Global stability of steady solutions for a model in virus dynamics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, p. 709-723

We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence - a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.

DOI : https://doi.org/10.1051/m2an:2003045
Classification:  34A34,  34G20,  70K20,  92D25,  92D10,  37A60
Keywords: virus dynamics, population dynamics, genetics, nonlinear integro-differential equations, nonlinear ordinary differential equations, dynamical systems in statistical mechanics, immunology, evolution theory
@article{M2AN_2003__37_4_709_0,
     author = {Frid, Hermano and Jabin, Pierre-Emmanuel and Perthame, Beno\^\i t},
     title = {Global stability of steady solutions for a model in virus dynamics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {4},
     year = {2003},
     pages = {709-723},
     doi = {10.1051/m2an:2003045},
     zbl = {1065.92013},
     mrnumber = {2018439},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_4_709_0}
}
Frid, Hermano; Jabin, Pierre-Emmanuel; Perthame, Benoît. Global stability of steady solutions for a model in virus dynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, pp. 709-723. doi : 10.1051/m2an:2003045. http://www.numdam.org/item/M2AN_2003__37_4_709_0/

[1] N. Bellomo and L. Preziosi, Modeling and mathematical problems related to tumors immune system interactions. Math. Comput. Model. 31 (2000) 413-452. | Zbl 0997.92020

[2] R. Bürger,The mathematical theory of selection, recombination and mutation. Wiley (2000). | MR 1885085 | Zbl 0959.92018

[3] M.A.J. Chaplain Ed., Special Issue on Mathematical Models for the Growth, Development and Treatment of Tumours. Math. Mod. Meth. Appl. S. 9 (1999). | Zbl 0929.00034

[4] E. De Angelis and P.-E. Jabin, Analysis of a mean field modelling of tumor and immune system competition. Math. Mod. Meth. Appl. S. 13 (2003) 187-206. | Zbl 1043.92012

[5] P. Degond and B. Lucquin-Desreux, The Fokker-Plansk asymptotics of the Boltzmann collision operator in the Coulomb case? Math. Mod. Meth. Appl. S. 2 (1992) 167-182. | Zbl 0755.35091

[6] O. Dieckmann and J.P. Heesterbeek, Mathematical Epidemiology of infectious Diseases. Wiley, New York (2000). | MR 1882991 | Zbl 0997.92505

[7] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, Adaptive dynamics without time scale separation. Work in preparation.

[8] A. Lins, W. De Melo and C.C. Pugh, On Liénard's equation. Lecture Notes in Math. 597 (1977) 334-357. | Zbl 0362.34022

[9] R.M. May and M.A. Nowak, Virus dynamics (mathematical principles of immunology and virology). Oxford Univ. Press (2000). | MR 2009143 | Zbl 1101.92028

[10] A.S. Perelson and G. Weisbuch, Immunology for physicists. Rev. modern phys. 69 (1997) 1219-1267.

[11] J. Saldaña, S.F. Elana and R.V. Solé, Coinfection and superinfection in RNA virus populations: a selection-mutation model. Math. Biosci. 183 (2003) 135-160. | Zbl 1012.92031

[12] C.H. Taubes, Modeling lectures on differential equations in biology. Prentice-Hall (2001).

[13] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of fluid mechanics, S. Friedlander and D. Serre Eds., Vol. 1. North-Holland, Amsterdam (2000) 71-305. | Zbl pre01942873

[14] D. Waxman, A model of population genetics and its mathematical relation to quantum theory. Contemp. phys. 43 (2002) 13-20.