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Discrete Dynamics in Nature and Society
Volume 2006, Article ID 58463, 13 pages
http://dx.doi.org/10.1155/DDNS/2006/58463

Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response

Département de Mathématiques et Informatique, Faculté des Sciences, Université Chouaib Doukkali, El Jadida BP 20, Morocco

Received 20 December 2005; Accepted 11 April 2006

Copyright © 2006 Radouane Yafia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Adam and N. Bellomo, A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, Massachusetts, 1996.
  2. L. Arlotti, N. Bellomo, and E. De Angelis, “Generalized kinetic (Boltzmann) models: mathematical structures and applications,” Mathematical Models & Methods in Applied Sciences, vol. 12, no. 4, pp. 567–591, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. N. Bellomo and L. Preziosi, “Modelling and mathematical problems related to tumor evolution and its interaction with the immune system,” Mathematical and Computer Modelling, vol. 32, no. 3-4, pp. 413–452, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. N. Bellomo and M. Pulvirenti, Eds., Modeling in Applied Sciences. A Kinetic Theory Approach, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Massachusetts, 2000. View at Zentralblatt MATH · View at MathSciNet
  5. N. Bellomo and M. Pulvirenti, Eds., Special issue on the modeling in applied sciences by methods of transport and kinetic theory, Mathematical and Computer Modelling \textbf{12} (2002), 909–990.
  6. M. Bodnar and U. Foryś, “Behaviour of solutions to Marchuk's model depending on a time delay,” International Journal of Applied Mathematics and Computer Science, vol. 10, no. 1, pp. 97–112, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Bodnar and U. Foryś, “Periodic dynamics in a model of immune system,” Applicationes Mathematicae, vol. 27, no. 1, pp. 113–126, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, 2000. View at Zentralblatt MATH · View at MathSciNet
  9. H. M. Byrne, “The effect of time delays on the dynamics of avascular tumor growth,” Mathematical Biosciences, vol. 144, no. 2, pp. 83–117, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. A. J. Chaplain, Ed., Special issue on mathematical models for the growth, development and treatment of tumors, Mathematical Models and Methods in Applied Sciences \textbf{9} (1999).
  11. K. L. Cooke and Z. Grossman, “Discrete delay, distributed delay and stability switches,” Journal of Mathematical Analysis and Applications, vol. 86, no. 2, pp. 592–627, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Desvillettes and C. Prevots, “Modelling in population dynamics through kinetic-like equations,” preprint n. 99/19 of the University of Orléans, Département de mathématiques.
  13. L. Desvillettes, C. Prevots, and R. Ferrieres, “Infinite dimensional reaction-diffusion for population dynamics,” 2003. View at Google Scholar
  14. O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, 2000. View at MathSciNet
  15. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations, vol. 110 of Applied Mathematical Sciences, Springer, New York, 1995. View at Zentralblatt MATH · View at MathSciNet
  16. T. Faria and L. T. Magalhães, “Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation,” Journal of Differential Equations, vol. 122, no. 2, pp. 181–200, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. U. Foryś, “Marchuk's model of immune system dynamics with application to tumor growth,” Journal of Theoretical Medicine, vol. 4, no. 1, pp. 85–93, 2002. View at Google Scholar · View at Zentralblatt MATH
  18. U. Foryś and M. Kolev, “Time delays in proliferation and apoptosis for solid avascular tumor,” preprint Institute of Applied Mathematics and Machanics, no. RW 02-10 (110), Warsaw University (2002).
  19. U. Foryś and A. Marciniak-Czochra, “Delay logistic equation with diffusion,” in Proceedings of the 8th National Conference on Application of Mathematics in Biology and Medicine, pp. 37–42, Łajs, 2002.
  20. M. Gałach, “Dynamics of the tumor-immune system competition—the effect of time delay,” International Journal of Applied Mathematics and Computer Science, vol. 13, no. 3, pp. 395–406, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. L. Greller, F. Tobin, and G. Poste, “Tumor hetereogenity and progression: conceptual foundation for modeling,” Invasion and Metastasis, vol. 16, pp. 177–208, 1996. View at Google Scholar
  22. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, 1993. View at Zentralblatt MATH · View at MathSciNet
  23. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981. View at Zentralblatt MATH · View at MathSciNet
  24. D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor-immune interaction,” Journal of Mathematical Biology, vol. 37, no. 3, pp. 235–252, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, and A. S. Perelson, “Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis,” Bulletin of Mathematical Biology, vol. 56, no. 2, pp. 295–321, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. H. Mayer, K. S. Zȧnker, and U. der Heiden, “A basic mathematical model of the immune response,” Chaos, vol. 5, no. 1, pp. 155–161, 1995. View at Publisher · View at Google Scholar
  27. M. A. Nowak and R. M. May, Virus Dynamics. Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. View at MathSciNet
  28. A. S. Perelson and G. Weisbuch, “Immunology for physicists,” Reviews of Modern Physics, vol. 69, no. 4, pp. 1219–1267, 1997. View at Publisher · View at Google Scholar
  29. J. Waniewski and P. Zhivkov, “A simple mathematical model for tumor-immune system interactions,” in Proceedings of the 8th National Conference on Application of Mathematics in Biology and Medicine, pp. 149–154, Łajs, 2002.
  30. R. Yafia, “Dynamics, Hopf bifurcation and stability analysis in a model for tumor-immune system competition with one delay,” submitted.
  31. R. Yafia, “Hopf bifurcation in a delayed model for tumor-immune system competition,” submitted.
  32. R. Yafia, “Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response,” to appear in Discrete Dynamics in Nature and Society.