It’s all about R. Lockdown critics don’t understand exponential mathematics

Quite early in the Covid-19 outbreak, policymakers told us that the critical thing to understand about managing the disease was something they called “R”. This is the reproduction rate: the average number of other people that anybody that catches the virus will infect. If it less than 1.0 the epidemic will fade away. More than one and the disease spreads exponentially. Discussion of R then dropped from view, with people focusing more on the absolute level of infection (usually given as cases per 100k). That’s a shame because R is central to understanding what has just happened here in UK, which also applies to most of the rest of Europe with small variations.

In those the dark days of March, R was a horrific 2-3. That meant the disease was spreading very rapidly through the population. For a moment people thought that might not be such a problem: that herd immunity would develop, and the disease would slow down as the number of potential carriers was diminished. But there are two major problems with this. The first is that if the disease spreads rapidly, then health services (and other infrastructure) become overwhelmed, and there s substantial collateral damage. And that is exactly what happened in the Spring. Hospitals started to fill up, requiring them to clear beds; people with other conditions suffered, and Covid-infected people were pushed into care homes to make room for others, with appalling results. The death rate across the country shot up, and not just because people were dying from Covid, though frightening numbers were. The second problem with the herd immunity strategy was that nobody knew whether, or for how long, catching the disease would confer immunity. It was quite an interesting example, incidentally, of how basing decisions on evidence is not quite as robust as it seems. What do you believe in the absence of evidence? In Britain people believed, absent evidence, that face masks were useless and that you would get long term immunity by catching the disease. There is no rhyme or reason to it being that way round. Too often people who ask where the evidence is are just sticking to their prejudices.

I digress. The only sensible response to an R of over 2 was strict lockdown, which was implemented in Britain on 23 March. This was remarkably successful (it is puzzling why some people suggest that it wasn’t). R came down to about 0.7, and the disease was beaten back to manageable levels in most places. But what next? The government eagerly implemented a general relaxation, while maintaining a certain number of rules on social distancing. By this time few people were talking about R, and it was hard to know what aim of it all was. I think it was felt that the relaxation would take the R back up to 1.0, but no further, thus keeping the virus at manageable levels. It is probable that a lot of faith was being put in the Track and Trace system to contain outbreaks as they occurred. And yet the system they built was not designed to achieve that level of rigour, which needs tough local leadership and timely data, both of which were practically designed out of the system from the start. The result was that R crept up to about 1.5. Much better than before, but also pretty useless in terms of managing the disease. If R is over 1, then the disease will rise up to overwhelming levels much more quickly than people will intuit, because of the exponential way in which the disease spreads.

The government’s next strategy was one of local local lockdowns. The hope was that these would reduce R to below 1 in areas were the prevalence had become high. Over the last few weeks two problems have emerged, though. First is that even with these stricter measures R is above 1, and health services are under imminent threat. Second is that R is rising scarily everywhere else, and to beyond 2 in some places. That meant that most of the rest of the country wasn’t in fact that far behind the hotspots. Just what was going on here will probably not be known for some time. Perhaps people in Tier 3 of the lockdown system felt bolshy because they were being singled out, and did not apply themselves properly to lockdown. Perhaps people in Tier 1 felt they could relax because they weren’t in trouble yet. Anyway, it is very clear that the regional response strategy has failed. And so we are back to national lockdown.

But schools an universities are still open, and weariness is creeping in as businesses fail and savings run out. The death rate is much lower than before (the rate of excess deaths remains negligible) and this is being used to suggest that we should just “live with” the virus. Some conservative newspapers (the Telegraph and Mail in particular) opposing lockdown, even though their recommendations would sentence many of their readers, more vulnerable than the average, to an untimely and horrible death, or perhaps just a long-term deterioration of health. Such critics have failed to understand the logic of R. It is not about choosing an acceptable level of disease and freezing it there: it is about stopping the disease before it overwhelms.

So how on earth do you live with the disease and retain a semblance of normal life? The only proven way is the Asian one (there are many variations, shown by China, South Korea, Taiwan, Japan and Australia and New Zealand). This requires the disease to be stamped out, mainly through strict lockdown, for inward travel to be heavily restricted, and for any outbreaks to be stamped on hard. For some reason Europeans, and not just the British, seem unable to do this (even the Germans are struggling). It’s even worse for the Americans. Asians, and interesting this includes Antipodeans of European heritage, seem much happier to comply with busybody regulations. Not all Asians of course: Indians struggle, as do Indonesians, Philipinos and a number of others.

Absent the Asian approach, thoughts turn to the use technologies that are not yet available. Vaccines may not be the silver bullet they are for diseases like polio, but they could still throw enough sand in the wheels of transmission to stop R getting above one. Mass testing, talked up by the Prime Minister Boris Johnson, offers ways of identifying infected people so that they can be isolated. That raises all sorts of questions.

Locally I have found the most informative source about the progress of the virus comes from the government’s interactive map. I have watched the infection rate steadily go up, with white ares (very low infections) moving to green, and green going moving to blue (over 100 per 100k); in my local patch it is 171. We’re keeping our heads down.

It’s all very depressing. The most important thing to understand about the spread of infectious diseases is that it is exponential. None of the lockdown critics I have heard or read seem to grasp that. And only a few understand that the critical thing isn’t the death rate but the stress on public infrastructure, including, but not limited to, hospitals. In the end it’s all about R.

4 thoughts on “It’s all about R. Lockdown critics don’t understand exponential mathematics”

  1. I seem to remember that we had a discussion in May on whether R was more than “a statistical abstraction”. It is, but often people can have a difficulty when terms like mathematical terms like “exponential” are used.

    The death rate is still going to be what matters in the end, though. The figure for total deaths caused by the Pandemic will be what is remembered. Just like it was in the 1919 Spanish Flu outbreak.

    Which is even more of a reason to try to get R under some sort of control and try “to flatten the curve.” If the NHS is overwhelmed many patients who would otherwise have been saved will die. The total death toll will be much higher than it would otherwise have been.

    That’s the stark reality.

  2. So how to get the public to understand the technicalities better? I commend the German and Swedish Models, where-by an expert government agency issues recommendations directly to all – the public, the local authorities the central government. With this type of institutional arrangements, we would have learnt five weeks ago that the expert advice was for an immediate circuit-breaker lockdown, thus cutting off before the exponential effect to which this post draws attention before it got such a serious grip on society. With a strong general lock down in place early, the restrictions could have been relaxed in a differential fashion, with safeguards for the vulnerable (such as a right not to go to work for the over 50s), and rewards for places such as the North East with better-than-average trends. In my view, we did not have to be in our present position – on the back foot.

  3. Having previously said that it is “all about R” and “exponentials” too I perhaps should qualify that by saying there is more than this to the mathematics of epidemics. Cases can’t rise exponentially for ever. Covid will burn itself out if we let it but the numbers aren’t pretty!

    There has been some suggestion that we’ve been here before except we didn’t know what a Coronavirus was at the time.

    This account of the “flu” from a Dublin physician of the time does sound remarkably like Covid. Especially the part of the patient describing a loss of taste.

    According to the Guardian article we had over 100k deaths in the 1890 -92 epidemic. I suspect that will be an underestimate because many people wouldn’t have had access to any medical help at all. Scale that up for an increased population, especially one with more old people and we can see how the current death toll could easily be in excess of 500k without a vaccine and without lockdowns.

    1. Yes quite right that the population size will limit the exponential transition – which is the idea behind herd immunity. The effects of herd immunity do look elusive, at least in developed economies. Countries which suffered badly in the first wave (including us, as well as Sweden) aren’t being spared in the second. probably because infection levels never got high enough – the Economists report some reduction in infections in some Italian towns, – but only those that lost 1% of their population to the disease.

      Very interesting about the outbreak in 19th century. It shows how much less tolerant we are of disease these days. Also that there is nothing especially modern about this sort of pandemic.

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