  In this paper we study the appearance of bifurcations of limit cycles in an
epidemic model with two types of aware individuals. All the transition rates
are constant except for the alerting decay rate of the most aware individuals
and the rate of creation of the less aware individuals, which depend on the
disease prevalence in a non-linear way. For the ODE model, the numerical
computation of the limit cycles and the study of their stability are made by
means of the Poincar\'e map. Moreover, sufficient conditions for the existence
of an endemic equilibrium are also obtained. These conditions involve a rather
natural relationship between the transmissibility of the disease and that of
awareness. Finally, stochastic simulations of the model under a very low rate
of imported cases are used to confirm the scenarios of bistability (endemic
equilibrium and limit cycle) observed in the solutions of the ODE model.
