When maths really matters
By Trevor Moore
Two hours after I wrote this article, the Chief Medical Officer referred to concern about a “very, very steep growth … [that] … is very concerning. The final graph below shows that.
I hadn’t realised until I spoke to a few people that the term “exponential growth” is not well understood. We have, over the last few weeks, been showered with numbers and new terms that are important to epidemiology.
Epidemiology of course is the study of how diseases spread in populations. It’s evidence-based and that means that the epidemiologist needs to collect data. Based on that data she can make projections by building a model that represents how things happen. Now, I am not an epidemiologist, but I am a mathematician and there are some basic parameters that the epidemiologist modeller needs to know so that the maths works.
I have had people (and more than one) say to me that COVID19 is like the ’flu. Now I am not a medic so I cannot comment on the physical symptoms but there are some basic differences between the epidemiology of the ‘flu and of COVID19. I learnt this morning, from Professor Hugh Montgomery who is the Professor and the Director of the UCL Institute for Human Health and Performance at University College London, that if I have “normal ‘flu” I will infect about 1.3 to 1.4 other people. This number is called the reproduction number for a disease and epidemiologists call it R0 (r-zero). Now R0 is determined by multiplying the “attack rate” by the number of contacts that the population has had. The “attack rate” is the chance that a contact will get the disease.
When I consult the Online Portal for COVID-19 Modeling Research I find several estimates for what they call the reproduction number. Montgomery says that the number is 3 and though I cannot find a source for that number, I assume (and it is an assumption) that it is based on up-to-date data. I have found a reference in the Lancet (published on 11 March 2020 but clearly based on much older, and only Wuhan, data that appears to be peer-reviewed (in other words other academics have looked at the work and given it a tick) that finds that the reproductive rate in Wuhan dropped from 2.35 to 1.05 after travel restrictions were introduced. Now if R0 is less than 1 the disease is going to die out. That’s why we have the restrictions we have: we need to get R0 down to less than 1. Unlike the ‘flu, for which is also more than 1 but only slightly so, we have no immunity to COVID19 and we have no vaccine.
Having said that epidemiological modelling is based on maths and statistics and as someone once said “there are lies, damned lies and statistics.” The saying is attributed to Disraeli because Mark Twain got it wrong in his autobiography. It seems that the quote is by Arthur James Balfour, 1st Earl of Balfour who was Prime Minister of the UK between 1902 and 1905. The epidemiological models provide a statistical analysis of a load of data and they help us form a view about what might happen. There is no certainty about any of the projections from a given model. Different models give different projections. But there are some underlying mathematical facts that are facts. Not false facts. Actual facts. And the term “exponential growth” is one such term.
So, here’s a graph what happens with R0 = 1.4 for the ‘flu. The horizontal (x) axis is simply a sequence of numbers, it could be dates but we assume that they are evenly spaced. You can see that the graph rockets up and to the right … and it gets quicker as it goes.
Now let us take our friend Prof Hugh Montgomery’s estimate of R0 = 3. Here is the graph for R0 = 3. You can see that it looks similar but the slope at the right is much steeper than the one for R0 = 1.4. You can also see that the numbers are significantly larger. With R0 = 1.4 and starting with 1 case and assuming that we step along the horizontal (x) axis in equal steps, after 17 steps (that number is not significant) we have 305 cases. The equivalent for R0 = 3 is 129,140,163 cases. Now that is far higher than the population of Australia. We simply do not have the resources to cope with R0 = 3. Again that is why you need to wash your hands and keep your distance. Reduce the attack rate.
I illustrate the difference in growth between R0 = 1.4 and R0 = 3 in this graph. The blue curve (it looks like a line) is R0 = 1.4. The orange line is R0 = 3. It makes a huge difference.
Now, before we look at the Australian numbers, we have one more piece of mathematics. I am going to assume that we all remember the logarithm (log) of a number. In my day at school we used to have little booklets of logs that we looked up: there were 4 figure logs and 7 figure logs, and they made multiplying big numbers easier. Now, when you put the log of an exponential curve you get a straight line. It looks like the figure below.
Now let’s look at the Australian numbers. These are sourced (24 March 2020) from Coronavirus (COVID-19) in Australia. The current number of cases is 1886 up from 709 five days ago and 249 ten days ago.
Now if we were reducing R0 then when we plotted the logarithm of the numbers, we would find a curve that falls downward as it moves from left to right. Unfortunately, that is not what we find. I have plotted the numbers on the following chart and shown the line that we should be falling under. We aren’t. The crossover point on this chart is 13 March 2020. So, something happened in the couple weeks before mid-March that caused things to deteriorate.
Post script: two hours after I wrote this the Chief Medical Officer referred to concern about a “very, very steep growth … [that] … is very concerning. The last graph shows that.
Wash your hands. Keep your distance … … do whatever it says on the Federal Government Health Website. And if you aren’t already doing it, then you better start now.