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Separable transition density in the hybrid model for tumor-immune system competition. (English) Zbl 1234.92026
Summary: A hybrid model of the competition of tumor cells in the immune system is studied under suitable hypotheses. An explicit form for the equations is obtained in the case when the density function of the transition is expressed as the product of separable functions. A concrete application is given starting from a modified Lotka-Volterra system of equations.

92C50 Medical applications (general)
37N25 Dynamical systems in biology
Full Text: DOI
[1] N. Bellomo, A. Bellouquid, and M. Delitala, “Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition,” Mathematical Models and Methods in Applied Sciences, vol. 14, no. 11, pp. 1683-1733, 2004. · Zbl 1060.92029 · doi:10.1142/S0218202504003799
[2] N. Bellomo and G. Forni, “Looking for new paradigms towards a biological-mathematical theory of complex multicellular systems,” Mathematical Models and Methods in Applied Sciences, vol. 16, no. 7, pp. 1001-1029, 2006. · Zbl 1093.92002 · doi:10.1142/S0218202506001443
[3] N. Bellomo and A. Bellouquid, “On the onset of non-linearity for diffusion models of binary mixtures of biological materials by asymptotic analysis,” International Journal of Non-Linear Mechanics, vol. 41, no. 2, pp. 281-293, 2006. · Zbl 1160.76403 · doi:10.1016/j.ijnonlinmec.2005.07.006
[4] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, “Multicellular biological growing systems: hyperbolic limits towards macroscopic description,” Mathematical Models and Methods in Applied Sciences, vol. 17, no. 1, pp. 1675-1692, 2007. · Zbl 1135.92009 · doi:10.1142/S0218202507002431
[5] N. Bellomo and M. Delitala, “From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells,” Physics of Life Reviews, vol. 5, no. 4, pp. 183-206, 2008. · doi:10.1016/j.plrev.2008.07.001
[6] N. Bellomo, Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach, Birkhäuser, Boston, Mass, USA, 2008. · Zbl 1140.91007
[7] C. Bianca and N. Bellomo, Towards a Mathematical Theory of Multiscale Complex Biological Systems, World Scientific, Singapore, 2010. · Zbl 1286.92003
[8] 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. · Zbl 1087.34028 · doi:10.1016/j.physd.2005.06.032
[9] A. d’Onofrio, “Tumor-immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy,” Mathematical Models and Methods in Applied Sciences, vol. 16, no. 8, pp. 1375-1401, 2006. · Zbl 1094.92040 · doi:10.1142/S0218202506001571
[10] A. d’Onofrio, “Tumor evasion from immune control: strategies of a MISS to become a MASS,” Chaos, Solitons & Fractals, vol. 31, no. 2, pp. 261-268, 2007. · Zbl 1133.92016 · doi:10.1016/j.chaos.2005.10.006
[11] 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. · Zbl 1148.92026 · doi:10.1016/j.mcm.2007.02.032
[12] 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. · Zbl 0789.92019 · doi:10.1007/BF02460644
[13] N. V. Stepanova, “Course of the immune reaction during the development of a malignant tumour,” Biophysics, vol. 24, no. 5, pp. 917-923, 1979.
[14] Y. Tao and H. Zhang, “A parabolic-hyperbolic free boundary problem modelling tumor treatment with virus,” Mathematical Models and Methods in Applied Sciences, vol. 17, no. 1, pp. 63-80, 2007. · Zbl 1115.35137 · doi:10.1142/S0218202507001838
[15] V. A. Kuznetsov and G. D. Knott, “Modeling tumor regrowth and immunotherapy,” Mathematical and Computer Modelling, vol. 33, no. 12-13, pp. 1275-1287, 2001. · Zbl 1004.92021 · doi:10.1016/S0895-7177(00)00314-9
[16] C. Cattani, A. Ciancio, and B. Lods, “On a mathematical model of immune competition,” Applied Mathematics Letters, vol. 19, no. 7, pp. 686-691, 2006. · Zbl 1278.92017 · doi:10.1016/j.aml.2005.09.001
[17] 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. · Zbl 1142.82019 · doi:10.1142/S0218202507001875
[18] C. Cattani and A. Ciancio, “Third order model for tumor-immune system competition,” in Proceedings of the 4th International Colloquium Mathematics in Engineering and Numerical Physics, pp. 30-37, Bucharest, Romania, 2007. · Zbl 1132.34319
[19] 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. · Zbl 1145.34303 · doi:10.1016/j.mcm.2007.07.005
[20] C. Cattani, A. Ciancio, and A. d’Onofrio, “Metamodeling the learning-hiding competition between tumours and the immune system: a kinematic approach,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 62-69, 2010. · Zbl 1201.34071 · doi:10.1016/j.mcm.2010.01.012
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