Oxford Mathematician explains SIR Disease Model for COVID-19 (Coronavirus)

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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[@TomRocksMaths]

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.

[@dptirkey]

This model is then simulated using system dynamics software like Vensim , or there are other softwares the Government uses ?

[@TR4zest]

Tom: thank you for this. I am no mathematician, but you helped me understand the model. Good work.

[@JohnDlugosz]

@@dptirkey See Numberphile's new video for solving and animating the curves. They used a free program called Georgebra.

[@UsmanAli-yo3nc]

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 KZbin. 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.

[@jarsoabdulkadir7801]

Thank you.

[@siddharthsethia5569]

If machine gun kelly pursued maths instead of rap. Thanks a lot for the video, helped me with a report.

[@TomRocksMaths]

I'm listening to his new album on repeat at the moment - loving it!

[@a_green_cat]

Haha i was thinking the same

[@alessandrobertarelli4499]

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 )

[@kalynhowes6151]

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.

[@TomRocksMaths]

Thanks Kalyn - glad it was helpful!

[@clark882]

Hi, I'm S. Korean Highschooler and honestly respect you. This was much significant to me.Thx

[@TomRocksMaths]

You're very welcome.

[@logicomix7976]

Thank you so much for sharing understanding and reason in this absolutly crazy time. Best wishes to you all from Munich :)

[@lynnmillerhandbells]

I’m so glad you are here. Stay safe, spread the math word. Thanks so much. Minnesota, USA

[@mathemaniac]

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!

[@plrc4593]

How did you use SIR model since real I and hence also R and S are unobservable?

[@mathemaniac]

@@plrc4593 I simply used the data of reported cases from Worldometer. See the spreadsheet linked in my latest video for more information.

[@plrc4593]

​@@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.

[@mathemaniac]

@@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.

[@plrc4593]

@@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.

[@soilsurvivor]

Brilliant, very accessible explanation. Thank you!

[@theultimatereductionist7592]

Differential algebraist here! Our entire lives/careers are devoted to searching for exact solutions to systems of differential equations like the SIR model!

[@lte23401]

Thanks so much for sharing your expert knowledge with general public!

[@mr.niemand6179]

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 :)

[@jackwilliams1468]

I was just building an SIR model to fit the parameters for COVID-19 in python! Fantastic video

[@plrc4593]

How did you use SIR model since real I and hence also R and S are unobservable?

[@montanariarthur]

@@plrc4593 S and R are actually observable variables (assuming you have a reliable measure of I)

[@plrc4593]

@@montanariarthur Hehe "assuming" :D

[@montanariarthur]

@@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!

[@AnisaKhatun]

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 :)

[@TomRocksMaths]

Awesome - I'm glad you enjoyed it!

[@leanneleaderman7453]

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!

[@TomRocksMaths]

You're very welcome Leanne!

[@TomRocksMaths]

Part 2 on extending the model to include the movement of populations is here: kzbin.info/www/bejne/q4Svd6iagKeheKs

[@abhiroopreddy8673]

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?

[@tobiasnn5792]

this video is pure gold, my final exam was an sir model based project, and your video helped me a lot

[@TomRocksMaths]

Glad it helped Tobias!

[@krishnapriyau4458]

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. 🙏👏

[@TomRocksMaths]

Thanks Krishna - I discuss a more advanced model with spatial dependence here: kzbin.info/www/bejne/q4Svd6iagKeheKs

[@hubertmallard7254]

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

[@Shamansdurx]

Good one, I wish many people took up statistics as a subject which would make people understand the spread of a pandemic.

[@lamyahomaydan6131]

Thank you !. Could not have done my assessment without this video.

[@veqv]

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.

[@TomRocksMaths]

This is brilliant - thanks for sharing Kelly!

[@Xeridanus]

I don't get all of them but I get enough to realise just how brilliant this is.

[@tomasstrnad2635]

I would love to see some of the more advanced models

[@pauvilimelisaceituno2893]

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

[@Lalit-yw2tb]

@@pauvilimelisaceituno2893 thanks for the insight and the extra information.

[@teresahanlon2205]

try Corona Virus & Mathematical Modelling KZbin. The Tutor Wizard Inc.

[@chandelgaming8975]

Please speak in hindi

[@akhyarr]

@@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..

[@OntologicalQuandry]

In this video I learnt about the SIR model for diseases and that vertical stripes really accentuate hip movements.

[@TomRocksMaths]

Don't wear stripes on camera...

[@OntologicalQuandry]

@@TomRocksMaths no, just make sure to make the hip movements accentuate the maths being described. There were several variables that had much more impact for a little tick of the hips.

[@klauscosmin]

Thank you very much! I finnaly understood the SIR model with all mathematical explanations you gave

[@TomRocksMaths]

That's great to hear :)

[@magtazeum4071]

I’m in love with Tom , he rocks

[@helenlin7839]

Thank you so much for making this video, it really helped my math modelling coursework, got 80+%🎉 ( I am in a UK uni so super happy with that ❤❤❤)

[@oxbmaths]

Excellent work! Very clearly explained. The wikipedia entry on the SIR model should include this video!

[@TomRocksMaths]

Thanks!! And glad you enjoyed it :)

[@noonesperfect]

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 :)

[@bassil6138]

such an worthy information..thanks👍🏼

[@jaffa3027]

Very good explanation of the SIR model.

[@dabeveryday9991]

Cheers for this, needed it for my uni essay!

[@afeefasmoideen2352]

Thankyou tom❤️. fantabulous presentation. Loved it and understood almost everything clearly. And this is my project topic.

[@TomRocksMaths]

Glad it was helpful!

[@suwelapaimvassoa7637]

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.

[@TomRocksMaths]

You're very welcome!

[@alp4119]

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..

[@nahidameghji1510]

@@alp4119 should I send you a sample IA I found on this same topic ?

[@snigdhaadhikary9870]

@@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?

[@lamaabdullah1937]

I'm so glad to find this video, thank you so much.

[@TomRocksMaths]

Glad it was helpful Lama!

[@krakingaming]

Wow ...so clearly explained.👌 I will be doing my masters project ( most probably ) in modelling pandemics. Please keep making such videos.

[@TomRocksMaths]

There's a whole series on Disease Modelling here: kzbin.info/aero/PLMCRxGutHqfmBoC2YyFradH8NqpvbovMt

[@beautylife310]

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!!

[@TomRocksMaths]

Awesome - thanks!!

[@brittanyleach1537]

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!

[@TomRocksMaths]

Thanks Brittany, I'm glad you found it useful!

[@PamweChete2503]

The differential equation im interested in is dC/dB = RTC^2. Where dC/dB is the rate of colour change of the black coat next the the chalk board.

[@ltonetto]

Outstanding explanations!! Thank you so much!!

[@TomRocksMaths]

You're very welcome Leonardo!

[@gouthamibhonsle9575]

Thank u for the valuable information 😊

[@munzeralseed]

Very informative yet simply explained! You really helped me understand more details about the SIR model. I've actually done dynamic modelling before, but all my DEs were linear. I'm wondering if we can get the equations of S(t), I(t) and R(t) (most probably they would be exponentials). Is that possible?

[@TomRocksMaths]

The issue is that the equations are coupled (they all depend on each other) and so the clever tricks I use in the video (such as combining two of the equations to get one that we can integrate) are how we try to solve them - rather than solving explicitly like you say. We actually have an explicit equation for I and S together in Q2.

[@asddsaasddsa1111]

@@TomRocksMaths But this is not exactly the max infected at a time right? This is the max for I(S) function not I(t). Could you please help me to understand?

[@o_jungio]

I am so glad to chance upon this video. Thank you.

[@TomRocksMaths]

Glad it was helpful Owen!

[@debalinaghosh6404]

Thank you for this wonderful video.

[@tommygeorge28]

This is fantastic. Thank you!

[@TomRocksMaths]

You're very welcome Tommy!

[@Rupamsetupati]

If you made video on intercept theorem thats grateful for me

[@nbknowsdeli]

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 😄

[@TomRocksMaths]

Awesome - thanks!

[@emiliahartmann5557]

Very Good video! It is very clear even for me who do not understand english so well. Thank you so much!

[@TomRocksMaths]

Glad it was helpful Emilia!

[@koroushbozorgmehr3951]

Hey Tom, Thank you very much indeed for sharing this very important topics with the world. I ran through some the calculations and I came up with a slightly different result than yours @ 13:46 Let me break down my calculations: We have I+S-(1/q)lnS=I0+S0-(1/q)lnS0, rearranging it will give us: I=I0+S0-(1/q)lnS0-S+(1/q)lnS. Now, in order to calculate the IMax, we plugin S=1/q into the above equation. So we get the following: IMax=I0+S0-(1/q)lnS0-(1/q)+(1/q)ln(1/q) ==> IMax=I0+S0-(1/q)[lnS0+1-ln(1/q)] ==> IMax=I0+S0-(1/q)[1+ln(S0/q)] The only difference is in "ln(S0/q) vs ln(S0q)" I am not sure if I am right or not. But, if we think about it S0 is considerably a large fixed number, so if as q increases the product of qS0 increases accordingly, and so does the natural log. I would highly and greatly appreciate it if you could kindly comment and clarify this. Thanking you in advance and I look forward to hearing from you. Yours Sincerely, Koroush

[@TomRocksMaths]

Hi Koroush, I've gone through your working and there's a slight error in the last step when you manipulate the logs. We have -ln(1/q) which is equal to +ln(q) by the properties of logs. then we have ln(S0)+ln(q) which we can put together using the properties of logs to get ln(qS0). Hope that helps!

[@koroushbozorgmehr3951]

@@TomRocksMaths Thank you very much indeed Tom. I really appreciate the time you took to reply. Stay safe and healthy :)

[@TomRocksMaths]

@@koroushbozorgmehr3951 And you!

[@Jaiviths]

Hi Tom, Thanks for uploading this video

[@TomRocksMaths]

You're very welcome.

[@pedrosanchez7111]

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 ?

[@TomRocksMaths]

Good point Pedro, and the answer is I guess we don't, but we can 'hope' that it is the case.

[@trendyanim3982]

simple and wonderful explanation

[@TomRocksMaths]

Thanks!!

[@juliettepreisig4672]

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 ?

[@onedabber6991]

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?

[@TomRocksMaths]

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.

[@andrewgonzales1359]

I work in COVID testing and I find this very helpful.

[@andrewgonzales1359]

This will give me an understanding of where to allocate resources when this happens again. I’ve taken college level calculus and biology - thank you helping integrating the knowledge and disseminating it. I’m a former UCLA student and University of Minnesota - Twin Cities graduate.

[@andrewgonzales1359]

The last time in England was over a decade ago, and when I have spoken to Brits who have visited the United States, they love the NHS.

[@andrewgonzales1359]

Why am I learning about this year later and I’m the only person on my team who tried to learn this? I have no idea.

[@andrewgonzales1359]

Despite my experience with a chi-square analysis, I was not offered a position with an employer. This is what I have been learning in my spare time.

[@ProfesorMartinV]

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.

[@TomRocksMaths]

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.

[@mirandagarcia7058]

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.

[@TomRocksMaths]

The main system of the 3 ODEs will be unchanged yes. You will just have different initial conditions.

[@txoutlawent.6515]

Is the sound of the chalkboard soothing to anyone else?

[@TomRocksMaths]

This.

[@StNick119]

Very much so. The gentle tap tap and scratch scratch, how pleasing.

[@nkosinamandlatakatshana7909]

Wish you could show how to estimate R naught using that formula on a curve fitting for a particular data and time

[@elynwu575]

Thank you this is so well explained

[@TomRocksMaths]

Glad it was helpful Elyn!

[@miguelpassosmorgado4719]

Hello Tom. First of all thank you for your clear explanation and congratulations for your work in teaching math concepts. I’m facing one difficulty with this model. I tried to implement it in excel using the variations on S, I and R and considering a So and a Io. So, I started with the initial values of S and I and then with the variations given by the model I obtained the new values of S, I and R for each period of time. And that was good. I obtained the graphics for S, I and R as I should (with I increasing, reaching a maximum and then decreasing). Then I tried to find the maximum I using the equation you reached on question 2. But the value I obtained using this equation is not the same I obtained using the equations of the model (dS/dt, dI/dt and dR/dt). This is normal or I’m doing some mistake? I thought it could be because of the value we choose for c (the constant of integration). Can you help me? Thank you.

[@TomRocksMaths]

You should be able to fix the constant of integration using the initial conditions. We know that at the start of the epidemic, when t=0, that S=1, R=0 and I=I0. These should allow you to fix any constants of integration by just substituting t=0 and the above initial values into your integrated equations.

[@plrc4593]

Do exist extensions of SIR model where parameters (r and a) are functions of time? Will you tell us about them?

[@arifamohammadi2698]

Thank you so much, it was awesome!

[@TomRocksMaths]

You're very welcome :)

[@adamvictorio566]

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!

[@TomRocksMaths]

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 :)

[@sereenamariyamsaji9866]

I got what i need for my project. Thank you soo much

[@TomRocksMaths]

You’re welcome 😊

[@pedrocordoeira5273]

Your video is just awesome!!

[@parisshopping-zg3ts]

Thank you so much

[@happykaunda3914]

does this video also explain how to derive the stability analysis??

[@istvanbody9519]

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.

[@curtiswfranks]

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.

[@Tipsi-mo7bl]

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.

[@TomRocksMaths]

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.

[@Tipsi-mo7bl]

@@TomRocksMaths I see, thanks for taking the time for answering my questions. Keep up your great work! :)

[@judysh2103]

@@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)???

[@TomRocksMaths]

@@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.

[@sschmachtel8963]

Do you have some PDE solver? I think SIR can be transferred into a reaction diffusion equation and that is a system to play with. Would like to but have to find some free solver. At least 1D for cylinder coordinates. Then you get a diffusion constant that you can play with, size of the area, influences of population densyties. etc. Nice bit about the max I !

[@TomRocksMaths]

There is some more information on how to this in the latest Numberphile video here: kzbin.info/www/bejne/oWfRfZl5l6atndE

[@TVWJ]

It is not reaction plus diffusion. It is an autocatalytic reaction sequence. Pde solver: use Python.

[@jamesq3896]

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

[@TomRocksMaths]

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.

[@zohaibmazhar4426]

I love your teaching way sir

[@TomRocksMaths]

Thanks Zohaib!

[@andrespatino7204]

Hello Tom great video! Can you help me with these two questions? How are defined the values of "r" and "a" coefficients? I understand the theoretical concept as contact rate and recovery rate respectively but no how their values are defined for obtaining q or R0, also for exploring the model in mathigon. The other question is about the contact ratio, is equal to q or to 1/q? Thanks!

[@TomRocksMaths]

Hi Andres, the values of the coefficients 'r' and 'a' need to be estimated from real-world data. These will of course be different in different environments, countries and scenarios so there's no easy answer I'm afraid. As I've defined it, the contact ratio is equal to q.

[@davidjani8180]

Thank you for the video l am an aspiring mathematician interested in disease modelling

[@osamaagamirashwan4783]

Unbelievable explanation that let's us think about Mathematical Modelling in our critical time and when we have no vaccine

[@TomRocksMaths]

Glad you enjoyed it Osama!

[@tianzexia2118]

Hello, can anyone explain how the right hand side equation at 12:38 formed? How is it related to the initial conditions?

[@lunasofia5754]

hey! loved your video, I am doing a project with the SIR model combined with Euler's method, any chance you know how to compute these set of equations to Excel?

[@TomRocksMaths]

not in excel, but you might find this interactive graph useful: mathigon.org/pandemic

[@MAx-gi1pn]

Thank you, this is a great video even for people like me that have no knowledge on this type of subjects. Although there is just one thing I didn´t understand and would apreciate so much if someone could answer me. When finding the equation for the max number of infected people at certain time, after integrating, what is the reasoning behind making it equal to the initial conditions?

[@TomRocksMaths]

When you integrate an expression you have an 'integration constant' which is normally represented as +c. This is here because whenever you differentiate a constant term you will always get zero. To solve for the value of this constant you can insert the value at any time (as it's constant it cannot change) so we use the initial values (t=0) as we know what these are.

[@aakankshaverma7947]

Best explanation..😊

[@yen_1190]

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

[@lukasgarly193]

Thank you for the amazing video! You cover a lot of really interesting things about the SIR model, but i didn't really understand why q is equal to r divided by a, i've tried figuring it out myself, but a can't find an expression for it... How did you do it? :)

[@TomRocksMaths]

Hi Lukas - glad you enjoyed the video. q is defined to be r/a, since this ratio appears in the analysis to question 1 about whether or not an epidemic will occur.

[@LionsYouth]

Awesome explanation! Thank you

[@TomRocksMaths]

Glad it was helpful Chris!

[@LionsYouth]

@@TomRocksMaths It really was! Rock on and happy holidays!

[@leoflores3080]

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?

[@TomRocksMaths]

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!

[@djnorm98]

I found the explanation on Numberphile also very helpful. He sent us over here for more info.

[@Xeridanus]

Here is that video: kzbin.info/www/bejne/oWfRfZl5l6atndE

[@Sperminator10000]

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.

[@lawrencejwinkler]

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.

[@nicolasgarcia7305]

Great presentation my friend!

[@TomRocksMaths]

Thanks Nicolas!

[@chadgidakis]

Hi Tom, I am attempting to use your model formulation for healthcare demand purposes... I am just wondering how you would go about finding a value for q based on (only) the government's social distancing tactics i.e. "If we reduce the contact by 50%, an infected person will pass it on to 1.25 people as opposed to 2.5? Or based on any other info out there for that matter?

[@TomRocksMaths]

Tricky question... the problem is the virus is so new that we just don't have enough data. Even something like 'seasonal flu' which has been around for hundreds of years isn't completely understood and requires some level of estimation of the parameter values. In practice, what we do is run the model for lots of different values of q to see which gives the best outcome, and then try to work out how to reach those particular values. The best suggestion I can give you is to try to find the same values for eg. seasonal flu and then increase them a little (as it seems COVID-19 spreads faster).

[@johnfalcon3335]

You're a Brit? How come I clearly get your words? I love it.

[@MrDivyanshu33]

Found this on wikipedia. For people interested in diving deeper: en.wikipedia.org/wiki/Compartmental_models_in_epidemiology

[@madvexing8903]

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?

[@TomRocksMaths]

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!

[@elshroomness]

Oh my god. I understand!!! i finally understand! thank you.

[@TomRocksMaths]

Glad it helped Melvin :)

[@gabrielsebastianperez3898]

Hello! Good video!! How would you estimate de numeric value for constants r and a?

[@TomRocksMaths]

Great question and unfortunately not one with an easy answer... the best we can do is try to estimate the values based on the data available to us. The value of a is a little easier as we can just look at the rate of recovery/death for the cases we currently have. The value of r, however, is much more difficult as it depends on so many different factors - social distancing measures, how often people wash their hands, cultural norms etc. A good starting point would be to take the value for something like seasonal flu and increase it a little. Not ideal, but it will at least tell us something and we can modify and improve the models from there to make them more accurate.

[@cyberdolt]

Tom Rocks Maths But couldn’t you reason: At the very outset before anybody has recovered or died, “a” isn’t a factor. And simply by observing that on Monday there was, say, 1 case and on Tuesday there were 7, take those initial values, and solve dI/dt=rI; and from that determine “r”? Then, take the given values for Ro, the “r”, the S, and use them to calculate the “a”?

[@plrc4593]

@@cyberdolt Where did you get dI/dt=rI from?

[@donharrold1375]

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.

[@TomRocksMaths]

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.

[@donharrold1375]

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.

[@SourodiptoOfficial]

Dr Tom, thanks a lot for making mathematics much more interesting. Initially I thought that you are an chemical or mechanical engineer as because your strong hold in Fluid Mechanics is quite remarkable, Sir. I am really relearning mathematical modelling after watching your videos despite doing major in chemical process modelling. I have a doubt Sir, is it possible to carryout disease modelling in MS-EXCEL. Can you apply or mimic chemical reaction modeling in this disease modeling ,a s because interaction between Susceptible and infection people are concerned.

[@TomRocksMaths]

you need to run the model on a computer since the equations are coupled. there is a nice interactive tool here: mathigon.org/pandemic

[@absolutelymath3399]

Great video!!!! Just a question about why we multiply S * I to model interactions between invectives and susceptible group?

[@TomRocksMaths]

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.

[@absolutelymath3399]

@@TomRocksMaths thanks!

[@daramuthmainnah4964]

Great video! I'm working on my assignment using SIR model. I have a question, what is the formula for a and r in the covid-19 case?

[@TomRocksMaths]

Hey Dara, there isn't an explicit formula as the values come from data and knowledge of the disease. The best we can do is to estimate them based on the current data available.

[@akshaygs4048]

@@TomRocksMaths Can you send a link for procedure/approaches to estimate the a and r values?

[@oxforduniversitychinesesoc6581]

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.

[@TomRocksMaths]

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.

[@oxforduniversitychinesesoc6581]

@@TomRocksMaths Thanks that's very clear now. Should've used my personal account tho lol

[@artursztuc9140]

I'm tempted to write a small Bayesian framework to model COVID-19. Do you know any resources describing more advanced pandemic models?

[@escuccim]

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.

[@TomRocksMaths]

This is the one being talked about in the UK: www.imperial.ac.uk/news/196234/covid-19-imperial-researchers-model-likely-impact/

[@michaelsommers2356]

Try this review: www.maths.usyd.edu.au/u/marym/populations/hethcote.pdf

[@atrumluminarium]

Is there a way to add a Brownian term to this? How would one go about it?

[@TomRocksMaths]

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!

[@achrafouaddou1348]

Tom Rocks Maths how we can do it ?

[@ello7034]

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)

[@atrumluminarium]

@@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

[@sebastianblackfyre]

well explained. Job well done

[@husseinkaraman60]

First I would like to say that I enjoyed you vedio very much however I have a question to you and I hope I have the answer. What would be the situation if I want to start my calculations from a point that 10% or 20% from the initial susceptible people already have the disease but they don't know and furthermore a part from them is recovered. Does this make effect on the answer of the proposed three questions?

[@TomRocksMaths]

Hi Hussein, the number of people at the start that have the disease is given by I0. If this is 10% or 20% of the population then this will just be 0.1*N or 0.2*N where N is the total population. In this model we don't have any recovered people at the start of the disease outbreak, and even if you did, since we are not modelling re-infection the number of recovered people would just be subtracted from the total population, ie. the starting population N = I0 + S0 would just be a bit smaller. The answers to the questions would still be the same, just with slightly different values for S0 and I0.

[@husseinkaraman60]

Is it difficult to incorporate a new term or a new assumption that can deal with this situation. Please forgive me for my questions as I am not a mathematician and thank you in advance

[@TomRocksMaths]

@@husseinkaraman60 both ideas that you've mentioned would be straight forward modifications to the model and could easily be incorporated. They wouldn't really change the mathematical results though, as only the values of the parameters would change, and I purposefully didn't include any specific values in the video.