If you haven't seen part 1 on the basic SIR model, I highly recommend giving it a watch so that you can get the most out of this new model. The first video is here: ruclips.net/video/NKMHhm2Zbkw/видео.html
Thanks for everything, Tom. I know you are Navier-Stokes guy, and I hope one day we can talk about it without these stressing events but I really appreciate what you are doing. Recently some Icelandic scientists claimed that near of 50% of infected (I) are in truth asymptomatic (Ia), so they keep the infection rates higher as vectors while symptomatic infected (Is) are the ones that really suffer the disease consequences and consume medical resources. Humans live with many microbial entities for ages, now inoffensive, but prehistoric and antique outbreaks adapted the Humans for such conditions. Michael Crichton's "Andromeda Strain" is a good reading, the first movie is also interesting. 》Is there any model that consider such infected distinction? I believe we can understand that the dead cannot spread diseases since they can't walk... except in Apocalyptic Zombies movies and TV series. However... 》Is there any model that consider such distinction between Dead (D) and actual Recovered (R)? 》Could a Severity of a Disease (DS) be calculated as a function of symptomatic infected (Is) and Dead (D, instead use the "actual" Recovered R) over the Susceptible population (S) as this: DS = f(Is, D)/S ? I don't know which kind of function could be since the set of Dead (D) was once part of the set of symptomatic infected (Is): {D} ς {Is} . The use of R₀ as reproduction number has been criticized by many, and the range of such number is too large to get some relative precision. R₀ is a "natural" (biological) number for reproduction of an organism _in natura_ , after the human reaction: monitoring, containment & treatment, no matter how effective, this number changes to a R₁ as a function of the efficacy of the reaction. A correct R₀ number is almost impossible to achieve. 》Is there any model that could consider an "actual" R₁ number or funtcion instead of an empirical R₀ ?
@@alexdemoura9972 all of these modifications can be (and most likely are) made to the basic SIR model, it just means things are much more complicated and so have to be solved by a computer. This is what actual researchers do - run the models with different parameter values to determine which best describe the situation we are seeing, and then plan accordingly. Of al of the things you mention, I think modelling R0 as a function rather than a fixed value is the 'best' idea in terms of creating something that better matches what we see in real life. R0 represents the number of people one individual will infect on average, which of course will depend on many factors such as social distancing measures, social norms, whether or not that person is asymptomatic etc.
Supposedly the R0 of COVID-19 is around 2.65. But, how it would be possible to add the different parameters into place such as the weather and social isolation to be used within these equations.
@@DeschaineFamilia those factors are incorporated into the transmission rate r, which in turn affects the contact ratio q and thus the reproductive number R0. There are no specific variables for them, but they will still have an effect on the system through the parameters mentioned.
@@TomRocksMaths one other thing i'm curious about is how reliable is the SIR model in light of Covid-19 reinfection being a possibility? If i'm not mistaken in the SIR model the Recovered is taken out of the equation as they are no longer capable to spread the virus due to either death or immunity development. But some papers have suggest that immunity may not develop post-infection, and that some recovered patients are might still be susceptible to reinfection. Perhaps adding the possibility for a percentage of the Recovered to be cycled back as a part of the Susceptibles might paint a better picture?
Thanks so much for taking the time to do this. It's the first video I've seen which goes beyond a simplistic SIR compartment model, and you made the maths quite approachable. Cheers mate, I appreciate it a lot.
Oh, you're killin' me already (only 1 minute into video). I have ocean engineering master's, and recall "celerity" (speed of propagation). How cool that "Deterministic Wave Theory" has a footing in disease propagation!! Wondering how fibonacci series or more specifically the "Golden Ratio" could play in this....each wave is 31% the magnitude (volume) of previous wave???
One question that I didn't get about the non-dimensionalizing: I can see why the constants r and a disappear, since you wrote them in terms of R_0. But I don't get what happens to the constant D, the coefficient in the diffusion part of the set of equations?
This is some brilliant explanation help me to understand how the model works instead mere exponential growth rate parameter (which is quiet baby in front of this one lol). Again the complexity of these model itself and real life situation which is more complex than model indeed great lesson to learn. Awesome video , you clear the most idea behind it. Question what is P in S=1-P ? (as Susceptible is now space and time dependent with variable of space "x" and "y" and variable for time is "t" so is it P is particular instance which is nothing but x,y,t dependent in non-dimensional frame?) Thanks
P is still represents the number of susceptibles, but you can think of it explicitly as the number of susceptibles that have caught the disease since the start of the outbreak - ie. it represents the decrease in the susceptibles, which is why we are subtracting it from the full value of 1.
Would you be able to suggest one or two textbooks that cover this? (Context: I did maths at college 30 years ago, which included a course on differential equations in fourth year, but it's all a distant memory now. I do remember that I didn't understand phase portraits at the time!)
Travelling waves - or overshoots in a 2nd order system. A simple dynamic model is w/(s^2 + 2.z.w.s + w^2) which is the Laplace transform of the 2nd order DE, where w is the natural frequency (about 5 months) and z is the damping factor z = 1/(root(2).R), R the reproduction number. When R = 1, the system is critically damped and the pandemic is under control, when R > 1, the system is underdamped resulting in overshoots (waves), and when R < 1, the system overdamped. This fits in roughly with what we have seen. The vaccines will effect R greatly. Your 2D (spacial) expansion is interesting, but like all models is only approximate and its conclusion is having terrible effects on the population and economy. Those people with the fake fraudulent "MA" degree form Oxford in Classics don't appreciate this fact. Its interesting that Sweden has a death rate of 0.13% of population (and no masks) whilst the UK has a death rate of 0.19% (and masks). Masks and social distancing have only a weak effect on R. Don't like your rainbow badge Tom, Yuk.
Hello Tom, thank you for your interesting videos! I have a question that is there any way to add some variations like people visit public places(e.g. grocery stores, pharmacies, ...) on certain frequency? For example, what if quarantined people visit grocery stores once a week and contact 15 people(which is an extremely higher number than they normally contact) with an expected transmission rate of 0.2? I just wanna see how this assumption could be shown on equations.
Hi Heesun, thanks for your question. Including such intricate details would require a much more advanced model than the one discussed here. While such things are certainly possible, there's always a balance with all mathematical models between how realistic they are, and how 'quickly' they can be solved. To run a model that included variations on an individual level like you suggest would take several months on a supercomputer, which is something that we unfortunately just don't have the time for right now.
If you feel like tackling this, there is an issue with the SIR model, in that if you put a part of the population in a quarantine, the number of infected never goes down to zero, or it goes down too slowly. How could that be fixed? There need to be an upper limit of a certain number of days for the -aI differential because the viral charge is not eternal in a person.
Hello Tom.Thanks for these two great videos! Please, is there a way to calculate the impact of social distancing in some scenarios. For example the measures taken in my country: closure of schools and uni, absolute lockdown from 6pm to 5am, transit restriction to men and women in specific days.
Hi Alex, it's very difficult to quantify specific scenarios in the model, but all of the things you mention will absolutely help to reduce the contact ratio q, which I've hopefully convinced you from these 2 videos is exactly what we want to do!
Good video - it was easily understandable. Do you have any articles or papers explaining your analysis, e.g. how the minimum wave speed was determined?
I got a bit lost, new to all of this! From a statistical standpoint, considering we've increased the complexity of the model, we'd expect the travelling wave model to show different (and better) predictions than a regular SIR model?
20:40 is the slope on the graph right compared to the distance of "1" on the x axis, to the y axis? If so, then the R0 calculated for Covid being between 2 and 4 gives us an S(end) nearly 0.4 to 0.1 or less Please correct me if I'm wrong, I'd like to know how bad this estimations are
The graph is just a sketch and is definitely not to scale so I wouldn't try to use it to get actual numbers. The idea was just to show the shape as q decreases towards 1.
We learn different things from each of them. The travelling wave model tells you about the speed of the disease spread, which we don't get from the basic SIR model.
Good question - because we have 'non-dimensionalised' c doesn't actually have any units in the form given. If we re-introduce units to the problem, I believe the units of speed will be D * S0^(1/2) * r^(1/2), where D is the diffusion coefficient, S0 the initial susceptible population and r the rate of transmission of the disease.
Thanks Tom but I think you need to define x, the space variable. Is it a simple distance? If so how is it measured? - miles, a fraction of the maximum distance, or what?
@@TomRocksMaths Really impressed you got back to me Tom. I thought it would be something amorphous like that but it if it can't be measured how can it represent reality. Sorry for this picky question but I really want to understand travelling waves to see how well it fits my ABM SIR spatial model. I tried to dip into the the academic papers but find it difficult to know where to start as they all seem to require prior knowledge of the theoretical basis. I then found your tutorial and it made good sense to me except that I couldn't see how to relate it to my measures. Any leads would be greatly appreciated.
Hey Tom, Can you give an example from the past when travelling wave was used and it's very hard to find accurate data on COVID-19 graphs and other disease graphs in general, It would be great if u could help me out in these areas, I am working on a report, so all the data would really help...
You're right that the most difficult part of using any disease model is gathering accurate data for the parameters. The best we can do is work with what's available and for some of the more complex models, it's even possible to account for the uncertainty in the parameter values by modelling them stochastically (meaning as random variables with some underlying probability distribution). Ultimately this introduces an element of randomness into the models which accounts for the potential inaccuracy of the data.
I actually really hate dimensionless analysis, even if it is easier. Having units around often shows you quickly when and where you've made a mistake. Hauling constants around is the price you pay for clarity of thought, in my book.
just thinking out loud......can you model the impact of social distancing on mental health, educational achievement and what has actually gone on in the spread of the disease now that there is quite a bit of data out there.
I agree, but they would be based on pure speculating at the moment as we just don't have enough information about the disease. I purposefully avoided using actual values as I want to concentrate on setting out the model, rather than making any far-fetched predictions.
@@TomRocksMaths this might be the coolest thing ever, thanks to this and s few other channels I got a grasp of how to initially model it. My past math professor connected me with the dean of the math dept and we're gonna get me going on a research project... all because of an impulsive move to model COVID out of boredom
S(end) could arguably need to be be made as small as possible in the long run because we want herd immunity, initially at the start of the disease we need to make S(end) large but assuming we can/could control the viral load per transmission event e.g. with a vaccine, wearing masks, physical distancing - what does modelling tell us about the alert level we should adopt to mange the spread vs stop it. Most DRs seem to agree that there are two ways out of this 1. herd immnity and 2 a Vaccine (they are both different version of the same thing). Stamping it out is unrealistic now (sadly). How would you model the impact of reduces alert levels assuming some level of contact tracing performance.
For herd immunity we might want S(end) to be small, BUT that would mean the vast majority of the population caught the disease and given the current death rates that is means far too many people that would die. Let's hope the vaccine works out!
Surely the infected population is the least mobile sector of the populace, not the most? Won't a good fraction of the I compartment be displaying symptoms and be locked down tight?
Very true, but the movement of the infectives is what we are interested in rather than the movement of S or R, as it's the infectives that cause the disease to spread.
It's via a change of variable. We let y = x-ct where c is a constant and then change the 'old' x and t derivatives into 'new' ones just involving the variable y. The calculation requires an application of the chain rule.
@@TomRocksMaths Can't get to exactly what you get, could you add some more explaination? Also do you have references from where you got some of your info? Thanks
But the susceptibles do move and so do the removed....oops nm most of this was way over my head anyway...been too long since I studied calc or diffeqs!
Check out the heat equation is derived, the second derivative comes out to model the diffusion of heat (or anything that flows according to a gradient) through a volume.
Diffusion or the 'spreading out' of something is modelled using a second order spatial derivative. The full explanation comes from considering the physics of the problem, and is the same as for the diffusion of heat or of a chemical.
@@leinadatinsoc I actually recorded the explanation but it was excluded from the final video due to time constraints. Whilst important, I felt it wasn't key to the main point of the video, which is getting to the results of the model.
@@TomRocksMaths the beauty of the video is not its result which is obvious, I mean we do not need differential mathematics to understand that reducing the contagion rate is a major issue (though we may counter argue here with herd immunisation etc) teh beauty of it is that everybody can really understand that maths can be applied to really concrete issues which concern us all, so math is beautiful. This is what you have brilliantly succeeded, congrats! Now, I come from Romania, so Eastern Europe, where maths is being intensively tought and learned but on a purely theoretical approach, this is why I liked it so much. Keep on and thank you in advance!
Sorry, but this video falls down for the same reason as my main objection to computer code: namely, unhelpful variable names. Computer programmers don't want excessive keystrokes so abbreviate variables to shortened names - unfortunately, excessively shortened names - that render them meaningless to those analysing the code. Mathematicians don't even bother with the courtesy of naming their variables because they are able to abstract that bit further: they simply choose the next letter in the alphabet with a passing reference to what it might be and then proceed as if that were normal. Remember that you are presenting to the public (badly, I might add). In these models, there are two variables of interest and for two different groups: the health authorities want the number of infected to plan hospital space, and the public and the media care about the number of dead, which you fail to take into account as they are a subset of one of your variables. Please, please, please, take the time to name your variables properly and take the time to attempt to estimate the values that people actually care about.
@@TomRocksMaths thanks for taking the time to reply. It's 50 years since I failed A-level maths but it was gratifying to realise that at least some of was still there in the old grey matter. Your presentation skills are admirable and perhaps if my teachers had been as good as you I may even have passed.
![The Most Useful Curve in Mathematics [Logarithms]](http://i.ytimg.com/vi/OjIwCOevUew/mqdefault.jpg)
If you haven't seen part 1 on the basic SIR model, I highly recommend giving it a watch so that you can get the most out of this new model. The first video is here: ruclips.net/video/NKMHhm2Zbkw/видео.html
Thanks for everything, Tom.
I know you are Navier-Stokes guy, and I hope one day we can talk about it without these stressing events but I really appreciate what you are doing.
Recently some Icelandic scientists claimed that near of 50% of infected (I) are in truth asymptomatic (Ia), so they keep the infection rates higher as vectors while symptomatic infected (Is) are the ones that really suffer the disease consequences and consume medical resources. Humans live with many microbial entities for ages, now inoffensive, but prehistoric and antique outbreaks adapted the Humans for such conditions. Michael Crichton's "Andromeda Strain" is a good reading, the first movie is also interesting.
》Is there any model that consider such infected distinction?
I believe we can understand that the dead cannot spread diseases since they can't walk... except in Apocalyptic Zombies movies and TV series. However...
》Is there any model that consider such distinction between Dead (D) and actual Recovered (R)?
》Could a Severity of a Disease (DS) be calculated as a function of symptomatic infected (Is) and Dead (D, instead use the "actual" Recovered R) over the Susceptible population (S) as this: DS = f(Is, D)/S ?
I don't know which kind of function could be since the set of Dead (D) was once part of the set of symptomatic infected (Is): {D} ς {Is} .
The use of R₀ as reproduction number has been criticized by many, and the range of such number is too large to get some relative precision. R₀ is a "natural" (biological) number for reproduction of an organism _in natura_ , after the human reaction: monitoring, containment & treatment, no matter how effective, this number changes to a R₁ as a function of the efficacy of the reaction. A correct R₀ number is almost impossible to achieve.
》Is there any model that could consider an "actual" R₁ number or funtcion instead of an empirical R₀ ?
@@alexdemoura9972 all of these modifications can be (and most likely are) made to the basic SIR model, it just means things are much more complicated and so have to be solved by a computer. This is what actual researchers do - run the models with different parameter values to determine which best describe the situation we are seeing, and then plan accordingly. Of al of the things you mention, I think modelling R0 as a function rather than a fixed value is the 'best' idea in terms of creating something that better matches what we see in real life. R0 represents the number of people one individual will infect on average, which of course will depend on many factors such as social distancing measures, social norms, whether or not that person is asymptomatic etc.
Supposedly the R0 of COVID-19 is around 2.65. But, how it would be possible to add the different parameters into place such as the weather and social isolation to be used within these equations.
@@DeschaineFamilia those factors are incorporated into the transmission rate r, which in turn affects the contact ratio q and thus the reproductive number R0. There are no specific variables for them, but they will still have an effect on the system through the parameters mentioned.
@@TomRocksMaths one other thing i'm curious about is how reliable is the SIR model in light of Covid-19 reinfection being a possibility? If i'm not mistaken in the SIR model the Recovered is taken out of the equation as they are no longer capable to spread the virus due to either death or immunity development. But some papers have suggest that immunity may not develop post-infection, and that some recovered patients are might still be susceptible to reinfection. Perhaps adding the possibility for a percentage of the Recovered to be cycled back as a part of the Susceptibles might paint a better picture?
Thanks so much for taking the time to do this. It's the first video I've seen which goes beyond a simplistic SIR compartment model, and you made the maths quite approachable. Cheers mate, I appreciate it a lot.
You're very welcome Stephen. Thanks for watching.
@@TomRocksMaths i am an amateur but for the first time i understand how difff equations actually work and how to set them up.
Applied mathematical modelling in action - Awesome!!!
Just discoverd this Chanel, love it
Welcome Tibor
Is Tom the Navier Stokes guy from Numberphile?
That is indeed me
That "I" letter magic @ 08:25
dammit, was hoping you wouldn't spot it...
@@TomRocksMaths, still better CGI than Black Panther v Killmonger fight scene.
Many thanks Tom, a great insight into how modelling is applied in real life situations. Stay safe.. cheers.
Glad you enjoyed it Owen!
Oh, you're killin' me already (only 1 minute into video). I have ocean engineering master's, and recall "celerity" (speed of propagation). How cool that "Deterministic Wave Theory" has a footing in disease propagation!! Wondering how fibonacci series or more specifically the "Golden Ratio" could play in this....each wave is 31% the magnitude (volume) of previous wave???
That’s great Tom, keep on going with these videos
This video is even cooler than the last!
OMG a travelling wave and epidemics... how lovely!
One question that I didn't get about the non-dimensionalizing: I can see why the constants r and a disappear, since you wrote them in terms of R_0. But I don't get what happens to the constant D, the coefficient in the diffusion part of the set of equations?
We scale the spatial variable x with r^(1/2) * S0^(1/2) / D^(1/2) so that it 'cancels' out with the double x derivative.
Great info for those kids AND old folks staying at HOME.
Thanks John.
Great Channel, i learned more about the SIR Model, thanks Tom Rocks Maths
Is there somewhere to find step for working out of both non-dimensionalisation equations
This is some brilliant explanation help me to understand how the model works instead mere exponential growth rate parameter (which is quiet baby in front of this one lol). Again the complexity of these model itself and real life situation which is more complex than model indeed great lesson to learn. Awesome video , you clear the most idea behind it.
Question what is P in S=1-P ? (as Susceptible is now space and time dependent with variable of space "x" and "y" and variable for time is "t" so is it P is particular instance which is nothing but x,y,t dependent in non-dimensional frame?)
Thanks
P is still represents the number of susceptibles, but you can think of it explicitly as the number of susceptibles that have caught the disease since the start of the outbreak - ie. it represents the decrease in the susceptibles, which is why we are subtracting it from the full value of 1.
@@TomRocksMaths ahh i see.... thank you.
Thank you for your easily understanding math videos! It's a big help!
You're very welcome Julie :)
Would you be able to suggest one or two textbooks that cover this? (Context: I did maths at college 30 years ago, which included a course on differential equations in fourth year, but it's all a distant memory now. I do remember that I didn't understand phase portraits at the time!)
The go to text is 'Mathematical Biology' by JD Murray.
this was the best explanation as to why we need social distancing I've seen since I even knew that was a thing.
Glad I could help!
For the second equation after change of variables, shouldn't it be the second d/dy of I, not S?
Yes - I fixed a few of them near the derivation part of the video, but there may be some erroneous s terms later on.
Travelling waves - or overshoots in a 2nd order system. A simple dynamic model is w/(s^2 + 2.z.w.s + w^2) which is the Laplace transform of the 2nd order DE, where w is the natural frequency (about 5 months) and z is the damping factor z = 1/(root(2).R), R the reproduction number. When R = 1, the system is critically damped and the pandemic is under control, when R > 1, the system is underdamped resulting in overshoots (waves), and when R < 1, the system overdamped. This fits in roughly with what we have seen. The vaccines will effect R greatly. Your 2D (spacial) expansion is interesting, but like all models is only approximate and its conclusion is having terrible effects on the population and economy. Those people with the fake fraudulent "MA" degree form Oxford in Classics don't appreciate this fact. Its interesting that Sweden has a death rate of 0.13% of population (and no masks) whilst the UK has a death rate of 0.19% (and masks). Masks and social distancing have only a weak effect on R. Don't like your rainbow badge Tom, Yuk.
Really excellent work, thanks Dr.
You're very welcome :)
Hello Tom, thank you for your interesting videos! I have a question that is there any way to add some variations like people visit public places(e.g. grocery stores, pharmacies, ...) on certain frequency? For example, what if quarantined people visit grocery stores once a week and contact 15 people(which is an extremely higher number than they normally contact) with an expected transmission rate of 0.2? I just wanna see how this assumption could be shown on equations.
Hi Heesun, thanks for your question. Including such intricate details would require a much more advanced model than the one discussed here. While such things are certainly possible, there's always a balance with all mathematical models between how realistic they are, and how 'quickly' they can be solved. To run a model that included variations on an individual level like you suggest would take several months on a supercomputer, which is something that we unfortunately just don't have the time for right now.
Tom Rocks Maths oh I see, thank you! Stay safe 🙏🏼
If you feel like tackling this, there is an issue with the SIR model, in that if you put a part of the population in a quarantine, the number of infected never goes down to zero, or it goes down too slowly. How could that be fixed? There need to be an upper limit of a certain number of days for the -aI differential because the viral charge is not eternal in a person.
The issue of quarantining and how to model it is actually the topic of my next video. Hopefully coming at the end of this week.
Hello Tom.Thanks for these two great videos! Please, is there a way to calculate the impact of social distancing in some scenarios. For example the measures taken in my country: closure of schools and uni, absolute lockdown from 6pm to 5am, transit restriction to men and women in specific days.
Hi Alex, it's very difficult to quantify specific scenarios in the model, but all of the things you mention will absolutely help to reduce the contact ratio q, which I've hopefully convinced you from these 2 videos is exactly what we want to do!
Great video Tom, thanks. You've got yourself another subscriber.
Happy to Paul - and thanks!
Good video - it was easily understandable. Do you have any articles or papers explaining your analysis, e.g. how the minimum wave speed was determined?
I got a bit lost, new to all of this! From a statistical standpoint, considering we've increased the complexity of the model, we'd expect the travelling wave model to show different (and better) predictions than a regular SIR model?
Yes that's correct. It is a 'slightly' more realistic model than the basic SIR one.
20:40 is the slope on the graph right compared to the distance of "1" on the x axis, to the y axis?
If so, then the R0 calculated for Covid being between 2 and 4 gives us an S(end) nearly 0.4 to 0.1 or less
Please correct me if I'm wrong, I'd like to know how bad this estimations are
The graph is just a sketch and is definitely not to scale so I wouldn't try to use it to get actual numbers. The idea was just to show the shape as q decreases towards 1.
Great Content, Love it. I wonder how can you apply phase plane to find c, Can you brief how to do it . Thank so much.
Sent you an email :)
Tom Rocks Maths This is my question too... I would appreciate it if you could clarify this to me too.🙏🏻🙏🏻🙏🏻
It would be nice if you could do a full video of this without jumping over details and also fix the errors
Just found your channel, thoroughly enjoyed the video. Thanks and Stay Safe.
Welcome Supriyo - have fun!
You are very clever and a good teacher thank you 👍
You are very welcome Sir Xenon!
Resources please on how to do phase plane analysis and linearization, because I am doing a report in a course in college about it..
The textbook 'Mathematical Biology' by J. D. Murray is excellent.
Thanks Tom. What are the differences between predictions by the two models?
We learn different things from each of them. The travelling wave model tells you about the speed of the disease spread, which we don't get from the basic SIR model.
Great explanation
is 'c' measured in m/s or something else?
Good question - because we have 'non-dimensionalised' c doesn't actually have any units in the form given. If we re-introduce units to the problem, I believe the units of speed will be D * S0^(1/2) * r^(1/2), where D is the diffusion coefficient, S0 the initial susceptible population and r the rate of transmission of the disease.
@@TomRocksMaths Going through the non-dimensionalization, I get that speed is normalized by sqrt(D*r)
Thanks Tom but I think you need to define x, the space variable. Is it a simple distance? If so how is it measured? - miles, a fraction of the maximum distance, or what?
It represents the spatial variable, but as the entire model is non-dimensionalised it has no units.
@@TomRocksMaths Really impressed you got back to me Tom. I thought it would be something amorphous like that but it if it can't be measured how can it represent reality. Sorry for this picky question but I really want to understand travelling waves to see how well it fits my ABM SIR spatial model. I tried to dip into the the academic papers but find it difficult to know where to start as they all seem to require prior knowledge of the theoretical basis. I then found your tutorial and it made good sense to me except that I couldn't see how to relate it to my measures. Any leads would be greatly appreciated.
The textbook ' Mathematical Biology' by J. D. Murray is really good for disease modelling in general.
@@TomRocksMaths Great lead Tom. Just beginning to read Chapter 13. Thanks again.
Hi sir how did you do the phase plane analysis?
It's pretty complicated - one of my interns Amie is working on a video explaining it all which will be out soon so watch this space :)
@@TomRocksMaths Sure!! Thanks!!
Suddenly at 11:13, the value of S changed from one to S naught and then at 11:57 back to 1
Well spotted. It should be 1 throughout.
Good lecture!
8:49 that I going over his hand caught me off guard.
nice spot
Thanks a lot Tom... I love it
You're very welcome Ari!
Hey Tom, Can you give an example from the past when travelling wave was used and it's very hard to find accurate data on COVID-19 graphs and other disease graphs in general,
It would be great if u could help me out in these areas, I am working on a report, so all the data would really help...
I'd suggest looking at the textbook 'Mathematical Biology' by J. D. Murray
@@TomRocksMaths OK thanks tom.
@@TomRocksMaths Which edition though tom?
SIR requires accurate inputs, how do you account for inaccurate inputs, rates of reinfection? In the States how would this work?
You're right that the most difficult part of using any disease model is gathering accurate data for the parameters. The best we can do is work with what's available and for some of the more complex models, it's even possible to account for the uncertainty in the parameter values by modelling them stochastically (meaning as random variables with some underlying probability distribution). Ultimately this introduces an element of randomness into the models which accounts for the potential inaccuracy of the data.
@@TomRocksMaths John von Neumann's comment about fitting an elephant comes to mind.
Bro I love this videos
Thanks Luis!!
Okay but I love your style🥺
I actually really hate dimensionless analysis, even if it is easier. Having units around often shows you quickly when and where you've made a mistake. Hauling constants around is the price you pay for clarity of thought, in my book.
just thinking out loud......can you model the impact of social distancing on mental health, educational achievement and what has actually gone on in the spread of the disease now that there is quite a bit of data out there.
Nice video.
It would be nice to see some estimates of the parameters based on actual data and see the trajectory S-I in different countries.
I agree, but they would be based on pure speculating at the moment as we just don't have enough information about the disease. I purposefully avoided using actual values as I want to concentrate on setting out the model, rather than making any far-fetched predictions.
@@TomRocksMaths well, you could do it with past epidemics: measles, smallpox, the plague in India...
Nobody:
Us Nerds: *Model of COVID-19*
Time for maths to shine.
@@TomRocksMaths this might be the coolest thing ever, thanks to this and s few other channels I got a grasp of how to initially model it. My past math professor connected me with the dean of the math dept and we're gonna get me going on a research project... all because of an impulsive move to model COVID out of boredom
@@insaiyantraining6035 amazing well done!!
brilliant
were can i get such a t-shirt. taking in cosideration that Im staying home )
i also wanna know haha let me know if you find out
I got it a while ago, but I 'think' it was from a UK store called Topman
It would be nice not to skip some steps just to understand where equations appear.
S(end) could arguably need to be be made as small as possible in the long run because we want herd immunity, initially at the start of the disease we need to make S(end) large but assuming we can/could control the viral load per transmission event e.g. with a vaccine, wearing masks, physical distancing - what does modelling tell us about the alert level we should adopt to mange the spread vs stop it. Most DRs seem to agree that there are two ways out of this 1. herd immnity and 2 a Vaccine (they are both different version of the same thing). Stamping it out is unrealistic now (sadly). How would you model the impact of reduces alert levels assuming some level of contact tracing performance.
For herd immunity we might want S(end) to be small, BUT that would mean the vast majority of the population caught the disease and given the current death rates that is means far too many people that would die. Let's hope the vaccine works out!
Gets insant flash back to wave equations at uni
Surely the infected population is the least mobile sector of the populace, not the most? Won't a good fraction of the I compartment be displaying symptoms and be locked down tight?
Very true, but the movement of the infectives is what we are interested in rather than the movement of S or R, as it's the infectives that cause the disease to spread.
@@TomRocksMaths Right. Thanks. Grabbling with a 6D ODE system would not be pleasant, I guess.
@@thermodynamics458 pretty much! With all modelling you have to balance the level of realism versus complexity.
8:04 how did those equations come ?
It's via a change of variable. We let y = x-ct where c is a constant and then change the 'old' x and t derivatives into 'new' ones just involving the variable y. The calculation requires an application of the chain rule.
@@TomRocksMaths Can't get to exactly what you get, could you add some more explaination? Also do you have references from where you got some of your info? Thanks
Really great!
Thanks Luca - glad you found it useful.
Is it possible that you make a video about the stochastic SIR Model? I think it is more closely to reality, don‘t you think?
But the susceptibles do move and so do the removed....oops nm most of this was way over my head anyway...been too long since I studied calc or diffeqs!
but they don't infect anyone, so the quantity S does not change when they move. Same with R.
I wonder how did you come to a second derivative in terms of space, I mean the explanation...
Check out the heat equation is derived, the second derivative comes out to model the diffusion of heat (or anything that flows according to a gradient) through a volume.
Diffusion or the 'spreading out' of something is modelled using a second order spatial derivative. The full explanation comes from considering the physics of the problem, and is the same as for the diffusion of heat or of a chemical.
Thank you to both of you. Maybe it would have been worth mentioning it in the video; it is not self evident. :)
@@leinadatinsoc I actually recorded the explanation but it was excluded from the final video due to time constraints. Whilst important, I felt it wasn't key to the main point of the video, which is getting to the results of the model.
@@TomRocksMaths the beauty of the video is not its result which is obvious, I mean we do not need differential mathematics to understand that reducing the contagion rate is a major issue (though we may counter argue here with herd immunisation etc) teh beauty of it is that everybody can really understand that maths can be applied to really concrete issues which concern us all, so math is beautiful. This is what you have brilliantly succeeded, congrats! Now, I come from Romania, so Eastern Europe, where maths is being intensively tought and learned but on a purely theoretical approach, this is why I liked it so much. Keep on and thank you in advance!
awesome
Thanks Niki - glad you enjoyed it!
Sorry, but this video falls down for the same reason as my main objection to computer code: namely, unhelpful variable names.
Computer programmers don't want excessive keystrokes so abbreviate variables to shortened names - unfortunately, excessively shortened names - that render them meaningless to those analysing the code.
Mathematicians don't even bother with the courtesy of naming their variables because they are able to abstract that bit further: they simply choose the next letter in the alphabet with a passing reference to what it might be and then proceed as if that were normal.
Remember that you are presenting to the public (badly, I might add). In these models, there are two variables of interest and for two different groups: the health authorities want the number of infected to plan hospital space, and the public and the media care about the number of dead, which you fail to take into account as they are a subset of one of your variables.
Please, please, please, take the time to name your variables properly and take the time to attempt to estimate the values that people actually care about.
oh, this is so nice :-)
travelling wave solution to a reaction diffusion problem . That you really have to first come up with. Magick
Glad you enjoyed it!
Excellent video! I’ll try to apply to Mexico’s case. You remain me Sex Pistol, that’s make it even better!
Awesome - thanks Alejandro!
Well, that was very interesting but I realise I have a "curates egg" of a brain. I understood it in parts.
Don't worry - I teach this to my third year maths undergraduates at Oxford and they struggle with it!
@@TomRocksMaths thanks for taking the time to reply. It's 50 years since I failed A-level maths but it was gratifying to realise that at least some of was still there in the old grey matter. Your presentation skills are admirable and perhaps if my teachers had been as good as you I may even have passed.
thats what i thought, yeah
Tattoos are great spreaders of diseases too like Hep-C!