The local behavior around switching planes in a mathematical model to chemoimmunotherapy

Abstract

In this paper we consider a piecewise smooth mathematical model to chemoimmunotherapy. The positive octant of the 3D space (N,I,Q) is partitioned into 4 regions separated by the planes I=I0 and N=N0. The plane I=I0 (respectively, N=N0) controls if the immunotherapy (respectively, chemotherapy) is being applied or not. As a consequence, there are 4 smooth vector fields modeling each one of the scenarios. By means of the theory developed by piecewise smooth vector fields we are able to show the kind of contact between the trajectories of the smooth vector fields with the switching planes, the sliding motion defined over the planes and even the sliding motion defined over the simultaneous tangency lines of two smooth vector fields and the switching planes. This description is very useful in the understanding of the behavior of the treatment proposed.