Dynamics and bifurcations in a simple quasispecies model of unstable tumorigenesis

Abstract

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behaviour, at least qualitatively, also allowing us to make predictions. Here we analyse a simplified quasispecies mathematical model given by ODEs describing the behaviour of tumor cells populations. We find the equilibrium points, also characterizing their stability and bifurcations focusing on the study of replication and mutation parameters. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells population. Such a bifurcation involves an scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient time for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response during mutagenic therapies.