The maths of zombies.
Previously people didn't care if you studied epidemiology through mathematics;
all they cared about was keep me healthy. And so the way you'd sell it to people is okay,
let's talk about zombieism. It's the same maths of flu, it's the same maths as
chickenpox, but it spices it up a little. And so what we're going to consider are some zombies,
what's a hallmark of the zombie? Well they move randomly and the zombie does
a random walk. So we can't say where a particular zombie will be at any time;
but a lot of zombies we can actually predict. We use a thing called the diffusion equation,
okay, so we have a population z that's how many zombies are at a location at a particular time.
The zombie are changing over time, zt means rate of change of zombies over time, that's
equal to Dz. Dz tells us how fast the zombies are moving, how fast they're spreading out.
Big Dz, very fast, they're riding motorbikes, small Dz, very slow,
their legs have been cut off. And then we have the Laplacian of z - upside down triangle -
very scary piece of mathematics but it just means random motion. Okay, and from this what you get is
a relation between time of first interaction and that's proportional to how far away you are from
them, space is being x here, and their rate of diffusion, Dz. So t and Dz are related. Time of
first meeting is approximately proportional to how far away I am from the zombie, x,
and how fast they're dispersing, Dz. So if I slow the zombies down, I put pits out, rubble - it's a
good reason for not cleaning your room I always say - so if I halve their rate of motion my time
doubles. Half the rate of motion, double my time. So if it takes 20 minutes for them to get to
me and I put a lot of things out that will stop them it'll take 40 minutes to get to me.
Okay that's that's that's useful, that's useful to know. But what's more useful to know
is x, because time and space are related quadratically - what does that mean? Well if I
double the distance between me and the zombies, because of this x squared time is increased by
four. So I double my distance, let's say 10 metres I go to 20 metres away, my time has gone from
10 minutes to 40 minutes. So it's much better to run away from a zombie than it is to try and slow
them down. I won't judge you! I will be at the head of the people- I know this! I've got a plan
to get out of the way, I'm not trying to stop them, I'm running away. So don't worry, I'll be
a chicken, you'll be a chicken we'll all be a chicken. - (Brady: That does kind of assume you have an)
(infinite ability to run away, whereas if you put an obstacle down an obstacle will be there forever.)
Very true, very true, and that's a good point - and you might also trip over that obstacle, very true.
I mean uh the UK isn't that big. You can keep running- in fact we wouldn't have very
far to run until we hit the the sea from from from where you're living here, Brady. So
I'd just like to say maybe think about moving if a zombie apocalypse happens? This is telling us about
the start of an apocalypse, okay, what should we do? Run away. But you're quite right, we now have to add
in interactions. So what can zombies and humans do when they meet? So we're gonna have zombies
meet humans, zombie plus human. What can come out of their interaction? We can have the zombie kill the
human. So zombie meets a human, zombie survives, not great. The human can kill a zombie, great!
We've survived! We've got a good outcome and a bad outcome, but that's certainly not the worst outcome.
What's the worst outcome? We get bitten and we end up with two zombies. So now what we can do
is rewrite this in terms of our equation. So now we have two populations to keep tabs on: we have the
zombies, they're diffusing, moving randomly, they don't know what they're doing, they're mindless.
And we're going to say that because of these interactions something can happen. We can increase
the zombie population at a rate proportional to what's happening, so let's give these three rates:
this can happen at a rate a, a rate b, and a rate c. All it really means is whichever's biggest is
the one that's most likely to happen. If b was really big then the zombies are ineffectual, you
know, they're flies you're just batting them out away, keep away, go away I don't want you. If c is
really big then zombieism is very infectious, you know, they- saliva can pass it on, very infectious,
we don't want that. So this is the equation we're going to write down from this set up here: zombies
are moving randomly and what we have is that the humans kill the zombies, so we lose some zombies
at a rate b, that's the amount that are killed by the humans; but we gain zombies
at a rate c, okay? That's all that happens, we gain zombies through zombification, we lose
zombies because of humans killing zombies. We have to also account for the humans. So human
population changing over time, I'm also going to consider the humans are moving randomly.
Now generally that's not true, you know where you're going to be most of the time.
But I'm going to suggest that if the dead start to rise you may not go to work that day, you may not
go to school, you might panic and try and escape. So that's what I'm saying here, that humans are
panicking, they're moving randomly, not many people have a zombie plan - the fools. And we have the same
idea here, we put on the kinetic. So what can happen? We can die at a rate a, and we die at the rate c.
So that's that's the entirety of a zombie population written down in two simple equations.
Simple equations - ha! So what do they tell us? Well the first thing we can do is is
simplify this bit of it, so let's have a look at the human equation. This term is always negative.
Now I shouldn't have to tell anyone who's seen a zombie film before - or anyone in fact - but it's
never beneficial for a human to meet a zombie. Yes we can kill them, but it doesn't increase
our population. We are assuming that our zombie pandemic occurs over a short time, because you
know we could have added in birth rates into this. But I'm going to say that most zombie films are
around an hour and a half; it takes nine months to give birth so I'm going to say that birth isn't
really going to affect this equation too much. Because of this term being negative we are always
losing human population - don't meet a zombie. Now let's look at the zombie equation, do this
idea of connect reaction terms. Random motion, and then we put (c minus b)Hz. Now this term can
either be positive or negative - what does that mean? If this is positive, c is larger than b.
The zombies are infecting more people than we're killing, they're very infectious.
If b is bigger than c, we are killing more zombies than they are infecting, this is
negative. So this- again tells us what situation we- to be able to survive we need to kill more zombies
than they are infectious, okay? Again this makes sense, you know, all I'm doing is converting our
knowledge of any good zombie film into mathematics. But this does tell us one more thing, because these
equations are- can be seen as a Fisher equation which is a type of equation in mathematics.
And that equation has a wave structure in it. So what we'll have is our city; we'll have the
zombies being green and we'll put them here, we'll put all of our zombies in a graveyard. And we're
going to say that the rest of our city, the humans are pretty well spread out along our city. If you run
these equations you will find that this spreads out like a wave, invading the blue, either eating
them or turning them. Some of the zombies may die but a lot more of the humans will be converted and
eaten. That's also a big problem, you may want to think about your strategies now. Essentially all of
the humans around you, if you're in a populated place, they're just potential zombies for you.
Hmm, think about that for a second. But what is the speed of this wave? So we'll call this v, it's
move- this wave is is charging through the human population at a rate v. And v is proportional to
the square root of Dz H0 and (c minus b). Okay, some of these we've seen before, some of
them we haven't. Dz we've seen before, that's just the spread of the zombies. You slow the
zombies down, you slow the speed down, makes sense. c minus b, that was this term here. The rate of zombie
production minus the rate of zombie killing. This term can be negative, but it's inside a square root.
That means if we make this negative this can't happen. We need to be more deadly than the zombies
are infectious, if that's possible we stop the infection spreading. Now h0, we've not spoken about
this parameter yet. h0 is the initial population of the humans. If you can isolate yourself, remove
anyone potentially around you, great, you don't have to worry. But if you're surrounded by humans,
all they are to you are potential infections. I'm not saying you should get rid of those humans,
I'm just saying what the maths says Brady. You may choose to interpret it however you so wish. - (Ah see)
(I interpreted that as, I just need to move out to the country, but you're saying I need to kill)
(the humans!) - No I specifically said- I did not say that. I've said you need to isolate
yourself in some way. However true- however- disclaimer, disclaimer, disclaimer! - This episode was supported by
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...and you get extra growth from it, and then you see that in the fact that the bigger
that bulge gets the bigger your black hole gets as well. - One fact I know about eyes, so this is
like where all the blood is flowing into it right? If you're God and you're designing an
eye you might put all of those blood vessels behind where it's actually receiving light?