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Qualitative analysis of second-order models of tumor-immune system competition. (English) Zbl 1145.34303
Summary: This paper deals with the qualitative analysis, existence of equilibria and asymptotic behavior of some second-order models of the competition between tumor and immune cells. The background model belongs to {\it A. d’Onofrio} [Physica D 208, No. 3--4, 220--235 (2005; Zbl 1087.34028); Math. Models Methods Appl. Sci. 16, No. 8, 1375--1401 (2006; Zbl 1094.92040)]. Various developments proposed in this paper are focussed on the hiding-learning dynamics, followed by the qualitative analysis.

34A34Nonlinear ODE and systems, general
92C50Medical applications of mathematical biology
92D25Population dynamics (general)
Full Text: DOI
[1] D’onofrio, A.: A general framework for modeling tumor--immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Physica D 208, 220-235 (2005) · Zbl 1087.34028 · doi:10.1016/j.physd.2005.06.032
[2] D’onofrio, A.: Tumor--immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. models methods appl. Sci. 16, 1375-1401 (2006) · Zbl 1094.92040 · doi:10.1142/S0218202506001571
[3] Bellomo, N.; Forni, G.: Dynamics of tumor interaction with the host immune system, Math. comput. Modelling 20, 107-122 (1994) · Zbl 0811.92014 · doi:10.1016/0895-7177(94)90223-2
[4] Kolev, M.: A mathematical model of cellular immune response to leukemia, Math. comput. Modelling 41, 1071-1082 (2005) · Zbl 1080.92042 · doi:10.1016/j.mcm.2005.05.003
[5] Byrne, H.; Alarcon, T. A.; Murphy, J.; Maini, P. K.: Modelling the response of vascular tumors to chemotherapy: A multiscale approach, Math. models methods appl. Sci. 16, 1219-1241 (2006) · Zbl 1094.92038 · doi:10.1142/S0218202506001522
[6] Kheifetz, A.; Kogan, Y.; Agur, Z.: Long-range predictability in models of cell populations subjected to phase-specific drugs: growth-rate approximation using properties of positive compact operators, Math. models methods appl. Sci. 16, 1155-1172 (2006) · Zbl 1094.92022 · doi:10.1142/S0218202506001492
[7] Wheldon, T. E.: Mathematical models in cancer research, (1988) · Zbl 0696.92002
[8] Stepanova, N. V.: Course of the immune reaction during the development of a malignant tumor, Biophysics 24, 917-923 (1980)
[9] Lollini, P. L.; Motta, S.; Pappalardo, F.: Modeling tumor immunology, Math. models methods appl. Sci. 16, 1091-1125 (2006) · Zbl 1099.92036 · doi:10.1142/S0218202506001479
[10] Galach, M.: Dynamics of the tumor--immune system competition: the effect of time delay, Int. J. Appl. math. Comput. sci. 13, No. 3, 395-406 (2003) · Zbl 1035.92019
[11] Sotolongo-Costa, O.; Morales-Molina, L.; Rodriguez-Perez, D.; Antonranz, J. C.; Chacon-Reyes, M.: Behavior of tumors under nonstationary therapy, Phys. D 178, 242-253 (2003) · Zbl 1011.92028 · doi:10.1016/S0167-2789(03)00005-8
[12] De Vladar, H. P.; Gonzalez, J. A.: Dynamic response of cancer under the influence of immunological activity and therapy, J. theoret. Biol. 227, 335-348 (2004)
[13] Tao, Y.; Zang, H.: A parabolic--hyperbolic free boundary problem modeling tumor treatment with virus, Math. models methods appl. Sci. 17, 63-80 (2007) · Zbl 1115.35137 · doi:10.1142/S0218202507001838
[14] Gatenby, R. A.; Vincent, T. L.; Gillies, R. J.: Evolutionary dynamics in carcinogenesis, Math. models methods appl. Sci. 15, 1619-1638 (2005) · Zbl 1077.92031 · doi:10.1142/S0218202505000911
[15] Brú, A.; Herrero, M. A.: From the physical laws of tumor growth to modelling cancer processes, Math. models method appl. Sci. 16, 1199-1218 (2006) · Zbl 1094.92037 · doi:10.1142/S0218202506001510
[16] Bellouquid, A.; Delitala, M.: Modelling complex multicellular systems -- A kinetic theory approach, (2005) · Zbl 1093.82016
[17] Cooke, K. L.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194
[18] Gourley, S. A.; Kuang, Y.: A stage structured predator--prey model and its dependence on maturation delay and death rate, J. math. Biol. 49, 188-200 (2004) · Zbl 1055.92043 · doi:10.1007/s00285-004-0278-2
[19] Villasana, M.; Radunskaya, A.: A delay differential equation model for tumor growth, J. math. Biol. 47, No. 3, 270-294 (2003) · Zbl 1023.92014 · doi:10.1007/s00285-003-0211-0
[20] Foryś, U.: Marchuk’s model of immune system dynamics with application to tumor growth, J. theor. Med. 4, No. 1, 85-93 (2002) · Zbl 1059.92031 · doi:10.1080/10273660290015215
[21] Galach, M.: Dynamics of the tumor--immune system competition: the effect of time delay, Int. J. Appl. comput. Sci. 13, No. 3, 395-406 (2003) · Zbl 1035.92019
[22] Cattani, C.; Ciancio, A.: Hybrid two scales mathematical tools for active particles modelling complex systems with learning hiding dynamics, Math. models methods appl. Sci. 17, 171-187 (2007) · Zbl 1142.82019 · doi:10.1142/S0218202507001875
[23] Bellomo, N.; Forni, G.: Looking for new paradigms towards a biological--mathematical theory of complex multicellular systems, Math. models methods appl. Sci. 16, 1001-1029 (2006) · Zbl 1093.92002 · doi:10.1142/S0218202506001443
[24] Bellomo, N.; Maini, P.: Preface, Math. models methods appl. Sci. 16, iii-vii (2006)
[25] Bellomo, N.; Bellouquid, A.: On the onset of nonlinearity for diffusion models of binary mixtures of biological materials by asymptotic analysis, Int. J. Nonlinear mech. 41, 281-293 (2006) · Zbl 1160.76403 · doi:10.1016/j.ijnonlinmec.2005.07.006
[26] Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J.: Multicellular biological growing systems -- hyperbolic limits towards macroscopic description, Math. models methods appl. Sci. 17 (2007) · Zbl 1135.92009 · doi:10.1142/S0218202507002431