Skip to main content
SpringerLink
Log in

Viral Immunology: Modeling and Analysis

  • Chapter
  • First Online:
Mathematical Modelling in Health, Social and Applied Sciences

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

Abstract

Viral immunology is a subdiscipline of immunology which studies the interactions between viruses and immune system. The aim of this work is to model and analyze these interactions by proposing two mathematical models that describe the role of adaptive immune response in infectious diseases caused by viruses such as the human immunodeficiency virus (HIV), hepatitis B virus (HBV) and hepatitis C virus (HCV). The first model is formulated by delay differential equations (DDEs) that include many special cases existing in the literature. The second model is formulated by partial differential equations (PDEs) in order to describe the evolution in time and space of viral infections. Also, the models integrate the two main modes of virus propagation that are virus-to-cell infection and direct cell-to-cell transmission. Furthermore, the dynamical behaviors of both models are rigorously analyzed. Moreover, biological findings of the analytical results are also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750 (2003)

    Article  Google Scholar 

  2. N. Yousfi, K. Hattaf, M. Rachik, Analysis of a HCV model with CTL and antibody responses. Appl. Math. Sci. 3(57), 2835–2845 (2009)

    MathSciNet  MATH  Google Scholar 

  3. N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptative immune response in HBV infection. J. Math. Biol. 63(5), 933–957 (2011)

    Article  MathSciNet  Google Scholar 

  4. K. Hattaf, N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response. Int. J. Dyn. Control. 4, 254–265 (2016)

    Article  MathSciNet  Google Scholar 

  5. Y. Yan, W. Wang, Global stability of a five-dimensional model with immune responses and delay. Discret. Contin. Dyn. Syst.-B 17(1), 401–416 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Y. Su, D. Sun, L. Zhao, Global analysis of a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate. Math. Methods Appl. Sci. 38(14), 2984–2993 (2015)

    Article  MathSciNet  Google Scholar 

  7. X. Wang, S. Liu, A class of delayed viral models with saturation infection rate and immune response. Math. Methods Appl. Sci. 36(2), 125–142 (2013)

    Article  MathSciNet  Google Scholar 

  8. Y. Zhao, Z. Xu, Global dynamics for a delayed hepatitis C virus infection model. Electron. J. Differ. Equ.S 2014(132), 1–18 (2014)

    MathSciNet  MATH  Google Scholar 

  9. K. Hattaf, M. Khabouze, N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response. Int. J. Dyn. Control. 3, 253–261 (2015)

    Article  MathSciNet  Google Scholar 

  10. M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet, O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes. J. Virol. 81, 1000–1012 (2007)

    Article  Google Scholar 

  11. H. Sato, J. Orenstein, D. Dimitrov, M. Martin, Cell-to-cell spread of HIV-1 occurs within minutes and may not involve the participation of virus particles. Virology 186, 712–724 (1992)

    Article  Google Scholar 

  12. J.M. Carr, H. Hocking, P. Li, C.J. Burrell, Rapid and efficient cell-to-cell transmission of human immunodeficiency virus infection from monocyte-derived macrophages to peripheral blood lymphocytes. Virology 265, 319–329 (1999)

    Article  Google Scholar 

  13. M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996)

    Article  Google Scholar 

  14. X. Wang, Y. Tao, X. Song, Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response. Nonlinear Dyn. 66(4), 825–830 (2011)

    Article  MathSciNet  Google Scholar 

  15. K. Hattaf, N. Yousfi, A. Tridane, Global stability analysis of a generalized virus dynamics model with the immune response. Can. Appl. Math. Q. 20(4), 499–518 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Y. Li, R. Xu, Z. Li, S. Mao, Global dynamics of a delayed HIV-1 infection model with CTL immune response. Discret. Dyn. Nat. Soc. 2011, 1–13 (2011)

    MathSciNet  MATH  Google Scholar 

  17. X. Li, S. Fu, Global stability of a virus dynamics model with intracellular delay and CTL immune response. Math. Methods Appl. Sci. 38(3), 420–430 (2015)

    Article  MathSciNet  Google Scholar 

  18. J. Wang, M. Guo, X. Liu, Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay. Appl. Math. Comput. 291, 149–161 (2016)

    MathSciNet  MATH  Google Scholar 

  19. A.M. Elaiw, A.A. Raezah, K. Hattaf, Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response. Int. J. Biomath. 10(5), 1–29 (2017)

    Article  MathSciNet  Google Scholar 

  20. A. Murase, T. Sasaki, T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics. J. Math. Biol. 51, 247–267 (2005)

    Article  MathSciNet  Google Scholar 

  21. S.F. Wang, D.Y. Zou, Global stability of in-host viral models with humoral immunity and intracellular delays. Appl. Math. Model. 36, 1313–1322 (2012)

    Article  MathSciNet  Google Scholar 

  22. T. Wang, Z. Hu, F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response. J. Math. Anal. Appl. 411, 63–74 (2014)

    Article  MathSciNet  Google Scholar 

  23. T. Wang, ZHuF Liao, W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity. Math. Comput. Simul. 89, 13–22 (2013)

    Article  MathSciNet  Google Scholar 

  24. H. Miao, Z. Teng, C. Kang, A. Muhammadhaji, Stability analysis of a virus infection model with humoral immunity response and two time delays. Math. Methods Appl. Sci. 39(12), 3434–3449 (2016)

    Article  MathSciNet  Google Scholar 

  25. X. Wang, S. Liu, A class of delayed viral models with saturation infection rate and immune response. Math. Methods Appl. Sci. 36, 125–142 (2013)

    Article  MathSciNet  Google Scholar 

  26. J. Lin, R. Xu, X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity. Appl. Math. Comput. 315, 516–530 (2017)

    MathSciNet  MATH  Google Scholar 

  27. A.M. Elaiw, A.A. Raezah, A.S. Alofi, Effect of humoral immunity on HIV-1 dynamics with virus-to-target and infected-to-target infections. AIP Adv. 6, 085204 (2016)

    Article  Google Scholar 

  28. K. Hattaf, N. Yousfi, Modeling the adaptive immunity and both modes of transmission in HIV infection. Computation 6(2), 1–18 (2018)

    Article  Google Scholar 

  29. J.K. Hale, J. Kato, Phase space for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11–41 (1978)

    MathSciNet  MATH  Google Scholar 

  30. Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, In: Lecture Notes in Math, vol. 1473, (Springer, Berlin, 1991)

    Google Scholar 

  31. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, San Diego, 1993)

    MATH  Google Scholar 

  32. J.K. Hale, S.M. Verduyn Lunel, in Introduction to Functional Differential Equations (Springer, New York, 1993)

    Google Scholar 

  33. K. Hattaf, N. Yousfi, Global stability for reaction-diffusion equations in biology. Comput. Math. Appl. 66, 1488–1497 (2013)

    Article  MathSciNet  Google Scholar 

  34. K. Hattaf, N. Yousfi, Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases, Mathematics Applied to Engineering, Modelling, and Social Issues (Springer, Cham, 2019), pp. 529–549

    Book  Google Scholar 

  35. K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response. Computation 7(2), 1–16 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre Régional des Métiers de L’Education Et de la Formation (CRMEF), 20340, Derb Ghalef, Casablanca, Morocco

    Khalid Hattaf

  2. Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955, Sidi Othman, Casablanca, Morocco

    Khalid Hattaf

Authors
  1. Khalid Hattaf
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hattaf, K. (2020). Viral Immunology: Modeling and Analysis. In: Dutta, H. (eds) Mathematical Modelling in Health, Social and Applied Sciences. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-2286-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation