The stochastic dynamics of early epidemics: probability of establishment, initial growth rate, and infection cluster size at first detection

Emerging epidemics and local infection clusters are initially prone to stochastic effects that can substantially impact the epidemic trajectory. While numerous studies are devoted to the deterministic regime of an established epidemic, mathematical descriptions of the initial phase of epidemic growth are comparatively rarer. Here, we review existing mathematical results, and derive new results to elucidate the early dynamics of an infection cluster started by a single infected individual. We cover the probability of establishment of an epidemic as a function of the distribution of secondary infections, the expectation of epidemic size as a function of time, and the convergence to a deterministic exponential regime. Stochasticity systematically accelerates the initial growth of an epidemic, because epidemics that do not get extinct have faster initial growth on average. This also affects the long-term epidemic growth by increasing the cumulative size by a constant factor. Applying these results, we compute the probability distribution of the first detection time of an infected individual in an infection cluster depending on the testing effort (assumed to be constant in time), and estimate that the SARS-CoV-2 variant of concern B.1.1.7, detected in September 2020, first appeared in the United Kingdom early August 2020. We estimate a minimal testing frequency to detect clusters before they exceed a certain threshold size, and we compute the detection rate of infected individuals during a single mass testing effort. For example, in a COVID-19-parameterized model with an effective reproduction number R=1.1, a fraction 1.3% of a population needs to be randomly tested each day for cluster size to not exceed 30 infected individuals. These results improve our theoretical understanding of early epidemics and are useful for the study and control of local infectious disease clusters.