Journal of Mathematical Sciences: Advances and Applications[1] L. J. S. Allen, An Introduction to Mathematical Biology, Upper
Saddle River, New Jersey, 2007.
[2] F. Brauer, Absolute stability in delay equations, Journal of
Differential Equations 69(2) (1987), 185-191.
DOI: https://doi.org/10.1016/0022-0396(87)90116-1
[3] B. S. Blumberg, Hepatitis B: The Hunt for a Killer Virus,
Princeton University Press, Princeton, 2002.
[4] S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson,
Modeling the mechanisms of acute hepatitis B virus infection, Journal
of Theoretical Biology 247(1) (2007), 23-35.
DOI: https://doi.org/10.1016/j.jtbi.2007.02.017
[5] W. A. Coppel, Stability and Asymptotic Behavior of Differential
Equation, Boston Health, 1965.
[6] B. Custer, S. D. Sullivan, T. K. Hazlet, U. Iloeje, D. L. Veenstra
and K. V. Kowdley, Global epidemiology of hepatitis B virus, Journal
of Clinical Gastroenterology 38(10) (2004), 158-168.
DOI: https://doi.org/10.1097/00004836-200411003-00008
[7] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a
delay model of hepatitis B virus infection with logistic hepatocyte
growth, Mathematical Biosciences and Engineering 6(2) (2009),
283-299.
DOI: https://doi.org/10.3934/mbe.2009.6.283
[8] 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(2) (2008), 140-153.
DOI: https://doi.org/10.1080/17513750701769873
[9] K. Hattaf and N. Yousfi, A numerical method for a delayed viral
infection model with general incidence rate, Journal of King Saud
University Science 28(4) (2016), 368-374.
DOI: https://doi.org/10.1016/j.jksus.2015.10.003
[10] K. Hattaf and N. Yousfi and A. Tridane, Mathematical analysis of
a virus dynamics model with general incidence rate and cure rate,
Nonlinear Analysis: Real World Applications 13(4) (2012),
1866-1872.
DOI: https://doi.org/10.1016/j.nonrwa.2011.12.015
[11] K. Hattaf and N. Yousfi, Hepatitis B virus infection model with
logistic hepatocyte growth and cure rate, Applied Mathematical
Sciences 5(47) (2011), 2327-2335.
[12] K. Hattaf, N. Yousfi and A. Tridane, A Delay virus dynamics model
with general incidence rate, Differential Equations and Dynamical
Systems 2(22) (2014), 181-190.
DOI: https://doi.org/10.1007/s12591-013-0167-5
[13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional
Differential Equations, Springer-Verlag, New-York, 1993.
[14] J. K. Hale, Theory of Functional Differential Equations,
Springer-Verlag, New-York, 1977.
[15] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of
a hepatitis B viral infection model with logistic hepatocyte growth,
Journal of Mathematical Biology 60(4) (2010), 573-590.
DOI: https://doi.org/10.1007/s00285-009-0278-3
[16] Y. Kuang, Delay Differential Equations with Applications in
Population Dynamics, Academic Press, San Diego, 1993.
[17] Ruy M. Ribeiro, Arthur Lo and Alan S. Perelson, Dynamics of
hepatitis B virus infection, Microbes and Infection 4(8) (2002),
829-835.
DOI: https://doi.org/10.1016/s1286-4579(02)01603-9
[18] M. Marsh and A. Helenius, Virus entry: Open sesame, Cell 124(4)
(2006), 729-740.
DOI: https://doi.org/10.1016/j.cell.2006.02.007
[19] C. C. McCluskey, Global stability for an SEIR epidemiological
model with varying infectivity and infinite delay, Mathematical
Biosciences & Engineering 6(3) (2009), 603-610.
DOI: https://doi.org/10.3934/mbe.2009.6.603
[20] L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic
virus infection model with application to HBV infection, Rocky
Mountain Journal of Mathematics 38(5) (2008), 1573-1585.
DOI: https://doi.org/10.1216/RMJ-2008-38-5-1573
[21] W. Mothes, N. M. Sherer, J. Jin and P. Zhong, Virus cell-to-cell
transmission, Journal of Virology 84(17) (2010), 8360-8368.
DOI: https://doi.org/10.1128/JVI.00443-10
[22] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas
and H. McDade, Viral dynamics in hepatitis B virus infection,
Proceedings of the National Academy of Sciences of the United States
of America 93(9) (1996), 4398-4402.
DOI: https://doi.org/10.1073/pnas.93.9.4398
[23] M. A. Nowak and R. M. May, Viral Dynamics, Oxford University
Press, Oxford, 2000.
[24] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune
responses to persistent viruses, Science 272(5258) (1996), 74-79.
DOI: https://doi.org/10.1126/science.272.5258.74
[25] R. Ouncharoen, S. Pinjai, Th. Dumrongpokaphan and Y. Lenbury,
Global stability analysis of predator-prey model with harvesting and
delay, Thai Journal of Mathematics 8(3) (2010), 589-605.
[26] B. Rehermann and M. Nascimbeni, Immunology of hepatitis B virus
and hepatitis C virus infection, Nature Reviews Immunology 5(3)
(2005), 215-229.
DOI: https://doi.org/10.1038/nri1573
[27] Q. Sattentau, Avoiding the void: Cell-to-cell spread of human
viruses, Nature Reviews Microbiology 6(11) (2008), 815-826.
DOI: https://doi.org/10.1038/nrmicro1972
[28] X. Tian and R. Xu, Global stability of a virus infection model
with time delay and absorption, Discrete Dynamics in Nature and
Society 20 (2011), 583-600.
DOI: https://doi.org/10.1155/2011/152415
[29] R. Thiemme, S. Wieland, C. Steiger, J. Ghrayeb, K. A. Reimann, R.
H. Purcell and F. V. Chisari, 
cells mediate viral clearance and disease
pathogenesis during acute hepatitis B virus infection, Journal of
Virology 77(1) (2003), 68-76.
DOI: https://doi.org/10.1128/JVI.77.1.68-76.2003
[30] P. van den Driessche and J. Watmough, Reproduction numbers and
sub-threshold endemic equilibria for compartmental models of disease
transmission, Mathematical Biosciences 180(1-2) (2002), 29-48.
DOI: https://doi.org/10.1016/S0025-5564(02)00108-6
[31] C. Vargas-De-Leon, Analysis of a model for the dynamics of
hepatitis B with noncytolytic loss of infected cells, World Journal of
Modelling and Simulation 8(4) (2012), 243-259.
[32] K. Wang, A. Fan and A. Torres, Global properties of an improved
hepatitis B virus model, Nonlinear Analysis: Real World Applications
11(4) (2010), 3131-3138.
DOI: https://doi.org/10.1016/j.nonrwa.2009.11.008
[33] World Heath Organization, Hepatitis B: Fact sheet: No. 204
(2015), Accessed 10 Oct 2015.
http://www.who.int/mediacentre/factsheets/fs204/en/
[34] C. Xu, Stability and bifurcation analysis in a viral model with
delay, Mathematical Methods in the Applied Sciences 36(10) (2013),
1310-1320.
DOI: https://doi.org/10.1002/mma.2688
[35] C. Xu and M. Liao, Bifurcation analysis of an autonomous epidemic
predator-prey model with delay, Annali di Matematica Pura ed Applicata
193(1) (2014), 23-38.
DOI: https://doi.org/10.1007/s10231-012-0264-z
[36] C. Xu, Z. Liu, M. Liao, P. Li, Q. Xiao and S. Yuan,
Fractional-order bidirectional associate memory (BAM) neural networks
with multiple delays: The case of Hopf bifurcation, Mathematics and
Computers in Simulation 182 (2021), 471-494.
DOI: https://doi.org/10.1016/j.matcom.2020.11.023
[37] C. Xu and Y. Wu, Bifurcation and control of chaos in a chemical
system, Applied Mathematical Modelling 39(8) (2015), 2295-2310.
DOI: https://doi.org/10.1016/j.apm.2014.10.030
