Global stability of steady solutions for a model in virus dynamics

Hermano Frid; Pierre-Emmanuel Jabin; Benoît Perthame

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 4, page 709-723
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Frid, Hermano, Jabin, Pierre-Emmanuel, and Perthame, Benoît. "Global stability of steady solutions for a model in virus dynamics." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.4 (2003): 709-723. <http://eudml.org/doc/245449>.

@article{Frid2003,
abstract = {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.},
author = {Frid, Hermano, Jabin, Pierre-Emmanuel, Perthame, Benoît},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {virus dynamics; population dynamics; genetics; nonlinear integro-differential equations; nonlinear ordinary differential equations; dynamical systems in statistical mechanics; immunology; evolution theory; Virus dynamics},
language = {eng},
number = {4},
pages = {709-723},
publisher = {EDP-Sciences},
title = {Global stability of steady solutions for a model in virus dynamics},
url = {http://eudml.org/doc/245449},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Frid, Hermano
AU - Jabin, Pierre-Emmanuel
AU - Perthame, Benoît
TI - Global stability of steady solutions for a model in virus dynamics
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 4
SP - 709
EP - 723
AB - 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.
LA - eng
KW - virus dynamics; population dynamics; genetics; nonlinear integro-differential equations; nonlinear ordinary differential equations; dynamical systems in statistical mechanics; immunology; evolution theory; Virus dynamics
UR - http://eudml.org/doc/245449
ER -

References

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  1. [1] N. Bellomo and L. Preziosi, Modeling and mathematical problems related to tumors immune system interactions. Math. Comput. Model. 31 (2000) 413–452. Zbl0997.92020
  2. [2] R. Bürger,The mathematical theory of selection, recombination and mutation. Wiley (2000). Zbl0959.92018MR1885085
  3. [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). Zbl0929.00034
  4. [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. Zbl1043.92012
  5. [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. Zbl0755.35091
  6. [6] O. Dieckmann and J.P. Heesterbeek, Mathematical Epidemiology of infectious Diseases. Wiley, New York (2000). Zbl0997.92505MR1882991
  7. [7] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, Adaptive dynamics without time scale separation. Work in preparation. 
  8. [8] A. Lins, W. de Melo and C.C. Pugh, On Liénard’s equation. Lecture Notes in Math. 597 (1977) 334–357. Zbl0362.34022
  9. [9] R.M. May and M.A. Nowak, Virus dynamics (mathematical principles of immunology and virology). Oxford Univ. Press (2000). Zbl1101.92028MR2009143
  10. [10] A.S. Perelson and G. Weisbuch, Immunology for physicists. Rev. modern phys. 69 (1997) 1219–1267. 
  11. [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. Zbl1012.92031
  12. [12] C.H. Taubes, Modeling lectures on differential equations in biology. Prentice-Hall (2001). 
  13. [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. Zbl1170.82369
  14. [14] D. Waxman, A model of population genetics and its mathematical relation to quantum theory. Contemp. phys. 43 (2002) 13–20. 

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