Modified quasispecies model: the analysis of a periodic therapy

We propose a modified mathematical model of the quasispecies type to analyze an unstable tumor progression evolution. In our study, we consider a heterogeneous population with different individuals, generated by the accumulation of successive mutations. Our model's main feature is that it allows for variable growth rates for each subpopulation and takes into account mutations from nonconsecutive types of mutants. Bifurcations and linear stability of the steady states are analyzed. We focus on two equilibria; one of them implies the coexistence of anomalous growth and genetically unstable cells. The other one yields the dominance of the anomalous growth population and the extinction of the malignant cells. However, linear stability analysis of the second equilibrium is inconclusive and suggests a suitable environment for the study of periodic therapy. This is carried out by introducing a small perturbation modeling the effect of a periodic medical treatment. As a result, a Zero-Hopf periodic orbit is identified, showing a cyclic behavior among the populations, with a strong dominance of the parental anomalous growth cell population.