Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay

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January 2012, 17(1): 297-302. doi: 10.3934/dcdsb.2012.17.297

Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay

Jinliang Wang ^1, and Lijuan Guan ^2,

1.	School of Mathematical Science, Heilongjiang University, Harbin, Heilongjiang 150080, China
2.	Department of Mathematics, Arts and Science College, Harbin Normal University, Harbin, Heilongjiang 150025, China

Received April 2011 Revised August 2011 Published October 2011

Abstract
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A recent paper [H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems - Series B, 12(2009), 511--524] presented a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. By combining the analysis of the characteristic equation and the Lyapunov-LaSalle method, they obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. In the present paper, we show that the global dynamics are fully determined for $\Re_1<1<\Re_0$ and $\Re_1>1$ (Theorem 4.2 and Theorem 4.3) without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov-LaSalle invariance principle.

Keywords: Lyapunov functional., Global stability, Intracellular delay, Immune response.

Mathematics Subject Classification: Primary: 92B05, 92C50; Secondary: 92D2.

Citation: Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

References:

[1]	R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 1347-1354. doi: 10.1098/rspb.2000.1149.
[2]	R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.
[3]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[4]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Science, 99, Springer-Verlag, New York, 1993.
[5]	A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.
[6]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academics Press, Inc., Boston, MA, 1993.
[7]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[8]	M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
[9]	K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020.
[10]	J. Wang, G. Huang, Y. Takeuchi and S. Liu, SVEIR epidemiological model with varying infectivity and distributed delays, Math. Biosci. Eng., 8 (2011), 875-888. doi: 10.3934/mbe.2011.8.875.
[11]	J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Medic. Bio. doi: 10.1093/imammb/dqr009.
[12]	H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 511-524.

show all references

References:

[1]	R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 1347-1354. doi: 10.1098/rspb.2000.1149.
[2]	R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.
[3]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[4]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Science, 99, Springer-Verlag, New York, 1993.
[5]	A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.
[6]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academics Press, Inc., Boston, MA, 1993.
[7]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[8]	M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
[9]	K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020.
[10]	J. Wang, G. Huang, Y. Takeuchi and S. Liu, SVEIR epidemiological model with varying infectivity and distributed delays, Math. Biosci. Eng., 8 (2011), 875-888. doi: 10.3934/mbe.2011.8.875.
[11]	J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Medic. Bio. doi: 10.1093/imammb/dqr009.
[12]	H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 511-524.

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