Abstract
Emerging epidemics and local infection clusters are initially prone to stochastic effects that can substantially impact the early 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 on the size of the epidemic over time, and derive new results to elucidate the early dynamics of an infection cluster started by a single infected individual. We show that the initial growth of epidemics that eventually take off is accelerated by stochasticity. As an application, we compute the distribution of the first detection time of an infected individual in an infection cluster depending on testing effort, and estimate that the SARS-CoV-2 variant of concern Alpha detected in September 2020 first appeared in the UK early August 2020. We also compute a minimal testing frequency to detect clusters before they exceed a given threshold size. These results improve our theoretical understanding of early epidemics and will be useful for the study and control of local infectious disease clusters.
Highlights
The emergence and spread of infectious diseases pose an increasing threat in an ever more interconnected world
We have collected key equations and derived new results to account for stochasticity during the early phase of epidemic trajectories
Taking into account stochastic effects during the early phase of an epidemic allowed us to compute a good description of the mean epidemic size for all times (equation (2.8))
Summary
The emergence and spread of infectious diseases pose an increasing threat in an ever more interconnected world. During the early phase of an epidemic in a local infection cluster stochastic effects cannot be neglected These stochastic effects are due to the initially low number of infected individuals, and to the inherent stochasticity of the transmission process. Epidemic size individuals becomes large, this stochastic model can be approximated by a deterministic partial differential equation describing the distribution of the time since infection of the host population. In addition to the added biological realism, a time-varying infectiousness of infected individuals can properly capture the dynamical consequences of abrupt changes in transmission rate [6,8] This is not possible with an ODE framework [9]. The reviewed and newly derived results can inform public health-related questions: how many importations will eventually result in a local infection cluster? How large is a local cluster once a first case is detected? When did a new variant—like Alpha, first detected in the UK—arise? How large is the detection rate of infectious individuals by a single mass testing effort? How many daily tests need to be conducted to detect local clusters before they exceed a certain size? We show how our theoretical results provide quantitative answers to these questions
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