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.
- 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.
- 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.)
- 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.
- 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.
- 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 which is . 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!