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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 1: 59-73. doi: 10.3934/mbe.2013.10.59
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Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells

  • 1.
    Department of Mathematics and Informatics, University of Messina, Viale F. Stagno d'Alcontres n.31, 98166 Messina
  • 2.
    Department I.C.I.E.A.M.A., University of Messina, Contrada Di Dio (S.Agata), 98166 Messina
  • Received: 01 April 2012 Accepted: 29 June 2018 Published: 01 December 2012

  • MSC : Primary: 92B05; Secondary: 35Q35.

  • In a previous paper a mathematical model was developed for thedynamics of activation and clonal expansion of T cells during theimmune response to a single type of antigen challenge, constructedphenomenologically in the macroscopic framework of a thermodynamictheory of continuum mechanicsfor reacting and proliferatingfluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.

    Citation: D. Criaco, M. Dolfin, L. Restuccia. Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells[J]. Mathematical Biosciences and Engineering, 2013, 10(1): 59-73. doi: 10.3934/mbe.2013.10.59

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  • In a previous paper a mathematical model was developed for thedynamics of activation and clonal expansion of T cells during theimmune response to a single type of antigen challenge, constructedphenomenologically in the macroscopic framework of a thermodynamictheory of continuum mechanicsfor reacting and proliferatingfluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.


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