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.
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