  Epidemic processes on random graphs or networks are marked by localization of
activity that can trap the dynamics into a metastable state, confined to a
subextensive part of the network, before visiting an absorbing configuration.
Quasistationary (QS) method is a technique to deal with absorbing states for
finite sizes and has played a central role in the investigation of epidemic
processes on heterogeneous networks where localization is a hallmark. The
standard QS method possesses high computer and algorithmic complexity for large
systems besides parameters whose choice are not systematic. However, simpler
approaches, such as a reflecting boundary condition (RBC), are not able to
capture the localization effects as the standard QS method does. In the present
work, we propose a QS method that consists of reactivating nodes proportionally
to the time they were active along the preceding simulation. The method is
compared with the standard QS and RBC methods for the
susceptible-infected-susceptible model on complex networks, which is a
prototype of a dynamic process with strong and localization effects. We
verified that the method performs as well the as standard QS in all
investigated simulations, providing the same scaling exponents, epidemic
thresholds, and localized phases, thus overcoming the limitations of other
simpler approaches. We also report that the present method has significant
reduction of the computer and algorithmic complexity than the standard QS
methods. So, this method arises as a simpler and efficient tool to analyze
localization on heterogeneous structures through QS simulations.
