## Like be an accurate representation of the spread of

Like any model, we can choose the complexity. To begin with, we start with the
simplest model where recovery is the only option. Using simple models to begin
with will allow us to make a good comparison between different simple models
and judge how much correlation there is to the real world. We will be using
examples of the spread of (A DISEASE) in each model to see how it changes.
We take the total population to be constant; no births or deaths are possible.
This is consistent with the course of an epidemic being short compared to an
individuals lifetime, however this is often not the case.

We must make assumptions with our simplest model in order for it to be an
mission of the disease and length of incubation period are made with most
models, no matter their complexity. Firstly, we must set our notation.

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We begin with assuming the disease can be split into three distinct classes:
susceptibles (S), infectives (I) and the removed class (R).

S represents the proportion of individuls susceptible to the disease.

I represents the proportion of infectious individuals who can pass on the disease.

R represents the proportion who have either had the disease and are recovered,
immune, isolted until recovered or dead.

With S(t), I(t) and R(t) as the number of indivuals in each class we can now
make our assumptions on the rate of change of our dependent variables.

The susceptibles are lost at the same rate as the gain in the infective class
which is at a rate proportional to the number of infectives and susceptibles.
Written rSI where r > 0 is a constant parametre.

The rate of removal of infectives to the removed class is proportional to
the number of infectives. Written aT where a > 0 is a contant. Hence 1/a
is a measure of the time spent in the infectious class.

Once a susceptible has caught the disease they are immediatly in the
infected class, i.e the incubation period of the disease is negligible.

A large assumption we take for this model is that all classes are uniformly mixed,
so the probability of each class interracting with the other is equal. In real life
this is very unlikely for most diseases, as often once one contracts the disease,
one distances themselves from others able to contract it.

The simplest model we start with is the IR model where we start off with a
certain number of infected people, I0, and we model their recovery. Here, we
are assuming that no transmission occurs as each individual is either infected
(I) or recovered (R).

An individual may only go through the stages as such: I ? R. For the model
to hold, we assume a constant, ?; the recovery rate, is independent for each
individual. This model can be described by the differential equation

dI = ??I.
dt

The rate at which individuals recover is given by ?I and the negative sign is
essential as recovery reduces the number of infected individuals.

We can solve this differential equation by initgration and using the initial con-
dition at t = 0,I = I0,

AIDS Example

Lets take the (unrealistic) case where there are no susceptibles to be infected
and model the recovery of people living with AIDS. As we have said, there is
no cure for AIDS however individuals on medication supressing the virus means
they do not infect others (if they take their medication as prescribed). To make
the model work simply we assume the medication is the recovery and not those
who have died from the disease. For this example we will take estimated values
from the present day and see how it models the future of the recovery for people
living with AIDS as if there were no susecptibles. From the equation

Black Death Example

For comparison we take the example of the Plague just in London so the popu-
lation is considered closed and we model the recovery. The recovery rate, ?, is
found to be in the region of 0.05. At this time there were no treatments able to
save people from the Plague, and so mortality rates were extremely high. I0 =
Number of living with the Plague

Hence we see that

I0 = 100000
I(t) = 100000 × e?0.05t.

Again, we are only interested when t > 0. In this case, because of the small
recovery rate the disease lasts a lot longer, around 60 years before there is no
one left with the Plague. Similarly to the AIDS example, it seems the number
of infectives reaches zero from the graph but again the model does not allow this.

It is not entirely fair to compare these two examples as one has a much smaller
population than the other and what is regarded as ‘recovery’ differs. However,
it does show that if we do see HIV patients on medication as ‘recovered’ modern
medicine has served us well; both are deadly epidemics that are very destructive
to individuals and society however by being in control of the epidemic and treat-
ing those infected we will be able to stop HIV from spreading. We know that it
will be at least 50 years before HIV will have been eradicated since it first found
fame, much higher than the 3 years shown by this very simplistic model. This
just shows how much of a difference the susecptible population makes to how
we control epidemics. Part of the reason it is so important to educate young
people about the spread of sexually transmitted diseases so that they are less
susceptible to contracting the virus. And logically if there are less people able
to contract the virus the faster the recovery time of the whole population.

For this model we assume that

the population is closed,

individuals can only be either susceptible or infectious,

there is no incubation period and an infected individual may never recover
or die.

An individual may only go through the stages as such: S ? I. This model can
be described by a couple of differential equations,

dS = ??S,
dt

dI = ?S.
dt

Recall, S denotes the number of susceptible individuals in the population and
I the number of infectious individuals. Here ? is called the force of infection,
the rate at which susceptible individuals become infected per unit time. 4 It
is dependent on three factors

the rate individuals make contact with others

the probability that a given encounter is with an infectious individual

the probability that a given encounter with an infectious individual results
in an infection of the susceptible in contact.

To keep the model as simple as possible, we assume the population is well mixed,
i.e any two individuals are equally likely to come in contact with one another.
This means that the rate susceptibles become infectives is proportional to both
frequencies of susceptibles and infectives,

and since the population is fixed,

where N is the size of the population and ? is called the infection parametre.
Using this, we can combine the two differential equations into one by using the
substitution S = N ? I so we then have

Roy Anderson and Roger May are both experts in epidemiology. Their book
Infectious Diseases in Humans (1992) is highly acclaimed and many of their
papers have been published in places such as Nature. They use different notation
than this project/paper as do many other authors in epidemiology. In the book
they use X, Y and Z for this model. They do however differ with the majority in
the fact that they use ?SI to be the infection term. Most mathematicians use,
as in this paper, ? (or alternative notation: ?). Using ?SI as the infection term
is called “mass action” and creates large differences. We will not be discussing
the differences in this paper as it is not hugely relavent to the aims of this
project.

This model is described as logistic growth. In logistic growth a population’s
per capita growth rate gets smaller and smaller as the population reaches its
maximum. Exponential growth is not very sustainable in most cases, it may
happen for a while but realistically the growth rate will plateau and create an
‘S’ shape.

Papers by Kermack and McKendrick (1927, 1932 and 1933) have had a major
influence on the development of the mathematics for modelling the spread of
disease. They are still very relevant in many epidemic situations. The first pa-
per works on the assumption that once a susceptible has contracted the disease
and recovered, the are now in the removed class. It later goes on to create a
model with less assumptions but the model we are about to discuss is often
referred to as THE Kermack-Mckendrick model. 4

A basic problem is to describe the spread of infection within the population
as a function of time. A question that we find is impotrant to ask, is when
to ascertain the epidemic has come to an end. Will this be when all suscepti-
bles have contracted the disease, or when the epidemic dies out by immunity,
recovery and/or mortality before all susceptibles are infected. It is important
to determine whether an infection will spread, and how to develops over time,
specifically when it will begin to decline. Schemtically, each individual goes
through the stages S ? I ? R.

We assume using the model, that all susceptibles in the population are equally
likely to contract the disease. We find the basis of each model as a function of
time:

The critical parametre ? = a/r is sometimes called the relative removal rate
and its reciprocal ?(= r/a) is the infection’s contact rate. From (1) we see that
dS/dt?0asr>0.AlsoS>0andI>0,S?S0 aswemuststartwithoneor
more suseptible individuals for the model to work. Looking at the case where
S0 a/r,

dI =I(rS?a)?0 forall t?0, (9)
dt

we see that for t ? ?, I(t) initally increases and so an epidemic occurs.
We can see that epedemic means that I(t) > I0 for some t > 0.
An important parameter when dealing with epidemics is the basic reproduction
rate of the infection. This is the number of secondary infections producted by
one primary infection given the population is purely formed by susceptibles. It
is given by

R0 = rS0 .
a

If more than one secondary infection is produced from just one primary infection;
R0 > 1 and so an epidemic occurs. We will see that in many epidemic models
R = 1 is the critical value where R 1 implies
an epidemic is possible.

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