R number

Population Structure and the Cyclical Pattern of Epidemic Waves

Regular readers will remember Philippe Lemoine from my interview with him back in August. For those who missed it: Lemoine is a PhD candidate in philosophy at Cornell with a background in computer science. During the pandemic, he’s written several interesting articles, including a particularly good one titled ‘The Case Against Lockdowns’.

Lemoine’s latest article is a zinger. It begins with the puzzle of why the effective reproduction number often fluctuates wildly in the absence of changes in aggregate behaviour. Or put another way: why do infections sometimes start falling, or start rising, for no apparent reason?

I actually noted this puzzle myself in article back in March (which Lemoine kindly cites). Specifically, I noted that case numbers in South Dakota began falling rapidly in mid November, despite almost no government restrictions and little change in people’s overall mobility.

There are at least two existing explanations for this phenomenon. The first is seasonality: the effective reproduction number may partly depend on variables like temperature, humidity and UV light. Yet as Lemoine points out, there are many examples where case numbers changed suddenly that seasonality can’t explain (like South Dakota).

The second is viral evolution: the effective reproduction number may suddenly rise when a new, more-transmissible variant emerges (such as Delta or Omicron). Once again, however, case numbers have undergone dramatic changes in the absence of new variants. And while viral evolution can explain the rises, it has harder time explaining the falls.

Lemoine’s explanation is different: population structure. Traditional epidemiological models, he notes, assume the population is ‘quasi-homogenous’. This means that your chance of infecting someone of the same age who lives next door is the same as your chance of infecting someone of the same age who lives on the other side of the country.

Not very realistic, of course, but models have to make simplifying assumptions. How much does this one matter? It matters a lot, Lemoine argues.  

Rather than assuming there’s one big quasi-homogenous population, imagine the population is divided into a large number of ‘subnetworks’. These could be based on location, age-group, behaviour or a combination of factors. For example, one subnetwork might be ‘school children and their parents in central London’.

Suppose that transmission occurs frequently within subnetworks but infrequently between them. So when a child within the school subnetwork catches the virus, it quickly spreads to other children and their parents. But what it doesn’t do is quickly spread to those outside the subnetwork.

German R Number Has “No Direct Connection” With Lockdown, Say Researchers

Since the start of the pandemic, Germany has seen the lowest level of excess mortality of any major European country – just 4% according to the World Mortality Dataset. Is this because the country effectively suppressed the virus using lockdowns? A new research note suggests not.

Annika Hoyer, Lara Rad and Ralph Brinks – three researchers at the Ludwig Maximilian University of Munich – sought to compare the epidemic’s trajectory to the timing of lockdown measures. (Their paper is written in German, but you can translate it using Google.)

Hoyer and colleagues begin by noting that, due to changes in testing, case numbers are unsatisfactory for tracing the epidemic over time. They note, “Varying test behaviour should be understood here as the fact that in the course of the epidemic… the execution of tests has changed and changed very strongly, both temporally and regionally.”

They argue that the R number (the average number of people an infected individual transmits the virus to) provides a better guide to the epidemic’s trajectory. According to the authors, “The estimation of the R-value also involves some statistical difficulties, such as the necessary nowcasting, but the main disadvantage of the dependency on test behavior, which can imply large fluctuations, does not apply.”

Hence they plotted the R number over time, and looked for major changes or “breaks” in the series – as shown in the chart below:

The R number decreased dramatically in March of 2020. It rose slowly over the following six months (the apparent spike in the summer may be a random fluctuation due to the small number of cases at the time). It then rose more quickly in September, only to fall again in October. It fluctuated around 1 during the winter months, and has fallen since the start of April.

Given that Germany’s first lockdowns (which varied from state to state) were imposed around March 22nd, it’s clear that the initial decline in infections preceded them. (The statistician Simon Wood has argued that the same thing happened in Britain – i.e., infections were already declining when lockdowns came into force.)

Hoyer and colleagues point out that changes in the R number don’t seem to be correlated with the timing of winter lockdowns either. They note, “there has been no direct connection with the measures taken since September – neither with the lockdown light on November 2nd and the tightening on December 16, 2020, still with the ‘Federal Emergency Brake’, which was decided at the end of April 2021”.

Their research note provides further evidence that lockdowns have little impact on the epidemic’s trajectory beyond the effects of voluntary social distancing and restrictions on large gatherings.