Articles, Blog

Exponential growth and epidemics

March 8, 2020


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

  • Reply Tihim Rahman March 8, 2020 at 8:36 pm

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

  • Reply Erville Wright March 8, 2020 at 8:36 pm

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

  • Reply Lunar Flu March 8, 2020 at 8:37 pm

    here from /Pol/

  • Reply Klaa2 March 8, 2020 at 8:38 pm

    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.

  • Reply William Hagler March 8, 2020 at 8:38 pm

    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.

  • Reply Mac Mcleod March 8, 2020 at 8:38 pm

    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.

  • Reply dess March 8, 2020 at 8:38 pm

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

  • Reply DJKokaKola March 8, 2020 at 8:39 pm

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

  • Reply MOONSKIN1976 March 8, 2020 at 8:40 pm

    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…

  • Reply MoonMage March 8, 2020 at 8:41 pm

    rip monetization on this video

  • Reply GWS March 8, 2020 at 8:42 pm

    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.

  • Reply Cody Smith March 8, 2020 at 8:43 pm

    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

  • Reply Barcel March 8, 2020 at 8:43 pm

    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.

  • Reply leo March 8, 2020 at 8:44 pm

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

  • Reply r0b0t slavic March 8, 2020 at 8:44 pm

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

  • Reply TankingShaman March 8, 2020 at 8:44 pm

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

  • Reply jeremy french March 8, 2020 at 8:44 pm

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

  • Reply Erblin Beqa March 8, 2020 at 8:44 pm

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

  • Reply Yaboylemon March 8, 2020 at 8:45 pm

    Nice.

  • Reply Izak Kanter March 8, 2020 at 8:45 pm

    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.

  • Reply 3dEmil March 8, 2020 at 8:46 pm

    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/

  • Reply GM 79 March 8, 2020 at 8:46 pm

    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

  • Reply Garaad Ugaas March 8, 2020 at 8:46 pm

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

    Albert einstein.

  • Reply x_factor March 8, 2020 at 8:47 pm

    Thank you. Amazing video!!!

  • Reply TuringBot March 8, 2020 at 8:47 pm

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

  • Reply AoooR March 8, 2020 at 8:47 pm

    we'll all die

  • Reply Ross G March 8, 2020 at 8:48 pm

    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.

  • Reply smurfyday March 8, 2020 at 8:48 pm

    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.

  • Reply Ghada Elbasty March 8, 2020 at 8:49 pm

    thank you

  • Reply Tom Wesstein March 8, 2020 at 8:49 pm

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

  • Reply MoonMage March 8, 2020 at 8:51 pm

    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

  • Reply Yunkyu Song March 8, 2020 at 8:51 pm

    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).

  • Reply Marnie Frenette March 8, 2020 at 8:52 pm

    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!

  • Reply Peter Brough March 8, 2020 at 8:52 pm

    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.

  • Reply Master of Entropy March 8, 2020 at 8:52 pm

    Dr Mike:
    -Alert not anxious!!!
    -Wash, wash, wash your hands!
    -Chest compression, chest compression, chest compression!!!!

  • Reply Kate Iry March 8, 2020 at 8:52 pm

    me: worrying about the epidemic

    my roommate: hippity hoppity traveling makes me so happy

  • Reply Joe Salimao March 8, 2020 at 8:53 pm

    Great video!

  • Reply Taunter Atwill March 8, 2020 at 8:53 pm

    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. :-))

  • Reply Libertas March 8, 2020 at 8:54 pm

    Didnt know about the growth factor. Thx!

  • Reply marco b March 8, 2020 at 8:54 pm

    Great video!!

  • Reply Charles Nelson March 8, 2020 at 8:54 pm

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

  • Reply ゼィオ先生二位 March 8, 2020 at 8:55 pm

    Nepnep!

  • Reply Lucas Agerskov March 8, 2020 at 8:56 pm

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

  • Reply Matei Stoian March 8, 2020 at 8:57 pm

    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.

  • Reply Smiso Nkosi March 8, 2020 at 8:57 pm

    Who are you?

  • Reply Jason Sebring March 8, 2020 at 8:57 pm

    "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.

  • Reply Rick Rubenstein March 8, 2020 at 8:59 pm

    I'm very interested in knowing what the curve currently looks like within China, where data suggests the growth rate is in fact already decreasing.

  • Reply aashish kapoor March 8, 2020 at 8:59 pm

    Roosevelt : Only thing to fear is fear itself
    Grant(yolo) : well yes but actually no

  • Reply iTeerRex March 8, 2020 at 9:03 pm

    Beautifully done!

  • Reply Master Dementer March 8, 2020 at 9:03 pm

    When epidemics get a mathematical view: (Video)

  • Reply Dan Nuttle March 8, 2020 at 9:03 pm

    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!

  • Reply Théo Léo March 8, 2020 at 9:04 pm

    Nice video man that was cool

  • Reply Thor P.N March 8, 2020 at 9:04 pm

    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.

  • Reply Gribbo9999 March 8, 2020 at 9:05 pm

    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.

  • Reply George Lindsey March 8, 2020 at 9:05 pm

    Fantastic explanation, thank you!!!!!

  • Reply WENKANG WANG March 8, 2020 at 9:05 pm

    Finally

  • Reply tomimn2233 March 8, 2020 at 9:05 pm

    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.

  • Reply vlogbrothers March 8, 2020 at 9:07 pm

    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

  • Reply BoomDot March 8, 2020 at 9:08 pm

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

  • Reply Y K March 8, 2020 at 9:09 pm

    “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.

  • Reply Ode March 8, 2020 at 9:09 pm

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

  • Reply Kalamares FTW March 8, 2020 at 9:09 pm

    siempre cetrero

  • Reply Kristian T March 8, 2020 at 9:09 pm

    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.

  • Reply bharat path March 8, 2020 at 9:09 pm

    thank you for a clear concise explanation

  • Reply whatthree16 March 8, 2020 at 9:10 pm

    wow, that virus is going viral!

  • Reply Blake unage March 8, 2020 at 9:10 pm

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

  • Reply Alberto Mora March 8, 2020 at 9:12 pm

    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

  • Reply Heba Heba March 8, 2020 at 9:12 pm

    I feel bad for the pi's 🙁

  • Reply Spam Email March 8, 2020 at 9:13 pm

    Man made.

  • Reply WesEd17 March 8, 2020 at 9:13 pm

    I love you 3blue1brown

  • Reply Louis Swanepoel March 8, 2020 at 9:13 pm

    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

  • Reply gene dietz March 8, 2020 at 9:14 pm

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

  • Reply Rob Lowery March 8, 2020 at 9:14 pm

    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!

  • Reply 3Blue1Brown March 8, 2020 at 9:15 pm

    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.

  • Reply FedericoLov March 8, 2020 at 9:15 pm

    I think the Big question is about detection biases

  • Reply SuperRayW March 8, 2020 at 9:15 pm

    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.

  • Reply nyanrlz March 8, 2020 at 9:17 pm

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

  • Reply jake liu March 8, 2020 at 9:20 pm

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

  • Reply yingming xu March 8, 2020 at 9:21 pm

    Hopefully everybody will be ok

  • Reply what happened March 8, 2020 at 9:21 pm

    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…

  • Reply Kartoonpanda March 8, 2020 at 9:21 pm

    Mr. Aiello says hi

  • Reply Jon Johnson March 8, 2020 at 9:22 pm

    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.

  • Reply Gregorious Maths March 8, 2020 at 9:22 pm

    ooh im early

    I love your videos btw

  • Reply AngryMurloc March 8, 2020 at 9:23 pm

    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!

  • Reply cente March 8, 2020 at 9:23 pm

    Case case cluster cluster boom

  • Reply D Man March 8, 2020 at 9:24 pm

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

  • Reply peter parker March 8, 2020 at 9:24 pm

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

  • Reply Hans Isbrücker March 8, 2020 at 9:27 pm

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

  • Reply Ahmed Helal March 8, 2020 at 9:28 pm

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

  • Reply You You March 8, 2020 at 9:28 pm

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

  • Reply Adam Filinovich March 8, 2020 at 9:29 pm

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

  • Reply radixvinni March 8, 2020 at 9:30 pm

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

  • Reply Larven Karlsson March 8, 2020 at 9:30 pm

    ray kurzweil and other futurists should watch this video

  • Reply h0ll0wm9n March 8, 2020 at 9:30 pm

    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.

  • Reply TTeh March 8, 2020 at 9:31 pm

    Not great, not terrible..

  • Reply Jeffrey Dedeyne March 8, 2020 at 9:32 pm

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

  • Reply TheDroidBay March 8, 2020 at 9:33 pm

    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.

  • Reply Zero Sum One March 8, 2020 at 9:33 pm

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

  • Reply Alexander Sanchez March 8, 2020 at 9:34 pm

    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.)

  • Reply Fish poem March 8, 2020 at 9:35 pm

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

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