Advertisement

Modeling the Memory and Adaptive Immunity in Viral Infection

  • Adnane Boukhouima
  • Khalid Hattaf
  • Noura Yousfi
Chapter

Abstract

The adaptive immunity is a complex system that involves memory and plays an important role in the control of viral infection such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), and hepatitis C virus (HCV). In this paper, we propose a mathematical model formulated by fractional differential equations in order to describe the role of the adaptive immunity in viral infections and take into account the memory effect. In the proposed model, the infection transmission is modeled by Hattaf–Yousfi functional response. We first show that the model is mathematically and biologically well-posed. Using appropriate Lyapunov functionals and LaSalle’s invariance principle, the global stability of the equilibria is established and characterized by five threshold parameters. Numerical simulations are presented to illustrate our theoretical results.

References

  1. 1.
    J. Velasco-Hern\(\acute {a}\)ndez, J. Garc\(\acute {i}\)a, D. Kirschner, Remarks on modeling host-viral dynamics and treatment, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods, and Theory, vol. 125 (2001), pp. 287–308Google Scholar
  2. 2.
    D. Wodarz, Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750 (2003)CrossRefGoogle Scholar
  3. 3.
    N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection. J. Math. Biol. 63, 933–957 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Hattaf, M. Khabouze, N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response. Int. J. Dynam. Control. 3(3), 253–261 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.M. Elaiw, Global stability analysis of humoral immunity virus dynamics model including latently infected cells. J. Bio. Dynam. 9(1), 215–228 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Maziane, K. Hattaf, N. Yousfi, Global stability for a class of HIV infection models with cure of infected cells in eclipse stage and CTL immune response. Int. J. Dynam. Control. 5(4), 1035–1045 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15–67 (1997)CrossRefGoogle Scholar
  8. 8.
    R.J. Marks II, M.W. Hall, Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29, 872–877 (1981)CrossRefGoogle Scholar
  9. 9.
    G.L. Jia, Y.X. Ming, Study on the viscoelasticity of cancellous bone based on higher-order fractional models, in Proceeding of the 2nd International Conference on Bioinformatics and biomedical Engineering (ICBBE’08) (2006), pp. 1733–1736Google Scholar
  10. 10.
    R. Magin, Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 13–77 (2004)Google Scholar
  11. 11.
    E. Scalas, R. Gorenflo, F. mainardi, Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)Google Scholar
  12. 12.
    L. Song, S.Y. Xu, J.Y. Yang, Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15, 616–628 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Capponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: modelling and control applications. World Sci. Ser. Nonlinear Sci. Ser. A 72 (2010)Google Scholar
  14. 14.
    K.S. Cole, Electric conductance of biological systems. Cold Spring Harb. Quant. Biol., 107–116 (1993)Google Scholar
  15. 15.
    C.M.A. Pinto, A.R.M. Carvalho, A latency fractional order model for HIV dynamics. J. Comput. Appl. Math. 312, 240–256 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    F.A. Rihan, M. Sheek-Hussein, A. Tridane, R. Yafia, Dynamics of hepatitis C virus infection: mathematical modeling and parameter estimation. Math. Model. Nat. Phenom. 12(5), 33–47 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A.A.M. Arafa, S.Z. Rida, M. Khalil, A fractional-order model of HIV infection: numerical solution and comparisons with data of patients. Int. J. Biochem. 7(4), 1–11 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate. Int. J. Differ. Equ. 2017, 1–8 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    C.V. De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inform. 26, 15–27 (2008)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Adnane Boukhouima
    • 1
  • Khalid Hattaf
    • 1
    • 2
  • Noura Yousfi
    • 1
  1. 1.Laboratory of Analysis, Modeling and Simulation (LAMS)Faculty of Sciences Ben M’sik, Hassan II UniversityCasablancaMorocco
  2. 2.Centre Régional des Métiers de l’Education et de la Formation (CRMEF)CasablancaMorocco

Personalised recommendations