  We study the dynamics of a spatially structured model of worldwide epidemics
and formulate predictions for arrival times of the disease at any city in the
network. The model is comprised of a system of ordinary differential equations
describing a meta-population SIR compartmental model defined on a network where
each node represents a city and edges represent flight paths connecting cities.
Making use of the linear determinacy of the system, we consider spreading
speeds and arrival times in the system linearized about the unstable disease
free state and compare these to arrival times in the nonlinear system. Two
predictions are presented. The first is based upon expansion of the heat kernel
for the linearized system. The second assumes that the dominant transmission
pathway between any two cities can be approximated by a one dimensional lattice
or homogeneous tree and gives a uniform prediction for arrival times
independent of specific network features. We test these predictions on a real
network describing worldwide airline traffic.
