Invariant Compact Sets of Two-dimensional Restrictions for a Cancer Tumour Growth Model

Abstract

A three-dimensional model of cancer growth showing the interaction of immune, tumor and host (healthy) cells is considered. This model was proposed by L.G. de Pillis and A. Radunskaya and is based on the earlier work of Kuznetsov et al. The equilibrium points of this system were obtained, and assumptions were made about restrictions on the model parameters that do not violate the meaningful sense of the problem. The main attention was paid to the construction of localizing sets of two-dimensional restrictions, characterized by the absence of immune, healthy or tumor cells.When considering the two-dimensional restriction of the model in the absence of immune cells, equilibrium points were obtained and their types were explored. The research was carried out taking into account the introduced restrictions on the system parameters. It was found that in the absence of immune cells, the localizing set consists of one point (the equilibrium point of the system), which is a stable node and indicates the presence of tumor formation.In the two-dimensional restriction of the model in the absence of host cells (healthy cells), a compact localizing set was obtained. This restriction has from two to four equilibrium points. For any values of the system parameters the equilibrium points are contained in the localizing set. Conditions were found on the model parameters under which the localizing set turns out to be a segment of coordinate axis and coincides with the maximal invariant compact set. When exploring the model restriction in the absence of tumor cells, the only stable equilibrium point was obtained that corresponds to a healthy organism. All trajectories in the positive octant tend to the stable equilibrium point.

Keywords

• invariant compact set, localization problem, localizing set, equilibrium point