Time-varying reproductive number estimation for practical application in structured populations

Objectives EpiEstim is a popular statistical framework designed to produce real-time estimates of the time-varying reproductive number, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> Rₜ . However, the methods in EpiEstim have not been tested in small, non-randomly mixing populations to determine if the resulting <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mo>̂</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> R̂ₜ estimates are temporally biased. Thus, we evaluate the temporal performance of EpiEstim <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mo>̂</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> R̂ₜ estimates when population structure is present, and then demonstrate how to recover temporal accuracy using an approximation with EpiEstim . Methods Following a real-world example of a COVID-19 outbreak in a small university town, we generate simulated case report data from a two-population mechanistic model with an explicit generation interval distribution and expression to compute true <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> Rₜ . To quantify the temporal bias, we compare the time points when true <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> Rₜ and estimated <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mo>̂</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> R̂ₜ from EpiEstim fall below the critical threshold of 1. Results When population structure is present but not accounted for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mo>̂</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:math> R̂ₜ estimates from EpiEstim prematurely fall below 1. When incidence data is aggregated over weeks the estimates from EpiEstim fall below the critical threshold at a later time point than estimates from daily data, however, population structure does not further affect timing differences between aggregated and daily data. Last, we show it is possible to recover the correct timing when by using the lagging subpopulation outbreak to estimate