Influence of the age structure on the stability in a tumor-immune model for chronic myeloid leukemia

Abstract

In this paper a model of tumor-immune response for chronic myeloid leukemia (CML) is proposed and analyzed. It is based on the ordinary differential equations’ models (ODE) studied in A. Besse, et al., Bull. Math. Biol. (2017) 1—27, G.D. Clapp, et al., Cancer Res. 75 (2015) 4053-4062. The proliferation of cells, their differentiation in the bone marrow and the interactions of leukemic and immune cells are described. The model is based on a non-monotonic immune response. At low levels immune response increases with the tumor load, whereas at high levels tumor is suppressing the effect of the immune system (immunosuppression). We consider that the age of cells is described by a continuous variable which we use to structure the system and obtain a partial differential equations’ model (PDEs). We analyze the stability of the equilibrium points of the model and compare it to the case of A. Besse, et al., Bull. Math. Biol. (2017) 1—27 where age was described as a discrete state. In particular, an equilibrium point describing remission, induced by a control of the immune system, is shown to be unstable in certain situations for the PDE model, whereas in A. Besse, et al., Bull. Math. Biol. (2017) 1-27 it was systematically stable.