The story of the exponentially growing virus

by George J. Dance

I have to admit that I am no statistician; rows of numbers usually bore me. By April, though, I figured that if I was to understand the coronavirus event, I had to learn enough to be able to read the numbers and get an idea of what they were saying. So I began looking at the statistics, and asking questions on social media.

The first thing I ran into was the assertion that the virus grew "exponentially". Unlike linear growth (at which something grows at a constant rate) exponential growth meant that, the more the virus grew, the faster it would grow, with infections and deaths both doubling at a constant rate. People were pushing this idea all over social media, of course, but in more academically respectable venues as well. For example, the infamous Imperial College Report 9 by Neil Ferguson et al modelled "an exponentially growing rate (with a doubling time of 5 days)," to forecast almost 250 million infections and over 2.2 million deaths in the United States by August.

Ferguson's predictions were instrumental in pushing both the U.S. and Britain into lockdown; and although they never came true, that had a simple explanation: the lockdowns save us. Consequently, the assumption of exponential growth has gone unchallenged; rather than study the virus's growth in real time, these days scientists are more interested in studying why "American participants mistakenly perceive the virus’s exponential growth in linear terms (conservatives more so than liberals)."

With the ending of lockdowns in some states, their easing in others, and the people increasingly ignoring them in others, many fear that the virus is again spreading exponentially. Not just in the United States; the mayor of Calgary, Alberta, announced today that Calgary is experiencing exponential growth. That got me concerned enough to finally look at the numbers.

I began with a quick look at doubling times of deaths in the United States, to see if there were any change. Thanks to extensive media coverage, it is easy enough to google the day of any number of deaths in the country; so I did a few quick searches for doubled values, and used the dates of the newspaper stories. Here are the results: 

Deaths - exponential (time to double)
2,500 - Mar. 30
5,000 -  Apr. 1 (2 days)
10,000 - Apr. 6 (5 days)
20,000 - Apr 11 (5 days)
40,000 - Apr 20 (9 days)
80,000 - May 11 (21 days)
160,000 -

That told me the disease was spreading exponentially in March and (though slower) in early April. However, after mid-April, the time it took to double increased: to twice as long, then four times as long. Then, in May, deaths stopped doubling. So I decided to check whether growth had gone linear instead. Using the same method as above, I began checking the time interval to add a further 10,000 deaths (the amount of the last true doubling).

Deaths  - linear (time to increase by 10,000)
10,000 - Apr. 6
20,000 - Apr 11 (5 days)
30,000 - Apr 15 (4 days)
40,000 - Apr 20 (5 days)
50,000 - Apr 24 (4 days)
60,000 - Apr 29 (5 days)
70,000 - May 5 (7 days)
80,000 - May 11 (6 days)
90,000 - May 17 (6 days)
100,000 - May 28 (11 days)
110,000 - June 6 (9 days)
120.000 - Jun 21 (15 days)
130,000 - July 6  (15 days)
140,000 - Jul 19 (13 days)

That was more informative. As I suspected, the virus continued to spread, in a linear fashion, through the lockdown period.  The surprising thing is that even after some states began ending lockdowns, deaths continued to grow in a linear fashion, and at an even lower rate. The recent shortening of the increase time, though, shows that the death rate is trending up again, and this may indicate the onset of exponential growth; it is too soon to say anything.

So I decided to check confirmed cases as well. Cases give a less reliable measure of the disease due to under- and over-counting, as well as testing capacity and differing test specificities; however, they give earlier information, which can indicate what the death toll will look like in the future. This time I used a more precise data source, the figures provided on the Worldometer site (a great source of data). Again I checked time to double for the earlier period:

Cases - exponential (time to double)
Feb. 15 - 15
Feb. 20 - 35 (5 days)
Feb. 27 - 60 (7 days)
Mar. 3 -  124 (5 days)
Mar. 6 -  319 (3 days)
Mar. 8 - 540  (2 days)
Mar. 10 - 994 (2 days)
Mar. 13 - 2,185 (3 days)
Mar. 16 - 4,611 (3 days)
Mar. 18 - 9,333 (2 days)
Mar. 20 - 19,608 (2 days)
Mar, 23 - 44,325 (2 days)
Mar. 26 - 86,669 (3 days)
Mar. 31 - 194,114 (3 days)
Apr. 7 - 411,116 (7 days)

When, once again, the time needed to double itself began doubling, I assumed linear growth, and began recording the time it took to add another 100,000 infections:

Cases - linear (time to increase by 100,000)
Mar 31 - 194,114 (3 days)
Apr. 4 - 319,444 (4 days)
Apr. 7 - 411,116 (3 days)
Apr 10 - 512,010 (3 days)
Apr 13 - 597,642 (3 days)
Apr 16 - 685,712 (3 days)
Apr 20 - 801,820 (4 days)
Apr 23 - 890,436 (3 days)
Apr 26 - 991,670 (3 days)
Apr 30 - 1,100,105 (4 days)
May  3 - 1,193,573 (3 days)
May 7  - 1,298,462 (4 days)
May 11 - 1,393,513 (4 days)
May 15 - 1,494,646 (4 days)
May 20 - 1,606.295 (5 days)
May 24 - 1,701,574 (4 days)
May 29 - 1,810,430 (5 days)
Jun. 2 - 1,899,666 (4 days)
Jun. 7 - 2,010,569 (5 days)
Jun 11 - 2,093,253 (4 days)
Jun 16 - 2,212,264 (4 days)
Jun 19 - 2,300,667 (3 days)
Jun 22 - 2,391,754 (3 days)
Jun 25 - 2,506,490 (3 days)
Jun 27 - 2,597,454 (2 days)
Jun 29 - 2,682,781 (2 days)
Jul. 1 - 2,781,217 (2 days)
Jul. 3 - 2,898,358 (2 days)
Jul. 5 - 2,994,393 (2 days)
Jul. 7 - 3,101,020 (2 days)
Jul. 9 - 3,224,892 (2 days)
Jul 10 - 3,297,170 (1 day)
Jul 12 - 3,417,795 (2 days)
Jul 13 - 3,483,584 (1 day)
Jul 15 - 3,621,637 (2 days)
Jul 16 - 3,695,025 (1 day)
Jul 17 - 3,770,012 (1 day)
Jul 19 - 3,898,550 (2 days)
Jul 21 - 4,028,569 (2 days)

The first thing I noticed is that cases are rising much faster than deaths – a good sign, as it means that the death rate is steadily decreasing. The second is that, while the amounts differ, both cases and deaths obey the same trajectory: increasing exponentially at the beginning, slowing to linear growth during lockdown, and then slowing even more before rising again later in the summer. The third is that cases and deaths change trajectory in lockstep, with deaths following cases by three weeks.

That allowed me to conclude that, though deaths will increase through mid-August, their growth will stay linear for now. "For now" is important; with both cases and deaths increasing, there is no guarantee that we will not suddenly be witnessing exponential growth. Although I found no evidence of exponential growth after the very earliest stage, that does not prove it cannot happen.

In addition, I realize that my conclusions could be wrong; as I said, I have not studied statistics, and I could very well be overlooking something. Which is why I rejected my earlier title for this piece, "The myth of the exponentially growing virus" - while my research shows no evidence for exponential growth, I cannot conclude at this time that exponential growth is only a myth.

Accordingly I welcome detailed, informative criticism of my observations. In addition, I intend to keep these tables updated,, and most likely do a few for other geographic areas.