Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks

Abstract: 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.