Persistent and susceptible bacteria with individual deaths

The aim of this paper is to study two models for a bacterial population subject to antibiotic treatments. It is known that some bacteria are sensitive to antibiotics. These bacteria are in a state called persistence and each bacterium can switch from this state to a non-persistent (or susceptible) state and back. Our models extend those introduced in [6] by adding a (random) natural life cycle for each bacterium and by allowing bacteria in the susceptible state to escape the action of the antibiotics with a fixed probability 1-p (while every bacterium in a persistent state survives with probability 1). In the first model we "inject" the antibiotics in the system at fixed, deterministic times while in the second one the time intervals are random. We show that, in order to kill eventually the whole bacterial population, these time intervals cannot be "too large". The maximum admissible length is increasing with respect to p and it decreases rapidly when p<1.