The health secretary impressed parliament last week when he said daily Omicron infections were estimated at about 200,000. This was classic “number theatre” – pulling a big statistic out of a hat without supporting evidence. Although the Health Security Agency (UKHSA) later briefed the press, it only publicly revealed its workings on Thursday: it had estimated 23,000 Omicron infections on 7 December, then assumed exponential growth with a doubling time of 1.9 days. We will have to wait for infection survey estimates to know if this is accurate.
Exponential growth simply means something increases in proportion to its current value – the bigger it gets, the more it increases each day. This does not necessarily mean “fast” – a savings account has this kind of growth, even with a 0.1% compound interest rate.
Doubling times are, unsurprisingly, how long it takes for a value to double. A popular trick in finance is to estimate the doubling time by dividing 72 by the annual interest rate, provided it is low. A 4% compound interest rate (remember that?) would mean your capital doubled in around 72 divided by four, or 18 years. Note: 72 is chosen as it is easy to divide and roughly equals 100 times the natural logarithm of two, as given by a standard mathematical model.
The UKHSA was assuming a somewhat faster doubling time for infections, of about two days, which implies the numbers of infections were increasing by 41% a day, since 1.41 times 1.41 is roughly double. This is an extraordinarily rapid and worrisome rise.
Exponential growth phases inevitably end, through changes in behaviour, prior immunity or simply running out of people to infect, although turning points remain hard to predict. UKHSA has warned against extrapolating this curve further – it would mean 18 million infections by Christmas Day.
There is an old joke about fitting an exponential curve to the number of Elvis Presley impersonators, reckoned to reach a third of the world’s population by 2019. This did not happen. Exponential growth can’t go on for ever, so treat such projections with suspicious minds.
• David Spiegelhalter is chair of the Winton Centre for Risk and Evidence Communication at Cambridge. Anthony Masters is statistical ambassador for the Royal Statistical Society
