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The problem with R-targeting
Why targeting reproduction numbers if problematic
Welcome to Plugging the Gap (my email newsletter about Covid-19 and its economics). In case you don’t know me, I’m an economist and professor at the University of Toronto. I have written lots of books including, most recently, on Covid-19. You can follow me on Twitter (@joshgans) or subscribe to this email newsletter here.
There has been a widespread education regarding some basics of epidemiology this past 10 months. One of the concepts that have been given attention is that of the reproduction number or R. Here is Germany’s science explainer in chief:
This is a core concept in epidemiology. It is something that can be estimated and hence, it is something that can be targeted. One goal is to get R as low as possible. But a threshold goal is to get R below 1. Do that and the cases start to fall. Interestingly, in many places, R has hovered around 1. It goes above it and then below it. When R is greater than 1, a combination of individual behaviour and government interventions reduce it. When it is below 1, both tend to relax. The evidence suggests it is the individual behavioural margin that matters the most.
R-targeting is common to think about policies. Here is one analysis of mask-wearing published in The Lancet yesterday that focuses on how the extent of mask-wearing, as well as their quality impacts on R. And here is how that study was reported:
Results – published in The Lancet Digital Health – suggest combining social distancing with a 10% increase in self-reported mask-wearing more than triples a community’s odds of keeping R below one.
And those studies come thick and fast. Here is another one that came with news reporting that masks could save $1 trillion. The catch was that required R = 0. Can masks get us to that? There is certainly nothing in the study that suggests that and I highly doubt it. I think people are getting carried away.
R-targeting can be useful but we have to remember that it is an aggregate statistic and also an average. Policy-makers do target such things. A famous example is inflation which is a measure of the average rise in prices. One of the reasons why inflation is a potentially reasonable target is because, at some level, it is driven by the amount of money there is available. More money eventually leads to higher prices because it has to go somewhere. There are bumps on that road but there is approximately a linear relationship.
However, treating pandemic policies based on a linear relationship with R can be misleading. R is a measure that can be backed out of the thing we really care about — are cases rising or falling. But it is also a measure that is derived from certain foundations — specifically, given current conditions, on average, how many people will one infected person, ‘successfully’ infect? We care a lot about that but we do not care about it as an average per se. We care about it for each and every infected individual.
The reason this is important is that variance matters. If every infected person was identical in terms of how many others they infected, then if we have an intervention that reduces R on average because it impacts on all infected people the same way, it maps one-to-one with what we care about. But we don’t have that. Some people and situations are much more likely to be infectious than others. In other words, the individual R’s vary considerably and that variation matters.
This is explained in a paper by Richard Holden and D.J. Thorton. The key figure is this one.
Notice here that, as an average, R is well less than 1. However, for a good number of people it is greater than 1. So go from one day to the next and the aggregate case numbers are falling. But that danger zone is a tinder box. If the infected people in that zone are still exposing others, there is potential for the case numbers to blow up again. That variance means that R, as an aggregate average, is potentially misleading in terms of telling us whether we have things under control or not.
The problem we face is that we have ways of estimating the aggregate average, R, using data from a region. but we don’t have good ways of estimating individual R’s and so getting a sense of this distribution. This lack of knowledge matters for policies.
Consider masks. Masks do not prevent susceptible people from interacting with infectious people. What they do is reduce the probability that those meetings result in transmission. On average, that is a good thing. But with respect to variance, maybe not. If everyone knew a meeting was, for sure, going to result in you infecting someone, then that would be bad but also the variance wouldn’t be changed if you had people wearing masks. But if it was, say 60:40 and the masks reduced it to 50:50, then while the average transmission might fall, the variance would actually rise. Half the meetings work out well while the other half do not. That increases variance.
The problem then becomes that this type of policy could actually make the aggregate average R we are targeting more misleading. The information gap is widening.
This is not true for all policies. For instance, policies that prevent meetings from happening in the first place, like lockdowns but also screening programs that result in isolation, do both reduce the average R and its variance. In other words, those policies do well and also make R a more reliable indicator of success.
The point is that we have to be mindful regarding what our policies might do the variance of R and not simply the average. This is because a policy that reduces the average may leave or increase the size of the danger zone which will impact on whether that policy results in sustained mitigation of the outbreak or not. As ever, the real metric is whether we bring cases down and keep them down.