  The massive employment of computational models in network epidemiology calls
for the development of improved inference methods for epidemic forecast. For
simple compartment models, such as the Susceptible-Infected-Recovered model,
Belief Propagation was proved to be a reliable and efficient method to identify
the origin of an observed epidemics. Here we show that the same method can be
applied to predict the future evolution of an epidemic outbreak from partial
observations at the early stage of the dynamics. The results obtained using
Belief Propagation are compared with Monte Carlo direct sampling in the case of
SIR model on random (regular and power-law) graphs for different observation
methods and on an example of real-world contact network. Belief Propagation
gives in general a better prediction that direct sampling, although the quality
of the prediction depends on the quantity under study (e.g. marginals of
individual states, epidemic size, extinction-time distribution) and on the
actual number of observed nodes that are infected before the observation time.
