Apocalypse averted

Sloughing off the parachute of lockdown

Mitigation’s misguided arithmetic

In my last article, I showed the first tentative steps on a path to modelling the spread of COVID and argued that simple arithmetic demonstrated the futility of attempting to get to high levels of immunity without overburdening the health service by containing the spread of the virus — the so-called “mitigation” strategy.

Adam Kucharski’s demonstration at the BMJ conference on COVID unknowns of the indistinguishability of COVID curves according to how they are suppressed

Cometh the hour, cometh the math

Only, however, slightly more sophisticated math. At least to answer the broad questions we faced in March. All the math you need is in my layman’s guide to the simplest epidemiological model, the Susceptible, Infected, Removed or SIR model. Why it’s enough, I’ll try and explain here.

  • If on average, infected people infect more than one other person before they’re no longer infectious (because they isolate, wind up in hospital or die) then the number of infected people will — at least to start — grow exponentially. (Eventually, you start running out of people to infect, but not before things get really ugly.)
  • If, on the other hand, they do not, then the contagion dies out

Exponential growth isn’t just very fast growth

They key to understanding what this means is to understand what exponential growth means mathematically, rather than just what it means in its everyday colloquial usage, i.e. just “really fast”.

How seriously should we take it given the shortcomings of the model?

When the UK and Denmark made their decision to lockdown (UK a couple of weeks later than DK), we could see what was happening in Italy and what happened in China. We could also see the characteristics of exponential growth in the statistics: case numbers were increasing by a constant factor every week, not by a constant number. So the clues were there.

Models for the moment

Many governments opted for more sophisticated models. Why shouldn’t they? The advanced models were there, together with the expertise to use them. The problem is that more advanced models require more data and a deeper understanding of the causal relations between components than we realistically had — for COVID-19 — in March.

  • Be open about the uncertainty and the current resolution of the model
  • Discuss the alternatives relative to that uncertainty
  • Present what you are doing to refine the decision in the future.

Making the most of the model of the moment

The trick is to focus on what the simple model can tell you — in this case that early exponential growth is real — rather than what it can’t (what it would take to get the reproduction number well down under 1).

Decisions under uncertainty with necessarily inadequate models and inevitably insufficient data. Aka Politics, for short.

We know we can’t cope with exponential growth and that early exponential growth is an absolutely stable feature of every possible model where the reproduction number is comfortably over 1 and the contagion has taken hold. We know that the long, flat curve leading to herd immunity is just too long, even if we can at all exert that level of control.



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Graeme Keith

Graeme Keith


Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.