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Fermi problems and public health

Enrico Fermi was a physicist well known for his ability to make good approximations to difficult questions. A well-known Fermi problem is ‘how many piano tuners are there in Chicago?’ Answering these sorts of questions involves using estimates of simpler quantities that, if correct, yield the correct answer when combined. Indeed, we could consider these calculations as arising from a simple model. Many public health policies can be approached in this way and yet, despite large amounts of journal space devoted to these issues, the debate usually revolves around the quality of the evidence. So why aren’t these sorts of calculations commonplace as a start to informing the debate? Consider the following two examples.


As we’ve noted before, e-cigarettes are a contentious public health issue. But we don’t want to wade into the vagaries of this debate, rather we’d just like to consider the benefits of maintaining a regulated e-cigarette market versus prohibition of e-cigarettes. If we assume that under a prohibition, ex-smokers who use e-cigarettes would switch back to smoking, and that the costs of enforcing a prohibition are balanced by the gain in tax revenue. Then the costs of prohibition and the benefits of regulation could be equivalent, and equal to the product of:

  • The number of switchers from cigarettes to e-cigarettes. Approximately 850,000 people.
  • Relative risk reduction of major smoking related illness. Let’s say 95% reduction.
  • The risk of a premature death. About 1.1% of smokers die prematurely each year.

This gives around 9,000 premature deaths avoided each year. And similarly at a cost of about £350 per person per year (£3b annual NHS costs divided by 8.8m smokers), a saving of approximately £14.5m per year. As a further cost to a regulated market we could add uptake of e-cigarette use by never smokers – about 56,000 people. Taking this into account, we get 8,600 premature deaths avoided and £13.5m saved. But, the benefits are likely to rise as public information increases and more people aim to quit smoking, which already seems to be the case (figure, left). Smoking consumption among e-cigarette using smokers (1.3m in the UK) may also be declining (figure, right).


Figure. Left – smoking prevalence by year in the UK. Right – daily cigarette consumption among smokers in the UK. Source data.

Seven day NHS

Seven day NHS services are a key aspect of current government healthcare policy. The aim is to reduce the so-called weekend effect: the increased risk of mortality associated with weekend admission versus weekday admission. Rachel Meacock described her paper on this blog that attempted to value those ‘excess’ deaths to see if the seven day policy met typical cost-effectiveness standards (they did not). But, as they and others have noted, not all of these deaths are preventable. We can attempt to come up with an estimate of how many of those deaths are preventable based upon what we know. So, we multiply:

  • Number of emergency weekend admissions: 1.23 million
  • Risk of a preventable adverse event: 4.5%
  • Relative risk increase of an adverse event associated with weekend admission: 20%
  • Probability an adverse event lead to death: 7.4%

Which gives 820 preventable deaths. This is 15% of the total number of deaths associated with weekend admission, which would suggest that the benefits of preventing these deaths are 15% of the size estimated by Meacock et al.

Informing the debate

Certainly, these are not thorough statistical analyses and do not capture the uncertainties involved. They are Fermi Problems and they are a simple attempt to inform the debate. The quality of the evidence is of course key to ensuring the estimates informing the calculations are correct. For example, the 95% reduction in risk associated with e-cigarettes is disputed by some. But, that figure would have to be close to zero, or there would have to be some very large unaccounted-for cost to regulation in order to favour prohibition.

There are, of course, moral, political, and legal dimensions to policy formation. Policy analysis simply provides the figures to inform that debate. In some cases no further evidence might be forthcoming, leaving such calculations as the only option. For example, policy aimed at mitigating pandemic influenza requires studies from a pandemic, and it is too late when one occurs. Even a back of the envelope calculation like this is preferable to basing policy on intuition and whim alone.


  • Health economics, statistics, and health services research at the University of Warwick. Also like rock climbing and making noise on the guitar.


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