  We study four discrete time stochastic systems on $\bbN$ modeling processes
of rumour spreading. The involved individuals can either have an active or a
passive role, speaking up or asking for the rumour. The appetite in spreading
or hearing the rumour is represented by a set of random variables whose
distributions may depend on the individuals. Our goal is to understand - based
on those random variables distribution - whether the probability of having an
infinite set of individuals knowing the rumour is positive or not.
