  This paper is concerned with the robustness of the sustained oscillations
predicted by an epidemic ODE model defined on contact networks. The model
incorporates the spread of awareness among individuals and, moreover, a small
inflow of imported cases. These cases prevent stochastic extinctions when we
simulate the epidemics and, hence, they allow to check whether the average
dynamics for the fraction of infected individuals are accurately predicted by
the ODE model. Stochastic simulations confirm the existence of sustained
oscillations for different types of random networks, with a sharp transition
from a non-oscillatory asymptotic regime to a periodic one as the alerting rate
of susceptible individuals increases from very small values. This abrupt
transition to periodic epidemics of high amplitude is quite accurately
predicted by the Hopf-bifurcation curve computed from the ODE model using the
alerting rate and the infection transmission rate for aware individuals as
tuning parameters.
