We continue our discussion of the calculation of the basic reproductive
number, this is part three, and we are now going to work out an example which
will compute the next generation matrix and the eigenvalues of this matrix the
example for the application of this method to compute R0 as the example
is provided by an epidemic model that we call the SEIR model. We were saying in the
title here this includes vital dynamics because we had seen a previous version
of this model in which mu is zero, in which mu is not part of the model
but here
mu denotes the rate of departure out of these population that sends me the notes
the death rate, so out of every compartment of SEIR is always the constant rate
mu that has been subtracted meaning that these are the rate at which individuals
in every compartment are dying due to national natural causes.
We also assume that the deaths, this is an epidemic model was that also is connected
with their model for birth and death
process. Here we have the deaths being
constant and a variable compartment and then we see that all birds all their
rivals into this population are susceptible, and assumed to be susceptible
in such a way that the term that appears here in a separate class denotes
the new arrivals the newborns and they all seem to be susceptible. So it is very
standard model. So the first thing we do is that would like to do the composition
in which we are going to choose according to the status which compartments are
disease compounds in this case the E and I compartment are both compartments for disease
that defines the same variable X and then the operator f, fancy f and fancy b
operators have to be fine in a way which we can derive differential equations
from disease compartments so that will see that in just one second
so here we write X as E, I
meaning that X1 denotes E and X2 denotes I and this way the
dynamics or the model for X is written as operator F minus operator V
meaning here is F here is V with this formulation we do what we do next is to
compute partial derivatives.
So the matrix F is computed
out of partial derivative from the operator F lets take a look at that
operator again, this operator fancy F as a vector has two components
which is Beta S times I/N I is denoted here as X2. What we do
like to do is this first component is considered F sub 1 so we want to
compute derivatives of every component with respect to x 1, and x2 derivative of this expression with
respect to x one will be 0 derivative of this expression with respect to X2 will be BS/N
derivatives of the second component they are both zero.
that's what leads to these two by two matrix one thing to note is that
disease-free equilibrium here is 00, X1, 0 , X2,0 and s=n. We evaluate
these type of Jacobian matrix of that disease free clear then we obtain
these two by two matrix. For the other expression of the other operator which
appears here, this operator we do the kind of same calculations
component computer parcel respect to x 1. First partial will be Kappa + mu, and the
the parcel of that expression with respect to x2 is zero and for the second
component and obtain constants. and when we evaluate those derivatives at the disease free equilibrium
that were obtained the matrix is reduced to this. So we have derived from the four
the SEIR model with births and deaths, we have derived the matrices F&B which are
used to form the next generation matrix. So we have to do that is perhaps no easy
step is that will now that we have the matrix V which is given by
these 2x2 matrix
we have to compute the inverse of that matrix. Again not a trivial step and once
we have that we can multiply F times V inverse and that's how we arrived at
these expressions. This is a 2x2 matrix and what we will do next is to
compute the eigen values of 2x2 to matrix. Ok, so we have, First we observe
that for these matrix called the next generation matrix the entry on row [1,2]
to that will be this entry here. Sorry this entry B over mu plus gamma
that entry 1, 2 is quantified, is the expected number
the average number of secondary infections produced in compartment E by
an individual in compartment I and this production is over the course of the
infection. Remember the first row of this matrix is connecting back to the
first state variable which is X1, which denotes E now the entry in row 1 column 1
that would be Kappa beta divided by quantity mu + kappa times mu plus gamma
so that 1,1 entry denotes
the average number of secondary infections produced in compartment E by an infected
individual in compartment E. So that's a note there
about the meaning of these entries and so the eigen values of this matrix K are zero
and the entry on the 1,1 location. So it is also straight forward to notice that
both of these eigenvalues have 0 imaginary part they are what we considered
real numbers. Precisely they are either 0 or simply positive so it's not
too hard to figure between these two Eigen values what is the modulus modulus
of zero is zero and the modulus of a positive real number is that real number in this
case modulus refuses to be an absolute value and to choose the one that is
largest in modulus is also straightforward that's what we want. We want the spectral
radius of the next iteration matrix and that spectral radius is precisely this
expression kappa times beta over both of these quantities that will be what we
consider the basic reproductive number for the SEIR model with vital dynamics.
And remember the meaning of this quantity is that we can establish at least with this
model this is a simple dynamics, we can establish a threshold so that when R0
exceeds one then we will have what we consider an epidemic and
when it doesn't exceed one then the epidemic dies out. So this is what is
illustrated very straightforward you can code this up R using L soda and
you can choose parameters, you can choose values of the model parameters such that R0
lets go back, you can choose values of kappa, beta, mu,
and gamma such that these when you plug them into this expression they are
greater than one. That is exactly what was done here setting parameter values such that
that expression exceeds 1. In this case it is 2.2 and these are the
numerical solutions of the SEIR model with births and deaths. You can see how all
state variables here are settling into some equilibrium levels that are positive
levels so they this is resembling a scenario that we would call coexistence.
We have used that terminology before and what happens when this is an area in
which the basic reproductive number exceeds one meaning that those that are
infectious are capable of replacing themselves, they're capable of producing
on average more than one secondary infection, but what happens when that
quantity is less than one. Here's an example values of parameters were chosen
such that
R0 is less than one then there is a complete failure on behalf of the
infected individuals to keep up. The infection dies off because they were never
able to have that critical mass of infected individuals so that transmission
really quickly dies off just for an illustration
illustrated here
the total population was chosen to be 10,000 and half of them were chosen to be
susceptible, half of them were chosen to be infected, but that didn't matter
because with parameter values such that R0 is less than one that the infected
population
quickly dies off, this is the red curve in this remake of solutions and
basically entire population settles on their constant population size. You can
also see how all these curves E, I and R the green the red and the blue are
all going to zero so this here is precisely the case in which the
disease free equilibrium is stable. E and I are zero and then solutions that are converging E of T and I of T are converging
are converging to 0 and S is converging to the value of in here that
was chosen to be 10,000. So then this illustrates the role that plays R0
in a very simple model and we have illustrated steps for the calculation of
these spectral radius of the next generation matrix and this concludes our
conversation on the calculation of R0