title: Collective versus hub activation of epidemic phases on networks.
abstract: 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 degree exponent, at odds with canonical theoretical approaches.
authors.affiliation: Departamento de Física, Universidade Federal de Viçosa, 36570-000 Viçosa, Minas Gerais, Brazil.
authors.forename: Silvio C
authors.initials: SC
authors.lastname: Ferreira
authors.affiliation: Departamento de Física, Universidade Federal de Viçosa, 36570-000 Viçosa, Minas Gerais, Brazil.
authors.forename: Renan S
authors.initials: RS
authors.lastname: Sander
authors.affiliation: Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain.
authors.forename: Romualdo
authors.initials: R
authors.lastname: Pastor-Satorras