The Mathematics of COVID-19

There has been so much coverage of COVID-19 that some of the language of epidemiological models used to forecast the virus’s evolution is now familiar to us. But for those who do not fully grasp how those models work, here is a little primer.

The reproduction number R

It’s not often that a mathematical symbol – even if it’s only a letter of the English alphabet – becomes common in the public media. But since the onset of COVID-19 we occasionally see the symbol R (or R0) – the reproduction number – mentioned in mass media articles.

R is the number of people that a person who already has the disease – an infectious person – will infect. (An infected person is assumed to be infectious; as we shall see, this assumption can have variations.) If R is much greater than one, the disease will spread rapidly. If R is less than one, the disease will gradually dissipate; the smaller R, the more rapid the dissipation.

Think of it this way. Suppose R is the number of children a typical woman has. In the 1960s R in the United States was about three. Population increased rapidly. In some countries in Africa and parts of South Asia, R can be as much as 5 or 6 – population increases very rapidly so long as the death rate is not also high.

In Italy and other countries in Europe, in Taiwan and Japan and other places, R is low – even below one. The population ages and dwindles.

The religious Shaker communities of the late 18th and 19th centuries had – at least in principle – an R of zero. They were celibate and did not believe in child-bearing. Therefore, unsurprisingly, they died out.

Another useful figure is the serial interval – the time from when a person gets infected until the next person that person infects becomes infected. The serial interval for SARS and COVID-19 has been estimated at seven days.

If the serial interval for human births were seven days instead of what it is, about 25 years – that is, if humans reproduced every seven days instead of 25 years on average – and the reproduction number were two, then in a year there would be two quadrillion new people on earth (2,000 trillion). That’s how fast it goes.