The SIR Model of Epidemics(jaydaigle.net)
jaydaigle.net
The SIR Model of Epidemics
https://jaydaigle.net/blog/the-sir-model-of-epidemics/
41 comments
Layman with absolutely no understanding about any of this.
I currently live in a country with absolutely no infrastructure to handle this. Our first confirmed case was reported almost three weeks ago. Within a week we ran out of testing kits.
Last night I started researching on the topic to see if there is a formula to calculate the current number of infected people in the country. I don't care about accuracy, it just needs to give an approximate.
The current official figure is less than ten and I feel that by having an estimate (from what I've been able to read, time is a bad indicator, it is what it is) it would help change what I believe to be a false sense of security if one is looking at the official figures.
Is there any such formula or should I use the SIR model? (which currently is a bit over my head)
I currently live in a country with absolutely no infrastructure to handle this. Our first confirmed case was reported almost three weeks ago. Within a week we ran out of testing kits.
Last night I started researching on the topic to see if there is a formula to calculate the current number of infected people in the country. I don't care about accuracy, it just needs to give an approximate.
The current official figure is less than ten and I feel that by having an estimate (from what I've been able to read, time is a bad indicator, it is what it is) it would help change what I believe to be a false sense of security if one is looking at the official figures.
Is there any such formula or should I use the SIR model? (which currently is a bit over my head)
Tomas Pueyo published an article: https://medium.com/@tomaspueyo/coronavirus-act-today-or-peop...
He wrote a model that roughly estimates infection rates in a population based on either deaths or confirmed cases: https://docs.google.com/spreadsheets/u/1/d/17YyCmjb2Z2QwMiRR...
The model is oriented around companies, but could be applied to any gathering of people. There may be more sophisticated or accurate models out there, but I found his relatively quick to understand and apply.
He wrote a model that roughly estimates infection rates in a population based on either deaths or confirmed cases: https://docs.google.com/spreadsheets/u/1/d/17YyCmjb2Z2QwMiRR...
The model is oriented around companies, but could be applied to any gathering of people. There may be more sophisticated or accurate models out there, but I found his relatively quick to understand and apply.
Thanks.
I'm a bit late to this, sorry; one of my students let me know this thread existed!
I want to encourage you not to try to use this formula or anything like it to make concrete predictions. You are, unfortunately, right that the numbers we have access to are crap. But that doesn't mean that our modeling will be any better; we'd need to use the public numbers as input. And those public numbers are crap.
There are people whose entire job is to figure out ways to make useful predictions from the information we have. Their estimates aren't great, in part because the numbers they're using as input are crap.
But the numbers they have for input are as good as what we have, so we're unlikely to manage better. And in the worst case we can easily Epstein ourselves and generate predictions that are wildly, embarrassingly off the mark.
I'm not saying you should trust the official or expert numbers. But they're almost certainly better than what you'll manage on your own.
I want to encourage you not to try to use this formula or anything like it to make concrete predictions. You are, unfortunately, right that the numbers we have access to are crap. But that doesn't mean that our modeling will be any better; we'd need to use the public numbers as input. And those public numbers are crap.
There are people whose entire job is to figure out ways to make useful predictions from the information we have. Their estimates aren't great, in part because the numbers they're using as input are crap.
But the numbers they have for input are as good as what we have, so we're unlikely to manage better. And in the worst case we can easily Epstein ourselves and generate predictions that are wildly, embarrassingly off the mark.
I'm not saying you should trust the official or expert numbers. But they're almost certainly better than what you'll manage on your own.
And you can play with the parameters yourself using scipy:
https://scipython.com/book/chapter-8-scipy/additional-exampl...
At the risk of getting hellbanned for sneaking in reposts of my largely ignored posts from yesterday, the projections for our current situation can be found online here:
http://covid19.healthdata.org/projections
paper here: http://www.healthdata.org/research-article/forecasting-covid...
http://covid19.healthdata.org/projections
paper here: http://www.healthdata.org/research-article/forecasting-covid...
Interesting, they're projecting the peak much earlier than other projections I've seen, especially considering they're factoring in social distancing measures. I'm not sure I buy it. For example, North Carolina currently has 91 patients in the hospital for COVID19. The model projected 300-1000 hospitalizations by this point, suggesting that their model is over-aggressive with the ramp-up time.
The models I've seen with the peak in April have been the truly apocalyptic scenarios. Most models that incorporate social distancing push the peak out into May and beyond.
The models I've seen with the peak in April have been the truly apocalyptic scenarios. Most models that incorporate social distancing push the peak out into May and beyond.
> North Carolina currently has 91 patients in the hospital for COVID19. The model projected 300-1000 hospitalizations by this point
The doubling time is in many places only 3 days. From my point of view the model which is off by only a week could still be quite correct.
The doubling time is in many places only 3 days. From my point of view the model which is off by only a week could still be quite correct.
The data backs you up.
The model projected 437 deaths for March 28th (tolerance bands of 380 - 508).
According to [0], the deaths were actually 264.
But the model predicted around that only two days prior ( 273 deaths on March 26th).
I’d say it’s pretty accurate - off only by a couple days.
[0] - https://www.worldometers.info/coronavirus/country/us
The model projected 437 deaths for March 28th (tolerance bands of 380 - 508).
According to [0], the deaths were actually 264.
But the model predicted around that only two days prior ( 273 deaths on March 26th).
I’d say it’s pretty accurate - off only by a couple days.
[0] - https://www.worldometers.info/coronavirus/country/us
Considering it was probably two doubling times (I've been seeing 2.77 days or so as well from the jhu site) to write the paper and get it out- working extremely hard.
So, you're just assuming other jurisdictions are going to have a curve that looks like Wuhan's?
https://en.wikipedia.org/wiki/William_Farr#Law_of_epidemics
https://en.wikipedia.org/wiki/William_Farr#Law_of_epidemics
The study was made by
"The Institute for Health Metrics and Evaluation (IHME)" launched in 2007 "with the goal of providing an impartial, evidence-based picture of global health trends to inform the work of policymakers, researchers, and funders. Main supporters are the Bill & Melinda Gates Foundation and the state of Washington."
Edit: once I read the claim "That methodology looks extremely dodgy" or "doesn't make it right" (in the answer below) I'd expect some specific information which would support that claim. Especially when the same person writes "you" to the poster posting the link to the study of an independent institute residing at the University of Washington.
"The Institute for Health Metrics and Evaluation (IHME)" launched in 2007 "with the goal of providing an impartial, evidence-based picture of global health trends to inform the work of policymakers, researchers, and funders. Main supporters are the Bill & Melinda Gates Foundation and the state of Washington."
Edit: once I read the claim "That methodology looks extremely dodgy" or "doesn't make it right" (in the answer below) I'd expect some specific information which would support that claim. Especially when the same person writes "you" to the poster posting the link to the study of an independent institute residing at the University of Washington.
That really doesn't answer the question. That methodology looks extremely dodgy, and merely having funding from the Gates foundation doesn't make it right.
It's curve-fitting. By real epidemiologists. It's not just based on Wuhan, but based on knowledge of how things like this spread. Wuhan fit the model, so there's hope.
But it's just curve-fitting. And it depends on certain things, like people being smart. You get your parameters from what you observe.
If you have a better way to do it please share.
But it's just curve-fitting. And it depends on certain things, like people being smart. You get your parameters from what you observe.
If you have a better way to do it please share.
They're getting flack from other epidemiologists for doing just that. Curve-fitting the AIDS epidemic suggested it would peter out in 1995.
We only have the same curves if we put in the same restrictions as Wuhan, or get R down as much. Yet we still have megachurches holding services.
And politicians are using these as worst-case scenarios already, which will make them complacent about the urgent need for more ventilators, masks and PPE supplies.
We only have the same curves if we put in the same restrictions as Wuhan, or get R down as much. Yet we still have megachurches holding services.
And politicians are using these as worst-case scenarios already, which will make them complacent about the urgent need for more ventilators, masks and PPE supplies.
They answer all that in the paper. It's not predicting behavior of people- we can still do stupid things and make this a lot worse.
But if we behave and practice social distancing and don't do stupid things, this is the estimate we have today (or a couple of days ago) of the timeline.
If I were a leader of the United States, I would challenge the population to beat these numbers.
But if we behave and practice social distancing and don't do stupid things, this is the estimate we have today (or a couple of days ago) of the timeline.
If I were a leader of the United States, I would challenge the population to beat these numbers.
How could the US possibly beat these numbers? Do you imagine shutdowns more severe than Wuhan happening here?
More worrying is that a lockdown that started recently will not be reflected immediately in any statistics -- it takes time for that. If I understand correctly, the lockdown of Wuhan started as they reported only 23 deaths and was more drastic than any other. The growth in cases however continued afterwards for around a month.
Some of the things they did initially were counterproductive; isolating carriers is actually much better than sending them home to infect family. I saw figure of 75% of cases being transmitted by close contacts.
All that said, the math is brutal and you're right about things not being immediately apparent in the statistics. There's a delay for symptoms, and then another for deaths.
Unless another country epicly fubars their response, the USA should maintain its #1 spot. All the more so if states don't encourage social distancing for another week.
All that said, the math is brutal and you're right about things not being immediately apparent in the statistics. There's a delay for symptoms, and then another for deaths.
Unless another country epicly fubars their response, the USA should maintain its #1 spot. All the more so if states don't encourage social distancing for another week.
I'm thinking of a challenge more like Sweden rather than an enforced lockdown. Make it a competition- a race to the moon. What happens in Sweden remains to be seen though.
More interesting to me is that we have a measure by which we can evaluate our leader's response. Deaths well north of 200K means he severely screwed up.
Of course we'll never convince his base of that should it come to pass.
And to be fair, if he keeps the number of deaths in the 100K range then he will have done a good job from the point at which he started taking action.
More interesting to me is that we have a measure by which we can evaluate our leader's response. Deaths well north of 200K means he severely screwed up.
Of course we'll never convince his base of that should it come to pass.
And to be fair, if he keeps the number of deaths in the 100K range then he will have done a good job from the point at which he started taking action.
We shouldn't let him anchor to those numbers. More than 10k is already a catastrophe, and difficult to justify given the advance warning.
Everything in excess of that is a shocking mix of incompetence, hubris and malice - including refusing help to states whose governors aren't sufficiently deferential.
Everything in excess of that is a shocking mix of incompetence, hubris and malice - including refusing help to states whose governors aren't sufficiently deferential.
Many key variables change the outcomes significantly. I've heard epidemiologists talk about reducing the "populations" down to things like schools. But then they assume "everyone interacts with everyone". Hence they predict very high infection rates. Michael Levitt takes into account our typically small social circles and predicted Chinese "end" accurately (2-3%). The message is of course, physical distancing works and with our personal circles rather small, it works quickly. I am looking for Levitt's published algo but no sign, I'm going by reports. He is a Stanford Prof so likely legit.
Numberphile did a great video on the SIR Model if you prefer a video explanation: https://www.youtube.com/watch?v=k6nLfCbAzgo
On the topic of SIR, check out the 3Blue1Brown video on this topic: https://www.youtube.com/watch?v=gxAaO2rsdIs
Are there alternative models? I read (not very reliable source) that SIR was pretty naive in the sense that it doesn't model the graph of social interactions. Also, Spanish flu didn't really follow SIR.
Tons - in addition to the split between stochastic (statistical) and deterministic (mathematical) formulation, there's individual vs. population averaged models, various kinds and types of spatial generalizations, an alphabet soup of additional compartments (SEIR and SEIRS being probably the most common), generalizations of the compartment membership time (check out PS-SEIR models), and much more.
Yes tons and tons. It's fairly easy to just imagine it as a sort of DFA/NFA. I personally like an SIR(D) model where from the "I" state you can either progress to recovered or dead. You can kinda take care of that in the regular "R" case with the SIR model, but the "D" state makes the fatality of the virus all the more visible. Additionally, you can add "Vital dynamics" to the model by basically adding a birth rate dependent on the total size of the reproducing population that feeds into the "S" state. You can also add a general baseline death rate (\mu) that takes away some population from each state in SIR. Lastly, like the sibling commenter mentioned, you can add an "E" state where there is a state between susceptible and infected where an individual is infected but not yet infectious. All that is added are more letters and computational time :)
> computational time
Understatement of the century lol, especially for fully Bayesian versions.
Understatement of the century lol, especially for fully Bayesian versions.
Interesting in the abstract but we know that even simpler model works well in the case of the epidemic we're sadly watching.
In this model, change in the number of people is function of the number of people infect, that's it. The number of potentially infectable people is so large that changes in it's value won't enter into consideration until the US is at WWII-invasion level of fatalities.
dS/dt = Sr, for some value of r.
A simple differential equation indeed, solved by S = e^(t*R); exponential growth as we see (note, I'm not claiming my r here is R0 or whatever).
In this model, change in the number of people is function of the number of people infect, that's it. The number of potentially infectable people is so large that changes in it's value won't enter into consideration until the US is at WWII-invasion level of fatalities.
dS/dt = Sr, for some value of r.
A simple differential equation indeed, solved by S = e^(t*R); exponential growth as we see (note, I'm not claiming my r here is R0 or whatever).
You're totally right, of course. And I even mention that in the post. If we assume that S ~ N then we get that exponential growth. But eventually (unfortunately) enough people will get sick that S is no longer close to N; that's why the epidemic peaks.
A friend who is a former epidemiologist, writing about some of his work in another place:
> Also, mine is a SIRD model (guess what the D stands for). People didn't use to include mortality in these because it depletes the population you are trying to simulate, but I've been able to run large enough numbers that you can include it as long as your disease is not too lethal ... and if it is that's also a result of sorts.
> Also, mine is a SIRD model (guess what the D stands for). People didn't use to include mortality in these because it depletes the population you are trying to simulate, but I've been able to run large enough numbers that you can include it as long as your disease is not too lethal ... and if it is that's also a result of sorts.
Depends on the type of model, but death and birth and everything in between has been put into these models. Feel free to peruse any of the literature.
How could one add undertesting to the model?
See the supplimentary material of this paper [1]
They split the I compartment into two, one for the documented infected and one for undocumented infected.
[1] Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2) Li et al. DOI: 10.1126/science.abb3221 https://science.sciencemag.org/content/early/2020/03/24/scie...
[1] Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2) Li et al. DOI: 10.1126/science.abb3221 https://science.sciencemag.org/content/early/2020/03/24/scie...
Testing isn't part of the model at all, it operates only on the true numbers of susceptible/infected/recovered. Testing only gives you an idea of what those might be.
That said, it does have the potential to impact the perception of danger to individuals. And perception of danger should influence the β factor the author mentions. Learning that a lot of people have it or a lot of people are dying for example may cause people to take it more seriously, and thus influence β.
[deleted]
Don't assume the data going into this model has anything to do with reality. There's a very, very good reason that the China data almost perfectly fits a curve. That's because they used a quadratic equation to generate the data.
The lines of hundreds/thousands of people waiting to recover the ashes of relatives from several cremation facilities, along with the documented arrival of roughly 45,000 urns to Wuhan alone belies the rotten data that even now claims only 3,300 total deaths in .cn
The lines of hundreds/thousands of people waiting to recover the ashes of relatives from several cremation facilities, along with the documented arrival of roughly 45,000 urns to Wuhan alone belies the rotten data that even now claims only 3,300 total deaths in .cn
If China is understating the damage from the virus they are only doing it by a constant factor at best. You cannot hide an exponentially growing epidemic, no matter how hard you try. That's just how it is on this bitch of a function.
Western propaganda has trained its people to think the Chinese communist party is as stupid as it is authoritarian. Quick quiz, if you don't know who Li Keqiang is then you more likely than not treat the CCP as a black box controlled by Winnie the Pooh.
Western propaganda has trained its people to think the Chinese communist party is as stupid as it is authoritarian. Quick quiz, if you don't know who Li Keqiang is then you more likely than not treat the CCP as a black box controlled by Winnie the Pooh.
[1] https://github.com/EpiModel/EpiModel-Gallery
[2] http://www.epimodel.org/
[3] https://github.com/statnet/EpiModel