Viral Exponential Trending, and Napier

by J Michael Buchanan

Words such as ‘exponential’ and ‘doubling times’ have been loosely banded about in relation to the COVID-19 pandemic. Time series graphs have been, without constraint, projected skywards, with little indication of uncertainties in forward projections.

In The Times (March 2020), non-linear graphs were displayed. No explicit explanation was given of their technical foundations, suitability, or limitations. They appeared to be log-linear plots, but there was an absence of annotations to confirm this. Some were very small and overcrowded.

We have the Scottish inventor of logarithms, John Napier (1550-1617), Laird of Merchiston, to thank for these. (see St Andrews in Focus, Issues 69 and 82).

A simple ‘compound interest rate’ or ‘constant doubling time’ results in a straight line on a log-linear plot. More generally, when the key variable ranges over several orders of magnitude (e.g. from 10 to a million), a compact presentation is possible (10 becomes 1, 1000 3, and 1 million 6).

Not all phenomena and models canoe straight-jacketed as an ideal single monotonic exponential function. This is especially true in pandemics, where the optimists expect rapidly increasing numbers to peak, then morph into a declining pattern that might be described by a ‘half-time’ akin to the half-life of radio-nuclides. In other cases, boring tabulation of key comparators may be safer than graphical presentations.

In the early days of personal electronic calculators (viz. the swinging late 1960s), at University, in Physics, I used a slide-rule, printed log tables, and a pad of log-linear graph paper routinely, in St Andrews, where Napier had been a teenage student around 1563. Today, all three tools are virtually museum pieces.

At my instigation, Napier now names a Reading Room in the redundant Church repurposed as the Richardson Research Library at Martyrs Kirk since 2016. His portrait proudly hangs on the wall. He also names the academic Chair of Astronomy from an earlier date.

Napier’s Magnum Opus on Logarithms appeared in 1614 after 50 years of self-financed development. After 30 years, in 1593, he published on his other passion, The Book of Revelation, the Last Judgement and The Apocalypse. Napier’s interests are spookily relevant to today’s global pandemic viral crisis. The Times made two passing references to John Napier, but in 2015 – late for 2014 (1614), and very early for 2017 (1617).

If only his insightful wisdom were reflected in the decisions and posturing of those who ‘lead’ us today.