A practical generation interval-based approach to inferring the strength of epidemics from their speed

Sang Woo Park; David Champredon; Joshua S. Weitz; Jonathan Dushoff

Estimating the reproductive number R is a key part of characterizing and controlling infectious disease spread. The initial value of R for an outbreak is often estimated by estimating the initial exponential rate of growth, and then using a generation-interval distribution to relate the two quantities [44, 39, 28, 40] . However, detailed estimates of the full generation interval are difficult to obtain, and the link between uncertainty in the generation interval and uncertainty in estimates of R are often unclear. Here we introduced and analyzed a simple framework for estimating the relationship between R and r, using only the estimated mean and CV of the generation interval. The framework is based on the gamma distribution. We used three disease examples to test the robustness of the framework. We also compared estimates based directly on estimated mean and variance of of the generation interval to estimates based on maximum-likelihood fits. The gamma approximation for calculating R from r was introduced by [29] , and provides estimates that are simpler, more robust, and more realistic than those from normal approximations (see Appendix). Here, we presented the gamma approximation in a form conducive to intuitive understanding of the relationship between speed, r, and strength, R (See Fig. 2) . In doing so, we explained the general result that estimates of R increase with mean generation time, but decrease with variation in generation times [44, 46, 37] . We also provided mechanistic interpretations: when generation intervals are longer, more infection is needed per generation (larger R) in order to produce 15 . CC-BY 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/312397 doi: bioRxiv preprint a given rate of increase r. Similarly, when variance in generation time is large, there is more early infection. As early infections contribute most to growth of an epidemic, faster exponential growth is expected for a given value of R. Thus a higher value of R will be needed to match a given value of r.