When accompanying the number of infected people in a given place it is easy to be shocked at how quickly the numbers rise after the apparent stability of the early phases. Understanding the laws that govern how a virus spreads is important to ensure that we control the panic but still take the actions necessary to minimize damage.
What is exponential growth?
Exponential growth means that every day the number of new infected is equal to the number of currently infected people multiplied by some constant value. For example, if this constant value is 2, and in a certain day we have 10 people infected, 20 new people will be infected in that day. These will be added up to the 10 we already had, meaning we now have a total of 30 people infected. The following day we will have 60 new infected, making a total of 90 with the ones we already had, and so on. If we want to describe this in a more mathematical way, we can write it like this:
Don’t let yourself be scared: being able to read this isn’t necessary to understand how exponentials work. If you want to, however, I is the number of infected people, t is how much time has passed, and C and τ are simply two parameters (numbers) we can adjust to make this equation describe any real life exponential growth. e is just a number called Neper’s constant. Any other number bigger than 1 would work, but this one is just more convenient for reasons that we don’t really care about here.
A virus spreads from contact between an infected person and a healthy one. If we say that each infected person has enough contact with healthy people so that they, on average, infect N people, then the total number of new infected will be N times the number of infected we had already. This is precisely what our exponential model describes, so we should be able to make some predictions with it.
If we draw the results in a graph, we get a shape like this one:
We can see that the more time has passed, the more vertical the line becomes, meaning the total number of infected grows faster and faster.
Looking at the data
Let us now see how our model does in the real world:
The blue dots are the real world data while the orange line is an exponential function adjusted to be as close to them as possible. A good way to examine the results is to plot them in a logarithmic scale as we will get a simple linear graph:
We can that our model seems to make a wonderful job at predicting how many infected people we will have. The number of infected seems to multiply itself by 10 about every 9 days.
It is very easy in these conditions to be very scared when in less than two weeks we have 10 times as many infected people as we did before; after all, if the trend keeps up, by day 71 we will have hit 7 498 982 066 people, which is more than the total world population.
In the real world, we run into a limitation: after some time, there are no more people to infect. In our exponential model, however, the numbers grow forever up to infinity; at some point it must deviate from real world data. In our simple description we failed to take into account things like the fact that people tend to live in communities with limited exposure to each other, that there is a limited number of people on the planet and, most important of all, that we can limit the contact we have with other people. The model used for this kind of conditions is called a logistic curve.
A logistic curve is in the beginning indistinguishable from an exponential curve, but it soon slows down and stabilizes. We can look at a real world example in China. In Hubei, the infection is in a much more advanced phase in which the halting is evident:
We’re left with a problem then: how do we know at what point of the curve we are? When are things going to slow down? To know we must find what is called the inflection point, which is that middle point where the graph goes from curving up to curving down. To find it we must simply monitor what is called the growth factor, which is very neatly explained here:
In the long term, the growth of the epidemic is extremely sensitive to how many people we are in contact with and the standards of hygiene we maintain. Even if the virus is mostly dangerous only to immunocompromised people, it is the responsibility of everyone to ensure that we do everything we can to curb the transmission. The point of social distancing is not to make sure you do not catch the virus, but to disrupt the domino effect at every possible point.