Flattening the curve is a pervasive and useful idea. It explains why it’s important to take steps to limit the spread of Coronavirus even if we don’t think we can stop it from spreading around the world: even if we can’t stop the epidemic, epidemic control measures can reduce the number of severely ill patients at the peak of the epidemic, so that we can take care of them with the limited resources (e.g., ICU beds) available.

But if you like to think mechanistically, you may wonder how (or if) it’s possible that interventions that can’t limit the long-term reach of the epidemic can limit the peak size.

Very simple models of disease spread, whose basic ideas go back a century, illuminate the sense in which this intuition is partly right but also importantly wrong.

It is right in the sense that the interventions such as social distancing will reduce transmission and thus flatten the epidemic peak. It is wrong in the sense that limiting transmission will not just flatten and slow the epidemic; it also limits the total size of the outbreak.

But #FlattenTheCurve is nonetheless important because epidemic control measures have a much stronger effect on peak height than on total epidemic size.

We can start by plotting the results from a simple epidemic model (for the technically minded, these are from an SIR model with a doubling time of 6 days and a maximum \({\cal R}_0\) of 2.5; simple individual-based models give similar results).1

What happens to both the total number of cases (proportional to the area under the curve) and the peak number of cases as we increase the strength of control?

While both peak and total cases decrease to zero when the epidemic is completely controlled, peak cases fall off much faster. For example: in this example (starting from \({\cal R}_0=2.5\)), a 20% reduction in transmission yields only a 11% decrease in total number of cases, but a 34% decrease in the epidemic peak.

source code: https://github.com/bbolker/bbmisc/blob/master/peak_I_simple.rmd

  1. Imposing 40% control makes the epidemic take a long time to die out; we need to hope that doing enough social distancing to make the peak flat and slow also gives us enough time to figure out how to finish off the epidemic for good, so that we don’t have to maintain social distancing for 250 days.↩︎