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2013, 10(1): 59-73. doi: 10.3934/mbe.2013.10.59

## 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, Italy, Italy 2 Department I.C.I.E.A.M.A., University of Messina, Contrada Di Dio (S.Agata), 98166 Messina, Italy

Received  April 2012 Revised  September 2012 Published  December 2012

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.
Citation: D. Criaco, M. Dolfin, L. Restuccia. Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 59-73. doi: 10.3934/mbe.2013.10.59
##### References:
 [1] M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells, Mathematical and Computer Modelling, 53 (2011), 314-329. [2] G. Boillat, "La Propagation des Ondes," $1^{st}$ edition, Gauthier-Villars, Paris, 1965. [3] G. Boillat, Ondes asymptotiques nonlineaires, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31-44 (in french). [4] D. Fusco, Onde non lineari dispersive e dissipative, Bollettino U.M.I, 16-A (1976), 450-458 (in italian). [5] A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation," $1^{st}$ edition, Academic Press, New York, 1964. [6] A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system, Z. A. M. P. 14 (1963), 31-314. [7] A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Review, 14 (1972), 582-653. [8] A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables, Arch. Rational Mech. Anal., 14 (1963), 27-37. [9] P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math., 6 (1983), 231-238. [10] A. Georgescu, "Asymptotic Treatment of Differential Equations," $1^{st}$ edition, Chapman and Hall, London, 1995. [11] A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September," $1^{st}$ edition, C. N. R. Press, Rome, 1987. [12] Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires, J. Math. Pures et Appl., 48 (1968), 117-158 (in french). [13] P. D. Lax, "Contributions to the Theory of Partial Differential Equations," $1^{st}$ edition, Princeton University Press, Princeton, 1954. [14] P. D. Lax, Hyperbolic systems of conservation law (II), Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [15] T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation, J. Phys. Soc. Japan, 24 (1968), 941-946. [16] V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media, Physica A, 132 (1985), 606-616. [17] V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media), Physica A, 131 (1985), 251-262. doi: 10.1016/0378-4371(85)90090-1. [18] V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory, Physica A, 142 (1987), 309-320. [19] A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves," $1^{st}$ edition, Pitman, London, 1976. [20] I. Muller, "Thermodynamics," $1^{st}$ edition, Pitman Advanced Publishing Program, Princeton, 1985. [21] I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," $1^{st}$ edition, Springer, Berlin-Heidelberg-New York, 1998. [22] R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients, Bullettin of Mathematical Biology, 53 (1991), 721-749. [23] D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Theoretical Biology, 81 (1979,) 475-503. [24] A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks, Mathematical Biology, 68 (2006), 1819-1836. doi: 10.1007/s11538-006-9071-2. [25] H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366. doi: 10.1093/imammb/20.4.341. [26] G. Carini, "Lezioni di Istituzioni di Fisica Matematica," edition, Springer, Mediterranean Press , 1989. [27] J. D. Murray, "Mathematical Biology, vol I," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. [28] J. D. Murray, "Mathematical Biology, vol II," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. [29] E. Hopf, The partial differential equation ut + uux = xx, Comm. Appl. Math., 3 (1950), 201-230.

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##### References:
 [1] M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells, Mathematical and Computer Modelling, 53 (2011), 314-329. [2] G. Boillat, "La Propagation des Ondes," $1^{st}$ edition, Gauthier-Villars, Paris, 1965. [3] G. Boillat, Ondes asymptotiques nonlineaires, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31-44 (in french). [4] D. Fusco, Onde non lineari dispersive e dissipative, Bollettino U.M.I, 16-A (1976), 450-458 (in italian). [5] A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation," $1^{st}$ edition, Academic Press, New York, 1964. [6] A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system, Z. A. M. P. 14 (1963), 31-314. [7] A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Review, 14 (1972), 582-653. [8] A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables, Arch. Rational Mech. Anal., 14 (1963), 27-37. [9] P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math., 6 (1983), 231-238. [10] A. Georgescu, "Asymptotic Treatment of Differential Equations," $1^{st}$ edition, Chapman and Hall, London, 1995. [11] A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September," $1^{st}$ edition, C. N. R. Press, Rome, 1987. [12] Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires, J. Math. Pures et Appl., 48 (1968), 117-158 (in french). [13] P. D. Lax, "Contributions to the Theory of Partial Differential Equations," $1^{st}$ edition, Princeton University Press, Princeton, 1954. [14] P. D. Lax, Hyperbolic systems of conservation law (II), Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [15] T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation, J. Phys. Soc. Japan, 24 (1968), 941-946. [16] V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media, Physica A, 132 (1985), 606-616. [17] V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media), Physica A, 131 (1985), 251-262. doi: 10.1016/0378-4371(85)90090-1. [18] V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory, Physica A, 142 (1987), 309-320. [19] A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves," $1^{st}$ edition, Pitman, London, 1976. [20] I. Muller, "Thermodynamics," $1^{st}$ edition, Pitman Advanced Publishing Program, Princeton, 1985. [21] I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," $1^{st}$ edition, Springer, Berlin-Heidelberg-New York, 1998. [22] R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients, Bullettin of Mathematical Biology, 53 (1991), 721-749. [23] D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Theoretical Biology, 81 (1979,) 475-503. [24] A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks, Mathematical Biology, 68 (2006), 1819-1836. doi: 10.1007/s11538-006-9071-2. [25] H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366. doi: 10.1093/imammb/20.4.341. [26] G. Carini, "Lezioni di Istituzioni di Fisica Matematica," edition, Springer, Mediterranean Press , 1989. [27] J. D. Murray, "Mathematical Biology, vol I," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. [28] J. D. Murray, "Mathematical Biology, vol II," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. [29] E. Hopf, The partial differential equation ut + uux = xx, Comm. Appl. Math., 3 (1950), 201-230.
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