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Maths often gets presented “without algebra”. People trying not to scare those who didn’t enjoy maths at school, or who are just glad it’s over. One reason mathematicians – and epidemiologists – use algebra is because it’s a convenient way to apply old maths to new numbers. 1/n
We’ve all got a bit of time at the moment, and there’s a bit of maths that we should be applying every day, whenever we hear the news. And to understand it properly, we’re going to need to look at some algebra. But there doesn’t need to be anything difficult about it. 2/n
The first formula you did see at school: y = mx. It describes a straight line on a graph. y is the number on the vertical axis; x the number on the horizontal axis. m is a number that stays the same: a constant. Our straight line is made by multiplying each value of x by m. 3/n
y = mx is called a “linear function”. It’s easy to see why: a “function” is something that takes one number (x) and turns it into another (y). And we’ve just seen that if we use this function to plot a graph, we get a straight line. 4/n
Linear functions are everywhere. Think about filling up your car with fuel: the price (y) is the price of the fuel (m) multiplied by the amount of fuel you buy (x). Two litres costs twice as much as one litre. Five litres costs five times as much as one litre. 5/n
But lots of functions aren’t linear. You are the result of a different type of function. When your parents’ sperm and egg fused to form one cell, you were conceived. Quickly, that one cell became 2, then 4, then 8, then 16, then 32 and so on. What function describes this? 6/n
Time for a quick detour into another piece of maths you remember. Square and cube numbers: You probably remember that “5 squared” is written as 5², and means 5×5 (which is 25); “5 cubed” is 5³ is 5×5×5 (which is 125). 7/n
Generally, we call these “powers”. 5² is “5 to the power of 2”. 5³ is “5 to the power of 3”. On Twitter, we use the ^ sign to show powers: 5² is 5^2. 5^4 means 5 to the power of 4: 5×5×5×5 (625); 5^6 = 5×5×5×5×5×5 = 15625. 5^10 = 9,765,625 With powers, numbers get big, fast. 8/n
So back to you in the womb. It doesn’t happen quickly – at least at first – but your cells double. First 2, then 4, then 8, 16… that’s 2^1, 2^2, 2^3, 2^4, and so on. By the time a baby is born, it has about 26 billion cells; a bit less than 2^25; or 2^26 for twins. 9/n
Now think back to our linear function, y = mx. In this function, each time x goes up one, y goes up by m; but for our developing embryo, each times x goes up by one, the value of y is multiplied by m. You should see that this function does the job: y = m^x. 10/n
To be conventional, we’ll use b instead of m. So in y = b^x, our constant is b. For the example of the cells in the embryo doubling, b=2. For x = 1, y = 2^1 = 2 For x = 2, y = 2^2 = 4. For x = 3, y = 2^3 = 8 And so on. Now look at the graph for this function: 11/n
We call this an exponential graph, and it often happens when things grow (like populations) or spread (like viruses). These all work by multiplying the number we already have. Parents, on average, are responsible for bringing two-point-something new people into the world. 12/n
When we also think about rate of dying, and how many people don’t reproduce, we can get a number for b. At the moment, for annual world population growth, b is about 1.012. To get the function for the next few years, we need to know our starting point, about 7.5 billion. 13/n
This gives us the full form of an exponential function, which actually has two constants: y = a×b^x, or y = ab^x. For our population growth, a is 7.5 billion. b is 1.012. and so the graph looks a bit like this. 14/n
So how do we control the population? We can’t do much about x: that’s time, in years starting now. An evil world dictator might reduce the size of the current population – reduce the value of a. But if (s)he did that, exponential growth would eventually bring it back up. 15/n
As we know, what we really need to do is reduce growth – that means reducing b. If b is over 1, the population grows exponentially. If b is 1, it stays constant. And if b is less than 1, it begins to shrink. 16/n
Now, if one person infected with a virus gives it to two other people, and they each give it to two more, you can see that the number of new infections grow like the cells in the new embryo. So what can we do to control the numbers? 17/n
Remember, our function is y = ab^x. Without a cure, we can’t do much about a. Social distancing and improved hygiene are aimed at reducing b; if we can get b under 1, we’re reducing the number of cases. 18/n
The value of b is some constant (bigger than 1). Before isolation, maybe 1.2. Isolation should make b a smaller constant. But when we hear on the news that “today was the biggest rise in new cases”, we shouldn’t be surprised: that’s exactly what exponential functions do. 19/n
What we need is for b to go down. When headlines tell us the big number, it’s bad news for each person and family involved. But it doesn't tell us much about the fight against the virus: for that we need to look at the graph, and to see whether we’re managing to change b. 20/20
Disclaimer: all of this is just about the maths, and the mathematical language we can use to discuss the spread of infections. Equipped with the maths, we can better understand what is being said by - and about - the experts who understand the disease.
(Also, I forgot to post the gif of the twins: born with 2^25 cells each, and 2^26 between them)
