Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

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.