A Parable About a Run on Log Paper

An astronomer discovers a weak radio communication emanating from an advanced and surprisingly close civilization. The transmission appears to be an internal communication leaking into space. Earth's inadvertent eavesdropping reveals a magnitude buried in the signal that is unmistakably intended to communicate a number of some kind. The number is updated regularly and is rapidly growing. Experts begin plotting the alien number on logarithmic paper. They hypothesize that because it is exponentially growing, the mysterious quantity may reflect contagious growth of some kind.

Possibly the civilization has discovered how to breed spaceships that breed more spaceships and the number might be an artifact of the civilization's actual growth. But a more plausible and popular theory emerges - the aliens are infected by a virus and are internally communicating the count.

News agencies cover little else. Logarithmic plots of this mysterious quantity enter our living rooms via the nightly news. The earlier data is a bit noisy but squinting, it looks a lot like a straight line on log paper. The world is transfixed. A run on logarithmic paper ensues, rapidly followed by a run on toilet paper and a race between countries to print money. The world is running out of paper. Despite the difficulty of finding paper, data scientists create more logarithmic plots than the world has seen in quite a while. It is observed that every person who creates a logarithmic plot inspires two more people to do the same.

The alien data is de-noised and backfilled somewhat, with contributions from experts across the planet. A nuance emerges. All the logarithmic curves bend down even though they are supposed to be straight. This is getting harder to deny, but it is widely believed, for reasons that are forgotten or never existed, that for some reason the growth must be exponential. The logarithm infects all statistical activity. Forecasters tinker however. They must. They model slowly changing doubling times. Periodic revisions to the slope are published. Reproduction numbers are updated. Forecasts extrapolated from lines on log paper are retracted, then updated, then retracted again.

Then comes a plot twist. A radio telescope engineer is able to determine one additional and crucial piece of information about the supposedly beleaguered alien civilization: their speed. The engineer notices that for several days in a row his approximate velocity measurements, however imperfect, seem related to changes in the mystical number the aliens broadcast. Inspired, the engineer takes another look at the historical alien numbers received by Earth - but this time he puts aside his logarithmic paper and instead, plots the square root of the mysterious indicator against time. Behold, a straight line !

The aliens have not been broadcasting the growth of a virus. The truth: several weeks ago they turned on the engines and began accelerating towards Earth. They have been communicating amongst the fleet a record of distance travelled, which of course rises quadratically not exponentially.

At this point the statistical insanity is supposed to end, but it does not. With the population mostly impervious to evidence and transfixed by logarithmic plots, a new debate erupts concerning the question of whether the aliens will ever reach Earth. No-one can deduce the mathematical form taken on the logarithmic plot - but they draw lines based on polynomials that indicate the curve will turn downwards soon. In a remarkable spin on Zeno's paradox, it is announced that the aliens will never reach us.

The problem has magically gone away.

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This story is continued in a technical note in which I exhibit a plausible trajectory for a pandemic that transitions from exponential to power law growth and back down, thus explaining some of the curvature you see on those bendy curves. I present a continuous agent model for disease spread on the plane, and show that population dynamics are described by an augmented compartmental model (differential equations) whose only deviation from the standard models is an infection attenuation function taking a simple form. Peak infection is driven not only by a herd effect but also the exhaustion of novel acquaintances. The range of outcomes is very different to the standard CIR or SEIR models and, due to the different nature of the equilibrium, there are qualitative differences in the way the model reacts to lifting of a lockdown.