Title: Relevance of backtracking paths in epidemic spreading on networks,
Abstract: The understanding of epidemics on networks has greatly benefited from the
recent application of message-passing approaches, which allow to derive exact
results for irreversible spreading (i.e. diseases with permanent acquired
immunity) in locally-tree like topologies. This success has suggested the
application of the same approach to reversible epidemics, for which an
individual can contract the epidemic and recover repeatedly. The underlying
assumption is that backtracking paths (i.e. an individual is reinfected by a
neighbor he/she previously infected) do not play a relevant role. In this paper
we show that this is not the case for reversible epidemics, since the neglect
of backtracking paths leads to a formula for the epidemic threshold that is
qualitatively incorrect in the large size limit. Moreover we define a modified
reversible dynamics which explicitly forbids direct backtracking events and
show that this modification completely upsets the phenomenology.