Hey, you won’t be surprised to hear that I’ve recently been thinking a bit more about epidemiology than physics. When I see daily news reports on COVID-19 [onscreen show total case numbers: “now 15000 cases in NY state!” etc], it’s really difficult for me to build a coherent picture of what’s actually going on, because the numbers are changing so quickly (which is exactly what you expect with exponential growth) that they’re almost immediately out of date. We know that epidemics tend to grow exponentially at first, and also that exponential growth is really really hard for our human brains to understand because of just how crazily fast it is.
My friend Grant Sanderson has a great overview video about exponential growth which I highly recommend. But regarding the news – I’d rather know where we’re headed, and if we’re making detectable progress. Are we winning or losing? Because of course, we can’t have exponential growth forever – at some point, the disease will run out of new people to infect, either because most people have already been infected, or because we as a society managed to get it under control. But – and this is the scary part to me – when you’re in the middle of an exponential, it’s essentially impossible to tell when it’s going to end. Are we in for 10 times as many cases as we currently have? Or 100 times as many? Or 1000?
Exactly when exponential growth ends is important, because it hugely determines how many people fall ill, yet so little reporting actually focuses at all on how to tell if exponential growth is ending (which would be a super positive sign!). After talking about this with my friend Aatish, he put together a new – and very useful – animated graph visualizing the COVID-19 epidemic on a global scale. This graph shows all countries travelling along the trajectory of exponential growth, and it makes it super obvious which ones have managed to stop the exponential spread of disease – they plummet downwards off the main sequence in a way that I find super compelling. And this figure also makes it abundantly clear that, even if a country doesn’t have lots of cases right now, covid-19 is probably going to follow this same trajectory there and end up spreading & spreading & spreading – until that country hits the emergency eject button. If you’re planning for the future and your country doesn’t have a lot of cases yet, it’s nevertheless a safe bet that you’re probably headed down a similar path. So how did we make this graph? Well, there are three key ideas: the first is to plot on a logarithmic scale, since that’s the natural scale for exponential growth – note that the tick marks grow by multiples of 10, so 10, 100, 1000, rather than 10, 20, 30. This scales up small numbers and scales down large numbers, making the growth equally apparent on all scales, and lets us compare the growth in countries with very different numbers of cases. Which brings us to the second idea: catch changes early, by looking at change itself.
For example, if you look at the growth of cases in South Korea, you can see that at first they’re exponential, and later, the growth slows down. But when you’re halfway up this curve, it’s hard to tell by eye that it’s slowing down – it still looks exponential. If instead you instead chart the number of new cases in the last week, in other words, the rate of growth, it’s much easier to see that the growth is starting to slow down. When the number of new cases each week flattens out or goes down, you’ve escaped the (scary) exponential growth zone. The third idea behind our graph is one from physics: don’t plot against time. Usually, when you see exponential growth, the number of cases is plotted versus time. But the spread of the disease doesn’t care if it’s March or April; it only cares about two things: how many cases there are, and how many new cases there will be – that is, the growth rate.
The defining feature of exponential growth is that the # of new cases is proportional to the # of existing cases, which means that if you plot new cases vs total cases, exponential growth appears as a straight line. So these are what we plotted on our graph: the number of new cases (aka the growth rate) is on the y axis, and the cumulative number of cases is on the x axis, both on logarithmic scales. [visual footnote on the graph about cumulative vs current # of cases] This gives us a beautiful-horrible graph that shows where all countries are in their COVID-19 journeys; it makes it obvious that the disease is spreading in the same manner everywhere – we’re all headed on the same trajectory, just shifted in time; and it makes it obvious where public health measures like testing, isolation, social distancing, and contact tracing have started to beat back the disease, and where they either aren’t working or haven’t had time to show up in the data. [graph with animation] In nearly every country (*so far), the number of cases grows at a roughly similar rate, until it doesn’t. And that’s what I feel like is missing from so much COVID-19 coverage: a sense of whether or not we can see the light at the end of the tunnel.
Are we still on the rocketship of contagion, or have we managed to hit the emergency eject button? And this graph does that; it gives us some sense of what’s actually happening in these uncertain times. That said, this graph also has a number of caveats & limitations – its main goal is to emphasize deviations from exponential growth – that is, to amplify the light at the end of the tunnel, so it may be less informative for other purposes. [Logarithmic scales distort] 10,000 looks really close to 1,000 on a log scale; this kind of distortion might allow people to take COVID-19 less seriously. Also, the log scale on the x axis makes it harder to see a resurgence of new infections after a significant downturn — a normal plot compared with time is better for that. [Time is implied] Also, unlike most other COVID graphs you’ve probably seen, time isn’t on the x axis, which might be confusing! Instead, time is shown through an animation. [“Confirmed Cases” =/= Infections] Another important caveat is that this graph (& basically every other COVID-19 graph that you’ve seen) is not actually showing the true number of cases, just the number of detected cases.
The true number of cases is unknown but certainly much higher than the number detected. [True Growth Rate vs Tested Growth Rate] In reality, COVID-19 cases spread at a slower rate than what the data implies. It’s a subtle idea, but the data reflect not just an increase in cases, but also an increase in the number of tests performed. [Imperfect Data] The data we’re using is incomplete, as it relies on daily reports from overburdened healthcare systems around the world. Also, different countries have dramatic differences in the resources that are available or dedicated for testing. [Slightly Delayed] Finally, the trends in this plot are delayed a few days, since we’re plotting the average growth rate over last week (there’s too much variability in the data to plot daily growth rates). This is actually kind of a good thing – it means that it’s a pessimistic graph, it doesn’t get too excited too soon, and so a downward trend on the graph is much more likely to be a real downward trend. And a real downward trend is what we want, for all countries! A lot of the daily news just reports recent data points.
Yet to understand where we’re headed, it’s not enough to know just where we are today – we need to be talking about the trends: how many new cases there are today relative to the number of new cases yesterday, or last week. Charting the rate of change empowers us all to more clearly see what the future holds.