  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 auto-correlation 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.
