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

Collective versus hub activation of epidemic phases on networks