If there is one thing this pandemic has taught us, then it is that few people truly understand exponential growth. This post is an attempt to explore this magical function with minimal prerequisites.

How do we know something will behave exponentially?

When you hear exponential growth, you probably picture a curve that starts increasing slowly and increases faster and faster. And that is how many people use the term: “Something that increases faster and faster”, even though the relation might not be exponential at all. And if that is how you think about exponential growth, it is quite natural to doubt that a disease should spread exponentially. I mean why should it spread faster and faster? This seems like it requires an explanation.

To actually understand why, it is helpful to forget this preconceived notion of exponential growth. Instead we are going to build it from the ground up. In other words, we are going to define exponential growth in a way which makes it obvious why disease should spread this way, and then we will show that this type of growth becomes faster and faster. Not the other way around.

So how can we model an infectious disease? Let us start with a single person getting infected. This person is going to meet a random number of other people and infect them with a certain probability. Let us say that on average, an infected person infects $0.7$ other people per day. Similarly the person might become healthy again, perhaps half of the infected people will become healthy every day.

So the change of infected people from one day to another is

Now you might find my particular selection of infection rates and recovery rates unrealistic. That is fine. Select other rates. The only thing I want you to realize is, is that both the number of newly infected people as well as the number of recovered people is proportional to the currently infected number of people. In other words: Our rate of change is just a constant multiplied with the current number of infected people

$$ \Delta \text{infected} = c\cdot\text{infected}. $$

This property

the rate of change is proportional to the current value

is how exponential growth should actually be defined. Now we just need to get back to the faster and faster growth bit. So consider the infections $I_d$ at day $d$

Iterating on this idea (“induction”) yields

$$ I_d = (1+c) I_{d-1} = (1+c)^2 I_{d-2}=\dots=(1+c)^d I_0. $$

So now we have a function to predict the number of infected people on day $d$ given a set number of people infected on day zero $I_0$

$$ I(d) = (1+c)^d I_0. $$

Faster and Faster Growth

For $c=0.2,$ as we have used in the motivation above, and one infected person on day zero $I_0=1,$ we get

After the first month only $236$ people are infected, which is less than a school. One month later it is a small town. One more month and more than $15\%$ of Germany is infected, 10 days later the entirety of Germany. And it would continue growing like that! Wait a minute… How can it continue to grow when the entire population is infected? Well we made a small mistake. You cannot infect an already infected person. So we need to subtract the number of already infected people from the number of people a person would have infected if there were no other infected people around.

But at the beginning of a pandemic, there are not that many infected people around, so the likelihood of “infecting an already infected person” is very low because you do not meet many infected people. In other words: if the number of infected people is small, then this correction we need to make is negligible. For this reason, exponential growth is a reasonable approximation to infectious behavior until the number of infected people becomes large enough such that it is likely that you pseudo infect some of them as a infected person. Only then does this problem actually start to have an effect.

Takeaways for Pandemics

An incidence rate of 2000 (per 100 000) might be high when it comes to hospitalizations and the strain on the healthcare system, but as a share of the population it is still “only” $2\%.$ Therefore the likelihood that a sick person meets another infected is only $0.02$ which is still quite small. Therefore exponential growth is still a good approximation. For smaller incidence rates this is even more so. So exponential growth is a good approximation during the entire pandemic. The most important thing here is the growth factor $c$. The equivalent quantity, the reproductive value $R=(1+c)$ is what was (and is) widely published during the pandemic, or to be more precise $R_0=(1+c)$ (which does not take already infected people into account).

In our example we had a relatively mild reproductive value of $1.2$ and yet the disease spread like wildfire. So when an $R$ value of $3$ was estimated at the beginning of the pandemic, anyone who understood how exponential functions work shook their heads when politicians argued “but it is just 3 people”. And if you were not before, I hope you are one of those next time.