  Epidemic threshold is one of the most important features of the epidemic
dynamics. Through a lot of numerical simulations in classic
Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS)
models on various types of networks, we study the simulated identification of
epidemic thresholds on finite-size networks. We confirm that the susceptibility
measure goes awry for the SIR model due to the bimodal distribution of outbreak
sizes near the critical point, while the simulated thresholds of the SIS and
SIR models can be accurately determined by analyzing the peak of the epidemic
variability. We further verify the accuracy of theoretical predictions derived
by the heterogeneous mean-field theory (HMF) and the quenched mean-field theory
(QMF), by comparing them with the simulated threshold of the SIR model obtained
from the variability measure. The results show that the HMF prediction agrees
very well with the simulated threshold, except the case that the networks are
disassortive, in which the QMF prediction is more close to the simulated
threshold.
