## International Journal of Differential Equations

### Mathematical Modeling of the Adaptive Immune Responses in the Early Stage of the HBV Infection

#### Abstract

The aim of this paper is to study the early stage of HBV infection and impact delay in the infection process on the adaptive immune response, which includes cytotoxic T-lymphocytes and antibodies. In this stage, the growth of the healthy hepatocyte cells is logistic while the growth of the infected ones is linear. To investigate the role of the treatment at this stage, we also consider two types of treatment: interferon-$\mathrm{\alpha }$ (IFN) and nucleoside analogues (NAs). To find the best strategy to use this treatment, an optimal control approach is developed to find the possibility of having a functional cure to HBV.

#### Article information

Source
Int. J. Differ. Equ., Volume 2018, Special Issue (2017), Article ID 6710575, 13 pages.

Dates
Revised: 18 November 2017
Accepted: 18 December 2017
First available in Project Euclid: 17 March 2018

https://projecteuclid.org/euclid.ijde/1521252047

Digital Object Identifier
doi:10.1155/2018/6710575

Mathematical Reviews number (MathSciNet)
MR3762165

Zentralblatt MATH identifier
06915959

#### Citation

Allali, Karam; Meskaf, Adil; Tridane, Abdessamad. Mathematical Modeling of the Adaptive Immune Responses in the Early Stage of the HBV Infection. Int. J. Differ. Equ. 2018, Special Issue (2017), Article ID 6710575, 13 pages. doi:10.1155/2018/6710575. https://projecteuclid.org/euclid.ijde/1521252047

#### References

• C. Ferrari, “HBV and the immune response,” Liver International, vol. 35, supplement 1, pp. 121–128, 2015.
• C. Ferrari, A. Penna, A. Bertoletti et al., “Cellular immune response to hepatitis B virus-encoded antigens in acute and chronic hepatitis B virus infection,” The Journal of Immunology, vol. 145, no. 10, pp. 3442–3449, 1990.
• B. Rehermann and M. Nascimbeni, “Immunology of hepatitis B virus and hepatitis C virus infection,” Nature Reviews Immunology, vol. 5, no. 3, pp. 215–229, 2005.
• J. Waters, M. Pignatelli, S. Galpin, K. Ishihara, and H. C. Thomas, “Virus-neutralizing antibodies to hepatitis B virus: The nature of an immunogenic epitope on the S gene peptide,” Journal of General Virology, vol. 67, no. 11, pp. 2467–2473, 1986.
• G. J. M. Webster, S. Reignat, M. K. Maini et al., “Incubation phase of acute hepatitis B in man: Dynamic of cellular immune mechanisms,” Hepatology, vol. 32, no. 5, pp. 1117–1124, 2000.
• E. Loggi, N. Gamal, F. Bihl, M. Bernardi, and P. Andreone, “Adaptive response in hepatitis B virus infection,” Journal of Viral Hepatitis, vol. 21, no. 5, pp. 305–313, 2014.
• D. R. Milich, M. Chen, F. Schödel, D. L. Peterson, J. E. Jones, and J. L. Hughes, “Role of B cells in antigen presentation of the hepatitis B core,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 94, no. 26, pp. 14648–14653, 1997.
• S. Hagiwara, N. Nishida, and M. Kudo, “Antiviral therapy for chronic hepatitis B: Combination of nucleoside analogs and interferon,” World Journal of Hepatology, vol. 7, no. 23, pp. 2427–2431, 2015.
• R.-N. Chien, C.-H. Lin, and Y.-F. Liaw, “The effect of lamivudine therapy in hepatic decompensation during acute exacerbation of chronic hepatitis B,” Journal of Hepatology, vol. 38, no. 3, pp. 322–327, 2003.
• L. J. Sun, J. W. Yu, Y. H. Zhao, P. Kang, and S. C. Li, “Influ-ential factors of prognosis in lamivudine treatment for patients with acute-on-chronic hepatitis B liver failure,” Journal of Gas-troenterology and Hepatology, vol. 25, no. 3, pp. 583–590, 2010.
• 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, vol. 6, no. 2, pp. 283–299, 2009.MR2532017
• 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, vol. 2, no. 2, pp. 140–153, 2008.MR2428891
• 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, vol. 60, no. 4, pp. 573–590, 2010.MR2587590
• 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 Acadamy of Sciences of the United States of America, vol. 93, no. 9, pp. 4398–4402, 1996.
• J. Pang and J.-A. Cui, “Analysis of a hepatitis B viral infection model with immune response delay,” International Journal of Biomathematics, vol. 10, no. 2, Article ID 1750020, 2017.
• A. Tridane, K. Hattaf, R. Yafia, and F. A. Rihan, “Mathematical modeling of hbv with the antiviral therapy for the immunocompromised patients,” Communications in Mathematical Biology and Neuroscience, vol. 2016, 31 pages, 2016.
• Y. Wang and X. Liu, “Dynamical behaviors of a delayed HBV infection model with logistic hepatocyte growth, cure rate and CTL immune response,” Japan Journal of Industrial and Applied Mathematics, vol. 32, no. 3, pp. 575–593, 2015.MR3426151
• N. Yousfi, K. Hattaf, and A. Tridane, “Modeling the adaptive immune response in HBV infection,” Journal of Mathematical Biology, vol. 63, no. 5, pp. 933–957, 2011.MR2844670
• A. Meskaf, K. Allali, and Y. Tabit, “Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses,” International Journal of Dynamics and Control, vol. 5, no. 3, pp. 893–902, 2017.MR3695044
• J. E. Forde, S. M. Ciupe, A. Cintron-Arias, and S. Lenhart, “Optimal control of drug therapy in a hepatitis B model,” Applied Sciences (Switzerland), vol. 6, no. 8, article no. 219, 2016.
• A. M. Elaiw, M. A. Alghamdi, and S. Aly, “Hepatitis B virus dynamics: Modeling, analysis, and optimal treatment scheduling,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 712829, 2013.
• P. Tchinda Mouofo, J. J. Tewa, B. Mewoli, and S. Bowong, “Optimal control of a delayed system subject to mixed control-state constraints with application to a within-host model of hepatitis virus B,” Annual Reviews in Control, vol. 37, no. 2, pp. 246–259, 2013.
• S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, G. Dusheiko, and A. S. Perelson, “The role of cells refractory to productive infection in acute hepatitis B viral dynamics,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 104, no. 12, pp. 5050–5055, 2007.
• G. K. Michalopoulos and M. C. DeFrances, “Liver regeneration,” Science, vol. 276, no. 5309, pp. 60–65, 1997.
• M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, London, UK, 2000.MR2009143
• S. A. Whalley, J. M. Murray, D. Brown et al., “Kinetics of acute hepatitis B virus infection in humans,” The Journal of Experimental Medicine, vol. 193, no. 7, pp. 847–853, 2001.
• M.-P. Bralet, S. Branchereau, C. Brechot, and N. Ferry, “Cell lineage study in the liver using retroviral mediated gene transfer: Evidence against the streaming of hepatocytes in normal liver,” The American Journal of Pathology, vol. 144, no. 5, pp. 896–905, 1994.
• R. A. Macdonald, “\textquotedblleftLifespan" of Liver Cells: Autoradiographic Study Using Tritiated Thymidine in Normal, Cirrhotic, and Partially Hepatectomized Rats,” JAMA Internal Medicine, vol. 107, no. 3, pp. 335–343, 1961.
• 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, vol. 247, no. 1, pp. 23–35, 2007.MR2306969
• M. A. Nowak and R. M. May, Virus Dynamics: Mathematics Principles of Immunology and Virology, Oxford University Press, London, UK, 2000.MR2009143
• S. M. Ciupe, R. M. Ribeiro, and A. S. Perelson, “Antibody responses during hepatitis B viral infection,” PLoS Computational Biology, vol. 10, no. 7, Article ID e1003730, 2014.
• R. Ahmed and D. Gray, “Immunological memory and protective immunity: Understanding their relation,” Science, vol. 272, no. 5258, pp. 54–60, 1996.
• W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, vol. 1, Springer, New York, NY, USA, 1975.MR0454768
• D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, NY. USA, 1982.MR668522
• L. Gollmann, D. Kern, and H. Maurer, “Optimal control problems with delays in state and control variables subject to mixed control-state constraints,” Optimal Control Applications and Methods, vol. 30, no. 4, pp. 341–365, 2009.MR2553382
• A. B. Gumel, P. N. Shivakumar, and B. M. Sahai, “A mathematical model for the dynamics of HIV-1 during the typical course of infection,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 3, pp. 1773–1783, 2001.
• J. Karrakchou, M. Rachik, and S. Gourari, “Optimal control and infectiology: application to an HIV/AIDS model,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 807–818, 2006.MR2292006
• L. Chen, K. Hattaf, and J. Sun, “Optimal control of a delayed SLBS computer virus model,” Physica A: Statistical Mechanics and its Applications, vol. 427, pp. 244–250, 2015.MR3318705
• K. Hattaf and N. Yousfi, “Optimal control of a delayed HIV infection model with immune response using an efficient numerical method,” ISRN biomathematics, vol. 2012, Article ID 215124, 7 pages, 2012.
• H. Laarabi, A. Abta, and K. Hattaf, “Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment,” Acta Biotheoretica, vol. 63, no. 2, pp. 87–97, 2015.
• L. Boglione, G. Cariti, G. Di Perri, and A. D'Avolio, “Sequential therapy with entecavir and pegylated interferon in a cohort of young patients affected by chronic hepatitis B,” Journal of Medical Virology, vol. 88, no. 11, pp. 1953–1959, 2016. \endinput