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Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 1967–1976 | Cite as

Reaction–Diffusion Equations in Immunology

  • G. A. Bocharov
  • V. A. Volpert
  • A. L. TasevichEmail author
Article
  • 2 Downloads

Abstract

The paper is devoted to the recent works on reaction–diffusion models of virus infection dynamics in human and animal organisms. Various regimes of infection propagation in tissues are described. In particular, it is shown that infection can spread in tissues of organs as a reaction–diffusion wave. Methods for studying the conditions of the existence of wave modes of the time and space dynamics of infections is discussed.

Keywords:

virus infection immune response mathematical modeling reaction–diffusion equations time and space dynamics 

Notes

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 17-01-00636; and by the Ministry for Education and Science of the Russian Federation within the program 5-100 for enhancing the competitiveness of the Peoples’ Friendship University of Russia among the leading science and education centers in 2016–2020.

REFERENCES

  1. 1.
    A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of CD4\(^{ + }\) T Cells,” Math. Biosci. 114 (1), 81–125 (1993).CrossRefzbMATHGoogle Scholar
  2. 2.
    M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science 272, 74–79 (1996). Google Scholar
  3. 3.
    A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D. D. Ho, “HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,” Science 271, 1582–1586 (1996).CrossRefGoogle Scholar
  4. 4.
    A. S. Perelson, “Modelling viral and immune system dynamics,” Nat. Rev. Immunol. 2, 28–36 (2002).CrossRefGoogle Scholar
  5. 5.
    D. Wodarz and M. A. Nowak, “Mathematical models of HIV pathogenesis and treatment,” BioEssays. 24, 1178–1187 (2002).CrossRefGoogle Scholar
  6. 6.
    S. Alizon and C. Magnus, “Modelling the course of an HIV infection: Insights from ecology and evolution,” Viruses. 4, 1984–2013 (2012).CrossRefGoogle Scholar
  7. 7.
    S. Gadhamsetty, T. Coorens, and R. J. de Boer, “Notwithstanding circumstantial alibis, cytotoxic T cells can be major killers of HIV-1 infected cells,' J. Virology 90, 7066–7083 (2016).CrossRefGoogle Scholar
  8. 8.
    M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, and H. McDade, “Viral dynamics in hepatitis B virus infection,” Proc. Natl. Acad. Sci. U.S.A. 93, 4398–4402 (1996).CrossRefGoogle Scholar
  9. 9.
    A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden, and A. S. Perelson, “Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-\(\alpha \) therapy,” Science 282, 103–107 (1998).CrossRefGoogle Scholar
  10. 10.
    R. J. De Boer, M. Oprea, R. Antia, K. Murali-Krishna, R. Ahmed, and A. S. Perelson, “Recruitment Times, proliferation, and apoptosis rates during the CD8\(^{ + }\) T-Cell responses to lymphocytic choriomeningitis virus,” J. Virology 75 (22), 10663–10669 (2001).CrossRefGoogle Scholar
  11. 11.
    R. J. De Boer, D. Homann, and A. S. Perelson, “Different dynamics of CD4\(^{ + }\) and CD8\(^{ + }\) T cell responses during and after acute lymphocytic choriomeningitis virus infection,” J. Immunol. 171, 3928–3935 (2003).CrossRefGoogle Scholar
  12. 12.
    C. L. Althaus, V. V. Ganusov, and R. J. De Boer, “Dynamics of CD8\(^{ + }\) T cell responses during acute and chronic lymphocytic choriomeningitis virus infection,” J. Immunol. 179, 2944–2951 (2007).CrossRefGoogle Scholar
  13. 13.
    G. I. Marchuk, SDelected Works, Vol. 5 Mathematical Modeling in Immunology and Medicine (Inst. Vychisl. Mat., Moscow, Ross. Alad. Nauk, 2018) [in Russian].Google Scholar
  14. 14.
    G. I. Marchuk, “Mathematical modelling of immune response in infectious diseases,” in Mathematics and Its Applications, Vol. 395 (Kluwer, Dordrecht, 1997).Google Scholar
  15. 15.
    G. A. Bocharov and G. I. Marchuk, “Applied problems of mathematical modeling in immunology,” Comput. Math. Math. Phys. 40, 1830–1844 (2000).MathSciNetzbMATHGoogle Scholar
  16. 16.
    G. I. Marchuk, R. V. Petrov, A. A. Romanyukha, and G. A. Bocharov, “Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction and parameters evaluation for hepatitis B,” J. Theor. Biol. 151, 1–40 (1991).CrossRefGoogle Scholar
  17. 17.
    G. I. Marchuk, A. A. Romanyukha, and G. A. Bocharov, “Mathematical model of antiviral immune response. II. Parameters identification for acute viral hepatitis B,” J. Theor. Biol. 151, 41–70 (1991).CrossRefGoogle Scholar
  18. 18.
    G. A. Bocharov, “Mathematical model of antiviral immune response. III. Influenza A virus infection,” J. Theor. Biol. 167, 323–360 (1994).CrossRefGoogle Scholar
  19. 19.
    G. A. Bocharov, “Modelling the dynamics of LCMV infection in mice: Conventional and exhaustive CTL responses,” J. Theor. Biol. 192, 283–308 (1998).CrossRefGoogle Scholar
  20. 20.
    G. Bocharov, P. Klenerman, and S. Ehl, “Modelling the dynamics of LCMV infection in mice: II. Compartment structure and immunopathology,” J. Theor. Biol. 221, 349–378 (2003).CrossRefGoogle Scholar
  21. 21.
    G. Bocharov, B. Ludewig, A. Bertoletti, P. Klenerman, T. Junt, P. Krebs, T. Luzyanina, C. Fraser, and R. M. Anderson, “Underwhelming the immune response: Effect of slow virus growth on CD8\(^{ + }\)-T-lymphocyte responses,” J. Virol. 78, 2247–2254 (2004).CrossRefGoogle Scholar
  22. 22.
    G. Bocharov, J. Argilaguet, and A. Meyerhans, “Understanding experimental LCMV infection of mice: The role of mathematical models,” J. Immunol. Res. No. 16, 1–10 (2015).Google Scholar
  23. 23.
    G. A. Funk, V. A. Jansen, S. Bonhoeffer, and T. Killingback, “Spatial models of virus-immune dynamics,” J. Theor. Biol. 233, 221–236 (2005).MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. C. Strain, D. D. Richman, J. K. Wong, and H. Levine, “Spatiotemporal dynamics of HIV propagation,” J. Theor. Biol. 218, 85–96 (2002).MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Beauchemin, “Probing the effects of the well-mixed assumption on viral infection dynamics,” J. Theor. Biol. 242, 464–477 (2006).MathSciNetCrossRefGoogle Scholar
  26. 26.
    X. Sewald, N. Motamedi, and W. Mothes, “Viruses exploit the tissue physiology of the host to spread in vivo,” Current Opinion in Cell Biol. 41, 81–90 (2016).CrossRefGoogle Scholar
  27. 27.
    W. Mothes, N. M. Sherer, J. Jin, and P. Zhong, “Virus cell-to-cell transmission,” J. Virol. 84, 8360–8368 (2010).CrossRefGoogle Scholar
  28. 28.
    F. Graw, D. N. Martin, A. S. Perelson, S. L. Uprichard, and H. Dahari, “Quantification of hepatitis C virus cell-to-cell spread using a stochastic modeling approach,” J. Virol. 89, 6551–6561 (2015).CrossRefGoogle Scholar
  29. 29.
    S. A. Prokopiou, L. Barbaroux, S. Bernard, J. Mafille, Y. Leverrier, C. Arpin, J. Marvel, O. Gandrillon, and F. Crauste, “Multiscale modeling of the early CD8 T-cell immune response in lymph nodes: An integrative study,” Computation 2, 159–181 (2014).CrossRefGoogle Scholar
  30. 30.
    R. Dunia and R. Bonnecaze, “Mathematical modeling of viral infection dynamics in spherical organs,” J. Math. Biol. 67, 1425–1455 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    G. Bocharov, A. Danilov, Yu. Vassilevski, G. I. Marchuk, V. A. Chereshnev, and B. Ludewig, “Reaction-diffusion modelling of interferon distribution in secondary lymphoid organs,” Math. Model. Nat. Phenom. 6, 13–26 (2011).MathSciNetCrossRefGoogle Scholar
  32. 32.
    A. Kislitsyn, R. Savinkov, M. Novkovic, L. Onder, and G. Bocharov, “Computational approach to 3D modeling of the lymph node geometry,” Computation 3, 222–234 (2015).CrossRefGoogle Scholar
  33. 33.
    E. L. Haseltine, V. Lam, J. Yin, and J. B. Rawlings, “Image-guided modeling of virus growth and spread,” Bull Math Biol. 70, 1730–1748 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    B. Su, W. Zhou, K. S. Dorman, and D. E. Jones, “Mathematical modelling of immune response in tissues,” Comput. Math. Meth. Medicine 10 (1), 9–38 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    O. Stancevic, C. N. Angstmann, J. M. Murray, and B. I. Henry, “Turing patterns from dynamics of early HIV infection,” Bull. Math. Biol. 75, 774–795 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ph. Getto, M. Kimmel, and A. Marciniak-Czochra, “Modelling and analysis of dynamics of viral infection of cells and of interferon resistance,” J. Math. Anal. Appl. 344, 821–850 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    M. Labadie and A. Marciniak-Czochra, “A reaction-diffusion model for viral infection and immune response,” 2011. \( < \)hal-00546034v2\( > \) Google Scholar
  38. 38.
    R. Bertolusso and M. Kimmel, “Spatial and stochastic effects in a model of viral infection,” Fund. Inform. 118, 327–343 (2012).zbMATHGoogle Scholar
  39. 39.
    G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert, “Spatiotemporal dynamics of virus infection spreading in tissues,” PlosOne, 2016. doi 10.1371/journal.pone.0168576Google Scholar
  40. 40.
    S. Trofimchuk and V. Volpert, “Traveling waves for a bistable reaction-diffusion equation with delay,” SIAM J. Math. Anal. 50, 1175–1190 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, and V. Volpert, “Modelling the dynamics of virus infection and immune response in space and time,” Int. J. Parallel, Emergent Distrib. Syst. 32, (2017).Google Scholar
  42. 42.
    G. Bocharov, B. Ludewig, A. Meyerhans, and V. Volpert, Mathematical Immunology of Virus Infections (Springer, 2018).CrossRefzbMATHGoogle Scholar
  43. 43.
    A. Bouchnita, G. Bocharov, A. Meyerhans, and V. Volpert, “Towards a multiscale model of acute HIV infection,” Computations, 5, 1–22 (2017). doi 10.3390/computation5010006Google Scholar
  44. 44.
    A. Bouchnita, G. Bocharov, A. Meyerhans, and V. Volpert, “Hybrid approach to model the spatial regulation of T cell responses,” BMC Immunol., 18 (2017).Google Scholar
  45. 45.
    V. Volpert, “Existence of reaction-diffusion waves in a model of immune response,” J. Fixed Points Appl., (2018), in press.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • G. A. Bocharov
    • 1
    • 2
  • V. A. Volpert
    • 1
    • 3
    • 4
    • 5
  • A. L. Tasevich
    • 1
    • 6
    Email author
  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Institute of Numerical Mathematics, Russian Academy of SciencesMoscowRussia
  3. 3.Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1VilleurbanneFrance
  4. 4.INRIA Team Dracula, INRIA Lyon La DouaVilleurbanneFrance
  5. 5.Poncelet Center, UMI 2615 CNRSMoscowRussia
  6. 6.Federal Research Center “Computer Science and Control,” Russian Academy of SciencesMoscowRussia

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