On a Mathematical Model of Adaptive Immune Response to Viral Infection

  • Mikhail Kolev
  • Ana Markovska
  • Adam Korpusik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


In this paper we study a mathematical model formulated within the framework of the kinetic theory for active particles. The model is a bilinear system of integro-differential equations (IDE) of Boltzmann type and it describes the interactions between virus population and the adaptive immune system. The population of cytotoxic T lymphocytes is additionally divided into precursor and effector cells. Conditions for existence and uniqueness of the solution are studied. Numerical simulations of the model are presented and discussed.


numerical simulations integro-differential equations nonlinear dynamics kinetic model 


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  1. 1.
    Abbas, A., Lichtman, A.: Basic Immunology: Functions and Disorders of the Immune System. Elsevier, Philadelphia (2009)Google Scholar
  2. 2.
    Arlotti, L., Bellomo, N., Lachowicz, M.: Kinetic equations modelling population dynamics. Transport Theory Statist. Phys. 29, 125–139 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Belleni-Morante, A.: Applied Semigroups and Evolution Equations. Oxford Univ. Press, Oxford (1979)zbMATHGoogle Scholar
  4. 4.
    Bellomo, N., Carbonaro, B.: Toward a mathematical theory of living systems focusing on developmental biology and evolution: a review and perspectives. Physics of Life Reviews 8, 1–18 (2011)CrossRefGoogle Scholar
  5. 5.
    Bellomo, N., Bianca, C.: Towards a Mathematical Theory of Complex Biological Systems. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  6. 6.
    Bianca, C.: Mathematical modelling for keloid formation triggered by virus: Malignant effects and immune system competition. Math. Models Methods Appl. Sci. 21, 389–419 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kolev, M., Korpusik, A., Markovska, A.: Adaptive immunity and CTL differentiation - a kinetic modeling approach. Mathematics in Engineering, Science and Aerospace 3, 285–293 (2012)zbMATHGoogle Scholar
  8. 8.
    Shampine, L., Reichelt, M.: The Matlab ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wodarz, D.: Killer Cell Dynamics. Springer, New York (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mikhail Kolev
    • 1
  • Ana Markovska
    • 2
  • Adam Korpusik
    • 3
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of Warmia and MazuryOlsztynPoland
  2. 2.Faculty of Mathematics and Natural SciencesSouth-West University “N. Rilski”BlagoevgradBulgaria
  3. 3.Faculty of Technical SciencesUniversity of Warmia and MazuryOlsztynPoland

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