  We study epidemic arrival times in meta-population disease models through the
lens of front propagation into unstable states. We demonstrate that several
features of invasion fronts in the PDE context are also relevant to the network
case. We show that the susceptible-infected-recovered model on a network is
linearly determined in the sense that the arrival times in the nonlinear system
are approximated by the arrival times of the instability in the system
linearized near the disease free state. Arrival time predictions are extended
to an susceptible-exposed-infected-recovered model. We then study a recent
model of social epidemics where high order interactions of individuals lead to
faster invasion speeds. For these pushed fronts we compute corrections to the
estimated arrival time in this case. Finally, we show how inhomogeneities in
local infection rates lead to faster average arrival times.
