Chapter 8 Modelling Epidemics by xiagong0815

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									Chapter 8 Modelling Epidemics
[First Draft CBPrice November 20th 2011]


8.1 Introduction

Disease results in more deaths than road-traffic accidents, wars and famine combined. The 14th Century
Black Death resulted in the death of about one third of the population in Europe at that time (around 85
million). Our current understanding on the causes of disease focus on various micro-organisms, such as
viruses, bacteria, parasites and fungi. In this chapter we describe one classical model for the population
dynamics of these “disease agents”. Such a model may serve to help us understand how the disease spreads
and also to engineer strategies for disease control, such as vaccination.

What is an “Epidemic”? The traditional definition of an epidemic is that the occurrence of a particular
disease is “in excess of normal expectancy”, for infectious diseases an epidemic is characterised by a sharp
increase in the number of cases over time. A good example is influenza where person to person contact
results in a rapid increase in the incidence of this disease.

In this chapter we shall take a dynamical systems approach, where we consider a population which is
exposed to infection and we shall study under what conditions the infected individuals changes with time.
Our definition of an “epidemic” is a situation where the number of infected individuals increases above the
initial number of infected individuals.

We shall discuss two models. First a model which does not include the contact between individuals in space.
This model focusses on the state of the individuals, whether they are in a “susceptible”, “infected” or
“recovered” (removed) state. The second model explicitly looks at the actual interactions between
individuals, and considers how the disease is spread through inter-individual contact in space.


8.2 The SIR Epidemic Model (Deterministic, Non-Spatial)

This model does not focus on actual interactions between people in space, but looks at a whole population of
people which may be in one of three states: Within this total population there are a number of distinct
classes: (i) The susceptibles S who can catch the disease, (ii) the infected I who have caught the disease, and
(iii) the removed R who may have had the disease and have become immune, or who have the disease and
have been isolated (until recovered). The movement of each individual through this model is indicated in
Figure 1.
This model involves setting up a set of differential equations to determine how the sub-populations S,I,R
vary with time, hence the “SIR Epidemic Model”. Let’s think about how to construct the differential
equations.

The rate at which the disease spreads depends on the contacts between the infected and the susceptible
individuals. We can write this mathematically as        , where r is a parameter. To understand this product,
consider the situations shown in Figure 2. In Fig2(a) we contrast two cases where we have just one infected
(red) but on the left a few susceptibles and on the right many susceptibles. Clearly the rate of infection will
be greater on the right since we have more susceptibles, so that’s why S appears in the expression      . Now,
looking at Fig.2(b) we contrast two cases where we have the same number of susceptibles. On the left we
have a few infected and on the right we have many. Clearly the rate of infection will be greater on the right
since there is more chance of the infection spreading.




That’s why I appears in the expression      . Therefore the change in time of the infected has a term      and
its expression starts off as




The removal of the infected (due to isolation or treatment) depends on the number of infected. This is given
by the term –aI where a is a parameter which is the “removal rate” of the infected. Adding this into the
above expression for the rate of change of infected gives us




Now we have to formulate two additional differential equations. First for the rate of change of the
susceptibles. Since susceptibles become infected at a rate , then the susceptibles must decrease with this
rate, in other words
Finally the increase of the removed is the same as the rate they are removed from the infected which is –aI.
So we have




In conclusion, the set of three coupled ordinary differential equations which describes the SIR model is




We can apply the Euler algorithm to these equations to give




which we can re-write as




which in turn could yield the computer code

               suscep += -r*suscept*infect*deltaT;
               infect += (r*suscept*infect – a*infect)*deltaT;
               recovd += a*infect*deltaT;


8.3 Analytical solutions to the SIR Epidemic Model

It is easy to obtain some useful mathematical analytical solutions to the above system of three ODEs. Let’s
first consider the conditions needed for an epidemic to be produced. Remember our definition of an
epidemic is that the number of infected should increase (at least initially) with time. The rate of change of
the infected is of course       , so if this increases then we need            . So from the differential
equation for I we conclude that
or in other words

In this case, I initially increases and we have an epidemic. This inequality makes sense. If a (the rate of
recovery) is large then we need more susceptibles for the infection to spread. If r (the rate of infection) is
large, then we need fewer suscpetibles for the infection to spread. This inequality is of high significance in
understanding how infections spread and how to prevent this.
A further analytical solution to the system of three ODEs can be found by dividing the second ODE by the
first. This eliminates time and allows us to obtain a relationship between infected I and susceptibles S at any
moment in time. We do not explain the maths here (phew!) but merely state the result of our analysis:




Here I and S refer to the infected and susceptibles at any time, and   and    refer to the initial numbers of
infected and susceptibles at the start of the simulation.


8.4 Phase Plane Description of the SIR Epidemic Model

Typically, a study of the SIR system of equations will include graphs of how S, I, and R change with time.
But we can make other plots, for example for the SIR system a useful graph is obtained by plotting I against
S. This is shown in Figure 3 for a two experiments.




Here the results of two simulation runs are shown. Simulation (1) (red) shows an epidemic. The initial
values of S and I are shown on the right. As the system evolves, it passes along the curve in the direction
indicated by the arrows. Clearly the number of infected initially rises until it reaches a maximum then
decays to a smaller value where I is zero and the epidemic has ended. Simulation (2)(green) shows the
evolution of S and I which corresponds to a non-epidemic. As you follow the curve (arrows) the number of
infected always decreases.

Remember that the initial value of S must be large enough for an epidemic to ensue.
[Next revision: Show the N = I + S line and a family of orbits]
8.5 The SIR Epidemic Model (Spatial, Stochastic)

The model presented in 8.2 above, and developed in 8.3 and 8.4 did not take into account the spatial
distribution of individuals involved in an epidemic. In other words there was no explicit modelling of the
transmission of the disease between individuals. In this section we address this issue and present a simple,
though effective model of the transmission of the disease between individuals. A typical model of disease
transmission involves the direct contact between individuals, and this occurs when the individuals are
present in the same place.

The model presented here divides continuous space (a room, a town, a country) into a grid of cells. Each cell
may contain a number of individuals, though to maintain simplicity, we assume here that each cell contains
one individual. The difference between continuous space and discrete space is shown in Figure 4.




The model assumes that at the beginning of the simulation, each individual in each cell is found in one of
three states, susceptible, infected or removed. A typical starting configuration is shown in Figure 5, where
green represents susceptible and red represents infected. The simulation advances in time by computing the
chance that an individual in a particular state may transit to a new state. For example, an individual who is
in the state susceptible may transit to the state infected if surrounded by two or more individuals who are in
the state infected. On the other hand an individual who is in the state susceptible and who is not surrounded
by any infected will surely remain in the state susceptible.

To develop this model consider a single susceptible individual. Let us denote the chance of infection by
having contact with one infected neighbour as . So the chance of NOT getting infected with this single
contact is (1 - . So if there are N infected neighbours the chance of NOT getting infected with all of these
is
       (1 -                                               . (N times)

ie

So the chance of getting infected is 1 – the chance of not getting infected which is simply




So if the individual is in a susceptible state then we move to the infected state with this chance. This is done
in the following lines of code.
     infectionProb = 1 - (1 - rho)**nrInfected;

     randNum = RandRange(0.0,1.0);

     if(infectionProb > randNum) nextState = 'infected';
     else nextState = 'susceptible';

The first line calculates the infection probability (chance). The second line returns a random number
between 0.0 and 1.0. Every random number has an equal chance of being returned. The if statement decides
whether or not to infect. Let’s see how this works. Say that randNum is 0.5 and infectionProb is 0.6. Then
we choose to infect. But if randNum is 0.7 and infectionProb is 0.6 then we choose not to infect. Since
randNum when called say 100 times will return a value between 0.0 and 1.0, then we expect 60 of these
values to lie below 0.6 and 40 of these values above. So the number of times we will infect is 60 out of 100,
which of course is the probability 0.6.

The removal of infected occurs with a fixed probability . This is done with the following lines of code

     removedProb = alpha;

     randNum = RandRange(0.0,1.0);

     if(rmovedProb > randNum) nextState = 'removed';

The same argument presented above applies to this code.



8.6 Correspondence between the Deterministic and Stochastic Models.

								
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