Special Issues

Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells

  • 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

    Related Papers:

  • 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.


    加载中
    [1] Mathematical and Computer Modelling, 53 (2011), 314-329.
    [2] $1^{st}$ edition, Gauthier-Villars, Paris, 1965.
    [3] Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31-44 (in french).
    [4] Bollettino U.M.I, 16-A (1976), 450-458 (in italian).
    [5] $1^{st}$ edition, Academic Press, New York, 1964.
    [6] Z. A. M. P. 14 (1963), 31-314.
    [7] SIAM Review, 14 (1972), 582-653.
    [8] Arch. Rational Mech. Anal., 14 (1963), 27-37.
    [9] Comm. Pure Appl. Math., 6 (1983), 231-238.
    [10] $1^{st}$ edition, Chapman and Hall, London, 1995.
    [11] $1^{st}$ edition, C. N. R. Press, Rome, 1987.
    [12] J. Math. Pures et Appl., 48 (1968), 117-158 (in french).
    [13] $1^{st}$ edition, Princeton University Press, Princeton, 1954.
    [14] Comm. Pure Appl. Math., 10 (1957), 537-566.
    [15] J. Phys. Soc. Japan, 24 (1968), 941-946.
    [16] Physica A, 132 (1985), 606-616.
    [17] Physica A, 131 (1985), 251-262.
    [18] Physica A, 142 (1987), 309-320.
    [19] $1^{st}$ edition, Pitman, London, 1976.
    [20] $1^{st}$ edition, Pitman Advanced Publishing Program, Princeton, 1985.
    [21] $1^{st}$ edition, Springer, Berlin-Heidelberg-New York, 1998.
    [22] Bullettin of Mathematical Biology, 53 (1991), 721-749.
    [23] Theoretical Biology, 81 (1979,) 475-503.
    [24] Mathematical Biology, 68 (2006), 1819-1836.
    [25] Mathematical Medicine and Biology, 20 (2003), 341-366.
    [26] edition, Springer, Mediterranean Press , 1989.
    [27] $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002.
    [28] $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002.
    [29] Comm. Appl. Math., 3 (1950), 201-230.
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1788) PDF downloads(445) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog