Multiple peaks patterns of epidemic spreading in multi-layer networks.

The study of epidemic spreading on populations of networked individuals has seen recently a great deal of significant progresses. A common point in many of past studies is, however, that there is only one peak of infected density in each single epidemic spreading episode. At variance, real data from different cities over the world suggest that, besides a major single peak trait of infected density, a finite probability exists for a pattern made of two (or multiple) peaks. We show that such a latter feature is distinctive of a multilayered network of interactions, and reveal that a two peaks pattern may emerge from different time delays at which the epidemic spreads in between the two layers. Further, we show that the essential ingredient is a weak coupling condition between the layers themselves, while different degree distributions in the two layers are also helpful. Moreover, an edge-based theory is developed which fully explains all numerical results. Our findings may therefore be of significance for protecting secondary disasters of epidemics, which are definitely undesired in real life.