Data from: 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.

新发传染病需实现从断续传播链到持续传播链的转变，但这一关键转变未必与系统进入超临界状态同步发生。换言之，即便将病原体引入超临界系统，也仅以一定概率引发人群中相当比例的个体感染。因此，明确新发传染病首次大规模暴发的等待时长，远比判断系统何时进入超临界状态更为复杂。我们将传染病暴发视为一种动态分岔现象，并借助分岔延迟的概念，分析系统进入超临界状态后的暴发等待时长。具体而言，我们构建了含时变传播项且存在境外随机感染输入的SIR模型（Susceptible-Infected-Recovered Model）。我们推导了延迟时长的解析密度函数，且总体而言，该函数与随机模拟结果吻合良好。研究发现，核心参数为病原体引入速率与基本再生数（basic reproductive ratio）的变化速率。上述研究结果有助于我们理解真实的传染病暴发事件，且可被纳入旨在预测疾病传播风险的早期预警系统之中。