The Puzzling Linearity of COVID-19

We all understand how the total number of cases of COVID-19 and the total number of deaths due to COVID-19 are expected to grow exponentially during the early phase of the pandemic — every infected individual is in contact with others, who are unlikely to themselves be infected, and on average infects more than one of them, leading to the number of cases growing by a fixed percentage every day. We also know that this can’t go on forever — at some point, many of the people in contact with an infected individual have already been infected, so they aren’t a source of new infections. Or alternatively, people start to take measures to avoid infection.

So we expect that on a logarithmic plot of the cumulative number of cases or deaths over time, the curve will initially be a straight line, but later start to level off, approaching a horizontal line when there are no more new cases or deaths (assuming the disease is ultimately eliminated). And that’s what we mostly see in the data, except that we haven’t achieved a horizontal line yet.

On a linear plot of cases or deaths over time, we expect an exponentially rising curve, which also levels off eventually, ultimately becoming a horizontal line when there are no more cases or deaths. But that’s not what we see in much of the data.

Instead, for many countries, the linear plots of total cases or total deaths go up exponentially at first, and then approach a straight line that is not horizontal. What’s going on?

Before trying to answer this question, I’ll first illustrate the issue with some plots taken from https://www.worldometers.info/coronavirus/.

Here’s the linear plot of cases in the UK:

We see how the initial exponential rise changes into a linear rise from about April 3. We can also look at the daily number of cases:

From around April 3, the initial exponential rise in new cases changes to an approximately constant number of new cases each day.

It’s the same for deaths, with a small time lag:

Browsing around worldometers shows that the plots for many (though not all) countries are similar.  For instance, Canada:

And the United States:

It’s not too hard to come up with a reason for the number of cases to rise linearly. One need only assume that the country has a limited, and fixed, testing capacity. Once the number of probable cases reaches this capacity, the number of confirmed cases can’t rise any faster than they can be tested, so even if the true number of cases is still rising exponentially, the number of reported cases rises only linearly. Now, one might instead think that testing capacity is being increased, or that cases are being reported as COVID-19 based on symptoms alone, without a confirming test, so this isn’t a completely convincing reason for linearity. But considering that the relationship of number of reported cases to number of actual cases may be rather distant, it’s maybe not too interesting to think about this further.

A linear rise in the number of deaths seems harder to explain.

The analogue of limited testing capacity would be limited hospital capacity. But for that to produce a linear rise in reported deaths, we’d need to assume not only that hospital capacity has been exceeded (which seems not to be true in most places) but also that only the deaths from COVID-19 that occur in hospitals are reported.

It’s possible to imagine situations where the true number of deaths rises exponentially at first, but then linearly. For example, people could live along a river. Infections start among people at the river’s mouth, and expand exponentially amongst them, until most of them are infected. It also spreads up the river, but only by local contagion, so the number of deaths (and cases) grows linearly according to how far up-river it has spread. This scenario, however, seems nothing like what we would expect in almost all countries.

Most countries have taken various measures to slow the spread of COVID-19. We might expect that in some countries, these measures are insufficient, and that the growth in total deaths (and daily deaths) is still exponential, just with a longer doubling time. We might expect that in some other countries, the measures are quite effective, so that the number of new deaths is now declining exponentially, with the plot of total deaths levelling off to a horizontal line. To get a linear growth in number of deaths, the measures taken would need to be just effective enough, but no more, that they lead to a constant number of deaths per day, neither growing or shrinking. Considering that various disparate measures have been taken, it seems like an unlikely coincidence for their net effect to just happen to be a growth rate of zero.

What’s left? Well, it could be that the growth in total deaths is not really linear, but is instead now growing exponentially with a very long doubling time, or is instead levelling off, but very slowly.

Certainly there are some countries where total deaths seem to be levelling off, such as Spain:

Maybe if we only looked at these plots up until about April 7, we’d see the growth in total deaths as being nearly linear (after about March 26), and the number of daily deaths as being almost constant.

But the number of countries where growth in total deaths is almost linear seems more than I’d expect. Could these countries have very precisely calibrated the magnitude of their infection control measures to keep the number of new cases constant, thereby avoiding overwhelming their health care systems at minimal social and economic cost? This seems unlikely to me. I’m puzzled.