  Discrete- and continuous-time approaches are frequently used to model the
role of heterogeneity on dynamical interacting agents on the top of complex
networks. While, on the one hand, one does not expect drastic differences
between these approaches, and the choice is usually based on one's expertise or
methodological convenience, on the other hand, a detailed analysis of the
differences is necessary to guide the proper choice of one or another approach.
We tackle this problem, by comparing both discrete- and continuous-time
mean-field theories for the susceptible-infected-susceptible (SIS) epidemic
model on random networks with power-law degree distributions. We compare the
discrete epidemic link equations (ELE) and continuous pair quenched mean-field
(PQMF) theories with the corresponding stochastic simulations, both theories
that reckon pairwise interactions explicitly. We show that ELE converges to
PQMF theory when the time step goes to zero. The epidemic localization analysis
performed reckoning the inverse participation ratio (IPR) indicates that both
theories present the same localization depence on the network degree exponent
$\gamma$: for $\gamma<5/2$ the epidemic is localized on the maximum k-core of
network with a vanishing in the infinite-size limit while for $\gamma>5/2$, the
localization happens on hubs what leads to a finite value of IPR. However, the
IPR and epidemic threshold of ELE depend on the time-step discretization such
that a larger time-step leads to more localized epidemics. A remarkable
difference between discrete and continuous time approaches is revealed in the
epidemic prevalence near the epidemic threshold, in which the discrete-time
stochastic simulations indicate a mean-field critical exponent $\theta=1$
instead of the value $\theta=1/(3-\gamma)$ obtained rigorously and verified
numerically for the continuous-time SIS on the same networks.
