  A Belief Propagation approach has been recently proposed for the zero-patient
problem in a SIR epidemics. The zero-patient problem consists in finding the
initial source of an epidemic outbreak given observations at a later time. In
this work, we study a harder but related inference problem, in which
observations are noisy and there is confusion between observed states. In
addition to studying the zero-patient problem, we also tackle the problem of
completing and correcting the observations possibly finding undiscovered
infected individuals and false test results. Moreover, we devise a set of
equations, based on the variational expression of the Bethe free energy, to
find the zero patient along with maximum-likelihood epidemic parameters. We
show, by means of simulated epidemics, how this method is able to infer details
on the past history of an epidemic outbreak based solely on the topology of the
contact network and a single snapshot of partial and noisy observations.
