 (1993), 81-125.
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), 1582-1586.
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), 90-99.
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), 1593-1610.
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), 1743-1750.
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), 511-524. |
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