  The metapopulation framework is adopted in a wide array of disciplines to
describe systems of well separated yet connected subpopulations. The subgroups
or patches are often represented as nodes in a network whose links represent
the migration routes among them. The connections have been so far mostly
considered as static, but in general evolve in time. Here we address this case
by investigating simple contagion processes on time-varying metapopulation
networks. We focus on the SIR process and determine analytically the mobility
threshold for the onset of an epidemic spreading in the framework of
activity-driven network models. We find profound differences from the case of
static networks. The threshold is entirely described by the dynamical
parameters defining the average number of instantaneously migrating individuals
and does not depend on the properties of the static network representation.
Remarkably, the diffusion and contagion processes are slower in time-varying
graphs than in their aggregated static counterparts, the mobility threshold
being even two orders of magnitude larger in the first case. The presented
results confirm the importance of considering the time-varying nature of
complex networks.
