by George J. Dance
Imagine an invasive species of water lily that takes over a pond, choking off all other life; it grows exponentially, doubling its area every day. In 60 days the pond is completely full of lilies. When was the pond half full? When was the pond 1% full?
I expect most readers got the first answer: The pond was half full on day 59. The second is a bit trickier, and I found it much harder to estimate. It looks intuitive that if the pond is full in two months, it must have been at least 1% full after one month. However, that intuition does not account for the phenomenon of exponential growth. If the pond was full on day 60, then it was half full on day 59 (as you figured out), 1/4 full on day 58, 1/8 full on day 57, and so on. On day 53, the pond was only 1/128 full of lilies; so it was not 1% full until day 54. For more than seven weeks, there was nothing to see; then suddenly, in less than a week, the infestation had ripped through the pond killing everything in its path.
I read that story on the web in March. It was meant to illustrate the scary potential of the novel coronavirus, which at the time appeared to be growing exponentially. The warning was clear: even though, outside of New York, the virus was not even noticeable on this side of the ocean, it was only a matter of time before we had as many sick and dead as New York. The numbers got even scarier when one projected exponential growth into the future: epidemiologists at the Imperial College of London calculated that, by August 2020, 250 million Americans would be sick and 2.2 million would die, most in the last month.
Imperial College's report panicked Britain and the United States into lockdowns; and it was only lockdowns, we are told, that prevented its nightmare scenario from coming true: Without lockdowns, those countries would have suffered exponential growth and almost 3 million dead. The conclusion was inescapable: if those countries abandon lockdown, they once again face exponential growth and close to 3 million dead.
However, American state lockdowns have proved to be unsustainable. While still hugely popular with voters, they are being increasingly defied and ignored; and state governments (with a few exceptions like California) have found it necessary to lift or at least relax them. Since then the virus has resurged, and if the lockdown theory is right we can expect it to now resume exponential growth. Which is exactly why it is important to establish whether the virus is and/or will grow exponentially, anywhere.
Last week I looked at the numbers for the United States as a whole. I found that, while the virus grew exponentially in March (when total numbers were small), by mid-April (with 43 American states under lockdown) growth had become linear instead; the virus still spread, but at a steady rate. Even more encouragingly, after states began lifting and easing lockdowns, growth stayed linear.
One helpful comment I received on that column suggested I look at the Republican states that have had newsworthy recent surges: Arizona, Florida, Georgia, and Texas. All four states have Republican governors, who have drawn extensive criticism, and lost some support with voters, for easing lockdowns already. If exponential growth were re-occurring anywhere, surely it would be there. So, leaving Georgia for another time, I did a similar dive into the numbers for those states, again using Worldometer's "United States" coronavirus pages for data. In addition, I used the numbers for California (a state that has has maintained a strick lockdown, but has also seen a surge) as a control. My tabulations appear at the end.
For the three "Republican" states, I found that while cases have risen dramatically in all of them, growth in all cases has stayed linear. Both Florida and Texas have seen a big surge in cases, to more than 10,000 a day; however, neither state's figures show the constant doubling that indicates exponential growth. Florida went from 50,000 cases to 100,000 in 30 days; from 100,000 to 200,000 in 13 days; from 150,000 to 300,000 in 15 days; and from 200,000 to 400,000 in 19 days - a big speedup, followed by a slowdown. For Texas, the comparable time intervals were 30, 18, 18, and 21 days. Arizona's cases doubled from 20,000 to 40,000 in 15 days; from 40,000 to 80,000 in 14; from 60,000 to 120,000 in 20; and from 80,000 to 160,000 in 25 days - a consistenty slower growth.
It was otherwise with California, the lockdown state. California, with a larger population than either Florida or Texas, has only a slightly higher number of cases than either; and while its death total is more than double that of either, it per capita death rate is below that of both Florida and Arizona. California's numbers seem low; but then, so did the number of lilies on the pond back on day 30. Given exponential growth, low numbers early on mean little.
And in California, exponential growth may be occurring. The state's caseload doubled from 50,000 to 100,000 in 26 days; from 100,000 to 200,000 in 29 days; from 150,000 to 300,000 in 26 days; and from 200,000 to 400,000 in 26 days. That steady increase, though happening slowly, looks like constant doubling and therefore exponential growth While it is too early to say for sure (and the pattern may change at any time), California could be in serious trouble by the fall.
California was the earliest state to lock down, on March 19. The state's strict lockdown has already been lauded by at least one academic study, a National Bureau of Economic Research paper published in April, which concluded that the lockdown had prevented "as many as 1,661" deaths from COVID-19 in its first month. One can only hope that those, and subsequent months', prevented deaths in the state remain prevented..
===Arizona===
Deaths (time to add 500)
Jun. 5 - 1,012 (28 days)
Jun 24 - 1,490 (19 days)
Jul. 9 - 2,038 (15 days)
Jul 16 - 2,492 (7 days)
Jul 22 - 2,974 (6 days)
Cases (time to add 20,000)
Jun. 1 - 20,123
Jun 16 - 39,097 (15 days)
Jun 24 - 59,974 (8 days)
Jun 30 - 79,215 (6 days)
Jul. 5 - 98,089 (5 days)
Jul 11 - 119,930 (6 days)
Jul 17 - 138,523 (6 days)
Jul 25 - 160,041 (8 days)
===California===
Deaths (time to add 1,000)
Apr 17 - 1,041
Apr 30 - 2,029 (13 days)
May 13 - 2,966 (13 days)
May 28 - 4,039 (15 days)
Jun 12 - 4,988 (15 days)
Jun 29 - 5,976 (17 days)
Jul 11 - 7,026 (12 days)
Jul 22 - 8,045 (11 days)
Cases (time to add 50,000)
Apr 30 - 50,347
May 26 - 100,208 (26 days)
Jun 13 - 150,921 (18 days)
Jun 24 - 194,415 (11 days)
Jul. 2 - 249,094 (8 days)
Jul. 9 - 303,323 (7 days)
Jul 15 - 355,285 (6 days)
Jul 20 - 399,898 (5 days)
Jul 25 - 453,121 (5 days)
===Florida===
Deaths (time to add 1,000)
Apr 23 - 987
May 18 - 1,997 (25 days)
Jun 16 - 2,996 (19 days)
Jul. 9 - 4,009 (23 days)
Jul 19 - 4,985 (10 days)
Jul 27 - 5,933 (8 days)
Cases (time to add 50,000)
May 23 - 50,127
Jun 22 - 100,217 (30 days)
Jun 30 - 152,434 (8 days)
July 5 - 200,111 (5 days)
Jul 10 - 244,151 (5 days)
Jul 15 - 301,180 (5 days)
Jul 19 - 350,047 (4 days)
Jul 24 - 402,312 (5 days)
===Texas===
Deaths (time to add 1,000)
May 6, - 1,006
Jun 14 - 1,996 (29 days)
Jul. 9 - 3,052 (25 days)
Jul 18 - 4,007 (9 days)
Jul 25 - 5,069 (7 days)
Cases (time to add 50,000)
May 18 - 49, 684
Jun 17 - 99,304 (30 days)
Jun 27 - 148,845 (10 days)
Jul. 5 - 202,146 (8 days)
Jul 10 - 254,319 (5 days)
Jul 15 - 302,817 (5 days)
Jul 20 - 347,135 (5 days)
Jul 26 - 397,992 (6 days)
Imagine an invasive species of water lily that takes over a pond, choking off all other life; it grows exponentially, doubling its area every day. In 60 days the pond is completely full of lilies. When was the pond half full? When was the pond 1% full?
I expect most readers got the first answer: The pond was half full on day 59. The second is a bit trickier, and I found it much harder to estimate. It looks intuitive that if the pond is full in two months, it must have been at least 1% full after one month. However, that intuition does not account for the phenomenon of exponential growth. If the pond was full on day 60, then it was half full on day 59 (as you figured out), 1/4 full on day 58, 1/8 full on day 57, and so on. On day 53, the pond was only 1/128 full of lilies; so it was not 1% full until day 54. For more than seven weeks, there was nothing to see; then suddenly, in less than a week, the infestation had ripped through the pond killing everything in its path.
I read that story on the web in March. It was meant to illustrate the scary potential of the novel coronavirus, which at the time appeared to be growing exponentially. The warning was clear: even though, outside of New York, the virus was not even noticeable on this side of the ocean, it was only a matter of time before we had as many sick and dead as New York. The numbers got even scarier when one projected exponential growth into the future: epidemiologists at the Imperial College of London calculated that, by August 2020, 250 million Americans would be sick and 2.2 million would die, most in the last month.
Imperial College's report panicked Britain and the United States into lockdowns; and it was only lockdowns, we are told, that prevented its nightmare scenario from coming true: Without lockdowns, those countries would have suffered exponential growth and almost 3 million dead. The conclusion was inescapable: if those countries abandon lockdown, they once again face exponential growth and close to 3 million dead.
However, American state lockdowns have proved to be unsustainable. While still hugely popular with voters, they are being increasingly defied and ignored; and state governments (with a few exceptions like California) have found it necessary to lift or at least relax them. Since then the virus has resurged, and if the lockdown theory is right we can expect it to now resume exponential growth. Which is exactly why it is important to establish whether the virus is and/or will grow exponentially, anywhere.
Last week I looked at the numbers for the United States as a whole. I found that, while the virus grew exponentially in March (when total numbers were small), by mid-April (with 43 American states under lockdown) growth had become linear instead; the virus still spread, but at a steady rate. Even more encouragingly, after states began lifting and easing lockdowns, growth stayed linear.
One helpful comment I received on that column suggested I look at the Republican states that have had newsworthy recent surges: Arizona, Florida, Georgia, and Texas. All four states have Republican governors, who have drawn extensive criticism, and lost some support with voters, for easing lockdowns already. If exponential growth were re-occurring anywhere, surely it would be there. So, leaving Georgia for another time, I did a similar dive into the numbers for those states, again using Worldometer's "United States" coronavirus pages for data. In addition, I used the numbers for California (a state that has has maintained a strick lockdown, but has also seen a surge) as a control. My tabulations appear at the end.
For the three "Republican" states, I found that while cases have risen dramatically in all of them, growth in all cases has stayed linear. Both Florida and Texas have seen a big surge in cases, to more than 10,000 a day; however, neither state's figures show the constant doubling that indicates exponential growth. Florida went from 50,000 cases to 100,000 in 30 days; from 100,000 to 200,000 in 13 days; from 150,000 to 300,000 in 15 days; and from 200,000 to 400,000 in 19 days - a big speedup, followed by a slowdown. For Texas, the comparable time intervals were 30, 18, 18, and 21 days. Arizona's cases doubled from 20,000 to 40,000 in 15 days; from 40,000 to 80,000 in 14; from 60,000 to 120,000 in 20; and from 80,000 to 160,000 in 25 days - a consistenty slower growth.
It was otherwise with California, the lockdown state. California, with a larger population than either Florida or Texas, has only a slightly higher number of cases than either; and while its death total is more than double that of either, it per capita death rate is below that of both Florida and Arizona. California's numbers seem low; but then, so did the number of lilies on the pond back on day 30. Given exponential growth, low numbers early on mean little.
And in California, exponential growth may be occurring. The state's caseload doubled from 50,000 to 100,000 in 26 days; from 100,000 to 200,000 in 29 days; from 150,000 to 300,000 in 26 days; and from 200,000 to 400,000 in 26 days. That steady increase, though happening slowly, looks like constant doubling and therefore exponential growth While it is too early to say for sure (and the pattern may change at any time), California could be in serious trouble by the fall.
California was the earliest state to lock down, on March 19. The state's strict lockdown has already been lauded by at least one academic study, a National Bureau of Economic Research paper published in April, which concluded that the lockdown had prevented "as many as 1,661" deaths from COVID-19 in its first month. One can only hope that those, and subsequent months', prevented deaths in the state remain prevented..
===Arizona===
Deaths (time to add 500)
Jun. 5 - 1,012 (28 days)
Jun 24 - 1,490 (19 days)
Jul. 9 - 2,038 (15 days)
Jul 16 - 2,492 (7 days)
Jul 22 - 2,974 (6 days)
Cases (time to add 20,000)
Jun. 1 - 20,123
Jun 16 - 39,097 (15 days)
Jun 24 - 59,974 (8 days)
Jun 30 - 79,215 (6 days)
Jul. 5 - 98,089 (5 days)
Jul 11 - 119,930 (6 days)
Jul 17 - 138,523 (6 days)
Jul 25 - 160,041 (8 days)
===California===
Deaths (time to add 1,000)
Apr 17 - 1,041
Apr 30 - 2,029 (13 days)
May 13 - 2,966 (13 days)
May 28 - 4,039 (15 days)
Jun 12 - 4,988 (15 days)
Jun 29 - 5,976 (17 days)
Jul 11 - 7,026 (12 days)
Jul 22 - 8,045 (11 days)
Cases (time to add 50,000)
Apr 30 - 50,347
May 26 - 100,208 (26 days)
Jun 13 - 150,921 (18 days)
Jun 24 - 194,415 (11 days)
Jul. 2 - 249,094 (8 days)
Jul. 9 - 303,323 (7 days)
Jul 15 - 355,285 (6 days)
Jul 20 - 399,898 (5 days)
Jul 25 - 453,121 (5 days)
===Florida===
Deaths (time to add 1,000)
Apr 23 - 987
May 18 - 1,997 (25 days)
Jun 16 - 2,996 (19 days)
Jul. 9 - 4,009 (23 days)
Jul 19 - 4,985 (10 days)
Jul 27 - 5,933 (8 days)
Cases (time to add 50,000)
May 23 - 50,127
Jun 22 - 100,217 (30 days)
Jun 30 - 152,434 (8 days)
July 5 - 200,111 (5 days)
Jul 10 - 244,151 (5 days)
Jul 15 - 301,180 (5 days)
Jul 19 - 350,047 (4 days)
Jul 24 - 402,312 (5 days)
===Texas===
Deaths (time to add 1,000)
May 6, - 1,006
Jun 14 - 1,996 (29 days)
Jul. 9 - 3,052 (25 days)
Jul 18 - 4,007 (9 days)
Jul 25 - 5,069 (7 days)
Cases (time to add 50,000)
May 18 - 49, 684
Jun 17 - 99,304 (30 days)
Jun 27 - 148,845 (10 days)
Jul. 5 - 202,146 (8 days)
Jul 10 - 254,319 (5 days)
Jul 15 - 302,817 (5 days)
Jul 20 - 347,135 (5 days)
Jul 26 - 397,992 (6 days)
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