The phrase “exponential growth” is familiar

to most people, and yet human intuition has a hard time really recognizing what it means

sometimes. We can anchor on a sequence of small seeming

numbers, then become surprised with suddenly those numbers look big, even if the overall

trend follows an exponential perfectly consistently. This right here is the data for recorded cases

of COVID-19, aka the Coronavirus, outside mainland China, at least as of the time I’m

writing this. Never one to waste an opportunity for a math

lesson, I thought this might be a good time for us all to go back to the basics on what

exponential growth is, where it comes from, what it implies, and maybe most pressingly,

how to know when it’s coming to an end. Exponential growth means as you go from one

day to the next, it involves multiplying by some constant. In our data, the number of cases each day

tends to be between 1.15 and 1.25 times the number of cases the previous day. Viruses are a textbook example of this kind

of growth because what causes new cases are the existing cases. If the number of cases on a given day is N,

and we say each individual with the virus is, on average, exposed to E people on a given

day, and each exposure has a probability p of becoming an infection, the number of new

cases each day is E*p*N. The fact that N itself is a part of this is what really makes things

go fast because as N gets big, the rate it grows also gets big. One way to think of this is that as you add

on these new cases to get the next day’s count, you can factor out the N, so it’s

just the same as multiplying by some constant bigger than 1. This is sometimes easier to see if we put

the y-axis on a logarithmic scale, meaning each step of a fixed distance corresponds

to multiplying by a certain factor; in this case, each step is another power of 10. On this scale, exponential growth looks like

a straight line. With our data, it took 20 days to go from

100 to 1,000, and 13 days to go from that to 10,000, and by doing a linear regression

to find the best fit line, you can look at the slope of that line to say it tends to

multiply by 10 every 16 days on average. This regression also lets us be more quantitative

about how close the exponential fit really is, and to use the technical jargon here,

the answer is that it’s really freaking close. It can be hard to digest what this really

means, if true. If you see one country with 6,000 cases, while

another has 60, it’s easy to think the second is doing 100 times better and, hence doing

fine. But if you’re in a situation where numbers

multiply by 10 every 16 days, another way to view the same fact is that the second country

is about a month behind the first. This is, of course, rather worrying if you

draw out the line. I’m recording this on March 6th, and if

the present trend continues, it would mean hitting 1M cases in 30 days (April 5th), hitting

10M in 47 days (April 22nd), 100M in 64 days (May 9th), and 1 billion in 81 days (May 26th). Needless to say, though, you can’t draw

out a line like this forever, it clearly must start slowing down at some point, but the

crucial question is when. Is it like the SARS outbreak of 2002 capped

out at about 8,000 cases, or more like the Spanish Flu in 1918 ultimately infected about

27% of the world’s population? In general, just drawing a line through your

data is not a great way to make predictions, but remember that there’s an actual reason

to expect an exponential here. If the number of new cases each day is proportional

to the number of existing cases, it means each day you multiply by some constant, so

moving forward d days is the same as multiplying by that constant d times. It is inevitable, though, that this factor

in front of N eventually decreases. Even in the most perfectly pernicious model

for a virus, which would be where every day, each person with the virus is exposed to a

random subset of the world’s population, at some point most of the people they’re

exposed to will already be sick, and so can’t become new cases. In our equation, this means the probability

of infection should include some factor to account for the probability that a person

you’re exposed to isn’t already infected, which for a random exposure model would be

(1 – the proportion of people in the world who are infected). When you include a factor like that and solve

for how N grows, you get what’s known as a logistic curve, which is essentially indistinguishable

from an exponential at the beginning, but ultimately levels upon approaching the total

population size, as you’d expect. True exponentials essentially never exist

in the real world, they’re all the beginnings of logistic curves. The point where this curve goes from curving

up to instead curving down is known as the “inflection point”. At that point, the number of new cases each

day, represented by the slope of this curve, is roughly constant, and will soon start decreasing. So one number that people will often follow

with epidemics is the “growth factor”, which defined as the ratio between the number

of new cases one day, and the number of new cases the previous day. So, just to be clear, if you were looking

at the totals from on day to the next, then tracking the changes between these totals,

the growth factor is the ratio between two successive changes. While you’re growing exponentially, this

factor will stay consistently above 1, whereas seeing a growth factor around 1 is a sign

you’ve hit the inflection. This can make for another counterintuitive

fact while following the data. Think about what it would look like for the

number of new cases one day to be about 15% more than the number of new cases the previous

day, and contrast that with what it would feel like for it to be about the same. Just looking at the totals, they really don’t

feel that different, but if the growth factor is 1, it could mean you’re at the inflection

point of a logistic, which means the total number of cases will max out around 2 times

wherever you are now. But a growth factor bigger than 1 means you’re

on the exponential part, which could imply orders of magnitude of growth still lie ahead

of you. While in the worst case this saturation point

would be the total population, it’s of course not true that people with the virus are randomly

shuffled around the world’s population like this, people are clustered in communities. But when you run simulations where there’s

even a little bit of travel between the clusters like these, the growth is not actually much

different. What you end up with is a kind of fractal

pattern, where communities themselves function like individuals. Each one has some exposure to others, with

some probability of spreading the infection, so the same underlying exponential-inducing

laws apply. Fortunately, saturating the whole population

is not the only thing that causes the growth factor to slow. The amount of exposure goes down when people

stop gather and traveling, and the infection rate goes down when people wash their hands

more. The other thing that’s counterintuitive

about exponential growth is how sensitive it is to this constant. For example, if it’s 15%, and we’re at

21,000 cases now, that means 61 days from now it’s over 100 million. But if through a bit less exposure and infection

it drops to 5%, it doesn’t mean the projection drops by a factor of 3, it actually drops

to around 400,000. So if people are sufficiently worried, there’s

much less to worry about, but if no one is worried, that’s when you should worry.

## 100 Comments

This is what should be taught in math lessons instead of calculating how many bananas jonathan ate

I don't think that the clowns in the MSM could explain it any better. Thank you sir.

here from /Pol/

When schools close for a month or more, like they have in Hong Kong, we'll see how unprepared the USA is for something like this.

Amazon delivered my bat soup today. I'm good till the inflection point. After that it's my stash of Chef Boyardee. See you at the next virus, if you make it.

When I was a child, we used to have measles and chickenpox sleepovers. When one child got sick, all the parents would take their children to that house and they would sleep over. Then the children would come home and be quarantine until they were well.

The odds of complications for children were small and this rapidly improved herd immunity. It also prevented schools from becoming breeding grounds for chicken pox and measles epidemic. We still vaccinated for polio religiously.

Lol I’m in a red zone, I can’t exit from my house

I love this because my entire research project has been involving logit curves and it's cool to see it used in real life!

Have a friend who had COVID-19…

He has a history of asthma and said the virus was the mildest cold he ever had…

…just… stop already…

rip monetization on this video

You have to also account for the fact that over time people get better and thus their chance of infecting people is reduced to 0, and their chance of being re-infected is probably significantly less even though this virus seems to be able to infect the same person multiple times.

Interventions can work. China hit their inflection point in early February, and by early March had a pretty textbook logistic curve:

https://upload.wikimedia.org/wikipedia/commons/e/e5/2020_coronavirus_patients_in_China.svg

I have a problem with the cluster model – once a cluster only (or mostly) has infected individuals, it's safe to assume that it will be quarantined, and individuals will not travel outside or inside that cluster.

Thank you for not including China's data, it creates a significantly more accurate model.

If the virus mutated the new population of possible infected will change tho

Eveybody calm down, the spreading will stop when we're all infected 🙂

This man just did all that math to tell me to wash my hands

Well, no one with money is worried about climate change.

Nice.

Rate of cases recorded isn't equal to the actual rate of the virus's spread. The exponentiality of the recorded cases graph is likely due to the exponential increase of people being examined.

This means that we don't actually know at what rate and speed the virus is spreading, which isn't good.

According to this video all people have to get sick or get worried and take serious precautions in order for the virus to diminish. Completely wrong premise.

Can the author of this video then explain why regular flu seasons are short lived and have an end without having such special attention as this new virus? During the severe 2017-2018 flu season in recent years, estimates indicate that more than 900,000 people were hospitalized and more than 80,000 people died from flu in the US. In comparison the damage of this corona virus is a small fraction to this.

In China the new cases are exponentially decreasing every day now without infecting all 58.5 million population of the Hubei China where the outbreak started. Recorded cases in all of China so far are 80,700 but active cases are 20,270 and new cases exponentially decreasing worldometers.info/coronavirus/

Like any other flu seasons and corona virus cycles, virus outbreaks have a short period of existence and they don't have to infect all people in the world, a country, or even a city in order to go away.

Also it is possible this virus has been around for years in all flu seasons but first examined this year by the Chinese.

If special and expensive kits are needed for testing that haven't been available before how can anybody proof that this virus have not been around every winter and people sick from it considered as having regular flu? Maybe the Chinese examined and discovered it as a new virus for the first time now because the number if sick people spiked more than usual but that doesn't mean its new. Any of the already flu strains can spike in a certain year more than usual and kill more people.

The only thing certain about this virus is that it has higher mortality rate than regular flu for older people especially those with underling conditions and only that aspect should be taken more seriously than regular flu everything else is fear mongering and hysteria. https://www.worldometers.info/coronavirus/coronavirus-age-sex-demographics/

If people are worried, then some people feel safe and don't worry because other people are worrying for them. The people who are worried will see the one who is not worried and start to not worry themselves, and then shit hits the fan, and things become dangerous. When everyone worries, Covid 2019 goes away

" You shouldnt worry about COVID-19 , death is inevitable".

Albert einstein.

Thank you. Amazing video!!!

You might want to check this out https://www.youtube.com/watch?v=jo7QNwaMSlM

we'll all die

Odd request: could you do a video on climate change? There is a massive void of content out there, and your brand of concise-yet-thorough is well-suited IMO.

Well, it's not a good sign that most people I know aren't changing their activities much, and the president of the United States keeps telling people this thing will die out next month.

thank you

Can you do a follow-up on this video maybe in some days/weeks? I’m really curious on how the numbers will change

people who get the infection get immune to it though dont they? so they can neither get reinfected or infect others and so can basically block the infection once a sufficient amount of them exist

Can you do a video on martingale in light of Corona virus?

My professor mentioned that we might be able to apply martingale to predict the spread with two types of random variables(one for general people, and one for super spreader).

There is now an ‘S’ strain and an ‘L’ strain. How virulence/infectivity is changed with mutations is impossible to foresee. Wash your hands, mutants!

I survived both the Corrupted Blood pandemic of '05 and the Zombie Plague of '08.

If you're really worried you can just log out until the developers do a hard reset.

Dr Mike:

-Alert not anxious!!!

-Wash, wash, wash your hands!

-Chest compression, chest compression, chest compression!!!!

me: worrying about the epidemic

my roommate: hippity hoppity traveling makes me so happy

Great video!

The growth factor is as trustworthy as the governments who are keeping track of the numbers. :-)) That means in China and the US you have to expect the numbers to be at least ten times higher. :-))

Didnt know about the growth factor. Thx!

Great video!!

Solid math but futile application to the present situation where there is no accurate information.

Nepnep!

That was a great video Grant! Always love it when you show us how to apply analysis to real life. 🙂

While overcaution is ideal from the simple perspective of minimizing the spread, you can't ignore it's effects on economic and societal behaviour. The mad rush by people to empty stores recently has honestly scared me quite a lot.

Who are you?

"If everyone is worried about it, then there's no reason to worry…but If no-one is worried about it…that's when you should worry ." That is pretty funny but true. At least fear-mongering news is actually a good thing in regards to this outbreak.

I'm very interested in knowing what the curve currently looks like

withinChina, where data suggests the growth rate is in fact already decreasing.Roosevelt : Only thing to fear is fear itself

Grant(yolo) : well yes but actually no

Beautifully done!

When epidemics get a mathematical view: (Video)

I really enjoyed this. Diligent efforts over the past several years to resurrect my math skills have paid off (and this channel has been a resource in doing that). The math here was very simple for me. Huzzah!

Nice video man that was cool

The mathematics/statistics used where somewhat less precise than most of the videos from this channel, here is roughly what I would change looking at 1:34 seconds in.

For this equation to be deterministic, ie. having no random terms, we need to redefine Nd as the expected number of cases on a given day. This might seem small, but without it this model could not fit any data observed except a perfect fit. I would also change the definition of E from an average to an expectation, averages are random, expectations are not.

Having defined Nd as an expectation it also makes the regression used later on make more sense.

Darwinism in action. Weeds out the weak and vulnerable. Leaves the rest stronger. Pops off the boomers disproportionately and leaves the rest to enjoy their pension funds later.

Fantastic explanation, thank you!!!!!

Finally

1 billion in 81 days, guys.three months. we have three months. take care of your parents–ESPECIALLY your parents and grandparents. if they already weak then the virus will hit them harder than most.

Man, we just began 2020 and we about to end.

I needed to see this video today. So do millions of others! Helpful, concise, unalarmist, and a good lesson in exponential growth to boot! Thank you. -John

This means ill be able to skip school soon? I hope so

“COVID-18” is what WHO that’s working for China is making people call it. China now started claiming it is possibly not from China, trying to p put blame on either Japan or South Korea. It is WUHAN VIRUS. Chinese Wuhan Virus.

Forgot to include time how long per person is able to infect more.

siempre cetrero

Your mathematical models and simulations didn’t consider a few important aspects – people who are infected (N) heal and loose their ability to infect new people. So there is an over-time force that places downward pressure on N. This is what quarantine is about in the real world – it’s about encircling an infected group, and preventing them from exposing to new potential victims until they are no longer infected. As the virus numbers go up, real world governments will take stronger quarantine measures and impose further travel bans.

thank you for a clear concise explanation

wow, that virus is going viral!

this was done on purpose people…nothing more just man made

How to get real news about COVID: Fit every day COVID's reports on this model. The closer you get (every day's new cases / new cases on the previous day) to 1, the closer you are to the end of the epidemia. Wash your hands to decrease (E * p) factor

I feel bad for the pi's 🙁

Man made.

I love you 3blue1brown

Just to add, this is just for infections, which does not mean death or lasting damage to health. Although, the psychological effect is probably most damaging factor

We are currently in an unknowable ratio of " if an infected person can be reinfected", and another factor of how many people are infected!

I'm worried, so I shouldn't worry that much, which means I should be worrying more…. I love your paradoxical statement. Great way to sum up a great video!

While the intent here is to give a lesson on exponential and logistic growth as general phenomena, with epidemics as a timely case study, there are a few notes worth adding when it comes to epidemics themselves. Probably the most important, mentioned only as a small on-screen note, is that these models should account for the amount of time someone with the virus remains infectious. Those who recover (or die) are no longer able to spread it, and so don't factor into the growth equation. The faster the growth, the less this matters, since at each point on the curve most people with the virus will have only contracted it recently, but especially in the long run or with slower growth, any realistic model has to consider this. The other factor, which I was hesitant to even get into here, is the extent to which reported cases reflect real cases.

Generalizing away from epidemics, though, the key upshot is to be aware of phenomena where the rate of growth is proportional to the size of the thing growing. Compound interest, technological progress, population growth, and many other things fit this pattern, and it's shocking how bad our intuitions can be at recognizing what it means.

I think the Big question is about detection biases

So what you're saying is, once everyone is infected, we'll stop getting new cases…brilliant. We'll just have to start tracking numbers for people getting it the 2nd, 3rd, 4th, etc. time.

I played Plague not long ago and it didn't go very well..

Yup, we are doomed. I feel nobody is worried here in LA.

Hopefully everybody will be ok

tbh some of the victims of COVID-19 had tat urge to travel around the world just for the cure, but they ended up by spreading the virus on said country and pretend they didn't did it.

oh wait…

Mr. Aiello says hi

COVID-19 is the disease, the SARS-CoV-2 is the virus. Like AIDS and HIV. "Coronavirus" itself describes the type of virus it is.

ooh im early

I love your videos btw

You might wanna add to this video how the data in the beginning somehow isn't quite exponential like the rest, and how you can detect data tampering from official stats like this (There is a nice reddit thread, that discusses how china faked their official numbers according to a well defined quadratic function to not look as bad)

If you're bold enough and/or have the time.

Thanks for your exceptional content!

Case case cluster cluster boom

Im not going to worry that nobody is worrying, that stupid.😷😂😂😂😂

Nice video. P.S. writing from Italy. Closed in my house, studying math as always 😁

For things like (technological) progress do you think it goes linear, exponential (like Ray Kurzweil thinks) or like a logistic curve? 🤔

https://justfadfada.blogspot.com/2020/03/the-relation-between-coronavirus-and.html

CORONAVIRUS AND MOBILE NETWORK

EVENT 201 GLOBAL PANDEMIC EXERCISE……you may not like the truth when you see it.

So you're telling me my worry should be inversely related to the average person's worry

So how many people will be infected? What is logistic curve saying about it?

ray kurzweil and other futurists should watch this video

I would somehow add "herd immunity" (say, H_i) as variable to the formulas presented in the video . In that case, NOT being exposed to pathogen (in an infected community) is a potential risk factor.

Not great, not terrible..

My Chromecast chokes constantly with audio, exclusively with projection of certain realities.

In my country, the UK, we're just pretending its not happening, as the only maths we seem to care about is the share price of our airlines. Fancy a flight to locked down Italy right now? 40 quid to you, mate.

Fake News panic propaganda is growing. Where is that graph?

There’s some irony in “exponential trends don’t really occur in the real world” (or whatever he said to that effect) due to the fact that cosmology (in particular, the growth of the universe) may be the only place we actually have real exponential growth. (Even then, it’s not clear we do—and probably don’t, technically speaking.)

Great video! Complex ideas presented lucidly. Sent it to high-IQ 13 yr-old grandkid.