Estimating the effective reproduction number; looking ahead

21 April 2020

Here are some of the most interesting posts, articles, papers, and resources I’ve come across recently, organised by theme.

How close are we to R=1?

You’ve probably heard about the basic reproduction number by now, R0 (R-naught). This is the average number of people infected by each infected individual a situation in which the entire population is susceptible (no immunity) and no measures have been taken to reduce the spread.

The number that matters most now is the effective reproduction number, often simply called R. This is the average number of people infected by each infected individual in a given population, at a given point in time, taking into account the actions taken to reduce the spread.

Studying the paths of different countries, and talking to epidemiologists, one very simple but very important rule is clear: If R<1 (and remains below 1) in a given population (say, a city or a country), the pandemic will gradually die out in that population. If R>1, the epidemic is still increasing exponentially.

Of course, it also matters if R is a little more than 1 or a lot more than 1.

One of the best explanations of how important this concept is, and how much depends exactly where we are relative to that threshold of 1, comes from none other than German Chancellor Angela Merkel. Below is an excellent video, subtitled in English, where she explains this clearly (and here is the link if the embed below isn’t working for you).

While there are more and more individual studies trying to assess what R is in a given location at a given time, the most useful general resource I’ve found for looking at this globally is the Epiforecasts site from the Centre for the Modelling of Infectious Diseases (CMMID). The good news is that R is trending down towards, near, or possibly even slightly below 1 in a number of countries (six shown here, many more available on the site):

Figure 2: Time-varying estimate of the effective reproduction number (light ribbon = 90% credible interval; dark ribbon = the 50% credible interval) in the regions expected to have the highest number of new confirmed cases. Estimates are shown up to the 2020-04-11. Confidence in the estimated values is indicated by translucency with increased translucency corresponding to reduced confidence. The dotted line indicates the target value of 1 for the effective reproduction no. required for control.

The site also helpfully presents the number of daily cases both with the date on which they were reported, and on a time-adjusted basis where they use some fancy mathematical tricks to try to map cases to the most likely date of infection; this can present a very different picture:

Looking across a number of countries (not just those reproduced above), a few things stand out:

  • Control measures have clearly been very effective at reducing R, albeit at high cost.
  • Few countries that experienced a large outbreak have gotten R significantly below 1. France, Austria, and South Korea stand out as having significantly reduced R below R.
  • A number of countries that imposed strict control measures continue to have R very near 1, including the US, the UK, Italy, Germany, and Spain. These countries may have trouble relaxing control measures significantly (as Merkel discusses).
  • Many countries that have not yet had large outbreaks have R meaningfully above 1; it may be that they have not yet felt the pressure to impose significant control measures, or that they have taken very effective preventative measures.

Still on the topic of R, and for geeky voyeuristic value, I particularly enjoyed this debate between Nobel Laureate in Economics Paul Romer (whom, for the record, I think is an extraordinary thinker), and mathematical epidemiologist Adam Kucharski, about whether a given set of measures would likely be sufficient to get R below 1. (Spoiler alert: Paul gracefully backed down from his claim that these measures would suffice after Adam publicly walked him through a simple framework that demonstrated that they likely would not.]. The important point, though, was that even assuming we had sufficient testing capacity, periodic testing (e.g., fortnightly) is unlikely to be sufficient to keep R below 1 given a high rate of asymptomatic transmission.

Finally, one recent study looking just at France estimates that R0 was 3.3 and that lockdown has taken France to an R of 0.5. Note that one result of slowing the spread of the epidemic so effectively, in order to preserve health care capacity and avoid Lombardy-type situations, is that we are very far from herd immunity. This study forecasts that only 5.7% of the total French population will have been infected (those currently infected as well as those who have recovered) as of 11 May, the target date for relaxing control measures.

Dancing towards the exit?

You may remember Tomas Pueyo’s early, influential Medium post from 10 March which warned what was coming, with great data and charts. His latest post, Learning How to Dance, takes an approach I like a lot: it looks at what we can learn from successes and failures around the world.

Most significantly, he compares Singapore and Japan, early success stories that originally contained the pandemic successfully and more recently have seen accelerating infection rates, with Taiwan, South Korea, and Hong Kong, who have managed to allow some degree of reopening without resurgence.

The detailed description of how Taiwan’s highly centralized, mandatory, police-state like approach to contact tracing and quarantine is impressive; but also depressing, as it’s hard to imagine that approach being accepted, let alone being implemented successfully, in the West.

Similarly, South Korea reportedly has tracked travel and contacts centrally, coupled with extensive testing and mandatory quarantine in government-run shelters for those who test positive.

By contrast, Singapore made critical mistakes: they allowed European travelers in for too long, permitted seeding of new cases; started with under-resourced manual contact tracing; and achieved low opt-in for automated contract tracing.

Apparently, adoption of masks has been another point of difference:

Until April 3rd, Singapore only recommended masks for the sick. As we saw before, that contrasts with both Taiwan (with masks managed centrally) and South Korea (with 98% of people wearing masks at least sometimes and 64% all the time outside).

If you’re a fan of Tomas’s essays, you’ll also want to read his extremely depressing post about the pandemic in the US. He concludes that the US will only succeed in having an effective response if it centralises many aspects of it (e.g., healthcare supplies, contact tracing, setting guidelines for social distancing measures, testing initiatives); exactly the opposite direction that the US is currently going.

More generally, I am a big fan of comparing the approaches taken in different countries to see what is effective. To that end, I found the Oxford COVID-19 Government Response Tracker very helpful. It provides a quantitative way to assess, report, and compare how stringent the measures being adopted by different countries are, and how those are evolving over time.

The data show that in most countries, responses have lagged (and presumably been in response to) outbreaks:



(I know correlation doesn’t prove causation, but I’m pretty sure that higher stringency doesn’t cause more cases, so I’ll go with escalating cases causing stringency.)

The year ahead, according to the NYT

The New York Times’ recent feature on the outlook for the next year, summarizing the views of around 20 experts, was excellent and (I’m sorry to keep using this word) sobering. Some key points:

  • Many experts are pessimistic about the ability to reopen the economy without a re-acceleration of the pandemic. “Until a vaccine or another protective measure emerges, there is no scenario, epidemiologists agreed, in which it is safe for that many people to suddenly come out of hiding. If Americans pour back out in force, all will appear quiet for perhaps three weeks.”
  • “Without a vaccine, the virus is expected to circulate for years, and the death tally will rise over time.”
  • Although the actions taken will reduce the death toll from what it would have been, COVID-19 is still likely to be the leading cause of death in the US and could kill more than the 420,000 Americans who died in WWII.
  • There are emerging templates for what it would take to reopen the economy safely, but their criteria seem very difficult to meet anytime soon: “Resolve to Save Lives, a public health advocacy group run by Dr. Thomas R. Frieden, the former director of the C.D.C., has published detailed and strict criteria for when the economy can reopen and when it must be closed. Reopening requires declining cases for 14 days, the tracing of 90 percent of contacts, an end to health care worker infections, recuperation places for mild cases and many other hard-to-reach goals.”
  • Those certified immune will have great advantages in their ability to participate in the economy, which creates perverse incentives to seek to be infected.
  • Mandatory quarantine of infected individuals (a cornerstone of some effective responses such as Taiwan and South Korea) is controversial in the US; and contact tracing efforts are not being scaled up rapidly.
  • Vaccines are unlikely to arrive soon.

Testing capacity?

Every proposed path to reopening relies on scaling testing capacity. But where will that capacity come from?

This epidemiologist from the Johns Hopkins Center for Health Security points out that the US has plateaued at 1M tests/week but needs 3.5M/week.

This article in StatNews discusses a global shortage of relevant reagents.

A long road ahead?

Here’s a small selection out of many articles suggesting that the path ahead is a long one. Harvard Professor of Epidemiology, Dr William Hanage, writing in The Guardian:

This crisis is not close to over, quite the reverse. The pandemic is only just getting started.

In order to get really depressed, read this article in Science, co-authored by Marc Lipsitch, projecting multiple years of recurrence. It takes the approach of assuming that, without a vaccine, we have no choice but to achieve herd immunity through infection and recovery, and tries to model what that will take, using as a constraint that we do not want to exceed critical care capacity. The paper is very clearly written and worth reading in full.

We projected that recurrent wintertime outbreaks of SARS-CoV-2 will probably occur after the initial, most severe pandemic wave. Absent other interventions, a key metric for the success of social distancing is whether critical care capacities are exceeded. To avoid this, prolonged or intermittent social distancing may be necessary into 2022.

Naturally, there is a lot of attention (as discussed above in relation to Tomas Pueyo’s post) on contact tracing and quarantine. Adam Kucharski points out that it would take extremely highly adopted tracing programmes and high compliance with quarantine for this to be sufficient to completely exit other control measures (though of course, even partial adoption and compliance would help). The key challenges are that even with much more testing, a lot of spreading would still happen prior to testing picking up new cases; and enormous effort would be required to manually trace large numbers of contacts (in the absence of near-universal adoption and acceptance of technical measures as in Taiwan).

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