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Alternative to SIR: Modelling coronavirus (COVID-19) with stochastic process [PART I] - YouTube
Channel: Mathemaniac
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A lot of channels have talked about the SIR
model, including my own, but there is a very
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different way to model coronavirus, which
boils down to how diseases really spread from
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person to person. This is generally known
as branching process, because every individual
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here branches out to some random number of
other individuals. Because this branching
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out is random, as it can have 2 branches,
3 branches, or even not branching out at all,
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we say this is a kind of stochastic process.
Stochastic being a fancier name for random.
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This video will focus on the simplest type
of branching process, known as Bienayme-Galton-Watson
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process, BGW for short. So what does this
BGW say about the spread of coronavirus?
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Before we move on, we need to introduce a
few terminologies. All people on the same
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vertical line would be in the same generation.
We call this orange column the first generation,
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with the number of individuals denoted as
X_1. Similarly, we denote X_2, X_3 and so
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on. Next up, we say these two individuals
from the same branch the offspring of the
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individual in the previous generation that
spreads the disease to them. The reason we
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use these terms is that this process was originally
studied in a different context related to
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reproduction, but that鈥檚 a story for another
time.
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Let鈥檚 focus on the first branching here.
We say that this is a random process, so actually,
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it could stop branching right away, or branch
to 1 individual, or 2, or 3 and so on. Because
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the number of branches is just X_1, the number
of offspring of this first patient, we say
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that X_1 follows a distribution. The distribution
can be more conveniently written as a table.
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The left column represents all the possible
values of X_1, and the right column represents
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the corresponding probability. So the probability
of X_1 being 0 is p_0 and so on. Because these
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are the probabilities, they have to be nonnegative,
and the sum of these probabilities would be
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1.
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The reason why this BGW process is the simplest
branching process is that it assumes two things:
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each branching that you see here are all the
same, which means it all follows the same
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distribution as the first branching. So they
are said to have the same offspring distribution.
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The second simplification is that it assumes
each individual in the same generation spreads
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the disease independently. Independence has
a specific meaning in probability theory,
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summarised in this formula. As a very quick
detour, let鈥檚 see why we have this formula
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for independence.
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To see why this formula has anything to do
with independence of two events, let鈥檚 consider
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this rectangular box as the entire sample
space. And the green and yellow ovals represent
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the events A and B respectively. Then the
overlapping region represents A and B happening
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together. For the intuitive sense of independence,
the probability of B should be the same regardless
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of whether A happens. So if we just focus
on the event A, the proportion of the overlapping
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region should be the same as the proportion
of the yellow bubble within the rectangular
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box itself. This means the ratio of probabilities
on the left side equals the probability of
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B. Just by rearranging, we get this definition
of independence. More generally, if we have
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n independent events, we will have the probabilities
of them happening together to be just the
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product of the individual probabilities.
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Going back to the BGW model, the two questions
that we want to ask is what is the number
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of individuals in the nth generation, but
because this is a random process, it can be
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0, 1 and so on. So actually what we are looking
for should be the distribution of X_n, detailing
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the probabilities of the different values
of X_n. The second question is what鈥檚 called
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the extinction probability. This is the probability
that at some point, the entire generation
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of patients just stop spreading coronavirus
entirely. But how do we get started on these
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two problems?
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The main issue here is that there seems to
be a lot of parameters in this offspring distribution
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to characterise the entire process. These
probabilities here can all change. So we want
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to turn this table to just one single thing.
We do this via something called a generating
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function, which is a way to encode an infinite
amount of data into just one thing. It does
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the encoding like this: the final product
will be a function of a variable, say z. This
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will be a power series in z. The constant
term would be p_0, the coefficient of the
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z term would be p_1, and so on. As a very
simple example to illustrate why this is useful,
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suppose the probabilities are just negative
powers of 2, then using the formula for geometric
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series, we can obtain the generating function
to be just a succinct formula.
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This hopefully explains why generating functions
are actually very useful. Basically, if you
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have all these probabilities in the distribution,
then you can encode it pretty easily to this
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generating function. What鈥檚 not obvious
is the decoding, but it can be done. If you
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are a bit concerned about the rigour of all
these, pause and read this.
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Anyway, another useful way to see this generating
function is that it is the weighted average
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of z^X_1. This is because p_0 of the time,
X_1 is 0, and so z^X_1 is just 1, and you
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get the constant term, and p_1 of the time,
X_1 is 1, and so z^X_1 is just z, and you
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get the z term to have the weight p_1, and
so on. This weighted average is usually called
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the expected value, denoted as E of z^X_1.
All these discussions would also be valid
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for a general X_n, the number of individuals
in the nth generation, just that the probabilities,
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hence the generating function or the weighted
average are all different, which are all exactly
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what we are after. So basically, if we are
given the offspring distribution, then we
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can encode it in the generating function,
which can be written in terms of weighted
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averages, then we will find some way to find
the generating function for X_n, which we
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then decode to find the distribution of X_n,
which is exactly what we want. This is a much
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easier approach than just directly going from
the offspring distribution.
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In this BGW process, we have X_n to be the
number of individuals in the nth generation.
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Because the entire generation comes from the
branchings of the previous generation, X_n
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is the sum of the different copies of X_1,
because each branching is just a copy of X_1.
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The number of copies is X_(n-1), the number
of people in the previous generation.
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So the generating function for X_n, written
in terms of the weighted average, can be written
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in terms of this, by rewriting X_n as the
sum of copies of X_1. This is pretty complicated
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because X_{n-1} is not a fixed value, because
it can vary. So let鈥檚 suppose X_{n-1} is
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a fixed number m, and then we will see what
we can do afterwards. Because we are calculating
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weighted averages, we have to sum up terms
of this form, with the weight being the probability
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that the copies of X_1 being a specific value.
Because we have assumed all the spreading
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here is independent, we can apply this more
general definition of independence. So we
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can change this weight into a product of probabilities
as shown here. Similarly, we can rewrite the
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value here, into a product of these powers.
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Rearranging, we get a product of terms of
the form of weighted values. So if we first
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sum up all the values of a_1, then only these
two terms will be affected. Summing terms
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of this form is precisely the weighted average
of z^(X_1), which is the generating function
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for X_1. Similarly, the same generating function
appears for the other pairs. Since there are
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m pairs, on the condition that X_(n-1) is
really m, the weighted average would be G(z)
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to the m. However, X_(n-1) is really a variable,
this temporary result only happens sometimes.
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More precisely, this G(z) to the m will happen
with exactly this probability. This is again
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in the form of weighted values. And so, this
can be expressed as a weighted average of
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these powers of G(z). This is just the generating
function for X_(n-1), but applied to G(z)
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as an argument. In other words, G_n(z) is
G_(n-1) applied to G(z). This is true for
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all n larger than 1, and so we can apply this
identity again to get G_(n-2) of G of G of
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z. Repeatedly applying this we get the generating
function for X_n is just generating function
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for X_1 iterated n times.
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This means that we have finally cracked the
first question, because given the offspring
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distribution, we can theoretically work out
the distribution of X_n by decoding that generating
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function. As a sanity check that this is correct,
let鈥檚 suppose there will always be 2 branches,
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which means that the offspring distribution
looks something like this, where the probability
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that X_1 equals 2 is 1, and all other probabilities
are 0. This means the generating function
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is z^2. By iterating it n times, the generating
function for the nth generation is z^(2^n),
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meaning that the number of individuals in
the nth generation is 2^n, which is what we
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expected!
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However, the second question will be discussed
in another video, which will be released in
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a few days. What about limitations? Or the
original context that prompted the three mathematicians
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to study this process? Don鈥檛 worry, I will
cover all these in the future.
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If you enjoyed this video, be sure to give
it a like and subscribe to the channel with
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notifications on! See you next time!
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