Abstract

VALLE, Paul A.; CORIA, Luis N.  and  STARKOV, Konstantin E.. Study of the Global Dynamics for a Tumor Immune-Evasion System. Comp. y Sist. [online]. 2014, vol.18, n.4, pp.773-786. ISSN 2007-9737.  https://doi.org/10.13053/CyS-18-4-1920.

In this paper we study the global dynamics for a Tumor Immune-Evasion model proposed by Arciero, Jackson and Kirschner [1], which describes the interaction between effector cells, cancer cells, and the cytokines IL -2 and TGF - β in the tumor site. This system describes different behaviors such as equilibrium points, periodic orbits, and stable limit cycles. By using the Localization of Compact Invariant Sets method, we define a domain where all the dynamics of the Immune-Evasion system are located. The localization of these sets is important because they provide information about the individual's health in the short and long term. The domain boundaries are expressed by inequalities depending on the system's parameters and represent the minimum and maximum values of the system variables. Furthermore, by taking a Lyapunov candidate function, we demonstrate that the localizing region is a positively invariant domain. This ensures that for any initial condition outside this domain, the trajectories of the system will not diverge. Finally, we present numerical simulations and realize an analysis of possible biological implications of our results.

Keywords : Compact invariant set; bounded positively invariant domain; Lyapunov function; cancer; biological system.