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2011, 8(3): 689-694. doi: 10.3934/mbe.2011.8.689

## A note for the global stability of a delay differential equation of hepatitis B virus infection

 1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China, China

Received  September 2010 Revised  October 2010 Published  June 2011

The global stability for a delayed HIV-1 infection model is investigated. It is shown that the global dynamics of the system can be completely determined by the reproduction number, and the chronic infected equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [S. A. Gourley,Y. Kuang and J.D.Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2(2008), 140-153].
Citation: Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
##### References:
 [1] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. doi: 10.1080/17513750701769873. [2] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004. [3] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functional for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [4] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [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] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [7] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS models epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75. [8] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1. [9] 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. [10] M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322. [11] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [12] 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. [13] L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky. Mount. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573. [14] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Nat. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398. [15] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

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##### References:
 [1] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. doi: 10.1080/17513750701769873. [2] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004. [3] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functional for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [4] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [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] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [7] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS models epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75. [8] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1. [9] 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. [10] M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322. [11] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [12] 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. [13] L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky. Mount. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573. [14] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Nat. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398. [15] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
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