Please poke holes in my Covid-19 math (SF Bay Area)
drodio.com
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When you're modeling exponential growth, most of your possible errors are just an additive time shift, a small one if the growth is rapid. For example, if the actual current cases are 4080, a 100% error in the estimate above of 2040, that just moves the time to reach those 43068 cases from 31 days away to 24 days away. But an error in the exponential growth rate is a multiplicative time distortion, leading to an exponentially large error at any given point in the future.
If you have a look at Germany who is having a very good overview of the spread of the pandemic since testing is free, available, and done rapidly (5h turnaround time ) capacity for 12.000 tests a day ) you get a close to 3 day case doubling time currently if you look at the official German numbers from the Robert Koch Institute. And we'll start seeing in a week or two how good the drastic containment measures they are starting now are impacting the growth or viral spread.
11.03.2020 1,288.00
12.03.2020 1,567.00
13.03.2020 2,369.00
14.03.2020 3,062.00
15.03.2020 3,795.00
16.03.2020 4,838.00
In a country like the US where the Author makes his calculations the government frankly has zero clue how many people are infected already because testing is not free, not available and not rapid ( current turnaround 3 -5 days ) for most citizens. There is a chance that the case doubling time is worse, or bigger, but if Germany who had a lot better reaction to the viral threat is having problems reaching a case doubling time of 6+ days and if you think about that China is at 80.000 cases now but put the most drastic containment measures in at only 600 confirmed cases there is a huge probability we are all in for a very wild ride.
That said every day counts because there are promising randomized control studies going on in China which hopefully give doctors access to a drug they can actually help people with. Right now all they can do is support your breathing and pump you full with a cocktail of meds they have no idea if it will work for the specific patient or not. Everyone who lands into ICU with Covid-19 is basically a guinea pig for the rest of us healthy people at this point.
In the end all that matters right now is that we don't know enough about the illness to treat it effectively, and we are in desperate need of time because no government was prepared for a pathogen like this that usually is a once in a century event.
P.S: This is a current snapshot. If the containment measures like in China are working the case doubling rate rises quite fast. China is at over 21+ days now. But I think it's important to take this day by day and re-evaluate the numbers. The "doomsdays predictions" will most likely be wrong and just a theoretical numbers game.
So, what accounts for the difference? The paucity of international travel (but then why is the spread so much slower in every other country — Germany is hardly famous for close physical contact with strangers)? Or is it just that Germany's comprehensive testing is allowing them to detect almost all the cases, while everybody else is underestimating by a factor of 10 to 50 or more — a constantly exponentially growing factor?
It seems unlikely that the pandemic was growing by 30% per day since December; it would have already infected 450 million people. Can we really impute that difference entirely to containment measures?
It all started with a small cluster around Munich that showed up around the end of January ( 27th ) as the first confirmed case in Germany came back from business travel to China that was contained rather quickly as everyone involved did a great job isolating and breaking the infection chain and everyone involved recovered without severe symptoms just like any other flu for most mid-aged and healthy people. Then for a good month there were no confirmed cases, so no more drastic containment measures were taken.
After that the Government carelessly allowed Carnival in Germany to happen thinking they contained the virus and severely underestimating the threat it could pose. Carnival is one of the biggest celebrations in Germany each year with millions of people from all over the country coming to see, especially the Cologne Carnival on the 24th of February. While in general you are not wrong when you say "Germany is hardly famous for close physical contact with strangers” the carnival season and Oktoberfest are the two occasions where Germans do voluntarily have a ton of contact with strangers and party heavily. The Rhineland is famous for their carnival all over Europe and in some parts of the world. If you look at pictures of it you’ll see what kind of nightmare this is if you have a potentially deadly pathogen spreading silently during the celebration from an epidemiologist point of view.
Three days later a couple from the region were the second and third confirmed cases that heavily participated in the celebrations while already being sick. Remember Covid-19 can be contagious up to 14 days before a patient shows symptoms. On top of that there was no chance to break the infection chain because the man is still in critical care and couldn't talk to officials and the woman had no idea where they could have been infected because they had contact with hundreds, possibly thousands of people over the course of the celebration days until the husband fell critically ill. So the lack of care from the Government together with gathering of hundreds of thousands of people from all over the country in close proximity, kissing, drinking and standing head to head in a crowd paired with that there were no tests done on people with covid-19 symptoms for a whole month between the munich and heinsberg cases is most likely the reason for the explosion in case numbers.
Also remember we are basically looking into the past up to 14 days as people who need hospitalization now have been infectious for up to that timeframe and just now start getting very sick. This is an incredible different pathogen to deal with and easy to underestimate because of this. To fight a pandemic you need to know the number of infected people, if you know someone is infected that gives you a fighting chance to break the infection chain if you isolate all people they have been in contact with. Unfortunately because of the huge celebrations this was impossible to do and we are where we are now.
"while everybody else is underestimating by a factor of 10 to 50 or more" this could very well be the case. With the numbers of infected people we see travelling outside Iran there are estimatons that a large part of the population is already infected. 7.5 Million people registered on an Iranian website that allowed for symptom screening. You usually don't register for such a site if you don't have any symptons that could be related to Covid-19. Just to show the difference between the real reported numbers and what could be likely.
If you ask about the containment measures, yes they work incredibly well, but at the cost of gigantic economic damage and right now I don’t want to be in the shoes of any leader in any country that basically has to decide between saving lifes now and that there also needs to be a good future for the survivors of this crisis afterwards. It’s horrible to think that up to 25% of a certain population and age could die. But it’s equally frightening to think that depending how long this takes millions of people and their families could lose their homes, jobs and potentially starve because we have an almost complete economic shutdown across all sectors.
A doubling time of 3 days corresponds to 26% growth per day (not 41%), because 1.26^3 ≈ 2.
You can see that here, all the others are linear in a log plot: https://studylib.net/coronavirus-growth
>>> (24747/7375.0)**(1/7.0)
1.188799592827802
More broadly, we're currently at 167676 confirmed cases worldwide; the first day we have numbers for is January 16th, when there were 45 cases. It's been 59 days since then (31-16+29+15) giving a 15% daily growth rate
>>> (167676/45.0)**(1/59.0)
1.1495551283620273
(If the Italians were to keep spreading the disease at this rate, by mid-May they'd have more cases than the entire rest of the world; unfortunately for the plausibility of this prediction, their case count would have to be over half a billion people at that point, requiring vast influxes of guest patients from the rest of the world to meet such a quota.)
I did projections at the beginning of February based on a 3.5-day doubling time (22% daily growth) which was what we seemed to be seeing at the time. They said it would infect the majority of the world's population in February. That didn't happen.
† Hey, I'm guilty of this myself; the last time I kissed somebody I'd just met that day was, like, Tuesday. Here in Argentina we can expect rapid contagion.
https://cmmid.github.io/topics/covid19/severity/diamond_crui...
and the current number of deaths in Italy: 1809 one could come to the estimate of actual number of infections in Italy being around 180K people. That big number is real only if Italians managed to treat that many cases as good as it was possible before the hospitals were overwhelmed. What's probably true, however, is that their hospitals already reached their limits and now the number of deaths per infection raises compared to the ideal case, lowering the number of actual infections.
But even not using that estimated "bigger number" Italy reports 24747 actually confirmed cases.
And assuming that the DNA analysis of this Italian researches is right, Italy's strain must have reached Italy not before January 19th:
https://www.reuters.com/article/us-health-coronavirus-italy-...
Even starting from one person at that date we get, for the number of confirmed cases a factor of close to 1.2 per day, or for the assumed cases (a bigger number) 1.24 per day. That gives doubling time between 3 and 4 days, and never longer than that.
Note that even a difference of using 24K or 180K infected can't give the factor you suggest: 1.15.
That's only observing Italy. I don't know if you are combining Chinese development, where they indeed contained the spread on their territory, with Europe and the rest of the world, where no comparable actions happened (and the actions taken still haven't significantly slowed down the growth), but if you do, that's not proper use of the data available.
If there really was a 10% "true" growth, and higher growth was just testing capacity ramping up, you would see very few straight lines in that plot. Yet that is what you see in all cases except for China, South Korea and Iran.
Also, you can't look at the world-wide growth rate since January like that, since that includes a period where the total number of cases was totally dominated by Chinese cases, but China had slowed already. Actually, the fact that you get 15% by using this approach which substantially underestimates the growth shows that 10% is way too low.
I think it's reasonable to expect that similar containment measures will be put in place in most countries soon, though, so perhaps it's not unreasonable to take them into account when trying to estimate future growth rates.
† https://en.wikipedia.org/w/index.php?title=Template:2019%E2%...
Here are some studies supporting the doubling time estimate of 6-8 days:
https://wwwnc.cdc.gov/eid/article/26/5/20-0146_article
https://www.nejm.org/doi/full/10.1056/NEJMoa2001316
https://www.thelancet.com/journals/lancet/article/PIIS0140-6...
>...
>80% of those will be "mild" which means "possibly as bad as having pneumonia but not needing a hostpital stay." 14% will require a hospital bed.
I have an issue with extrapolating the number of serious cases this way. I assume the 80% figure for non-serious cases is based on the number of formally tested patients.
Therefore the 80% figure doesn't include all the community-spread undiagnosed cases that must also be non-serious (or they would have been in hospital).
The percentage of serious cases amongst the true number of cases is presumably much smaller than 20%?
That said I think you can put an upper bound on the number of undiagnosed cases out there by looking at deaths and hospitalizations (in places where that data is reasonably accurate).
There's at least a few ways of getting ballpark figures:
- https://cmmid.github.io/topics/covid19/severity/diamond_crui... (EDIT: this gives some estimates by age correcting the data from the diamond princess cruise which I think is one of the best data sets we have since most passengers got quarantined/tested/monitored).
- looking at data from countries with reasonably strong testing regimes and responses. E.g. Israel or Canada are two I'm following. In Canada 12% of the cases so far required hospitalization (EDIT: https://torontosun.com/news/provincial/ontario-sees-spike-of... )
- Surveillance test data for countries that do that.
20% does sound on the high side from what I've seen read. My mental ballpark is 5%-10%. I'm sure there are local variations (smokers, air pollution, age distribution, general health...)
EDIT: The US has 62 reported deaths. Just estimating based on 0.5% ifr that gives 12,400 cases. Then you have to back-adjust this. Let's say by 2 weeks? That's maybe on the order of 100k+. Obviously this will have a very large error bar but it gives us some intuition. Feel free to poke holes at my math ;)
http://weekly.chinacdc.cn/en/article/id/e53946e2-c6c4-41e9-9...
I also have no idea the basis for the 10-50x assumption r.e. confirmed/unconfirmed cases.
Using the same math from feb 26 instead, which is the date I saw a CDC article published about first suspected community transmission, the figures are instead 15 * 10 * 2^(18/3) = 9600, which is more plausible. In any case, the calculations are extremely sensitive to the growth rate which, as you say, is also a thing we can control by modulating behavior. The extent to which we will actually do so is debatable, of course. As far as I’m aware I’m still expected to show up on site for work tomorrow.
This is why the spread of a virus like this is a logistic curve, not an exponential. An exponential is a good model at first, but extrapolating too far doesn't work like this.
More on this by 3Blue1Brown: https://www.youtube-nocookie.com/embed/Kas0tIxDvrg
"true exponentials are basically non-existent in real life"
1. What can we actually measure with confidence to relate to a model of interest 2. At what level of sophistication is it feasible to actually model things
Neither question is particularly straightforward in the face of ongoing epidemics.
This isn't to say that modelling isn't useful, but that forecasting a novel infection while it's ongoing is really, really, really hard.
Edit: Also, I’d be interested to see a comparison of the fit between a typical logistic, four parameter logistic, Gompertz etc. Not all sigmoids are the same.
Ultimately for such complex things like spread with barriers Tsalis statistics are good. That could mean q-Weibull distribution.
1) Number of cases is not flat. Even by now.
2) There was a slowed growth in late cases.
It fits Weibull statistics. Late spread is random but responds to distance. (A'la Rayleigh.)
Poke -- lots of measures are already being put into place (Stanford is effectively shut down, the big tech companies are making everyone work from home) etc.
Edit oh you have this right after: "Or, multiply it by less if you want to take into account various amounts of social distancing we're all doing. I'll cut that number above down by a huge amount – 90% – on the assumption that we all learn to stay home and self-isolate immediately, just to be super aggressive on my assumptions about how humanity will rise to the occasion"
Still, kinda weird to start with assumption of no measures when there clearly are some already.
It seems plausible that social distancing can reduce human contact for the average person by 50%, and for people with some flu symptoms by 80%. All this is to agree, the growth rate is by far the biggest factor in getting an estimate of peak impact.
That should already be helping, but I wonder if spreading might follow a power law pattern where "super-spreaders" account for a vastly disproportionate share of transmissions. If that's the case, it might require a strict lockdown to meaningfully reduce the exponent?
Dropping 90% is not a great way to calculate this. Look at the math for 61 days out (from March 8): https://youtu.be/Kas0tIxDvrg?t=465
I don't think our social distancing and hand washing will be near enough to prevent millions of deaths, but it's important to recognize that a small change in the growth rate does make a huge difference.
Estimating how much we can affect the growth rate is hard, but I think models should be based on a iffy guess on that value rather than an iffy guess on the resulting count. (Expert models certainly are.)
Edit: The math guy being 3Blue1Brown in his video linked elsewhere in this thread.
That may be correct... this could all be a fever-induced hallucination. It's more likely, though, that the articles assumptions are idiotic.
I don't think that it was accounted for in the calculation for number of beds in hospital
I'm struggling to find a name for the group that I participated in an exercise with, but in my volunteer Search and Rescue days I was a 'victim' for a practice exercise of deploying a tent hospital for a combined human and animal pandemic.
The ability to set up these hospitals is there, what isn't there is the skilled nursing staff for ICU patients, or ventilators for patients who needed them. I'm really not sure how China handled that portion of caring for those that fell ill.
Because there are so many variables, no one knows yet what the numbers will work out to be. The only safe thing to do is to expand capacity as much as possible by taking extreme measures.
Probably not unless you have popup / prefab hospitals.
> Have a look at the state of capacity in Italy: it's closed to maxed out with lots of improvised wards.
Italy is not "close to maxed out", it's way overcapacity. It might seem like it's at capacity if you count the total number of ICU beds in the entire country, but most of the cases for now are in the north, so the actual radio is way higher. Lombardia was at ~200% capacity circa wednesday.
Source?
Mine is:
https://www.giornaledibrescia.it/italia-ed-estero/coronaviru...
Google Translate:
"«We have very few free places in intensive care, now we are in the order of 15 or 20 available. Every day we get someone new, tomorrow 3 more arrive and San Raffaele is creating an area with 14 seats which will be ready, however, in a week. Today we recover them by closing the operating rooms, where there are respirators that can also be used to support the breath »."
The only real hope here is to prevent infections via quarantine measures, there is absolutely no way to build out the kind of health care capacity that would be needed in a pessimal outbreak.
Er, ah, uh, .... Multiply 204 cases by an estimated ratio of confirmed to unconfirmed cases of 10 to 50 gives 2040 - 10,200 estimated current cases in the Bay Area.
"Just because if I multiply by a 50x ratio, it returns a number larger than the 7.8MM total residents in the Bay area."
As you just discovered, a raw exponential increase cannot continue for very long, if only because infected individuals begin to have difficulty finding uninfected individuals to infect.
"80% of those will be "mild" which means "possibly as bad as having pneumonia but not needing a hostpital stay." 14% will require a hospital bed."
You may want to knock down the severity numbers a bit, since it seems likely that severe cases will be reported and confirmed in larger numbers than less severe cases.
"Potentially, even assuming aggressive social distancing, the Bay area needs 11x more beds than it has available in the next 30 days."
You probably should do this calculation in terms of bed-days: how long a given case occupies a bed on average before release/death.
Finally, as far as I know, the severity for old and infirm cases is much, much higher than the severity for those of you who are young and healthy. You may want to modify your model for local demographics.
It's more important to look at ICU bodies and deaths. Those aren't as affected by supply issues.
But the Bay Area does have MUNI and BART, it's not a no public transportation situation.
the long-winded introduction, use of caps and sense of urgency it piggybacks on top of and the obvious straw man. i mean if you don't know how to do the math, then following the steps as presented in the article (ie. "add this, then times it by that.." etc) you're not actually reaching your own conclusions; that's just hand waving credence to the conclusion that the reader cannot fairly evaluate.
EDIT: writing my originally short comment i noticed many other subtle cues as well. for example:
the author's prediction of the article going viral (establishing trust because they've been "right" before)
the open invitation to criticize the article (suggesting that what you're reading is essentially the result of consensus)
the claim that it's being written despite any criticism it might receive because the subject is so important (moral authority)
all of these points are presupposed by the author, and not actually derived from the work. the more i think about it the more subtle and manipulative it appears. i hope im wrong, poke holes in it! but given the subject we're discussing it's really rather unsettling. we are indeed in a crisis, drodio be damned.
I don't have a lot of exponential-math insight, but those rates seem low if the virus has spread like crazy already. How does that reconcile with the math in this article?
now, I know, today's number's aren't fully in yet, and there are day to day fluctuations. but, if you trust the numbers for the last 3 days, it looks like the number of cases is not increasing exponentially. I know the actual case number is much higher, but, what matters is the relative percent increase or decrease from day to day, and it does not seem to be increasing in the last 3 days. am i wrong?
Starting a few days ago, almost no one is going out, except to do their big shopping. so how can the virus still spread?
I've been following their data for a while and I admire their work, but I've stopped believing their numbers without double-checking. (Yes, I've filed bugs.) In addition, the last day on the graph has always made it look like cases are tapering off because it's partial data.
daily new cases (at the very bottom) shows there's very little increase from the 13th to the 14th.
I think this is a very localized view of behavior. There are many large regions, pockets, demographic slices of the country where life is going on roughly as it was a few weeks ago.
Check out the Nashville music and party scene... It's hopping right now. I think #NashvilleUndefeated is their slogan. In Portland OR, I've observed a slight decrease in activity, but only by 5-10% from what I normally see.
I think for this part you meant "actual number of cases"? Since the confirmed number was 1/10th of that.
If I take Td / (Td + Tr), I get:
China: 0.05
Italy: 0.44
Iran: 0.14
South Korea: 0.08
Germany: 0.19 (Germany's numbers for both are so low I don't believe this is useful.)
What's up with that?
2. Italy, because of the size of the epidemic event we are facing, is sampling only people that go to the hospital in serious conditions. Other countries are sampling the more general population, finding a lot of positive cases with mild symptoms or asymptomatic at all. So there is a selection bias.
3. It also depends on the way you count people. In Italy we are counting as a COVID19 victim even people severely ill that once dead are positive for the virus.
TLDR: you can't compare such numbers.
The numbers are only so low in Germany/Switzerland because its a) growing fast and b) they are testing aggressively. More precisely (and morbidly), the death rate is low because not enough time has passed yet.
In Germany and Switzerland if all the serious cases are deaths then the death rate is 0.1-0.5%.
Two weeks ago, Germany had ~100 cases. Today, Germany has 11 deaths.
Germany has a total of 57 cases with outcomes (46 of 57 recovered).
South Korea has ~900 cases with outcomes (~75 dead). China has ~70k cases with outcomes (~3k dead).
Let's wait a few weeks before reaching conclusions about the mortality rate in Germany and using that as evidence to prove covid-19 is just a mild flu.
Deaths / cases with outcomes is an overestimate (19% for Germany). Deaths / total cases is an underestimate (0.2% for Germany). Somewhere in between is the true mortality rate.