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The simple mathematics of social distancing

As we continue to batten the hatches in response to the coronavirus threat, we have all become familiar with the term 'flattening the curve'. Most of us have seen illustrative curves and have a grasp of the basic concept. And in particular, we accept the need to flatten the curve to control the spread of the virus, which we are told is best achieved by 'social distancing'.

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

Let's look at some simple calculations

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 at between 2.0 and 3.0 for COVID-19, 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 the virus 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.

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% probability that each interaction will pass on the virus. That means on average, the carrier of the virus will infect,

4.5 x 10% = 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 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 a couple of weeks ago.

Where does 'social distancing' come in?

Without a vaccine, the 10% 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).

The one parameter that we can materially manipulate is the number of interactions that people have with each other.

Suppose 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% = 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% 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 two weeks ago, we have actually seen the growth rate fall to less than 5% per day, so the early signs of enforcing distancing strategies are encouraging.

And if close people interactions could be reduced to just two, then R becomes 1.0 (= 2 x 10% 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.

Whoever feasted on bat pie in Wuhan several months ago probably had the most expensive meal in history.

 

Tony Dillon is a freelance writer and former actuary. This article is general information and does not consider the circumstances of any investor.

 

6 Comments
Chris Holmes
April 12, 2020

The cost to the government and our economy is not little. The cost to an infected patient is possibly extreme. Would anyone expect UNDER KILL.
When the history of this event is written with all the poor reporting of numbers out of China corrected, which I assume confused more than the WHO [and poor poor Trump still thrashing around out of his depth], started by Chinese officials falling over themselves to coverup the situation out of a false sense of loyalty with little interest in the people, we may be able to appreciate the need for extreme action on so limited information.
We may be more than thankful for overkill.

Graeme Bullock
April 12, 2020

I agree with Alanb and others that the restrictions have been poorly thought through, inconsistent to the point of absurdity, and the punitive measures grossly out of step with the potential level of harm. But I have to go one step further. It is fact that in the 2017 influenza season of 19 weeks in the US, 30 million were estimated to be infected and 80,000 people died (CDC numbers), and there was no economic or social lockdown, and very little publicity until after the event. OK, Fear of the Unknown (is that FOU?) created a different dimension to COVID-19, but there was plenty of statistical info coming out of China. Even our own Chief Medical Officer suggested that for the vast majority of infected people it was a very mild disease (though he has been strangely silent since).
So, I can walk jog or bike ride on busy paths, I can go and collect firewood in the bush if I need it, but I can't solo fish (for food), or solo canoe or kayak (for exercise) in splendid isolation. Ho hum, guess I'll just wander off to Bunnings.

AlanB
April 11, 2020

The fact is that in Australia to date only 57 people have died from this flu.
The destructive measures and removal of civil liberties implemented to stop its spread are far out of proportion with its severity.
The problem with this type of modelling is that it fails to account for differences between populations and assumes the same spread rate for diverse populations in completely different.
For example, this model does not factor in differences in age demographics (countries/regions/cities with proportionately older populations), differences in housing, lifestyle or quality of healthcare. This is why a flu will always kill more people in countries (like Italy) or cities (NY) with older populations, cramped, unsanitary living conditions and an under-resourced heath system.
This model therefore totally over-estimates the danger for Australia, as evidenced by our relatively lower infection and death rate and should not be relied on for social, economic and public policy decisions impacting millions adversely right now and into the future.

AlanB
April 11, 2020

The fact is that in Australia to date only 53 people have died from this flu.
The destructive measures and removal of civil liberties implemented to stop its spread are far out of proportion with its severity.
The problem with this type of modelling is that it fails to account for environmental differences between populations and assumes the same spread rate for diverse populations.
For example, this model does not factor in differences in age demographics (countries/regions/cities with proportionately older populations), differences in housing, lifestyle or quality of healthcare. This is why a flu will always kill more people in countries (like Italy) with older populations, cramped, unsanitary living conditions and an under-resourced heath system.
This model therefore totally over-estimates the danger for Australia, as evidenced by our relatively lower infection and death rate and should not be relied on for social, economic and public policy decisions impacting millions adversely right now and into the future.

Reader comment
April 11, 2020

The Government Response is Absurd overkill. A bad diet will compromise your immune system. The vaccine will not work. Depression from no work and no money will kill more than the Virus.

Simon Taylor
April 09, 2020

Excellent overview. Many people just don't seem to get it and this explains it well. The advice to just 'stay at home' is clearly directed at this basic principle of reducing the number of INTERACTIONS between people. Going out to do anything just increases the chance of a social interaction or occupying a space where someone with the infection has been.
The other observation is that we are NOT expecting eradication. Flattening the curve will PROLONG the duration of the spread of infection, but make it manageable in our health system and reduce the crippling impact on health services for non-Covid reasons. We need this to develop more widespread immunity in the community - otherwise the nation will continue to be at risk from visitors from overseas and further community spread.

 

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