Exponential Growth and R0 rates for the uninitiated.
Exponential growth for people who don’t understand it. Warning, this will probably take about ten minutes to read and I know on social media people don’t like to read anything which takes more than five seconds to read. The trouble is, some things take more than five seconds to understand.
Just for fun I wrote something for a friend while trying to explain the R rate and exponential growth, why this figure is far more important the the mortality rate distributed by conspiracy theorists and why because of this it is the number combined with the health effects which have driven countries into unprecedented lockdown measures. It’s lengthy by social media standards so if you aren’t interested I won’t be offended, I only think it might make a nice boot-camp for people who still think the survival rate, not the R rate, is the important thing, for illustrative purposes.
As you may well be aware R0 is the rate of growth, but what does it tell us and how can it help to make national health forecasts?
For any R rating you can calculate the number of infections on day i with the formula R^i. On a pocket calculator this formula often appears as a cursive letter x followed by a cursive y in superscript just off to the top right of the x. If you can find that key then you can go through the numbers too. We’ll see the explanation for why R0=1, which we’ve heard the Prime Minister speak about, is the magic number on which the future of lockdown hinges .
You see, when you have an R rating of exactly 1, nothing much happens, the infections stay stable, we can show this mathematically and you can try it yourself: 1^i always = 1, no matter what value you use for y. This means the number of infected people will remain stable over time.
Remember, when reading these numbers below, they are: “R rate”^”days”=”infections on that day”. With R values less than 1, the number of infected people falls over time, let’s see this in action with an R rate of 0.9 after 3 then 40 days:
With R values above 1, the number of new infection on any one day increases. Let’s use an R of 1.1 and see how it works out on day 3 then day 4:
This is why Boris Johnson explained he wants to keep the R rating below 1. That hinge point is where the 1 figure comes from.
So what makes epidemiologists jumpy is the same thing that makes mathematicians and computer programmers jumpy when they see exponents appear in their figures. R in each of these cases signifies the rate of growth of an exponential curve. The problem being, exponential curves almost always hide sleeping giants. They look only moderately frightening at first, but they can turn really nasty very quickly.
Let’s illustrate that by returning to what the R rate forecasts again, ie. the total number of new cases on any one day and we’ll see what happens if we change the R value. Remember first, with an R0 rate of 1.1 there will be 1.4ish new cases after 4 days, shown above.
How about we complicate things a bit this time and bring in mortality-rate too? This will instead create an illustrative example of where we might be heading looking forward by way of human deaths. Let’s give ourselves a nice realistic 1% mortality rate, the figure liberally shared on social media as the real issue nobody is talking about. So, for every 100 people who get it, 99 of them will live. So now in our improved outbreak simulation we are testing an R0 rate of 1.1 and a mortality rate of 1%, much as it was said to be before the lockdown. I’m going through this slowly so as to hopefully reduce the chance I’ll be accused of falsehood: I’m actually using the figures which have emerged from the published statistics, and we can change them, you can change them yourself if you think my figures are off.
So given these numbers of an R of 1.1 with the new mortality rate of 1% we can now predict the total number of deaths from the cases any number of days ahead with R0-rate ^ days * survival rate, so on day 50 with 99% survival and R=1.1:
1.1⁵⁰*.01 = 1.17
It says one infected person, with an R of 1.1 and a mortality rate of 1% we lose one person from the 117 new cases on the 50th day. I think many people will say this is nothing to be alarmed by. We can compare it to the background average: About 1,400 people die each day in the UK so our number is a fraction of the expected real-life total death figure.
The problem with the sleeping giant I mentioned is it falls in the fact that obtaining an accurate estimate for the R0 value on an ongoing basis is somewhere between hard and impossible, even if you are an expert. With the method we are using nationally we are not testing a fraction of the people we will need to, nor are we implementing the contact tracing necessary, to approach an accurate estimate for what the R0 value really is.
So what if we get it wrong?
Well we’ve seen, in strictly numerical terms, with a 1% mortality rate and a R of 1.1 in 50 days we’ll see few deaths.
What happens if we had thought the R rating was 1.1 but it isn’t. Instead of infecting a little over one person per day, as we calculated, the true number is a little closer to 2 per day. So let’s change the R rating from the 1.1 we thought it was to 1.7. We will keep the same 1% mortality. How many deaths do we get on day 50 if we have an R0 of 1.7?
We get 3.3 billion. Half the population of planet earth, on one day. With an R of 1.7 and a mortality of 1% the entire human population of the earth will have died before the 50th day. All of us. Everyone. Verify it on your calculator before moving on that the numbers I have given you are accurate.
This is of course hypothetical and in real life it would not happen exactly that way, but I do hope this illustrates from an epidemiological point of view why it’s an accurate understanding of the R rating and *NOT* the mortality rate which is important.
One more thing before I go. You’ve seen a difference in R rate of a little over 50% changes the death statistics from being manageable to being devastating. What happens if we change the much discussed mortality rate instead to see how that changes things? That’s easy. We’ll even rig the figures to favour the mortality rate. We’ll compare the difference of doubling the R0 rate to the difference made by multiplying the mortality rate by 10, beginning with a single infection:
R0=1, mortality rate=1%, 32 day ahead forecast:
1³²*0.1 = No deaths (0.01 of a person average).
R0=1, mortality rate =10%, 32 day ahead forecast:
1³²*0.1 = No deaths (0.1 of a person average)
R0=1, mortality rate =100%, 32 day ahead forecast:
1³²*0.1 = 1 death.
R0=2, mortality =1%
2³²*.01=43 million deaths.
It is NOT the mortality rate which makes the difference, it’s the infection rate. its’ simple: If nobody gets the disease nobody dies, even if the mortality rate is 100%.
Ask yourself, is even a 1 in 100 chance of dying if you contract the disease a gamble worth taking for you right now?
Lock-down doesn’t change the mortality rate. The point of lockdown is it reduces the number of people who contract the disease from the people who have it.