COVID math question
March 16, 2020 8:35 AM Subscribe
If there are 7 billion people on the planet, and we have 164,000 reported cases of COVID, that’s .0023% right? Since the virus has been “in the wild” since, say, December, I’m unclear on the epidemiology risk. Is it that the numbers are vastly underreported? Or have infections not increased exponentially as anticipated? Trying to wrap my head around this.
It's that exponential refers to a growth pattern or trend line, not to a certain number. In early days of the epidemic they observed that cases were doubling about every 7 days. Absent human intervention, that's expected to continue until it can't anymore, generally because a disease runs out of new population to infect.
People often use exponential colloquially to mean big. But exponential means that the growth rate is proportional to how much of the thing already exists. It's like the parable about the man who asked the king to pay him by taking a chessboard, putting one grain of rice on the first square, two on the second square, four on the third square, and continuing to double it each square. The first couple of squares seem small - 8, 16, 32, 64, 128 - but then it starts to climb. 256, 512, 1024, 2048, 4096... The 20th square is 1 million. 25th square 32 million.
But if you know what an exponential curve looks like, you can notice the pattern when you still have 64, 128, 256 grains/cases, and act then.
posted by Lady Li at 8:48 AM on March 16, 2020 [13 favorites]
People often use exponential colloquially to mean big. But exponential means that the growth rate is proportional to how much of the thing already exists. It's like the parable about the man who asked the king to pay him by taking a chessboard, putting one grain of rice on the first square, two on the second square, four on the third square, and continuing to double it each square. The first couple of squares seem small - 8, 16, 32, 64, 128 - but then it starts to climb. 256, 512, 1024, 2048, 4096... The 20th square is 1 million. 25th square 32 million.
But if you know what an exponential curve looks like, you can notice the pattern when you still have 64, 128, 256 grains/cases, and act then.
posted by Lady Li at 8:48 AM on March 16, 2020 [13 favorites]
The growth IS exponential, roughly. The current number of cases doesn't tell you anything about the growth rate.
The insidious thing about exponential growth is that it starts off seemingly slowly, but the growth rate increases rapidly. It may have taken from December until now to get the first 164,000 cases, but the next 164,000 will probably happen in a matter of weeks.
This graph shows the number of diagnosed cases over time. You can see that the growth rate is increasing rapidly.
posted by mekily at 8:50 AM on March 16, 2020 [5 favorites]
The insidious thing about exponential growth is that it starts off seemingly slowly, but the growth rate increases rapidly. It may have taken from December until now to get the first 164,000 cases, but the next 164,000 will probably happen in a matter of weeks.
This graph shows the number of diagnosed cases over time. You can see that the growth rate is increasing rapidly.
posted by mekily at 8:50 AM on March 16, 2020 [5 favorites]
Please remember that TESTED POSITIVE, something that requires human intervention and technology and is what these sites show, is not the same as the number of people who are or have been INFECTED.
posted by seanmpuckett at 8:56 AM on March 16, 2020 [35 favorites]
posted by seanmpuckett at 8:56 AM on March 16, 2020 [35 favorites]
Additionally, yes, as seanmpuckett implies, it's likely that the actual number of infected people is much higher than the number of confirmed cases.
posted by mekily at 8:58 AM on March 16, 2020
posted by mekily at 8:58 AM on March 16, 2020
The number is vastly under-reported. Hugely so. This isn't me stating a conspiracy theory, all the science commentators are saying it and it is an underlying assumption of government planning.
Especially here in the West the tests are being highly targeted towards those presenting with serious symptoms, so that amongst that group treatment can be applied most effectively.
We know that the young are having less symptoms, and are hence less likely to be tested under our system. We do not know how many older people also have lesser symptoms and hence don't get counted.
To give the specific example of the UK where I am - we have limited ability to process tests daily and are not testing people deemed possible cases as a matter of policy[1]. Last I read they were aiming to raise the test capacity to 10,000 a day. The best way to be sure how much of the population has been infected (and possibly recovered) would be population sampling or extrapolating from what we do know (eg deaths) if there is a well known relationship (it isn't well known now). This data is critical for calculating the "herd immunity" element of beating the virus so we'll need to get a handle on it eventually. But at the moment we don't have the capacity to test seemingly healthy people.
Everything I've read is based on the idea that a "small percentage" of the population is infected with a "high impact" illness. Non of this is to say it isn't growing exponentially. Even an exponential growth can be a small percentage of your population for weeks on end when it starts from 0.
[1] 2 weeks ago the advice was "if your symptoms are x call NHS, we bring you in and look at you". Then it became "if your symptoms are x call NHS, we will triage you by phone", now it is "if you symptoms are x do not call NHS, we are overloaded with enquires, you should self triage using our website checklist and then contact us as a last resort". (I'm paraphrasing, this is not medical advice and is slightly exaggerated to make a point - which is that we're not testing a LOT of suspect cases)
posted by samworm at 8:58 AM on March 16, 2020 [5 favorites]
Especially here in the West the tests are being highly targeted towards those presenting with serious symptoms, so that amongst that group treatment can be applied most effectively.
We know that the young are having less symptoms, and are hence less likely to be tested under our system. We do not know how many older people also have lesser symptoms and hence don't get counted.
To give the specific example of the UK where I am - we have limited ability to process tests daily and are not testing people deemed possible cases as a matter of policy[1]. Last I read they were aiming to raise the test capacity to 10,000 a day. The best way to be sure how much of the population has been infected (and possibly recovered) would be population sampling or extrapolating from what we do know (eg deaths) if there is a well known relationship (it isn't well known now). This data is critical for calculating the "herd immunity" element of beating the virus so we'll need to get a handle on it eventually. But at the moment we don't have the capacity to test seemingly healthy people.
Everything I've read is based on the idea that a "small percentage" of the population is infected with a "high impact" illness. Non of this is to say it isn't growing exponentially. Even an exponential growth can be a small percentage of your population for weeks on end when it starts from 0.
[1] 2 weeks ago the advice was "if your symptoms are x call NHS, we bring you in and look at you". Then it became "if your symptoms are x call NHS, we will triage you by phone", now it is "if you symptoms are x do not call NHS, we are overloaded with enquires, you should self triage using our website checklist and then contact us as a last resort". (I'm paraphrasing, this is not medical advice and is slightly exaggerated to make a point - which is that we're not testing a LOT of suspect cases)
posted by samworm at 8:58 AM on March 16, 2020 [5 favorites]
Also, it isn't evenly distributed worldwide. So dividing cases/general world population is deceptive.
posted by argybarg at 9:00 AM on March 16, 2020 [1 favorite]
posted by argybarg at 9:00 AM on March 16, 2020 [1 favorite]
If you look at the graphs, its definitely growing exponentially and that's just confirmed cases.
The confirmed cases are just a fraction of the actual infections though. It varies by country but most people are not getting tested. If 80% of cases are "mild" - ie. don't require hospital treatment then up to 80% of cases have no reason to even get tested. Around here the advice here is to stay home if you have any symptoms, don't got to the doctor or the pharmacist so those people will not be tested or reported as infected
When you also factor in asymptomatic carriers, I've seen suggestions that the actual number infected is 5-10x higher than the reported number of cases.
posted by missmagenta at 9:01 AM on March 16, 2020
The confirmed cases are just a fraction of the actual infections though. It varies by country but most people are not getting tested. If 80% of cases are "mild" - ie. don't require hospital treatment then up to 80% of cases have no reason to even get tested. Around here the advice here is to stay home if you have any symptoms, don't got to the doctor or the pharmacist so those people will not be tested or reported as infected
When you also factor in asymptomatic carriers, I've seen suggestions that the actual number infected is 5-10x higher than the reported number of cases.
posted by missmagenta at 9:01 AM on March 16, 2020
Taking action, whether centralized or collective, blunts the exponential curve.
China took extreme measures early, which kept cases from hitting the 100,000 mark in that country. South Korea's curve is likewise flattening out, and they do broad testing, so we assume intervention is working.
The worry is that countries will be unable or unwilling to act decisively, or for a long enough duration, and growth will continue unchecked (exponentially). Italy has passed the 10K mark, and is poised to pass 100K in a manner of weeks if cases continue on the same trajectory.
posted by RobotVoodooPower at 9:07 AM on March 16, 2020
China took extreme measures early, which kept cases from hitting the 100,000 mark in that country. South Korea's curve is likewise flattening out, and they do broad testing, so we assume intervention is working.
The worry is that countries will be unable or unwilling to act decisively, or for a long enough duration, and growth will continue unchecked (exponentially). Italy has passed the 10K mark, and is poised to pass 100K in a manner of weeks if cases continue on the same trajectory.
posted by RobotVoodooPower at 9:07 AM on March 16, 2020
And you have to keep in mind, with the 2 week incubation period (ie ~2.5 doublings), that you're looking at with "new case" numbers is really "the infections that happened 2 weeks ago". The number of infections happening today is much, much higher. That's why people were freaking out this weekend about St. Patrick's day and the airports. That was a bloodbath. The numbers are gonna spike major on mar 27-30, maybe a few days sooner.
posted by sexyrobot at 9:12 AM on March 16, 2020 [3 favorites]
posted by sexyrobot at 9:12 AM on March 16, 2020 [3 favorites]
It’s vastly underreported, or rather, underconfirmed. Italian doctors assume there are tens of thousand of milder cases that haven’t been tested, some of which will be counted only if they become severe and the patient needs the hospital.
posted by lydhre at 9:38 AM on March 16, 2020 [1 favorite]
posted by lydhre at 9:38 AM on March 16, 2020 [1 favorite]
Best answer: Is it that the numbers are vastly underreported?
Yes. US numbers are underreported by a factor of 50x or more. I can't speak for other countries, but massive testing fiascoes here mean we're literally turning away symptomatic patients at the door unless they need hospitalization. (Literally. They sent out a PDF to print and stick to the door of the clinic.)
After all, if you don't take a temperature, you can't find a fever.
posted by basalganglia at 9:45 AM on March 16, 2020 [9 favorites]
Yes. US numbers are underreported by a factor of 50x or more. I can't speak for other countries, but massive testing fiascoes here mean we're literally turning away symptomatic patients at the door unless they need hospitalization. (Literally. They sent out a PDF to print and stick to the door of the clinic.)
After all, if you don't take a temperature, you can't find a fever.
posted by basalganglia at 9:45 AM on March 16, 2020 [9 favorites]
I work in a high risk industry with its own health system, and we have been told that they are rationing tests to those who fit the profile. I went in with a very bad cough but no fever and was not tested. I'm not saying I actually had it - based on the full symptom list I PROBABLY didn't - but I don't actually KNOW. I'm far from the only one.
posted by showbiz_liz at 9:50 AM on March 16, 2020 [1 favorite]
posted by showbiz_liz at 9:50 AM on March 16, 2020 [1 favorite]
Best answer: Epidemiologist chiming in to say, about the exponential growth situation, the math is pretty straightforward and we've done calculations that show how, in about eight or nine weeks, the US alone might be seeing more than 800,000 new cases of transmission per week absent significant isolation efforts to slow the spread.
And, yes, we have only (literal) guesses at the true rate of transmission, number of cases, etc. right now. A lot of that work is done retrospectively. For instance, at some point we'd like to sample the blood of people who never felt ill to see how common asymptomatic carriers are/were--those are numbers that can really change the speed with which we go from a few cases to more cases than we can handle effectively.
So, short answer, there are a lot of unknowns but the knowns are plenty concerning. Everyone should be voluntarily locking their shit down, right now. Keep your distance, wash your hands well, and, as best you can, chill out and don't panic buy everything. Call and facetime and email your friends and family and neighbors and help convince them to do the same. Public messaging from governments only goes so far. You're the best person to convince people in your circle about what behavior is prudent.
posted by late afternoon dreaming hotel at 9:50 AM on March 16, 2020 [11 favorites]
And, yes, we have only (literal) guesses at the true rate of transmission, number of cases, etc. right now. A lot of that work is done retrospectively. For instance, at some point we'd like to sample the blood of people who never felt ill to see how common asymptomatic carriers are/were--those are numbers that can really change the speed with which we go from a few cases to more cases than we can handle effectively.
So, short answer, there are a lot of unknowns but the knowns are plenty concerning. Everyone should be voluntarily locking their shit down, right now. Keep your distance, wash your hands well, and, as best you can, chill out and don't panic buy everything. Call and facetime and email your friends and family and neighbors and help convince them to do the same. Public messaging from governments only goes so far. You're the best person to convince people in your circle about what behavior is prudent.
posted by late afternoon dreaming hotel at 9:50 AM on March 16, 2020 [11 favorites]
Best answer: Sources that I have seen recently addressing your question include:
> Mar. 14, 2020: "“We’re about to experience the worst public health disaster since polio,” said Dr Martin Makary, professor at Johns Hopkins University’s Bloomberg School of Public Health, speaking to Yahoo Finance. “Don’t believe the numbers when you see, even on our Johns Hopkins website, that 1,600 Americans have the virus. No, that means 1,600 got the test, tested positive. “There are probably 25 to 50 people who have the virus for every one person who is confirmed. I think we have between 50,000 and half a million cases right now walking around in the United States.”"
> Mar. 14, 2020: "At this point we may already have tens of thousands of infections in the United States — no one knows, because testing has been catastrophically bungled — and the number of cases is probably doubling every six days or so. In these circumstances, stopping a few new cases from Europe may not matter so much. [...] We’ll be honest: We worried that the clean lines in the graphics here risk suggesting a false precision. None of us know what lies ahead. But the wise uncertainty of epidemiologists is preferable to the confident bluster of television blowhards. The one thing we can be confident of is that enormous risks lie ahead — including a huge loss of life — if we don’t take aggressive action."
> Mar. 16, 2020: "As The New York Times reported last week, epidemiologists at the Centers for Disease Control and Prevention recently prepared four scenarios. Their calculations showed a large range of possible fatalities in the United States: between 200,000 and 1.7 million Americans over the course of Covid-19, assuming minimal efforts to contain it. [...] Places where a flood of sick patients have overwhelmed hospital capacity have had higher death rates than places where everyone who needs medical care can get it. [...] [Tom Frieden, who was director of the Centers for Disease Control and Prevention in the Obama administration,] has published his own estimates for the potential death toll. The top of his range assumes no more than half the U.S. population becomes infected. He puts the fatality rate at 1 percent, resulting in about 1.6 million deaths, though he noted that public health efforts could reduce that number. "
> Mar. 12, 2020: "A crucial thing to understand about the coronavirus threat — and it’s playing out grimly in Italy — is the difference between the total number of people who might get sick and the number who might get sick at the same time. Our country has only 2.8 hospital beds per 1,000 people. That’s fewer than in Italy (3.2), China (4.3) and South Korea (12.3), all of which have had struggles. More important, there are only so many intensive care beds and ventilators. It’s estimated that we have about 45,000 intensive care unit beds in the United States. In a moderate outbreak, about 200,000 Americans would need one. [...] The total number of infected people isn’t what scares many epidemiologists. It’s how many are infected at the same time."
posted by katra at 9:56 AM on March 16, 2020 [7 favorites]
> Mar. 14, 2020: "“We’re about to experience the worst public health disaster since polio,” said Dr Martin Makary, professor at Johns Hopkins University’s Bloomberg School of Public Health, speaking to Yahoo Finance. “Don’t believe the numbers when you see, even on our Johns Hopkins website, that 1,600 Americans have the virus. No, that means 1,600 got the test, tested positive. “There are probably 25 to 50 people who have the virus for every one person who is confirmed. I think we have between 50,000 and half a million cases right now walking around in the United States.”"
> Mar. 14, 2020: "At this point we may already have tens of thousands of infections in the United States — no one knows, because testing has been catastrophically bungled — and the number of cases is probably doubling every six days or so. In these circumstances, stopping a few new cases from Europe may not matter so much. [...] We’ll be honest: We worried that the clean lines in the graphics here risk suggesting a false precision. None of us know what lies ahead. But the wise uncertainty of epidemiologists is preferable to the confident bluster of television blowhards. The one thing we can be confident of is that enormous risks lie ahead — including a huge loss of life — if we don’t take aggressive action."
> Mar. 16, 2020: "As The New York Times reported last week, epidemiologists at the Centers for Disease Control and Prevention recently prepared four scenarios. Their calculations showed a large range of possible fatalities in the United States: between 200,000 and 1.7 million Americans over the course of Covid-19, assuming minimal efforts to contain it. [...] Places where a flood of sick patients have overwhelmed hospital capacity have had higher death rates than places where everyone who needs medical care can get it. [...] [Tom Frieden, who was director of the Centers for Disease Control and Prevention in the Obama administration,] has published his own estimates for the potential death toll. The top of his range assumes no more than half the U.S. population becomes infected. He puts the fatality rate at 1 percent, resulting in about 1.6 million deaths, though he noted that public health efforts could reduce that number. "
> Mar. 12, 2020: "A crucial thing to understand about the coronavirus threat — and it’s playing out grimly in Italy — is the difference between the total number of people who might get sick and the number who might get sick at the same time. Our country has only 2.8 hospital beds per 1,000 people. That’s fewer than in Italy (3.2), China (4.3) and South Korea (12.3), all of which have had struggles. More important, there are only so many intensive care beds and ventilators. It’s estimated that we have about 45,000 intensive care unit beds in the United States. In a moderate outbreak, about 200,000 Americans would need one. [...] The total number of infected people isn’t what scares many epidemiologists. It’s how many are infected at the same time."
posted by katra at 9:56 AM on March 16, 2020 [7 favorites]
Here's a nice mathematical explanation for why you should be worried about the lack of worry.
posted by Poldo at 11:22 AM on March 16, 2020 [3 favorites]
posted by Poldo at 11:22 AM on March 16, 2020 [3 favorites]
Someone reminded me of this math problem the other day, which might help with perspective:
Q: You have have an algae that doubles itself every day, and you know for a fact it will fill its pond in 24 days. On what day is the pond half full?
A: Day 23
On day 22 it was a quarter full, day 21 it was one eighth. By the time you get back to day 17 it's a tiny green patch in a tiny corner of the pond.
This is what makes exponential growth so scary. You can go for a very long time with small numbers that seem completely manageable and then suddenly, literally over the course of a few days everything is wildly out of control.
Obviously Coronavirus isn't doubling itself daily, but as long as it is spreading exponentially the end game is going to look the same. "Nothing ... nothing ... nothing ... HOLY CRAP!"
posted by Tell Me No Lies at 1:44 PM on March 16, 2020 [3 favorites]
Q: You have have an algae that doubles itself every day, and you know for a fact it will fill its pond in 24 days. On what day is the pond half full?
A: Day 23
On day 22 it was a quarter full, day 21 it was one eighth. By the time you get back to day 17 it's a tiny green patch in a tiny corner of the pond.
This is what makes exponential growth so scary. You can go for a very long time with small numbers that seem completely manageable and then suddenly, literally over the course of a few days everything is wildly out of control.
Obviously Coronavirus isn't doubling itself daily, but as long as it is spreading exponentially the end game is going to look the same. "Nothing ... nothing ... nothing ... HOLY CRAP!"
posted by Tell Me No Lies at 1:44 PM on March 16, 2020 [3 favorites]
See this article for a pretty in-depth explanation of the growth rates in different countries. Spoiler: it is growing exponentially.
posted by number9dream at 3:43 PM on March 16, 2020
posted by number9dream at 3:43 PM on March 16, 2020
Ditto on Poldo's link for the math part. It's a 3Blue1Brown video explaining exponential growth and how in the real world it eventually turns into logistical growth. It's the shape of the problem. The hard part is actually getting the numbers from testing and such.
posted by zengargoyle at 8:12 PM on March 16, 2020
posted by zengargoyle at 8:12 PM on March 16, 2020
I just came across this: World statistics | Coronavirus COVID-19 Observer which uses data from the Johns Hopkins University Center for Systems Science and Engineering. It's nice to look at China and see the hump peaking and then falling. Most other countries are far behind and still on the climbing part of the curve. You have to take into account that it started in China and grew for a good while before it managed to leave the country and start spreading around the world. The rest of the world is still in the going up rapidly phase.
posted by zengargoyle at 10:06 PM on March 16, 2020
posted by zengargoyle at 10:06 PM on March 16, 2020
Response by poster: So many great answers here. Thank you everyone.
posted by Gusaroo at 7:38 AM on March 17, 2020
posted by Gusaroo at 7:38 AM on March 17, 2020
This thread is closed to new comments.
posted by knownfossils at 8:45 AM on March 16, 2020