The identification of a first case (e. g. by a disease-related death or hospitalization event) raises the question of the actual size of a local outbreak. Quick estimates of the outbreak size are required to assess the necessary testing, contact tracing and potential containment effort. Using general branching processes and assuming that epidemic parameters (including the basic reproductive number) are constant over time, we characterize the distribution of the first hospitalization time and of the epidemic size at this random time. We find that previous estimates either overestimate or largely underestimate the actual epidemic size. In addition, we provide upper and lower bounds for the number of infectious individuals of the local outbreak over time. The upper bound is the cumulative epidemic size, and the lower bound is a constant fraction of it. Lastly, we compute the number of detectable cases if one were to test the whole local outbreak at a single point in time. In a growing epidemic, most individuals have been infected recently, which can strongly limit the detection of infected individuals when there is a delay between an infection and its potential detection. Overall, our analysis provides new analytical estimates about the epidemic size at identification of a first disease-related case. This piece of information is important to inform policy makers during the early stages of an epidemic outbreak .
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 .
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The identification of a first case (e. g. by a disease-related death or hospitalization event) raises the question of the actual size of a local outbreak. Quick estimates of the outbreak size are required to assess the necessary testing, contact tracing and potential containment effort. Using general branching processes and assuming that epidemic parameters (including the basic reproductive number) are constant over time, we characterize the
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
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hospitalization time and of the epidemic size at this random time. We find that previous estimates either overestimate or largely underestimate the actual epidemic size. In addition, we provide upper and lower bounds for the number of infectious individuals of the local outbreak over time. The upper bound is the cumulative epidemic size, and the lower bound is a constant fraction of it. Lastly,
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
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number of detectable cases if one were to test the whole local outbreak at a single point in time. In a growing epidemic, most individuals have been infected recently, which can strongly limit the detection of infected individuals when there is a delay between an infection and its potential detection. Overall, our analysis provides new analytical estimates about the epidemic size at identification of a first disease-related case. This piece of information is important to inform policy makers during the early stages of an epidemic outbreak
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