Mathematical Models for Self-Adaptive Response to Cancer Dynamics

We consider two minimal mathematical models for cancer dynamics and self-adaptation. We aim to capture the interplay between the rapid progression of cancer growth and the possibility to leverage and enhance self-adaptive defense mechanisms of an organism, e.g., motivated by immunotherapy. Yet, our two models are more abstract and generic encapsulating the essence of competition between rapid cancer growth and the speed of adaptation. First, we propose a four-dimensional ordinary differential equation model on a macroscopic level. In this model only for a combination of external input and speeding up adaptation, cancer does not reach critical levels. To study the adaptation speed-up in the initial phase, we switch to a microscopic probabilistic model. The microscopic model undergoes a sharp transition under variation of the self-adaptation probability. It is shown that a combination of temporal memory and rare stochastic positive adaptation events is crucial to move the sharp transition point to a desired regime.