Compound infections

This graphic beautifully illustrates exponential growth: the stem splits into two branches, which each split into two branches and so on. Soon there are so many branches, they form a dense canopy.

Out of the many things 2020 has done, one was to teach us about exponential growth. You can easily see why epidemics can grow exponentially. If one infected person infects, say, 2 others, then a single infection gives rise to $2$ new ones, which in turn spawn $2\times 2=4$ new infections, which go on to create $2 \times 2\time0\2 \times 2 =8$ new infections, and so on. After $n$ generations we have $2^ n$ new infections.

The expression $2^ n$ grows incredibly fast as the exponent $n$ grows. And if an infected person infects other people within a relatively short period of time, then the epidemic will also grow incredibly fast in time. This explosive growth will continue until transmission is sufficiently reduced by one means or another.

When epidemiologists predict how a disease will spread in the shortish term, they do indeed use an exponential model, but it looks a little different. Imagine that now, at time 0, we have $N$ infectious cases. Then the exponential model predicts that in $t$ days the number $c(t)$ of cases will be

$c(t)=Ne^{rt},$

where $r$ is the growth rate of the disease per day. The growth rate obviously depends on the particulars of the disease and situation you are dealing with. However, $e$ does not depend on the disease or situation at all — it's a mathematical constant, also known as Euler's number, roughly equal to 2.7. Whatever disease you are trying to model, be it COVID-19, the flu, measles, or HIV, the base of the exponential expression in the model is always $e \approx 2.7.$

But where does the $e$ come from? What makes this number so universal?

Counting things

Compound interest is a blessing for saving, but a curse for debt.

The answer to this question is purely mathematical — and one way of explaining it involves the same calculations you'd use if you wanted to work out how the money (or debt) in your bank account grows through compound interest. To separate out the maths from epidemiological, virological, medical, etc, questions we will phrase our mathematical explanation in general terms.

Imagine that you have $N$ things (e.g. infectious people), each of which produces a fixed number $r$ of new things (e.g. new infections) a week. The question is, how many things will there be after some number $t$ of weeks?

To find an answer, let’s start counting. There are $N$ things to start with. Since each of them produces $r$ new things during the first week, there will be $rN$ new things after the first week. Adding this to the $N$ things that already exist, we see there will be

$N+rN=N(1+r)$

things after the first week.

During the second week, the number of things grows by a factor of $r$ again, meaning there are

$rN(1+r)$

new things in addition to the $N(1+r)$ things that already exist. Thus, at the end of the second week there are

$N(1+r)+rN(1+r)=N(1+r)(1+r)=N\left(1+r\right)^2$

things.

Similarly, after the third week the number of things will be the $N\left(1+r\right)^2$ that already exist plus $rN\left(1+r\right)^2$ , so that’s

$N\left(1+r\right)^2+rN\left(1+r\right)^2=N\left(1+r\right)^2(1+r)=N\left(1+r\right)^3.$

If you think you can spot a pattern here, then you are right. Continuing this calculation, you can show that after $t$ weeks there will be

$N\left(1+r\right)^ t$

things.

The graph below shows how the number of things grows, according to this calculation, for $N=1$ over a period of ten weeks. Move the slider on the left to change the value or $r$ .

One thing that immediately meets the eye is that this graph is like a staircase rather than a smooth curve. That’s because in the graph we implicitly assumed that the new things appear at the very end of each week, at the stroke of midnight on Sunday.

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But this can lead to an underestimate of the number of things! We ignored the possibility that an existing thing A could spawn one of its offspring, thing B, earlier during the week, rather than at the end of Sunday. This new thing B would then not only exist earlier than we thought, it could also produce its own offspring before the end of the week. Our calculation above didn’t account of this potential early offspring of B, and may therefore give too low an estimate.

Counting things better

But, never fear, we can improve on this. Let’s allow the possibility that each thing spawns some offspring every day, at the rate of $r/7$ things per day (so it’s still $r$ things overall within a week).

How many things would you have at the end of the first week? Well, the same calculation as above, applied to days rather than weeks and with $r$ replaced by $r/7$ , shows that at the end of the first week there will be

$N\left(1+\frac{r}{7}\right)^7$

things.

At the end of $t$ weeks (which are $7t$ days) there will be

$N\left(1+\frac{r}{7}\right)^{7t}$

things.

If new things are being spawned all the time, then this gives a better estimate than our first calculation. However, just as the first calculation assumed things spawn new things only at the point of midnight on Sunday, this calculation assumes that things spawn new things only on the point of midnight at the end of each day. This could again lead to an underestimate (things could spawn offspring at midday who then go on to spawn their own offspring the same day).

The next thing to do, then, is to switch our basic time period from days to hours or, even better, to minutes or seconds.

Actually, we can go even further and be more general. Let’s divide our week into $n$ time periods of equal length, where $n$ is some very large number, so the individual time periods are very short. By making $n$ larger, you can make the time periods as short as you like. We assume that each thing produces new things in each of these short time intervals at the rate of $r/n$ things per interval. The same reasoning we used above then shows that at the end of $t$ weeks the number of things is

$\begin{equation} N\left(1+\frac{r}{n}\right)^{nt}\end{equation}$

(1)

To the limit

The question is, what happens to this expression as the number $n$ of time periods gets larger and larger (so we are allowing for the possibility that things are spawned at shorter and shorter time intervals)?

Let’s focus on just a part of our expression, namely

$\left(1+\frac{r}{n}\right)^ n.$

As $n$ grows larger and larger, this expression converges to

$e^{r},$

where $e$ is Euler’s number from above. This means that, as $n$ grows,

$\left(1+\frac{r}{n}\right)^ n$

gets increasingly closer, indeed arbitrarily close, to

$e^{r}.$

(See here for a formal definition of a convergent sequence.) It's because of this convergence that the number $e$ turns up in models of exponential growth, where we assume that the growth can happen continuously as time passes.

Now, returning to our formula (1) for the number of things after $t$ weeks, we have

$N\left(1+\frac{r}{n}\right)^{nt}= N\left(\left(1+\frac{r}{n}\right)^ n\right)^ t.$

So as $n$ grows larger and larger the total number of things converges to

$N\left(e^{r}\right)^ t=Ne^{rt}.$

You can see a graph of this expression below for $N=1$ (together with the step function we considered earlier). Use the slider on the left to vary the value of $r$ .

A final issue to note concerns the variable $t$ , which we have taken to measure weeks, and the parameter $r$ , which in our example gives the growth rate of new things per week. There was no need for $t$ to represent weeks here. We could equally well have taken $t$ to measure days and started our calculation from there, adjusting $r$ to represent the growth per day. In the end, our calculations would still have yielded an expression involving $e$ as above. Similarly, we could have taken $t$ to measure hours, minutes, or seconds. What time frame you use depends on what you are dealing with: money (for which interest rates often refer to the period of a year), populations of bacteria (which can grow very quickly), or epidemics (whose speed of growth depends on how quickly an infected person infects others).

What if things disappear?

In our argument so far, we have assumed that new things appear each week. But what if things can also disappear (e.g. in an epidemic where people recover from a disease or, sadly, die)? Luckily, this doesn’t change the calculations, as long as we assume that the net effect is the same every week: that the creating and disappearing together amount to there either being $r$ new things per thing each week (which is what we already considered above), or $r$ fewer things per thing each week. The possibility of there being $r$ fewer things can be captured by taking $r$ to be a negative number. As it turns out, the calculations still work for $r$ negative, again leading to the formula

$Ne^{rt}.$

As you can see from the graph below, for $r$ a negative number, the number of things now decreases exponentially. We have what mathematician call exponential decay. (Use the slider to vary the value of $r$ .)

Caveats

Like all mathematical models, the exponential model we have just developed has its limits and comes with caveats.

As far as epidemics are concerned, one issue is that as people recover from the disease and (hopefully) become immune for a time, or die, the pool of people available to become infected shrinks, so the disease can't go rampaging about at the same rate. This is why the exponential model gives good predictions only over a suitable period of time (a few weeks for COVID-19) while there are still lots of people around to infect. It doesn't give good predictions indefinitely.

There are many other things our exponential model also doesn't take into account, for example, that people don't necessarily become infectious as soon as they have caught the disease, that not everyone in the population is the same, and that the growth rate could vary over time for all sorts of reasons. To take account of such issues, and to get longer-term predictions, you need to build more sophisticated models (find out more here).

That said, the exponential model is powerful tool to get handle on how much of a threat an epidemic poses over a suitable period of time. And it demonstrates the beauty and power of the number $e$ .

About the author

Marianne Freiberger is Editor of Plus.