Oxford Mathematician explains SIR Disease Model for COVID-19 (Coronavirus)
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- Опубликовано: 2 окт 2024
- The SIR model is one of the simplest disease models we have to explain the spread of a virus through a population. I first explain where the model comes from, including the assumptions that are made and how the equations are derived, before going on to use the results of the model to answer three important questions:
1. Will the disease spread? 6:32
2. What is the maximum number of people that will have the disease at one time? 11:00
3. How many people will catch the disease in total? 16:55
The answers to these questions are discussed in the context of the current COVID-19 (Coronavirus) outbreak. The model tells us that to reduce the impact of the disease we need to lower the ‘contact ratio’ as much as possible - which is exactly what the current social distancing measures are designed to do.
The second video explaining Travelling Wave solutions to the SIR model is here: • Oxford Mathematician e...
The third video including an Incubation Time in the SIR disease model is here:
• Oxford Mathematician e...
Produced by Dr Tom Crawford at the University of Oxford.
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Just to clarify, the idea of this video is to inform people about how maths is used to model disease spread and to show why social distancing is SO important in reducing the impact of an outbreak. I am not giving medical advice, just informing you all of some of the background behind the advice given by the experts. Stay safe everyone and remember to keep your distance.
This model is then simulated using system dynamics software like Vensim , or there are other softwares the Government uses ?
Tom: thank you for this. I am no mathematician, but you helped me understand the model. Good work.
@@dptirkey See Numberphile's new video for solving and animating the curves. They used a free program called Georgebra.
Thank you so much Tom. I am a medical doctor and hence do not have much understanding of mathematics. I think your video is best to explain the SIR model and it's formulas on RUclips. I still cannot understand the concept of IMAX calculation and R calculation. Can you explain it a bit more that how log comes into play? And how equations are derived. Secondly any suggested readings for beginners like me? Million time thanks.
Thank you.
If machine gun kelly pursued maths instead of rap. Thanks a lot for the video, helped me with a report.
I'm listening to his new album on repeat at the moment - loving it!
Haha i was thinking the same
This video is great and the material is explained so well! This is exactly what I needed for my Calc III report on COVID-19. I'm glad to see some "real" math rather than the same graph showing the "flattening of the curve" over and over again.
Thanks Kalyn - glad it was helpful!
I used the statistics on the values of S, I, R, and the changes in S, I, and R each day, and plug it into the SIR model, and the current estimate is that the total number of infected individuals (I + R) can reach *a quarter of* the world population.
We do need to decrease the value of R_0 drastically to stop the spread of the disease. Great video with a great message!
How did you use SIR model since real I and hence also R and S are unobservable?
@@plrc4593 I simply used the data of reported cases from Worldometer. See the spreadsheet linked in my latest video for more information.
@@mathemaniac But how did you use them? If for example Italy reports today they've got say 1000 new cases it means those patients fell ill say 5 days ago, not today :D And they infected other people throughout all these days. Moreover they're now in hospital so they're removed from the system and don't infect other people any longer. Other than that in addition to those 1000 known cases another say 10 000 people fell ill 5 days ago but simply didn't present symptoms. But they still infect other people. Did you think about all these questions? :D I bet you didn't.
@@plrc4593 It's not that I didn't consider these. There are lots of caveats shown in both the pinned comment and the spreadsheet, which include some of your concerns (maybe you haven't checked those out?). This is kind of the limitation of the SIR model though. It assumes quite a lot, so that we can get the big picture as well as making it easier to understand.
@@mathemaniac You mean this spreadsheet: docs.google.com/spreadsheets/d/14XWEmLefkh-jRiHMeWc8kvM3sk9Q-Mx4EYHHlOSC_1Q/edit#gid=1994039230 ? There is no formula. At least I don't see any. Just few sentences.
Hi Dr Crawford, I've discovered your channel recently and, as a maths fan, I wanted to say that I think you did an amazing job at explaining the SIR model in a clear and understandable way.
In fact, I've liked this video so much that I decided to add Italian captions so that I can share it even with people here in Italy who don't understand a lot of English. Hopefully my captions will soon be available and other languages will be added too by other people because I believe your videos deserve them :)
(Also perhaps I will add captions for other videos of yours soon )
this video is pure gold, my final exam was an sir model based project, and your video helped me a lot
Glad it helped Tobias!
Hi, I'm S. Korean Highschooler and honestly respect you. This was much significant to me.Thx
You're very welcome.
I’m in love with Tom , he rocks
Is the sound of the chalkboard soothing to anyone else?
This.
Very much so. The gentle tap tap and scratch scratch, how pleasing.
I would love to see some of the more advanced models
Just a quick comment on some of the limitations of this model, particularly relating to the structure of interactions between people:
The calculation of q requires having an average of people infected by patient. However, this assumes that we do have an average that converges to the expected value. This would be true if most people were to meet people regularly and randomly, but this is not necessarily the case. Humans interact in ways that often follow a heavy tailed distribution, meaning that few people have a very large number of connections, while most people have few. Similarly, people interact most days with a small sample of individuals and every now and then they have contact with a ton of individuals (ex: football match).
In more mathematical terms: assume that the distribution of contacts per person follows a power law distribution (if we talk about the network of contacts this is often called scale-free networks), meaning that the probability of an individual having contact with k people on a given day is
p(k) = k^-g
where g> 0 and it is often between 2 and three. In that case, the moments of this distribution are
mth moment = E[k^m]= integal [k^m p(k)] from k_min to infinity = (g-1)/(g-1-m) k_min ^m
The problem with this is that for m=2, this gives us the variance, and it does not converge. In practical terms, this means that new samples would continuously change our average, hence prediction is difficult and needs further assumptions or "weaker" results.
Naturally, if there are no large gatherings and very connected individuals do not touch a lot of people the distribution of contacts might shift to non-heavy tailed.
Ref: R. Pastor-Satorras & A. Vespignani (2001). "Epidemic spreading in scale-free networks". Physical Review Letters.
Other issues include: clustering (two of my friends have a high probability of being friends), or adding delays into the equations (since the number of recoveries does not depend directly on the number of infected people, but on the number of infected people days before). Ref: HW Hethcote, P van den Driessche - Journal of Mathematical Biology
For further reading I would check the labs of Alessandro Vespignani or Victoria Colizza
@@pauvilimelisaceituno2893 thanks for the insight and the extra information.
try Corona Virus & Mathematical Modelling RUclips. The Tutor Wizard Inc.
Please speak in hindi
@@pauvilimelisaceituno2893 SIR is a basic model, with deterministic approach. We can develop (and make it more complicated 😁) by using stochastic solution, adding Expose compartment for delaying infection, etc..
Part 2 on extending the model to include the movement of populations is here: ruclips.net/video/uSLFudKBnBI/видео.html
Hey tom how do we calculate the transmission rate? and if so, what is the rate of contact/transmission rate for current the current COVID-19 pandemic?
If I can make a suggestion, please keep the important equations, such as "q" written down, I had to go back to look it up every time. You have a whole left side of the board you're not using.
Noted - thanks for the feedback :)
@@TomRocksMaths Hi, i was programming the SIR model in python, but im stuck at the moment where i have to give values to the variable you call "r" and "a" into your video. What are the values for the covid19 and where/how to find them ? Im not looking for the exacts values cause i know it's impossible, but im looking for realistic values. On wikipédia i've seen that Ro = infection rate * average number of people met (per day) * number of days. Is it good to deduce that r = (Ro/people met)/number of days ? It looks logic but it gives stranges results. Results become more realistic when i consider that r = (Ro/population of the country)/number of days, but i don't understand why the population of the counrty would be in the equation. Also, for the value "a", do i have to consider that it equals to 1/number of days ? Or does is correspond to the recovered/infected ratio ? Nice video btw ;)
@@ggldmrd5583 these are very difficult values to obtain which is why I purposefully avoided including them in the video. I would suggest starting with the definitions I give in the video and then trying to interpret any available data that you can find to give approximate values. The value of a should be obtainable by looking at death/recovery rates for the disease. The value of r is more difficult but you could perhaps look up the same value for something like seasonal flu and increase it a little?
@@TomRocksMaths Thank you for your answer, about r im still in trouble, but your answer helped me for the other value. I also found that : www.lewuathe.com/covid-19-dynamics-with-sir-model.html
which helped me for the other value. Indeed a = 1/time as i guessed. The source also mentions that r = Ro*a but it gives the results that i told you before, which is extremly low for a Ro (Ro = 0.0012 for Italia for exemple in this source, using this sort of low numbers i can do a good simulation, but using normal Ro values such as 3 or 4, it just doesn't work). Sorry for my english btw, it is not my primery langage.
@@ggldmrd5583 I think you need a variable R0. It's 2 or more at the start, but social distancing reduces it to maybe 1.4
Right now I'm in 8th grade, and I have no idea what's going on but I still find everything quite intriguing. By the way, I came from Mike's channel :D
Hopefully, these videos will help me in the future!
Awesome - and welcome :)
Thank you so much for this! I happen to have been given a class project to evaluate the SIR model and this video was incredibly helpful - thank you again!
You're very welcome Leanne!
First of all i would like to say thank you for sharing your worthy information with us.
As a mathematics student i think that,by familiarising the applications in mathematical field we tend more close to this field.Moreover,by explaining safety measures of this epidemic spread in a mathematical model is way more accessible.
I would love to see more advanced models if that's something you are interested as well.Spread the math word and thank you once again. 🙏👏
Thanks Krishna - I discuss a more advanced model with spatial dependence here: ruclips.net/video/uSLFudKBnBI/видео.html
Thank you so much for sharing understanding and reason in this absolutly crazy time.
Best wishes to you all from Munich :)
Sir you just explained SIR model so very well, much learning when we go through various equations developed in model to scenario and everyone's perspective on it....i always found videos of yours explaining all the minute details.... keep up the good work :)
I was just building an SIR model to fit the parameters for COVID-19 in python! Fantastic video
How did you use SIR model since real I and hence also R and S are unobservable?
@@plrc4593 S and R are actually observable variables (assuming you have a reliable measure of I)
@@montanariarthur Hehe "assuming" :D
@@plrc4593 Well, the system is still observable, the problem is that a Luenberger observer is not going to asymptotically converge to the true value of S and R!
my math teacher assigned for us to watch over the “break” we have, we were suppose to take notes. Do you think that as an eighth grade class, we can understand this because I become lost many times?
It's certainly possible if you watch it several times and look up anything that you don't understand, but this is what I teach to my second year university students so don't worry at all about getting stuck!
I found the explanation on Numberphile also very helpful. He sent us over here for more info.
Here is that video:
ruclips.net/video/k6nLfCbAzgo/видео.html
this would be pretty tough for your age....13? Instead of worrying about all the equations. You can still follow how changes in Q, change outcomes.
If instead of differential equations, use difference equations. This uses discrete time steps but the equations are doable by hand.
S(t+1) = S(t) - a I(t) S(t)
I(t+1) = I(t) + a I(t) S(t) - b I(t)
R(t+1) = R(t) + b I(t)
So, no calculus, only algebra.
Thanks from France... retired, coming from physics, I like when the universal mathematics language explains, especially from young people ... in physics we use spring, daspot, mass in "our" equation... the first simple non linear solution of such differential equation is the Weibull form, perhaps it could be inserted to see or smoothe the non symetric responses and anticipate that the queue is always wider... a special day as Christmas is only a "little" mass in the equation system etc... give a report of your explanations and it could be translated in other langage to young people every where
Oh man, I'm so happy I found this video! You're explanations were spot on!
Here's a poem I wrote that you might enjoy:
For Nietzsche a horse was the abyss that he gazed
Cantor's Menge left his Geist behind
For Frege a letter: set-foundation ablaze
Russell's type too simple, unrefined
Old Gödel starved from independence
Turing tested apples, halting just the same
Bourbaki brought austerity to their descendants
But Coxeter grouped beauty with this formalist game
Now,
we know that
S increased when Boltzmann hanged his brain and
Noether's tumor was 'only' topological
Grothendieck schemed so much
and then
he went insane
Riemann and Ramanujan left before their prime
But it's their heartiness and flux
that let them pass the test of time
Ehrhart taught high-school
And Erdős was homeless
In the end
Euler went blind
Little Andrew Weil beat the devil with a deal
Weierstrass had demons too, concerning with the reals
The foundations still shudder
The Crisis, incomplete
Brouwer's points don't fix or feel
The shaking at our feet.
This is brilliant - thanks for sharing Kelly!
I don't get all of them but I get enough to realise just how brilliant this is.
Good one, I wish many people took up statistics as a subject which would make people understand the spread of a pandemic.
Thankyou tom❤️. fantabulous presentation. Loved it and understood almost everything clearly. And this is my project topic.
Glad it was helpful!
Conclusion: Please stay at home..
very informative, had a complete idea about SIR model related to any pandemic/epidemic
Found this on wikipedia. For people interested in diving deeper:
en.wikipedia.org/wiki/Compartmental_models_in_epidemiology
Absolutely amazing video! However I found your conclusions a little bit unfounded. One of the basic hypotheses of this model is that the population perfectly mixing. If we fit this model to the real life data from countries where the infection is far gone, for exemple China, South Korea, or Italy, it's obvious that S(0) is way smaller, then the populations of these countries, even if we count with the undiagnosed cases. In fact, S(0) is rather the size of a fictive population in wich we can more or less assume perfect mixing. Speaking of South Korea S(0) is quiet low, around 8000 according my calculations based on fresh data.
This shows us even more the importance of social distancing, because it's not just lowers the contact number, but also can isolate different groups of the society before the hole population get involved in the pandemic.
My impression is that even a little bit of mixing causes the spread to basically match these predictions, just on their own timescales in each otherwise-isolated subpopulation. And, while we are at it, we should note that it is fractalic in nature: spread at the international level is similar to spread at the subnational regional level, which is similar to the spread at community level, which is similar to spread between households. So, any movement between 'populations' at any level in this heirarchy can kick off similar spreads throughout the entire hierarchy for the receiving population.
Brilliant, very accessible explanation. Thank you!
Sería bueno que tenga subtitulos para que mas personas puedan entender el video :/
I completely agree Martha. Unfortunately I don't speak Spanish but if you know anyone that does that would be willing to help out please let me know :)
thank you for the explanation and it is great model!
i have one question, the equation at minute 12:45, how did you get the equation I + S - 1/q(ln S) = I0 + S0 - 1/q(ln S0) ?
I have big confuse how to get that equation. Thanks in advance!
Hi Adam, this follows from integrating the dI/dS equation with respect to S. We have dI/dS = -1 + 1/qS so this integrates to give I = -S + (1/q)*ln(S) + c. The constant c is then found by substituting in the initial conditions to give the equation you asked about :)
Is there a way to add a Brownian term to this? How would one go about it?
Yes, and it gets really really complicated really really quickly. I've purposefully started with the basic SIR model, but feel free to add to it!
Tom Rocks Maths how we can do it ?
I have a question about the math. We already know the data on least likely to die and most likely to die.
Why aren’t we using that data to build up a herd immunity that target specifically “those least likely to die”
Protect the vulnerable and start building up a herd immunity using the data.(which is highly favourable for young healthy people)
@@ello7034 I think the reason we don't is because it will be impossible from a social psychology stand-point. People objecting, breaking rules, thinking they know better, not really willing to intentionally infect themselves, etc. There's also the economic aspect where a sick person means a person not working.
Hopefully with the vaccines, a similar effect will be achieved without all that hassle
I never paid attention to maths like this dude.
The change of camera angle at 04:48 is awful. If you want to swap it about a bit how about a close-up camera watching you write but from the same angle? Otherwise you get that talking into space zombie look that videographers use all the bloody time lately and as viewers we're left thinking "Who got his "Jeremy Clarkson's Best Videographer In The World Camera Angle Bonus Pull Out" in this month's *MAG!*"" (Yes, I closed both sets of quote marks because as a mathematician I knew you'd be checking!)
Great comment. I can't watch videos that do this. It's like sitting at the far side of a theater while the actors ignore you.
At 12:32 when integrating the equation -1+(1/qS), wouldn't we get -S + (1/q) ln S + C? What were the initial values used, to find the value of C to be equal to I? Thanks in advance your videos are awesome!
Yes that is correct. We then solve for C by substituting in the initial values of S=1 at t=0 to give C=1.
Thanks! Your video helped me a lot.
I want to ask you: what is the difference between q and R₀?
I learned that βS₀/γ=R₀ but S₀ is considered as 1 so R₀ is same as β/γ
What is the difference with them?
12:23 If I make the equation dI/dS=-1+1/R₀S, is it wrong? help me ToT
Congratulations Tom, excellent tutorial.
I have a question.
You mention a function f (x) please What would be the independent variable x?
I tell you this because everything you have indicated as f (x) contains only constants.
The variable is the contact ratio q. It is treated as a constant in the model, but we can affect the value in our daily lives by practicing measures such as social distancing. The purpose of treating it like a variable was to show what we can do to help to reduce the impact of the disease.
thank you so much for making this video!! i was really interested in how the COVID-19 outbreak could be modelled, and as someone who recently finished differential equations in A Level Further Maths, your explanation was great and i enjoyed following along!! it's keeping me into maths while college is shut :)
Awesome - I'm glad you enjoyed it!
Thank you for this nice video !
I have a question: Normally, in the definition of reproduction number there is not the term S_0 because it is just the average infective period multiplied with the average number of individuals one infective will infect in unit of time. Are you using a slightly modified definition ?
Error evaluating the inequality of the first question (When does the desease spread).
Hi Frank - I addressed this in a comment below: In the video I explain that for the disease to spread we must have dI/dt being positive. If the RHS of the dI/dt equation is positive when S=S0 (ie. at the very beginning), then the number of infectives, I, will increase initially which is what we use as the definition of disease spread.
@@TomRocksMaths
The error is at 8:40. If rS0- a changes sign from negative to positive then you don't get an estimate from below for dI/dt.
Using S weakly less than S0 can only be used as estimate in one direction.
The right way to argue is to consider the behaviour only at t= 0.
In fact - and that is the side of your presentation that I like, the condition S = a/r for the maximum of I as funktion of S can be seen already here to motivate the proof as a solution for the extremal value problem given later.
This is fantastic. Thank you!
You're very welcome Tommy!
@Tom: First of all thanks to your brilliant video. One question I am stuck with is the following.
Your implicit equation to find the answer to question 3 is given as:
S_end - 1/q ln(S_end) = I_0 + S_0 - 1/q ln(S_0)
which you then plug into R_end = - S_end + I_0 + S_0
However, when I use, e.g. GeoGebra to visualise the implicit function: S_end - 1/q ln(S_end) = I_0 + S_0 - 1/q ln(S_0)
I don't receive that S_end is near zero. Ma result is that the curve has an asymptote towwards: I_0 + S_0. Could you please expand on that?
That would be great. Thank you!
Best wishes Holger
Hi Holger, if you plot y - (1/x)ln(y) = 1 - (1/x)*ln(1/2) then you should get a curve like the one I showed in the video (I just plotted the lower branch of the curve as it is the only one physically possible for populations). Here we have scaled by the total population I0+S0 and taken S0 to be 1/2 of the population size. The S0 value can be changed but must remain between 0 and 1 (and will most likely be very close to 1 for COVID-19). Hope that helps!
@@TomRocksMaths Hey Tom, thank a lot. That helps! Cheers Holger
BTW, do you have any mathematical reference which gives a ballpark for q, i.e. x in this case?
Cheers Holger
Dear Tom, I f I may ask one more question. Okay, just plotted your function: y - (1/x)ln(y) = 1 - (1/x)*ln(SO) For SO= 0.97. Again the problem occurs that the curve has an asymptote of y=1, which wouldn't mean that S_end is not small, but rather the full population, or? And thus S_end would be 0.
However there is a second branch of this implicit curve which does converge to 0. How do I know which branch of the implicit curve to consider?
Cheers Holger
Tom, your definition of the basic reproduction number is slightly different to the one given by wikipedia, however they are the same if you set S0 = N
How do you you account for the birth rate and no infected deaths rate of a population in your susceptible population? Do they change the rate of infection?
In this model we assume the population size is unchanged. Given the timescales being discussed its quite a reasonable assumption in fact.
Thanks for this great and energizing explanation! :)
However, I've got some questions regarding it:
At 8:30, where you write dI/dt < I r(S0-a), you say that the number of inections increases if the right side is larger than zero. But why's that? I mean, in order to have an increasing number of infections we would need dI/dt > 0 (instead of that upper limit for dI/dt)?
At 12:38, how do you come up with the right hand side of your equation? What we know from the initial values is that R + S + I = I0 + S0. So where does that equation come from?
Lastly, when determining the maximum value for I at 13:25, you say we make dI/dS equal zero. But I and S are time-dependent, wo what we are looking for would be the point in time where dI/dt=0. Did you skip some integration by separation of variable there or how would it work?
I'm very interested in the answers and would appreciate your thoughts. Thanks in advance.
If the RHS of dI/dt < I r(S0-a) is negative then the function cannot increase. If the RHS is positive, even only for a small amount of time, then an increase is possible. You're right in that it doesn't necessarily mean that dI/dt is postive, but if the maximum is negative then it can never be positive. So the RHS being positive somewhere is a necessary condition for an increase, but not sufficient.
When we integrate the dI/dS equation with respect to S we get I + S - (1/q)ln(S) = c where c is the integration constant. We fix this constant by plugging in the values of I and S initially since we know the values then. This gives c = I0 + S0 - (1/q)ln(S0) which we then plug back in to the get the equation in the video.
For I(max), we cannot just differentiate I with respect to t and set it equal to zero because we do not have an explicit formula for I in terms of t only. (You could of course find this point using graphing software on a computer). Instead, we have I as a function of S and so look for the maximum of I with respect to S. This point will change in time, which is why the I(max) value we have found is NOT the overall maximum, but the maximum AT ANY GIVEN POINT IN TIME. This is still valuable information as we want this to be as low as possible to prevent hospitals from being overwhelmed etc.
@@TomRocksMaths I see, thanks for taking the time for answering my questions. Keep up your great work! :)
@@TomRocksMaths Hi thank you for video and your comments, but I have one more question about yout answer-!! I wonder why constant C is ( I0+S0-1/qlnS0) , not -(I0+S0-1/qlnS0)???
@@judysh2103 Hi Judy - the constant is just fixed by the variables it represents. So when we integrate the dI/dS equation with respect to S we get I + S - (1/q)ln(S), and then substituting in the initial values we get c. If we move c over to the LHS of the equals sign then the sign will of course change (as it does in the video), but the original sign is just given by replacing the variables by their initial conditions.
Pause at 23:20, could you explain the graph a little bit please? Why are S0 and S are 5.1 and 3.1 respectively?
These are arbitrary values that showed a 'nice' looking graph that worked in the video. The values can be changed to anything and the shape of the graph is unchanged, it just stretches a little.
Tom Rocks Maths Ah I see, thanks!
Great video! I have one question... In finding the final equation for I max, would you be able to clarify how you got from the substitution of S = 1/q to I0 + S0 - (1/q)(1+ln(q*S0)) ??? Thank you!
I was also wondering that I got completely lost at that part
Hi Allissa, this comes from rearranging the logs in the integrated equation. After integrating and applying the initial conditions we have I + S - (1/q)lnS= I0 + S0 - (1/q)lnS0. Now we plug in S=1/q and have I + 1/q - (1/q)ln(1/q) = I0 + S0 -(1/q)lnS0. Leaving the I term only on the LHS we get I = - 1/q + (1/q)ln(1/q) + I0 + S0 -(1/q)lnS0. Taking out a factor of -(1/q) gives I = - 1/q(1 - ln(1/q)+ lnS0). Then -ln(1/q) = ln(q) by log properties and ln(q) + ln(S0) = ln(qS0) also by log properties and we are done.
@@TomRocksMaths omg this makes so much sense now thank you!!!!
@@TomRocksMaths didn't you forget the "R" group when setting the initial conditions equal to the new found expression with q?
Hi Tom. I understand this may be too late to ask, but I was hoping to make a model of this myself. How can I work out the rates of infection, or rather, where can I go to get information about the rates of infection - i.e. the constant r that is in dS/dT, or the constant a, that is in dR/dT?
Tricky question... the transmission rate r and the recovery rate a need to be estimated from the available data. They represent a 'rate' so must have units 'per day' or 'per month' etc. The best you can do is to try to estimate reasonable values based on published data about infection and recovery rates worldwide - good luck!
I don't get one thing, if q < 1 is what we want, not only we have to reduce transmission but also improve recovery. We just can't sit it out. Our government is spending virtually nothing on this. So yes I get it that social distancing must be there, but shouldn't the first priority now should be to make sure the governments lose their pockets and invest in healthcare?
You're not wrong Swagato, but as we don't really understand the virus yet, its much easier and effective to impose social distancing. The effects are more or less immediate, whereas investigating the disease will take years.
Great video and explanation! I would love a video on one of the more advanced models if that's something you're interested in as well :)
Could you please help me understand? It seems to me that when you are finding the max number of infected people you are finding the max of I(S) function which is not same I(t)max. What am I missing? Thanks for the video by the way, it rocks really!!!
You are correct. We are finding the maximum value of I at a given point in time. This will vary depending on what point in time we are considering, eg. the start, middle or end of the epidemic, but the equation will still be valid. The maximum value of I over all time, can be seen on the graph of I (shown in other videos/models) as the 'peak' of the curve. It's very difficult to calculate explicitly here because it depends on the other parameters.
By the assumption, the function s = S(t) is strictly monotone decreasing and thus invertible where t = T (s) for some differentable function T.
So you can write I(t) = I(T(s)) to obtain a rigorous descripition of what is going on considering I as a Function of S. Here the chain rule for differentiation applies.The presentation rightfully skips some of the formalities to better concentrate on the model itself but is correct here.
Hi, loved your video. I'm a bit confused by Q2: "What is I_max at any time". If we're looking for a time dependence, shouldn't we find the stationary value for dI/dt instead of dI/dS? Or you're saying that I_max is the largest infected number possible during a single outbreak. Apologise if I misunderstood.
Hi - thanks and great question. We can't calculate dI/dt explicitly because we don't have know how I depends on t explicitly. Therefore, we instead consider the maximum I at a given point in time, rather than the maximum over all points in time. It's a subtle difference but allows us to say something useful.
@@TomRocksMaths Thanks that's very clear now. Should've used my personal account tho lol
Hi, great explanation. I have a project on this, I was wondering if the equations would be the same if my R0 is not 0.
The main system of the 3 ODEs will be unchanged yes. You will just have different initial conditions.
Best explanation ! Very good ! I have one question : the end of the outbreak is when I = 0, how do we know it will happend at finite time ?
Good point Pedro, and the answer is I guess we don't, but we can 'hope' that it is the case.
Python code for a SIR model without a vaccine ie p= 0
Beta = 1/ 2 days
Gamma = 1/14 days
Lamda = Beta * (I/N)
# LOAD THE PACKAGES:
library(deSolve)
library(reshape2)
library(ggplot2)
# MODEL INPUTS:
# Specify the total population size
N
Hey Dr. Tom so fascinating even though I was a D- Geometry student. How would you calculate the amount of cases such as in the U.S. where there is little testing using the present amount of cases 17000, and deaths 229. Cheers
Good question James - for that you would need Statistics and a more detailed knowledge of the disease itself (which I do not have unfortunately). I've seen in the UK the total number of cases is believed to be at least 10x that of the known amount according to the government's chief scientific advisor.
does this video also explain how to derive the stability analysis??
For example take any graph, contains 9vertices and 14 edges, Transmission rate for S to I and I to R is 0.5 at initial infected node is 6 after two steps how many total infected nodes using SIR model. How to find sir? Can you give any idea?
Cheers for this, needed it for my uni essay!
sir this model is basic epidemic model..so we can apply it on any epidemic na?not just for covid19?
Yes it can be applied to any infectious disease that spreads through contact between someone that has the disease and someone that is susceptible.
How about the NO-B-S model as an alternative?
Please don't swap camera angles like that unless you're also going to always look at the active camera.
Very very very very very very nice
I am find any times then show your video very very nice
Hey Tom, this was a really detailed and great video! I really enjoyed it. What would you say is the PURPOSE of the SIR model? Like why model a disease using a SIR disease model in the first place?
Thanks - and really great question. We use this technique in mathematical modelling all of the time: start with the simplest possible model you can for a given situation, understand it, add an extra layer of complexity to make it more realistic, understand it, add another layer of complexity/realism, understand it, etc. Eventually the hope is that you get to a balance between something that you can understand/solve, AND is realistic enough to inform decision making. The SIR model was one of the first ever used to model disease spread and has now evolved into the incredibly complex and realistic computer-based models being used in the current pandemic.
I am so glad to chance upon this video. Thank you.
Glad it was helpful Owen!
I’m so glad you are here. Stay safe, spread the math word. Thanks so much. Minnesota, USA
Loved the video but would it be possible to site any sources?
The model I explain is based on the Mathematical Biology course I teach at Oxford. The accompanying textbook is ‘Mathematical Biology’ by J D Murray
Thank you so much
We will be in good hand after Trump understand this model. Can someone forward this to him ?
Actually he has an IQ of 156
Thanks so much for sharing your expert knowledge with general public!
Yes crawfmeister lad, best expln of where R0 comes from. Miss Else would be proud/ jelous/ a little turned on. I hope she sees this
My favourite RUclips comment of all time.
If you made video on intercept theorem thats grateful for me
Do you think you can model coronavirus?
The removed have a limit on:
(a) patients: 1-2% (the ICU Population limit).
(b) recovery: 50% (the number of ICU patients who will die even if they get care).
(b) safe handling of the dead: 2% (the bodybags available for the population).
The infectors have an individually varying infectiousness that is determined by the amount of time it takes to detect their infectiousness (the max value on their logarithmic curve means they died before you detected their infectiousness).
Hey, your equations are wrong! For dI_dt, the unit of the left term is people^2, while that of the right term is people, it doesn't make sense to subtract them together. The correct left term and dS_dt should be -rS(I/N), where r is the no. of contact per susceptible per unit time times the probability of being infected, (I/N) tells the percentage of whom they contected are being infected. It also doesn't make sense if you say 1/N is absorbed in r, because every country has different N, but r should apply globally.
This is a dimensionless model where all variables S, I and R have been scaled with the total population N. So, they represent a proportion rather than an absolute number.
why can't this guy be my math teacher?
bad idea. Those equations are dimensionally incorrect... look for a better teacher/modeler
Ty for making this video,if I have not seen this video I don’t even understand my parents since I don’t really understand the language I talk ty so much u explained it really well keep ur good work 😄
Awesome - thanks!
How does dI/dS = 0 gaurantee that dI/dt will also be zero?
It's something that has been considered but I don't see why that would always be the case.
dI/dS = dI/dt * dt/dS. From here we can say that it will be when dS/dt also not zero at that time. But is there any other way of telling the same?
I'm not entirely sure where I talked about this in the video Arka? Could you give me the timestamp please so that I can better answer your question?
could you further explain how you got 1/q(1+ln(qSo)? i want to use this for a paper i have to write and don´t know how to obtain that value for Imax. Thank you in advance
Hi Judith, this comes from rearranging the logs in the integrated equation. After integrating and applying the initial conditions we have I + S - (1/q)lnS= I0 + S0 - (1/q)lnS0. Now we plug in S=1/q and have I + 1/q - (1/q)ln(1/q) = I0 + S0 -(1/q)lnS0. Leaving the I term only on the LHS we get I = - 1/q + (1/q)ln(1/q) + I0 + S0 -(1/q)lnS0. Taking out a factor of -(1/q) gives I = - 1/q(1 - ln(1/q)+ lnS0). Then -ln(1/q) = ln(q) by log properties and ln(q) + ln(S0) = ln(qS0) also by log properties and we are done.
@@TomRocksMaths you just saved my life thank you so much!!!
Hi Tom are you from numberphile?
Sure am!
I want to know how to calculate the constants (the rates of recovery and the probability of an individual spreading the disease)?
They have to be estimated from the data. Johns Hopkins has a really good catalogue of infection rates etc. that is a good place to start.
@@TomRocksMaths Thank you!
Wow. Thank you so much! I know your aim was to emphasize the importance of the precaution measures for COVID-19, but this video was very explanatory for my IB Mathematics SL Internal Assessment.
You're very welcome!
Hey, what do you investigate in your IA? I'm doing the same subject and I'm sure that I will use this topic but I couldnt figure out how..
@@alp4119 should I send you a sample IA I found on this same topic ?
@@nahidameghji1510 hey even I wanted to use the SIR model for maths IA but I don’t really understand how. Could you please send me the sample IA?
Hello, can anyone explain how the right hand side equation at 12:38 formed? How is it related to the initial conditions?
Great video!!!! Just a question about why we multiply S * I to model interactions between invectives and susceptible group?
Thanks. One of the assumptions of the model is that the disease can only spread by passing from an infective to a susceptible and so the process of transmission will depend on the numbers of each population. For example, if there is only 1 infective and 1000 susceptibles we do not expect the disease to spread. This wouldn't be represented if we had only an I or S term - we need both of them.
@@TomRocksMaths thanks!
Thank you for this math. Clearly it is aimed at the college level. Not a problem. One piece of advice: for difficult concepts we need to make sure the listener does not have to suspend listening while he decodes a pronoun we use. Whenever possible we should repeat the noun rather than using the pronoun. You do pretty well at this.
I had this problem in college.. some teachers or GSIs would use so many pronouns. Others knew better. The lectures were so difficult to understand when many pronouns were used, since when you're trying to first grasp a concept, the fact that "it" can refer to three different things is insane.
Thanks so much for the video. I have been asked to work on Covid modelling recently, but didn't really understand the meaning behind R0, you explain it very well, helps a lot! (Plus, you look very cool, not like the traditional math teacher at all ;) Thanks and keep up the good work!!
Awesome - thanks!!
Thank you for your reply, but I don't get it why Ln(S0)=0? Because when you fix the initial condition, the equation becomes I0+S0-1/q*Ln(S0)=c. If c= S0+I0, then Ln(S0) must equal 0
You are correct that I0+S0-1/q*Ln(S0)=c but we also have I+S-1/q*Ln(S)=c and so they are both equal, giving: I+S-1/q*Ln(S) = I0+S0-1/q*Ln(S0) which is the equation in the video.
Hey Tom!
This is an awesome video for anyone who is trying to understand the SIR model.
I'm working on SIR and SIRS model, and was wondering how can we write an update rule for the SIR model using Gradient Descent and Newton's Method. Could you please help me out with this? Thanks!
Thanks Siddhant. No doubt these methods can be used to solve the system of equations, but as I am not a Mathematical Biologist myself, I'm going to stick with what I know well.
I'm lost after the 12:30 mark, not sure where all the log etc came in and how it was derived
The log comes from integrating the function 1/S with respect to S.
well explained. Job well done
Simply wonderful video!
I graduated ten years ago with my bachelor's in math and I've ended up not really using it. (Partially because of a long bout with disability)
BUT had I been taught about this connection to the medical field and mathematical modeling I may have continued my pursuit of mathematics further. I love math and medicine, but honestly hated differential equations because I had a terrible professor.
Your real world application makes it very interesting even if completely understanding the details concerns me a bit more than the average person.
Thanks for sharing!
Thanks Brittany, I'm glad you found it useful!
I'm tempted to write a small Bayesian framework to model COVID-19. Do you know any resources describing more advanced pandemic models?
This paper does just that science.sciencemag.org/content/early/2020/03/13/science.abb3221 Unfortunately they don't really provide any details as to the structure of their model. They use an SEIR model rather than SIR, and the code (in Matlab) and data are provided.
This is the one being talked about in the UK: www.imperial.ac.uk/news/196234/covid-19-imperial-researchers-model-likely-impact/
Try this review: www.maths.usyd.edu.au/u/marym/populations/hethcote.pdf
Differential algebraist here! Our entire lives/careers are devoted to searching for exact solutions to systems of differential equations like the SIR model!
Unbelievable explanation that let's us think about Mathematical Modelling in our critical time and when we have no vaccine
Glad you enjoyed it Osama!
Bruh my eighth grade math teacher assigned this
My brain is melting slightly
I'm not surprised - this is what I teach to my second year university students here at Oxford!! You might find my video on exponential growth easier to follow: ruclips.net/video/xUVERo2xpH4/видео.html
Thank you for the video l am an aspiring mathematician interested in disease modelling
please how we can find beta and gamma mathematically ( by linear algebra ...) ?
they need to be estimated from the real-world data
Maybe it's just me, but this seems pretty analogous to a Hamiltonian system. Any research out there where Hamiltonian mechanics methods are used to describe population dynamics?
An interesting thought... not something I've heard of myself but a quick google suggests such ideas do exist if you want to pursue a little more.
@@TomRocksMaths Thank you very much for your reply! I will look into it. Loved the video BTW.
Thank you for this sharing ,please if there are others Variants of thsi models and itwould be great if it coapled with exemples to be more effectve thank you in advance Tom great
I explain another 2 versions of the model - with spatial variance here: ruclips.net/video/uSLFudKBnBI/видео.html and with an incubation period here: ruclips.net/video/r7zKzvAS7Ig/видео.html
How does this map to the existing data on the ground - China, Italy, US etc. ?
I purposefully didn't use actual data in the video as to map the 'real-world' situation we would use a more advanced model than the one presented here. I believe this one would still capture the major trends however, eg. the exponential growth in infectives for example.
How do you build in for the effectiveness of NPIs (Non Pharmaceutical Intervention)? The simplest way seems to be to adjust the infection rate ”r”. That feels like a bit of a fiddle as actually what's happening is the contact frequency between S and I are being reduced. What are sensible values for ”r” and ”a” for COVID? I am solving the differential equations numerically (I'm an engineer - we always find a simple ways to manipulate maths) and setting ”r” to get a doubling of cases every 3-4 days and then adjusting ”a” to show cases recovering or being removed in around 5-6 days following infection. That approximates to an R0 of 3.65. In reality recovery is taking 10-14 days so the model tends to overestimate R (Removed) relative to experience? I was going to set dR/dt=0 for the first 6 days of the simulation to try and get a better approximation of reality. A bit of an Engineer’s fudge once again but that should be ok I think? Analysing the results, I am perplexed about any forecast of relaxing NPIs in April, May, June or July. Reducing the rate of infection slows down the spread and manages hospital beds but actually prolongs the pandemic extending it far into the future (12-18 months?). The only way to stop it in its tracks seems to be absolute lockdown to get R0 below 1 and that requires lockdown well into the autumn with universal testing and rapid isolation of new cases and their immediate contacts thereafter.
Hi Don, thanks for your (very detailed!) response. You're right that reducing the rate of infection will indeed prolong the epidemic, but that is what we are aiming for at the moment as it buys us more time. More time means we have a chance of creating a vaccine, or just understanding more about what this virus is and how it works so that we can better implement methods to stop its spread. At the moment its all so new and unknown and the best thing we can do is to get time on our side.
Tom Rocks Maths Thanks for the reply. I've always loved mathematics and simulations in particular. It's often hard for non scientific people to appreciate that some of us actually use mathematics to solve real world problems and make our living doing so. Hardly a day has passed in my long career where some aspect of my mathematical education hasn't been invaluable. It's great that guys like yourself are trying communicate this type of stuff to give a wider audience a better appreciation of mathematical applications.
I think that dI/dS is missleading in the sense that the dependence on S really is of the form I(S(t)) that means you cant really compute I on arbitary values of S they need to be values that S achieves as a solution to the ODE we dont really divide the equations we manipulate them so that the equation that you wrote holds along a solution, along an integral curve.
Yes, you're right that there is still time dependence which is why we only consider the resultant equation at specific points in time - ie. at the end of the epidemic, or at the (unknown) time of maximum infectives. The analysis will still be valid.
@@TomRocksMaths yeap it just annoys me to cancel dt's and i was in retrospect somewhat rude, sorry :P the video was on point and very informative thank you and greetings from Greece