Two golden times in two-step contagion models: A nonlinear map approach

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. Using an approach based on a nonlinear mapping, here we find that there exist two types of golden times in the two-step contagion model, which scale as [Formula: see text] and [Formula: see text] with the system size [Formula: see text] on Erdős-Rényi networks, where the measured [Formula: see text] is slightly larger than [Formula: see text]. They are distinguished by the initial number of infected nodes, [Formula: see text] and [Formula: see text] , respectively. While the exponent [Formula: see text] of the [Formula: see text]-dependence of the golden time is universal even in other models showing discontinuous transitions induced by cascading dynamics, the measured [Formula: see text] exponents are all close to [Formula: see text] but show model-dependence. It remains open whether or not [Formula: see text] reduces to [Formula: see text] in the asymptotically large- [Formula: see text] limit. Our method can be applied to several models showing a hybrid percolation transition and gives insight into the origin of the two golden times.