### Exponential Growth must come to an End

In its initial stages, the Covid-19 pandemic grew at an exponential rate. What does this mean? The number of infected people in a country is growing exponentially if it increases by a fixed multiple R each day: if N people are infected today, then R times N are infected tomorrow. The size of the growth-rate R determines how rapidly the virus is spreading. An example should make this clear [TM185 or search for “thatsmaths” at irishtimes.com]. “Flattening the curve” [image from ECDC].

Suppose that, on day zero, there is one infected person; one is enough to start an epidemic! We assume, for simplicity, that an individual is infective for just one day. Suppose each infected person transmits the virus to two others and then recovers. After one day, there are 2 infected people, after two days there are 4, after three days 8, with the number doubling each day. That is exponential growth, with a doubling time of one day.

From the initial case, the daily infection numbers continue with 2, 4, 8, 16 and 32. The numbers are small, but not for long. The tenth term is about a thousand and the twentieth term is about a million. Under these (highly simplified) conditions, the entire population of Ireland has been infected within 22 days.

Geometric Series

Some of us recall school mathematics lessons on geometric series. Each term of such a series is multiplied by a number R to get the next term. We may even remember a formula for the sum of the first N terms of the series; it involves R raised to the power N, that is, R multiplied by itself N times. The terms of a geometric series grow exponentially; the power N to which the growth-rate R is raised is called the exponent. If each term is double the one before (so that R = 2) and the first term is 1, then the tenth term is about a thousand and the twentieth about a million.

Of course, exponential growth cannot continue indefinitely. Since the population is finite, and the vast majority of people recover, the graph of the number of infected people is bell-shaped, growing to a maximum and then declining. If the growth-rate R, related to what epidemiologists call R-nought – the basic reproduction number – can be reduced, the bell-shaped curve can be flattened and extended so we can buy time by reducing the peak burden on the health service, and save lives.

The Origin of Chess

Legend has it that, long ago in the Gupta Empire, a great-but-greedy mathematician, Grababundel, presented to the Maharaja a new board-game that he had devised, called Chaturanga; we call it chess. The Maharaja was so pleased that he asked Grababundel to name his reward, expecting that he would ask for the hand of the Crown Princess. But no! Grababundel said simply: “give me one grain of rice for the first square on the board, two for the second, four for the third, and so on, doubling each time until the 64th square.”

This is exponential growth with a vengeance: the total amount of rice is about 18 million million million grains. If a grain is spherical with radius 2 mm, the packing density is 3/4, and the rice is stacked in a conical heap of height H and radius H, the total volume of the heap is about 800 cubic kilometres and the height of the cone is 9.3 km, higher than Mount Everest (8,848 m).