Estimating Generation Time Of Omicron

Generation Time of the Omicron Variant based on case counts in England in 2021

Alex Selby / / @alexselby1770

Abstract

There has been speculation that part of the advantage the Omicron variant has over Delta is that it has a significantly shorter generation period. It is important to establish whether or not this is the case because models of the epidemic with this as a default assumption will make substantially different predictions from those without. A shorter generation time would mean, for example, that Omicron's transmission advantage is lower than it otherwise would be which in turn would mean that restrictions (NPIs, reduction of contacts) ought to have a disproportionately successful effect on reducing spread. It would also suggest that a person's infectious period starts earlier and is shorter than otherwise expected, which would have implications for contact tracing and quarantine. Another consequence of a lower generation time is that we would expect the epidemic, if unmitigated by restrictions, to be over more quickly than it otherwise would be.

In this article we consider the sequence of Delta and Omicron daily case counts and highlight a particular striking pattern of growth in new cases per day: the daily growth rates, $$g_\Delta(t)$$ and $$g_O(t)$$ of Delta and Omicron respectively in England from 25 November - 24 December 2021 are related to a good approximation by $$g_\O(t)=0.483+3.391g_\Delta(t)$$ (95% CI for the gradient: 2.788 - 3.663). One possible explanation for the gradient of this line being so much larger than 1 is that the generation time of Omicron is much smaller than that of Delta.

If we use the generation time distribution of Delta from Hart et al, Oct 2021, then the above linear relationship suggests that the generation time of Omicron is 46% (38% - 60%) that of Delta, or 2.12 (1.75 - 2.77) days, and that the relative transmissibility in England in December 2021, $$R_t(O)/R_t(\Delta)$$, was 2.28 (2.03 - 2.68), lower than some early estimates that did not consider a different generation time. We then consider alternative possible explanations for the observed pattern of growth.

Note that here and throughout, confidence intervals only show statistical variation within the given model. It is possible there are larger sources of error due to an incorrect modelling assumption, as discussed in limitations below.

Linkage of growth rates and generation times

If $$\Delta_0, \Delta_1, \Delta_2, \ldots$$ and $$O_0, O_1, O_2, \ldots$$ are the respective daily time series of Delta and Omicron confirmed case counts - or otherwise estimated infection numbers - in some location, then a very simple model of change is that they grow at respective time-varying continuous rates $$g_\Delta(t)$$ and $$g_O(t)$$ which differ by a constant, $$h$$, say. The growth rates' time dependence is meant to account for changes in people's propensity to test, changes in case ascertainment rates (proportion of infections that are confirmed by a positive test) and changes in behaviour leading to increases or reductions in contacts, where in each case such changes are assumed to affect each variant equally. These assumptions lead to the identity $$O_t/\Delta_t=Ce^{ht}$$ for some constant, $$C$$ and this approximation has proved successful in the past in determining the relative growth rate, $$h$$, between two variants, by simply plotting $$\log(O_t/\Delta_t)$$ and finding the slope of (some version of) the best-fit line.

However, in past variant transitions such as Alpha-Delta in May 2021, the straight line relationship has sometimes appeared to bend over slightly as time has gone on, or otherwise slightly deviated from a single straight line. In the present case of the Delta-Omicron transition, there is a quite marked bending, as illustrated in Fig. 1 which is constructed from the ratio of estimates of Omicron to Delta on each day, using the SGTF proxy (which is known to be highly accurate for Omicron, apart from a small background rate of SGTF that is noticeable early on in Omicron's appearance, and to some extent has been adjusted for here). Fig. 2 goes a bit further and makes use of case counts in addition to variant proportions to estimate the growth rates of Delta and Omicron separately. This also suggests that $$g_O(t)-g_\Delta(t)$$ is positively correlated with $$g_\Delta(t)$$, though case counts during the last week of December should be interpreted with more caution owing to unusual testing patterns and possible limitations of testing capacity. (Note that the analysis in Fig. 2 is just for illustration/motivation here and is not relied upon in this article.)

Fig. 1 (click to enlarge)

Fig. 2 (click to enlarge)

This effect, a positive correlation between $$g_O(t)-g_\Delta(t)$$ and $$g_\Delta(t)$$, is what one would expect to see if Omicron's generation period were lower than Delta's. That is because instead of $$g_O(t)-g_\Delta(t)$$ being constant, it is more natural to suppose that $$R_t(O)/R_t(\Delta)$$ is constant: for example, if you halve the number of people you meet, then other things being equal, you are on average likely to halve the number of people you infect, which is to say, reduce $$R_t(O)$$ or $$R_t(\Delta)$$ by the same factor. Similarly, other forms of "noise", such as changes in propensity to test, would also be expected to have a proportional effect on the apparent values of $$R_t$$. In the simplest possible approximation that allows for different generation times, we'd have $$R_t(\Delta)=e^{T_\Delta g_\Delta}$$ where $$T_\Delta$$ is the generation time of Delta, and a similar equation for Omicron, which implies $$T_\Delta g_\Delta-T_O g_O$$ is constant.

A linear relationship between $$g_\Delta$$ and $$g_O$$, though with different coefficients, would still be expected to hold under slightly more general conditions, and this is what we consider in this article. The standard relationship between $$R$$, $$g$$ and generation time distribution $$T$$ is $$R\mathbb E[e^{-gT}]=1$$, and a popular and convenient family of positive distributions with two free parameters to use for $$T$$ is the Gamma family. In shape-scale notation, if $$T\sim\Gamma(k,\theta)$$, then $$R=(1+g\theta)^k$$. If we suppose $$T_\Delta$$ and $$T_O$$ have the same shape parameter, $$k$$, then $$R_t(O)/R_t(\Delta)$$ being constant would still lead to a linear relationship between $$g_\Delta(t)$$ and $$g_O(t)$$. Provided we have a decently varied supply of pairs $$(g_\Delta(t),g_O(t))$$ of reasonable quality, we should be able to discover this linear relationship, and hence get a relationship between $$k$$, $$\theta_O$$, $$\theta_\Delta$$ and $$R_t(O)/R_t(\Delta)$$. If we are additionally willing to make use of previously estimated values of $$\theta_\Delta$$ and $$k_\Delta$$, then we can estimate $$\theta_O$$ and $$R_t(O)/R_t(\Delta)$$, hence arriving at an estimate for the generation time of Omicron and its current transmission advantage over Delta.

There is no reason in principle why we can't estimate parameters in the wider situation of $$k_O\neq k_\Delta$$, but there is a question of how much information the data is powered to yield, and trying for an extra parameter like this is probably pushing it. Furthermore, the relationship between $$g_O$$ and $$g_\Delta$$ does look pretty linear (see Fig. 3 below), so there is probably not much advantage/justification in positing this particular extra parameter right now.

Making estimates of growth pairs

To estimate the generation time in the manner suggested above, we just collect enough examples of pairs $$(g_\Delta(t),g_O(t))$$ and then apply some suitable inference to find the relationship, though this requires a little care as $$g_\Delta(t)$$ and $$g_O(t)$$ are not directly measurable. In the end, the cleanest/most powerful method the author has found is to use instances of $$(\frac17\log(\Delta_{t+7}/\Delta_t), \frac17\log(O_{t+7}/O_t))$$, where these are taken separately over the nine regions of England, and $$t+7