We study two rumor processes on N, the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site 0 and ignorants at all the other sites of N, but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left.We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on N can be related to a suitably chosen discrete renewal process.Date: December 18, 2013.

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Abstract. We consider stochastic growth models to represent population subject to catastrophes. We analyze the subject from different set ups considering or not spatial restrictions, whether dispersion is a good strategy to increase the population viability. We find out it strongly depends on the effect of a catastrophic event, the spatial constraints of the environment and the probability that each exposed individual survives when a disaster strikes.

We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour, which started from the root of a tree, spreads out through an infinite number of its vertices. We present lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with Galton-Watson branching trees (homogeneous and non-homogeneous) and spherically symmetric trees which includes homogeneous and k−periodic trees.

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