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.

A major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing, disease-free state at sufficiently long times due to the finite size of the networks where epidemics evolves. In the present work, we compare different quasistationary (QS) simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved. We analyze the standard QS (SQS) method, where the sampling is constrained to active configurations, the reflecting boundary condition (RBC), where the dynamics returns to the pre-absorbing configuration, and hub reactivation (HR), where the most connected vertex of the network is reactivated after a visit to an absorbing state. We apply the methods to the contact process (CP) and susceptible-infected-susceptible (SIS) models on regular and scale free networks. The investigated methods yield the same epidemic threshold for both models. For CP, that undergoes a standard collective phase transition, the methods are equivalent. For SIS, whose phase transition is ruled by the hub mutual reactivation, the SQS and HR methods are able to capture localized epidemic phases while RBC is not. We also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite-size scaling exponents for the critical relaxation time for the investigated methods. Finally, we verify the equivalence between RBC method and a weak external field for epidemics on networks.

We instigate the properties of the threshold contact process (TCP), a process showing an absorbing-state phase transition with infinitely many absorbing states, on random complex networks. The finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field (HMF) approximation and compared with extensive simulations, particularly in the case of heterogeneous scale-free networks. We observe that the TCP exhibits the same critical properties as the contact process (CP), which undergoes an absorbing-state phase transition to a single absorbing state. The accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a HMF theory is a universal feature of absorbing state phase transitions in complex networks, depending only on the locality of the interactions and independent of the number of absorbing states. The conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible-infected-susceptible epidemic spreading model, which in fact belongs to a different universality class in complex networks.

Abstract. We present quasi-stationary simulations of three-dimensional models with a single absorbing configuration, viz. the contact process (CP), the susceptibleinfected-susceptible (SIS) and the contact replication process (CRP). The moment ratios of the order parameters for DP class in three dimensions were set up using the well established SIS and CP models. We also show that the mean-field exponents in d = 3 reported previously for CRP [Ferreira SC 2005 Phys. Rev Quasi-stationary simulations of DP class in d = 3 2 IntroductionPhase transitions to a single absorbing configuration, a state in which the system can not scape from, are nowadays a topic in the frontier of Nonequilibrium Statistical Physics [1,2]. Concomitantly with the increasing interest on absorbing/active phase transitions in complex topologies [3,4,5,6,7], there are still a lot of open problems being intensively investigated on regular lattices such as the effects of quenched disorder [8,9,10], diffusion [11], as well the modeling of predator-prey systems [12], and clonal replication [13,14].Under the renormalization group point of view, it is expected [1,15,16] that the absorbing phase transitions in models with a positive one-component order parameter, short-range interactions and without additional symmetries or quenched disorder belong generally to the universality class of directed percolation (DP). This conjecture is known as Janssen-Grassberger criterion [1]. It is worthwhile to mention, the interest on this kind of phase transitions was raised by the recent experimental observation of the DP class in absorbing-state phase transitions [18,19]. On the other hand, while DP is considered the most robust universality class of the absorbing-state phase transitions, the precise numerical determination of the critical exponents of a specific model can be masked by factors like diffusion [11] and weak quenched disorder [10].The contact process (CP), the standard example of the DP universality class, is a toy model of epidemics [20]. ‡. More recently, a novel variation of the CP was introduced for the modeling of clonal (copies of themselves) replication, the contact replication process (CRP) [13,14]. Since neither additional symmetries nor long-range interactions were included, the CRP fulfils the requirements of the Janssen-Gassberger criterion. However, the first dynamic spreading analysis of CRP in d = 1 − 3 dimensions, reported in [13,14] intriguingly classified the model in the DP universality class in one and two, but not in three dimensions. Surprisingly, in d = 3 the reported spreading exponents were those predicted by the mean-field approach [14].In the present work we applied spreading analysis and the method of quasistationary simulations [22,23] in three-dimensional models that fulfill the JanssenGrassberger criterion. Particularly, we turned back to the CRP model and showed that the mean-field behaviour observed previously in d = 3 [14] is a transient associated to the closeness between critical creation and annihilation events. ...