Pandemic blog 11: Why do curves bend?

When you plot the number of reported deaths from COVID on a log scale you get pictures that look like this one, by John Burn-Murdoch at the Financial Times:

A straight line represents exponential growth, which is what one might expect to see in the early days of a pandemic according to baby models. You’ll note that the straight line doesn’t last very long, thank goodness; in just about every country the line starts to bend. Why are COVID deaths concave? There are quite a few possible reasons.

  1. Suppression is working. When pandemic breaks out, countries take measures to suppress transmission, and people take their own measures over and above what their governments do. (An analysis by Song Gao of our geography department of cellphone location data shows that Wisconsinites median distance traveled from home decreased by 50% even before the governor issued a stay-at-home order.) That should slow the rate of exponential growth — hopefully, flip it to exponential decay.
  2. Change in reporting. Maybe we’re getting better at detecting COVID deaths; if on day 1, only half of COVID deaths were reported as same, while now we’re accurately reporting them all, we’d see a spuriously high slope at the beginning of the outbreak. (The same reasoning applies to the curve for number of confirmed cases; at the beginning, the curve grows faster than the true number of infections as testing ramps up.)
  3. COVID is getting less lethal. This is the whole point of “flattening the curve” — with each week that passes, hospitals are more ready, we have more treatment options and fuller knowledge of which of the existing treatments best suits which cases.
  4. Infection has saturated the population. This is the most controversial one. The baby model (where by baby I mean SIR) tells you that the curve bends as the number of still-susceptible people starts to really drop. The consensus seems to be we’re nowhere near that yet, and almost everyone (in the United States, at least) is still susceptible. But I guess one should be open to the possibility that there are way more asymptomatic people than we think and half the population is already infected; or that for some reason a large proportion of the population carries natural immunity so 1% of population infected is half the susceptible population.
  5. Heterogeneous growth rate. I came across this idea in a post by a physicist (yeah, I know, but it was a good post!) which I can’t find now — sorry, anonymous physicist! There’s not one true exponential growth rate; different places may have different slopes. Just for the sake of argument, suppose a bunch of different locales all start with the same number of deaths, and suppose the rate of exponential growth is uniformly distributed between 0 and 1; then the total deaths at time t is \int^1_0 e^{\alpha t} d \alpha which is (1/t)(e^t - 1). The log of that function has positive second derivative; that is, it tends to make the curve bend up rather than down! That makes sense; with heterogeneous rates of exponential growth, you’ll start with some sort of average of the rates but before long the highest rate will dominate.

I’m sure I’ve skipped some curve-bending factors; propose more in comments!

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8 thoughts on “Pandemic blog 11: Why do curves bend?

  1. How many local maxima (of the US graph) do you guess there will be?

  2. JSE says:

    My hope is there will be a lot but all but that all but the first will be at 1!

  3. Allen Knutson says:

    You rightly left out the Richard Epstein theory that the virus is mutating in realtime to be less lethal, the better to spread. (Not that AIDS or Spanish Flu did that…)

  4. Noah Snyder says:

    A problem with the first possibility in 4 is that if the number of asymptomatic people is much higher than expected, then that also means R0 is much higher than expected, and so 1-1/R0 herd immunity is farther away.

  5. adannenberg says:

    to get concavity from saturation you don’t need SIR. you just need for a population of N people with I infected (and perhaps recovered) that dI = r*I*(N-I) so that f(t) = f0*exp(rt)/(1+f0*[exp(rt)-1]) where f=I/N so the curve starts to flatten near f~0.5)

  6. Wurd Bro says:

    For weeks I’ve been seeing a word that starts with ‘a’, ends with ‘c’ has a ‘y’,’m’,’p’ some ‘t’s and some vowels in the middle, and, reading it as asymptotic. Now for once the error is not mine ;)

  7. JSE says:

    Ha, I fixed it, now your comment makes no sense!

  8. @Noah Snyder: Is that really a logical conclusion? We can simply assume that the outbreak started earlier. Not that R0 is higher. For New Yorkeven even a start in late January is consistent with the estimate in this article:

    Lobardy has 10000 deaths so far and a population of 10 million. Let’s assume that the mortality rate is 0.005. If it takes ~2 weeks to die, then one had 2 million infected 2 weeks ago. That makes the first possibility in 4 not too unrealistic in my opinion.

    For Belgium (where I live) I am at least open to this option. For instance end of March a hospital in Brussels CTed the lungs of all its incoming non-COVID-19 patients and 8% has symptoms (of course maybe there are other things that make the lungs look like that):

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