The mathematics behind social distancing

But what exactly is the curve, and what is the maths behind the "social distancing" theory, in layman's terms?

A key determinant of the curve is the "basic reproduction number". This number is usually denoted by R. It is an estimate of the average number of people who will catch the virus from a single infected person, at the outbreak of an infectious disease.

It has been estimated to be between 2.0 and 3.0 for the coronavirus, meaning each infected person infects between two and three others in a largely uninfected population. A figure greater than 1.0 gives rise to exponential growth in the number of infected cases, whilst a figure of 1.0 or less means the virus will quickly run out of puff.

And so to contain COVID-19 in the absence of a vaccine, R must be significantly reduced. The corresponding figure for influenza is about 1.3, above 1.0 but not out of control - and besides, we have a vaccine to help keep it under control.

But what drives the reproduction number, R? Let's use some simple numbers to explain (assumptions are for illustrative purposes only).

Assume someone who unknowingly has the virus, has on average, four and a half close interactions with uninfected people in a day (that is, an average nine interactions every two days). Assume also that there is a 10 per cent probability that each interaction will pass on the virus. That means on average, the carrier of the virus will infect:

4.5 x 10 per cent = 0.45 people per day.

Assume also that the average duration that an infected person remains infectious is five days. Then on average:

0.45 x 5 = 2.25 people will receive the virus from the carrier.

Which is within the range of where the COVID-19 R factor is estimated to be.

The value of R in turn drives the exponential growth rate, the rate at which the number of infected cases grow. The maths is complex, but under certain modelling, we can determine the growth rate approximately from the basic reproduction number R.

Now if someone with the virus remains infectious for five days, then the inverse must be the removal rate of the population exposed to the risk of contracting the virus. What that means is that for every person who becomes infected, after five days there is one less person in the population who can become infected, or 0.2 people per day (0.2 being the inverse of 5).

We have then 0.45 infection transmitting contacts per day, and 0.2 contacts removed per day, leaving a growth infection rate of:

0.45 - 0.2 = 0.25 per day.

That is, a growth rate of 25 per cent per day would see case numbers double approximately every three days, which is a rampant virus. In fact, that was approximately the exponential rate of growth we were experiencing in Australia just last month.

So where does "social distancing" come in?

Without a vaccine, the 10 per cent probability of passing on the virus with close contact, and the five-day duration of being infectious, cannot really be altered behaviourally (though quarantining high-probability carriers can reduce how long people can spread the virus). But the one parameter that we can materially manipulate is the number of interactions that people have with each other.

Suppose therefore, that with social distancing restrictions put in place, the number of close daily interactions between people has been reduced by a third to three. Then a virus carrier will infect on average:

3 x 10 per cent = 0.3 people per day.

And the R factor has reduced to:

0.3 x 5 = 1.5, implying a less rampant virus.

And significantly, the growth rate of infections has been reduced to:

0.3 - 0.2 = 0.1 per day.

That is, 10 per cent growth in the number of infections per day, which slows the rate at which the number of cases doubles, to 7.25 days.

In this instance, the growth curve has indeed been flattened, and that may be enough to avoid an overwhelmed hospital system. At an R of 1.5, the virus continues to spread, but not at unmanageable proportions.

Since social distancing restrictions were ramped up in Australia several weeks ago, we have actually seen the growth rate fall to less than 5 per cent per day, so the early signs of enforcing distancing strategies are encouraging.

And if close interactions could be reduced to just two, then R becomes 1.0 (= 2 x 10 per cent x 5) and an important threshold would be reached. The growth rate would be zero (= 0.2 - 0.2), so there would be no increase in cases per day, and the virus would quickly fizzle out.

The significance therefore, of controlling the exponential growth rate cannot be underestimated, and is the key to dampening the speed of the spread of infectious diseases.

Without social distancing, the virus would not be subdued, and the hospital system would not cope. Infections would still taper off at some point, as the virus runs out of population to infect, but not before damage to the health system has been incurred.

It is so important, therefore, that we adhere to the government's restrictions on socialising, with the health and economic costs of the virus already substantial, running into trillions of dollars globally.

Meaning whoever it was who feasted on bat pie in Wuhan several months ago has probably had the most expensive meal in history.