  Diseases spread through host populations over the networks of contacts
between individuals, and a number of results about this process have been
derived in recent years by exploiting connections between epidemic processes
and bond percolation on networks. Here we investigate the case of two pathogens
in a single population, which has been the subject of recent interest among
epidemiologists. We demonstrate that two pathogens competing for the same hosts
can both spread through a population only for intermediate values of the bond
occupation probability that lie above the classic epidemic threshold and below
a second higher value, which we call the coexistence threshold, corresponding
to a distinct topological phase transition in networked systems.
