Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 430486, 11 pages
http://dx.doi.org/10.1155/2013/430486
Research Article

A Class of Solutions for the Hybrid Kinetic Model in the Tumor-Immune System Competition

1Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
2Depatment of Mathematics and Computer Science, University of Messina, Viale Ferdinando d'Alcontres 31, 98166 Messina, Italy

Received 6 March 2013; Accepted 7 April 2013

Academic Editor: Ezzat G. Bakhoum

Copyright © 2013 Carlo Cattani and Armando Ciancio. 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. N. Bellomo, Modeling Complex Living Systems—Kinetic Theory and Stochastic Game Approach, Springer, Boston, Mass, USA, 2008. View at Zentralblatt MATH · View at MathSciNet
  2. A. Bellouquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Springer,, Boston, Mass, USA, 2006. View at Zentralblatt MATH · View at MathSciNet
  3. C. Bianca and N. Bellomo, Towards a Mathematical Theory of Multiscale Complex Biological Systems, World Scientific, Singapore, 2010.
  4. C. Cattani and A. Ciancio, “Qualitative analysis of second-order models of tumor-immune system competition,” Mathematical and Computer Modelling, vol. 47, no. 11-12, pp. 1339–1355, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. Cattani and A. Ciancio, “Hybrid two scales mathematical tools for active particles modelling complex systems with learning hiding dynamics,” Mathematical Models and Methods in Applied Sciences, vol. 17, no. 2, pp. 171–187, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Cattani and A. Ciancio, “Separable transition density in the hybrid model for tumor-immune system competition,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 610124, 6 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Abrams, “The evolution of predator-prey systems: theory and evidence,” Annual Review of Ecology, Evolution, and Systematics, vol. 31, Article ID 79105, 2000. View at Google Scholar
  8. H. P. de Vladar and J. A. González, “Dynamic response of cancer under the influence of immunological activity and therapy,” Journal of Theoretical Biology, vol. 227, no. 3, pp. 335–348, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. d'Onofrio, “A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,” Physica D, vol. 208, no. 3-4, pp. 220–235, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. d'Onofrio, “Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,” Mathematical and Computer Modelling, vol. 47, no. 5-6, pp. 614–637, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. U. Fory's, “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
  12. M. Galach, “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
  13. R. A. Gatenby, T. L. Vincent, and R. J. Gillies, “Evolutionary dynamics in carcinogenesis,” Mathematical Models & Methods in Applied Sciences, vol. 15, no. 11, pp. 1619–1638, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. A. Gourley and Y. Kuang, “A stage structured predator-prey model and its dependence on maturation delay and death rate,” Journal of Mathematical Biology, vol. 49, no. 2, pp. 188–200, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Kolev, “A mathematical model of cellular immune response to leukemia,” Mathematical and Computer Modelling, vol. 41, no. 10, pp. 1071–1081, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. 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 Google Scholar · View at Scopus
  17. N. V. Stepanova, “Course of the immune reaction during the development of a malignant tumour,” Biophysics, vol. 24, no. 5, pp. 917–923, 1979. View at Google Scholar · View at Scopus
  18. O. Sotolongo-Costa, L. Morales Molina, D. Rodríguez Perez, J. C. Antoranz, and M. Chacón Reyes, “Behavior of tumors under nonstationary therapy,” Physica D, vol. 178, no. 3-4, pp. 242–253, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. E. Wheldon, Mathematical Models in Cancer Research, Hilger, Boston, Mass, USA, 1988.
  20. H. R. Thieme, Mathematics in Population Biology, Princeton, NJ, USA, 2003. View at MathSciNet