John Dagpunar’s Discussion Contribution to Papers in Session 1 of the Royal Statistical Society’s Special Topic Meeting on COVID-19 Transmission: 9 June 2021

• 1.
  In the excellent paper by Parag, Thompson and Donnelly, $Rt^=1MW-rt^$ gives the instantaneous reproduction number in terms of the instantaneous growth rate. Wallinga and Lipsitch (2007) derive $Rt=1MW-r$⁠, where $MW.$ is the moment generating function of the generation interval $W$ and $r$ is a constant growth rate. But if $r$ is time dependent, then the renewal equation for (smoothed) incidence $It$ is

  $It=Rt∫0∞It-ugWudu$

  and an assumed exponential growth of

  $It=It-ue∫t-utrxdx$

  leads to

  $Rt=1∫0∞e-∫t-utrxdxgWudu$

  which is slightly different from the $Rt^=1MW-rt^$⁠. I wonder whether this difference does matter, as the growth rate can change over a generation interval, for example through increasing vaccination roll-out and changing non-pharmaceutical interventions. If that is the case, I think the answer is to use the renewal equation itself, which implicitly captures changing growth rates.
• 2.

  Most studies seem to assume that the distribution of the generation interval does not change with time. But changes in quarantining, isolation and effectiveness of contact tracing might lead to shortened generation intervals. So, is it not the case, that an assumed stationary distribution of generation interval, can easily lead to biased estimates of reproduction number? If so, that is an argument in favour of growth rate rather than reproduction number.

• 3.
  It is not easy to get good estimates of the distribution of the generation interval and therefore of reproduction number and that can lead to uncertainty. For example, for a gamma distributed generation interval, with mean $μ$ and shape parameter $α$⁠, just suppose the growth rates $r1t,r2t$ of two competing variants (Alpha and Delta) were reasonably constant over a generation interval. Then we could use

  $Rt1=1+μr1tαα$

  $Rt2=1+μr2tαα.$

The increased transmissibility of Delta over Alpha is then

In Dagpunar (2021), I show how sensitive the increased transmissibility and reproduction numbers are to the chosen distribution of generation interval. In this case the renewal equation method rather than the moment generating function method was used to ensure that changes in growth rate are captured.

Reproduction numbers and increased transmissibility as of 15 May 2021 (day 49)

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