Mathematical Modelling of Infectious Disease: A Stochastic Approach

Universityof Wolverhampton
October
2014
Mathematical Modelling
of Infectious Disease: A
Stochastic Approach
Matthew Bickley
0903642

1
Faculty of Science & Engineering
School of Mathematics & Computer Science
Dissertation Title: Mathematical Modelling of Infectious
Disease: A Stochastic Approach
Student Name: Matthew Bickley
Student ID: 0903642
Supervisor: Nabeil Maflahi
Award Title: MSc Mathematics
Presented in partial fulfilment of the assessment requirements for the above award.
This work or any part thereof has not previously been presented in any form to the University
or to any other institutional body whether for assessment or for other purposes. Save for any
acknowledgements, references and/or bibliographies cited in the work, I confirm that the
intellectual content of the work is the result of my own efforts and of no other person.
It is acknowledged that the author of any dissertation work shall own the copyright. However,
by submitting such copyright work for assessment, the author grants to the University a
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perform or show or play the work in public; to broadcast the work or to make an adaptation of
the work).
Signature:
Date: 03/10/2014

2
i. Dissertation Declaration
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Signature: Date: 03/10/2014
Print Name: MATTHEW BICKLEY Student ID: 0903642
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ii. Acknowledgments
I would like to thank my family for putting up with me during my academic career so far, in
the last five years since I started my undergraduate degree at Keele University, but especially
the past two years during my PGCE and postgraduate studies which has been difficult for not
just me, but my family as a whole. I would also like to thank Nabeil Maflahi for supervising
this dissertation and making sure my ideas stay on track and that I get to the point of what I
am trying to show, alongside putting up with my random chats and lengthy meetings when I
am worrying that all is wrong. I would also like to thank the other lecturers at the University
of Wolverhampton who have taught me various high-level topics in the past year to add on to
my always improving mathematical knowledge. I would like to thank my lecturers at Keele
University, specifically Martin Parker, David Bedford, Neil Turner and Douglas Quinney,
amongst others who all taught me the main areas of mathematics that I love. Finally, I would
also like to thank my Bro, Ian. He has been there for me since I meet him at Keele throughout
all of my trials and tribulations. He has been and still is in a similar position to me since we
met, on a personal, academic and career basis. We have been through a lot together and he has
always been there when things were awesome and when they were not so great. Without him,
I may not have had the confidence and vision to make sure I get what I want from my career.
He is my Bro and that will never change. Thanks, Bro. Maybe Ellen too…

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iii. Abstract
Mathematical models are very useful for representing a real world problem. In particular,
these models can be used to simulate the spread of an infectious disease through a population.
However, standard approaches using deterministic rates for the flow of individuals from state
to state can pose problems when the disease in question becomes more complex. Numerical
methods can be used to help analyse a system, but this still does not take away from the fact
that some factors are ignored to allow for solutions to be found.
Stochastic processes have inherent properties that allow for random events to take place.
Markov chains, specifically, allow modelling of an individual within a system where
transitions are described by probabilities of changing state. The probabilities can be calculated
from real world data and due to this, will incorporate a multitude of extra influences that the
deterministic model had to assume insignificant.
After analysing three diseases using standard ordinary differential equations, these three
diseases were then also modelled using a Markov chain. Comparing the results of these
analyses, we find that the stochastic approach did not fundamentally give better or worse
results than that of the traditional method. Although the stochastic method does include
random effects and other factors not explicitly accounted for before, it appears as if the
original assumptions made were justified and made no real different to the results. One of the
adapted models was hugely inaccurate with respect to real life cases, but this was due to the
fact that the original model was also poor. Further work would be to stochastically model
diseases with Markov chains which cannot be solved analytically using standard techniques
and incorporating more and more complexity to each model to allow for the most accurate
results possible.

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iv. Contents
i. Dissertation Declaration......................................................................................................2
ii. Acknowledgments ...............................................................................................................3
iii. Abstract............................................................................................................................4
iv. Contents ...........................................................................................................................5
1. Introduction .........................................................................................................................8
2. Deterministic Modelling – an overview............................................................................10
3. Deterministic Modelling – analysis...................................................................................12
3.1. Model 1 – Chickenpox...............................................................................................12
3.1.1. Compartmental model ........................................................................................12
3.1.2. Assumptions .......................................................................................................13
3.1.3. Representing the model ......................................................................................14
3.1.4. Finding the steady states.....................................................................................15
3.1.5. Finding 𝑹𝟎..........................................................................................................17
3.1.6. Analysing the data ..............................................................................................17
3.2. Model 2 – Measles .....................................................................................................20
3.2.2. Assumptions .......................................................................................................22
3.2.5. Finding 𝑹𝟎..........................................................................................................26
3.3. Model 3 – H1N1 ........................................................................................................30
3.3.2. Assumptions .......................................................................................................32
3.3.5. Finding 𝑹𝟎..........................................................................................................37
4. Overview of deterministic models.....................................................................................41

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5. Stochastic Processes – an overview ..................................................................................44
6. Stochastic Modelling – adaptations and analysis ..............................................................48
6.1. Model 1 adaptation – Chickenpox.............................................................................48
6.1.1. Stochastic model.................................................................................................48
6.1.2. Transition matrix ................................................................................................49
6.1.3. 𝒏-step transition..................................................................................................53
6.1.4. Limiting distribution...........................................................................................54
6.2. Model 2 adaptation – Measles ...................................................................................56
6.3. Model 3 adaptation – H1N1.......................................................................................67
7. Results and model comparisons ........................................................................................77
7.1. Model 1 comparisons.................................................................................................79
8. Conclusions .......................................................................................................................82
8.1. Conclusion and critical evaluation.............................................................................82
8.2. Further work...............................................................................................................84
9. References .........................................................................................................................86
10. Bibliography ..................................................................................................................92
11. Appendices ....................................................................................................................93
11.1. Appendix A ............................................................................................................93
11.1.1. Appendix A1.......................................................................................................93
11.1.2. Appendix A2.......................................................................................................94
11.1.3. Appendix A3.......................................................................................................95
11.1.4. Appendix A4.......................................................................................................97
11.1.5. Appendix A5.......................................................................................................98

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11.1.6. Appendix A6.....................................................................................................100
11.1.7. Appendix A7.....................................................................................................101
11.1.8. Appendix A8.....................................................................................................102
11.1.9. Appendix A9.....................................................................................................103
11.1.10. Appendix A10...............................................................................................104
11.1.11. Appendix A11...............................................................................................105
11.2. Appendix B...........................................................................................................106
11.2.1. Appendix B1.....................................................................................................106
11.2.2. Appendix B2.....................................................................................................109
11.2.3. Appendix B3.....................................................................................................112
11.3. Appendix C...........................................................................................................114

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1. Introduction
Mathematical models are a widely used system to represent problems in the real world. They
can be implemented represent almost anything from elements of the natural sciences; such as
physics, engineering and biology (PHAST 2011, Adeleye 2014), to computer science (Neves
and Teodoro 2010) and social sciences including business and economics (Reiter 2014) and
sociology (Inaba 2014). They can also represent how parts of language can be used (Pande
2012) and how education can adapt (Stohlmann et al. 2012). Mathematical models are
primarily used to study the effect of specific components and parameters involved in the
models, and use this and further analysis to predict behaviour and outcomes of some initial
input.
Models that represent biological situations can again be specified into particular areas; one of
which has been and will continue to be vitally important to the real world – mathematically
modelling infectious disease. Diseases have existed as long as life itself and will only
continue to be a problem to humans and other organisms for a long time to come. It is
therefore imperative that diseases can be outsmarted – that is to say we need to be able to
know, or at least predict with some certainty, exactly how a disease behaves. This is where
mathematical models come in. Using past data and specific discovered or known traits of a
particular disease, we can apply modelling knowledge to calculate how the disease will
spread, how infectious or deadly it may be, and how best to ensure the disease has the
minimal effect on the population.
However, these infections do not always behave the same from one outbreak to another, even
when looking at the same specified strain of the disease (Schmidt-Chanasit 2014, in Hille
2014). Other factors which may be very difficult or impossible to model can affect the
solutions in an unpredictable way. These may be but are not limited to environmental,
temporal, climate or population effects. This is where stochastic processes can be applied.
Stochastic processes are methods which inherently involve some amount of probability or
randomness, which diseases can show signs of when a small number of individuals infected

9
cause an outbreak (Spencer 2007). In this way, the solution to a stochastically-involved
system gives probabilities that certain events will have occurred, and exactly what this means
in the long-term. Although this can initially give the impression of misleading outcomes, it in
fact shows the variety that the system can produce, giving us a better overview of the situation
and then hopefully allowing us to pick the best real-world solution.
This study will look at a variety of diseases, from those which are not deadly but can cause
uncomfortable living, to those which have high mortality rates and or can reoccur in the same
individual over a period of time. At first, a standard albeit complex deterministic approach
will be used for each disease – a standard mathematical model. Each model will then be
adapted to include a stochastic basis for the disease in question, hopefully showing a similar
or if possible, better picture of the disease’s traits. Every model will be compared to real-
world data to see the appropriateness of each model and then to each other, to see overall if a
stochastic approach is better and if so, to what extent compared to the complexity increase
from standard systems. This will bring together knowledge about mathematical modelling (of
diseases) with that of stochastic models and processes. This application assumes that
stochastic properties can be applied to diseases that are usually modelled using standard
deterministic approaches which can help either represent the models better, in an alternative
way giving the same or similar results or give entirely different information which could be
used advantageously.
For the analysis of disease in this research, three diseases will be picked that can be
represented by a standard, deterministic mathematical model. The diseases were picked from
a list that can be modelling using standard techniques (CMMID 2014). Chickenpox, which
causes no (or few) deaths from the disease was picked to start with, followed by measles, a
disease with deaths and maternal immunity, and then the final disease, the H1N1 virus was
picked to allow for disease reoccurrence in the model and also to allow modelling and
comparisons of a disease which is usually almost non-existent at most times, but then
becomes very prevalent within populations when outbreaks occur.

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2. Deterministic Modelling – an overview
“A model is a simplified version of something that is real.” (Schichl 2004 p.28). Taking this
and applying it to a mathematically specific interpretation gives that a mathematical model is
a simplified version of a real-world event that helps us to solve an existing problem.
Mathematical models of infectious diseases are those which help us see what a disease is
doing and how best to combat the infection. William Hamer and Ronald Ross were the
earliest known pioneers of infectious disease modelling, giving a basis to today’s current
models in the early part of the twentieth century. Within the next two decades, further
research, data and scientific knowledge had allowed them, along with other researchers to
develop their own epidemic model which showed a clear and logical relationship between
individuals in a population – they could be susceptible, infected or immune. This is the root of
the susceptible-infected-recovered (SIR) model (Weiss 2013). This simplistic model is
effectively just three states, where an individual fits into one and only one. Movement of
individuals between the states is described by deterministic parameters that help calculate the
rate of change of the sizes, or proportions, of these states. Analysing this gives an easily
interpretable system which shows the dynamics of individuals moving through the system,
which in turn, shows the basic properties that a disease exhibits to cause an outbreak.
This model can be applied to a wide variety of diseases, but mainly those which transfer when
humans, or any other animals, directly come into contact with each other via touch or close-
proximity sneezing and coughing, for example (Weiss 2013). So modelling this way does not
(easily) allow for diseases which are purely airborne or waterborne, for example, to be
simulated. Adaptations would be needed, making new types of systems to represent these
types of infections. Focusing just on those diseases which can transmit when the host in
question is coming into contact with another allows for much simpler analysis yet efficacy.
In the last century, these models have been developed to include many more compartments,
extra parameters that depend on disease traits and far more complicated systems of equations
due to this. Extra states have been added in certain systems such as ‘exposed’, where

11
individuals belong who have the disease but are not infectious yet, ‘carrier’, where individuals
belong who can pass on the disease but never themselves suffer from it (and likely keep the
disease for life) and ‘quarantine’, where individuals who have the disease are placed so as to
no longer pass on the disease but still need time to recover. Limitations on these states can
also be introduced, such as a space limit in quarantines so that once the state is ‘full’, no more
individuals can enter and must stay put in these rest of the system. Parameters involved in the
system normally depend on the compartments considered, but some are added to existing
models such as a vaccination rate or a separate birth rate that takes into account all individuals
born with the disease.
These representations can also be extended from a standard system of linear ordinary
differential equations with deterministic, explicit parameters, to more complex systems such
as partial differential equations which look at spatial variances and systems that have implicit
or time-based parameters. The more complicated the original disease is, the more complex the
system of equations must be to accurately represent the virus. However, analytical solutions
are only feasible on the simpler of models. Numerical methods must be used on the most
complex, which have their advantages and disadvantages. Some converge to solutions very
quickly but the sizes of errors during the calculations are larger than others. Methods with the
smallest of errors are the best possible approximation to the solutions of these systems, but
convergence is slow and the method, albeit doable, is intricate.

12
3. Deterministic Modelling – analysis
3.1. Model1 – Chickenpox
To start with, modelling a disease than can be represented with a very simple model will give
a strong basis for the later, adapted model and comparison. Chickenpox, otherwise known
medically as varicella, which is a relatively basic and mild disease caused by the varicella-
zoster virus (VZV) spreads quickly and easily between individuals (NHS 2014a) but is rather
simplistic to model – it only very rarely (directly) causes deaths in the human population.
Symptoms of chickenpox are similar to that of influenza and often include sickness, high
temperatures, aching muscles, headaches, loss of appetite and most obviously, rashes of spots
which causes itching and irritation (NHS 2014b). These spots are the main way of the
infection spreading from person to person – and open spots are especially infective. Close
contact with individuals who have chickenpox themselves is the main cause of mass spread of
the disease, so reducing contact will lower the possibility that any individual will be able to
pass it on.
3.1.1. Compartmental model
Assuming the simplicity of this model, we can represent it with one of the simplest
compartmental models; that being, one that only includes one extra state besides the usual
three. This is the ‘susceptible-exposed-infectious-recovered’ model, or the SEIR model. This
is an adaptation of the ‘susceptible-infected-recovered’ model, or the SIR model, as it allows
for individuals who gain chickenpox to harbour the disease before being able to spread it
(exposed). Also, the ‘infected’ state is renamed ‘infectious’, to lessen the confusion as to what
individuals are capable of doing with the disease in each state. This fits with chickenpox, as
individuals will go through this stage before infecting others. After looking at how the
infection spreads, we can construct the first standard black-box model (Fig. 3.1. and Fig.
3.2.):

13
Figure 3.1.: Compartmental model for chickenpox
Or more mathematically:
Figure 3.2.: Compartmental model for chickenpox with mathematical symbols
3.1.2. Assumptions
The following assumptions can be made about this model with this disease. The aim of these
assumptions is to make the model viable and to enable a solution to be found, when at the
same time not taking away too much from the original traits of the disease and general
complexity of the disease transmission.
Development
rate
Transmission
rate
Susceptible Exposed
Births
Natural
Deaths
Natural
Deaths
Infectious
Natural
Deaths
Recovery
rate
Recovered
Natural
Deaths
𝜎𝛽
𝑆 𝐸
𝜇
𝜇𝜇
𝐼
𝜇
𝛾
𝑅
𝜇

14
 No individual is born with the virus (all individuals initially enter the susceptible
state).
 The population has a life expectancy of 𝜇, so that the natural mortality rate is
1
𝜇
.
 A closed population, so that the birth rate is also
1
𝜇
.
 The virus has a latency period of
1
𝜎
.
 A constant proportional population implies the individuals in the system add up to 1
(individuals either are susceptible, infected or have recovered, and nothing else):
𝑆 + 𝐸 + 𝐼 + 𝑅 = 1 (1)
where 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑆( 𝑡) = 𝑆, 𝐸( 𝑡) = 𝐸, 𝐼( 𝑡) = 𝐼 and
𝑅( 𝑡) = 𝑅.
3.1.3. Representing the model
Using the above assumptions, with 𝛽, the transmission rate being the product of the contact
rate and probability of a successful transmission, 𝛾, the recovery rate, and 𝜎, the development
rate, we can construct the following ordinary differential equations (ODEs) to represent the
rate of change of each compartment in the model:
𝑑𝑆
𝑑𝑡
= 𝜇 − 𝛽𝑆𝐼 − 𝜇𝑆
(2)
𝑑𝐸
𝑑𝑡
= 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸
(3)
𝑑𝐼
𝑑𝑡
= 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼
(4)
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 − 𝜇𝑅
(5)
where 𝛽, 𝜎, 𝛾, 𝜇 > 0.

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3.1.4. Finding the steady states
At the steady states to this system of equations, there will be no change to the proportions
represented by 𝑆, 𝐼 and 𝑅 as time continues to pass. Therefore, the rates of change are equal to
zero. Hence:
𝑑𝑆
𝑑𝑡
= 0,
𝑑𝐸
𝑑𝑡
= 0,
𝑑𝐼
𝑑𝑡
= 0,
𝑑𝑅
𝑑𝑡
= 0
Therefore:
𝜇 − 𝛽𝑆𝐼 − 𝜇𝑆 = 0 (6)
𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 = 0 (7)
𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 = 0 (8)
𝛾𝐼 − 𝜇𝑅 = 0 (9)
Rearranging all of (6) to (9) to make 𝐼 the subject gives:
𝐼 =
𝜇(1 − 𝑆)
𝛽𝑆
(10)
𝐼 =
( 𝜎 + 𝜇) 𝐸
𝛽𝑆
(11)
𝐼 =
𝜎𝐸
𝛾 + 𝜇
(12)
𝐼 =
𝜇𝑅
𝛾
(13)
If 𝐼 = 0 (no virus present), (10) ⟹ 𝑆 = 1
(12) ⟹ 𝐸 = 0
(13) ⟹ 𝑅 = 0
Checking in (1), 𝑆 = 1, 𝐸 = 0, 𝐼 = 0 and 𝑅 = 0 clearly satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 = 1.

16
This gives the trivial steady state:
( 𝑆, 𝐸, 𝐼, 𝑅) = (1,0, 0, 0) (14)
If 𝐼 ≠ 0 (virus present), then Maple confirms the following solution (along with the trivial
steady state):
>
Checking in (1), 𝑆 =
( 𝜎+𝜇)( 𝛾+𝜇)
𝛽𝜎
, 𝐸 =
𝜇( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇))
𝛽𝜎(𝜎+𝜇)
, 𝐼 =
𝜇( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇))
𝛽(𝜎+𝜇)(𝛾+𝜇)
and 𝑅 =
𝛾( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇))
𝛽(𝜎+𝜇)(𝛾+𝜇)
satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 = 1 (manually checked).
This solution along with some manual working leads to the non-trivial endemic steady state:
( 𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗) =
(
( 𝜎 + 𝜇)( 𝛾 + 𝜇)
𝛽𝜎
,
𝜇( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇))
𝛽𝜎(𝜎 + 𝜇)
,
𝜇( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇))
𝛽(𝜎 + 𝜇)(𝛾 + 𝜇)
,
𝛾( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇))
𝛽(𝜎 + 𝜇)(𝛾 + 𝜇) )
(15)

17
3.1.5. Finding 𝑹 𝟎
For a closed population (which we have here), the critical value at which the virus becomes an
epidemic is when it exceeds
1
𝑆
. So, for this model:
𝑅0 =
1
𝑆
=
𝛽𝜎
( 𝜎 + 𝜇)( 𝛾 + 𝜇)
If 𝑅0 is above 1, the theory suggests that an epidemic starts:
𝑅0 =
𝛽𝜎
( 𝜎 + 𝜇)( 𝛾 + 𝜇)
> 1
⟹ 𝛽𝜎 > ( 𝜎 + 𝜇)( 𝛾 + 𝜇) (16)
Interpreting this inequality, by reducing the transmission rate or increasing the mortality rate
or recovery rate, we can cause a disease to die out in this instance. Increasing the mortality
rate is unethical within most populations, especially for those diseases affecting humans
(culling of other species can be introduced on a case by case basis), and the recovery rate is
generally unchangeable due to traits of the virus, along with the latency period due to the
properties of the disease. Hence reducing 𝛽, so in turn, reducing the contact rate is the best
way to prevent an epidemic in this model. Keeping those affected by the chickenpox away
from others is by far the most effective way of reducing the spread.
3.1.6. Analysing the data
Assuming a standard scenario for chickenpox (an average rate of spread), we take an estimate
for the basic reproduction number, 𝑅0, as equal to 3.83 for England and Wales (Nardone et al.
2007). It is stated that this is just an estimate, with a 95% confidence interval (CI) given as
(3.32– 4.49). By taking a mid-range value, we can model for the average spread in the
United Kingdom (UK). The latency period,
1
𝜎
, is given as ten to fourteen days (Knott 2013),
so we take the average of twelve days. The recovery period,
1
𝛾
, is also reported as ten to
fourteen days (Lamprecht 2012), so again, we take the average of twelve days. Average life

18
expectancy at birth in the UK is approximately 81 years (WHO 2012), so this is taken to be
the value of
1
𝜇
. All parameters are rates, so all must be converted to the same units. Numerical
analysis will be taken at daily time periods with step size ℎ =
1
5
over the period of 365 days,
so all parameters are multiplied or divided as needed to give each value in terms of days. We
can substitute these values into 𝑅0 =
𝛽𝜎
( 𝜎+𝜇)( 𝛾+𝜇)
and rearrange, giving us a 𝛽 value
(transmission rate) of 0.319426 (six significant figures). This system has been analysed using
the 4th-order Runge-Kutta method for a more robust approach, with an initial susceptible
individual proportion, 𝑆(0) of 0.999 and infected proportion, 𝐼(0) of 0.001 as a starting point
(see Appendix B1 for data).
Figure 3.3.: Chickenpox 4th-order Runge-Kutta numerical analysis (with steady state
values)
As Fig. 3.3. shows, the chickenpox outbreak eventually subsides but does produce a worrying
epidemic initially. After around forty days, the exposed and infectious proportion starts to rise
rapidly and then both peaking just before day 100 at around 20%, giving approximately 40%
of the population having chickenpox at this point. This would most certainly be classified as
an epidemic if the outbreak spread across a national population, such as the entire UK. The
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150 200 250 300 350 400
Proportionofpopulation
Time (days)
S
E
I
R
S*
E*
I*
R*

19
data again shows that a majority of people have the virus as early as day 100, as half the
population have entered the recovered state at this point. From the year-long graph, it show
the levels of all four compartments fluctuate somewhat, with 𝑆 ever decreasing and 𝑅 ever
increasing, and 𝐸 and 𝐼 showing the peak where the outbreak is at its worst before lowering
back down again. The 𝑆 and 𝑅 states eventually level out in the long-term, moving toward the
respective dotted lines shown, giving a steady proportion of the population who are present in
each state.
By substituting in the values for 𝛽, 𝜎, 𝛾 and 𝜇 in (𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
) and using Maple to work
them out (see Appendix A1), we can find out the behaviour of the model analytically as 𝑡
tends to infinity (also shown in Appendix B1 as 𝑡 → ∞):
(𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
) = (0.261097,0.000299789,0.000299667,0.738304)
So eventually, there is a constant 26.1% of the population that are susceptible, 73.9% that
have had chickenpox at some point and recovered, and a negligible amount suffering from
chickenpox at any time (< 0.06%).

20
3.2. Model2 – Measles
Now, a disease with an extra state is introduced to try to see if the adaptation later will work
with a different basis model. Measles, otherwise known as morbilli, English measles or
rubeola, and not to be confused with rubella (German measles), is a disease caused by a
paramyxovirus. This disease is a little more complex to model as individuals are more often
than not born with maternal immunity to measles which lasts sometime into the second year
of life (Nicoara et al. 1999) and can cause deaths directly from the disease in the human
population. Measles has a relatively high 𝑅0 value, meaning it is more likely to cause
epidemics but spread can be reduced significantly by use of vaccinations (CDC 2014).
Symptoms of measles are similar to that of a cold, with red eyes, high temperatures and spots
appearing in the mouth and throat (NHS 2013). These spots are the main cause of the spread
of measles; tiny droplets are ejected from the nose or mouth when sneezing or coughing
respectively (NHS 2013), and then infect others close by or land on surfaces or objects. In
turn, these are then touched the hands of others and through improper and irregular hand-
washing, this causes the infection to successfully transmit. Being in close proximity with
individuals who have measles is the main cause of mass infection, so removing contact and
introducing vaccinations will lower the possibility that any one individual will be infected.
We can represent measles with a slightly more complex compartmental model; that being, one
that only includes one extra state besides the previously used four. This is the ‘maternal
immunity-susceptible-exposed-infectious-recovered’ model, or the MSEIR model. This is an
adaptation of the SIR model, as it allows for individuals who gain measles to harbour the
disease before spreading it (exposed). Again, the ‘infected’ state is renamed ‘infectious’, to
lessen the confusion. Also, a maternal immunity state is added. Individuals born enter this
category initially before be able to contract measles (being susceptible). This fits with
measles, as individuals will have immunity for a small time before losing it and entering a
standard SEIR part of the model. Vaccinations are also allowed in this model; individuals
who are vaccinated are only those in the susceptible state and will move into a recovered

21
position after the immunisation, preventing them catching the disease (all vaccinations are
assumed to be 100% effective). After looking at how the infection spreads, we can construct
the second standard black-box model (Fig. 3.4. and Fig. 3.5.):
Figure 3.4.: Compartmental model for measles
Figure 3.5.: Compartmental model for measles with mathematical symbols
Recovery
rate
Transmission
rate
Exposed Infectious Recovered
Maternal
Immunity
Susceptibl
e
Births (with
passive
immunity)
Natural
Deaths
Natural
Deaths
Natural
Deaths
Virus
Deaths
Natural
Deaths
Virus
Deaths
Natural
Deaths
Development
rate
Loss of
passive
immunity
Gain of active immunity
(vaccination)
𝛾𝛽𝐼
𝐸 𝐼 𝑅𝑀 𝑆
𝜇
𝜇 𝜇 𝜇
𝜒
𝜇
𝜒
𝜇
𝜎𝛿
𝜅
𝑋

22
When mathematically representing this model, we need to create a new state. A new death
state, 𝑋, is introduced to account for those who die due to the disease only.
3.2.2. Assumptions
The following assumptions can be made about this model with this disease. Again, the aim of
these assumptions is to make the model viable and to enable a solution to be found, when at
the same time not taking away too much from the original traits of the disease and general
complexity of the disease transmission.
 No individual is born with the virus, but all are born with maternal immunity for some
length of time to it (all individuals initially enter the maternal immunity state).
1
𝜇
.
1
𝜇
. This only includes individuals born
that replace those who die naturally, so the birth rate relates to every state except 𝑋 (so
in the first ODE, 𝜇 multiplies every state but 𝑋 – see below).
 The virus’ mortality rate is
1
𝜒
, so individuals who die from the virus directly enter the
𝑋 state at this rate.
1
𝜎
.
 Vaccinations are only administered to susceptible individuals, not those who may have
the virus but are not showing symptoms or capable or passing the virus on (exposed)
nor infants with passive immunity from their mother (maternal immunity).
 Vaccinations are assumed to be 100% effective from the moment they are
administered (individuals who receive it cannot catch measles at any point in the
future).
 Deaths caused by the virus have the same prevalence in both the exposed and
infectious individuals (virus mortality is the same whether you are exposed or
infectious).

23
 To allow for the constant proportionality of the model, those who die of the disease
must be modelled to be allowed to ‘die again’ naturally. Although, this would not
happen in real life, it allows the model to maintain the same amount of individuals in
the system and those who do ‘die again’ from the virus death state will ‘re-enter’ as an
individual with maternal immunity. The low rate of virus and natural deaths within the
length of a long-term analysis makes this error relatively small, albeit still present.
(individuals either have maternal immunity, are susceptible, exposed, infectious, have
recovered or have died from measles, and nothing else):
𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (17)
where 𝑀( 𝑡), 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡), 𝑋( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑀( 𝑡) = 𝑀, 𝑆( 𝑡) = 𝑆,
𝐸( 𝑡) = 𝐸, 𝐼( 𝑡) = 𝐼, 𝑅( 𝑡) = 𝑅 and 𝑋( 𝑡) = 𝑋.
Using the above assumptions, again with 𝛽, the transmission rate being the product of the
contact rate and probability of a successful transmission, 𝛾, the recovery rate, and 𝜎, the
development rate, but additionally 𝛿, the rate at which individuals lose their (passive
maternal) immunity, and 𝜅, the constant number of individuals immunised, we can construct
the following ODEs to represent the rate of change of each compartment in the model:
𝑑𝑀
𝑑𝑡
= 𝜇 − 𝛿𝑀 − 𝜇𝑀
(18)
𝑑𝑆
𝑑𝑡
= −𝛽𝑆𝐼 + 𝛿𝑀 − 𝜅𝑆 − 𝜇𝑆
(19)
𝑑𝐸
𝑑𝑡
= 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸
(20)
𝑑𝐼
𝑑𝑡
= 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼
(21)
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 + 𝜅𝑆 − 𝜇𝑅
(22)
𝑑𝑋
𝑑𝑡
= 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋
(23)

24
where 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒, 𝜅 > 0.
Again, at the steady states to this system of equations, there will be no change to the
proportions represented by 𝑀, 𝑆, 𝐸, 𝐼, 𝑅 and 𝑋 as time continues to pass. Therefore, the rates
of change are equal to zero. Hence:
𝑑𝑀
𝑑𝑡
= 0,
𝑑𝑆
𝑑𝑡
= 0,
𝑑𝐸
𝑑𝑡
= 0,
𝑑𝐼
𝑑𝑡
= 0,
𝑑𝑅
𝑑𝑡
= 0,
𝑑𝑋
𝑑𝑡
= 0
Therefore:
𝜇 − 𝛿𝑀 − 𝜇𝑀 = 0 (24)
−𝛽𝑆𝐼 + 𝛿𝑀 − 𝜅𝑆 − 𝜇𝑆 = 0 (25)
𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 = 0 (26)
𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 = 0 (27)
𝛾𝐼 + 𝜅𝑆 − 𝜇𝑅 = 0 (28)
𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 = 0 (29)
Rearranging (24) to make 𝑀 the subject gives:
𝑀 =
𝜇
𝛿 + 𝜇
This shows that 𝑀 is independent of any other states, and is only dependent on the parameters
given from the virus itself. More specifically, 𝑀 is independent of 𝐼, whether directly or
indirectly through another state, so the 𝑀 state will be the same for both steady states,
regardless whether an infection is present (𝐼 ≠ 0) or not (𝐼 = 0).

25
Rearranging all of (25) to (29) to make 𝐼 the subject and substituting 𝑀 gives:
𝐼 =
𝛿𝜇 − ( 𝛿 + 𝜇)( 𝜅 + 𝜇) 𝑆
𝛽( 𝛿 + 𝜇) 𝑆
(30)
𝐼 =
( 𝜎 + 𝜇 + 𝜒) 𝐸
𝛽𝑆
(31)
𝐼 =
𝜎𝐸
𝛾 + 𝜇 + 𝜒
(32)
𝐼 =
𝜇𝑅 − 𝜅𝑆
𝛾
(33)
𝐼 =
𝜇𝑋 − 𝜒𝐸
𝜒
(34)
If 𝐼 = 0 (no virus present), (30) ⟹ 𝑆 =
𝛿𝜇
( 𝛿+𝜇)( 𝜅+𝜇)
(32) ⟹ 𝐸 = 0
(33) ⟹ 𝑅 =
𝜅𝑆
𝜇
=
𝛿𝜅
( 𝛿+𝜇)( 𝜅+𝜇)
(34) ⟹ 𝑋 =
𝜒𝐸
𝜇
= 0
Checking in (17) (using Maple), 𝑀 =
𝜇
𝛿+𝜇
, 𝑆 =
𝛿𝜇
( 𝛿+𝜇)( 𝜅+𝜇)
, 𝐸 = 0, 𝐼 = 0, 𝑅 =
𝛿𝜅
( 𝛿+𝜇)( 𝜅+𝜇)
and
𝑋 = 0 satisfy 𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1:
>
>
>
>
>
>
>

26
( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = (
𝜇
𝛿 + 𝜇
,
𝛿𝜇
( 𝛿 + 𝜇)( 𝜅 + 𝜇)
, 0, 0,
𝛿𝜅
( 𝛿 + 𝜇)( 𝜅 + 𝜇)
, 0)
(35)
This steady state can be simplified even further – no infection in the population makes
vaccinations unnecessary. So we can set 𝜅 = 0, causing everyone to stay in the 𝑀 or 𝑆 states,
with no-one moving to the 𝑅 state via the infection or vaccination. This makes the steady state
become:
( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = (
𝜇
𝛿 + 𝜇
,
𝛿
𝛿 + 𝜇
, 0, 0, 0, 0)
steady state) (solution omitted due to size, see Appendix A2):
>
Checking in (17), 𝑀 = 𝑀∗
, 𝑆 = 𝑆∗
, 𝐸 = 𝐸∗
, 𝐼 = 𝐼∗
, 𝑅 = 𝑅∗
and 𝑋 = 𝑋∗
(see Appendix A2)
satisfy 𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (manually checked).
This solution along with some manual working leads to the non-trivial endemic steady state
(see Appendix A2):
( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝑀∗
, 𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗) (36)
For a closed population (which again, we have here), the critical value at which the virus
becomes an epidemic is when it exceeds
1
𝑆

27
𝑅0 =
1
𝑆
=
𝛽𝜎
𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒)
𝑅0 =
𝛽𝜎
𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒)
> 1
⟹ 𝛽𝜎 > 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) (37)
Interpreting this inequality, again, the simplest way of reducing the epidemic is to reduce the
transmission rate (as this only appears on the left hand side so reducing this quantity will not
affect the right hand side) or increasing the mortality rate or recovery rate (as these only
appear on the right hand, so similar to above). As before, increasing the mortality rate is
unethical and the recovery rate is generally unchangeable, along with the latency period in
this new model due to the properties of the disease. Hence reducing 𝛽, so in turn, reducing the
contact rate is the best way to prevent an epidemic in this model. Notice that the vaccination
number does not appear in the value of 𝑅0, so the rate at which we vaccinate people has no
effect of preventing the actual outbreak of an epidemic. It appears as if this value only helps
prevent the spread once an epidemic has begun. Keeping those affected by measles away from
other individuals in the population is by far the most effective way of reducing the spread.
Assuming a standard scenario for measles (an average rate of spread), we take an estimate for
the basic reproduction number, 𝑅0, as between 12 and 18 for the United States (CDC 2014).
By taking the mid-range value of 15, we can model for the average spread in the United
States (US). The latency period,
1
𝜎
, is given as seven to fourteen days (CDC 2009c), so we
take the average of 10.5 days. The recovery period,
1
𝛾
, is reported as three to five days plus a
few days for the virus to completely subside (CDC 2009c), so again, we take the average of
approximately seven days. Average life expectancy at birth in the US is approximately 79
years (WHO 2012), so this is taken to be the value of
1
𝜇
. Individuals lose maternal immunity
sometime between twelve and fifteen months (Nicoara 1999), so we take the average of 13.5

28
months. Individuals are vaccinated at approximately 91.9% coverage per year (CDC 2013)
with a CI of (90.2% − 92.0%), so we can then work out a rate per day for 𝜅. Approximately
one or two individuals per every thousand die directly due to measles (CDC 2009b) so we
take the average of 1.5 per thousand and then convert this for the parameter 𝜒. All parameters
are rates, so all must be converted to the same units. Numerical analysis will be taken at daily
time periods with step size ℎ =
1
5
over the period of 365 days, so all parameters are multiplied
or divided as needed to give each value in terms of days. We can substitute these values into
𝑅0 =
𝛽𝜎
𝜎𝛾+( 𝜇+𝜒)( 𝜎+𝛾+𝜇+𝜒)
and rearrange, giving us a 𝛽 value (transmission rate) of 2.14431
(six significant figures). Again, this system has been analysed using the 4th-order Runge-Kutta
method, with an initial maternally immune proportion, 𝑀(0) of 0.1, an initially susceptible
proportion, 𝑆(0) of 0.899 and infected proportion, 𝐼(0) of 0.001 as a starting point (see
Appendix B2 for data).
Figure 3.6.: Measles 4th-order Runge-Kutta numerical analysis (with steady state values)
As Fig. 3.6. shows, like chickenpox, the measles outbreak eventually subsides but again
produces a worrying epidemic initially, and this time, far worse than chickenpox. After
around only ten days, the exposed and infectious proportion starts to rise rapidly and then
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150 200 250 300 350 400
Time (days)
M
S
E
I
R
X
M*
S*
E*
I*
R*
X*

29
both peaking just before day 25 or so at around 45% and 20% respectively, giving
approximately 65% of the population having measles at this point. This would most certainly
be classified as another epidemic in the population. The data again shows that a majority of
people have the virus or have been vaccinated as early as day 40, as half the population have
entered the recovered state at this point. From the year-long graph, it shows the levels of all
compartments fluctuate somewhat, with 𝑀 always decreasing, 𝑆 mostly decreasing, 𝑅 mostly
increasing and 𝑋 always increasing but only slightly, and 𝐸 and 𝐼 showing the peak where the
outbreak is at its worst before lowering back down again. Like before and as predicted by the
steady state, the all states eventually level out in the long-term, moving toward the respective
dotted lines shown, giving a steady proportion of the population who are present in each state.
By substituting in the values for 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒 and 𝜅 in (𝑀∗
, 𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗
) and using
Maple to work them out (see Appendix A3), we can find out the behaviour of the model
analytically as 𝑡 tends to infinity (also shown in Appendix B2 as 𝑡 → ∞):
(𝑀∗
, 𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗
)
= (0.0111266,0.0666666,0.000159502,0.000106306,0.921910,0.0000314982)
So eventually, there is a constant 6.7% of the population that are susceptible, 92.2% that have
had chickenpox at some point and recovered or have been vaccinated, and a negligible
amount suffering from chickenpox at any time (< 0.026%). Approximately 0.0031% of the
population will have died from measles in a constant long-term population and about 1.1%
will have maternal immunity at any time.

30
3.3. Model3 – H1N1
Finally, a disease with a standard SEIR basis is proposed for analysis but with added loss of
natural immunity; that is, individuals can ‘loop’ in the model and gain the disease more than
once after contracting the virus or being vaccinated. H1N1, known fully as influenza A
(Haemagglutinin type 1-Neuraminidase type 1) or popularly (and incorrectly) as swine flu
(Flu.gov 2014), has exactly this property and exhibits a once-per-lifetime (on average) mass-
outbreak rate, which allows for easier analysis. The last two recorded major outbreaks were in
2009 and 1918 (known as Spanish flu). This disease is still complex to model as individuals
can gain H1N1 twice or more and like measles, it can directly cause deaths in the human
population. Symptoms of measles are that of seasonal flu, but much more extreme (Flu.gov
2014). The main contagious ways of passing the disease on is by coughing or sneezing, which
directly enters another person’s body or is transmitted via surfaces and improper hygiene
routines (CDC 2009d). Being in close proximity with individuals who have flu or related
symptoms is the main cause of spread, so removing contact and making sure individuals get
regular flu vaccinations will lower the chance that anyone will become sick.
We can represent measles with the most complex compartmental model presented in this
research; that being, one that includes one extra state besides the standard three but allows for
‘looping’. This is the SEIR model again, but includes a rate that allows recovered individuals
can become susceptible again, making it a slightly adjusted model known as the SEIRS
model. This fits with H1N1, as individuals will have immunity from vaccinations or earned
immunity from catching the disease for a time; individuals who are vaccinated are only those
in the susceptible state and will move into a recovered position after the immunisation (again,
all vaccinations are assumed to be 100% effective at the time of immunisation). After looking
at how the infection spreads, we construct our final black-box model (Fig. 3.7. and Fig. 3.8.):

31
Figure 3.7.: Compartmental model for H1N1
Figure 3.8.: Compartmental model for H1N1 with mathematical symbols
When mathematically representing this model, we need to create three new states. Firstly, just
as before, a new death state, 𝑋, appears to account for those who die due to the disease only.
Secondly, a ‘bin’ state, 𝐵, is created on the right of the model to account for individuals who
lose their active immunity and get H1N1 more than once. We could continue to represent this
with a loop (as in the previous diagram) but it would be impossible to analyse exactly how
𝛿
𝜅
𝛾𝜎𝛽𝐼
𝜒
𝑆 𝐸 𝐼
𝜇
𝑅
𝜇
𝜒
𝜇 𝜇 𝜇
𝛿
𝑋
𝐷 𝐵
Loss of active immunity
Gain of active immunity (vaccination)
Recovery
rate
Development
rate
Transmission
rate
Virus
Deaths
Susceptible Exposed Infectious
Births
Recovered
Natural
Deaths
Natural
Deaths
Natural
Deaths
Virus
Deaths
Natural
Deaths

32
many cases had occurred, let alone how many individuals got the disease at some point (at
least once). The individuals who enter the ‘bin’ state are replaced by another identical
individual in the system from a ‘dummy’ state, 𝐷, entering the susceptible state where they
are vulnerable to the disease again. The rate that individuals leave the 𝑅 state to enter 𝐵 is
exactly the same at which they leave the 𝐷 state and enter 𝑆, and is modelled to only allow
each individual to leave 𝐷 and enter 𝑆 when the corresponding individual has left 𝑅 to enter
𝐵. The loop at the top is therefore removed, and the corresponding rate, 𝛿, is placed on the
arrows in question. The dotted line along the bottom of the diagram represents that 𝐵 affects
𝐷 but no individual actually travels between the two states. No deaths occur in these states (as
the individuals who die when in the ‘bin’ state are accounted for elsewhere in the system by
another state) and the two new states are not needed to be included in the sum for constant
proportionality (see final assumption in section 3.3.2.).
3.3.2. Assumptions
The following assumptions can be made about this model with this disease. Again, the aim of
these assumptions is to make the model viable and to enable a solution to be found, when at
the same time not taking away too much from the original traits of the disease and general
complexity of the disease transmission. This is the most complex of the three models (due to
the possibility of ‘cycling’ round the model via loss of active immunity) so the assumptions
below are comparatively more simple.
 No individual is born with the virus (all individuals initially enter the susceptible
state).
1
𝜇
.
1
𝜇
.
 The virus’ mortality rate is
1
𝜒
.
1
𝜎
.

33
 Vaccinations are only administered to susceptible individuals, not those who may have
the virus but are not showing symptoms or capable or passing the virus on (exposed).
 Vaccinations are assumed to be 100% effective at the moment they are administered
(however individuals who receive it could lose immunity and catch H1N1 at another
point in the future).
 Deaths caused by the virus have the same prevalence in both the exposed and
infectious individuals (virus mortality is the same whether you are exposed or
infectious).
 As before, to allow for the constant proportionality of the model, those who die of the
disease must be modelled to be allowed to ‘die again’ naturally. Although, this would
not happen in real life, it allows the model to maintain the same amount of individuals
in the system and those who do ‘die again’ from the virus death state will ‘re-enter’ as
an individual with maternal immunity. The low rate of virus and natural deaths
compared to the overall length of a long-term analysis makes this error relatively
small, albeit still present.
(individuals either are susceptible, exposed, infectious, have recovered or have died
from H1N1, and nothing else, with those in the bin or dummy state being accounted
for somewhere else in one of the compartments):
𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (38)
where 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡), 𝑋( 𝑡), 𝐷( 𝑡), 𝐵( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑆( 𝑡) = 𝑆, 𝐸( 𝑡) = 𝐸,
𝐼( 𝑡) = 𝐼, 𝑅( 𝑡) = 𝑅, 𝑋( 𝑡) = 𝑋, 𝐷( 𝑡) = 𝐷 and 𝐵( 𝑡) = 𝐵.
Using these assumptions, again with 𝛽, the transmission rate being the product of the contact
rate and probability of a successful transmission, 𝜎, the development rate, 𝛾, the recovery rate,
𝛿, the rate at which individuals lose their (active) immunity, and 𝜅, the constant number of
individuals immunised, we can construct the following ODEs to represent the rate of change
of each compartment in the model:

34
𝑑𝑆
𝑑𝑡
= 𝜇 − 𝛽𝑆𝐼 + 𝛿𝑅 − 𝜅𝑆 − 𝜇𝑆
(39)
𝑑𝐸
𝑑𝑡
= 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸
(40)
𝑑𝐼
𝑑𝑡
= 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼
(41)
𝑑𝑅
𝑑𝑡
= 𝛾𝐼 − 𝛿𝑅 + 𝜅𝑆 − 𝜇𝑅
(42)
𝑑𝑋
𝑑𝑡
= 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋
(43)
𝑑𝐷
𝑑𝑡
= −𝛿𝑅
(44)
𝑑𝐵
𝑑𝑡
= 𝛿𝑅
(45)
where 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒, 𝜅 > 0.
The final two equations here ((44) and (45)) are not included in the constant proportionality
equation (38) nor are they included in the solutions to the system (using Maple or otherwise).
This is because setting those equations to zero would imply the 𝑅 state must be zero at any
trivial or endemic state found. Although this does make sense in long term as everyone would
eventually leave the 𝑅 state one way or another, either by dying naturally and then being
replaced by a new individual in the system to account for the equal birth and death rates, or by
losing their active immunity and ‘looping’ back to enter the system as a susceptible
individual. As the terms in (44) and (45) are repeated from other equations, the system still
holds that the sum of equations (39) to (43) is one. The system would also be unsolvable for
𝐷 or 𝐵 and these states do not appear explicitly in the system of ODEs (an infinite number of
solutions). These final two equations are only included when solving the system numerically
using the 4th-order Runge-Kutta method (see Appendix B3). The values of 𝐷 and 𝐵 will be a
fraction of 𝑅 (multiplied by negative and positive delta respectively) and 𝐵 specifically will
show the proportion of individuals having the possibility of getting the H1N1 virus more than
once.

35
Again, at the steady states to this system of equations, there will be no change to the
proportions represented by 𝑆, 𝐸, 𝐼, 𝑅 and 𝑋 as time continues to pass. Therefore, the rates of
change are equal to zero. Hence:
𝑑𝑆
𝑑𝑡
= 0,
𝑑𝐸
𝑑𝑡
= 0,
𝑑𝐼
𝑑𝑡
= 0,
𝑑𝑅
𝑑𝑡
= 0,
𝑑𝑋
𝑑𝑡
= 0
Therefore:
𝜇 − 𝛽𝑆𝐼 + 𝛿𝑅 − 𝜅𝑆 − 𝜇𝑆 = 0 (46)
𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 = 0 (47)
𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 = 0 (48)
𝛾𝐼 − 𝛿𝑅 + 𝜅𝑆 − 𝜇𝑅 = 0 (49)
𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 = 0 (50)
Rearranging all of (46) to (50) to make 𝐼 the subject gives:
𝐼 =
𝜇 + 𝛿𝑅 − ( 𝜅 + 𝜇) 𝑆
𝛽𝑆
(51)
𝐼 =
( 𝜎 + 𝜇 + 𝜒) 𝐸
𝛽𝑆
(52)
𝐼 =
𝜎𝐸
𝛾 + 𝜇 + 𝜒
(53)
𝐼 =
( 𝛿 + 𝜇) 𝑅 − 𝜅𝑆
𝛾
(54)
𝐼 =
𝜇𝑋 − 𝜒𝐸
𝜒
(55)

36
If 𝐼 = 0 (no virus present), (53) ⟹ 𝐸 = 0
(55) ⟹ 𝑋 =
𝜒𝐸
𝜇
= 0
Rearranging (54) to give 𝑅 in terms of 𝑆 and then substituting into (51) gives:
𝑅 =
𝜅𝑆
𝛿+𝜇
⟹ 0 =
𝜇+𝛿(
𝜅𝑆
𝛿+𝜇
)−( 𝜅+𝜇) 𝑆
𝛽𝑆
⟹ 𝑆 =
𝛿+𝜇
𝛿+𝜅+𝜇
⟹ 𝑅 =
𝜅
𝛿 + 𝜅 + 𝜇
Checking in (38) (using Maple), 𝑆 =
𝛿+𝜇
𝛿+𝜅+𝜇
, 𝐸 = 0, 𝐼 = 0, 𝑅 =
𝜅
𝛿+𝜅+𝜇
and 𝑋 = 0 satisfy 𝑆 +
𝐸 + 𝐼 + 𝑅 + 𝑋 = 1:
>
>
>
>
>
>
( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = (
𝛿 + 𝜇
𝛿 + 𝜅 + 𝜇
, 0 ,0,
𝜅
𝛿 + 𝜅 + 𝜇
, 0)
(56)
Again, this steady state can be simplified further – as before, no infection in the population
makes vaccinations unnecessary. So we can set 𝜅 = 0, causing everyone to stay in the 𝑆 state,
with no-one moving to the 𝑅 state via the infection or vaccination. This makes the steady state
become:
( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = (1, 0, 0,0, 0)

37
steady state, solution omitted due to size, see Appendix A4):
>
Checking in (38), 𝑆 = 𝑆∗
, 𝐸 = 𝐸∗
, 𝐼 = 𝐼∗
, 𝑅 = 𝑅∗
and 𝑋 = 𝑋∗
satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 =
1 (manually checked).
This solution along with some manual working leads to the non-trivial endemic steady state
(see Appendix A4):
( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗) (57)
For a closed population (which again, we have here), the critical value at which the virus
becomes an epidemic is when it exceeds
1
𝑆
𝑅0 =
1
𝑆
=
𝛽𝜎
𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒)
𝑅0 =
𝛽𝜎
𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒)
> 1
⟹ 𝛽𝜎 > 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) (58)

38
This gives the exact same 𝑅0 value as the previous measles model, due to similar traits of the
diseases and models (vaccination and immunity loss). Interpreting this inequality, again, by
reducing the transmission rate or increasing the mortality rate or recovery rate, we can cause a
disease to die out in this instance. Just as before, increasing the mortality rate is unethical and
the recovery rate is generally unchangeable, along with the latency period in this new model
due to the properties of the disease. Hence reducing 𝛽, so in turn, reducing the contact rate is
the best way to prevent an epidemic in this model. Notice here that again the vaccination
number does not appear in the value of 𝑅0, so the rate at which we vaccinate people has no
effect of preventing the actual outbreak of an epidemic. It appears as if this value only helps
prevent the spread once an epidemic has begun. Keeping those individuals with the H1N1
virus away from others is clearly yet again the most effective way of reducing the spread of
any endemic that may occur.
Assuming a standard scenario for H1N1 (an average rate of spread with one epidemic per
lifetime), we take an estimate for the basic reproduction number, 𝑅0, as 2.6 for the US (Barry
2009). The latency period,
1
𝜎
, is given between one and four days (Balcan et al. 2009, CDC
2010b), so we take the average of 2.5 days. The recovery period,
1
𝛾
, is reported as three to five
days (Asp 2009), so again, we take the average of approximately four days. Average life
expectancy at birth is the same as before in the US; 79 years (WHO 2012), so this is taken to
be the value of
1
𝜇
. Individuals lose active immunity at around 5% per year (Greenberg et al.
2009), so we take this for our value of 𝛿. Individuals are vaccinated for H1N1 and flu at
approximately 91 million vaccinations per year (Drummond 2010), so accounting for a daily
rate and the population size of the US, we can then work out a rate per day for 𝜅.
Approximately 12270 individuals died due to the 2009 outbreak (CDC 2010a), so again, we
can find the rate of death per day for the parameter 𝜒. All parameters are rates, so all must be
converted to the same units. Again, numerical analysis will be taken at daily time periods with
step size ℎ =
1
5
over the period of 365 days, so all parameters are multiplied or divided as
needed to give each value in terms of days. We can substitute these values into 𝑅0 =

39
𝛽𝜎
𝜎𝛾+( 𝜇+𝜒)( 𝜎+𝛾+𝜇+𝜒)
and rearrange, giving us a 𝛽 value (transmission rate) of 0.650147(six
significant figures). Again, this system has been analysed using the 4th-order Runge-Kutta
method, with an initial maternally immune proportion, 𝑆(0) of 0.999 and infected proportion,
𝐼(0) of 0.001 as a starting point (see Appendix B3 for data). This analysis includes 𝐵 and 𝐵∗
to show the rate at which individuals loop round the system.
Figure 3.9.: H1N1 4th-order Runge-Kutta numerical analysis (with steady state values)
As Fig. 3.9. shows, like the two diseases before, the H1N1 influenza outbreak eventually
subsides but again, a worrying epidemic ensues. After around twenty days, the exposed and
infectious proportion starts to rise rapidly and then both peaking at day 40 at around 12% and
18% respectively, giving approximately 20% of the population having this flu at this point.
This would certainly be classified as another epidemic in the population. The data shows that
a majority of people have the virus or have been vaccinated as early as day 50, as half the
population have entered the recovered state at this point, however, the level of recovered
people quickly drops back below 50% by about day 65. This is due to the fact that
individuals are constantly losing active immunity to H1N1. From the year-long graph, it
shows the levels of all compartments fluctuate somewhat but all have hit their respective
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150 200 250 300 350 400
Time (days)
S
E
I
R
X
B
S*
E*
I*
R*
X*
B*

40
steady level by about day 130. This type of disease is a faster-hitting but quicker-settling one,
due to the traits of the virus and how it interacts within a population.
By substituting in the values for 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒 and 𝜅 in (𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗
) and using Maple
to work them out (see Appendix A5), we can find out the behaviour of the model analytically
as 𝑡 tends to infinity (also shown in Appendix B3 as 𝑡 → ∞):
(𝑆∗
, 𝐸∗
, 𝐼∗
, 𝑅∗
, 𝑋∗
) = (0.384615,0.0574299,0.0918750,0.465119,0.000961296)
Therefore eventually, there is a constant 38.5% of the population that are susceptible (through
not having the virus or having it at least once and losing their active immunity), 46.5% that
have had H1N1 at some point and then either recovered or have been vaccinated, but this time
a significant amount of people suffering from it at any time (1.5%). Approximately 0.096%
of the population will have died from H1N1 in a constant long-term population, proving it to
be the most deadly of the three diseases in this research.

41
4. Overview of deterministic models
All three models show stable, long-term, analytical solutions. In comparison to the actual data
from real life outbreaks, two of the three models hold up particularly well (Table 7.1. and
Table 7.2.) and the latter two models predict some amount of death which can be used to help
discuss the extent at which vaccination or removal from the population needs to be
implemented. However, can these models be improved upon to provide better estimates? Or
are there other ways to represent the same information to give the same output? Are there any
advantages to doing this?
All dynamical systems that are represented in a similar way to those above are subject to the
‘Evolution Rule’. That is, if the state currently occupied in the system has pre-defined rate at
which the next state is entered, we define the system as deterministic. Whereas, if the change
in state is given as a probability, so that there is a chance that the subject in question can go to
one state or another, or even more than that, and not split off into both or any subset of them,
this is defined as a stochastic (or random) system (Meiss 2007). For the same initial starting
point, analysed at the same point in discrete time (whether a finite time period or an infinite
time steady state value), the standard deterministic model will always give the same output,
every time. This is due to the defined rates of the system, the parameters calculated before
analysis and the very same ones used in the analytical solution to the ODEs. The stochastic
model can, however, account for random processes which can occur during analysis. This is
done by calculating non-deterministic probabilities which are then applied as transition
probabilities between each state. Some of these may be one – forcing a state change in that
situation, whereas some could be zero – meaning there is no chance of moving to the state in
question. Other probabilities must between these two values, which is where the random
chance element comes into play. For example, if an individual is in a state which has two
exits; one to a state with probability of a third and the other with transition probability of two
thirds, it is unknown at the next step which state the individual will be in. Of an overall
population, though, the differences between analyses can be cancelled out to give a rather
(hopefully) accurate description of the overall behaviour of a system on a large scale.

42
One of the main problems with deterministic models is as stated before; the lack of
accountability for chance from the same starting point. During different time periods, diseases
can have different rates of infection and this makes it difficult to decide on exactly what the
rate involved will be (Yorke and London 1973). This also is a problem when analysing
diseases which have quiet periods and then sudden epidemics or those which have not
occurred in a long time. Current deterministic models of some populations exhibiting certain
diseases have problems when the population dynamics are not standard; for example, feral pig
population dynamics in Australia vary over time and in space due to the populations searching
for optimal feeding ground (Dexter 2003). This highly affects the outcomes of a deterministic
system, such as when this population is suffering from food and mouth disease (FMD) and a
model needs to account for exactly what point the dynamics are at and how they may change.
This can be represented by probabilistic elements, hence the application of a stochastic model.
However, applying these elements into the standard setup of deterministic ordinary
differential equations by Pech and Hone (1988) and Pech and McIlroy (1990) turned out to be
too complex – a stable equilibrium point was never reached for the model (Dexter 2003). This
was mainly due to the fact that the feral pig dynamics vary wildly depending on two other
factors – vegetation density and kangaroo density, alongside their own population’s density.
Simulation of these densities when random processes can affect the outcome left a constant
outbreak of FMD present in the system. Evaluation of the system leads to a suggestion that
the combination of both deterministic and probabilistic elements side-by-side causes complex
behaviour that is difficult to model and get meaningful and clear results from. Dexter agrees
that the untested assumptions of his model could substantially change the behaviour in the
dynamics – an intrinsic problem of all deterministic models. This is why I wish to test a
purely stochastic approach. This way, I will have two comparable models on each disease,
where any overlapping features that could cause common errors or deviations between the
models are eliminated as much as possible. This will allow a comparison of which method
seems better or easier, without underlying factors that are inherent in any sort of hybrid model
becoming significant.
Höhle et al. (2005) state that they wanted to take an SEIR model and extend it stochastically
to analyse previously undertaken disease transmission experiments in a more detailed way.
They do this by taking existing work on diseases involving Markov chains and applying it to

43
data from Belgian classical swine fever virus (CSFV). Like Dexter (2003), but this time with
a more standard SEIR model comparable to those presented earlier in this research, spatial
elements are considered to account for dynamics within the populations affected by CSFV.
This model gets very complex very quickly, and two models are proposed – one that deals
with data with missing time elements and one that only looks at data with complete entries for
all fields. The first model has stochastic elements which allow for contact heterogeneity; that
is that not all contacts between pigs are the same and an element of probability is involved in
whether the disease transmits or not. The results are positive – the contact heterogeneity
model has a better statistical test outcome than the standard one proving that the added
complexity has provided better reliability. The second model also shows a promising
improvement on the standard deterministic-only model, but lacks the robustness of the first
model; some data had to be estimated or based off less applicable data than the first model.
Although the results show a good and efficient outcome in comparison to data, I argue that the
complexity of both models outweigh the effectiveness of the results. That is, is the extra data
and analysis required worth the relatively small improvements in reliability of data?
Obviously, there is a place for these models – anything that improves on something
previously existing is worth using, but sometimes, simplicity can be the best approach.
Following on from my previous work, I wish to use only probabilistic elements to model the
flow of an individual around a system. This will hopefully provide a similarly simple model
to the deterministic ones above, if not even simpler, yet still show the essence of the disease
that they are modelling.

44
5. Stochastic Processes – an overview
Stochastic processes are similar to traditional mathematical models, in the way that they
attempt to represent some real-world problem in a way that can be understood, adapted and
solved to provide a solution to the initial question. However, they differ in a major way – the
analysis of the system. Where traditional deterministic models allow the parameters in the
system to depend on time and or space as required, stochastic and probabilistic systems do not
rely on time intrinsically, instead conditional on just the probabilities of any one event
randomly occurring to an individual at any one time. Here, the time must be split up into
distinct, discrete chunks to allow for the analysis of the system at any given point in time.
Any system which must be or is chosen to be analysed using probabilistic theory in some way
is classified as stochastic (Nelson 1985), and many standard application revolve around
quantum theory; the underlying rules of physics which are based entirely on probabilistic
chance. This leads to a few definitions which will allow the implementation of this method to
analyse disease transmission:
A random walk (RW) is defined as (Urwin 2011b):
“A random process in discrete time steps { 𝑋 𝑛: 𝑛 = 0,1,2,… } such that it is only possible to
move forward, backward or remain in the same state, always with the same probability.”
A Markov chain (MC) is further defined as (Urwin 2011b):
“A stochastic process in discrete time steps { 𝑋 𝑛: 𝑛 = 0,1,2, … } with either a finite or infinite
state space, which has both the Markov property (MP) and the stationarity property (SP).”
The Markov property (MP) is that the currently occupied state is only dependant on the
immediately occupied previous state, and the stationarity (or homogeneity) property (SP) says
that the probability of transitioning from any one state to another (or to itself) stays the same
no matter which step the process is at (Urwin 2011b).

45
Therefore, any model consisting of a finite number of states, with constant probabilities and
state transitions only relating between the originating and target states in question can be
represented using a Markov chain. Looking at the original three compartmental models above,
if the natural and virus mortality arrows become targets for new states alongside if the
deterministic parameters that link each compartment are replaced with an assumed constant
probability (adding in self targetting probability arrows) for the transition between the two
states, then this model becomes a Markov chain. No state is needed for births as this method
looks at each individual separately, so all individuals can be assumed to be born and
susceptible to the disease from the outset.
Each compartmental model needs one compartment for each ‘alive’ state; 𝑀, 𝑆, 𝐸, 𝐼 and 𝑅 are
examples used in this research, and models not considered here that involved other
compartments would need extra (such as for the carrier state 𝐶 in models which account for
individuals who can pass on an infection without ever suffering from it themselves).
Compartments are also needed for each ‘death’ state. These death states could be merged into
one as once an individual enters a death state, they cannot leave, but for easier reading of
results later, the death states are kept separate (except for the model presented in 6.3.). This is
so the results clearly show the percentage of individuals who died having had the disease at
one point, alongside rather than merged with the percentage who died never having caught it.
If there are 𝑚 states, let 𝑝𝑖𝑗 be the probability of moving from state 𝑖 to state 𝑗 in a specified
time interval, where 𝑖, 𝑗 ∈ {1,2, … , 𝑚}. This implies that the total of the values given on all of
the arrows leaving any state must add exactly to one (Urwin 2011a):
∑ 𝑝𝑖𝑗 = 1, ∀𝑖
𝑚
𝑗=1
Some states are known as absorbing states. These are states that once entered cannot be left.
This implies two things for an absorbing state 𝑖:
𝑝𝑖𝑖 = 1
and:
𝑝𝑖𝑗 = 0, ∀𝑖 ≠ 𝑗

46
In the following models, all death states are clearly absorbing – once and individual enters
them, they will stay in them throughout the entirety of the rest of the analysis (Urwin 2011a).
Each compartmental model can be represented by something called a transition matrix, 𝑃.
This is a square matrix made up of 𝑚 rows and 𝑚 columns, representing each of the finite 𝑚
states. By definition, just as above, each row of 𝑃 must add to one. Each element of the matrix
gives the probability of leaving a state and entering a new one (or staying in the same state) in
one discrete time step. That is, the element in the 𝑖 𝑡ℎ
row and 𝑗 𝑡ℎ
column of 𝑃 is the
probability of leaving state 𝑖 and entering state 𝑗 (Urwin 2011a). The transition matrix for
each model will look like this:
𝑃 = (
𝑝11 𝑝12 ⋯ 𝑝1𝑚
𝑝21 𝑝22 ⋯ 𝑝2𝑚
⋮ ⋮ ⋱ ⋮
𝑝 𝑚1 𝑝 𝑚2 ⋯ 𝑝 𝑚𝑚
)
To measure the probability that any individual in this system is in state 𝑗 after 𝑛 steps given
that they started in state 𝑖 can be calculated using the transition matrix. Matrix multiplication
allows us to find any probability we wish. If 𝑃 is raised to the 𝑛𝑡ℎ
power, the following matrix
is produced:
𝑃 𝑛
= (
𝑝11 𝑝12 ⋯ 𝑝1𝑚
𝑝21 𝑝22 ⋯ 𝑝2𝑚
⋮ ⋮ ⋱ ⋮
𝑝 𝑚1 𝑝 𝑚2 ⋯ 𝑝 𝑚𝑚
)
𝑛
=
(
𝑝11
( 𝑛)
𝑝12
( 𝑛)
⋯ 𝑝1𝑚
( 𝑛)
𝑝21
( 𝑛)
𝑝22
( 𝑛)
⋯ 𝑝2𝑚
( 𝑛)
⋮ ⋮ ⋱ ⋮
𝑝 𝑚1
( 𝑛)
𝑝 𝑚2
( 𝑛)
⋯ 𝑝 𝑚𝑚
( 𝑛)
)
Now, 𝑝𝑖𝑗
( 𝑛)
is not the same as 𝑝𝑖𝑗
𝑛
, the probability 𝑝𝑖𝑗 raised to the 𝑛𝑡ℎ
power. 𝑝𝑖𝑗
( 𝑛)
is denoted as
the probability of being in state 𝑗 after 𝑛 steps given that the individual started in state 𝑖. If we
wish to know what state someone will be in after 2 steps, for example, we calculate 𝑃2
and
then look at the appropriate element. Each of the following models will work out the
probability an individual has the disease in question after one year, or 365 days. Each of the
probabilities in 𝑃 will be calculated based on a daily probability, so each transition matrix will

47
be raised to the 365 𝑡ℎ
power to work out 𝑃365
(Urwin 2011b). Further multiplication will
then be carried out to find out the limiting distribution (Urwin 2011c).
The limiting distribution is the transition matrix 𝑃 raised to the 𝑛𝑡ℎ
power, as 𝑛 → ∞ (Urwin
2011c). This will produce a matrix where any further matrix multiplication by any power of 𝑃
will not change the answer you get from the previous step:
𝑃 𝑛
∗ 𝑃 𝑎
= 𝑃 𝑛+𝑎
= 𝑃 𝑛
, ∀𝑎 ≥ 1
This limiting distribution shows us when the disease will be steady in the population; where
people will catch the disease at exactly the same rate as others are recovering. Obviously, in
this model in this limiting matrix, each individual will end up dead in the long-term (end up in
an absorbing state). But depending on which dead state they end up in shows us exactly what
the probability of someone having the disease at some point was, and therefore, what the
long-term steady proportion of any given population having the disease in question was.

48
6. Stochastic Modelling – adaptations and analysis
6.1. Model1 adaptation– Chickenpox
6.1.1. Stochastic model
The start of this method of infectious disease modelling includes a compartmental model,
much in a way similar to the original deterministic model given in 3.1.1. However, instead of
parameters given for each arrow, a probability of entering the state in question is given.
This gives an initial compartmental model for chickenpox (Fig. 6.1. and Fig. 6.2.):
Figure 6.1.: Stochastic model for chickenpox
Susceptible Exposed Infectious
Susceptible
Deaths
Exposed
Deaths
Infectious
Deaths
Recovered
Recovered
Deaths
Probability of
staying susceptible
Probability of
staying exposed
Probability of
staying infectious Probability of
staying recovered
Probability of
infection
Probability of
development Probability of
recovery
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
natural death
Probability of
natural death
Probability of
natural death
Probability of
natural death

49
Figure 6.2.: Stochastic model for chickenpox with mathematical symbols
Arrows that are missing (such as 𝑝𝑆𝐼 ) imply that the probability of leaving state 𝑖 and entering
state 𝑗 is zero (so 𝑝𝑆𝐼 = 0, for example). If at any time, an individual in state 𝑖 has the chance
of staying in state 𝑖 (𝑝𝑖𝑖 > 0), then this is represented by a circular arrow that has the state it
left as its target.
6.1.2. Transition matrix
This can be represented in a square matrix, where each element corresponds to the relative
probability of leaving the state and entering a new one, where the element in the 𝑖 𝑡ℎ
row and
𝑗 𝑡ℎ
column is the probability of leaving state 𝑖 and entering state 𝑗. Here, we need an eight-by-
eight (8 × 8) matrix as we have eight states, so this gives us the matrix 𝑃 as follows:
𝑆 𝐸 𝐼
𝑊 𝑋 𝑌
𝑅
𝑍
𝑝𝑆𝑆
𝑝 𝐸𝐸 𝑝𝐼𝐼
𝑝 𝑅𝑅
𝑝𝑆𝐸 𝑝 𝐸𝐼 𝑝𝐼𝑅
𝑝 𝑌𝑌𝑝 𝑋𝑋
𝑝 𝑊𝑊 𝑝 𝑍𝑍
𝑝𝑆𝑊 𝑝𝐼𝑌𝑝 𝐸𝑋 𝑝 𝑅𝑍

50
𝑃 =
(
𝑝𝑆𝑆 𝑝𝑆𝐸 𝑝𝑆𝐼 𝑝𝑆𝑅 𝑝𝑆𝑊 𝑝𝑆𝑋 𝑝𝑆𝑌 𝑝𝑆𝑍
𝑝 𝐸𝑆 𝑝 𝐸𝐸 𝑝 𝐸𝐼 𝑝 𝐸𝑅 𝑝 𝐸𝑊 𝑝 𝐸𝑋 𝑝 𝐸𝑌 𝑝 𝐸𝑍
𝑝𝐼𝑆 𝑝𝐼𝐸 𝑝𝐼𝐼 𝑝𝐼𝑅 𝑝𝐼𝑊 𝑝𝐼𝑋 𝑝𝐼𝑌 𝑝𝐼𝑍
𝑝 𝑅𝑆 𝑝 𝑅𝐸 𝑝 𝑅𝐼 𝑝 𝑅𝑅 𝑝 𝑅𝑊 𝑝 𝑅𝑋 𝑝 𝑅𝑌 𝑝 𝑅𝑍
𝑝 𝑊𝑆 𝑝 𝑊𝐸 𝑝 𝑊𝐼 𝑝 𝑊𝑅 𝑝 𝑊𝑊 𝑝 𝑊𝑋 𝑝 𝑊𝑌 𝑝 𝑊𝑍
𝑝 𝑋𝑆 𝑝 𝑋𝐸 𝑝 𝑋𝐼 𝑝 𝑋𝑅 𝑝 𝑋𝑊 𝑝 𝑋𝑋 𝑝 𝑋𝑌 𝑝 𝑋𝑍
𝑝 𝑌𝑆 𝑝 𝑌𝐸 𝑝 𝑌𝐼 𝑝 𝑌𝑅 𝑝 𝑌𝑊 𝑝 𝑌𝑋 𝑝 𝑌𝑌 𝑝 𝑌𝑍
𝑝 𝑍𝑆 𝑝 𝑍𝐸 𝑝 𝑍𝐼 𝑝 𝑍𝑅 𝑝 𝑍𝑊 𝑝 𝑍𝑋 𝑝 𝑍𝑌 𝑝 𝑍𝑍 )
Clearly, the probability of staying dead when already in a dead state is one, so:
𝑝 𝑊𝑊 = 𝑝 𝑋𝑋 = 𝑝 𝑌𝑌 = 𝑝 𝑍𝑍 = 1
And the probability of moving out of a dead state to any other must be zero, so:
𝑝 𝑊𝑆 = 𝑝 𝑊𝐸 = 𝑝 𝑊𝐼 = 𝑝 𝑊𝑅 = 𝑝 𝑊𝑋 = 𝑝 𝑊𝑌 = 𝑝 𝑊𝑍 = 0
𝑝 𝑋𝑆 = 𝑝 𝑋𝐸 = 𝑝 𝑋𝐼 = 𝑝 𝑋𝑅 = 𝑝 𝑋𝑊 = 𝑝 𝑋𝑌 = 𝑝 𝑋𝑍 = 0
𝑝 𝑌𝑆 = 𝑝 𝑌𝐸 = 𝑝 𝑌𝐼 = 𝑝 𝑌𝑅 = 𝑝 𝑌𝑊 = 𝑝 𝑌𝑋 = 𝑝 𝑌𝑍 = 0
𝑝 𝑍𝑆 = 𝑝 𝑍𝐸 = 𝑝 𝑍𝐼 = 𝑝 𝑍𝑅 = 𝑝 𝑍𝑊 = 𝑝 𝑍𝑋 = 𝑝 𝑍𝑌 = 0
There is no chance in this model that an individual can jump straight from being susceptible
to being infectious, being susceptible to having recovered or from being exposed to having
recovered, so:
𝑝𝑆𝐼 = 𝑝𝑆𝑅 = 𝑝 𝐸𝑅 = 0
Individuals also cannot travel backwards in the model, so:
𝑝 𝐸𝑆 = 𝑝𝐼𝑆 = 𝑝𝐼𝐸 = 𝑝 𝑅𝑆 = 𝑝 𝑅𝐸 = 𝑝 𝑅𝐼 = 0
Each compartment that is not a dead state has its own respective dead state; individuals who
naturally die whilst susceptible go into the 𝑊 state, those who naturally die whilst exposed go
into the 𝑋 state, those who naturally die whilst infectious go into the 𝑌 state and those who
naturally die after having recovered go into the 𝑍 state. Therefore, for example, the
probability of a susceptible individual dying and going into states 𝑋, 𝑌 or 𝑍 is zero (similar
for states 𝐸, 𝐼 and 𝑅), so:
𝑝𝑆𝑋 = 𝑝𝑆𝑌 = 𝑝𝑆𝑍 = 0

51
𝑝 𝐸𝑊 = 𝑝 𝐸𝑌 = 𝑝 𝐸𝑍 = 0
𝑝𝐼𝑊 = 𝑝𝐼𝑋 = 𝑝𝐼𝑍 = 0
𝑝 𝑅𝑊 = 𝑝 𝑅𝑋 = 𝑝 𝑅𝑌 = 0
The reasons for having separate dead states is to make the analysis slightly easier later – those
who end up in states 𝑋, 𝑌 or 𝑍 must have had the infection at some point to end up in there,
whereas those who end up in state 𝑊 have not.
The remaining eleven probabilities have values which are neither zero (impossible) nor one
(certain). These can be calculated (estimated) as follows (all probabilities are rounded only at
the final answer and are rounded to six significant figures):
The average life expectancy of an individual in the United Kingdom (UK) is currently 81
years (WHO 2012). As each step of this model accounts for one day, 81 years in days is 81
years ∗ 365 days per year = 29565 days. Hence, the reciprocal of this gives the probability of
dying naturally each day during the analysis of this model. An assumption to be made here is
that the chances of dying naturally are equal regardless of whether the individual in question
is currently susceptible, exposed, infectious or has recovered from chickenpox, so:
𝑝𝑆𝑊 = 𝑝 𝐸𝑋 = 𝑝𝐼𝑌 = 𝑝 𝑅𝑍 =
1
29565
≅ 0.0000338238
Annually in the UK, there are approximately 57 cases of chickenpox per 10,000 people
presented to and recorded by doctors and practice nurses (Fleming et al. 2007 p.14). Although
this report is not taken from all hospitals, over 52 million patients were analysed (Fleming et
al. 2007 p.9) which is a high proportion of the UK population so it is representative.
Extrapolating this up to the entire population of the UK will give us the estimated number of
cases in one year. So there are approximately 57 ∗
64100000
10000
= 365370 cases each year.
Dividing this number by 365 will give the number of cases per day, so
365370
365
≅ 1001.01
cases per day. This number further divided by the population size will give the probability of
an individual contracting chickenpox on any given day, so:

52
𝑝𝑆𝐸 =
1001.01
64100000
≅ 0.0000156164
The latency period of chickenpox is given to be somewhere between ten and fourteen days
(Knott 2013), so taking the average of twelve days estimates the value of the reciprocal of
𝑝 𝐸𝐼 , so:
𝑝 𝐸𝐼 =
1
12
= 0.083̅ ≅ 0.0833333
The recovery period of chickenpox once contagious is also given to be somewhere between
ten and fourteen days (Lamprecht 2012), so again, taking the average of twelve days estimates
the value of the reciprocal of 𝑝𝐼𝑅 , so:
𝑝𝐼𝑅 =
1
12
= 0.083̅ ≅ 0.0833333
Now, all rows of the matrix must add up to one, so the final stationary probabilities can all be
worked out using the values we already know (given to full decimal places):
𝑝𝑆𝑆 = 1 − 0.0000156164 − 0.0000338238 = 0.9999505598
𝑝 𝐸𝐸 = 1 − 0.0833333 − 0.0000338238 = 0.9166328762
𝑝𝐼𝐼 = 1 − 0.0833333 − 0.0000338238 = 0.9166328762
𝑝 𝑅𝑅 = 1 − 0.0000338238 = 0.9999661762
Combining all of the above results give the following final transition matrix (given to
appropriate significant figures):
𝑃 =
(
0.999951 1.56 × 10−5
0 0 3.38 × 10−5
0 0 0
0 0.916633 0.0833 0 0 3.38 × 10−5
0 0
0 0 0.916633 0.0833 0 0 3.38 × 10−5
0
0 0 0 0.999966 0 0 0 3.38 × 10−5
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 )

53
6.1.3. 𝒏-step transition
The square of this matrix will give the probability that an individual will be in any given state
after one day. That is, the element in the 𝑖 𝑡ℎ
row and 𝑗 𝑡ℎ
column of 𝑃2
, denoted 𝑝𝑖𝑗
(2)
, gives the
probability that an individual will be state 𝑗 at the end of the second day, if they started in
state 𝑖. Following on from this, working out 𝑃365
will give all the probabilities that an
individual will end in any desired state, given any initial starting state, at the end of the 365 𝑡ℎ
day, or one year. Hence, using Maple (omitted due to size, see Appendix A6), this gives the
365-step matrix (given to appropriate significant figures):
𝑃365
=
(
0.982116 0.000184080 0.000184114 0.00524607 0.0122353 2.22 × 10−6
2.14× 10−6
3.04 × 10−5
0 1.6 × 10−14
5.28 × 10−13
0.987730 0 0.000405721 0.000405556 0.0114587
0 0 1.6 × 10−14
0.987730 0 0 0.000405721 0.0118643
0 0 0 0.987730 0 0 0 0.0122700
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 )
Now, assuming we start with a susceptible individual, we can just concentrate on the first row
of 𝑃365
. The chance of any susceptible individual having had chickenpox at any point in the
year of analysis is the sum of the following probabilities:
𝑝𝑆𝐸
(365)
+ 𝑝𝑆𝐼
(365)
+ 𝑝𝑆𝑅
(365)
+ 𝑝𝑆𝑋
(365)
+ 𝑝𝑆𝑌
(365)
+ 𝑝𝑆𝑍
(365)
The first three probabilities of these six represent those who were susceptible but have ended
day 365 being exposed, infectious or having recovered, respectively. The last three of these
probabilities represent those who were susceptible but have ended the year having passed
away naturally (not due to chickenpox directly) after being exposed, infectious or having
recovered, respectively. 𝑝𝑆𝑆
(365)
and 𝑝𝑆𝑊
(365)
are not included in this sum as these represent those
who were susceptible and ended the year still being susceptible (not having had chickenpox)
and those who started susceptible but have died naturally before contracting the disease at any
point.

54
Hence, the probability that any individual in the UK contracts chickenpox in this year is:
𝑝𝑆𝐸
(365)
+ 𝑝𝑆𝐼
(365)
+ 𝑝𝑆𝑅
(365)
+ 𝑝𝑆𝑋
(365)
+ 𝑝𝑆𝑌
(365)
+ 𝑝𝑆𝑍
(365)
= 0.000184080+ 0.000184114+ 0.00524607+ 0.00000221723
+ 0.00000214164+ 0.0000303784 ≅ 0.00564900
This shows that each individual has a 0.56% probability of contracting chickenpox during
year one.
6.1.4. Limiting distribution
If we take 𝑃 to higher and higher powers, we can get towards the limiting distribution of this
matrix system. 𝑃365
gives the probability of an individual, hence the percentage of a
population, that will get the disease in one year. Raising this new matrix to the power 𝑛 will
give the percentage of a constant population that will contract the disease at some point during
𝑛 years. As 𝑛 → ∞, the matrix ( 𝑃365) 𝑛
= 𝑃365∗𝑛
→ 𝑃 𝑛
. So raising the original matrix to
exceptionally high powers will give the steady proportion of a population (based on UK data)
that will contract chickenpox.
After some manual checking (using integer powers of 10 for 𝑛), both 𝑃365∗100000
=
𝑃36500000
and 𝑃365∗1000000
= 𝑃365000000
give the same matrix (using Maple, omitted due to
size, see Appendix A7). This implies that the limiting distribution, 𝑃 𝑛
, is (to six significant
figures):
𝑃 𝑛
=
(
0 0 0 0 0.684136 0.000128153 0.000128101 0.315608
0 0 0 0 0 0.000405721 0.000405556 0.999189
0 0 0 0 0 0 0.000405721 0.999594
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 )

55
Some rounding errors have occurred during the calculation process due to the sheer size of the
power the matrix 𝑃 has been raised to. However, rounding to significant figures shows that all
rows of 𝑃 𝑛
still add up to one.
From this, the result here shows that given enough time, all individuals will pass away (all
elements in the first four columns are zero so no-one ends up in the 𝑆, 𝐸, 𝐼 or 𝑅 states), which
is obvious. It also shows a steady percentage of 𝑝𝑆𝑊
( 𝑛)
= 68.4% of the population will have
died directly from the 𝑆 state, so 68.4% of the population will never contract chickenpox. The
sum of the other three non-zero values in this row gives the percentage of people who will get
chickenpox (this can also be worked out using 1 − 𝑝𝑆𝑊
( 𝑛)
as all rows must add to one):
𝑝𝑆𝑋
( 𝑛)
+ 𝑝𝑆𝑌
( 𝑛)
+ 𝑝𝑆𝑍
( 𝑛)
= 0.000128153 + 0.000128101 + 0.315608 = 0.315864254
So 31.6% of the population will contract chickenpox in a steady population that stays
constant for many years.

56
6.2. Model2 adaptation– Measles
6.2.1. Stochastic model
Just like the previous adaptation, the start of this model includes a compartmental model,
much in a way similar to the original deterministic model given in 3.2.1., with a probability of
entering the state in question is given instead of deterministic parameters.
This gives an initial compartmental model for measles (Fig. 6.3. and Fig. 6.4.):
Figure 6.3.: Stochastic model for measles
Probability of
staying susceptible
Probability of
staying exposed
Probability of
staying infectious
Probability of
staying
recovered
Probability of
staying immune
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Probability of
staying dead
Prob. of
natural death
Prob. of
natural death
Prob. of
natural death
Prob. of
natural death
Prob. of virus
death
Prob. of
natural
death
Prob. of
virus
death
Probability
of recovery
Probability
of infection
Exposed Infectious RecoveredMaternal
Immunity
Susceptible
Probability of
development
Probability of
loss of passive
immunity
Probability of gain of active immunity
(vaccination)
Infectious
Virus
Deaths
Recovered
Deaths
Maternal
Deaths
Susceptible
Deaths
Infectious
Natural
Deaths
Exposed
Natural
Deaths
Exposed
Virus
Deaths

57
Figure 6.4.: Stochastic model for measles with mathematical symbols
Again, arrows that are missing (such as 𝑝 𝑀𝐸 ) imply that the probability of leaving state 𝑖 and
entering state 𝑗 is zero (so 𝑝 𝑀𝐸 = 0, for example). If at any time, an individual in state 𝑖 has
the chance of staying in state 𝑖 (𝑝𝑖𝑖 > 0), then as before, this has been added and is
represented by a circular arrow that has the state it left as its target.
6.2.2. Transition matrix
This can be represented in a square matrix, where each element corresponds to the relative
probability of leaving the state and entering a new one, where the element in the 𝑖 𝑡ℎ
row and
𝑗 𝑡ℎ
column is the probability of leaving state 𝑖 and entering state 𝑗. Here, we need a twelve-
by-twelve (12 × 12) matrix as we have twelve states (five ‘alive’ states, five ‘natural death’
states and two ‘virus death’ states), so this gives us the matrix 𝑃 as follows:
𝑝𝑆𝑆 𝑝 𝐸𝐸 𝑝𝐼𝐼
𝑝 𝑅𝑅𝑝 𝑀𝑀
𝑝 𝑉𝑉 𝑝 𝑍𝑍
𝑝 𝑊𝑊 𝑝 𝑋𝑋 𝑝 𝑇𝑇 𝑝 𝑌𝑌 𝑝 𝑈𝑈
𝑝 𝑀𝑉 𝑝𝑆𝑊 𝑝 𝑅𝑍𝑝 𝐸𝑋 𝑝𝐼𝑈𝑝𝐼𝑌𝑝 𝐸𝑇
𝑝𝐼𝑅𝑝𝑆𝐸
𝐸 𝐼 𝑅𝑀 𝑆
𝑝 𝐸𝐼𝑝 𝑀𝑆
𝑝𝑆𝑅
𝑈 𝑍𝑉 𝑊 𝑌𝑋 𝑇

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