  The dynamics of epidemic spreading is often reduced to the single control
parameter $R_0$, whose value, above or below unity, determines the state of the
contagion. If, however, the pathogen evolves as it spreads, $R_0$ may change
over time, potentially leading to a mutation-driven spread, in which an
initially sub-pandemic pathogen undergoes a breakthrough mutation. To predict
the boundaries of this pandemic phase, we introduce here a modeling framework
to couple the network spreading patterns with the intra-host evolutionary
dynamics. For many pathogens these two processes, intra- and inter-host, are
driven by different selection forces. And yet here we show that even in the
extreme case when these two forces are mutually independent, mutations can
still fundamentally alter the pandemic phase-diagram, whose transitions are now
shaped, not just by $R_0$, but also by the balance between the epidemic and the
evolutionary timescales. If mutations are too slow, the pathogen prevalence
decays prior to the appearance of a critical mutation. On the other hand, if
mutations are too rapid, the pathogen evolution becomes volatile and, once
again, it fails to spread. Between these two extremes, however, we identify a
broad range of conditions in which an initially sub-pandemic pathogen can break
through to gain widespread prevalence.
