Numerical Modelling of Cellular Immune Response to Virus

  • Mikhail K. Kolev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


We present a mathematical model of cellular immune response to viral infection. The model is a bilinear system of integro-differential equations of Boltzmann type. Results of numerical experiments are presented.


Numerical modelling kinetic model integro-differential equations nonlinear dynamics virus immune system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arlotti, L., Bellomo, N., De Angelis, E., Lachowicz, M.: Generalized Kinetic Models in Applied Sciences. World Sci., New Jersey (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comput. Modelling 23, 11–29 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellomo, N., Maini, P.: Preface in: Cancer Modelling (II). Math. Models Methods Appl. Sci. 16(7b), iii–vii (2006) (special issue)Google Scholar
  5. 5.
    Bellomo, N., Sleeman, B.: Preface in: Multiscale Cancer Modelling. Comput. Math. Meth. Med. 20(2-3), 67–70 (2006) (special issue)CrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Boston (1997)zbMATHGoogle Scholar
  9. 9.
    Kolev, M.: Mathematical modelling of the interactions between antibodies and virus. In: Proc. of the IEEE Conf. on Human System Interactions, pp. 365–368. Krakow (2008)Google Scholar
  10. 10.
    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
  11. 11.
    Kuby, J.: Immunology. W.H. Freeman, New York (1997)Google Scholar
  12. 12.
    Lollini, P.L., Motta, S., Pappalardo, P.: Modeling tumor immunology. Math. Models Methods Appl. Sci 16(7b), 1091–1125 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lydyard, P.M., Whelan, A., Fanger, M.W.: Instant Notes in Immunology. BIOS Sci. Publ. Ltd., Oxford (2000)Google Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shampine, M.W., Reichelt, M.W.: The Matlab ODE suite. SIAM J. Sci. Comput. 18, 1–22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Volkov, E.A.: Numerical Methods. Hemisphere/Mir, New York/Moscow (1990)zbMATHGoogle Scholar
  17. 17.
    Wodarz, D.: Killer Cell Dynamics. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  18. 18.
    Wodarz, D., Bangham, C.R.: Evolutionary dynamics of HTLV-I. J. Mol. Evol. 50, 448–455 (2000)CrossRefGoogle Scholar
  19. 19.
    Wodarz, D., Krakauer, D.C.: Defining CTL-induced pathology: implication for HIV. Virology 274, 94–104 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

Personalised recommendations