From Outbreak to Endemicity or Control: Tracking First Passage Time in Infectious Diseases

This study investigates the transition from outbreak to endemic states or endemic control by analyzing the effective reproduction number R(t) within a stochastic epidemic framework. We derive the transition and stationary probability density functions of R(t), which are then used to characterize the first passage time (FPT) of R(t) across a boundary Z. These are later used to calculate the distribution of the first time that infection will dominate (or be controlled) in a population. Analytical expressions for some properties of the distribution provide quantitative measures of when endemic transition is likely to emerge under stochastic fluctuations. For a fixed threshold Z, the crossing time marks the moment R(t) first signals sustained endemic behavior. We further extend the analysis to a time-varying threshold Z(t) representing a moving barrier shaped by dynamic epidemiological or policy conditions. In this setting, crossing corresponds to the moment the disease process keeps pace with a shifting target. By linking epidemic dynamics with stochastic first-passage theory, this framework highlights how fluctuations and dynamic thresholds jointly determine the timing of epidemic transitions. The results offer probabilistic tools for anticipating epidemic resurgence, assessing intervention durability, and designing adaptive public health strategies. Applied to a rhinovirus calibration (daily units), our result demonstrates that reducing the per-capita transmission rate effectively suppresses the amplification of stochastic fluctuations and accelerates the transition of the system into a non-epidemic regime. We calibrate the framework, using region-specific parameter estimates for each of the ten U.S. Department of Health and Human Services (HHS) regions over a late-winter to early-summer window, to analyze the weekly percent-positive of rhinovirus/enterovirus (RV/EV) activity in the United States. Our result shows that the modal FPT typically precedes the crest of the percent-positive curve while the mean FPT aligns with the beginning of the sustained decline or plateau.