The Coronavirus Curve - Numberphile

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Ben Sparks explains (and codes) the so-called SIR Model being used to predict the spread of cornavirus (COVID-19).
More links & stuff in full description below ↓↓↓
National Health Service (UK) advice on Coronavirus: www.nhs.uk/conditions/coronavirus-covid-19/
Ben Sparks: www.bensparks.co.uk
Use the Geogebra file Ben created for this video: www.geogebra.org/m/nbjfjtpv
Another good file courtesy of Juan Carlos Ponce Campuzano: www.geogebra.org/m/utbemrca
3blue1brown on the exponential growth of epidemics: ruvid.net/video/video-Kas0tIxDvrg.html
Tom Crawford on the SIR Model: ruvid.net/video/video-NKMHhm2Zbkw.html
Kurzgesagt on COVID-19: ruvid.net/video/video-BtN-goy9VOY.html
Washington Post simulator: www.washingtonpost.com/graphics/2020/world/corona-simulator/
Extended presentation by Nick Jewell for MSRI: ruvid.net/video/video-MZ957qhzcjI.html
More videos with Ben Sparks: bit.ly/Sparks_Playlist
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outreach/science-sandbox/
And support from Math For America - www.mathforamerica.org/
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Science & Technology

Published on


Mar 25, 2020




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Comments 80
Numberphile 2 months ago
Be sure to check out the full video description (click SHOW MORE)- lots of useful links there.
Jonathan Bone
Jonathan Bone Month ago
@nacoran The beta is acutally the number of social interactions. So lets say for example we interact with 5 people. The assumption is that you will continue to have 5 interactions but as the proportion of susceptible population decreases and infected population increases, it is far more likely that the some of the 5 people you meet will be either infected or recovered so the number of new cases declines.
M. Hui
M. Hui Month ago
According to this model, roughly 80% of the population is susceptible?! How is this even possible?? Did I misinterpreted the model or what? Glad if someone could be so kind as to explain this to me!
Christian Evans
Christian Evans Month ago
Is there an expanded definition of susceptible here concerning those with weaker immune systems or since WHO declared the youth aren't invincible?
Mahdi B59
Mahdi B59 Month ago
Please answer this HOW TRANSM=3.2 AND RECOV=0.23??
Jolliebearfor Christ
xin zou
xin zou Hour ago
i hate assignment
Yash Sehrawat
Yash Sehrawat 2 days ago
Which Application are you using??
Lena Araneda
Lena Araneda 7 days ago
Hi, this might be a really naive question but how can the equations S', I' and R' produce curves if they are linear expressions?
Colin Harris
Colin Harris 8 days ago
Can you please provide details of what Geogebra application you use for this?
Hunc HancHoc
Hunc HancHoc 8 days ago
22nd May - Perhaps the govt should have used some free graphing software rather than Neil Ferguson's model...
always happy
always happy 10 days ago
Awsome explanation. This is exactlaty what I was looking for to help with my diff project
Bedio Soro
Bedio Soro 10 days ago
What software did you use
Rob Garrett
Rob Garrett 20 days ago
I like the video but you leave out other variables. What about modeling this disease and using herd immunity as a variable, or put in parameters such as not closing businesses but making rules to allow them to stay open, such as physical distancing within? There are a few states in the US that have done this with minimal to no increase in infection rates.
Badger Beater
Badger Beater 23 days ago
Does anybody know, using GeoGebra, how you then integrate these solved derivatives?
David Edwards
David Edwards 24 days ago
This video would have been reasonable if there was a thorough discussion of the assumptions made in the model. But without this it was very irresponsible.
Johann Sedrak
Johann Sedrak 26 days ago
Rest in Peace Sir Conway
Amy Beth
Amy Beth 28 days ago
I had understood Rnaught to be equal to transmission rate, not a ratio of transmission to recovery. Rnaught, per my understanding, is the average number of people infected by a person with the illness. (EG - measles' Rnaught is very high, like 13, I think? So one person with measles can infect 13 other people, just on average)
Dee Jarrett
Dee Jarrett Month ago
I got the bit when he said I'm going to use these colours.
Sarath Krishnan
Sarath Krishnan Month ago
Please forgive me :coronavirus
Sarath Krishnan
Sarath Krishnan Month ago
Show this video to corona virus they will run away after hearing this lecture
Yossi Lampert
Yossi Lampert Month ago
Yeah I can't see the difference between the colors you used but ok I grasped the math nevertheless
Easy Mathematics
Small "fun fact": In mathematics a "Corollary" is a trivial gift from a theorem. And this word "Corollary" is a derivative from the word "Corona".
This is a really nice math video. Special times but nice to see how math is able to predict parts of our world.
Charles Duggan
Charles Duggan Month ago
Completely off question but what program is Ben using to solve and plot this system of DEs? I've tried GeoGebra, Desmos, and CoClac and even when using the same logic and denomination as in the video, they all freak and act up. I'm trying to do a project which uses a system and would like to see a working plot and equation
David Kearney
David Kearney Month ago
way too geeky
Cooper Gates
Cooper Gates Month ago
Notice something important and expected, that S', R', and I' sum to zero. I thought he was going to define I' as -R' - S' or the analogous equation for one of the other derivatives.
cena kp
cena kp Month ago
Thank you for the videos. They are great, you have got a great channel.
Ankush Arya
Ankush Arya Month ago
wonderfully explained, kept it really simple, I can now implement it in python .. brilliant work .. thanks , hope I could be Mark's student
Peter Vanderwaart
I have a problem with the definition of S'. If you send 100 infected people out into a population of 1,000,000 people, they will infect the same number as if you sent them out into a population of 2,000,000.
Anjana Ashokkumar
Fellow mathematicians and statisticians. Please check out our contribution to curb this pandemic. Search for - Team 112 MHRD Mega Challenge. Please watch and like.
Ngocbich Cao
Ngocbich Cao Month ago
I hate corona :(
Eraj Qurishi
Eraj Qurishi Month ago
what software are u using bro!
thierry dagaeff
thierry dagaeff Month ago
@Numberphile: Dear Numberphiles, I have a demand.. Please, can you simulate the case where, from the peak of I(t) we try to increase transm with time? The challenge is to get a function transm(t) so that the peak becomes a flat horizontal line once it is reached. Interpreting the max of the peak as the number of cases the system can manage (e.g. depending on the hospital capacity), the stake is then to understand how we can go back to normal as fast as possible (increase transm with time) without going over the max. Would Transm(t) be linear, quadratic...?
thierry dagaeff
thierry dagaeff Month ago
And BTW, by keeping I(t) constant at the max for a given amount of time will allow to reduce the length of the tail...
Selim Month ago
great vid but the cardboard background is so ugly
interminas08 Month ago
Tried implementing it in MATLAB (potentially Octave?) Please find the code below and let me know if I made a mistake. N = 1; Istart = 0.01; Sstart = 0.99; Rstart = 0; transm = 3.2; recov = 0.23; maxT = 1; % define S',I',R' % S is y(1) % I is y(2) % R is y(3) %so here fun = @(t,y)[S';I',R']; fun = @(t,y)[-transm*y(1)*y(2);(transm*y(1)*y(2))-(recov*y(2));recov*y(2)]; % Provide the starting parameters Y0 = [Sstart; Istart; Rstart;]; % Define the range of t tspan=[0 maxT]; % Magic happens and matrix Y contains S,I,R [T,Y]=ode45(fun,tspan,Y0); % Plot plot figure plot(T,abs(Y(:,1)),'b-') % S hold on plot(T,abs(Y(:,2)),'r-') % I plot(T,abs(Y(:,3)),'g-') % R xlim([0 23])
KS_NZG Month ago
Hi! I'm trying to use this as a Geogebra lesson in 7th grade and I'm wondering what's the difference of the transmission rate (transm) and the R-nought, you're referring to. Please note I'm Danish, so something might be lost in translation, but as far as I've googled myself into, R-nought is also referred to as transmission rate.
Jade Smith
Jade Smith Month ago
Perhaps you should include more variables: economy, vaxinne. From previous models like Mers and Sars, you can some extrapolated values. The longer the lock down, the slower the spread, more job losses and more indirect deaths by jobless people without homes. I think S is difficult to determine as even young healthy people can get sick. S and R are basically each other's mirror image. I might seem to flatten and you can predict the end, but as people see an improvement, they think everything is okay again, which will result in a new spike.
Vlad Antipov
Vlad Antipov Month ago
Thank you for setting this up. I was able to replicate your example in geogebra. I would like to understand how I can inject real numbers into this example. I have a month of real data that I would like insert into this model. The scale for the X axis seems too compressed.
Michael Popovich
i hate you
LowApe Month ago
Once you've recovered are you susceptible to infection again? It seems that these models assume that you're not, which I don't think necessarily is the case
nab 6215
nab 6215 Month ago
I got sick. I programmed on the side, but I got sick. Now I do volunteer work. I am learning HTML5, CSS, while rebuilding a 10+-year-old web COPD website. I do stats and charts whenever I can. The lung folks are always Impressed.
Cassandra Church
I really enjoyed that, thanks. I’d be interested to see what happens to the curve if variables change over time. Ie, if we stop social distancing at some point.
Yilkal Negesse
Yilkal Negesse Month ago
Thank you for your terrific work and clear explanation. But I have question on the way that you calculate the transmission and recovery rate parameters. is there any criteria to calculate them? since I'm working on it and I can't determine them, please help me!!!
You mean fitting this model to real data ? Won't work. It's way too simple. Transmission rate is time dependent due to government measures. Also, I think recovery equation is wrong. As @Erumaaro pointed out, R`=recov * I assumes that recovery is random, i.e., some constant part of infected people recover every day. In real life, I think equation would be R`(t)=integral[ recov(tau)*I`(t-tau) d tau, from tau=0 to tau =t ], where recov(tau) is zero for some 10 days, then gaussian-like for some another 10 days, and then zero for the rest.
CJ Kruegs
CJ Kruegs Month ago
The Coronavirus : *Exists* Area 51 Nerds : *ZOMBIE APOCALYPSE*
Steve Lenores
Steve Lenores Month ago
All models have assumptions which are basically guesses. Computer modelers are modern version of court astrologers. They are no more reliable then reading animal entrails. You never have enough data to make accurate predictions. You occasionally get the shape right but almost never the quantity correct. This what gets officials into a panic taking draconian measures. A lesson for those who too much faith in climate models. Best application for models is preparation but never mitigation. A big key for me above is the sliding bar for infection rates which is BIG QUESTION MARK. It is dependent on other factors like social behavior, the virus itself, cleanliness of environment, and a number of factors we won't know for years o come. Shutting down is not the solution, changing behavior is. Changing behavior will help with every infectious disease not just coronavirus. Of course humans are resistant to change unless they learn the hard way.
John Paul
John Paul Month ago
I must be reading this wrong! If the transmission rate is pushed up to 10 the peak falls off so rapidly that the whole thing is over in 2 days. Does this imply that herd immunity may actually take place? or is it just that lock down may not work. Could Trump be right?
Rubén Alba
Rubén Alba Month ago
How to make a model which considers containment measures taken once the disease has begun to grow?
Garima Gupta
Garima Gupta Month ago
Hi, which software has been used in this project?
GeoGebra. How did you overlook this ?
Z_Krom Month ago
What if the recovery rate is 0?
Axel Andru
Axel Andru Month ago
You can't use infection rate data from China. Other countries must eat rats and dogs for the data to be considered.
Benzina Moktar
Benzina Moktar Month ago
What is the software you're using ?
Trade Tech
Trade Tech Month ago
Yeah...right..... Only one major problem. These models make absolutely terrible "assumptions" and that IS being seen in just how way off so many 'models' have been when put up against real world data! For example, models don't take into account cultural differences, average age of population & population density per country etc etc. My proof: Compare Italy to Sweden! Massive cultural differences on something as simple as personal greetings, age...(Italy has second oldest pop behind Japan) Now compare Japan to Italy, or New Zealand to Australia, or US to China's obvious BS case figures.... Keep mind here, this virus is essentially the same in every nation. It hasn't "mutated" enough yet to make up for the differences in cases vs actual death rate Another example of some terrible modelling that just made a mockery of it: New Zealand models were predicted to have 80,000 deaths if no lockdown, that was revised to 27,000; then 14,000....and the reality thus far? 1209 cases & 1 death "with" covid19.....NOT "from". That person was elderly and had serious preexisting health issues.... So a mortality rate of .0827
Floyd Barber
Floyd Barber Month ago
19:25 Flattens the curve by merely adjusting the spatial proportion on the X-axis lol...
Duane Barry
Duane Barry Month ago
Thank you, this is wonderfully clear explanation. What happens if the transmission rate is reduced enough to appear to flatten the curve but then is allowed to increase again before 80 or 90% of the population has recovered? Social distancing, if strongly implemented, should stop almost all transmission, but if a large majority of people have never been infected and restrictions are lifted because the curve seems to have flattened (number of cases is decreasing), couldn't the small number of people still infected lead immediately to another spike in cases and deaths?
David Bloomfield
Hi, Thank you for an excellent video. One comment - whilst "playing" with the variables you run the "infected" curve from peaking at 80% down to peaking at around 10%. In reality, the current worldwide level of (tested) infection is reported as 0.018% and for Spain (one of the highest) 0.3 % of the population. So even allowing for a 10-fold higher number of unreported/untested infections we are still at infection levels of just a few % of the population. So I think it would be better to start with a model running in that regime and look at possible developments rather than show peaks at such high levels as if we were already there.
Jessica Month ago
which Geogebra app is this? I'd love to try this on my own
Benzina Moktar
Benzina Moktar Month ago
I noticed that you can't show the animation of all graphs simultaneously online
Benzina Moktar
Benzina Moktar Month ago
The downloaded app works better than online
Kameron Briggs
Kameron Briggs Month ago
You can do complicated things with math, but its fundamentally simple. I wish more people realized that.
Horinius Month ago
It would have been interesting to plot I+R which is the (cumulated) confirmed cases. That could show people what exponential growth and logistic curve are.
Adam Kučera
Adam Kučera Month ago
can you debunk evolution using mathemics? look into stephen c meyer for this kind of info
Timothy Douglas
Timothy Douglas Month ago
What about deaths?
Michael Klopfer
Michael Klopfer Month ago
Thanks for this fantastic video to show people the basics of this type of modelling!
M. Hui
M. Hui Month ago
I am desperately in need of someone to explain to me how the values a and b can be found. Many many thanks in advance!
Mahdi B59
Mahdi B59 Month ago
simplicjusz Month ago
Ben looks like the Russell Crowe of maths!
Val Entin
Val Entin Month ago
It is 'prime', not 'dash'. first derivative of a function is df/dt or f', read f-prime.
Arsenio Buck
Arsenio Buck Month ago
Numberphile = pedaphile
Tab Conwell
Tab Conwell Month ago
Thanks guys!
Brian Odom
Brian Odom Month ago
So flattening the curve, optimistically, we’re looking at a 0.25 infection rate. I live in the United States with a population of about 330 million. So best case, we’re looking at about 82.5 million infections in the United States alone. Wow.
jtcouch Month ago
By treating early, at the onset of fever, coughing and other symptoms, Covid-19 is less likely to progress to pneumonia and is also less likely to invade other organs such as the heart and kidneys. The last thing you want is to wind up in the ICU on a breathing machine. Lack of a quick and easy test has led to 1,000s of needless deaths. That quick test is now available from Abbott Labs, but there remains a distribution problem and a local doctor awareness problem.
Fabian Ram
Fabian Ram Month ago
what sim are you using in this video
Nicolás Baquero
Does somebody knows how to calculate the transmission rate from the data of total cases and cases per day? I'm trying to make my own model for my country and I would like to approximate it the best I can. Thank you in advance if someone can help me! Stay safe everyone
Pat Trainor
Pat Trainor Month ago
Always awesome. But this begs the question, "Where can one get the actual/real numbers of the variables?"
ßri Month ago
You recycled the graph that everyone is using and it tells you nothing about what is actually going on. The only number that matters is the multiplayer and its rate of increase or decrease. Take the number of new cases today divided by the total number from yesterday and you get the daily multiplayer. keep track of that over time to see if you are getting better or worse . in Kentucky we have gone from a .3 multiplier to .14 in the last 15 days so we are getting better even though there are 607 more cases now than then.
FifaKing HD
FifaKing HD Month ago
Ben used to be a teacher at my school, really nice to see him again!
Miles Pulford
Miles Pulford Month ago
comes for a coronavirus video nobody: Ben: "differential equations"
ColGadarby Month ago
One thing I have noticed is the number of people in UK classed as totally recovered at 135 against 33,718 total number of cases makes less than 0.5% then I look at South Korea with 5,828 totally recovered against 9,976 making a recovery of 58.4% So surely we should be looking at their protocol and methodology of places like S Korea to see what they are doing right and/or what we are doing wrong. Many countries trying different strategies and some working better than others - this is where numbers can help us. (As @ 3rd April 2020)
Bishop Polycarp
Bishop Polycarp Month ago
So, after the lock down when the disease is not eradicated the spike begins all over again? Just delaying the inevitable?
Povl Besser
Povl Besser Month ago
This is awesome. I think there was a lot of explaining of the inherent assumptions in much of the maths that this channel is about. Like, why does that variable have a negative value, why does that constant have this value, and so on.
Robert Ballow
Robert Ballow Month ago
inhaling ethanol cures respiratory infection
Nima Nikuie
Nima Nikuie Month ago
Thank you for the wonderful video. What is the name of the math simulator you're using? And is this model predicting that, (short of a cure) 80 to 100% of population will eventually get this virus?
Matthew Acuff
Matthew Acuff Month ago
How about creating a curve that represents what's happening in the real world? Then we'd know the variables--may vary between countries.
Black Knight
Black Knight Month ago
R = resolved would be more accurate and less confusing.
Kevin McCarthy
Kevin McCarthy Month ago
Is there a video/tutorial for replicating this work in Geogebra from scratch?
Dicky Bannister
Dicky Bannister Month ago
If you open Geogebra Classic out of the box. the sliders should be on the tool bar. just click 3 of them onto the paper. the 'input' box will be at the bottom. follow the input given that is it. right click on the paper and use zoom. also use xAxis:yAxis to, say, 10:1 (i.e. to get the right relative scale between x and y). that is about it really
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