Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-Containing Capsids and Cure Rate

  • Adil Meskaf
  • Karam Allali


We present in this paper a delay-differential equation model that describes the interactions between hepatitis B virus (HBV) with DNA-containing capsids and liver cells (hepatocytes). Both the treatments, the intracellular delay and the cure rate of infected cells, are incorporated into the model. The first treatment represents the efficiency of drug treatment in preventing new infections, whereas the second stands for the efficiency of drug treatment in inhibiting viral production. Existence for the optimal control pair is established, Pontryagin’s maximum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are established to show the role of optimal therapy in controlling viral replication.


HBV infection Delay Optimal control Numerical simulation 


  1. 1.
    S.M. Ciupe, R.M. Ribeiro, P.W. Nelson, A.S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection. J. Theor. Biol. 247(1), 23–35 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Min, Y. Su, Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection. Rocky Mt. J. Math. 38, 1573–1585 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M.A. Nowak, S. Bonhoeffer, A.M. Hill, R. Boehme, H.C. Thomas, H. McDade, Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. 93(9), 4398–4402 (1996)CrossRefGoogle Scholar
  4. 4.
    K. Wang, A. Fan, A. Torres, Global properties of an improved hepatitis B virus model. Nonlinear Anal. Real World Appl. 11(4), 3131–3138 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Manna, S.P. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: modeling and analysis’. Commun. Nonlinear Sci. Numer. Simul. 22, 383–395 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. Manna, S.P. Chakrabarty, Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids’. J. Differ. Equ. Appl. 21, 918–933 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Manna, S.P. Chakrabarty, Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids’. Comput. Appl. Math. 36, 525–536 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. Bruss, Envelopment of the hepatitis B virus nucleocapsid. Virus Res. 106, 199–209 (2004)CrossRefGoogle Scholar
  9. 9.
    D. Ganem, A.M. Prince, Hepatitis B virus infection: natural history and clinical consequences. N. Engl. J. Med. 350, 1118–1129 (2004)CrossRefGoogle Scholar
  10. 10.
    J. Danane, A. Meskaf, K. Allali, Optimal control of a delayed hepatitis B viral infection model with HBV DNA-containing capsids and CTL immune response. Optimal Control Appl. Methods 39(3), 1262-1272 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Hattaf, N. Yousfi, Dynamics of HIV infection model with therapy and cure rate. Int. J. Tomogr. Stat. 16(11), 74-80 (2011)zbMATHGoogle Scholar
  12. 12.
    X. Zhou, X. Song, X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate. J. Math. Anal. Appl. 342,(2), 1342–1355 (2008)Google Scholar
  13. 13.
    W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, New York, 1975)CrossRefGoogle Scholar
  14. 14.
    D.L. Lukes Differential Equations: Classical to Controlled. Mathematics in Science and Engineering (Academic Press, New York, 1982), p. 162Google Scholar
  15. 15.
    L. Göllmann, D. Kern, H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Appl. Methods 30, 341–365 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Hattaf, N. Yousfi, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomath. 2012 (2012).
  17. 17.
    H. Laarabi, A. Abta, K. Hattaf, Optimal control of a delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor. 63(2), 87–97 (2015)CrossRefGoogle Scholar
  18. 18.
    L. Chen, K. Hattaf, J. Sun, Optimal control of a delayed SLBS computer virus model. Phys. A 427, 244–250 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Manna, Global properties of a HBV infection model with HBV DNA-containing capsids and CTL immune response. Int. J. Appl. Comput. Math. (2016).
  20. 20.
    A. Meskaf, K. Allali, Y. Tabit. Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses. Int. J. Dyn. Control (2016).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Adil Meskaf
    • 1
  • Karam Allali
    • 2
  1. 1.Department of SEG, Polydisciplinary FacultyUniversity Chouaib DoukkaliEl JadidaMorocco
  2. 2.Department of Mathematics, Faculty of Sciences and TechnologiesUniversity Hassan IIMohammediaMorocco

Personalised recommendations