Source:  New York Times, “Coronavirus Deaths by U.S. State and Country Over Time:  Daily Tracking”, downloaded March 25, 2020, with deaths reported as of 8:20am on March 25.

As in any epidemic where disease spreads person to person, the number of deaths from the Covid-19 coronavirus has exploded at exponential rates.  An important question is how long it will continue to grow like this.  It cannot continue forever, as one cannot infect more than 100% of the population, and most such diseases are turned around well before that.  But where it will turn around, with a leveling off in the cumulative number of cases, depends not only on the characteristics of the disease (how easily it spreads) but also on policy.  Since there is not yet a vaccine nor a treatment that will always work, the spread of the disease and the number of deaths from it depends on the effectiveness of social distancing measures, so there is less person to person contact and therefore less spread of the virus.  So far, this has been the only effective means to reduce the number of those catching the virus.

And there now appears to be some early evidence that such social distancing measures have helped.  The chart at the top of this post is an excellent graphic prepared by the New York Times, updated daily, which presents on a semi-log scale the cumulative number of deaths from Covid-19 by country, plotted against the number of days since that country’s 25th death.  The Financial Times also presents (and updates daily) a very similar chart, although it presents the results for each country in terms of the number of days since the 10th death.  Both of these news sources are making available this material, and all of their coronovirus coverage, free to anyone, including non-subscribers.  I very much encourage everyone to examine these postings, as there is a good deal of interesting further material (including charts on the number of deaths by national sub-regions, as well as the cumulative number of confirmed cases).  The New York Times charts are also interactive, where they present (for any individual country or region chosen) the rate of growth over the most recent 7 days, i.e. how fast it is growing now.

By presenting the numbers on a semi-log scale (where the vertical axis is logarithmic, while the horizontal axis is in regular linear terms), a path that is a straight line will indicate a constant rate of growth, and the slope of that line will indicate what that rate of growth is (where the steeper the line the higher the rate of growth).  [If you are not familiar with this, review your high school algebra textbook or read through a web post such as this one.]  The faint, gray, straight lines on the chart then show what the total number of deaths would be (for some number of days since the 25th death) if the number of deaths doubled each day, or doubled every 2 days, or doubled every 3 days, or every week, or every month.

There are several interesting findings one can draw from this:

a)  What I found most interesting, and the reason I titled this post as a “First Sign to be Hopeful”, is that for most of the countries (with the US a notable exception), the paths start out quite steep, with doubling times of between every day and every 2 days, but that they then begin to bend over to the right.  That is, they shift over time to a flatter slope, meaning a slowdown in the rate of growth in deaths.  Mainland China, which was hit first, is now (as I write this) at an almost completely flat slope, which means close to a zero rate of growth.  The curve for Italy has also bent over so that it now hits the gray line for doubling every 3 days.  But that does not mean deaths are doubling every 3 days in Italy right now.  Rather, cases were growing at a faster rate in Italy earlier (doubling at about every 2 days at first), and have decelerated to the point where the cumulative number of cases in Italy are now where they would be had they doubled every 3 days throughout.  But the slope now is a good deal less than what it was in the early days.  The New York Times interactive chart indicates that at the pace of the last 7 days, the number of deaths in Italy is now growing at a rate of doubling every 5 days.  And one sees that flattening out in a number of other cases as well, including hard-hit Iran and Spain.

b)  Japan and to a lesser degree South Korea are exceptions in that their recent rates of growth have not fallen.  But both are also exceptions in that their rates of growth, while steady (the path lines are close to straight), have also been a good deal less than that of other countries.  Their doubling times over the last 7 days (as I write this) are both low at 11 days for South Korea and 12 days for Japan.

c)  The US is also a notable exception.  The pace of growth was relatively low for the US for the first 10 days (from when the 25th death was recorded in the US).  Unlike any other country, the pace then accelerated in the US to a doubling time of every 3 days.  Or more precisely, the number of deaths in the US grew from 60 as of day 10 (from when the 25th death was recorded) to 728 deaths on day 19 (March 24).  That is a 32.0% rate of growth per day, or an increase of 2.3 times every 3 days.

Why was the pace of growth relatively modest in the US at first, and then picked up?  That is not clear from these aggregate figures, but might be because the US is a large country, where there have been several centers of outbreak and those centers are relatively distant from each other.  The earliest center was the State of Washington, and the second (and to a more limited degree at first) in California.  Then New York was hit, followed now by major centers in Michigan, Illinois, Florida, Louisiana, and Georgia.  Adding up these varied impacts by locale across the country as a whole (and where doubling times may also vary by locale, especially for New York) may explain the US curve that starts relatively slow, but then accelerates.

Furthermore, with the Trump administration unwilling or unable to provide direction and management at the national level, each state and locality has been left to enact measures on their own and at their own pace.  These have been primarily social distancing measures, but with major differences in how strict they have been structured.  And most of the measures have been enacted reactively, to local cases being confirmed, rather than preemptively.

What is perhaps most disturbing of all for those of us in the US is that there is no indication as yet of the aggregate US curve beginning to bend over to the right.  It has been close to a straight upward line for the last 9 days, growing at a rate of 32% a day.  The only encouraging sign is that the curve for Washington State alone (shown at the New York Times posting) does bend over to the right, similar to what is seen in other countries.  Washington was hit first with the virus, with the initial deaths there, and early on had a doubling time of every 3 days.  But it was then the first state to put in place relatively strict social distancing measures, and the pace of doubling has now dropped to every 9 days (based on deaths over the most recent week).

Overall, the US was late to enacting social distancing measures, with most only put in place over the last week to week and a half.  Their impact on the number of deaths from the virus will then only be seen two or three weeks later, due to the lag from when one catches the virus to when they start to show symptoms (about one week), to when their cases become serious and lead, for some, to death (a further week or two).

Watching whether the US curve starts to bend to the right soon, in the next week or two, will certainly be of interest.