EFFECT OF THE REED-FROST COMPUTATIONAL MODEL

ON A NUMERICALLY BASED SURVEILLANCE SYSTEM

USED IN CONTINUOUS MONITORING OF INFECTIOUS

DISEASES

BY

BASASON HUNVOUNOPWA BERNARD

M.Sc. MATHEMATICS

NOVEMBER, 2014

Effect of the Reed-Frost Computational Model on a Numerically Based

Surveillance System Used in Continuous Monitoring of Infectious Diseases

By

Basason Hunvounopwa Bernard

Matriculation Number

NSU/NAS/M.Sc./006/10/11

B.Sc. Statistics (Unimaid)

Supervisor

Prof. Peter Adewumi Osanaiye

A dissertation submitted in partial fulfillment of the requirement for the

award of masters degree in mathematics of the school of postgraduate

studies of Nasarawa State University, Keffi.

September, 2014

ii

DECLARATION

I hereby declare that this thesis has been written by me and it is a report

of my research work. It has not been presented in any previous application for

M.Sc. Mathematics. All quotations are indicated and sources of information

specifically acknowledged by means of references.

BASASON HUNVOUNOPWA BERNARD

iii

CERTIFICATION

This dissertation ”Effect of the Reed-Frost Computational Model on a Numer-

ically Based Surveillance System Used in Continuous Monitoring of Infectious

Diseases” meets the regulations governing the award of Masters Degree in

Mathematics, faculty of Natural Science of Nasarawa State University, Keffi.

Prof. Peter Adewumi Osanaiye Date

(Supervisor)

Dr. Mohammed Yau Date

(Internal Examiner)

Prof. Agwu Nnnanna Nwojo Date

(Head of Department)

Prof. Shilgba Date

(External Examiner)

(Dean, School of Postgraduate Studies) Date

iv

ACKNOWLEDGMENT

First and foremost, I offer my sincerest gratitude to God Almighty for the

opportunity afforded me to carry out this research. At some point it seemed

to me an insurmountable task, but at last I can heave a sigh of relief and

attribute all success to the Almighty God.

I particularly want to say a special thanks to my wonderful supervisor Profes-

sor Osanaiye Peter, who has been a great support throughout this study, with

his patience and knowledge whilst allowing me the room to work in my own

way. I attribute the level of success to his fatherly encouragement and effort

without which this report, too, would not have been completed or written.

One simply could not wish for a better or friendlier supervisor.

I equally express my profound gratitude to the entire staff and students of

Mathematics department, with their unwavering support, cooperation, and

commitment. Especially those, I for one was able to tap from their wealth

of knowledge including Prof. Onumanyi Peter, Prof. Shehu S. Farinwata,

Prof. Fatokun Johnson O., Prof. Anande R. Kimbir, and Prof. Nwojo Agwu

Nnanna etc. I also appreciate Dr. Nweze and Mr. Chaku for their encour-

agements.

v

I count myself fortunate to be with such an excellent and caring team of

friends and colleagues. My thanks to Mr. Phillip (our class representative),

Mrs Amuno, Mrs Zainab, Mal. Badau, Mal. Farouk. Special thanks to Mr.

Chidi Nwosu who has made a huge impact on me even outside the academic

environment and whose calls, gifts, care, concern, and support I cannot quan-

tify because you went the extra mile; Mr. Alasa Aliyu (my H.O.D) I truly

appreciate your encouragement because you were willing to create a vacancy

for me; Mr. Ioorpu, Mr. Said Bello etc. I sincerely value all your friendship

because you showed what true friends are. I cannot appreciate you enough

Mr. Benard Alechenu, rest assured I highly esteem you.

I salute all my exceptional colleagues in the office, Daniel Oghojafor, Tayo

Babalola, Oluwajuwonlo Oluwole and my main man Andrew Achille for all

your dedication and friendship.

Samson Patrick, Gilbert Nanyiso, and my dearest friend Donatus Onoja,

without your influence and support in my life, it would have been much more

difficult for me to finish this work. God bless all of you more abundantly.

Finally, I wish to thank my ever-there family members Dakaimi, Doreen and

especially Mr. and Mrs. David Malik, only God Almighty will bless all your

kindness and love to me throughout my entire stay with you. My superhero

and Huboshi as well as the rest of my siblings, I cannot wish for a better

family.

vi

DEDICATION

It is with a mix-blend of bitter-sweet memories I dedicate this research work

to the Basasons’ who are my greatest inspiration.

vii

ABSTRACT

This research work is aimed at studying the effect of the use of mathematical

method in conjunction with existing numerically based surveillance system

at monitoring available routinely collected patient records that could lead

to early detection of periods of deteriorating standards. The methodologies

employed in this study include the use of the Fourth Order Runge-Kutta and

Cumulative Sum (CUSUM) control charts.The findings of this study suggest

that after using the Reed-Frost epidemic model to model data collected on

cholera outbreak, a combination of Runge-Kutta Fourth Order and CUSUM

was more effective in early detection of the epidemic outbreak than applying

only CUSUM on the raw data. This study contributes to existing knowledge

in the area of monitoring epidemic outbreak, especially to obtain relevant

information regarding a departure from an acceptable pattern considered as

epidemic requiring some intervention. In such cases, the early detection of

shifts in rates of outbreak is critical since it will result in prompt investigation

of the cause and procedural changes.

viii

Contents

DECLARATION iii

CERTIFICATION iv

ACKNOWLEDGMENT v

DEDICATION vii

ABSTRACT viii

1 INTRODUCTION 1

1.1 Background of the study . . . . . . . . . . . . . . . . . . . 1

1.2 Statement of the problem . . . . . . . . . . . . . . . . . . 2

1.3 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Significance of the study . . . . . . . . . . . . . . . . . . . 3

1.6 Research questions . . . . . . . . . . . . . . . . . . . . . . . 4

1.7 Statement of the hypothesis . . . . . . . . . . . . . . . . . 4

1.8 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.9 Scope of the study . . . . . . . . . . . . . . . . . . . . . . . 5

1.10 Definition of some terms . . . . . . . . . . . . . . . . . . . 5

ix

1.11 Organization of the study . . . . . . . . . . . . . . . . . . 6

2 LITERATURE REVIEW 7

2.1 Infectious Disease Dynamics . . . . . . . . . . . . . . . . . 7

2.2 The Simple Kermack-McKendrick Model . . . . . . . . 9

2.3 Mathematical Challenges . . . . . . . . . . . . . . . . . . . 9

2.4 The Reed-Frost Epidemic Model . . . . . . . . . . . . . . 10

2.5 Solution of the Reed-Frost epidemic model . . . . . . . 12

2.6 Endemic Steady State . . . . . . . . . . . . . . . . . . . . . 14

3 METHODOLOGY 17

3.1 Common fourth order Runge-Kutta methods . . . . . . 17

3.2 Derivation of the Runge-Kutta fourth order method . 19

3.3 Explicit and Implicit Iterative Methods . . . . . . . . . 19

3.4 Explicit Runge-Kutta methods . . . . . . . . . . . . . . . 19

3.4.1 Transformation of the Model into State-Space . . . . . 24

3.4.2 Function Creation and Invoking the ODE Solver . . . 24

3.4.3 Setting Error Tolerance of Scheme in Mat Lab . . . . . 25

3.5 Cumulative Sum (CUSUM) Chart . . . . . . . . . . . . . 25

3.6 One-Sided Decision Interval CUSUMs . . . . . . . . . . 26

3.7 Choice of Scheme Parameters . . . . . . . . . . . . . . . . 26

3.7.1 How To Predict The Behaviour of CUSUMs . . . . . . 27

3.7.2 Possible ”Enhancements” . . . . . . . . . . . . . . . . 27

4 RESULTS 28

4.1 Modeling the Epidemic . . . . . . . . . . . . . . . . . . . . 28

x

4.2 Transformation of the Epidemic Model and Runge-

Kutta Simulation . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Choice of CUSUM Scheme Parameters . . . . . . . . . . 30

4.4 Surveillance System . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Comparing Performance of the Schemes . . . . . . . . . 34

5 DISCUSSION, CONCLUSION AND RECOMMENDATION 35

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . 38

REFERENCES 39

APPENDIX 45

xi

CHAPTER 1

INTRODUCTION

1.1 Background of the study

Process monitoring means different things to different people; the princi-

pal aim of process monitoring is to furnish relevant information to ascertain

whether or not the output from the process is in accordance to specification,

or identify promptly some departure from specifications requiring some inter-

vention. This intervention may take many forms, as output may trigger an

investigation for any assignable causes of deterioration in quality. In many

applications the information from the data will be required to estimate the

amount of such departure.

The cumulative sum (CUSUM) chart methodology was initially developed

by Page (1954) for industrial problems where monitoring of the production

process is of interest. In the industrial setting, CUSUM charts have been

shown to be ideally suited to detecting small persistent process change (Mont-

gomery, 1991). In the medical context, CUSUMs have been proposed to mon-

itor procedures in clinical chemistry (Nix et al., 1986) and to monitor rare

congenital malfunctions (Gallus et al., 1986).

1

The need to formally monitor surgical outcomes has been brought to the

forefront in some recent well-publicized cases (Treasure et al., 1997; Waldie,

1998) where undesirable high rates of surgical complications remained un-

detected for an undue length of time. In such cases, early detection of de-

terioration in surgical performance is critical since it will result in prompt

investigation of the cause and procedural changes. Epidemics can impose

significant challenges on societies, not only by affecting the health of the gen-

eral population, but also by causing negative trends in the economy (medical

treatments, absenteeism from work, missed business opportunities, etc). The

ongoing epidemics of AIDS and tuberculosis provide some revealing examples.

In the absence of an effective cure against many diseases, the best approach

to mitigate an epidemic outbreak (malicious or natural) resides in the devel-

opment of capability for its early detection and for prediction of its further

development. Such a capability would allow making any countermeasures

(quarantine, vaccination, medical treatment) much more effective and less

costly.

1.2 Statement of the problem

Early pioneers in infectious disease modeling were William Hamer and Ronald

Ross(in the early twentieth century), who applied the law of mass action to

explain epidemic behavior. Lowell Reed and Wade Hampton Frost developed

the Reed-Frost epidemic model which described the relationship between sus-

ceptible, infected and immune individuals in a population.

2

1.3 Aim

To study the effect of the use of a developed mathematical (computational)

modeling in conjunction with a numerically based surveillance system in con-

tinuous monitoring of infectious diseases, using routinely collected patient

records that could lead to early detection of periods of deteriorating stan-

dards.

1.4 Objectives

I. To model the epidemics.

II. Use the modeled epidemics in conjunction with CUSUM chart at moni-

toring the infectious disease

III. Use CUSUM independently to monitor the outbreak of the infectious

disease

IV. Determine the effect of the use of modeled epidemic in conjunction with

CUSUM at monitoring the infectious disease.

1.5 Significance of the study

The significance of this research work is to study epidemic outbreaks where

undesirable high rates of vaccination complications remain undetected for an

undue length of time. In such cases, the early detection of deterioration in

vaccination performance is critical since it will result in prompt investigation

of the cause and vaccination changes.

3

1.6 Research questions

I. What effect does a vaccination have?

II. How could the model take different age categories into account?

III. Does every infected actually become infectious?

IV. What effect will early detection have on an epidemic?

1.7 Statement of the hypothesis

The dynamics using an ODE-system can be resolved in state-space matrix

realization or numerical methods to reduce our Mathematical abstractions of

real world phenomena.

1.8 Assumptions

Models are only as good as the assumptions on which they are based. If

a model makes predictions which are out of line with observed results and

the mathematics is correct, the initial assumptions must change to make the

model useful.

I. Homogeneous mixing of the population, i.e., individuals of the population

under scrutiny, interact with one another at random and do not mix mostly

in a smaller subgroup. This assumption is often well justified, when dealing

with a country such as Nigeria.

II. Rectangular age distribution: Typically found in developed countries

where there is low infant mortality and much of the population lives to the life

expectancy. In developing countries like Nigeria this assumption is often not

4

well justified. However, Rectangular age distribution is a standard assumption

to make the mathematics tractable.

1.9 Scope of the study

This study is limited only to the modeling of infectious diseases using the

Reed-Frost computational model employing numerical methods.

1.10 Definition of some terms

• ODE - Ordinary differential equation.

• Ro - The basic reproduction number. Is the average number of other

individuals each infected individual will infect in a population that has

no immunity to the disease.

• S= S(t)= Susceptible people - Is the proportion of the population who

are susceptible to the disease (neither immune nor infected).

• I= I(t)= Immune people

• R= R(t)= Recovered people

• IVP - Initial value problem.

• CUSUM - Cumulative sum

• ARL - Average Run Length

5

1.11 Organization of the study

This study is generally organized into 5 chapters.

Chapter 1 presents an introduction to the study, which covers the back-

ground of the study, statement of the problem, aim and objectives of the

study, the significance of the study as well as definition of some key terms

used in the study.

Chapter 2 is on literatures reviews of some related research covering studies

of the population dynamics, structure and evolution of infectious diseases of

plants and animals, including humans. The focus is then narrowed to the

Reed-Frost compartmental epidemic model which the study seeks to employ

though a brief on the endemic steady state and some non-manufacturing

applications of CUSUM is made.

Also, chapter 3 highlights the methodology employed by the study, which

includes the use of the fourth order Runge-Kutta as well as the cumulative

sum chart (CUSUM). Both methods are brought to bare as to the usage and

application in this study whilst using Mat Lab and Microsoft Excel as tools

to achieve results.

Furthermore, chapter 4 presents the results of the study with graphical

solutions to show the effects of the methods in monitoring epidemic diseases.

Finally, chapter 5 discusses the study and makes recommendations based

on the study findings while provoking further study in the areas the study

could not cover.

6

CHAPTER 2

LITERATURE REVIEW

2.1 Infectious Disease Dynamics

In recent years, it has become obvious that there is need to accommodate

the growing integration of quantitative methods with the increasing volume

of data being generated on host-pathogen interactions. This has resulted in

a growing body of research covering quantitative or theoretical studies of the

population dynamics, structure and evolution of infectious diseases of plants

and animals, including humans.

Brauer and van Den Driessche, (2001) studied epidemic models with infective

immigrants, Shim (2006) studied epidemic models with infective immigrants

and vaccination, while Piccolo and Billings (2005) analyzed the effect of vac-

cination in an immigrant Susceptible, Infectious and Recovered (SIR) model.

Iwami et al. (2006) proposed and interpreted a mathematical model of the

spread of avian influenza from the bird world to the human world, where two

types of outbreaks of avian influenza may occur if the humans do not prevent

its spread. Their result suggests that in order to prevent the spread of avian

7

influenza in the human world, we must take the measures not only for the

birds infected with avian influenza to exterminate but also for the humans

infected with mutant avian influenza to quarantine when mutant avian in-

fluenza has already occurred.

Casagrandi et al. (2006) developed a simple ordinary differential equation

model to study the epidemiological consequences of the drift mechanism for

influenza A viruses. Improving over the classical SIR approach, they intro-

duce a fourth class for the cross-immune individuals in the population, i.e.,

those that recovered after being infected by different strains of the same viral

subtype in the past years. Their SIRC model predicts that the prevalence of

a virus is maximum for an intermediate value of the basic reproduction num-

ber. Nishiura (2007) examined the time variations in transmission potential

with regard to pandemic influenza and suggests methods to be explored to

construct effective non-pharmaceutical interventions such as household quar-

antine and mask-wearing. Vaccination and treatment are two elements of the

international strategy to forestall a pandemic, applying them concurrently

have been shown to be most effective as public health measures in curtailing

disease outbreak (Rwezaura et al.2009).

Such a model is more suitable for developing countries where access to medi-

cal care is difficult, only a few may get vaccinated, some may have access to

treatment depending on the proximity of the health services. The combina-

tion of these strategies with non-pharmaceutical individual countermeasures

which are crucial for poor resource settings, especially in developing coun-

8

tries (World Health Organization Writing Group 2006), provides a better

way to curtail the epidemic in rural communities, where household quaran-

tine and mask wearing could offer hope for development of these effective

non-pharmaceutical interventions.

2.2 The Simple Kermack-McKendrick Model

One of the early triumphs of mathematical epidemiology was the formulation

of a simple model by Kermack and McKendrick in 1927 whose predictions are

very similar to the behavior, observed in countless epidemics, of diseases that

invade a population suddenly, grow in intensity, and then disappear leaving

part of the population untouched. The Kermack-McKendrick model is a com-

partmental model based on relatively simple assumptions on the rates of flow

between different classes of members of the population.

The Severe Acute Respiratory Syndrome (SARS) epidemic of 2002-3 revived

interest in epidemic models, which had been largely ignored since the time

of Kermack and McKendrick, in favor of models for endemic diseases. More

recently, the threat of spread of avian flu raised in 2005 and the H1N1 in-

fluenza a pandemic of 2009, have provided a continuing source of important

modeling questions.

2.3 Mathematical Challenges

Mathematical abstraction of real world phenomena! Equations:

• No outbreaks are similar! (stochasticity).

9

• Different modes of disease transmission: (person-to-person, air-borne,

Water-borne, food-borne and vector-borne).Direct and indirect trans-

mission

• Populations heterogeneity (e.g. different places of residence, contact be-

havior, susceptibility) needs to be taken into account.

• Conflict between observation frequency and speed of the epidemic, (time

unit of a model). Not all relevant events for the course of the epidemic

are observable! Partial observability.

2.4 The Reed-Frost Epidemic Model

The SIR compartmental model tracks the numbers of susceptible (S), in-

fected (I) and recovered (R) individuals during an epidemic with the help of

ordinary differential equations (ODE). A major assumption of many mathe-

matical models of epidemics is that the population can be divided into a set

of distinct compartments. These compartments are defined with respect to

disease status as described above.

Susceptible- Individuals that are susceptible have, in the case of the ba-

sic SIR model, never been infected, and they are able to contract the disease.

Once they contract the disease they move into the infected compartment.

Infected- Infected individuals can spread the disease to susceptible individu-

als. The time they spend in the infected compartment is the infectious period,

after which they enter the recovered compartment.

Recovered- Individuals in the recovered compartment are assumed to be

10

immune for life.

The SIR model is easily written using ordinary differential equations (ODEs),

which implies a deterministic model (no randomness is involved, the same

starting conditions give the same output), with continuous time (as opposed

to discrete time).

We assume that encounters between infected and susceptible individuals oc-

cur at a rate proportional to their respective numbers in the population. The

rate of new infections can thus be defined as [βSI], where β is a parameter of

infection. Infected individuals are assumed to recover with a constant prob-

ability at any time, which translates into a constant per capita recovery rate

that we denote with γ, and thus an overall rate of recovery γI. Based on

these assumptions, the scheme of the model can be translated into a set of

ordinary differential equations:

dS(t)d(t) = −S(t)I(t)...(1)

dI(t)d(t) = S(t)I(t)− I(t)...(2)

dR(t)d(t) = I(t)...(3)

Equation (1), Contact Rate: describes the consumption rate of susceptibles

(S), which is due to infection (I).

Equation (2), Basic Reproduction Rate: describes the rate of contacts made

by one infected individual in a susceptible population per unit time that leads

to an infection. i.e. the total number of secondary infection produced by one

11

infected individual in a virgin host population.

Equation (3), Recovery Rate: describes the rate at which infected individuals

are expected to recover with a constant probability which equals the overall

number of immune in the population.

The above system of differential equations is a set of mathematical equa-

tions describing the relationship existing between the susceptible, infected

and recovered compartments in a closed population during an epidemic out-

break.

The presence of carriers usually complicates the dynamics and prevention

of a disease. They are not recognized as disease cases themselves unless they

are screened and they usually spread the infection without them being aware.

We argue that this has been one of the major causes of the spread of human

immunodeficiency virus (HIV) S.D. Hove-Musekwaa and F. Nyabadzab.

If R0 < 1 the number of infected is expected to fade out right after intro-

duction. On the other hand, if R0 > 1 an epidemic will result in a simple

SIR model, while the final size of an epidemic will be R0 = N . In a closed

population, the number of susceptible can only decrease with the introduction

of an infected individual as displayed by the solution of the system of ODEs.

2.5 Solution of the Reed-Frost epidemic model

The following code identifies the contagious and recovery parameters, β and

γ, from several observations of the percentage of infected and recovered in

12

the populations. The system of differential equations is discretized to form

an over-determined algebraic system. The least squares method is used to

approximate the two parameters. (Please refer to the appendix for the code)

13

2.6 Endemic Steady State

An infectious disease is said to be endemic when it can be sustained in a

population without the need for external inputs. This means that, on aver-

age, each infected person is infecting exactly one other person (any more and

the number of people infected will grow exponentially and there will be an

epidemic, any less and the disease will die out). In mathematical terms, that

is:

R0 × S = 1.

The basic reproduction number (R0) of the disease, assuming everyone is

susceptible, multiplied by the proportion of the population that is actually

susceptible (S) must be one (since those who are not susceptible do not fea-

ture in our calculations as they cannot contract the disease). Notice that this

relation means that for a disease to be in the endemic steady state, the higher

the basic reproduction number, the lower the proportion of the population

susceptible must be, and vice versa; a mathematical basis for a result that

might have been intuitively obvious. The first assumption (above) lets us say

that everyone in the population lives to age L and then dies. If the average

age of infection is A, then on average, individuals younger than A are suscep-

tible and those older than A are immune (or infectious). Thus the proportion

of the population that is susceptible is given by:

S = A/L

But the mathematical definition of the endemic steady state can be rear-

ranged to give:

S = 1/R0

Therefore, since things equal to the same thing are equal to each other:

14

1/R0 = A/L⇒ Ro = L/A

This provides a simple way to estimate the parameter R0 using easily available

data. For a population with an exponential age distribution, Ro = 1 + L/A

This allows for the basic reproduction number of a disease given A and L in ei-

ther type of population distribution. dydx = y′ = f(x, y), a ≤ x ≤ b[x ∈ (a, b)] is

known as classical initial value problem (CIVP). Classical initial value prob-

lems satisfying the integral equation y(x) = y(0)+ ∈xa f [s, y(s)]ds where

y(a) = y(0), Exists iff;

(a) That function is continuous. (b) f must be differentiable at least once.

(c) f must be bounded i.e |f | < M where M is finite i.e. |f | ≤ M < ∞ (d)

f must satisfy the Lipschitz condition |f(x1, y1)-(x2, y2)| ≤ L|y1, y2| for every

point on the Cartesian plane.

• Tayor’s method

• Adams-Moulton (Implicit)

• Simpsons (Explicit)

• Forward Euler (Explicit)

• Backward Euler (Implicit)

• Trapezoidal method of order two

• Linear multi-step methods

Anyone of the above listed numerical schemes can be employed to resolve the

REED-FROST epidemic model given below;

dS(t)d(t) = −S(t)I(t)...(1)

15

dI(t)d(t) = S(t)I(t)− I(t)...(2)

dR(t)d(t) = I(t)...(3)

However, this study will only employ fourth order Runge-Kutta method while

further research could be done using the aforementioned numerical schemes.

16

CHAPTER 3

METHODOLOGY

3.1 Common fourth order Runge-Kutta methods

Runge-Kutta methods are an important family of implicit and explicit it-

erative methods for the approximation of solutions of ordinary differential

equations. These techniques were developed around 1900 by the German

mathematicians C. Runge and M.W. Kutta.

One member of the family of RungeKutta method that is so commonly used

is often referred to as ”RK4” , ”classical RungeKutta method” or simply as

”the RungeKutta method”. Let an initial value problem be specified as fol-

lows.

y′= f(t, y), y(to) = yo

In words, what this means is that the rate at which y changes is a function of

y itself and of t (time). At the start, time is to and y is yo . In the equation,

y may be a scalar or a vector.

The RK4 method for this problem is given by the following equations: yn+1 =

yn + 16(k1 + 2k2 + 2k3 + k4)

tn+1 = tn+h where yn+1

17

is the RK4 approximation of y(tn+1) , and k1 = hf(tn, yn),

k2 = hf(tn + 12h, yn + 1

2k1)

k3 = hf(tn + 12h, yn + k2)

k4 = hf(tn + h, yn + k3)

(Note: the above equations have different but equivalent definitions in differ-

ent texts). Thus, the next value (yn+1) is determined by the present value

(yn) plus the weighted average of four increments, where each increment is

the product of the size of the interval, (h) , and an estimated slope specified

by function f on the right-hand side of the differential equation.

• k1 is the increment based on the slope at the beginning of the interval,

using (yn) , (Eulers method);

• k2 is the increment based on the slope at the midpoint of the interval,

using (yn + 12k1) ;

• k3 is again the increment based on the slope at the midpoint, but now

using (yn + 12k2) ;

• k4 is the increment based on the slope at the end of the interval, using

(yn + k3) .

In averaging the four increments, greater weight is given to the increments

at the midpoint. The weights are chosen such that if f is independent of y ,

so that the differential equation is equivalent to a simple integral, then RK4

is Simpsons rule. The RK4 method is a fourth-order method, meaning that

the error per step is on the order of (h5) , while the total accumulated error

has order (h4).

18

3.2 Derivation of the Runge-Kutta fourth order method

In general a RungeKutta method of order 4 can be written as:

yt+h = yt + h∑si=1 aiki + o(hs+1)

are increments obtained evaluating the derivatives of yt at the ith order.

We develop the derivation for the RungeKutta fourth order method using

the general formula with s=4 evaluated, as explained above, at the starting

point, the midpoint and the end point of any interval (t,t+h) , thus refer to

International Journal of Numerical Methods and Applications (2009).

3.3 Explicit and Implicit Iterative Methods

Numerical solution schemes are often referred to as being explicit or implicit.

When a direct computation of the dependent variables can be made in terms of

known quantities, the computation is said to be explicit. When the dependent

variables are defined by coupled sets of equations, and either a matrix or

iterative technique is needed to obtain the solution, the numerical method is

said to be implicit.

3.4 Explicit Runge-Kutta methods

The family of explicit RungeKutta methods is a generalization of the RK4

method mentioned above. It is given by

yn+1 = yn +∑si=1 biki,

where

k1 = hf(tn, yn),

k2 = hf(tn + c2h, yn + a21k1),

19

k3 = hf(tn + c3h, yn + a31k1 + a32k2)

(Note: the above equations have different but equivalent definitions in

different texts). To specify a particular method, one needs to provide the

integer (s) (the number of stages), and the coefficients (aij) for (1 ≤ j < i ≤ s)

,bi for (i = 1, 2, ...s) , and ci for (i = 1, 2, ..., s) . The matrix [aij] is called

the Runge-Kutta matrix, while the bi and ci are known as the weights and

the nodes. These data are usually arranged in a mnemonic device, known

as a Butcher tableau (after John C. Butcher): s The RungeKutta method is

consistent if∑i−1j=1 aij = ci for (i=2,3,...,s). There are also accompanying requirements if

we require the method to have a certain order (p) , meaning that the local

truncation error [T.E] is 0(hp+1) . These can be derived from the denition of

the truncation error itself. For example, a 2-stage method has order 2 if

b1 + b2 = 1 , b2c2 = 12 , and a21 = c2. This is a typical Runge-Kutta method

tableau

• Runge-Kutta method with one stage

However, the simplest RungeKutta method is the (forward) Euler method,

given by the formula yn+1 = yn + hf(tn, yn) . This is the only consistent

explicit RungeKutta method with one stage. The corresponding tableau

is

• Second-order Runge-Kutta methods with two stages

An example of a second-order method with two stages is provided by the

midpoint method

yn+1 = yn + hf(tn + 12h, yn + 1

2hf(tn, yn)).

The midpoint method is not the only second-order RungeKutta method

20

with two stages. In this family,[α = 12 ] gives the midpoint method and

[α = 1] is Heuns method.

As an example, consider the two-stage second-order RungeKutta method with

α = 23 with the corresponding equations

k1 = f(tn, yn),

k2 = f(tn + 23h, yn + 2

3hk1),

yn+1 = yn + h(14k1 + 34k2).

This method is used to solve the initial-value problem y′

= tan(y) + 1,

y(1) = 1, t ∈ [1, 1, 1] with step size h=0.025 , so the method needs to take

four steps.

The method proceeds as follows:

t0 = 1 :

y0 = 1

t1 = 1.025 :

y0 = 1,

k1 = 2.5574077;

k2 = f(t0 + 23h, y0 + 2

3hk1) = 2y1 = y0 + h(14k1 + 34k2) = 1.066869388

t2 = 1.05 :

y1 = 1.066869388,

k1 = 2.813546,

k2 = f(t1 + 23h, y1 + 2

3hk1), y2 = y1 + h(14k1 + 34k2) = 1.141332181

t3 = 1.075 :

y2 = 1.14132181,

k1 = 3.1835366,

k2 = f(t2 + 23h, y2 + 2

3hk1), y3 = y2 + h(14k1 + 34k2) = 1.227417567

21

t4 = 1.10 :

y3 = 1.227417567, k1 = 3.7968665,

k2 = f(t3 + 23h, y3 + 2

3hk1)y4 = y3 + h(14k1 + 34k2) = 1.335079087

The numerical solutions correspond to the underlined values.

• Adaptive Runge-Kutta methods The adaptive methods are designed to

produce an estimate of the local truncation error of a single Runge-Kutta

step. This is done by having two methods in the tableau, one with

order[p] and one with order. The lower-order step is given by y∗n+1 =

yn +∑si=1 = b∗iki ,where the ki are the same as for the higher order

method. Then the error is

en+1 = yn−1−y∗n+1 = h∑si=1(bi−b∗i )ki , which is 0(hp) . The RungeKuttaFehlberg

method has two methods of orders 5 and 4 . However, the simplest adap-

tive RungeKutta method involves combining the Heun method, which is

order 2 , with the Euler method, which is order 1 .The error estimate is

used to control the stepsize. Other adaptive RungeKutta methods are

the BogackiShampine method (orders 3 and 2 ), the CashKarp method

and the DormandPrince method (both with orders 5 and 4 ).

• Implicit RungeKutta methods

All RungeKutta methods mentioned up to now are explicit methods.

Unfortunately, explicit RungeKutta methods are generally unsuitable for

the solution of stiff equations because their region of absolute stability is

small; in particular, it is bounded. This issue is especially important in

the solution of partial differential equations. The instability of explicit

RungeKutta methods motivates the development of implicit methods.

An implicit RungeKutta method has the form

22

yn+1 = yn +∑si=1 biki

where ki = hf(tn + cih, yn +∑si=1 aijkj), i = 1, ..., s

The difference with an explicit method is that in an explicit method, the

sum over j only goes up to (i1) . For an implicit method, the coefficient

matrix is not necessarily lower triangular: The consequence of this dif-

ference is that at every step, a system of algebraic equations has to be

solved. This increases the computational cost considerably. If a method

with s stages is used to solve a differential equation within components,

then the system of algebraic equations has ms components. In contrast,

an implicit s-step linear multi-step method needs to solve a system of

algebraic equations with only s components.

Example. The simplest example of an implicit RungeKutta method is

the backward Euler method:

yn+1 = yn + hf(tn + h, yn+1) .

Another example for an implicit RungeKutta method is the trapezoidal

rule. The trapezoidal rule is a collocation method (as discussed earlier).

All collocation methods are implicit RungeKutta methods, but not all

implicit RungeKutta methods are collocation methods. The GaussLe-

gendre methods form a family of collocation methods based on Gauss

quadrature. A Gauss-Legendre method with s stages has order 2s (thus,

methods with arbitrarily high order can be constructed).

23

3.4.1 Transformation of the Model into State-SpaceX1

X2

X3

=

−X(1)X(2)

X(1)X(2)−X(2)

X(2)

If we let F (t, [X1, X2, X3]T ) = [(X1X2), (X1X2X2), X2]

T andX = [X1, X2, X3]T

, then X′

= [X′

1, X′

2, X′

3]T . The system of equations transformed in matrix

state space realization takes the form

X′= F (t, x)

3.4.2 Function Creation and Invoking the ODE Solver

X(0) =

X1(0)

X2(0)

X3(0)

=

1

2

3

function Xprime=F(t,x)

Xprime=Zeros(3,1)

Xprime(1)=–X(1)*X(2);

Xprime(2)=X(1)*X(2)-X(2);

Xprime(3)=X(2);

>>[t,x]=ode45(@F,[0,20],[1;2;3],[]);

>>plot(t,u);

>>Title(’A solution to the Reed-Frost Epidemic Model’)

>>xlabel (’t’), ylabel (’S,I,R’)

>>legend (’S’,’I,’R’), grid.

Idea from Fatokun J. (2011)

24

3.4.3 Setting Error Tolerance of Scheme in Mat Lab

• Relative-Tolerance (RelTol)

A relative error tolerance that applies to all components of the resid-

ual vector, is a measure of the residual relative to the size of f(x,y). It

corresponds to a positive scalar with {1e - 3} 0.1% accuracy. The com-

puted solution S(x) is the exact solution of S′(x) = F (x, S(x) + res(x)

on each sub-interval of the mesh, the residual satisfies; ||res (i)(max(AbsF (i))′

,

AbsTol(i)(RelTol) ||≤RelTol

• Absolute-Tolerance (AbsTol)

Absolute error tolerance that apply to the corresponding components

of the residual vector. AbsTol(i) is a threshold below which the values

of the corresponding components are unimportant. If a scalar value is

specified, it applies to each component.

3.5 Cumulative Sum (CUSUM) Chart

Interest lies in a small, sustained shift in a process, a practical example is given

with the game of golf, for each hole in a round of golf, there are a specified

number of times in which one should strike the ball, until it eventually drops

into the hole. For example, on a par 4, if you strike the ball 4 times and it

falls into the cup, then you held par. If you were able to do this task with only

three shots (a birdie) then you are ”1 under par” hence your cumulative sum

is -1. This is continued throughout the course, the ultimate winner therefore

having the lowest CUSUM. Picture a golfer who is holding par for the first

13 holes, then suddenly hits form and has five successive birdies towards the

25

end of the round. The final CUSUM is therefore 5, though from viewing a

CUSUM chart it would be clear to see when the process shifted. If one is

interested in detecting a small and sustained shift in a process, a CUSUM

chart is a useful vehicle to obtain such process knowledge.

3.6 One-Sided Decision Interval CUSUMs

• High-Side CUSUM

Ui = max[0, (Qik1) + Ui1]

• Low-Side CUSUM

Li = min[0, (Qik2) + Ui1]

All of k1 and/or k2 , U0 and/orL0

are (appropriately chosen) parameters of the scheme.

• Raw CUSUMs with a restart feature

• Reference values k1 and k2 are typically chosen above and below an ideal

Q

• Out-of-control signals derive from decision levels h and h

3.7 Choice of Scheme Parameters

Common choice of starting values is U0 = 0 and L0 = 0

For a given choice of k1 and/or k2 and a normal all-OKdistribution for Q

, the table below can be used to pick h providing a desired meantime between

false-alarms (a desired all-OK ARL)

26

Table 3.1: Choice of Scheme Parameters

K =0.25 0.50 0.75 2 1.00 1.25 1.50

8.01 4.77 3.34 2.52 1.99 1.60

To enter the table one must standardize by

K = k1−µQσQ or K = µQ−k2

σQ

and read out H .

Then set h = HσQ

For simultaneous high and low side schemes with k1+k22 = µQ and all-OK

ARL=370, Optimal choice of k1 and/or k2 is possible (for given all-OK ARL

and potential shift in mean Q )

For detecting a shift in mean Q of size δ and/or , approximately optimal

reference values are kopt1 = µQ+ δ2 k

opt2 = µQ− δ

2

3.7.1 How To Predict The Behaviour of CUSUMs

For a given k1 and/or k2 and h, with U0 = 0 and L0 = 0 and normal Q, it is

possible to find (not all-OK) ARLs (mean times of detection)

We can use the above formula to find parameters to enter the table for

one-sided schemes (one inputs the mean and standard deviation of Q) and

for combined high-and low-sided schemes.

3.7.2 Possible ”Enhancements”

Fast initial response CUSUMs

U0 = h2 and/or L0 = −h

2

27

CHAPTER 4

RESULTS

4.1 Modeling the Epidemic

Let N be the total size of the population and we assume homogeneity, that

is, each individual in the population has an equal probability of contacting

the disease with a rate of β.

• The number of contacts made by one infectious to transmit the disease

in the population is βN per unit time.

• The fraction of contacts by one infected individual with a susceptible is

SN

• The number of infectious is I, therefore the consumption rate of suscep-

tibles, which is due to infection, is βN( SN )(I)=(βSI)

• Basic reproduction number: R0 = βNγ

βN is the number of contacts made by one infective in an otherwise

susceptible population per unit time that leads to an infection.

1γ represents infectious period.

28

Therefore R0 describes the total number of secondary infections produced

when one infected individual is introduced into a host virgin population.

dS(t)d(t) = −S(t)I(t)...(1)

dI(t)d(t) = S(t)I(t)− I(t)...(2)

dR(t)d(t) = I(t)...(3)

where;

S - the number of susceptibles

I - the number of infectious

R - the number of recoveries

β - contact rate

γ - recovery rate

4.2 Transformation of the Epidemic Model and Runge-

Kutta Simulation

X(0) =

X1(0)

X2(0)

X3(0)

=

1

2

3

function Xprime=F(t,x)

Xprime=Zeros(3,1)

Xprime(1)=–X(1)*X(2);

Xprime(2)=X(1)*X(2)-X(2);

Xprime(3)=X(2);

>>[t,x]=ode45(@F,[0,20],[1;2;3],[]);

>>plot(t,u);

>>Title(’A solution to the Reed-Frost Epidemic Model’)

29

>>xlabel (’t’), ylabel (’S,I,R’)

>>legend (’S’,’I,’R’), grid.

4.3 Choice of CUSUM Scheme Parameters

The data in the table 3.1 is entered into Matlab with the following commands

entered in the command window:

� X = [0.25 0.50 0.75 1.00 1.25 1.50]

� Y = [ 8.01 4.77 3.44 2.52 1.99 1.60]

For instance:

We can plot the data in the form of vectors using the plot command:

� plot(x, y)

� plot(x,′ o−′)

� holdall

� plot(y,′ ∗−′)

� holdoff

� gridon

� xlabel(′K1opt′)

� ylabel(′K2opt′)

� legend(′Kopt′)

Fig.1; Optimal reference values

Problem 1.

Process monitoring with Q = x̄ based on n=4 all-OK process parameters

µ = 9.0, and σ = 1.6 (so all-OK distribution of Q = x̄ has µQ = 9.0 and

σQ = σn = 1.6

4 = 0.8) all-OK ARL=370 desired. Quickest possible detection of

a change of size 2.0 in the process mean (and therefore in mean Q) desired.

30

• How to Set-up a combination of high and low side scheme.

• First choose the reference values k1 = 9.0+ 2.02 = 10.0 and k2 = 9.0− 2.0

2 =

8.0

• Next choose h

K = k1−µQ

σQ= 10.0−9.0

0.8 = 1.25

From table 4.3, H=1.99

So take h = HσQ = (1.99)(0.8) = 1.592

• Use starting values U0 = 0 and L0 = 0

• For a given k1 and/or k2 and h, with U0 = 0 and L0 = 0 and normal Q,

it is possible to find (not all-OK) ARLs (mean times of detection).

• ”Fast initial response” CUSUMs

U0 = h2 and/or L0 = −h

2

• We can use the above formula to find parameters to enter the table for

one-sided schemes (one inputs the mean and standard deviation of Q)

and for combined high-and low-sided schemes.

Problem 2. ONE-SIDED CUSUM

The table 4.1 is called a ONE-SIDED CUSUM with k = 1 and Cusumi =

(Qi − k) + Cusumi−1

31

Problem 3. TWO-SIDED CUSUM

The table 4.2 is called a TWO-SIDED CUSUM which compares the result

of Runge-Kutta + CUSUM in contrast to ordinary CUSUM applied on the

raw data, with parameters k1 = 1, k2 = −1, Ui = max[0, (Qi − k1) + Ui − 1]

and Li = min[0, (Qi − k2) + Li − 1].

4.4 Surveillance System

Suppose Q = x and process standards are µ = 0 and σ = 1 . If quick

detection of a change in process mean of size δ = 0.5 is of importance, and

all-OK ARL=370, our interest is to set up a combined low-and high-side

CUSUM scheme. (what will be the appropriate values of k1 , k2 and h?)

• Given all-OK parameters µ = 0 and σ = 1 from a distribution Q = x

based on any sample size n

• Our all-OK distribution of Q = x has µQ = 0 and σQ= σ√n

= 1 all-OK

ARL=370 desired

• Quickest possible detection of a change of size σ = 0.5 in the process

mean (and therefore in mean Q) desired.

k1 = µQ + σQ2 = 0 + 1

2 = 0.5 and

k2 = µQ − σQ2 = 0− 1

2 = −0.5

K = k1−µQ

σQor K = µQ−k1

σQ= 0.5− 0

2 = 0.5

From table 3.1 on choice of parameters, H = 4.77

so h = HσQ = (4.77)(1) = 4.77

Fast Initial Response CUSUMs

32

i Qi Qi − k CUSUMi

0 0 -1 0

1 1 0 -1

2 -3 -4 -5

3 0 -1 -6

4 1 0 -6

5 20 20 13

6 -5 -6 7

7 0 -1 -6

8 1 0 0

Table 4.1: ONE-SIDED CUSUM

i Qi k1 k2 Qi − k1 Qi − k2 Qi − k1 + Ui − 1 Qi − k2 + Li − 1 CUSUM Runge−Kutta+ CUSUM

0 0 1 −1 −1 1 −1 1 0 1

1 1 1 −1 0 2 0 3 −1 3

2 −3 1 −1 −4 −2 −4 1 −5 1

3 0 1 −1 −1 1 −5 2 −6 2

4 1 1 −1 0 2 −5 4 −6 2

5 20 1 −1 19 21 14 25 13 24

6 −5 1 −1 −6 −4 8 21 7 20

7 0 1 −1 −1 1 7 22 6 21

8 1 1 −1 0 2 7 24 0 23

Table 4.2: TWO-SIDED CUSUM

33

U0 = h2 and L0 = −h

2

4.5 Comparing Performance of the Schemes

To observe the performance of Runge-Kutta+CUSUM as well as ordinary

CUSUM on the small sustained changes in the SIR population from table

4.2, the following command was entered into Mat Lab

� x = [0 −1 −5 −6 −6 13 7 6 0]

� y = [1 3 1 2 3 24 20 21 23]

� plot(x,′ o−′)

� holdall

� plot(y,′ ∗−′)

� holdoff

� gridon

� xlabel(′period′)

� title(′ComparisonoftheperformanceofRunge−Kutta+CUSUMwithordinaryCUSUMontheReed−

Frostepidemicmodel′)

� ylabel(′cases′)

� legend(′CUSUM ′,′Reed− Frost′)

34

CHAPTER 5

DISCUSSION, CONCLUSION AND

RECOMMENDATION

5.1 Discussion

The findings of this study clearly suggests that applying Runge-Kutta fourth

order and CUSUM on the raw data yielded a faster means of detecting an epi-

demic outbreak. At every node in successive segments, the combined approach

was more rapid in detecting deteriorating standards than only CUSUM. In

figure 2, the (h) and (-h) represents the upper and lower control limits re-

spectively, signifying whether or not there is a deviation from an expected

standard which has already been predetermined by investigating an epidemic

outbreak over a specified period of time. You will observe that the combined

approach of Runge-Kutta and CUSUM surpassed the upper control limit (h)

meaning that there is a deviation from acceptable deteriorating standards

which must be checked to avoid an epidemic outbreak. In other words, there

is an improvement in the rate of rapid detection of small sustained changes

in the process as compared with ordinary CUSUM. This study is particularly

35

important in monitoring epidemic outbreaks where undesirable high rates of

vaccination complications remained undetected for an undue length of time.

In such cases, the rapid detection of deterioration in vaccination performance

is critical since it will result in prompt investigation of the cause and vacci-

nation changes.

James M. Hyman and E. Ann Stanley used mathematical models to under-

stand the AIDS epidemic, in the same vein this study equally used a math-

ematical model (Reed-Frost SIR epidemic model) in understanding epidemic

outbreaks.

Then to achieve a control of the epidemic, the existence of an optimal control

system was formulated and solved numerically using Runge-Kutta fourth or-

der procedure, which is very similar to Kar TK, Batabyal A.

Consequently, P.A. Osanaiye and C. O. Talabi in detection of outbreak of

an epidemic used the CUSUM choosing an acceptable mean level of a disease

µ. In accordance to the administration’s desire the values of the parameters

will be revised periodically to determine the level of rejection of the epidemic.

What distinguishes this study from those highlighted, is that this study is

a combination of ideas from all the aforementioned studies. This study em-

ployed Runge-Kutta fourth order as well as CUSUM to achieve a control of

epidemic. Future research should employ a different numerical scheme with a

surveillance system to see if it will be more effective in detection of outbreaks

36

of epidemics.

5.2 Conclusion

In conclusion, from the analysis using fourth order Runge-Kutta, the test as

carried out established that there are differences in the mean level in succes-

sive segments, i.e it indicates some significant changes in retrospective sample

number. Finally, it can be seen from the results that a combination of CUSUM

scheme which was applied to the fourth order Runge-Kutta method is more

sensitive at detecting small shift in a process mean than ordinary CUSUM

applied to the raw data collected. This study contributes to existing knowl-

edge in the area of monitoring epidemic outbreak especially to obtain relevant

information regarding a departure from an acceptable pattern considered an

epidemic requiring some action. In such cases, the rapid detection of deteri-

oration in treatment is critical since it will result in prompt investigation of

the cause and procedural changes.

37

5.3 Recommendation

It is then suggested that the Federal Ministry of Health and other Health

Organizations/Agencies engaged in providing health services should increase

their earnest effort by creating more surveillance systems for continuous mon-

itoring of infectious diseases tests, using routinely collected data for rapid

detection of deteriorating standards; to see to it that Nigeria hospitals are

under proper check and safe for all users in order to reduce the high rate of

epidemic outbreaks which leads to avoidable loss of lives.

38

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APPENDIX

Code used to describe the graphical solution of a simple SIR model. % This

code uses least squares to identify two parameters in the SIR model:

% St = −βSI; It = βSI − γI; Rt = γI where

% a = ”contagious” coefficient and

% b = ”recovery” coefficient.

% The data is given in the vectors Sd, Id and Rd, and they are adjusted by

a random variable.

% The data is used in the finite difference approximation of the above:

% (Si + 1− Si − 1)/(2dt) = −βSiIi% (Ii + 1− Ii − 1)/(2dt) = βSiIi − γIi and

% (Ri + 1−Ri − 1)/(2dt) = γIi.

% Least squares is used to compute the linear polynomial coefficients.

% The first six data points are used.

% function [ty] = sirid

% global oldβ oldγ

% y0 = [99, 1, 0];

% t0 = 0;

% tf = 50;

% [ty] = ode45(′ypsirid′,[t0tf ], y0);

% function ypsirid = ypsirid(t, y)

% global oldβ oldγ

% ypsirid(1) = −oldβ ∗ y(1) ∗ y(2);

% ypsirid(2) = oldβ ∗ y(1) ∗ y(2)− oldγ ∗ y(2);

%ypsirid(3) = oldγ ∗ y(2);

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% ypsirid = [ypsirid(1)ypsirid(2)ypsirid(3)]′;

clear; clf(figure(1))

global oldβ oldγ

oldβ = 0.010; oldγ = 0.100;

td = [0.001.0772.1363.0924.1715.3726.6958.0209.015...10.29311.27115.03920.590];

% Sd = [99.097.292.984.868.142.820.08.34.32.01.10.20.04];

Id = [1.002.596.3913.6528.2048.7664.0066.9364.38...58.8554.1537.8821.85];

Rd = [0.000.280.631.553.748.3715.9724.7631.3039.19...44.7261.9278.11];

numdata = 13;

rvec = rand(1, numdata);

Id(2 : numdata) = Id(2 : numdata) + .1 ∗ rvec(1, 2 : numdata)− .05;

rvec = rand(1, numdata);

Rd(2 : numdata) = Rd(2 : numdata) + .1 ∗ rvec(1, 2 : numdata)− .05;

Sd = 100− Id−Rd;

fori = 2 : 1 : numdata− 1

ii = (i− 1) ∗ 3;

d(ii) = (Sd(i+ 1)− Sd(i− 1))/(td(i+ 1)− td(i− 1));

d(ii+ 1) = (Id(i+ 1)− Id(i− 1))/(td(i+ 1)− td(i− 1));

d(ii+ 2) = (Rd(i+ 1)−Rd(i− 1))/(td(i+ 1)− td(i− 1));

A(ii, 1) = −Sd(i) ∗ Id(i);A(ii, 2) = 0;

A(ii+ 1, 1) = Sd(i) ∗ Id(i);A(ii+ 1, 2) = −Id(i);

A(ii+ 2, 1) = 0.0;A(ii+ 2, 2) = Id(i); end

% meas = 6;

m = 3 ∗meas+ 1;

x = A(2 : m, :)(.2 : m)’;

% [oldβ oldγ]

[x(1)x(2)]

plot(td(1 : 1 : meas + 1), Sd(1 : 1 : meas + 1),′ ∗′, td(1 : 1 : meas +

1), Id(1 : 1 : meas + 1),′ o′, ...td(1 : 1 : meas + 1), Rd(1 : 1 : meas +

1),′ s′, td, Sd,′ x′, td, Id,′ x′, td, Rd,′ x′)

oldβ = x(1);

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oldγ = x(2);

[ty] = sirid;

hold on

plot(t, y(:, 1),′ b′, t, y(:, 2),′ g′, t, y(:, 3),′ r′)

47