Coronavirus - the basic maths

This is a distribution known as the Normal Curve.

It has other names and is also known as a Bell Curve (because it looks like a bell) or Maxwell Distribution. Both ends tail off into infinity but never actually reach the zero baseline I have deliberately chosen a diagram with no numbers on it because the numbers do not matter very much at this stage. It is used to model all sorts of things. The peak of the curve often represents the mean of whatever data is collected (adult height or weight, perhaps). This curve also models the number of people infected during an epidemic at any given point if left unchecked. To start with the numbers infected are very low and do not increase much or very quickly, then it grows rapidly and then it peters out. The reasons for this are fairly obvious when it comes down to it. Just imagine 6 people. To start with 1 of them will have the virus and there are 5 possible targets for infection. Once one person is infected 2 have the virus with new 4 targets, then 3 and 3, then 4 and 2, then 5 and 1, then 6 and 0. Nobody left to infect that hasn’t had it and those who have had it will now have natural anti-bodies to protect them against re-infection. However in the UK we have around 65m people not 6. The same logic applies but the numbers are far bigger. Once 1 person infects another there are 2 people spreading the disease. Those 2 might infect another 2. Now we have 4 people. Those 4 people infect another 4 making 8. Those 8 infect another 8 and it is now 16 and so on. So the number of new infections with each passing day is 1, 2, 4, 8, 16… Before long the number of new cases is huge. This is known as exponential growth and an exponential curve looks like this:

This next bit is quite technical. I’ll mention it because it fundamentally underpins all that follows. After which I will abbreviate massively so skip to that bit if you don’t understand what follows. The curve above is based on 2. 2 is a real number. It’s just 2. Like 2 legs or eyes or feet. 2. There is another number a little greater than 2 but less than 3. It is an irrational number meaning it can never be written down with exact accuracy (just like π or √2) and so has a symbol: e. If I replace 2 with e in the graph above the curve looks much the same but, crucially, the gradient of the curve is now also exactly the same as the value of the function (equation) that produces it.

This can be used to model all sorts of naturally occurring phenemona such as radioactive decay and much else besides. Don’t worry too much about e. Suffice it to say e is approximately 2.72. What matters here is the shape of the curve. It represents something that grows ever more quickly. It can also be used to model the spread of infection during an epidemic. There is family of curves which are all vaguely similar and are all known collectively as exponential functions. The exponential function, essentially the Daddy of them all, as already mentioned is y = e^x. The general form for all of the exponential functions is y = ae^bx + c where a, b and c are the particular constants (i.e. just numbers like 2, e, 9, 131, π…) which define the particular curve and x represents the time variable. In terms of viral spread a, b and c are unique to each virus. If you like they are the control factors. One of them is rate of contagion (just how contagious is it?), another is the number of interactions between people (which is why we are being urged to self-isolate) and another is the number of people who already have it. Nothing to be done about the passage of time, nothing to be done about those who already have it and nothing to be done to stop the exponential growth but it can be slowed down if we reduce the number of interactions (hence self-isolation) r the likelihood of it spreading (hence the hand washing). We are currently witnessing an absolutely classical example of exponential growth:

There you go. That’s where we are. I could model the curves and deduce the exact equation that defines them but that’s a level of detail unnecessary. The plan recently announced by our government was to suppress the rate of spread, move the peak forward as far as possible and so allow our medical capacity some chance of coping and not being completely overwhelmed. They used this graph to illustrate it.

Finally there is talk of “acting at the right time” and “maximising the effect” etc. This is mathematically incoherent. The mathematics of maximisation is known as calculus and you need to do something called differerentiation. This works well with some curves with quadratics being the easiest example to understand (you can see where the maximum is immediately).

…but if you differentiate an exponential function you just get another exponential function. There is no maximum anywhere on the curve. There is no exponential part of the curve because it is all exponential. The best way make a difference - to slow it down - is to act early. That’s it. I hope that this helps you to understand the underlying mathematics at least a little better than you did before.     And where we are headed is not good.