  The evolution of network structure and the spreading of epidemic are common
coexistent dynamical processes. In most cases, network structure is treated
either static or time-varying, supposing the whole network is observed in a
same time window. In this paper, we consider the epidemic spreading on a
network consisting of both static and time-varying structures. At meanwhile,
the time-varying part and the epidemic spreading are supposed to be of the same
time scale. We introduce a static and activity driven coupling (SADC) network
model to characterize the coupling between static (strong) structure and
dynamic (weak) structure. Epidemic thresholds of SIS and SIR model are studied
on SADC both analytically and numerically with various coupling strategies,
where the strong structure is of homogeneous or heterogeneous degree
distribution. Theoretical thresholds obtained from SADC model can both recover
and generalize the classical results in static and time-varying networks. It is
demonstrated that weak structures can make the epidemics break out much more
easily in homogeneous coupling but harder in heterogeneous coupling when
keeping same average degree in SADC networks. Furthermore, we show there exists
a threshold ratio of the weak structure to have substantive effects on the
breakout of the epidemics. This promotes our understanding of why epidemics can
still break out in some social networks even we restrict the flow of the
population.
