Advertisement

Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy

  • Pensiri Yosyingyong
  • Ratchada ViriyapongEmail author
Original Research

Abstract

In this paper, a nonlinear mathematical model describing the relationship between hepatitis B virus (HBV), the immune response and the drug therapy is studied. Two main equilibrium points (infection-free and endemic) are obtained. The basic reproduction number is also determined and becomes the threshold for equilibrium point stabilities. We show that when the basic reproduction number is less than one, the infection-free equilibrium point is both locally and globally stable whereas when it is greater than one, the system is uniformly persistent i.e. the virus is endemic and the endemic equilibrium point is globally asymptotically stable. The sensitivity analysis is carried out to seek for potential parameters that could reduce overall HBV infection. Further, by using Pontryagin’s minimum principle, the optimal control problem is constructed with two drug therapy controls. Finally, the numerical simulations are established to show the role of these optimal therapies in controlling viral replication and HBV infection. Our results show that the treatment by inhibiting viral production gives more significant result than the treatment by blocking new infection, however the combination of both treatments is the best strategy to reduce overall HBV infection and the concentration of free virus.

Keywords

Cytotoxic T lymphocytes Drug therapy Geometric approach Hepatitis B virus Hepatocyte Sensitivity 

Mathematics Subject Classification

37N25 62P10 

Notes

Acknowledgements

This work has been supported by Department of Mathematics, Faculty of Science, Naresuan University, Thailand. Pensiri Yosyingyong has been funded by DPST scholarship from the Thai government.

References

  1. 1.
    WHO (World Health Organiaztion): Hepatitis B fact sheet no. 204. The World Health Organisation, Geneva (2017). Retrieved January 2 (2017), from. http://www.who.int/mediacentre/factsheets/fs204/en/
  2. 2.
    Long, C., Qi, H., Huang, S.H.: Mathematical modeling of cytotoxic lymphocyte-mediated immune responses to hepatitis B virus infection. J. Biomed. Biotechnol. 38, 1573–1585 (2008).  https://doi.org/10.1155/2008/743690 Google Scholar
  3. 3.
    Bertoletti, A., Ferrari, C.: Innate and adaptive immune responses in chronic hepatitis B virus infections: towards restoration of immune control of viral infection. Gut 61, 1754–1764 (2012).  https://doi.org/10.1136/gutjnl-2011-301073 CrossRefGoogle Scholar
  4. 4.
    Dandri, M., Locarnini, S.: New insight in the pathobiology of hepatitis B virus infection. Gut 61, i6–i17 (2012).  https://doi.org/10.1136/gutjnl-2012-302056 CrossRefGoogle Scholar
  5. 5.
    Goyal, A., Ribeiro, R.M., Perelson, A.S.: The role of infected cell proliferation in the clearance of acute HBV infection in humans. Viruses 9(11), 1–17 (2017).  https://doi.org/10.3390/v9110350 CrossRefGoogle Scholar
  6. 6.
    Lannacone, M., Sitia, G., Guidotti, L.G.: Pathogenetic and antiviral immune responses against hepatitis B virus. Future Virol. 1, 189196 (2006).  https://doi.org/10.2217/17460794.1.2.189 Google Scholar
  7. 7.
    Suslov, A., Boldanova, T., Wang, X., Wieland, S., Heim, M.H.: Hepatitis B virus does not interfere with innate immune responses in the human liver. Gastroenterology 154, 1778–1790 (2018)CrossRefGoogle Scholar
  8. 8.
    Tsui, L.V., Guidotti, L.G., Ishikawa, T., Chisari, F.V.: Posttranscriptional clearance of hepatitis B virus RNA by cytotoxic T lymphocyte-activated hepatocytes. Proc. Natl. Acad. Sci. USA 92, 12398–12402 (1995)CrossRefGoogle Scholar
  9. 9.
    Guidotti, L.G., Ishikawa, T., Hobbs, M.V., Matzke, B., Schreiber, R., Chisari, F.V.: Intracellular inactivation of the hepatitis B virus by cytotoxic T lymphocytes. Immunity 4, 2536 (1996)CrossRefGoogle Scholar
  10. 10.
    Guidotti, L.G., Rochford, R., Chung, J., Shapiro, M., Purcell, R., Chisari, F.V.: Viral clearance without destruction of infected cells during acute HBV infection. Science 284, 825–829 (1999)CrossRefGoogle Scholar
  11. 11.
    Phillips, S., Chokshi, S., Riva, A., Evans, A., Williams, R., Naoumov, N.V.: CD8(+) T cell control of hepatitis B virus replication: direct comparison between cytolytic and noncytolytic functions. J. Immunol. 184, 287–295 (2010).  https://doi.org/10.4049/jimmunol.0902761 CrossRefGoogle Scholar
  12. 12.
    Pei, R.J., Chen, X.W., Lu, M.J.: Control of hepatitis B virus replication by interferons and toll-like receptor signaling pathways. World J. Gastroenterol. 20, 1161811629 (2014).  https://doi.org/10.3748/wjg.v20.i33.11618 CrossRefGoogle Scholar
  13. 13.
    Xia, Y., Protzer, U.: Control of hepatitis B virus by cytokines. Viruses 9, 8 (2017).  https://doi.org/10.3390/v9010018 CrossRefGoogle Scholar
  14. 14.
    Guidotti, L.G., Chisari, F.V.: To kill or to cure: options in host defense against viral infection. Curr. Opin. Immunol. 8, 478–483 (1996)CrossRefGoogle Scholar
  15. 15.
    Chisari, F.V.: Cytotoxic T cells and viral hepatitis. J. Clin. Invest. 99, 1472–1477 (1997).  https://doi.org/10.1172/JCI119308 CrossRefGoogle Scholar
  16. 16.
    Bartholdy, C., Christensen, J.P., Wodarz, D., Thomsen, A.R.: Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infected with lymphocytic chroriomeningitis virus. J. Virol. 74, 10304–10311 (2000)CrossRefGoogle Scholar
  17. 17.
    Wodarz, D., Christensen, J.P., Thomsen, A.R.: The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23, 194–200 (2002)CrossRefGoogle Scholar
  18. 18.
    Bocharov, G., Ludewig, B., Bertoletti, A., Klenerman, P., Junt, T., Krebs, P., Luzyanina, T., Fraser, G., Anderson, R.M.: Underwhelming the immune response: effect of slow virus growth on CD8+-T-lymphocytes responses. J. Virol. 78(5), 2247–2254 (2004)CrossRefGoogle Scholar
  19. 19.
    Wang, K., Wang, W., Liu, X.: Global stability in a viral infection model with lytic and nonlytic immune responses. Comput. Math. Appl. 51, 1593–1610 (2006).  https://doi.org/10.1016/j.camwa.2005.07.020 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lampertico, P., Aghemo, A., Vigan, M., Colombo, M.: HBV and HCV therapy. Viruses 1, 484–509 (2009).  https://doi.org/10.3390/v1030484 CrossRefGoogle Scholar
  21. 21.
    Hagiwara, S., Nishida, N., Kudo, M.: Antiviral therapy for chronic hepatitis B: combination of nucleoside analogs and interferon. World J. Hepatol. 7(23), 2427–2431 (2015).  https://doi.org/10.4254/wjh.v7.i23.2427 CrossRefGoogle Scholar
  22. 22.
    Hadziyannis, S.J., Tassopoulos, N.C., Heathcote, E.J., Chang, T.T., Kitis, G., Rizzetto, M., Marcellin, P., Lim, S.G., Goodman, Z., Wulfsohn, M.S., et al.: Adefovir dipivoxil for the treatment of hepatitis Be antigenNegative chronic hepatitis B. N. Eng. J. Med. 348, 800–807 (2003).  https://doi.org/10.1056/NEJMoa021812 CrossRefGoogle Scholar
  23. 23.
    Erik, D.C., Geoffrey, F., Suzanne, K., Johan, N.: Antiviral treatment of chronic hepatitis B virus (HBV) infections. Viruses 2(6), 1279–1305 (2010)CrossRefGoogle Scholar
  24. 24.
    Nowak, M.A., Bonhoeffer, S., Hill, A., Boehme, R., Thomas, H., McDade, H.: Viral dynamics in hepatitis B infection. Proc. Natl. Acad. Sci. USA 93, 4398–4402 (1996)CrossRefGoogle Scholar
  25. 25.
    Ciupe, S.M.: Modeling the dynamics of hepatitis B infection, immunity, and drug therapy. Immunol. Rev. 285, 38–54 (2018)CrossRefGoogle Scholar
  26. 26.
    Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996).  https://doi.org/10.1126/science.272.5258.74 CrossRefGoogle Scholar
  27. 27.
    Koonprasert, S., Moore, E.J., Banyatlersthaworn, S.: Sensitivity and stability analysis of hepatitis B virus model with non-cytolytic cure process and logistic hepatocyte growth. Glob. J. Pure Appl. Math. 12, 2297–2312 (2016)Google Scholar
  28. 28.
    Ciupe, S.M., Ribeiro, R.M., Nelson, P.W., Dusheiko, G., Perelson, A.S.: The role of cells refractory to productive infection in acute hepatitis B viral dynamics. Proc. Natl. Acad. Sci. USA 104, 5050–5055 (2007).  https://doi.org/10.1073/pnas.0603626104 CrossRefGoogle Scholar
  29. 29.
    Ciupe, S.M., Ribeiro, R.M., Nelson, P.W.: Modeling the mechanisms of acute hepatitis B virus infection. J. Theor. Biol. 247, 23–35 (2007).  https://doi.org/10.1016/j.jtbi.2007.02.017 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hews, S., Eikenberry, S., Nagy, J.D., Kuang, Y.: Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth. J. Math. Biol. 60, 573–590 (2010).  https://doi.org/10.1007/s00285-009-0278-3 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yous, N., Hattaf, K., Tridane, A.: Modeling the adaptive immune response in HBV infection. J. Math. Biol. 63(5), 933–957 (2011).  https://doi.org/10.1007/s00285-010-0397-x MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Elaiw, A.M., Almuallem, N.A.: Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells. Appl. Math. Comput. 265, 1067–1089 (2015).  https://doi.org/10.1016/j.amc.2015.06.011 MathSciNetzbMATHGoogle Scholar
  33. 33.
    Mboya, K., Makinde, D.O., Massawe, E.S.: Cytotoxic cells and control strategies are effective in reducing the HBV infection through a mathematical modelling. Int. J. Prevent. Treat. 2015 4(3), 48–57 (2015).  https://doi.org/10.1155/2018/6710575 Google Scholar
  34. 34.
    Tridane, A., Hattaf, K., Yafia, R., Rihan, F.A.: Mathematical modeling of HBV with the antiviral therapy for the immunocompromised patients. Commun. Math. Biol. Neurosci. ISSN:2052–2541 (2016)Google Scholar
  35. 35.
    Lewin, S., Ribeiro, R., Walters, T., Lau, G., Bowden, S., Locarnini, S., Perelson, A.: Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatology 34, 1012–1020 (2001).  https://doi.org/10.1053/jhep.2001.28509 CrossRefGoogle Scholar
  36. 36.
    Ribeiro, R.M., Lo, A., Perelson, A.S.: Dynamics of hepatitis B virus infection. Microbes Infect. 4(8), 829–835 (2002)CrossRefGoogle Scholar
  37. 37.
    Hattaf, K., Rachik, M., Saadi, S., Yousfi, N.: Optimal control of treatment in a basic virus infection model. Appl. Math. Sci. 3(17–20), 949–958 (2009)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Elaiw, A.M., Alghamdi, M.A., Aly, S.: Hepatitis B virus dynamics: modeling, analysis, and optimal treatment scheduling. Discrete Dyn. Nat. Soc. 2013, 1–9 (2013).  https://doi.org/10.1155/2013/712829 MathSciNetzbMATHGoogle Scholar
  39. 39.
    Forde, J.E., Ciupe, S.M., Cintron-Arias, A., Lenhart, S.: Optimal control of drug therapy in a hepatitis B model. Appl. Sci. 6, 219 (2016).  https://doi.org/10.3390/app6080219 CrossRefGoogle Scholar
  40. 40.
    Allali, K., Meskaf, A., Tridane, A.: Mathematical modeling of the adaptive immune responses in the early stage of the HBV infection. Int. J. Differ. Equ. Article ID 6710575 (2018)Google Scholar
  41. 41.
    Chenar, F.F., Kyrychko, Y.N., Blyuss, K.B.: Mathematical model of immune response to hepatitis B. J. Theor. Biol. 447, 98–110 (2108)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    van den Driessche, P., Watmough, J.: Reproductive numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math. Biosci. 180, 29–48 (2002).  https://doi.org/10.1016/S0025-5564(02)00108-6 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    LaSalle, J.P.: The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1976)CrossRefGoogle Scholar
  44. 44.
    Li, M.Y., Muldowney, J.S.: A geometric approach to global-stability problems. SIAM J. Math. Anal. 27(4), 10701083 (1996).  https://doi.org/10.1137/S0036141094266449 MathSciNetCrossRefGoogle Scholar
  45. 45.
    Li, M.Y., Muldowney, J.S.: On Bendixsons criterion. J. Differ. Equ. 106(1), 2739 (1993).  https://doi.org/10.1006/jdeq.1993.1097 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Freedman, H.I., Ruan, S., Tang, M.: Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equ. 6(4), 583600 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96(3), 42530 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Samsuzzoha, M.D., Singh, M., Lucy, D.: Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. Appl. Math. Model. 37, 903–915 (2013).  https://doi.org/10.1016/j.apm.2012.03.029 MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ngoteya, F.N., Gyekye, Y.N.: Sensitivity analysis of parameters in a competition model. Appl. Comput. Math. 4(5), 363–368 (2015).  https://doi.org/10.11648/j.acm.20150405.15 CrossRefGoogle Scholar
  50. 50.
    Ciupe, S.M., Ribeiro, R.M., Perelson, A.S.: Antibody responses during hepatitis B viral infection. PLOS Comput. Biol. 10(7), e1003730 (2014).  https://doi.org/10.1371/journal.pcbi.1003730 CrossRefGoogle Scholar
  51. 51.
    MacDonald, R.A.: Lifespan of liver cells. Autoradio-graphic study using tritiated thymidine in normal, cirrhotic, and partially hepatectomized rats. Arch. Intern. Med. 107, 335–343 (1961)CrossRefGoogle Scholar
  52. 52.
    Bralet, M.P., Branchereau, S., Brechot, C., Ferry, N.: Cell lineage study in the liver using retroviral mediated gene transfer. Am. J. Pathol. 144, 896–905 (1994)Google Scholar
  53. 53.
    Eikenberry, S., Hews, S., Nagy, J.D., Kuang, Y.: The dynamics of a delay model of HBV infection with logistic hepatocyte growth. Math. Biosci. Eng. 6, 283–299 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Whalley, S.A., Murray, J.M., Brown, D., Webster, G.J.M., Emery, V.C., Dusheiko, G.M., Perelson, A.S.: Kinetics of acute hepatitis B virus infection in humans. J. Exp. Med. 193, 847–853 (2001)CrossRefGoogle Scholar
  55. 55.
    Nowak, M.A., May, R.M.: Viral Dynamics. Oxford University Press, Oxford (2000)Google Scholar
  56. 56.
    Ahmed, R., Gray, D.: Immunologycal memory and protective immunity. Understanding their relation. Science 272, 5460 (1996)CrossRefGoogle Scholar
  57. 57.
    Pontryagin, L.S.V., Boltyanskii, G.R., Gamkrelidze, V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Gordon and Breach Science, New York (1986)zbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand

Personalised recommendations