  Most previous studies of epidemic dynamics on complex networks suppose that
the disease will eventually stabilize at either a disease-free state or an
endemic one. In reality, however, some epidemics always exhibit sporadic and
recurrent behaviour in one region because of the invasion from an endemic
population elsewhere. In this paper we address this issue and study a
susceptible-infected-susceptible epidemiological model on a network consisting
of two communities, where the disease is endemic in one community but
alternates between outbreaks and extinctions in the other. We provide a
detailed characterization of the temporal dynamics of epidemic patterns in the
latter community. In particular, we investigate the time duration of both
outbreak and extinction, and the time interval between two consecutive
inter-community infections, as well as their frequency distributions. Based on
the mean-field theory, we theoretically analyze these three timescales and
their dependence on the average node degree of each community, the transmission
parameters, and the number of intercommunity links, which are in good agreement
with simulations, except when the probability of overlaps between successive
outbreaks is too large. These findings aid us in better understanding the
bursty nature of disease spreading in a local community, and thereby suggesting
effective time-dependent control strategies.
