Identifying epidemic threshold by temporal profile of outbreaks on networks

Identifying epidemic threshold is of great significance in preventing and controlling disease spreading on real-world networks. Previous studies have proposed different theoretical and numerical approaches to determine the epidemic threshold for the susceptible-infected-recovered (SIR) model, but the numerical study of the critical points on networks by utilizing temporal characteristics of epidemic outbreaks is still lacking. Here, we study the temporal profile of epidemic outbreaks, i.e., the average avalanche shapes of a fixed duration. At the critical point, the rescaled average terminating and nonterminating avalanche shapes for different durations collapse onto two universal curves, respectively, while the average number of subsequent events essentially remains constant. We propose two numerical measures to determine the epidemic threshold by analyzing the convergence of the rescaled average nonterminating avalanche shapes for varying durations and the stability of the average number of subsequent events, respectively. Extensive numerical simulations demonstrate that our methods can accurately identify the numerical threshold for the SIR dynamics on synthetic and empirical networks. Compared with traditional numerical measures, our methods are more efficient due to the constriction of observation duration and thus are more applicable to large-scale networks. This work helps one to understand the temporal profile of disease propagation and would promote further studies on the phase transition of epidemic dynamics.