Exponential growth with epidemics

Yves, that was a major cliffhanger for me. With a vaccine at least 18 months away and with the current growth rate projecting the full world population to be infected somewhere at the beginning of June, the question is: what makes it stop. Is there some external factor that precipitates the inflection point? If so, what would/could that be in this case.

Jan Prummel

I really liked the take-away of being able to estimate the half-way point based on the daily new cases. Does the second derivative/difference have enough consistency to give useful duration estimates at the 1st derivative/difference inflection points (perhaps with some averaging/smoothing)? It seems like there could be a lot of confounders that would impact accuracy.

Jeremy

Another Excellent Production. Well done.

I love the modeled spread between groups! Thanks for putting this out.

Programmable Spacecraft

I ended up writing this in the pinned comment. At the moment, I'm tentatively thinking of making a follow on video that more directly discusses epidemic modeling.

3blue1brown

Indeed, it was demonetized. It does make me feel grateful for the fact that this channel has the community support that it does, so we don't have to worry about incentives like that.

3blue1brown

That is such a great video - thanks for making it. By the way, I read that youtube has been demonetising all videos mentioning the COVID-19, I wonder if that has affected this video also? Thank goodness you have Patreon which doesn't make silly policy like this.

Chang Yang Yew

I agree that talking about "taking preventive measures" is a better way to nudge people to get to the more optimistic numbers. Maybe there is a place for worry in the video, but as you are currently using it, I don't think you are making your message less clear.

Raul

Well done, I always appreciate your work.

No, the version I saw also said 1.9 billion cases. He appears to have updated it since.

I think you ought to say somewhere that it's not aiming to be a model of the coronavirus outbreak, but something that covers the principles of such outbreaks, as well a bit on E and p and how they are important to our current concerns

Steve

Actually it seems that some people have had the current illness twice.

Steve

It looked great to me. I concur that publishing it now is better, because the sooner people understand how their behavior affects E and p, the better off everyone is. Even a day or half a day matters in exponentials, right?

jason black

Great point! Would I be right in saying it'd be better to do it on the daily new cases (which should also look linear on a log scale)?

3blue1brown

All great, but I would just point out you can't really run a linear regression on the cumulative cases like that. It breaks the assumption of linear regressions that each observation is independently random from the rest (a*x + b + epsilon with normal distribution). Each observation is in fact the previous observation plus some new cases plus some randomness. That's not a small thing. If you have a true random distribution of new cases like you described, the linear regression of a cumulative number of those will *necessarily* approach a regression with R^2 = 1, because your washing out all the randomness by adding it up. So saying that the regression has a R^2 >0.9 doesn't really say anything. That of course was quite minor for the point of the video, which was a great explanation of how those dynamics work. I did my master thesis on technology diffusion, and it's weird to see how diseases follow basically the same equations :p

Leo Barlach

A flaw in the model is it does not account for people that recover and can no longer infect. That is Nd is the total number of infections, not the current number of infections.

I think there is an error in your Spanish Flu figure, you say 1.9 billion cases, but from what I can find online I think that is 1.9 billion was the total world population at the time. Seems like there were an estimated 500 million cases.

I showed this to a friend of mine who specialises in Infection Prevention and Control and who is working on the UK response to the outbreak. She loves it and is very keen to see it published. I think you should be prepared for a lot of interest.

Steve

I totally agree☝️☝️☝️

That is beautiful! Simple, elegant, clear, and meaningful. Vote for Grant to head CDC Communications...

Richard Hackathorn

Great little video - I know that coronavirus is the appropriate relevant subject of the hour, but It'd be great to expand this to populations and resources, or interest and debt, or any important subject area that includes types of exponential growth. People in general just don't have a good intuitive sense for exponential relationships, to our detriment.

RHall

My current thinking is to treat it just as a lesson on exponentials and sigmoids, motivated by a current event but hopefully not _too_ anchored on it, and to leave people to draw their own conclusions as they see the numbers and read the news. I honestly have no idea how to think about incorporating China numbers, as it's unclear how much their policies and reporting accuracy would be representative of the rest of the world's.

3blue1brown

Haha, it's original, but I don't think Twain-level.

3blue1brown

Thanks!

3blue1brown

Worry doesn't equate action. I'd just talk about making preventive measures instead. Nice video though, I really like it.

Liz Av

Best to publish ASAP- it will really help educate the public!

First time commenting for me since I became a patron. This video is a good example of why I decided to contribute. It is always a thrill for me to see mathematics applied to a real world situation. Keep up the good work!

Is that your quote. 'if people are sufficiently worried there is nothing to worry about, but if no one is worried that's when you should be worried'. Because it is the best quote I have heard about this and people should start quoting you when questioning if the media is over-hyping, etc. Puts things in better context. I will start using it myself. Better copyright it before someone attributes it to Mark Twain.

I agree with the comments above. China seems to have finished the sigmoid, so I would use that data to talk about the leveling off instead of leaving it all speculative. You can still talk about avoiding transmission, since China did enforce quarantine. The other interesting part is that even if the ceiling is the same, slowing down the transmission lowers the number of new cases per day, which is the key factor for treatment/shortages, etc.

Gabe

Actually, yes: https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 But it's as you say... if you can trust the data.

Christian Leichsenring

Really nice presentation. The big unknown is the number of cases not diagnosed. Notably India, but also Africa and South America, are not reporting many cases. However, the same problem there exists as in China where the first cases are just considered flu or pneumonia and there is no testing.

white beard geek

If you focus on just China, is there data that already shows the growth slowing down? That might be a nice addition to the video (if you can trust the data).

tdh

similarly at 6:15

Oh! and the overall tone is nice too... It's difficult not to incite panic, but you get the message across in a nice way, while providing context to government action... well done

I like how you use a very contemporary subject to enlight us on the basic mathematical concepts that drive them.

Thank You for this video. Around 5:50, I think the fractions on the right should be inverted : 2964/2159 and 3530/2964.

At 5:52 there seems to be a third line that begins to fade in but doesn't (Can only make out =1.15)

JacksonG

Nicely done, Grant. Helps me better understand why the Italian government is announcing a travel lockdown in the province where I live (Padova, near Venice) and across northern Italy today. Crossing my fingers we hit that inflection point sooner rather than later!

Steve Muench

Another interesting sub-topic would be how epidemics eventually fizzle out far sooner than they infect the total population. Out of the scope of this video, I know.

Dragi Raos

It may also be desirable to talk about specific real-world measures that health authorities actually take to combat epidemics (e.g. quarantines, public outreach, etc.), and how they directly impact the growth factor. For example, hand washing is not just a feel-good measure, it is one of the most straightforward and effective ways that a regular person can reduce the spread of infectious diseases. The video does very briefly mention that, but more emphasis would be nice.

Kevin

I think that for this simple treatment one can ignore this. But yes, it would be interesting to see how the math look like if one includes the fact that a vector ceases to be one after a while.

Dragi Raos

As usual, I think you did a great job illustrating the basic math, and the parts on helping the intuitions behind that basic math. In this case, I do worry that leaving the basic math undisclaimed could be misleading. We don't get to see Nd. We see some fraction of N(d-k) because symptoms take time to manifest, and not every infection gets confirmed. Perhaps you should mention that when talking about how folk might react to the day-by-day numbers.

Anthony Bailey

Very nice video with much-needed explanation of math behind the models of epidemics spread. I wonder whether you could spend a few more sentences why actually limiting the growth factor is so important, as it follows that reducing its value makes people becoming infected anyhow, but only later. It is important because slowing the rate of growth of the number of infected persons relieves a pressure on the health care system and social services, see e.g. this article in The Lancet: https://www.thelancet.com/lancet/article/s0140-6736(20)30567-5 It is essential that people understand why communities may decide, and probably will decide, to introduce measures like travel limitation, school closures up to what actually has been done in the Wuhan region.

Lech Mankiewicz

The video seems to suggest there are 2 kinds of people... Uninfected people waiting to get sick, and infected people waiting to infect the 1st group. But there's a 3rd group... those who are no longer infectious and are now immune. Your video made it seem like at some point, nearly everybody is a vector. But, since the infectious time for Covid19 seems to be on the order of 3 weeks, I'd expect over time the growth function would change. Essentially, the "no longer infectious" group in sub-populations is also growing exponentially after a few weeks. Fortunately, since survival rate is still relatively high, the "no longer alive" group probably doesn't change the curve much.

Ron Goodman

Cool, I love how you almost give a predicted date for ten billion people infected, and then go on to talk about logistic curves.

Rosuav

Great job as usual, Grant. Many thanks!

Thanks! Looks like I slipped in the wrong number there. This is why we do early releases :)

3blue1brown

I think at 3:37 it should read ~500 million cases for Spanish flu (out of 1.9 billion world population at that time this would be about 27%).

Nicely done!

Brian Michalowski