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| 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-alert-infected models without awareness decay. 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 this second epidemic threshold is not a generic property of this class of models. We show that the continuum of equilibria that characterizes the slow die-out dynamics collapses into a unique equilibrium when a constant 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. |
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