A major hurdle in the simulation of the steady state of epidemic processes isthat the system will unavoidably visit an absorbing, disease-free state atsufficiently long times due to the finite size of the networks where epidemicsevolves. In the present work, we compare different quasistationary (QS)simulation methods where the absorbing states are suitably handled and thethermodynamical limit of the original dynamics can be achieved. We analyze thestandard QS (SQS) method, where the sampling is constrained to activeconfigurations, the reflecting boundary condition (RBC), where the dynamicsreturns to the pre-absorbing configuration, and hub reactivation (HR), wherethe most connected vertex of the network is reactivated after a visit to anabsorbing state. We apply the methods to the contact process (CP) andsusceptible-infected-susceptible (SIS) models on regular and scale freenetworks. The investigated methods yield the same epidemic threshold for bothmodels. For CP, that undergoes a standard collective phase transition, themethods are equivalent. For SIS, whose phase transition is ruled by the hubmutual reactivation, the SQS and HR methods are able to capture localizedepidemic phases while RBC is not. We also apply the auto-correlation time as atool to characterize the phase transition and observe that this analysisprovides the same finite-size scaling exponents for the critical relaxationtime for the investigated methods. Finally, we verify the equivalence betweenRBC method and a weak external field for epidemics on networks.
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