A Mathematical Model of the Competition between Acquired Immunity and Virus

  • Mikhail K. Kolev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)

Abstract

A mathematical model describing the interactions between viral infection and acquired immunity is proposed. The model is formulated in terms of a system of partial integro-differential bilinear equations. Results of numerical experiments are presented.

Keywords

numerical modelling kinetic theory active particles partial integro-differential equations nonlinear dynamics virus acquired immune system 

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References

  1. 1.
    Abbas, A.K., Lichtman, A.H.: Basic Immunology. In: Functions and Disorders of the Immune System. Elsevier, Philadelphia (2004)Google Scholar
  2. 2.
    Arlotti, L., Bellomo, N., De Angelis, E., Lachowicz, M.: Generalized Kinetic Models in Applied Sciences. World Sci., New Jersey (2003)MATHGoogle Scholar
  3. 3.
    Bellomo, N.: Modelling Complex Living Systems. Birkhäuser, Boston (2007)Google Scholar
  4. 4.
    Bellomo, N., Bellouquid, A.: On the onset of nonlinearity for diffusion models of binary mixtures of biological materials by asymptotic analysis. Internat. J. Nonlinear Mech. 41(2), 281–293 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bellomo, N., Bianca, C., Delitala, M.: Complexity analysis and mathematical tools towards the modelling of living systems. Physics of Life Reviews 6, 144–175 (2009)CrossRefGoogle Scholar
  6. 6.
    Bellomo, N., Delitala, M.: From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells. Physics of Life Reviews 5, 183–206 (2008)CrossRefGoogle Scholar
  7. 7.
    Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comput. Modelling 20(1), 107–122 (1994)MATHCrossRefGoogle Scholar
  8. 8.
    Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modelling: Selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bellomo, N., Maini, P.: Preface in: Cancer Modelling (II). In: Math. Models Methods Appl. Sci. 16(7b) (special issue), iii–vii (2006)Google Scholar
  10. 10.
    Bellomo, N., Sleeman, B.: Preface in: Multiscale Cancer Modelling. Comput. Math. Meth. Med. 20(2-3) (special issue), 67–70 (2006)CrossRefMathSciNetGoogle Scholar
  11. 11.
    De Angelis, E., Lodz, B.: On the kinetic theory for active particles: A model for tumor-immune system competition. Math. Comput. Modelling 47(1-2), 196–209 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    De Lillo, S., Salvatori, M.C., Bellomo, N.: Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity. Math. Comput. Modelling 45(5-6), 564–578 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Forbus, K.D.: Qualitative Modeling. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Represantation, Foundations of Artificial Intelligence, vol. 3, pp. 361–393. Elsevier, Amsterdam (2008)Google Scholar
  14. 14.
    Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Boston (1997)MATHGoogle Scholar
  15. 15.
    Kagi, D., Seiler, P., Pavlovic, J., Ledermann, B., Burki, K., Zinkernagel, R.M., Hengartner, H.: The roles of perforin-dependent and Fas-dependent cytotoxicity in protection against cytopathic and noncytopathic viruses. Eur. J. Immunol. 25, 3256 (1995)CrossRefGoogle Scholar
  16. 16.
    Klienstein, S.H., Seiden, P.E.: Simulating the immune system. Computer Simulation, 69–77 (July-August 2000)Google Scholar
  17. 17.
    Kolev, M.: Mathematical modelling of the interactions between antibodies and virus. In: Proc. of the IEEE Conf. on Human System Interactions, Krakow, pp. 365–368 (2008)Google Scholar
  18. 18.
    Kolev, M.: Mathematical modelling of the humoral immune response to virus. In: Proc. of the Fourteenth National Conf. on Application of Mathematics in Biology and Medicine, Leszno, Poland, pp. 63–68 (2008)Google Scholar
  19. 19.
    Kolev, M.: Numerical modelling of cellular immune response to virus. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 361–368. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Kolev, M.: The immune system and its mathematical modelling. In: De Gaetano, A., Palumbo, P. (eds.) Mathematical Physiology Encyclopedia of Life Support Systems (EOLSS). Developed under the Auspices of the UNESCO, Eolss Publ., Oxford (2010), http://www.eolss.net Google Scholar
  21. 21.
    Kuby, J.: Immunology. W.H. Freeman, New York (1997)Google Scholar
  22. 22.
    Lollini, P.L., Motta, S., Pappalardo, P.: Modeling tumor immunology. Math. Models Methods Appl. Sci 16(7b), 1091–1125 (2006)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lydyard, P.M., Whelan, A., Fanger, M.W.: Instant Notes in Immunology. BIOS Sci. Publ. Ltd., Oxford (2000)Google Scholar
  24. 24.
    Marchuk, G.I.: Mathematical Modeling of Immune Response in Infectious Diseases. Kluwer Academic Publ., Dodrecht (1997)Google Scholar
  25. 25.
    Nowak, M.A., May, R.M.: Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford Univ. Press, Oxford (2000)MATHGoogle Scholar
  26. 26.
    d’Onofrio, A.: Tumor–immune system interaction and immunotherapy: Modelling the tumor–stimulated proliferation of effectros. Math. Models Methods Appl. Sci. 16(8), 1375–1402 (2006)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Perelson, A.S., Weisbuch, G.: Immunology for physicists. Rev. Mod. Phys. 69(4), 1219–1267 (1997)CrossRefGoogle Scholar
  28. 28.
    Pinchuk, G.: Schaum’s Outline of Theory and Problems of Immunology. McGraw-Hill, New York (2002)Google Scholar
  29. 29.
    Shampine, M.W., Reichelt, M.W.: The Matlab ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Volkov, E.A.: Numerical Methods. Hemisphere/Mir, New York/Moscow (1990)MATHGoogle Scholar
  31. 31.
    Wodarz, D.: Killer Cell Dynamics. Springer, Heidelberg (2007)MATHCrossRefGoogle Scholar
  32. 32.
    Wodarz, D., Bangham, C.R.: Evolutionary dynamics of HTLV-I. J. Mol. Evol. 50, 448–455 (2000)Google Scholar
  33. 33.
    Wodarz, D., Krakauer, D.C.: Defining CTL-induced pathology: implication for HIV. Virology 274, 94–104 (2000)CrossRefGoogle Scholar
  34. 34.
    Wodarz, D., May, R., Nowak, M.: The role of antigen-independent persistence of memory cytotoxic T lymphocytes. Intern. Immun. 12, 467–477 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mikhail K. Kolev
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Warmia and MazuryPoland

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