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The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?

1. Bauman Moscow State Technical University, 2-aya Baumanskaya ul., 5, Moscow 105005, Russia
2. Instituto Politecnico Nacional, CITEDI, Avenida IPN N 1310, Nueva Tijuana, Tijuana 22510, B.C., Mexico

In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters $s_{1, 2}$ under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on $s_{1, 2}$ under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.

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[1] J. C. Arciero,T. L. Jackson,D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dynamic. Syst. Ser. B, 4 (2004): 39-58.

[2] D. Kirschner,J. Panetta, Modelling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998): 235-252.

[3] D. Kirschner,A. V. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009): 573-583.

[4] A. P. Krishchenko, Estimations of domains with cycles, Comput. & Math. Appl., 34 (1997): 325-332.

[5] A. P. Krishchenko, Localization of invariant compact sets of dynamical systems, Differential Equations, 41 (2005): 1669-1676.

[6] A. P. Krishchenko,K. E. Starkov, Localization of compact invariant sets of the Lorenz system, Phys. Lett. A, 353 (2006): 383-388.

[7] A. P. Krishchenko,K. E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Commun. Nonlin. Sci. Numer. Simul., 33 (2016): 174-183.

[8] F. Salazar-Onfray, Interleukin-10: A cytokine used by tumors to escape immunosurveillance, Medical Oncology, 16 (1999): 86-94.

[9] K. E. Starkov, On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Phys. Lett. A, 382 (2018): 387-393.

[10] K. E. Starkov,S. Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016): 1059-1075.

[11] K. E. Starkov,D. Gamboa, Localization of compact invariant sets and global stability in analysis of one tumor growth model, Mathematical Methods in the Applied Sciences, 37 (2014): 2854-2863.

[12] K. E. Starkov and L. Jimenez Beristain, Dynamic analysis of the melanoma model: from cancer persistence to its eradication Internat. J. Bifur. Chaos Appl. Sci. Engrg. , 27 (2017), 1750151, 11pp.

[13] K. E. Starkov,A. P. Krishchenko, On the global dynamics of one cancer tumor growth model, Commun. Nonlin. Sci. Numer. Simul., 19 (2014): 1486-1495.

[14] K. E. Starkov,A. P. Krishchenko, Ultimate dynamics of the Kirschner-Panetta model: Tumor eradication and related problems, Phys. Lett. A, 381 (2017): 3409-3416.

[15] K. E. Starkov,A. Pogromsky, Global dynamics of the Owen-Sherratt model describing the tumor-macrophage interactions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013): 1350020, 9pp.

[16] K. E. de Visser,W. M. Kast, Effects of TGF-ß on the immune system: Implications for cancer immunotherapy, Journal of Immunotherapy, 20 (1997): 165-177.

[17] S. Wojtowicz-Praga, Reversal of tumor-induced immunosuppression: A new approach to cancer therapy, Journal of Immunotherapy, 20 (1997): 165-177.

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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