Open AccessApproximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells
Author(s) -
D. Criaco,
Marina Dolfin,
Liliana Restuccia,
D. Criaco,
Marina Dolfin,
Liliana Restuccia
Publication year - 2013
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2013.10.59
Subject(s) - burgers' equation , singular perturbation , mathematics , perturbation theory (quantum mechanics) , physics , mathematical analysis , partial differential equation , classical mechanics , quantum mechanics
In a previous paper a mathematical model was developed for the dynamics of activation and clonal expansion of T cells during the immune response to a single type of antigen challenge, constructed phenomenologically in the macroscopic framework of a thermodynamic theory of continuum mechanics for reacting and proliferating fluid 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.