  Host mobility plays a fundamental role in the spatial spread of infectious
diseases. Previous theoretical works based on the integration of network theory
into the metapopulation framework have shown that the heterogeneities that
characterize real mobility networks favor the propagation of epidemics.
Nevertheless, the studies conducted so far assumed the mobility process to be
either Markovian or non-Markovian with a fixed traveling time scale. Available
statistics however show that the time spent by travelers at destination is
characterized by wide fluctuations, ranging between a single day up to several
months. Such varying length of stay crucially affects the chance and duration
of mixing events among hosts and may therefore have a strong impact on the
spread of an emerging disease. Here, we present an analytical and computational
study of epidemic processes on a complex subpopulation network where travelers
have memory of their origin and spend a heterogeneously distributed time
interval at their destination. Through analytical calculations and numerical
simulations we show that the heterogeneity of the length of stay alters the
expression of the threshold between local outbreak and global invasion, and,
moreover, it changes the epidemic behavior of the system in case of a global
outbreak. Additionally, our theoretical framework allows us to study the effect
of changes in the traveling behavior in response to the infection, by
considering a scenario in which sick individuals do not leave their home
location. Finally, we compare the results of our non-Markovian framework with
those obtained with a classic Markovian approach and find relevant differences
between the two, in the estimate of the epidemic invasion potential, as well as
of the timing and the pattern of its spatial spread.
