The MATH of Pandemics | Intro to the SIR Model

How do organizations like the WHO and CDC do mathematical modelling to predict the growth of an epidemic? In this video we introduce the Susceptible- Infected-Recovered or SIR model. This is a simple system of differential equations that qualitatively behalves in a reasonable way. Studying this near the beginning gives a value called R_0 which governs the early growth rate which is exponential growth and is a standard metric in epidemilogy. Indeed real world data shows the current epidemic is well modeled by this kind of exponential growth.
PART II here: • The MATH of Epidemics ...
0:00 Assumptions of the SIR Model
3:10 Derivation of the SIR Model
6:44 Graphing the SIR Model
8:58 Finding R0
13:43 Real World Data
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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
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[@DrTrefor]

**TYPO** At 5:55 when R' is introduced it should be bI, the same b as showed up earlier, not a new symbol r. Also, thanks for making this vid go viral, stay safe everyone!

[@siddharthsethia5569]

Thank you for the clarification, Enjoyed the video

[@surbhi01]

excellent video...one query....how are the constants a and b calculated?

[@mnada72]

I thought that b is different from r because some of the infected are unfortunately dead.

[@realbangbang]

Me in 2014: screw this! I'll never use this information in calculus class! Me in 2020: *taking notes watching this video*

[@copernicofelinis]

Nothing like dying by drowning in your own fluid can motivate a student more.

[@julianocamargo6674]

Me in 2014: blah blah blah Me in 2020: OMG, how do I estimate R0 ?!? I NEED IT FAST!

[@ralphparker]

@@julianocamargo6674 I've been estimating R0 by taking the daily growth rate of the infected (over about a week) and calculating what it needs to be to match that growth rate. But, I'm sure your comment was a jest.

[@anadiru4382]

not in calculus but you will need this in differential equations

[@camelproductions4029]

I wanted to thank you real quick. I'm working on my Maths IA and this video really helped me understand the topic so thank you very much!

[@edayangin7955]

OMG ME TOO

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@jennyzhang2810]

heyy same!!! Were u doing hl or sl?

[@camelproductions4029]

@@jennyzhang2810 Hey! Wow, it's almost been a year now, huh. Thankfully I'm doing SL, I just don't think I would have survived HL lmao.

[@jennyzhang2810]

lmaoooo im dying in HL, just wanted to ask how did your IA turn out cuz I heard lots of people saying this topic is overused and wont score high:((

[@gracie4394]

Hi, I literally cannot begin to describe to you how helpful this video was. I've spent a solid 8 hours today trying to wrap my head around SIR models by reading academic journals but it was as if it was in a foreign language. I'll literally come back to this video to basically write my entire Maths IA for me - I'm trying to model the Ebola outbreak in Guinea. So yeah I just wanted to say thank u smmmm for this you're a lifesaver !!!

[@gracie4394]

@Raggul Kanakasabapathy infection rate is mortality rate of disease / susceptible population. i think i really simplified it but my teacher said it was right sooo

[@rohinikayal4440]

Hi I am doing the same, were you HL or SL Math? Would you say this topic was over-complicated or doable? Also, if you don't mind could share some other resources besides this video that you might have used?

[@cherylesper7033]

hello IB students! same doing my math IA~ it is super hard to read thorugh all those papers, but this video rly pointed out a clear way to success!

[@dharamsarhembram9413]

when I was teenage I used to love my maths a lot. After spending 4 yrs in Engg. without learning anything I thought that I couldn't do anything. this videos helps me to love maths again.

[@user-tv7jx9pw1q]

I studied all possible math modules in my supply chain master and business bachelor, learned a lot of interesting and crazy stuff... But at the end, I don't feel capable to use it properly... But sometimes, like you, I fall back to that love again :)

[@paofmacedon]

Thank you very much for the analytical and explicit presentation!

[@kwongkh6880]

Clear n precise explaination. I m a teacher, teaching Higher School Cert. Exam( Equivalents to A-level exam in UK). I m going to use this pandemic problem but with more simplify modification as a set induction to the topic of different equation to attract students attention.

[@ryanjensen4232]

You rock bro, used your vids almost exclusively for discrete.

[@nadija1389]

It all makes sense now! Thank you, I was struggling to understand this in my lecture!

[@tonymac96]

For Chemical Engineers, the SIR model is like when you have A==>B==>C or reactions in series in a batch reactor. Excellent Presentation!! Thanks.

[@osculocentric]

I work in Quantum Optics, I recently was using similar prey predator lotka volterra equations to predict if the photons will completely transfer to another system :-D.

[@sayantanmondal6461]

solution of SIR model by stochastic cellular automata kzbin.info/www/bejne/mXSTe4eqgKllZ8U

[@brett6376]

You are so correct. I'm also chemical engineer and I was also thinking of a batch reactor while he was explaining it. He's demonstrating a perfect system. I think I'm gonna check out my old notes and try to use the data that we got from the corona virus. Seeing that I'm in lockdown, I think I gonna try it. Perhaps make a simulation also on chemcad or something

[@rupe-huzz]

Timely and awesome! I just love this stuff! Trefor just has the knack for clearing the mist on pretty complex ideas. Big Thanks!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@florinaliu5489]

I simply think you explain fantastically this issue. I am new scholar in this issue, since i am implementing SIR into social sciences. I was able to understand everything from your video.

[@tonyf1163]

Awesome stuff. I used your videos for Calc II and they helped a bunch. I’m rocking Diff Eq now but I can wait to go back to your videos when I take Calc III over the summer

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@marwanbasem193]

Really thanks for that amazing video, I've never dealt with modelling before but you made it so intuitive. You also encouraged me to try and add some other factors into the model, like the effect of vaccination. Thank you so much.

[@fablun99]

This sort of models (ordinary differential equation-based, deterministic approach) are well suited to explain and understand basic concepts and dynamics for ideal cases (uniform homogeneous populations with homogeneous interaction dynamics). This type of models, for instance, does not consider that recovered individuals might infect others or could get infected again. In certain type of epidemics, dead individuals could also infect living ones. A better approach is to consider a stochastic epidemic model considering non-uniform, heterogeneous populations (with subpopulations in mind; for instance, one approach is stochastic network-based epidemics, there are other approaches). The model should also consider measures to fight epidemics to flatten the spread of the infection.

[@peterjackson7348]

Really interesting! I am an IB teacher and I am trying to find ideas for my students Maths exploration

[@ElPasoJoe1]

Really good video. Gonna go back and watch the video, take notes, and fill in the details...

[@Anooshehk]

I absolutely love your channel. Thank you so much.

[@user-yf1ep9zo8r]

Please don't delete this video. It fits really well with the information I've been looking for. I hope the video that enlightened me is on this channel for a long time. Please leave it😢😢.I'll subscribe!

[@atebbe1]

This was awesome, as an engineer I've been looking for covid data to play with and these are exactly the type of equations I've been looking for. Thank you!

[@BlindManBert]

Austin… You might also want to investigate the SEIR model, which introduces a fourth ‘Exposed’ population representing those folks who have been exposed but are not infectious yet. So the four populations might be termed Susceptible, Exposed, Infectious (not infected!) and Removed (recovered or dead individuals). The aSI/N term is thus subtracted from dS/dt and added to dE/dt, to represent the rate of infection. Then you add an additional rate, let's call it c (or gamma) that determines the transition from E to I, and you subtract cI from dE/dt and add it to dI/dt. Then the final rate of transition from I to R is the same, subtracting bI from dI/dt and adding it to dE/dt. So in summary the new ODEs are: dS/dt = -a S I / N [the division by N is required to get the dimensional analysis right] dE/dt = a S I / N - c E dI/dt = c E - b I dR/dt = b I The addition of the E group lets you model the delay from being exposed to becoming infectious. I’ve read that the incubation period of COVID-19 is averages 5.25 days, ranging from 2 to 14 days maximum. But that is the time from exposure to first symptoms appearing. The time to becoming infectious may be much shorter, perhaps only hours or a day or two (guessing). In any case, the SEIR model is more appropriate when there is a delay from exposure to infectiousness. And then you can do all sorts of fancy things like representing age groups in different pools with different transmission rates between the groups, illness times, time before isolation (removal), age dependent mortality, and so on. I wrote up a survey of mathematical modeling techniques including the SIR, SEIR, and actor (stochastic) models on the ScienceVsUniverse group on Facebook, which I can’t directly link to here. But if you look for “A partial survey of epidemiological models” you’ll find that article with links to a few good pro-grade SEIR and actor models. I also maintain via the same group a public-domain OpenOffice spreadsheet where time-series CDC data can be extracted, or you can use time-series data from the John Hopkins University CSSE GitHub site, or CSV files you can download from OurWorldInData(dot)org as the time-series data. I find it helpful as well to pass the time at home running models to double check the ‘sanity’ of models thrown about in and outside of scientific circles. Good luck!!

[@marvellousolatunji9719]

Interesting video, Kudos. I look forward to other videos on the model.

[@damercy]

Such nice explanation. Loved it! It's amazing to learn about new mathematical models, especially when it's taught beautifully like you did. Keep going!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@krisellcainicela129]

Gracias por la explicación, fue estupenda !, amé esos subtitulos!

[@falehalyazidi956]

I’ve been reading to understand this model and clear all misunderstandings. However, your way of presenting this model, the content and examples were very beneficial and amazing. Thank you very much..

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@elderaarondavis1]

Wow. As an Econ major, we use partial derivatives and DE in our model too.

[@arunhuddar]

Excellent explanation. Did not expect Maths to be helping saving lives.

[@giacomogavelli4095]

It is a pity a lot of people are afraid of mathematics almost as they are afraid of this Virus. Having the interest and the capability (which for me is relatively low) to understand this approach help me a lot in being aware of what is happening around me. I'm from Italy (and I'm sorry if my English is not so good) and I'm seeing a lot of people that don't understand why some precautions are taken and they think we are all going to die. P.S. thank you for the video. The content is very well explained

[@thevapingpig]

I'm a Canadian, the whole western world is in panic. We watched the growth, while our politicians for the last couple of months this thing was growing have been fucking around. Now we are in a situation where we are watching it explode in our country, but we've all went under mass quarantine to try to decrease our transmission ratio. I've been obsessed with learning the modelling and try to make my own predictions for how it'll effect my city, and country. Hope everything eventually improves in your country, people over here are super scared of calculus just like this virus.

[@floridaman318]

You're English is actually fine. What's most frustrating about this virus is the ignorance, if not downright arrogance, displayed by so many people. If they understood how damn cost effective it is to actually aggressively contain an outbreak in its early stages, they wouldn't be worried about "looking stupid" or "falling for the media sensationalism." You guys in Italy will be getting better in coming weeks, you huys reacted almost a month quicker than us. The US numbers aren't looking good.

[@BlindManBert]

@@floridaman318: I’m just chuckling at “You’re English is actually fine.” [English language tutor; not a grammar Nazi.]

[@floridaman318]

@@BlindManBert for a non native speaker, yeah. I've seen much worse writing from native speakers.

[@BlindManBert]

@@floridaman318: It might be that you missed what I was chuckling at; the choice of “You’re” as opposed to “Your.”

[@denisbaranoff]

Awesome deterministic version! Awaiting for stochastic 😊

[@maxdemuynck9850]

great video! really interesting and informative

[@terrywang7984]

Excellent, informative, and clearly explained. Subscribed

[@mishasulikashvili1215]

Thanks for great video, Trefor ❤️

[@Jabert0001]

awesome explanation, thank you for this video.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@hereusername]

awesome video

[@biaschatterjee9836]

I was trying to understand this model, your video made it crystal clear. Very helpful, excellent video. Definitely gonna watch your all videos 😊👍👍👌👌

[@andrenuber8852]

Great video that lets me understand how to get to the basic equations. Furthermore, you clearly demonstrate the connection to the exponential growth. Don't know if someone mentioned it before: Why do you use another constant (b) for the lost infected people than for the gained recovered people (r)?

[@asherstackhouze1107]

More clear than my teacher does. Thank you sir.

[@xmaster622]

Hey joe I gotta love your life I

[@abraga123]

Excelent! Thank you very much.

[@neelasingh9578]

You don't know , but there is an exam called UGEE here , and one of their questions looks directly lifted from your video , now this video gonna help class 12 students also 👍

[@user-pt-au-hg]

I won't study calculus even if my life depended on it, I use to say,... oh, oh, I guess it does, back to studying calculus, my teacher was right, I might want to know this someday, thanks, teacher. :)

[@maurocruz1824]

The explanation of the R_0 is brilliant. Thanks professor.

[@osamaagamirashwan4783]

Thanks for letting me know different application for the system of first order DE as I use it in the circuit with loops including capacitors, coils, Resistances, and DC source.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@ekwan16]

Thanks for a very clear introduction!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@ismaelabdulrahman7295]

I really like the way you showed the problem. I enjoyed the video. Thanks.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@MrFranciss]

Thanx for you great explanations. I could understand everythingeven with my arts formatted brain.

[@MrFranciss]

Lol, in the next life maybe.

[@duygukorkmaz9433]

You clear up the main of model. Thanks for your excellent explanation 💐.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@FloriUchiha789]

You're the best, thank you for the big help!

[@dr.natrajdravid7786]

Beautiful.

[@thiagocastrodias2]

I loved the video, but I think I should watch it again some times to better understand it. But it's very fascinating for sure

[@sayantanmondal6461]

solution of SIR model by stochastic cellular automata kzbin.info/www/bejne/mXSTe4eqgKllZ8U

[@Leila0S]

I like you Dr. Your dedicated explanation is attractive. The more I meet PHDs, the more I am convinced I must go for it.

[@23andrelopes]

Such a great video

[@gen1l]

People often say math is useless and boring. They wouldn't, once they know how to use it correctly for their life better. Thank you.

[@alexz2411]

Great Video

[@brandonkim4920]

This was very helpful. Thank you so much for making this wonderful video.

[@DrTrefor]

Glad it was helpful!

[@iarasouza8029]

Amazing!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@melkdoria]

Very clear and pleasant exposition. More could have been done, though 15 min is not enough and shouldn”t take longer. The singularities and the death could have been explored. Great job👏

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@jiteshbohra6164]

Great video

[@ahmedmagdyghouneimy7574]

Wooooow ♥ You make things intuitive. Thank you so much for this fascinating video!

[@johnnysparkleface3096]

I would give anything if I could understand just a tiny fraction of what he said.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@MrArunavadatta]

Well explained..Thanks

[@aibee5794]

Thank you Guruji

[@pikapuff123]

Hey Trefor, I'm really loving your content! Could you really briefly describe you process for making these videos: programs you use for generating the png plots and overlaying all of the elements of your videos. Thanks!

[@pipertripp]

This is great. I'm just starting to teach myself ODEs and this vid was a nice presentation on a practical (sadly) application. Talk about being relevant...

[@pipertripp]

@@DrTrefor for sure. I bought Boyce and Prima back in early march and I'm currently in chapter 2. I love modeling stuff with python so I already have a little bit of experience with numerical techniques (Euler-Cromer). I'll definitely check out your content. The text book is really excellent so far and I love the fact that both odd AND even number problems have solutions in the back. That's very helpful when working in isolation. That all said, content like your and others is essential as you guys kind serve like the tutor who puts things into conversational english and you provide a lot of context... that is something that I think is under appreciated in science and maths education. I find that concepts and information sticks with me much better when I already have some context in the subject in question. Thanks again. Without people like you, my road would be both rougher and steeper.

[@SpiderSN]

In my country, Greece, they imposed us on Lock down quite early. and we managed to get out without much damage. But they opened up tourism and parties everywhere, and he came back again. This model assumes that there are no imported cases, and that human behavior is stable. It would be extremely interesting to study the behavior of di / dt in imported case conditions, as is the case in reality. Also the use of mask, the social distancing etc. So this model can be specialized with more parameters, further to a and b.

[@alexdabruh]

Thanks Trefor, I missed your MAT223 class back in UofT

[@alexdabruh]

Trefor Bazett Lol, if it is in the 2016S semester I might be in it

[@kamalkhalil118]

Thank you for the well explanation. But what do you think if we consider the space diffusion in the phenomenon (in that case we cannot get ODEs anymore but a system of coupled reaction-diffusion equations). My question is about the expression of R_0 ?

[@Joscquin]

Hi, and thank's for this video. I just uploaded a french translation, which I hope should be available soon.

[@Joscquin]

Your video corresponds exactly to the mathematical level of my biology students, but their english level could prevent them from understanding everything. I'm sure they will be very interested in your video, as we in France have a very serious situation concerning Covid19.

[@MardkoMBR]

Joseph Nicolas, merci beaucoup pour la traduction en français !! P.S. Spain, where I live, as a much worse problem with Covid19 than France unhappily. Once again, thanks a lot for your translation. Keep safe !!

[@sanjaykrish8719]

I never expected a diff eqn being formulated, though i hv solved many. Math is nice

[@francaciari944]

Beautiful video. Thanks !

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@taibimath7092]

goooood presentation

[@johnmartin7346]

I think there is a stochastic variation of SIR model. It would be fine to see something about that. Thanks.

[@Laowu1898]

great work！

[@TheLvpatel]

great video

[@hughmcmanus2134]

Thanks!

[@spicemasterii6775]

Amazing video. Amazing explanation. Subscribed!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@AntonyDova]

Thank you so much sir!

[@SareBear2000]

Nice, I just read an article on the SIR model

[@presse55]

Awesome presentation! Ron from Montreal B.Math 78!

[@8829david]

There is one thing at 4:50, is not more probable to get infected if S=10 and I=100 than if S=100 and I=10 Because 100 infecteds will infect the last 10 non infected with a more probability than 10 infected to infect the helthy 100 but the rate is the same. Same happens in chemistry with rate of reaction in kinetics that each concentration is powered to some exponent

[@chmeland]

Great video. I stumbled over it in my search for how to calculate R from the growth factor. Could you explain that?

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@mohammedmalik7992]

Very informative. Thank you

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@aapsarna4198]

Thank you sir

[@cilou-]

Hello sir. I'd like to use this video to as a base to present a school project, it's a really interesting topic and model and obviously very well-explained ! I was wondering where the graph from the beginning of the video came, if you had the data written down somewhere that I could use - if it's not too much to ask ! If I could get the data, it would be really helpful ! Thank you very much !!

[@sjj_02]

What a great explanation! So can I use this video as a reference for my high school assignments?

[@andrespatino7204]

Hello Trefor! Great video Can you help me with this question? How are defined the values of "a" and "b" coefficients? I understand the theoretical concept as contact rate and recovery rate respectively but not how their values are defined or obtained?

[@imanelshair7746]

very informative and clear explanation , thanks a lot

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@YOURFRIENDMATHEMATICS]

Nice communication skill. I fully support you. Lets grow together

[@josephacolletti]

Bro, very nice on many levels. Format/style, info, clarity, simplicity ... you’re a genius

[@izikarasu5896]

Useful explanation. Just a technical question: the first 2 differential equations alone can be combined into a second order nonlinear equation for S alone. In this case R does not seem to play a role but it should. Would dI/dt perhaps also depend on R in some way?

[@amitabharolkar3560]

Very well explained 🙏

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@finn8205]

Epic thankyou

[@duddud2430]

Awesome video, extremely helpful! Will there be any more videos on the topic explaining the SIR model further or maybe introducing the SEIR model or Logistics curves?

[@duddud2430]

@@DrTrefor Awesome! I am currently writing a high school level paper on the different models, so your video was really helpful to rephrase in much more accessible terms what the academic papers I am reading are explaining. Do you have any rough idea on when more videos might come out? I totally get that the Laplace Transform series has to be finished first, you have got some very informative videos on there too

[@radio2712]

It is really a nice video😀😀

[@0xsuperman]

I'd like to ask a few follow-up questions. 1) For forecasting future cases using the SIR model, do you try to fit the SIR curves with existing, actual data? If so, what systematic approach or evaluation standard is accepted to identify the parameters which give the best fit? 2) How do you incorporate additional information/variable into the SIR models? I.e., if lock-down happens between day 20 - day 60, and the SIR curves are affected by it, is there a way to incorporate this lock-down variable so that we can estimate its potential impact into the future? Please explain or if you know any textbooks/papers, I will be appreciated to dig deeper.

[@countrycountry88]

Hi, I am interested in a different view of this video. It is such high-quality video. That is how the video is set up. Could you please tell me what camera, light, microphone do you use and how do you set up it? Thank you.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@geraldiothanius24]

Thankyou so much for the video,, i can use the lesson for my assignment...

[@sayantanmondal6461]

solution of SIR model by stochastic cellular automata kzbin.info/www/bejne/mXSTe4eqgKllZ8U

[@mahmoudafouna1592]

Great modeling, thanks for the video. I wanted to ask is there a methodical way to determine a,b, and r? may be tie it to actual data up until now? some sort of distribution fitting for example ? I could really use this tip. Thanks !

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐👍👍

[@roddiepiper5559]

The term "-bI" seems rather simplistic, as it assumes that a constant percentage of sick people will get well or die on any given day. Most acute illnesses seem to run a fixed-length course, so in a more realistic model, some percentage would die two weeks after infection and the rest would recover three weeks after infection. E.g. dI/dt = a*S*I - d*I[t-14] - (1-d)*I[t-21] where d is the death rate and I with a negative subscript is zero.

[@sardonnayuen2860]

Hi, I love this video! Thank you! I was wondering in ds/dt does the 'd' stand for delta for change?

[@sardonnayuen2860]

@@DrTrefor thank you!

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@ralphparker]

12:10, regarding coefficient b. If you consider recover more than getting well, but removing an infected person from the ability to infect others, then quarantining effectively increases b. Maybe a better term would be considered in this thought. In my thoughts, and attempt to model this epidemic, recovery is not proportional but a delay. That is, those who were infected at time T recovered at time, T+R_t, where R_t is recovery time or time that the infected person is removed from the infected pool. I would really like to see what the CDC and upper echelon Health experts are doing in their models.

[@anfarahat]

Thanks for the insightful video. Just the graphs give the impression that susceptibles go down to zero, which is not the case as S(infinity) could be a positive number.

[@beoptimistic5853]

kzbin.info/www/bejne/joKsk6FobMmCoKc 💐💐

[@bubelevakalisa7313]

Great stuff man.

[@beoptimistic5853]

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