Global stability and tumor clearance conditions for a cancer chemotherapy system. (English) Zbl 1510.92104
Summary: In this paper we study the global dynamics of a cancer chemotherapy system presented by L. G. de Pillis et al. [Math. Biosci. 209, No. 1, 292–315 (2007; Zbl 1122.49020)]. This mathematical model describes the interaction between tumor cells, effector-immune cells, circulating lymphocytes and chemotherapy treatment. By applying the localization method of compact invariant sets, we find lower and upper bounds for these three cells populations. Further, we define a bounded domain in \(\mathbb{R}_{+, 0}^4\) where all compact invariant sets of the system are located and provide conditions under which this domain is positively invariant. We apply LaSalle’s invariance principle and one result concerning two-dimensional competitive systems in order to derive sufficient conditions for tumor clearance and global asymptotic stability of the tumor-free equilibrium point. These conditions are computed by using bounds of the localization domain and they are given in terms of the chemotherapy treatment. Finally, we perform numerical simulations in order to illustrate
our results.
Citations:
Zbl 1122.49020References:
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