  We consider a general criterion to discern the nature of the threshold in
epidemic models on scale-free (SF) networks. Comparing the epidemic lifespan of
the nodes with largest degrees with the infection time between them, we propose
a general dual scenario, in which the epidemic transition is either ruled by a
hub activation process, leading to a null threshold in the thermodynamic limit,
or given by a collective activation process, corresponding to a standard phase
transition with a finite threshold. We validate the proposed criterion applying
it to different epidemic models, with waning immunity or heterogeneous
infection rates in both synthetic and real SF networks. In particular, a waning
immunity, irrespective of its strength, leads to collective activation with
finite threshold in scale-free networks with large exponent, at odds with
canonical theoretical approaches.
