# American Institute of Mathematical Sciences

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

 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

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.
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.

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##### 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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