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We study the impact on shape parameters of an underlyingBienaym\'e-Galton-Watson branching process (height, width and first hittingtime), of having a non-spatial branching mechanism with infinite variance.Aiming at providing a comparative study of the spread of an epidemics whosedynamics is given by the modulus of a branching Brownian motion (BBM) we thenconsider spatial branching processes in dimension d, not necessarily integer.The underlying branching mechanism is then either a binary branching model orone presenting infinite variance. In particular we evaluate the chance p(x) ofbeing hit if the epidemics started away at distance x. We compute the large xtail probabilities of this event, both when the branching mechanism is regularand when it exhibits very large fluctuations.

On the spread of a branching Brownian motion whose offspring number has infinite variance