Critical behavior of a two-step contagion model with multiple seeds.

A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of O(N) where N is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here we show analytically and numerically that when the density of infectious seeds is relatively small but O(1), the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.