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ANALYSIS OF A NEW MODEL FOR TUMOR-IMMUNE SYSTEM COMPETITION INCLUDING LONG-TIME SCALE EFFECTS

LOBNA DERBEL

Laboratoire d'Ingénierie et Mathématiques, École Polytechnique de Tunisie, BP 743, Rue El-Khawarezmi 2078 la Marsa, Tunisia

Département de Mathématiques et Applications, École Normale Supérieure de Paris, 45 rue d'ULM, 75230 Paris, Cedex 05, France

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https://doi.org/10.1142/S0218202504003738Cited by:33 (Source: Crossref)

Abstract

This paper deals with the competition between the immune system and an agressive host such as tumor. We are particularly interested in the asymptotic behavior of the system when the time goes to infinity: we prove that this competition will absolutely finish either by the suppression of the harmful substances or by the inhibition of the immune system. The model which we introduce is suitable to describe the disease evolution for a long time scale, because it takes the capacity of the body to repair cells damage into account: body cells produce new cells to replace dead ones. We also examine the evolution of the solutions if we assume that the immune system has a normal healthy state that it always tries to reach.

Keywords:

AMSC: 35B40, 35F25, 82C40, 92D25

References

J. Adam and N. Bellomo , A Survey of Models on Tumor Immune Systems Dynamics ( Birkhäuser , 1996 ) . Crossref, Google Scholar
N. Bellomo, A. Bellouquid and M. Delitala, Math. Mod. Meth. Appl. Sci. 14, 1683 (2004). Link, Web of Science, Google Scholar
L. Arlotti, N. Bellomo and E. De Angelis, Math. Models Meth. Appl. Sci. 12, 567 (2002). Link, Web of Science, Google Scholar
N. Bellomo and L. Preziosi, Math. Comp. Model. 31, 413 (2000). Crossref, Web of Science, Google Scholar
N. Bellomo and M. Pulvirenti (eds.) , Modelling in Applied Sciences: A Kinetic Theory Approach ( Birkhäuser , 2000 ) . Crossref, Google Scholar
N. Bellomo and M. Pulvirenti (eds.), Math. Comp. Model. 12, 909 (2002). Google Scholar
R. Bürger , The Mathematical Theory of Selection, Recombination and Mutation ( Wiley , 2000 ) . Google Scholar
E. De Angelis and P. E. Jabin, Math. Mod. Meth. Appl. Sci. 13, 187 (2003). Link, Web of Science, Google Scholar
E. De Angelis and L. Mesin, Math. Mod. Meth. Appl. Sci. 11, 1619 (2001). Google Scholar
L. Desvillettes and C. Prévost , Modelling in population dynamics through kinetic-like equations ( ) . Google Scholar
L. Desvillettes , C. Prévost and R. Ferrieres , Infinite dimensional reaction-diffusion for population dynamics Prépublication du CMLA . Google Scholar
O. Dieckmann and J. P. Heesterbeek , Mathematical Epidemiology of Infectious Diseases ( Wiley , 2000 ) . Google Scholar
O. Dieckmann et al. , Adaptative Dynamics without Time Scale Separation ( ) . Google Scholar
G. Forni et al. , Cytokine Induced Tumor Immunogeneticity ( Academic Press , 1994 ) . Google Scholar
H. Frid, P. E. Jabin and B. Pertame, Math. Model. Numer. Anal. 4, 709 (2003). Web of Science, Google Scholar
L. Greller, F. Tobin and G. Poste, Invasion and Metastasis 16, 177 (1996). Google Scholar
R. M. May and M. A. Noak , Virus Dynamics (Mathematical principles of immunology and virology) ( Oxford Univ. Press , 2000 ) . Google Scholar
A. S. Perelson and G. Weisbuch, Rev. Mod. Phys. 69, 1219 (1997). Crossref, Web of Science, Google Scholar
J. Saldana, S. F. Elana and R. V. Solé, Math. Biosci. 183, 135 (2003). Crossref, Web of Science, Google Scholar
C. H. Taubes , Modelling Lectures on Differential Equations in Biology ( Prentice-Hall , 2001 ) . Google Scholar
C. Villani, Handbook of Fluid Mechanics 1, eds. S. Friedlander and D. Serre (North-Holland, 2000) pp. 71–305. Google Scholar
D. Waxman, Contemp. Phys. 43, 13 (2002). Crossref, Web of Science, Google Scholar