Waiting time to infectious disease emergence

Emerging diseases must make a transition from stuttering chains of transmission to sustained chains of transmission, but this critical transition need not coincide with the system becoming supercritical. That is, the introduction of infection to a supercritical system results in a significant fraction of the population becoming infected only with a certain probability. Understanding the waiting time to the first major outbreak of an emerging disease is then more complicated than determining when the system becomes supercritical. We treat emergence as a dynamic bifurcation, and use the concept of bifurcation delay to understand the time to emergence after a system becomes supercritical. Specifically, we consider an SIR model with a time-varying transmission term and random infections originating from outside the population. We derive an analytic density function for the delay times and find it to be, in general, in agreement with stochastic simulations. We find the key parameters to be the rate of introduction of infection and the rate of change of the basic reproductive ratio. These findings aid our understanding of real emergence events, and can be incorporated into early-warning systems aimed at forecasting disease risk.