  Metapopulation models provide the theoretical framework for describing
disease spread between different populations connected by a network. In
particular, these models are at the basis of most simulations of pandemic
spread. They are usually studied at the mean-field level by neglecting
fluctuations. Here we include fluctuations in the models by adopting fully
stochastic descriptions of the corresponding processes. This level of
description allows to address analytically, in the SIS and SIR cases, problems
such as the existence and the calculation of an effective threshold for the
spread of a disease at a global level. We show that the possibility of the
spread at the global level is described in terms of (bond) percolation on the
network. This mapping enables us to give an estimate (lower bound) for the
pandemic threshold in the SIR case for all values of the model parameters and
for all possible networks.
