Now we are going to turn our attention to
a quantity that is extremely important in
the mathematical modeling of infectious
diseases. This quantity is called the reproductive
number and there is certainly a
depth of technical level to carry out calculations
to compute this quantity from a mathematical
model described by
a system of ordinary differential equations.
But before we
take that route of computing and calculating
this quantity, I would like to spend some
time motivating with the context
of... and in many compartmental models this
calculation reduces to a competition that
involves the competition of eigenvalues of
a matrix.
for models that are describing continuous
time there are generalizations on the definition
of their basic reproductive
number and this would involve sophisticated
integrals and especially when you have models
with H structure, you know, on a continuous
scale
and some other type of model described by
integrals or kernels or things of that sort.
So there are generalizations
on the definition of the basic reproductive
number but I would first like to motivate
with something that is very concrete in the
context of matrix models, okay? So, before
we talk in specific detail on an operator,
we are going to spend some time connecting
with material
we brief... that you were briefed on recently
that has to do with matrix models, okay?
So now, we're going to use a matrix model
to describe the evolution of infection in
a population and we're going to make a distinction
as to...
we want to consider stages of infection okay
so I think originally we had used
when we wrote the model it looked like this;
S going into I, right, and we only had one
type, one compartment where we were collecting
all individuals that are carrying the pathogen,
the pathogen that produces the infection,
and that one compartment was called I, was
the letter I.
So now imagine demanding, ok, this is good
for the beginning but now imagine that our
compartment is I sub 1, I sub 2, I sub 3,
and
there is a lot of them
all the way through I sub m. OK, so now imagine
that we can group individuals that are infected
but we can label the levels of infection in
something that I'm calling here infection
stages, okay, and then we can
label those stages with, I don't know, numbers
one through M, discrete numbers, okay? So,
if you only had two of those stages, that's
a very common example, if there is only two
of those then you could say this stage here
has to do with incubation
okay our sometimes this is what you call the
stage in which its individuals are infected...
they actually
are hosting the pathogen. They have been exposed
to the pathogen or infectious agent however
they first undergo this
this stage in which, while they have the pathogen,
they are still not capable of passing on the
pathogen and that capability comes
at another stage, another stage that is called
the infectious. So, here... infections, okay?
They are infectious over there. It almost
doesn't fit. Anyway! So,
this would be I sub one is what we would call
exposed and then I sub 2 is what is called
the infectious stage, right? So, this is an
example
in which there are two of these. In general,
you can say there are m infection stages.
When m equals two, I think we briefly
discussed a model... we briefly discussed
a model before that looked like S E I R, and
then we were saying you can go from S to E,
from E to I, from I to R.
okay and so definitely in this model these
two stages are stages in which individuals
are infected but is the second
stage here that we know that is a group of
individuals capable of passing on infection
okay?
So, that's the context we're working with
and now we're going to formulate this in terms
of a matrix model where every...
okay, so let's suppose that we have a matrix.
Here this matrix is denoted by the letter
K. So this matrix is square, it has M rows
and M columns. So here is K
and K looks like a square, ok? So there are
m rows and m columns. Typically you say that
any entry here in this matrix... well, that
looks... maybe I
should do another entry. So maybe one entry
here that looks kind of like this, ok? You
can say this is the i column ... excuse me...
this is the i row and this...
that is the j column. So you would say this
entry then in general is K sub ij. OK? That's
a very common way of describing entries of
a matrix and we call this matrix
the next generation matrix. OK, and you will
see why in just a second. We are interested
in using this matrix for a model that has
this form. Basically the next...
here the time steps are denoted by n. n denotes
the time unit and this model describes the
time evolution of infected individuals. So
this is telling us...
this is telling us that we can approximate
the time evolution of the infected population
of all stages of infection. We can approximate
that process with a
simple matrix model that we basically know
everything there is to know about this model.
So here the entries of the next generation
matrix denote the average number of new infections,
okay?
Every entry on row... on row i column j denotes
the number of new... average number of new
infections in state...
in stage i from infections that are originated
from stage j, ok? So this, this labeling of
the ij entry of the next generation
matrix has a meaning, okay? It has the meaning
that that number of new infections on a particular
location is produced or is
generated from infections at... at another
stage, ok?
So 
the time scale here... the time step is denoted
with n, right, and these are discrete time
steps and the fact that they are
discrete... the fact that they are integer
numbers and they have step size... ok, so
the time step here, it's going to be connected
to the
so-called generation but... So, you would
like to think of this process in terms of
in terms of offspring, okay? You like to think
of this process as the population
of infected individuals which are now distributed
or grouped into these stages of infection
and how
those stages of infection are progressing
with time but, in the context of infection,
the offspring right, if this was... if this
were in the context of a population of
animals and these were describing this...
this type of matrix models are very common
to describe population...
populations of species or animals for example
where the stages, you know 1, 2, 3 stages
are a simple way of describing three types
of, say, age groups,
for example, right? So you can think of babies
and then juveniles and then adults. Maybe
four groups; you can have elders,
right? This is a very common way of describing,
I don't know, populations of whales or some
type of reptiles and stuff, things of that
sort.
So, what matters in that context of ecology,
of theoretical ecology, is to simply describe
the time evolution on the female
of that species because the females are the
only ones capable of producing offspring and
the offspring is what keeps a
population evolving over time. So here the
analogy is very similar instead of describing
the evolution of reptiles or whales or
turtles or something like that, we are paying
attention... we're making the centerstage
here. This state variable X is keeping track
of the number
of infected individuals on... along all stages,
right? So this x at time n, or time n + 1,
is a vector that has m... m entries... m components,
okay, and so
the evolution here is described not necessarily
on typical time units such as, you know, days
or minutes or weeks or years but rather on
generations of infected individuals
and so you would like to, with this type of
model, think of this process as infected individuals
producing babies, but they're not literally
babies,
they are offspring of infections. So every
time an infected individual in a particular
stage, j for example, is capable of producing
an infected
individual in stage i, that production is
considered infection offspring and that's
precisely what this staff matrix model
describes and is actually a very common way
of describing that type of evolution, ok?
And as... there are some technical parts here
about the generation time of infection and
the the scale here is not necessarily you
know every one day or every 60,
I don't know, every 60 minutes or every 24
hours. These models are... the conceptual
understanding here, the heuristic understanding
is that the time scale is describing
generations of infections and this is not
necessarily
of the same length but the magnitude should
be consistent with the infectious period.
So what is the infectious period? It is the
length of time an individual remained capable
of... remains capable of passing on infection,
okay? For certain diseases, this infectious
period is something in the order of days,
two or three days, some other diseases would
have an infectious period that may be in the
order of weeks, maybe one week or maybe two
weeks, okay, but certainly
the infectious period is not going to be...
well, anyway, so that gives you an idea of
the magnitude, okay, and so what we are saying
with this type of
formulation here is that when you iterate
this model, right, the iteration of the model
means that you take the matrix and multiply
the matrix by a vector.
So you have the vector from the previous time
step, right? So this... what the operation
is going on there is a matrix operation. You
have a square matrix K.
You multiply that matrix by a vector and this...
remember this is m rows m columns and this
vector is m by one. This is giving you a vector
which is M by one and
this vector is your new update. That vector
would be the vector with the number of individuals
in every stage of infection at the next time
step.
But what is the meaning of that vector x of
n, or x of n+1. Well, the jth component, okay,
so the jth component, in other words, x sub
j at time n, that jth component is describing
the
number of infection cases in stage j at generation
n.
OK? J takes values between 1 through m but
here n is the step, is the time step, is the
generation so you have generation zero,
which are the initial conditions, generation
one, generation three, generation four, so
think about generation versus when we're talking
about time, think about yourself, your family,
right, and think about generation versus all
other
time units. For example, I am 39 years old,
so every year I celebrate my birthday. That's
a celebration remembering when I was born.
So there is the time
scale of three-hundred and 65 days, 52 weeks,
twelve months, they're all equal to one year
and every one year, basically now for
thirty-nine times in a row, then I remembered
that I was born on a particular date. However,
I am a member of
one generation. In my family, I'm a member
of three. My mother had offspring equal to
3 individuals, right? So, I have one sister
and one brother. The three of us belong to
one generation, okay? So, even though I am
39 years old, my sister is 35, and my brother
is 30,
and this is a substantial number of years
for each of us, we are... it doesn't matter
how many years, I mean, this is 39 but
if you say it in days, it's 39 times 365;
that's a large number, right, and similar
for 35 x 365 or
30 times 365. Those are large numbers and
if you multiply that by minutes, I mean, this
is a
substantial numbered amount of time. However,
I only count for one generation in my family.
So, my mother
and my father could be considered generation
zero.
They are offspring, my sister, brother, and
myself. We are generation one. My brother
has one daughter and my sister has
one daughter so those are my nieces and they
are members of generation two. Twenty years
from now, my nieces, or maybe there will be
more, you know,
twenty something years from now my children
and my nieces and nephews will have other
children. They will be members of
generation three. So we have generation zero,
generation one, generation two, and generation
three, as you can see
three generations can really extend over time
units that are quite significant, right? There
could be stretches of 10, 20, 30, 40 years,
right,
and all those stretches are within one generation,
okay so just keep that in mind.
Now, so we're going to talk about some mathematics
here. Some of these things are fascinating.
We had already seen in action the power of
eigenvalues of a matrix
and how relevant eigenvalues in a matrix are
for computing powers of a matrix and how relevant
these eigenvalues are in matrix models. So
this is basically going back to that, some
of those facts. Now,
and these are technical considerations, right,
this is something that you really want to
review here, your
linear algebra, okay? These concepts of eigenvalues
and eigenvectors, you really want to be fresh
and up-to-date on those concepts. Another
concept you really want to be fresh
about is this thing; linear independence.
When an n.... here is another buzz word here...
singularity... or non-singularity...
So when a matrix is not singular, there is
a number, there is a plethora of properties
what that are
equivalent. What it means to be non-singular
versus have a nonzero determinant versus two
reduce to the identity matrix versus
to have an inverse versus to have full rank
which means the matrix itself has columns
that are linearly independent
and that means that those columns, linearly
independent columns, are a basis of a subspace,
you know, there is a number,
there is a number of things there. And so
if we make an assumption, if we say suppose
that the next generation matrix is non-singular,
that is a strong
assumption; that is no - there is no free
lunch, okay? So, that is something that that
it is substantial, okay? So,
what what I'm going to discuss, I chose to
discuss this, making this assumption, but
bear in mind that
there is a whole... there is a whole branch
of mathematics called linear algebra and then
there is another branch called numerical
linear... numerical linear algebra, which
is also fascinating. It is very hands-on and
is motivated by very
realistic real-world problems... numerical
linear algebra.
So, not every matrix is non-singluar. There
is certainly... there is certainly a lot of
matrices that are non-singular. Some matrices
are near singular, okay, and
but anyways so if we assume for the sake of
this argument that we are going to make that
I'm going to go through. Let's assume that
the next generation matrix
is non-singular meaning that among many properties
the next generation matrix is going to have
eigenvectors that are linearly independent.
That means that it is possible to write that
matrix in the factorization. In other words,
we will be writing that K, which itself is
square,
can be written as a square matrix times a
diagonal matrix times a square matrix, OK?
That is the other factorization, this this
factorization is beautiful. It's this square
matrix here stores the columns of this matrix
S,
the eigenvectors, and so the columns of S
inverse... well anyway... so you compute S
inverse...
but then the matrix in the middle, this lambda,
is a diagonal matrix, diagonal means there
are zero's here.
There are zero's over there. Now it looks
like a percentage sign or something but
this is... this is the structure, okay? So,
what happens if this is the assumption we
make? Well, if that is the assumption, if
K can be a
factor in terms of the matrix, storing eigenvectors,
which I'm going to call the eigenvector matrix,
and in terms of the eigenvalue matrix, the
matrix storing the
eigenvalues on the diagonal, if that is the
case, then we can write the solution of the
matrix model, so the matrix model is this...
we iterate, right, at every time step we
take the matrix and multiply that matrix by
a column vector and that matrix column vector
multiplication itself gives a new vector and
that is the updated vector we're interested
in,
right, so that model, the matrix model, with
this initial condition, has this solution,
OK?
is the nth power of the next generation matrix
multiplied by the initial condition, the initial
condition being a vector, ok?
Now, what happens? Well, if the matrix is
diagonalizable, meaning that you can write
K as the multiplication of S times lambda
times S inverse, then taking the powers of
K, this is something we have previously discussed,
it reduces to simply
taking the powers here of each of those eigenvalues,
ok? Taking the powers of the entire matrix
is basically reduced
to raising to a power each of the eigenvalues,
OK?
So, what we would like to argue here, so let
me just switch gears, what we would like to
argue is the following. So,
let's say we have... we have this model...
so we're new with this so you're going to
have to bear with me,
so, we're saying is we have X at (n+1) is
K times X... k times X at n and we're saying
that we know the solution of this
being K to the n times initial condition but
that we can write this as S times lambda...
capital lambda to the n
times S inverse times the initial condition...
this being x bar 0, something like that, OK?
So here the important thing is that lambda
n is a diagonal matrix that has lambda 1 to
the n, lambda 2 to the n, all the way through
lambda m to the n.
In other words, this matrix looks like lambda
1 to the n, and then 0, 0, 0, 0, and then
lambda 2 to the n, and then
another bunch of zero's here and across the
diagonal this becomes here an entry m, m,
this is lambda m
raised to the nth power and there are all
zeros, all zeros above the diagonal, all zeros
below the main diagonal,
okay, that's what that lambda to the nth looks
like. So the important thing... the important
thing...
well, one important thing is to ask what is
the role played by... what is the role played
by
lambda sub j, okay, what is the role played
by the distance, by the modulus of lambda
sub j
in X (n)? What we are interested in is on
the solution, this is the solution,
of the matrix model describing the process.
Here is the process of infection,
okay? So, the behavior of this solution is
influenced by the nth powers of the eigenvalue
matrix and so we're wondering, okay,
if we know information on the modulus of every
eigenvalue, so I said something about lambda
j here and maybe I should say
that j goes from 1 through m.
Right, that's what n e j, so what is the role
played by the modulus of the eigenvalue on
the solution of that matrix model,
okay? So, that means... that means that we're
going to look at... I wanted to start visualizing
this in the following way.
Let's suppose, I mean we have m eigenvalues.
I'm just drawing here... I'm just drawing
a picture...
suppose m is equal to 4, ok, so that we can
look at this picture. The eigenvalues in general...
in general every lambda j is a complex number.
Every lambda j lives, resides on the complex
plane,
okay? In general. Now, this is a scenario
that we have here, a sketch of the location
of lambda 1, lambda 2, and
lambda 3 in the complex plane, so the complex
plane... the complex plane which is C with,
looks like a moon or something,
C with a fancy... this is... this means the
complex plane.
You can think of this plane, you have the
horizontal axis which contains... that horizontal
axis here is called real
axis and you have the vertical axis called
the imaginary axis so
every element of this plane, it's say for
example here is lambda 1,
that lambda 1 has the real part of lambda
1 on the horizontal axis and then the imaginary
part
of lambda 1 on the vertical axis. So, every
element of this plane is going to have a coordinate
on the real axis and another
coordintate on the imaginary axis. So, here
is lambda 3, and here is lambda 4.
so each of those, lambda 1, lambda 2, and
lambda 3 are elements of the complex plane
and they each have a component in the real
axis and a component in the imaginary axis.
Now, the modulus. So the modulus is what is
measuring, we use the modulus, for example,
for lambda 1
the distance from the origin in this plane
is measured by the modulus. This would
be the modulus of lambda 1, we now take
here and measure the distance of lambda 2
through the origin so this would be modulus
of
lambda 2, ok, and say this here will be the
modulus of lambda 3,
and from here we measure the distance to lambda
4, this would be the modulus of lambda 4,
okay? So, the modulus is a generalization
of the so-called absolute values. I'm going
to write the note here.
The modulus, in general we say lambda j and
you can.... if you didn't know this was a
complex number, you would say that is the
absolute value of lambda j, well,
in this context we don't call it absolute
value, it's absolute value in spirit, the
same spirit so
that here we call this modulus. And the modulus
of lambda j is measuring... is quantifying
the distance...
the distance... ok, the distance between the
complex number and the plane to the origin
so this is the distance from
the origin and the origin is (0+i) times zero
in other words 0,0… that origin there has
a real part equals zero and has an imaginary
part equals zero.
OK, that's, that is the so-called origin.
So, as you could argue, in this sketch here,
is that with this visualization, that's why
only, I only draw
four eigenvalues but you can argue that even
with this representation and even with I'm
not giving you exactly any precise
information as to what is exactly the real
part, what is exactly the imaginary part but
just by looking
at this figure we can pin down... we can say
which is the
eigenvalue that has the largest modulus and
in this case the eigenvalue with the largest
modulus
and I'm going to maybe draw on top of this
again,
according to my sketch here that eigenvalue
would be lambda 4, so this distance here,
even though it looks very
similar to this, it looks very similar to
the distance of lambda 2 and lambda 3, this
distance for lambda 4 is largest,
okay, that distance is largest. That's going
to be important. So,
lambda 4 has in this example, in this pictorial
representation, this distance is the largest
of the four,
okay, and that is going to be important because
with the largest distance centering... taking
center at... I'm sorry,
I'll begin again, taking center at the origin,
you can poicture as we can draw
a circle, OK? So we can draw a circle, not
the best circle of all, but this certainly
is not something that would fit here in my...
the sketch that I'm doing but it's... would
be something that looks like that... so this
is a...
this is a representation of a circle with
a radius... a modulus of lambda 4, okay, and
cente... center
at 0, Okay? This circle with radius... radius
equal to the modulus of lambda 4 and center
at zero, this circle
contains all eigenvalues, ok? So, if we identify
which eigenvalue has the largest modulus,
then we have this
notion of encircling, okay, taking center
at the origin and taking a radius equal to
the distance... to the largest distance from
the origin and
that guarantees that the circle now will contain
all the eigenvalues.
OK, it will contain all the eigenvalues. So,
there is in this context there is something
called spectral... spectrum, okay,
and more importantly, spectral radius, the
Spectral radius.
Remember, we're working with the diagonal
matrix lambda, which looks like this, this
is lambda 1 all the way
through lambda m, right, there are zeros over
there and then we're going to define this
with this notation
rho of lambda, rho of lambda is going to be
the maximum of the modulus of all eigenvalues.
So, we... and this is a notation, nonmathematical
notation, this is describing what we just
sketched previously, so you consider the modulus
of every eigenvalue and find the one eigenvalue
that has largest,
so you're really maximizing modulus and identify
which of those has largest value.
Modulus is distance so... modulus is a non-negative
real number
and so with this definition we're going to
say that rho of lambda, the spectral rho of
lambda, the maximum or the largest
modulus, this is called the spectral radius
and the reasons for calling this spectral
radius are precisely on
the previous illustration which if you now
define this as the radius of a circle that's
centered at the origin,
then that circle will contain the entire spectrum
which is another name for the set of eigenvalues,
okay?
So, we're going to take... I think this this
conversation will be continued. We have to
talk about matrix norms
but we're going to pause here for a minute.
