Mathematical Modelling of Infectious Disease: A Stochastic Approach

Mathematical Modelling of Infectious Disease: A Stochastic Approach

0903642

October 20

University of Wolverhampton

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.

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the work).

Signature:

Date: 03/10/2014

1

i. Dissertation Declaration

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3

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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i. 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.

5

ii. Contents

i. Dissertation Declaration......................................................................................................3

ii. Acknowledgments...............................................................................................................4

iii. Abstract............................................................................................................................5

iv. Contents...........................................................................................................................6

1. Introduction.........................................................................................................................9

2. Deterministic Modelling – an overview............................................................................11

3. Deterministic Modelling – analysis...................................................................................13

3.1. Model 1 – Chickenpox...............................................................................................13

3.1.1. Compartmental model.........................................................................................13

3.1.2. Assumptions........................................................................................................14

3.1.3. Representing the model.......................................................................................15

3.1.4. Finding the steady states.....................................................................................16

3.1.5. Finding R 0..........................................................................................................18

3.1.6. Analysing the data...............................................................................................18

3.2. Model 2 – Measles.....................................................................................................21


3.2.2. Assumptions........................................................................................................23



3.2.5. Finding R 0..........................................................................................................27


3.3. Model 3 – H1N1.........................................................................................................31


3.3.2. Assumptions........................................................................................................33



3.3.5. Finding R 0..........................................................................................................38


4. Overview of deterministic models.....................................................................................42

6

5. Stochastic Processes – an overview...................................................................................45

6. Stochastic Modelling – adaptations and analysis..............................................................49

6.1. Model 1 adaptation – Chickenpox.............................................................................49

6.1.1. Stochastic model.................................................................................................49

6.1.2. Transition matrix.................................................................................................50

6.1.3. n-step transition...................................................................................................54

6.1.4. Limiting distribution...........................................................................................55

6.2. Model 2 adaptation – Measles....................................................................................57





6.3. Model 3 adaptation – H1N1.......................................................................................68





7. Results and model comparisons........................................................................................78

7.1. Model 1 comparisons.................................................................................................80



8. Conclusions.......................................................................................................................83

8.1. Conclusion and critical evaluation.............................................................................83

8.2. Further work...............................................................................................................85

9. References.........................................................................................................................87

10. Bibliography..................................................................................................................93

11. Appendices.....................................................................................................................94

11.1. Appendix A.............................................................................................................94

11.1.1. Appendix A1.......................................................................................................94

11.1.2. Appendix A2.......................................................................................................95

11.1.3. Appendix A3.......................................................................................................96

11.1.4. Appendix A4.......................................................................................................98

11.1.5. Appendix A5.......................................................................................................99

7

11.1.6. Appendix A6.....................................................................................................101

11.1.7. Appendix A7.....................................................................................................102

11.1.8. Appendix A8.....................................................................................................103

11.1.9. Appendix A9.....................................................................................................104

11.1.10. Appendix A10...............................................................................................105

11.1.11. Appendix A11...............................................................................................106

11.2. Appendix B...........................................................................................................107

11.2.1. Appendix B1.....................................................................................................107

11.2.2. Appendix B2.....................................................................................................108

11.2.3. Appendix B3.....................................................................................................109

11.3. Appendix C...........................................................................................................110

8

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

9

randomness, which diseases can show signs of when a small number of individuals infected

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.

10

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

11

due to this. Extra states have been added in certain systems such as ‘exposed’, where

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

1.1. Model 1 – 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.

1.1.1. 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

1.1.2. 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

14

Natural Deaths

Recovered

Recovery rate

Natural Deaths

Infectious

Natural Deaths

Natural Deaths

Births

ExposedSusceptible

Transmission rate

Development rate

μ

Rγ

μ

I

μ μ

μ

ESβ σ

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 (all individuals initially enter the susceptible

state).

The population has a life expectancy of μ, so that the natural mortality rate is 1μ

.

A closed population, so that the birth rate is also 1μ

.

The virus has a latency period of 1σ

.

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.

1.1.3. 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:

dSdt

=μ−βSI−μS (2)

dEdt

=βSI−σE−μE (3)

dIdt

=σE−γI−μI (4)

dRdt

=γI−μR (5)

15

where β ,σ , γ , μ>0.

1.1.4. 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:

dSdt

=0 ,dEdt

=0 ,dIdt

=0 ,dRdt

=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= σEγ+μ

(12)

I=μRγ

(13)

If I=0 (no virus present), (10) ⟹S=1

16

(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.

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:

17

( 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)

1.1.5. Finding R0

For a closed population (which we have here), the critical value at which the virus becomes an

epidemic is when it exceeds 1S

. So, for this model:

R0=1S= 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.

18

1.1.6. Analysing the data

19

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, 1σ

, is given as ten to fourteen days (Knott 2013), so we

take the average of twelve days. The recovery period, 1γ

, is also reported as ten to fourteen

days (Lamprecht 2012), so again, we take the average of twelve days. Average life

expectancy at birth in the UK is approximately 81 years (WHO 2012), so this is taken to be

the value of 1μ

. 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=15

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).

20

0 50 100 150 200 250 300 350 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

S

E

I

R

S*

E*

I*

R*

Time (days)

Pro

port

ion

of p

opul

atio

n

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

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 %).

21

1.2. Model 2 – 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

22

are vaccinated are only those in the susceptible state and will move into a recovered 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


23

Gain of active immunity (vaccination)

Loss of passive

immunity

Development rate

Natural Deaths

Virus Deaths

Natural Deaths

Virus Deaths

Natural Deaths

Natural Deaths

Natural Deaths

Births (with passive

immunity)

Susceptible

Maternal Immunity

RecoveredInfectiousExposed

Transmission rate

Recovery rate

X

κ

δ σ

μ

χ

μ

χ

μμμ

μ

SM RIEβI γ

Figure 3.5.: Compartmental model for measles with mathematical symbols

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.

1.2.2. 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).


.


. 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 1χ

, so individuals who die from the virus directly enter the

X state at this rate.


.

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).

24

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).

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:

dMdt

=μ−δM−μM (18)

dSdt

=−βSI+δM−κS−μS (19)

dEdt

=βSI−σE−μE− χE (20)

25

dIdt

=σE−γI−μI− χI (21)

dRdt

=γI+κS−μR (22)

dXdt

= χE+ χI −μX (23)

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:

dMdt

=0 ,dSdt

=0 ,dEdt

=0 ,dIdt

=0 ,dRdt

=0 ,dXdt

=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:

26

M= μδ+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).

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= σEγ+μ+ χ

(32)

I=μR−κSγ

(33)

I=μX− χEχ

(34)

If I=0 (no virus present), (30) ⟹S= δμ(δ+μ ) (κ+μ )

(32) ⟹ E=0

(33) ⟹ R=κSμ

= δκ(δ +μ ) (κ+μ )

(34) ⟹ X= χEμ

=0

Checking in (17) (using Maple), M= μδ+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:

27

>

>

>

>

>

>

>


( M , S , E , I , R , X )=( μδ+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 )=( μδ+μ

,δ

δ +μ, 0 , 0 ,0 , 0)


steady state) (solution omitted due to size, see Appendix A2):

>

28

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)

1.2.5. Finding R0

For a closed population (which again, we have here), the critical value at which the virus

becomes an epidemic is when it exceeds 1S


R0=1S= 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

29

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, 1σ

, is given as seven to fourteen days (CDC 2009c), so we take the

average of 10.5 days. The recovery period, 1γ

, 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 1μ

. Individuals lose maternal immunity

sometime between twelve and fifteen months (Nicoara 1999), so we take the average of 13.5

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=15

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).

30

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

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.

31

0 50 100 150 200 250 300 350 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M

S

E

I

R

X

M*

S*

E*

I*

R*

X*

Time (days)

Pro

port

ion

of p

opul

atio

n

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.

32

1.3. Model 3 – 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.):

33

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

34

BD

X

δμμμ

χ

μ

R

μ

IES

χ

βI σ γ

κ

δ

Natural Deaths

Virus Deaths

Natural Deaths

Natural Deaths

Natural Deaths

Recovered

Births

InfectiousExposedSusceptible

Virus Deaths

Transmission rate

Development rate

Recovery rate

Gain of active immunity (vaccination)

Loss of active immunity

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

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.).

1.3.2. 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).


.


.

35

The virus’ mortality rate is 1χ

.


.

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,

36

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:

dSdt

=μ−βSI +δR−κS−μ S (39)

dEdt

=βSI−σE−μE− χE (40)

dIdt

=σE−γI−μI− χI (41)

dRdt

=γI−δR+κS−μR (42)

dXdt

= χE+ χI −μX (43)

dDdt

=−δR (44)

dBdt

=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

37

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.


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:

dSdt

=0 ,dEdt

=0 ,dIdt

=0 ,dRdt

=0 ,dXdt

=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)

38

I= σEγ+μ+ χ

(53)

I=(δ+μ ) R−κS

γ(54)

I=μX− χEχ

(55)

If I=0 (no virus present), (53) ⟹ E=0

(55) ⟹ X= χEμ

=0

Rearranging (54) to give R in terms of S and then substituting into (51) gives:

R= κSδ +μ ⟹0=

μ+δ( κSδ+μ )−( κ+μ ) S

βS

⟹S= δ+μδ+κ+μ

⟹ R= κδ +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:

>

>

>

>

>

>

39


( S , E , I , R , X )=( δ +μδ+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 )


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)

1.3.5. Finding R0

For a closed population (which again, we have here), the critical value at which the virus

becomes an epidemic is when it exceeds 1S


40

R0=1S= 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)

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, 1σ

, 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, 1γ

, 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

41

be the value of 1μ

. 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=15

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

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.

0 50 100 150 200 250 300 350 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

S

E

I

R

X

B

S*

E*

I*

R*

X*

B*

Time (days)

Pro

port

ion

of p

opul

atio

n

Figure 3.9.: H1N1 4th-order Runge-Kutta numerical analysis (with steady state values)

42

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 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.

43

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.

44

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

45

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.

46

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 { Xn: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 { Xn: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

47

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).

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):

∑j=1

m

pij=1 , ∀ i

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:

48

pij=0 , ∀ i≠ j

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 ith row and jth 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 ⋯ p1 m

p21 p22 ⋯ p2 m

⋮ ⋮ ⋱ ⋮pm1 pm2 ⋯ pmm

)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:

Pn=(p11 p12 ⋯ p1 m

p21 p22 ⋯ p2 m

⋮ ⋮ ⋱ ⋮pm1 pm2 ⋯ pmm

)n

=(p11

(n ) p12(n ) ⋯ p1m

(n )

p21(n ) p22

(n ) ⋯ p2m(n )

⋮ ⋮ ⋱ ⋮pm1

(n ) pm2( n) ⋯ pmm

(n ) )Now, pij

( n) is not the same as pijn, 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

49

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

be raised to the 365th 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:

Pn∗Pa=Pn+a=Pn ,∀ 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.

50

6. Stochastic Modelling – adaptations and analysis

1.4. Model 1 adaptation – Chickenpox

1.4.1. 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

51

Probability of natural death




Probability of staying dead

Probability of staying dead Probability of

staying deadProbability of staying dead

Probability of recovery

Probability of developmentProbability of

infection

Probability of staying recovered

Probability of staying infectious

Probability of staying exposedProbability of

staying susceptible

Recovered Deaths

Recovered

Infectious Deaths

Exposed Deaths

Susceptible Deaths

InfectiousExposedSusceptible


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.

1.4.2. 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 ith row and jth

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:

52

pRZpEX p IYpSW

pZZpWW

pXX pYY

p IRpEIpSE

pRR

p IIpEE

pSS

Z

R

YXW

IES

P=(pSS pSE pSI pSR pSW pSX pSY pSZ

pES pEE pEI pER pEW pEX pEY pEZ

p IS p IE p II pIR p IW p IX p IY p IZ

pRS pℜ pRI pRR pRW pRX pRY pRZ

pWS pWE pWI pWR pWW pWX pWY pWZ

p XS pXE pXI pXR pXW pXX pXY p XZ

pYS pYE pYI pYR pYW pYX pYY pYZ

pZS pZE pZI pZR pZW pZX pZY pZZ

)Clearly, the probability of staying dead when already in a dead state is one, so:

pWW=pXX=pYY =pZZ=1

And the probability of moving out of a dead state to any other must be zero, so:

pWS=pWE=pWI=pW R=pWX=pWY=pWZ=0

pXS=pXE=pXI=pXR=pXW=pXY=pXZ=0

pYS=pYE=pYI=pYR=pYW=pYX=pYZ=0

pZS=pZE=pZI=pZR=pZW=pZX=pZY =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=pER=0

Individuals also cannot travel backwards in the model, so:

pES=p IS=p IE=pRS=pℜ=pRI=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:

53

pSX=pSY=pSZ=0

pEW=pEY=pEZ=0

p IW=p IX=p IZ=0

pRW=pRX=pRY=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 =pEX=p IY=pRZ=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

54

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:

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 pEI,

so:

pEI=1

12=0.083≅ 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 p IR, so:

p IR=1

12=0.08 3≅ 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

pEE=1−0.0833333−0.0000338238=0.9166328762

p II=1−0.0833333−0.0000338238=0.9166328762

pRR=1−0.0000338238=0.9999661762

Combining all of the above results give the following final transition matrix (given to

appropriate significant figures):

P=¿

55

(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 00 0 0.916633 0.0833 0 0 3 .38 ×10−5 00 0 0 0.999966 0 0 0 3.38 × 10−5

0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

)1.4.3. 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 ith row and jth 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 365th

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 0.01145870 0 1.6 × 10−14 0.987730 0 0 0.000405721 0.01186430 0 0 0.987730 0 0 0 0.01227000 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 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)

56

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.

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.

1.4.4. 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 → Pn. 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.

57

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, Pn, is (to six significant figures):

Pn=(0 0 0 0 0.684136 0.000128153 0.000128101 0.3156080 0 0 0 0 0.000405721 0.000405556 0.9991890 0 0 0 0 0 0.000405721 0.9995940 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

)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 Pn 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.

58

1.5. Model 2 adaptation – Measles


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

59

Exposed Virus

Deaths

Exposed Natural Deaths

Infectious Natural Deaths

Susceptible Deaths

Maternal Deaths

Recovered Deaths

Infectious Virus

Deaths

Probability of gain of active immunity (vaccination)

Probability of loss of passive

immunityProbability of development

SusceptibleMaternal Immunity

RecoveredInfectiousExposed

Probability of infection


Prob. of virus death

Prob. of natural death

Prob. of virus death












Probability of staying immune Probability of

staying recovered


Probability of staying exposed

Probability of staying susceptible


Figure 6.4.: Stochastic model for measles with mathematical symbols

Again, arrows that are missing (such as pME) imply that the probability of leaving state i and

entering state j is zero (so pME=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.




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:

60

TX YWV ZU

pSR

pMS pEI

SM RIEpSE

p IR

pET p IY p IUpEX pRZpSWpMV

pUUpYYpTTpXXpWW

pZZpVV

pMM pRR

p IIpEEpSS

P=(pMM pMS pME pMI pMR pMV pMW pMX pMT pMY pMU pMZ

pSM pSS pSE pSI pSR pSV pSW pSX pST pSY pSU pSZ

pEM pES pEE pEI pER pEV pEW pEX pET pEY pEU pEZ

pℑ p IS p IE p II p IR pIV p IW p IX p¿ p IY p IU pIZ

pRM pRS pℜ pRI pRR pRV pRW pRX pRT pRY pRU pRZ

pVM pVS pVE pVI pVR pVV pVW pVX pVT pVY pVU pVZ

pWM pWS pWE pWI pWR pWV pWW pWX pWT pWY pWU pWZ

pXM pXS pXE pXI pXR pXV pXW p XX pXT pXY pXU pXZ

pTM pTS pTE pTI pTR pTV pTW pTX pTT pTY pTU pTZ

pYM pYS pYE pYI pYR pYV pYW pYX pYT pYY pYU pYZ

pUM pUS pUE pUI pUR pUV pUW pUX pUT pUY pUU pUZ

pZM pZS pZE pZI pZR pZV pZW pZX pZT pZY pZU pZZ

)Like before, each probability needs to be worked out. Clearly, the probability of staying dead

when already in a dead state is one, so: pVV =pWW=pXX=pTT=pYY=pUU=pZZ=1


pVM=pVS=pVE=pVI=pVR=pVW=pVX=pVT=pVY=pVU=pVZ=0

pWM=pWS=pWE=pWI=pWR=pWV=pWX=pWT=pWY=pWU=pWZ=0

pXM=pXS=pXE=pXI=pXR=pXV=pXW=pXT=pXY=pXU =pXZ=0

pTM=pTS=pTE=pTI=pTR=pTV=pTW=pTX=pTY=pTU=pTZ=0

pYM=pYS=pYE=pYI=pYR=pYV=pYW=pYX=pYT=pYU=pYZ=0

pUM=pUS=pUE=pUI=pUR=pUV =pUW=pUX=pUT=pUY=pUZ=0

pZM=pZS=pZE=pZI=pZR=pZV=pZW=pZX=pZT=pZY =pZU=0

There is no chance in this model that an individual can jump straight from having maternal

immunity to being exposed, infectious or having recovered, from being susceptible to being

infectious, or from being exposed to having recovered, so:

pME=pM I=pMR=pSI=pER=0

Individuals also cannot travel backwards in the model, so:

61

pSM=pEM=pES=pℑ=p IS=pIE=pRM=pRS=pℜ=pRI=0

Each compartment that is not a dead state has its own respective dead state; similar to before,

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, but also,

individuals who naturally die whilst having passive immunity go into the V state. Individuals

can only die from the disease whilst exposed or infectious, so the according states after a virus

death can only being reached from these states too. Therefore, for example, the probability of

an immune individual dying and going into states W , X , T , Y , U or Z is zero (similar for

states S, E, I and R), so:

pMW=pMX=pMT=pMY=pMU=pMZ=0

pSV =pSX=pST=pSY=pSU=pSZ=0

pEV=pEW=pEY=pEU=pEZ=0

p IV=p IW=p IX=p¿=p IZ=0

pRV=pRW =pRX=pRT=pRY=pRU=0

Again, 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 states V or W have not.

The remaining seventeen probabilities have values which are neither zero nor one. These can

be calculated as follows (as before, 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 States (US) is currently 79 years

(WHO 2012). Similar to before, 79 years in days is 79 years ¿365 days per year ¿28835 days.

The reciprocal of this gives the probability of dying naturally each day. An assumption

continuing to be made here is that the chances of dying naturally are equal regardless of

62

whether the individual in question is currently young with maternal immunity, susceptible,

exposed, infectious or has recovered from measles, so:

pMV=pSW=pEX=p IY=pRZ=1

28835≅ 0.0000346801

Annually in the US, there are only approximately 55 cases of measles (CDC 2009a). Simply

dividing by the US population will give the probability of contracting measles each year, and

then further dividing by 365 will give the chance per day:

pSE=55

316148990∗365≅ 0.000000000476626

The chance of dying from measles if an individual has caught it is approximately one or two

in a thousand, for children over the entire length of the disease (17.5 days) (CDC 2009b). As

most adult individuals in the US will have already been vaccinated, it is safe to assume that

any individual catching the disease is relatively young. This allows the child virus mortality

rate to be used in this model (taking the average of one and a half out of one thousand and

dividing by the length of having the disease). An assumption that dying due to measles

whether the individual is exposed or infectious is equivalent in the two states is made here:

pET=p IU=1.5

1000∗17.5≅ 0.0000857143

Again, looking specifically at the child vaccination rate as most individuals in the US will be

immunised at a young age, the data shows that a mean of 91.9 % of children across the US are

immunised per year. The ages in question are well after children are estimated to have lost

their maternal immunity so it is safe to assume that all individuals immunised in this model

are in the susceptible state, and not the maternal immunity state. So, dividing the value by 365

will give the probability per day:

pSR=0.919365

≅ 0.00251781

The latency period of measles is somewhere between seven and fourteen days (CDC 2009c),

so taking the average of ten and a half days gives the value of the reciprocal of pEI, so:

pEI=1

10.5≅ 0.0953381

63

The recovery period of measles after becoming contagious is approximately seven days; about

four days for a rash to form and another three for it to be completely gone and have fully

recovered (CDC 2009c), so again, taking this value and finding the reciprocal gives p IR, so:

p IR=17=0. 142857≅ 0.142857

Maternal immunity is basically fully lost by the age of twelve to fifteen months (Nicoara

1999). So, if an individual has a 100 % chance after these 13.5 months (taking the average),

converting this into years (1.125 years), and dividing by the 365 days per year gives the

probability of losing the conferred immunity:

pMS=1

1.125∗365≅ 0.00243531

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):

pMM=1−0.00243531−0.0000346801=0.9975300099

pSS=¿

1−0.000000000476626−0.00251781−0.0000346801=0.997447509423374

pEE=1−0.0953381−0.0000346801−0.0000857143=0.9045415056

p II=1−0.142857−0.0000346801−0.0000857143=0.8570226056

pRR=1−0.0000346801=0.9999653199



P=¿

64

(0.997530 0.00243531 0 0 0 3.47 ×10−5 0 0 0 0 0 0

0 0.997448 4.77 ×10−10 0 0.00251781 0 3.47 ×10−5 0 0 0 0 00 0 0.904542 0.0953381 0 0 0 3.47 × 10−5 8 .57 × 10−5 0 0 00 0 0 0.857023 0.142857 0 0 0 0 3.47 × 10−5 8.57 × 10−5 00 0 0 0 0.999965 0 0 0 0 0 0 3.47 ×10−5

0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1


As before, 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 one year. Hence, using Maple

(omitted due to size, see Appendix A8), this gives the 365-step matrix (given to appropriate

significant figures):

P365=¿

(0.405489 0.355942 1.77 ×10−9 1.18× 10−9 0.225990 0.00834728 0.00312797 1.50 × 10−11 3.70×10−11 9.70 ×10−12 2.40× 10−11 0.00110342

0 0.393431 2.02 ×10−9 1.37×10−9 0.593991 0 0.00824132 4.04 ×10−11 9.99×10−11 2.66 ×10−11 6.58× 10−11 0.004337360 0 1.25 × 10−16 2.50 × 10−16 0.985943 0 0 0.000363300 0.000897922 0.000242251 0.000598739 0.01195500 0 0 3.48 × 10−25 0.986829 0 0 0 0 0.000242557 0.000599495 0.01232870 0 0 0 0.987421 0 0 0 0 0 0 0.01257870 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

)Now, differing from the previous disease, if we assume we start with a maternally immune

individual (as this model is based around children catching measles and being vaccinated at

an early age), we can just concentrate on the first row of P365. The chance of any individual

65

having had measles at any point in the first year of analysis (hence in the first year of their

life) is the sum of the following probabilities:

pME(365)+ pMI

(365)+ pMR(365)+ pMX

(365)+ pMT(365)+ pMY

(365)+ pMU(365)+pMZ

(365)

The first three probabilities of these eight represent those who were immune to start with but

have ended day 365 being exposed, infectious or having recovered, respectively. The last five

of these probabilities represent those who were immune but have ended the year having

passed away naturally after being exposed, infectious or having recovered, or died due to

measles whilst they were exposed or infectious.

Hence, the probability that any individual in the US contracts measles in this year is:

pME(365)+ pMI

(365)+ pMR(365)+ pMX

(365)+ pMT(365)+ pMY

(365)+ pMU(365)+pMZ

(365)=0.00000000177157+0.00000000117831+0.225990+0.0000000000149744+0.0000000000370103+0.00000000000969920+0.0000000000239723+0.00110342≅ 0.227093

This shows that each initially immune (i.e. someone who has just been born) individual has a

22.7 % probability of contracting measles during year one. However, due to vaccinations, not

all of these 22.7% of the population will have caught measles. Some will have become

immune due to vaccination – approximately 91.9 % in this year. So to account for these, we

need to take away 91.9 % from 100 % to find the proportion that were not vaccinated, and

then multiply the two resulting percentages together:

(1−0.919 )∗0.227093=0.081∗0.227093=0.018394533

Hence, an individual with maternal immunity will have a 1.84 % probability of catching

measles in the first year of life.

66

However, if we assume we start with a susceptible individual (someone who has already lost

their maternal immunity but has not yet contracted any form of measles), we can look at the

second row of P365. The chance of any of these individuals having had measles at any point in

the first year of analysis is the sum of the following probabilities:

pSE(365)+ pSI

(365)+ pSR(365)+ pSX

(365)+ pST(365)+ pSY

(365)+ pSU(365)+pSZ

(365)

Again, the first three probabilities of these eight represent those who were susceptible to start

with but have ended day 365 being exposed, infectious or having recovered, respectively. The

last five of these probabilities similarly represent those who were susceptible but have ended

the year having passed away naturally after being exposed, infectious or having recovered, or

died due to measles whilst they were exposed or infectious.

Hence, the probability that any susceptible individual in the US contracts the disease in this

year is:

pSE(365)+ pSI

(365)+ pSR(365)+ pSX

(365)+ pST(365)+ pSY

(365)+ pSU(365)+pSZ

(365)=0.00000000201838+0.00000000137033+0.593991+0.0000000000404158+0.0000000000998904+0.0000000000266171+0.0000000000657859+0.00433736≅ 0.598328

This shows that each initially susceptible individual has a 59.8 % probability of contracting

measles during the year. Again, however, due to vaccinations, not all of these 59.8 % of the

population will have caught measles. Some will have become immune due to vaccination –

91.9 % in this year. To account for these, we again need to multiply 8.1% by the other

percentage:

(1−0.919 )∗0.598328=0.081∗0.598328≅ 0.0484646

Hence, an unvaccinated but susceptible individual will have a 4.85 % probability of catching

measles.

67


Like before, 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, just as we had with the previous disease. As n → ∞, the matrix

( P365 )n=P365∗n → Pn. So raising the original matrix to exceptionally high powers will give the

steady proportion of a population (based on US data) that will contract measles.

After some manual checking (using integer powers of 10 for n), just like before both

P365∗100000=P36500000 and P365∗1000000¿ P365000000 give the same matrix (using Maple, omitted due

to size, see Appendix A9). This implies that the limiting distribution, Pn, is (to appropriate


Pn=¿

(0 0 0 0 0 0.0140406 0.0133960 6.69 × 10−11 1.65 ×10−10 4.46×10−11 1.10× 10−10 0.9725630 0 0 0 0 0 0.0135868 6.78 × 10−11 1.68 ×10−10 4.52×10−11 1.12×10−10 0.9864130 0 0 0 0 0 0 0.000363300 0.000897922 0.000242251 0.000598739 0.9978980 0 0 0 0 0 0 0 0 0.000242557 0.000599495 0.9991580 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

)As before, 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 Pn still add up to one.

68

From this, the result here shows that given enough time, all individuals will pass away (all

elements in the first five columns are zero so no-one ends up in the M , S, E, I or R states),

which is obvious. By adding pMV( n) + pMW

(n ) , it also shows a steady percentage of the population

who will have died directly without ever getting the disease or being vaccinated (assuming

immune start), so:

pMV( n) + pMW

(n ) =0.0140406+0.0133960=0.0274366

So 2.74 % of the population will never contract measles even when not vaccinated. By taking

pMZ( n) and multiplying by 0.919 to find out the percentage of those who are vaccinated

(assuming the rate of 91.9 % per year is kept steady), we get:

pMZ( n) ∗0.919=0.972563∗0.919=0.893785397

So 89.4 % of the population will have been vaccinated and be safe from measles at any point.

This seems counter-intuitive if a constant 91.9 % are being vaccinated, however, this value

accounts for the fact that individuals who have been vaccinated can die as others are being

immunised, and others can be born and lose their maternal immunity. The difference between

pMZ( n) and this answer gives the percentage of people who do contract measles but eventually

recover from it:

0.972563−0.893785397=0.078777603

So 7.88 % of those who wind up in the recovered state (or associated death state) will have

contracted measles and lived (with no prior vaccination). The sum of this value and the other

four non-zero values in this row gives the percentage of people who will get measles at some

point in their life (whether they survive from it or not):

pMX( n) + pMT

( n) + pMY( n) + pMU

(n ) +0.078777603=0.0787776033870348

69

So 7.88 % of the population will contract measles in a steady population that stays constant

for many years. As this value is very similar to our previous value, most people who contract

measles will live. The difference will give the overall exact percentage who will die:

0.0787776033870348−0.078777603=0.0000000003870348

So 0 .000000039 % of the long-term population will die from measles.

70

1.6. Model 3 adaptation – H1N1


Again, much in a way similar to the original deterministic model given in 3.3.1., the

probabilistic model starts with a compartmental model, with a probability of entering the state

in question is given instead of deterministic parameters.

This gives an initial compartmental model for H1N1 (Fig. 6.5. and Fig. 6.6.):

Figure 6.5.: Stochastic model for H1N1

71

Probability of loss of active immunity

Exposed Virus

Deaths

Exposed Natural Deaths

Infectious Natural Deaths

Susceptible Deaths

Recovered Deaths

Infectious Virus

Deaths

Probability of gain of active immunity (vaccination)

Probability of development

Susceptible RecoveredInfectiousExposed

Probability of infection


Probability of virus death

Probability of natural

death

Probability of virus death










Probability of staying recovered


Probability of staying exposed

Probability of staying susceptible


Figure 6.6.: Stochastic model for H1N1 with mathematical symbols

Here, we are trying a simplification on this final model by merging together the exposed and

infectious death states but keeping natural and virus deaths separate. This will make the

calculations simpler later as we have two fewer states, hence two fewer rows and columns in

the transition matrix P. This is considered as this model loops and cannot be simplified

further in any other way. No ‘dummy’ or ‘bin’ states are added here – a dummy state would

be pointless as you can only leave it and never enter it (when dealing with each individual

separately as per this model, we are only interested in the states that can be entered), and the

bin state would be an absorbing state, so the individual cannot leave. However, as opposed to

the traditional model for H1N1 where anyone entering the bin would be ‘replaced’ by another

72

individual from the dummy state, this would not occur here. The model here, though, is

accepted as a valid comparison to that of above.

Again, 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 as before, this has been added and is represented by a

circular arrow that has the state it left as its target.




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 (four ‘alive’ states, three ‘natural death’ states and

one ‘virus death’ state), so this gives us the matrix P as follows:

P=(pSS pSE pSI pSR pSW pSX pSY pSZ

pES pEE pEI pER pEW pEX pEY pEZ

p IS p IE p II pIR p IW p IX p IY p IZ

pRS pℜ pRI pRR pRW pRX pRY pRZ

pWS pWE pWI pWR pWW pWX pWY pWZ

p XS pXE pXI pXR pXW pXX pXY p XZ

pYS pYE pYI pYR pYW pYX pYY pYZ

pZS pZE pZI pZR pZW pZX pZY pZZ

)Again, the probability of staying dead when already in a dead state is one, so:

pWW=pXX=pYY =pZZ=1


pWS=pWE=pWI=pWR=pWX=pWY=pWZ=0

73

pXS=pXE=pXI=pXR=pXW=pXY=pXZ=0

pYS=pYE=pYI=pYR=pYW=pYX=pYZ=0

pZS=pZE=pZI=pZR=pZW=pZX=pZY =0

There is no chance in this model that an individual can jump straight from being susceptible

to being infectious or from being exposed to having recovered, so:

pSI=pER=0

Individuals also cannot travel backwards in the model in most cases, so:

pES=p IS=p IE=pℜ=pRI=0

Each compartment that is not a dead state has its own respective dead state (with exposed and

infectious merged); individuals who naturally die whilst susceptible go into the W state, those

who naturally die whilst exposed or infectious go into the Y state and those who naturally die

after having recovered go into the Z state. Similarly, those who die due to H1N1 whilst

exposed or infectious go into the X 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

pEW=pEZ=0

p IW=p IZ=0

pRW=pRX=pRY=0

As before, 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. However, it is valid to merge the

death states for exposed and infectious individuals as when summing up whether an

individual had died whilst affected by H1N1 (whether due to the virus or not), these states

would be added together anyway.

74

The remaining fifteen probabilities have values which are neither zero nor one. These can be

worked out as follows (all probabilities are rounded only at the final answer and are rounded

to six significant figures):

As already stated, the average life expectancy of an individual in the United States (US) is

currently 79 years (WHO 2012). So, 79 years in days is 79 years ¿365 days per year ¿28835

days. The reciprocal of this gives the probability of dying naturally each day. An assumption

continuing to be made here is that the chances of dying naturally are equal regardless of

whether the individual in question is susceptible, exposed, infectious or has recovered from

H1N1, so:

pSW =pEY=p IY=pRZ=1

28835≅ 0.0000346801

In the US during the 2009 H1N1 epidemic, there were approximately 60000000 (sixty

million) cases of H1N1 influenza (CDC 2010a). So the chance of contracting it during this

year is one in sixty million, so dividing by 365 will give the chance per day:

pSE=1

60000000∗365≅ 0.0000000000456621

Again, during the 2009 epidemic, the US had 12270 deaths in the year (CDC 2010a). So, the

reciprocal of this quantity further divided by 365 will give the chance of dying from H1N1 if

you have the virus on any given day. An assumption that dying due to H1N1 whether the

individual is exposed or infectious is equivalent regardless of the state they are in is made

here:

pEX=p IX=1

12270∗365≅ 0.000000223287

The US government administered a figure of 91000000 (ninety-one million) (Drummond

2010) vaccinations during the year of the outbreak. That works out to approximately 249315

vaccines performed each day. The reciprocal of this gives the chance of being vaccinated each

day:

pSR=1

249315=0.000004 010989≅ 0.00000401099

75

The latency period of the virus is approximately 2.5 days (Balcan et al. 2009, CDC 2010b),

and taking the reciprocal of this gives pEI, so:

pEI=1

2.5=0.4

The recovery period after becoming contagious is approximately four days on average (Asp

2009), so again, taking this value and finding the reciprocal gives p IR, so:

p IR=14=0.25

The final probability we need to calculate from data is the chance of someone who has

recovered (after either having H1N1 or having the vaccination) loses their active immunity

and becomes susceptible again. Approximately 5 % of individuals who recover or are

vaccinated lose their immunity at some point in the year or do not gain immunity from the

vaccination in the first place (Greenberg et al. 2009). This implies our value for pRS is 5 %

divided by 365:

pRS=0.05365

≅ 0.000136986

Just as before, like the other stochastic models, 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=1−0.0000000000456621−0.00000401099−0.0000346801=0.9999613088643379

pEE=1−0.4−0.000000223287−0.0000346801=0.599965096613

p II=1−0.25−0.000000223287−0.0000346801=0.749965096613

pRR=1−0.000136986−0.0000346801=0.9998283339



76

P=¿

(0.999961 4.57×10−11 0 4.01×10−6 3.47×10−5 0 0 0

0 0.599965 0.4 0 0 2.23×10−7 3.47× 10−5 00 0 0.749965 0.25 0 2.23×10−7 3.47× 10−5 0

0.000136986 0 0 0.999828 0 0 0 3.47× 10−5

0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1


As before, P365 gives the probabilities that an individual will end in any desired state, given

any initial starting state, at the end of one year. Hence, using Maple (omitted due to size, see

Appendix A10), this gives the 365-step matrix (given to appropriate significant figures):

P365=¿

(0.986012 1.13 ×10−10 1.80 ×10−10 0.00140919 0.0125697 2.37 × 10−14 3.68 ×10−12 9.01×10−6

0.0472912 5.36 ×10−12 8.48 ×10−12 0.940129 0.000296913 1.45 ×10−6 0.000225382 0.01205640.0476128 5.40 ×10−12 8.54 ×10−12 0.939808 0.000301039 8.93 ×10−7 0.000138701 0.01213890.0481270 5.46 ×10−12 8.64 ×10−12 0.939294 0.000307686 5.72× 10−16 8.89 ×10−14 0.0122710

0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

)Like the first stochastic model, if we assume all individuals start off susceptible, we can just

concentrate on the first row of P365. Approximately 5 % of all individuals who are in the

recovered state lose their immunity and if 91 million people per year are immunised, this

gives a vaccination proportion in the first year of 91000000

316148990≅ 0.287857. This allows us to

work out the proportion of pSR(365) and pSZ

(365) that actually have had the virus at some point and

77

are not in those states because of vaccination and to account for those who loop round the

system:

pSE(365)+ pSI

(365)+(1−0.287857+0.05 )∗pSR(365 )+ pSX

( 365 )+ pSY(365 )+ (1−0.287857+0.05 )∗pSZ

( 365 )=pSE(365 )+ pSI

( 365 )+0.762143∗pSR( 365 )+ pSX

(365 )+ pSY(365 )+0.762143∗pSZ

(365)

The first three probabilities of these six represent those who were susceptible to start with but

have ended day 365 being exposed, infectious or having recovered, respectively. The

coefficient of pSR(365) accounts for looping and loss of immunity, as not all individuals in the

recovered state after one year will have necessarily had H1N1. The last three represent those

who were susceptible but have ended the year having passed away naturally after being

exposed, infectious or having recovered, or died due to the virus whilst they were exposed or

infectious. Again, the coefficient of pSZ(365) acts similarly to above.

Hence, the probability that any individual in the US contracts H1N1 in this year is:

pSE(365 )+ pSI

( 365 )+0.762143∗pSR( 365 )+ pSX

(365 )+pSY(365 )+0.762143∗pSZ

(365 )=0.000000000112559+0.000000000180098+0.762143∗0.00140919+0.0000000000000236918+0.00000000000367973+0.762143∗0.00000900923≅ 0.00108087

Hence, an individual in this population will have a 0.108 % probability of contracting H1N1

in this year of life.

Individuals who die directly due to H1N1 in this year can be estimated as follows. All

individuals who die due to the virus must end up in the X state. Once in there, they cannot

leave, so the probability of dying from the virus is simpler to see than the cases of H1N1:

pSX(365 )=0.0000000000000236918


Like before, 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

78

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, just as we had with the previous disease. As n → ∞, the matrix

( P365 )n=P365∗n → Pn. So raising the original matrix to exceptionally high powers will give the

steady proportion of a population (based on US data) that will contract H1N1.

After some manual checking (using integer powers of 10 for n), just like before both

P365∗100000=P36500000 and P365∗1000000¿ P365000000 give the same matrix (using Maple, omitted due

to size, see Appendix A11). This implies that the limiting distribution, Pn, is (to appropriate


Pn=¿

(0 0 0 0 0.977168 0.00000000000186701 0.000000000289977 0.02283190 0 0 0 0.779583 0.00000145112 0.000225382 0.2201900 0 0 0 0.779651 0.000000893025 0.000138701 0.2202090 0 0 0 0.779760 0.00000000000148983 0.000000000231396 0.2202400 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

)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

again, is obvious. By adding pSX( n) + pSY

(n ) , it shows a steady percentage of the population who

will have caught H1N1 and died whilst infected (naturally or due to the virus). Then adding a

proportion of those who ended up dying whilst susceptible or recovered will give us the total

number of cases. This proportion is worked out as follows:

As before, 0.762143∗pSZ(n ) would give the correct proportion for this compartment. We need to

work a similar one out for the pSW( n) element. Now, (1−0.762143 )∗pSW

( n) would give those who

79

have been round the system before, so working these two values out and then adding to pSX( n)

and pSY( n) gives:

pSX( n) + pSY

(n ) +0.762143∗pSZ(n )+ (1−0.762143 )∗pSW

(n ) =0.00000000000186701+0.000000000289977+0.762143∗0.0228319+0.237857∗0.977168≅ 0.249904

So 25.0 % of the population will contract H1N1 at some point during their life, assuming a

long-term outbreak is sustained.

Deaths from the virus are much simpler to work out, it is just the value of pSX( n) :

pSX( n) =0.00000000000186701

So 0.000000000187 % of the population will contract H1N1 and go on to die directly due to

the virus, in a steady population that stays constant for many years. As this value is very

small, we can assume a low death rate from H1N1 if an outbreak or epidemic is sustained.

80

2. Results and model comparisons

Using the results above from all six models, presented here is an overview and comparison of

how accurate each model was. Although comparisons are made between each model and the

number of cases, the main comparison is that of how much the models differ in their

predictions in the long-term (i.e. non-trivial endemic steady states against limiting

distributions). Note that for chickenpox and measles, actual cases and actual deaths are

multiplied by the life expectancy in the relevant country as these diseases are constantly

around and never truly flare up or outbreak under normal circumstances. However, for H1N1,

there is approximately one outbreak in each lifetime (based on the space between the previous

two epidemics) so actual cases per lifetime are just based off a single outbreak. The UK

population is taken to be 64.1 million (ONS 2014) and the US population as 316148990 (U.S.

Census Bureau 2013). These errors are shown in Table 7.1., below.

As the maximum positive error in Table 7.1. is theoretically infinite, yet the maximum

negative error is only −100 % (as the lowest prediction is clearly 0, so

0actual cases /deaths

−1=−100 % for all cases or death values), it could be seen that all

under-predictions are closer to the actual values than any over-predictions. To counter this

unwanted effect, taking the logarithm (base 10) of each of the error calculations plus one (i.e.

just the fractional part) will allow simpler and more visual comparisons, and permit each error

to be seen as to how many orders of magnitude difference they are away from the real number

and each other’s predictions. These calculations are also shown in Table 7.2., below.

81

Disease Chickenpox (UK) Measles (US) H1N1 pandemic (US)

Deterministic

model cases

prediction per

lifetime:

0.7389033941048∗64100000≅ 473637080.07497197730196∗316148990≅ 237023150.1967780042823∗(1+0.02325592285 )∗316148990≅ 63657945

Stochastic model

cases prediction

per lifetime:

0.315864254∗64100000≅ 202468990.0787776033870348∗316148990≅ 249054590.249904∗316148990≅ 79006897

Actual number of

cases per

lifetime:

365370∗81=29594970 55∗79=4345 60000000 per outbreak∗app . 1 outbreak per

lifetime=60000000

Deterministic

model deaths

prediction per

lifetime:

N/A 0.00003149821716∗316148990≅ 99580.0009612958123∗316148990≅ 303913

Stochastic model

deaths prediction

per lifetime:

N/A 0.0000000003870348∗316148990≅ 0.1223610.00000000000186701∗316148990≅ 0.000590253

Actual number of

deaths per

lifetime:

N/A 55∗79∗1.51000

≅ 712270 per outbreak∗app .1 outbreak per

lifetime=12270

Deterministic

model case error:

4736370829594970

−1≅+60.0 %error23702315

4345−1≅+5454 %error

6365794560000000

−1≅+0.061 % error

Stochastic model

case error:

2024689929594970

−1≅−31.6 %error24905459

4345−1≅+5731 %error

7900689760000000

−1≅+31.7 % error

Deterministic

model death error:

N/A 99587

−1≅+1422 % error30391312270

−1≅+23.8 % error

Stochastic model

death error:

N/A 0.1223617

−1≅−98.3 %error0.000590253

12270−1≅−100 %error

Table 7.1.: Comparison of models to actual cases and deaths plus errors in predictions

Disease Chickenpox (UK) Measles (US) H1N1 pandemic (US)

82

Deterministic

model case error:log10( 47363708

29594970 )≅ 0.2042log10( 237023154345 )≅ 3.7368 log10( 63657945

60000000 )≅ 0.0257

Stochastic model

case error:log10( 20246899

29594970 )≅−0.1649log10( 249054594345 )≅ 3.7583 log10( 79006897

60000000 )≅ 0.1195

Deterministic

model death

error:

N/Alog10( 9958

7 )≅ 3.1531 log10( 30391312270 )≅ 1.3939

Stochastic model

death error:

N/Alog10( 0.122361

7 )≅−1.7575log10( 0.00059025312270 )≅−7.3178

Table 7.2.: Logarithmic model errors in predictions

2.1. Model 1 comparisons

As can be seen from the tables above, the results for each disease vary, especially when

comparing each model with the same disease. Taking chickenpox for the first comparison, it

can be seen that the traditional model over-predicts the number of cases by approximately

60%, whereas the stochastic model under-predicts them by just over 30%. Looking at the

logarithmic figures directly, both predictions are less than an order-of-magnitude difference

away from the real-world case number (within one either side of zero), but more importantly,

both the predictions themselves are less than an order-of-magnitude difference away from

each other (|0.2042−(−0.1649 )|=0.3691<1). This implies that the basis for the adaptation to

construct the stochastic model for chickenpox was good and provided a good yet novel model.


For measles, the results are not as reliable. Both the standard model and the stochastic one

predict case values well above that of real life, giving huge over-predictions of well into the

83

order of the five-thousand percentage range. Looking at the logarithmic figures, both

predictions are huge, being an order-of-magnitude of almost four away. However, like for

chickenpox, the adapted model does follow the original model quite well; both the predictions

themselves are again, less than an order-of-magnitude difference away from each other (

|3.7368−3.7583|=0.0215≪1). This error is likely due to the inclusion of a maternally

immune state. In most models, individuals born enter the susceptible state but here, they do

not. This had added huge complexity to the models, such as how many individuals should

start maternally immune during numerical analysis and which row or combination of rows of

the limiting distribution matrix should be used from the stochastic approach. Deaths from the

disease are included in this model analysis too, yet both are wildly inaccurate. The standard

model over-predicts by well over one-thousand percent and the adapted model under-predicts

virtually no deaths (less than one person would die). The logarithmic values do differ here, as

they are an order-of-magnitude of almost five away from each other (

|3.1531−(−1.7575 )|=4.9106). This implies that the mortality number predictions for both

models are flawed, not only in comparison to the actual deaths from measles but in

comparison to each other.


Finally, for the H1N1 virus, the results are better. Both the standard model and the stochastic

one predict case values only a little more than that of real life, giving slight over-predictions

in both approaches. The errors are similar, with the standard model being closer at a lowly

error of 0.061% and the stochastic model not far behind at 31.7 %. Although percentage point

wise, these look vastly different, looking at the logarithmic figures, both predictions are very

small, and are well within an order-of-magnitude difference away from the real-life cases.

Compared to each other, they give the closest comparison of all parts of all accurate model

adaptations, having a tiny difference in relative errors (|0.0257−0.1195|=0.0938≪1). Like

measles, deaths from H1N1 are included in this model analysis too, however just as before,

both are inaccurate. The standard model does well and only over-predicts by approximately

five times the actual value. The stochastic adaptation does far worse, having an even smaller

value for mortality numbers than measles did, even though more individuals would die from

84

H1N1 in real life. The logarithmic values differ hugely, as they are an order-of-magnitude of

almost nine away from each other (|1.3939−(−7.3178 )|=8.7117), giving our worst prediction

for any part of any model, contrasting the accuracy of the case prediction for this disease. This

implies that the mortality number predictions for both the standard and stochastic models are

inconsistent yet again, in comparison to the actual deaths in the real world and to each other.

85

3. Conclusions

3.1. Conclusion and critical evaluation

In conclusion, the stochastic approaches attempted in this paper worked well only when the

original deterministic model was relatively accurate. The models for chickenpox and the

H1N1 virus both estimated very accurate case numbers, as seen in Table 7.2., and the

following analysis showed that both models were accurate to each other as the difference

between their respective logarithmic values were well within one. The models for measles,

both deterministic and probabilistic, were not accurate to real life, well over-predicting the

actual number of cases. This was due to the complexity of the model and the fact that not all

individuals start out maternally immune in real life. That is, starting analysis at any one point,

a certain (low) proportion of the population would be maternally immune (the long-term

Runge-Kutta analysis suggests this as approximately 1.1 %). However, even when starting

with M (0 ) at around that point, the analysis was still not close to approaching a real life case

value.

The mortality modelling parts for the latter two models which did include it was also not

accurate for both types of models. This implies that the setup of the mortality sections was

flawed from the start; the deterministic model did not have the correct structure for modelling

deaths due to the disease, so when adapted and changed into the stochastic method, this

suffered the same fate. However, the stochastic model predicts very few deaths (even as low

as a fraction of a person), whereas the deterministic models well over-predicted the mortality

from disease. This makes sense, however, as the rate in the deterministic model would slowly

‘build up’ the deaths and they will never fall, where the stochastic model has a constant an

unchanging probability each day due to the fact it has been modelled as a simple Markov

chain.

86

Comparing the models directly to each other shows the main point of this research; stochastic

modelling using Markov chains is feasible, relatively simple and provides results as accurate

as those of standard deterministic modelling using systems of ordinary differential equations.

The initial models are built on the same state-space within each disease, and the same

characteristics of each disease are used between the deterministic and stochastic models. But

the steps each model takes in the analysis differ based upon fixed rates and fixed probabilities,

respectively. This allows for the most reliable comparison between two completely different

methods, negating any effects besides the direct comparisons intended to be made here. To

improve the robustness of each model, more data over longer period of time (i.e. longer than

one year) could be used; however, this is unlikely to be accurate or available for most diseases

where a lifetime comparison is to be completed (as data would need to stretch back

approximately 80 years).

Furthermore to the issues encountered whilst deciding on reliability of the results,

timekeeping and planning ahead during the entire dissertation was difficult. I had originally

planned to get all of the analysis of all of the models finished with over a month to go, thus

allowing me time to read many more articles that I did and take my time during the write-up

and critical evaluation of other people’s work and my own. However, unsurprisingly, my

plans had to change and adapt accordingly after my models became complicated from the

outset. This meant my final model was only finished with two weeks to go, halving the

amount of time I had available. Obviously, the write-up and evaluations had been going on

along the way and had been started well before this, but some elements could not be fully

included until my own analysis was complete. My time schedule was shorter than anticipated

and involved more work than originally planned, but I feel I coped with the demand and

workload well over the period since I started this dissertation.

Overall, although the stochastic approach did not reduce the reliability or precision of the

cases predicted, it did not improve on them either. That way, presented here is not a

progression of the original model into a better one, but more of an alternative approach which

can be used in the right circumstances if the researcher believes it to be more beneficial. This

87

then poses a new question; is the novel stochastic method worth implementing in place of an

already established method that is used by mathematicians and biologists worldwide?

3.2. Further work

Models that use differing rates and probabilities of infection for each time period could be

used. These models would, however, be far more complex, possibly even too complex to

apply to a disease analysis. Constant rates and probabilities were used throughout this

research; firstly to keep the simplicity of each model and remove over-complicating the

problem, and secondly to allow for the use of a Markov chain in the adaptations. Markov

chains have a property that requires constant probabilities for every step of the analysis,

regardless of the point the analysis is at. If the probability was variable over time, the limiting

distribution could not be found and hence, this research would not be possible. More work

could be undertaken where probability changes. This would either require a completely

different stochastic process to be used or Markov chains with a theoretically infinite state-

space where the infectious state is split up into states representing each day of a disease. The

states would have probabilities associated with them that the individual could either pass on to

the next day (an average case) or recover and leave the infectious part of the model. These

states could be limited in number (say N ) but this would force any individual in the system to

recover within N stages (for example, N days) and would not realistically and truly give an

appropriate view of the disease’s traits. Besides these points, a transition matrix of order N

would need to be used; where small errors in this method would be enhanced to unreliable

levels after hundreds of iterations of matrix multiplication. If another stochastic process was

used besides Markov chains, probability density functions could be introduced, but as stated

before, the complexity of this system would shoot up and make it unviable for use instead of a

deterministic model. It should be noted that over-fitting of the data to new models can occur if

too many factors are allowed for and including in working out the probabilities associated.

This would present a perfect model for the problem at hand but later use of the model will

likely be questionable.

88

Further work would need to be attempted to model deaths more accurately from diseases,

possibly even as far to say as developing specific and separate ‘mortality model’ for the

disease in question where the entire population is assumed to be infected and we are

effectively ‘zooming in’ on just the virus deaths part to each model. Demography in this

model would not represent births and natural deaths, but more like the contraction and

recovery from the disease. Constant proportionality could be used in this model if the

assumption is made that the long-term steady state has already been reached so I (t ) will be

unchanged over time.

Spatial elements would be the next logical step from this research; using systems of partial

differential equations to represent a disease flow within a population and show how exactly

the disease spreads. The space in the model could represent ‘hotspots’ where a disease

initially flares up, so that certain individuals in the model are technically immune until the

individuals with the disease spread to areas nearby. However, without extensive

simplifications, the stochastic method using matrices may struggle to keep up with the

relatively ease of use and application of the traditional modelling methods. Another type of

‘grouping’ could be established, such as specific age ranges being more susceptible to certain

diseases or individuals becoming more or less susceptible at certain times of the year or

period of analysis. The easiest way to analyse age ranges may be to have separate models for

each age group and then combine the results at the end, using real life age distributions to

fully evaluate the data. Different parameters or probabilities could then be used to account for

certain ages being more likely to catch a disease. The seasonal effects of a disease could be

modelled similarly; possibly four models representing each season or even twelve

representing each month. The initial values of one model would have to be the final values in

the analysis of the previous one, meaning the differing parameters in the next model could try

to accurately account for temperature or other environmental factors.

89

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Weiss, H. (2013) The SIR model and the Foundations of Public Health. MATerials

MATemàtics (MAT2), 2013(3).

World Health Organization (WHO) (2012) Global Health Observatory Data Repository

[online]. WHO. [Accessed 12 September 2014]. Available at:

<http://apps.who.int/gho/data/node.main.688?lang=en>

Yorke, J.A. and London, W.P. (1973) Recurrent outbreaks of measles, chickenpox and

mumps: II. Systematic differences in contact rates and stochastic effects. American Journal of

Epidemiology, 98(6), pp.469-482.

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5. Bibliography

Breathing earth

Age differences H1N1 http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=19344

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6. Appendices

6.1. Appendix A

Omitted Maple outputs

1.1.1. Appendix A1

Endemic steady state values for deterministic model 1 – Chickenpox

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1.1.2. Appendix A2

Endemic steady state for

deterministic model 2 – Measles

(with trivial steady state shown

first)

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1.1.3. Appendix A3

Endemic steady state values for deterministic model 2 – Measles

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1.1.4. Appendix A4

Endemic steady state for deterministic model 3 – H1N1 (with trivial steady state shown

second)

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1.1.5. Appendix A5

Endemic steady state values for deterministic model 3 – H1N1

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1.1.6. Appendix A6

365-step matrix for stochastic model 1 – Chickenpox

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1.1.7. Appendix A7

Limiting distribution (both matrices equal) for stochastic model 1 – Chickenpox

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1.1.8. Appendix A8

365-step matrix for stochastic model 2 – Measles

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1.1.9. Appendix A9

Limiting distribution (both matrices equal) for stochastic model 2 – Measles

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1.1.10. Appendix A10

365-step matrix for stochastic model 3 – H1N1

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1.1.11. Appendix A11

Limiting distribution (both matrices equal) for stochastic model 3 – H1N1

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1.2. Appendix B

4th-order Runge-Kutta numerical analysis using Microsoft Excel

1.2.1. Appendix B1

4th-order Runge-Kutta numerical analysis of deterministic model 1 – Chickenpox (first and

last pages of analysis only – see Appendix C for full data)

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1.2.2. Appendix B2

4th-order Runge-Kutta numerical analysis of deterministic model 2 – Measles (first and last

pages of analysis only – see Appendix C for full data)

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1.2.3. Appendix B3

4th-order Runge-Kutta numerical analysis of deterministic model 3 – H1N1 (first, stage when

the steady state values are reached and last pages of analysis only – see Appendix C for full

data)

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1.3. Appendix C

Compact disc (CD) containing full data for all model analysis (4th-order Runge-Kutta

numerical analysis using Microsoft Excel) and full electronic copy of dissertation

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