We have previously discussed the next iteration matrix and the basic
reproductive number and now this is a continuation on this topic and in fact
we're going to be very specific in learning one of the algorithms or one
of the methods used for the calculation of this quantity. Something that applies for
compartmental models, models that are described by systems of ordinary
differential equations. So here we're going to see how this
calculation relies on computing eigenvalues of a matrix and kind of the signs here
is putting together this matrix, the so-called next generation matrix, but
before we jump in to these technical details, we start this lecture with
nothing less than this, let me just read this to you,
These are not my own words, these are the
wards Bob May and I figured that perhaps you like to share
opening this lecture with his words. The literature on the basic
reproductive number is extensive. This is a key quantity
and the mathematical modeling context of infectious disease and epidemics and I
mentioned here in this slide briefly two main references you will see in some of
the recommended reading some of the required reading that there are more
references. So we had seen this definition before, but let's make sure
we are on the same page. The basic reproductive number R0
is an average the average number of secondary infections
produced by an infective that is introduced or immersed into a completely
susceptible population. Sometimes this infective who starts the outbreak of the
epidemic is referred to as the index case. You may find these exact words, this is a
quote from a paper by van den Driessche
and Watmough from 2008 and another paper by Heesterbeek, there is also reference
to the context of R0 in demography and ecology where R0 is interpreted as
the expected number of female offspring born to one female during her entire lifetime
in many of these mathematical models in ecology the time evolution of a
particular species relies only on modeling the females. Since they are the only
ones that give offspring and therefore that's what keeps the species going. So,
but, these are very, two very relevant definitions to our discussion today so
what we would like to keep in mind is that this quantity serves as a quantification
on the number of secondary cases but it also provides a threshold condition.
And this threshold condition is around one so when R0 is less than one we are going to
think of the scenario which the infectious individuals in fact the
infection is incapable of taking off. This is an area in which the epidemic
dies off, there is actually no outbreak, no epidemic, and what's happening is
that these infectious individuals that are passing on the pathogen, they
failed to replicate or replace themselves. You would like to think of
this as the offspring of the infectious individuals and if on average they
don't produce more than one secondary infection then that's
the idea of being destined to die off. However, when R0 is exceeding one then the
number of infected individuals will increase and this is the scenario we
typically call an epidemic or an outbreak. So this is a threshold
parameter in the context of mathematical models for infectious diseases but it's
important to remark that for the invasion of disease organism the
invasion the sense of how this pathogen would basically invade or take over
a population that is naive in the sense of being completely susceptible, this idea of
R0 it's something that applies exactly in that type of window when the
invasion is being attempted but once the disease has begun to spread the
conditions favor and spread will change and R0 is is no longer a good
measure of transmission. Instead, one must turn to other quantifications of
transmission that there is indeed a named defendant in before we dive into
some these details involved in matrices and eigenvalues let's quote some
extremely simple mathematical model for keeping track of the hope of one
population so we use here in st to the north side of time to use some
population and we are going to drive expression for the rate of change in
population size here we rely on hiring in the mall term for birth
term for death rate this is the simplest description of the rate of change of
population size the notion is you considered the birds and you subtract
the birth the birds are going to have a positive effect on the rate of change
and the deaths are going to have a central detrimental effect birds like
new arrivals into a population and the day at departures right so we can ride a
mathematical expressions of this sort would we hear as him that this is the
simplest way of formulating this type of this type of vehicle dynamics then we
would say be the notes per capita birth great indeed announced per capita they
have great and I writing these expressions were seen they had these
processes the birth process and the death process each of them is
proportional to the total population size great we'll keep this in mind
because the willingness Enix is going to relate to these very simple formulation
so in the spirit of caring how the combination of these are not we have to
convert the epidemiological model which are really well assuming we're working
with it
deterministic described by a system of ordinary differential equations where
there are compartments compartments define in terms of the infection process
or the epidemic process so we need to do is to greeleyville and agree on
basically two sets of compliance one of the sets is going to include
compartments for diseases disease compartments where individuals in those
compartments are hosting the party and therefore they're infected in these
compartments are going to include individuals that have been exposed or
are undergoing latency that means they are infected with HIV infections then
Wednesday progress into the states of being capable of transmitting button on
the battery in the Admin see their infectious and then they're all
individuals the show symptoms or individuals that do not show symptoms
vary all these individuals will be considered part of the compartments of
their disease components and then you can have the rest of individuals into an
apartment so they're not deceased semi-automatic me and this is where we
start throwing out Asian women say all right we're going to sort this into in
this is compartments in a.m. on this is complains so we're going to denote X we
use X here in vector form saying OK access stores in compliance they are
those who is this these different stages of disease and then we stock everything
else into the variable Y now we are going to be fine and operator think of
operators as somehow functions or a higher dimensional space or functions
define spaces of functions and so on so an operator here if he's going to denote
the rate of secondary infections so that this is keeping track of an increase on
the eye disease compartment nvm ok so we are heading towards something that makes
a complete analogy with birth and their prices we just discussed we just
mentioned the birth of their that we have birds as a driver and there as the
partners right so when we are able to separate segregated decompose the
American more into these compartments now would like to keep track for these
variable X have a notion of arrivals and departures
in those arrivals are going to be related to the increment on secondary
infections in being so that was going to be the gain for those this is compliant
in here the operator V is going to denote different types of progression
their quarry and so on
ok so these allows to ride the original into a new form where we have stayed
very wants in terms of the disease compartments X 1023 all the way through
exit in in all those components can be written in terms of if minors but these
FB an operator and then we stack everything else here into these
typically nonlinear
operator denoted by G ok so this is the mall we had originally by other
reorganize or 3 returned so these operators f you know fans here from the
fancy villa's I would call them they are assembling gains and losses and this is
compliance and that's the analogy with the earth and their prices so the
composition of the dynamics of them into
and the and this nation has infected or affected me not be unique so that's
something to keep in mind
we're gonna take a little pass here and we will continue this conversation on
the next slide