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