  The existence of a die-out threshold (different from the classic
disease-invasion one) defining a region of slow extinction of an epidemic has
been proved elsewhere for susceptible-aware-infectious-susceptible models
without awareness decay, through bifurcation analysis. By means of an
equivalent mean-field model defined on regular random networks, we interpret
the dynamics of the system in this region and prove that the existence of
bifurcation for this second epidemic threshold crucially depends on the absence
of awareness decay. We show that the continuum of equilibria that characterizes
the slow die-out dynamics collapses into a unique equilibrium when a constant
rate of awareness decay is assumed, no matter how small, and that the resulting
bifurcation from the disease-free equilibrium is equivalent to that of standard
epidemic models. We illustrate these findings with continuous-time stochastic
simulations on regular random networks with different degrees. Finally, the
behaviour of solutions with and without decay in awareness is compared around
the second epidemic threshold for a small rate of awareness decay.
