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

Alexander P. Krishchenko; Konstantin E. Starkov

doi:10.3934/mbe.2018057

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2018, Volume 15, Issue 5: 1243-1254. Doi: 10.3934/mbe.2018057

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

Alexander P. Krishchenko^1, and
Konstantin E. Starkov^2, ,

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

* Corresponding author: kstarkov@ipn.mx; konstarkov@hotmail.com
* Corresponding author: kstarkov@ipn.mx; konstarkov@hotmail.com

Received: January 03, 2017

Revised: March 06, 2018

Published: October 2018

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Abstract

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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Mathematics Subject Classification: Primary: 58F35, 58F17; Secondary: 53C35.

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