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Comparative Coronavirus Case Growth
Yesterday morning, our friend Kozak posted a comment about Coronavirus case growth by country. It included a chart showing case growth in a number of Western nations (Italy, Germany, France, US, Switzerland, UK, and Japan), but not including S. Korea. His impression was that “[w]e are on the exact same trajectory as Italy,” and noted that “Korea quickly moved to identify and isolate cases and do aggressive contact tracing to limit the spread of the disease. As a result, they kept their medical system from being overwhelmed.”
I was skeptical of this, so I did a bit of data analysis on my own. I will copy the graph from Kozak’s comment later on, but I don’t want to bias your impression. My contention is that, at present, it is not possible to tell whether we are on the (somewhat troubling) path of Italy or the (much more reassuring) path of S. Korea.
I did not find daily data for Switzerland, but I was able to easily find it at the Worldometer site (here) for other relevant countries. My methodology was to start data reporting on the first day in which each country reported over 200 cases. The countries, and dates on which they exceeded 200 cases, are as follows:
- S. Korea — 209 total cases on Feb. 21
- Italy — 229 total cases on Feb. 24
- Germany — 203 total cases on Mar. 3
- France — 212 total cases on Mar. 3
- Spain — 228 total cases on Mar. 4
- US — 221 total cases on Mar. 5
- UK — 209 total cases on Mar. 7
Here is the graph, by country, for the first 10 days. Note that this graph uses a logarithmic scale, as did the one posted by Kozak (still to come).
Notice that all six countries follow a very similar pattern. There is little difference, over this period, between S. Korea (medium blue) and Italy (orange), and the US (green) is quite similar to both.
The US is only 9 days into this analysis, with 3,110 reported cases. At 9 days in, S. Korea was at 3,736 reported cases, and Italy was at 3,089.
Accordingly, I contend that based on reported cases to date, it is not possible to determine whether the US is following the trajectory of Italy or the trajectory of S. Korea.
Kozak is correct to note the difference between the experience of Italy and S. Korea. Here is the full graph, through the latest case reports for yesterday (Mar. 14), though I’m switching to a linear scale for this graph:
In S. Korea, there are 22 days of information since it first passed 200 reported cases on Feb. 21, while in Italy, there are 19 days of information since it passed 200 reported cases on Feb. 24.
Note that the linear scale properly displays the significant rate of growth in total cases, especially in Italy. It has the disadvantage of stretching the vertical scale, making it more difficult to distinguish between the various countries at the early stages of the spread of the WuFlu.
Notice the rapid rise of WuFlu in Spain (light blue). Spain appears to be on a trajectory significantly higher than any of the other countries included in this graph. If this trend continues, it would not be surprising if the focus of news coverage switches from Italy to Spain in the next few days.
Based on my (admittedly limited) research in the area, epidemiologists generally expect the number of cases to follow an “S-curve.” The S. Korean figures follow this curve — an initial, rapid increase that appears to show exponential growth, but reaches an inflection point, after which the rate of increase declines and approaches an asymptote. For many such S-curves, the inflection point appears to occur at about half the level of the asymptote.
For S. Korea in particular, the inflection point appears to have occurred around Day 11 or 12 (Mar. 3 or 4), when the total number of cases were 5,186 and 5,621, respectively. This suggests that S. Korea’s asymptote will be around 10,000 to 11,000.
With this explanation, here is the graph posted by our friend Kozak yesterday (here, comment #21). Again, I emphasize that this is not Kozak’s graph, but is (presumably) something that he found online:
I have several criticisms of this graph. First, notice that it does not include S. Korea, but does include Japan. Inclusion of S. Korea — as shown in my initial graph above — demonstrates that the other countries are on a trajectory that cannot be distinguished from that of Italy or S. Korea. However, the inclusion of Japan gives an inaccurate impression that the progression of the WuFlu in Europe is significantly different than in Asia. This impression would have been eliminated by the inclusion of S.Korea in the graph.
Second, note the inclusion of a “33% daily increase” line. This line is characteristic of exponential growth. Notice that it appears to be a straight line on this graph, because of the logarithmic scale.
Third, note that this graph includes data only through Mar. 9. This is not inherently deceptive. To the contrary, I assume that the graph was prepared around Mar. 10 or 11, with the latest data available.
However, the design of the graph above gives the inaccurate impression that ongoing, 33% daily growth is what we should expect from WuFlu.
This is not true even of Italy — though through Mar. 9, and especially if we use a logarithmic scale, it appears to be pretty accurate, as shown by the graph on the right.
I have calculated the daily growth rates for reported WuFlu cases in Italy, over the past 22 days, with the following averages:
- Feb. 24-Feb. 29 (5 days): Daily increase 37.5%
- Feb. 29-Mar. 5 (5 days): Daily increase 27.9%
- Mar. 5-Mar. 10 (5 days): Daily increase 21.3%
- Mar. 10-Mar. 14 (4 days): Daily increase 20.2%
Note the decline in the percentage rate of increase, which is what we would expect from an S-curve, but not what we would expect from an exponential curve.
If you’ve read this far, I thank you for your attention. The big reveal is coming.
Looking at the last couple of graphs: the one from Kozak’s comment, and mine, both seeming to show that a 33% daily increase is a reasonable projection for the growth of WuFlu in Italy — and, by extension, for the US and other countries. But here is how the 33% daily increase graph compares with the actual number of reported cases in Italy through yesterday, Mar. 14, using a linear scale:
See the problem?
An estimated growth rate that seemed perfectly reasonable just 5 days earlier, particularly when viewed on a logarithmic scale, is revealed to be completely wrong.
The estimate for Mar. 14 for Italy, using the 33% daily growth model, is 51,644 cases. The actual number of reported cases in Italy, as of Mar. 14, was 21,157.
Thus, the 33% daily growth model, suggested by the graph posted just yesterday by Kozak, is already overstating cases in Italy by 144%, just 5 days later.
I want to emphasize, again, that my goal is not to pick on Kozak. We have very poor information about what is going on, and this seems true even for medical providers.
I have very little idea how the WuFlu is going to progress. I am quite confident that it will not progress with ongoing 33% daily increases, as shown in the graph on the left.
Just eyeballing this graph, and assuming that the number of cases is going to follow an S-curve, it appears that Italy is probably pretty close to its inflection point. If today is the inflection point, we’d expect a total of about 40,000-45,000 cases. Perhaps it will be a bit higher, both in Italy and elsewhere.
As noted previously, I’m particularly concerned about Spain, if its present trend continues.
It would be helpful if the authorities would release some sort of reasonable projections. What we are getting are “worst-case scenario” projections — like this from Dr. James Lawler at an American Hospital Association webinar on Feb. 27, projecting 96 million cases and 480,000 deaths in the US, or this reporting yesterday on a CDC projection of 160-214 million cases over the next year in the US, and 200,000 to 1,700,000 deaths.
What I worry about is that these projections are like the 33% increase projection debunked above with respect to Italy.
Time will tell, I suppose. God bless, everyone, and fear no darkness.
Published in General
Nice plot. Please update it regularly for us.
It seems like the other countries we are comparing the virus spread with have a much more condensed population. Will that also be a factor when trying to decide if we are more like Italy or S. Korea? It seems like densely populated cities by definition cause the greatest risk. Of course the U.S. has that too but rural areas don’t seem to carry the same risk factors to me. Am I correct?
Again. South Korea early on aggressively tested and contact traced. They identified and isolated as many of the infective as possible and their contacts, resulting in the slowing of growth and the “bending of the curve” away from exponential growth. Italy, on the other hand did very little initially to try and contain the infection.
The Korean Clusters
They missed one person, Patient 31 and she spread it to 1,160 people, resulting in the large outbreak in Daegu.
The Koreans have tested over 200,000 people in a population of 51 million.
To do an equivalent job in the US we would have to have tested 1.3 million people.
We have tested less than 10,000.
We have given up on contact tracing because, it’s too late. It’s now pointless.
Yeah epidemiological curves follow an “S” shape eventually. For example when the population gets saturated and there is no new people to infect.
So by what miracle do you think the US is going to avoid the continued exponential growth of Covid infections in the US when we have taken essentially no action?
Maybe we will get lucky and the infections will fall when the weather turns. If that happens it will be luck and Gods Grace and not because we did anything to deserve it.
It’s still growing exponentially. (and the only way you get a perfect curve fit with biologic systems is like what happened in China where they lied their A##s of about the numbers)
And the health care system is at the breaking point, we can expect the death rate to sharply increase, which is exactly what Korea avoided.
It will depend on how widespread it gets. Also we are a very mobile nation.
With a 14 day incubation period it can get spread pretty widely.
That said, I would say the worst place to be is going to be in a large urban area, but not necessarily because of infection alone.
It’s already not growing exponentially in Italy, at least based on reported cases, which is all that we have to go on.
We’ve taken unbelievable action. School is cancelled. The NCAA, NBA, and NHL seasons are over. Now I’m reading that Ohio is closing restaurants.
I suspect that this disease does not spread nearly as rapidly as people fear. I don’t know this for sure, and I don’t think that anyone does.
What I am seeing are projections that are falsified in a matter of a few days. I see projections of 100-200 million cases, when the country in the worst shape — Italy — seems to be heading toward something in the 50,000 range. Even if this is off by a factor of 2, it is a far cry from tens of millions.
What type of growth does that graph show?
You’re all forgetting the most important thing:
Trump is the President, so everything is his fault, and the experts say we’re all going to die. Don’t question science. How dare you.
Italy quarantined over 16 million people in a population of 60 million.
Now the entire country is in lock down.
And we shut down the NBA.
I am extremely concerned about the people in Vatican City.
According to the site you gave
in the most recently reported day, there were 892,045 new deaths in Vatican city. The 2020 population is 801. This means that each resident has, on average, died 1113 times, in just one day.
I don’t see how the Pope will be able to perform his duties if the death rate gets much higher.
[EDIT: (being serious, now) The site has posted this interesting message:
We apologize for the temporary disservice that you may have experienced. For about 20 minutes, our site showed clearly incorrect data due to a malicious act. We have investigated the issue and we’re now implementing protective measures to prevent this from happening again. The other day we got hit with a big DDoS attack. Now this. We’ll continue with our daily efforts and we’ll not give up.]
But on the plus side, there are negative 324,045 active cases.
I admit that I should not have focused on the bad news. I am no medico, but I assume that having -324,045 active cases in the City means that we could move 324,045 active cases into Vatican City and have no active cases at all.
So as Sheila would say, let’s all take of off our Grumpy Pants.
[EDIT: In case you missed my EDIT to an earlier comment: after the Saint and I posted our good-natured satirical comments, the website publishers explained the wacky Vatican City data. Turns out that they were hit by malicious intruders, and they fixed the bad data very promptly.
I suppose I might-should feel some slight shame making them the butt of a joke when, like most subjects of comedy, they had suffered real pain through no fault of their own. But that’s a theological question for my fellow comic.]
Below is a chart which indicates what may be happening in Italy. In green is Lodi, the first city in Italy to implement a shutdown on February 23 and has slowed case growth. In red is Bergamo which did not shut down until March 8.
The paper the chart is from can be found here. Still making my way through it but it makes some points about the demographic issues around the varying fatality rates in countries.
One of the conclusions is:
I am thinking this spreads much *faster* than most people realize, but that it is milder. Closing restaurants is crazy. That is not flattening the curve, it is scorched earth.
I have seen this term used to describe the curve “Gompertz Curve”.
Well, they already closed the schools a few days ago, so now they need to close something else to show they’re on top of things.
BTW, first-hand reports I’ve read from places like Rome, Italy seem to indicate that many retail establishments are required to limit the number of customers inside at any time. If you want to enter one and it’s full up to its allowed limit, you have to wait until someone leaves before you can enter. I don’t know if that’s happening with restaurants, though.
Mark, I know of no empirical way that I can take interventions into account. What I can do is to report the data on cases, observe the shape of the curves, and debunk the idea that the growth is exponential in Italy, which it is not.
Now you’re getting into individual cities in Italy. What is your point? Do you think that this demonstrates the effectiveness of extreme measures?
Perhaps it does. I’ve never heard of either of these cities — but looking them up, I find that Lodi has a population of about 46,000 and Bergamo has a population of about 122,000. So perhaps it’s not surprising that a city almost 3 times larger has a greater number of cases.
Not exponential. I already demonstrated that. I will do so again, using your data.
Changing the type of the graph to a bar graph — which does not show what an exponential curve would actually look like — does not disprove my point. As I demonstrated above, the curve in Italy looks like a Sigmoid curve heading toward a peak in the 40,000-45,000 range. The reduction in the rate of increase today, back down to 17% after yesterday’s upward tick to 20%, is good news.
I have used your data, since Feb. 24 when the number of cases was 227 (in your data). The calculated daily growth rate from Feb. 24 through Mar. 15 is 26.5%. The 33% daily growth graph, suggested by the chart referenced in the OP, is off by a whopping 175% now, predicting 68,086 cases today, while the actual number is only 24,747. Here are your figures in a line graph:
Notice that the 33% daily growth line is far above actual cases. The 26.5% daily growth line is forced to the same current figure, but the trajectory of the 26.5% daily growth line does not match the actual figures — it lies below the actual figures, so it is likely to be significantly overstating the number of cases in 4-5 days.
Kozak, you seem a bit upset about this. This is good news. The numbers are showing that Italy is not suffering exponential growth. The curve is starting to level off.
Bergamo is a charming city north of Milano, with an ancient town on a hill and a modern town in the valley below.
Mrs Doctor Robert and I spent three wonderful days there in 2014 studying an antique English horn at the Museo Donizettiano, drinking wine and having all sorts of fun. Opera composer Gaetano Donizetti
was born here
and returned in his final illness to die in another home on the same street.
My attempts to drive in the old city led me onto a cobblestone via which turned into a pedestrian mall. The actions of a local woman, who opened the door of our rented Fiat, just about dragged me out and took over the driving, kept us from killing someone or being arrested.
Bergamo is also home to San Pellegrino sparkling water, which I enjoy most everyday.
It pains me to learn of how the city is now suffering.
The rate of new cases in Bergamo has been steady at around 2 per thousand of population for the last 6 days.
For the same period, Lodi where the intervention started sooner has been more than triple that: steady at around 6.5 per thousand.
This seems to contradict Gumby Mark’s point.
(I may be misinterpreting the data; please correct me if so.)
You are apparently the only statistician in the wold who has discovered this fact.
Be fair, @kozak! He’s a lawyer.
Yeah, this says something. It says something rather disturbing.
I actually am a statistician, in a sense, though not a professional one. I took a year of math graduate school, once upon a time, focused in probability and statistics. I am certainly not in the category of a PhD in the area, but I probably know more than 99.9% of people in this area.
Advanced statistics is not required in this instance, just Algebra II, coupled with basic competence in Excel. This allowed me to demonstrate that the graph of reported WuFlu cases in Italy is not exponential.
In fact, I’m not the only one making this point. Here is a paper by an economics professor in Geneva, making the same point, posted on March 12.
It’s also demonstrated in this article in the Lancet, posted on March 13. The Lancet article seems to be a strange mix of good and bad modeling. They start by using an exponential curve to forecast the number of cases in Italy, with results similar to mine. They predicted about 32,000 cases on March 15, while Kozak’s data above reported 24,747. (32,000 is based on their Fig 1B; the text only says “over 30,000”.)
Thus, the Lancet exponential projection was wrong by about 30-35% just two days after it was published.
On the bright side, the Lancet article also included a comparison to the Hubei Province figures, which demonstrated how inaccurate the exponential projection could turn out to be. Here is their Fig. 3:
The Lancet article then states:
This is pretty close to my claim. My estimate is that Italy is near its inflection point, and that it’s curve will level off and reach an asymptote, as the Hubei Province graph above indicates.
[Cont’d]
My initial estimate was that the ultimate number of cases in Italy would be in the 40,000-45,000 range, about double the then-present level. Looking at the Chinese graph, I would modify this slightly, as the asymptote appears to be a bit more than twice the level of the inflection point. Accordingly, the data seems to indicate that Italy will end up with about 60,000 cases.
I continue to be puzzled at the hostility to the idea that WuFlu case growth is not exponential, particularly from Kozak, who I know to be a really good guy. There is a tendency, among docs, to be annoyed by questioning by non-experts, reflected in the old joke:
I was told that joke by my dad, who was a doctor himself (of medicine, and not a lowly doctor of laws like me).
There’s a bit of additional good news in the Lancet article. It reports that the percentage of active cases requiring ICU care is around 10% consistently, in Italy (“consistently between 9% and 11% of patients who were actively infected”). Italy has about 5,200 ICU beds, about half of which are available for WuFlu patients, according to the Lancet article.
Remember that “active cases” is a lower figure than “total cases.” As of today, Italy has 24,747 total cases and 20,603 active cases. The figures for the US are 3,680 total cases and 3,553 active cases. (The active case percentage is high in the early period, as very few cases have yet resolved.)
I started the OP by arguing that we cannot tell whether the US is following the trajectory of Italy, or the trajectory of South Korea. Even if we are following the trajectory of Italy, this suggests that we are heading toward about 60,000 total cases (not all of which will be active at the same time).
To be very cautious, let’s double this figure and assume that the US will get up to 100,000 active cases. This would require 10,000 ICU beds. Fortunately, we have about 95,000 ICU beds (source here) — almost 19 times as many as Italy. If we use the same figure as Italy — that about half of the ICU beds can be made available for WuFlu patients — then we have no problem. Our capacity is about 45,000-50,000, and our expected case load — even assuming a trajectory about twice as bad as in Italy — is only around 10,000.
Amazing coincidence that the case number count goes up when testing increases. Sort of like the way crime goes down when there are more people incarcerated. How can this possibly happen?
There can be a connection between increased testing and reduced incidence of disease- but the connection is not necessarily dispositive. Testing does not prevent transmission or cure disease, it can only contribute to better isolation of infectious carriers. If you do a good job of isolation and quarantine, it makes little difference if you never tested a soul; that is why the comparisons to the RoK (where testing 200K out of 60 million is 0.03%, that is a ratio of .0003) are relatively meaningless.
The big problem we face is that draconian measure to curb spread are appropriate- but they should be embraced as panic-reduction measures; they are good news, not bad. Instead, people- driven by media hype- act as though the preventive measures that make things better and less risky increase the most problematic (elderly and ill who get the worst strain) case load instead.
We have a good friend who is a geriatric medicine specialist (to give you and idea how good a friend, she spent the night on our couch just last summer when visiting our state), but just a few months ago after more than a decade in practice went to work for the state dept of health, and is on the governor’s COVID-19 task force. Last weekend, she and her husband (an MD who works for the VA) had a few people over (including a set of late 60’s parents). They made sure that no one on the guest list was ill, and sanitized the house, the sensible precautions. And then had a nice time.
Calm down, people.
We are unlikely to reenact South Korea’s trajectory because we have not tested the way South Korea tested. Unfortunately, with Trump announcing that testing will concentrate on the symptomatic, that is not changing.
Let’s pray that Otto von Bismarck’s famous aphorism still holds true: “G-d has a special providence for fools, drunks, and the United States of America.”
But depending on miracles isn’t a good public health strategy.
Can you work your magic on the US figures? I’ve been making a point looking those “US -new” case numbers and they are not coming close to the “R2 and R3” infection projections we should be seeing.
In all of my reading and research, the bolded text above is never talked about. I think that the 14 day incubation period means this thing is everywhere. I personalty know of 8 people who have all the symptoms and are quarantined, but the docs won’t test because they are mild cases. The last I looked there were 3 confirmed cases in this county. I realize those two numbers are too low to be statistically valid, but it still tells a story. the infection rate probably is closer to that 33% chart, we just don’t know it.
Spin, good point. I suspect that the infection rate is much higher, perhaps 5-10 times higher, but that the rate of serious/critical cases and death is much lower, in inverse proportion.
I would expect that the serious and critical cases are being identified, as these are the people reporting for medical care.
It will probably turn out to make no difference, in the following sense. Imagine that, based on current figures, we project 60,000 total cases in Italy, 10% needing ICU care (6,000). It it turns out that the actual number of cases is 10 times higher (600,000), but the percentage needing ICU care is 10 times lower (1%), then we end up at the same place in terms of ICU needs (6,000).
I wonder if resturants will have the added cost of spoiled food. I don’t imagine take out increase a lot because everyone will be home eating soup and pasta. Just kidding. One think leads to another. If it keeps growing exponentially we are all dead. I agree that the fear has grown like a wildfire.