Two golden times in two-step contagion models

The two-step contagion model is a simple toy model for understanding pandemic outbreaks that occur in the real world. The model takes into account that a susceptible person either gets immediately infected or weakened when getting into contact with an infectious one. As the number of weakened people increases, they eventually can become infected in a short time period and a pandemic outbreak occurs. The time required to reach such a pandemic outbreak allows for intervention and is often called golden time. Understanding the size-dependence of the golden time is useful for controlling pandemic outbreak. Here we find that there exist two types of golden times in the two-step contagion model, which scale as $O(N^{1/3})$ and $O(N^{\zeta})$ with the system size $N$ on Erd\H{o}s-R\'enyi networks, where the measured $\zeta$ is slightly larger than $1/4$. They are distinguished by the initial number of infected nodes, $o(N)$ and $O(N)$, respectively. While the exponent $1/3$ of the $N$-dependence of the golden time is universal even in other models showing discontinuous transitions induced by cascading dynamics, the measured $\zeta$ exponents are all close to $1/4$ but show model-dependence. It remains open whether or not $\zeta$ reduces to $1/4$ in the asymptotically large-$N$ limit.