  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.
