STABILITY ANALYSIS OF

TUBERCULOSIS MODEL

BY

AKANDE, KAZEEM BABATUNDE07/55EB015

Supervisor:

Prof. IBRAHIM M.O.

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF

MATHEMATICS, FACULTY OF SCIENCE, UNIVERSITY OF ILORIN,

NIGERIA.

IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE

AWARD OF MASTER OF SCIENCE DEGREE IN MATHEMATICS

SEPTEMBER, 2013.

i

CERTIFICATION

I certify that this dissertation was carried out by AKANDE, Kazeem Babatunde with

Matriculation Number: 07/55EB015 in the Department Mathematics, University of Ilorin

for the award of Master of Science Degree in Mathematics.

......................................... ............................

PROF. IBRAHIM M.O. DATE

(Supervisor)

......................................... ............................

PROF. IBRAHIM M.O. DATE

(Head of Department)

......................................... ............................

PROF. A.W. GBOLAGADE DATE

EXTERNAL EXAMINER

ii

DEDICATION

This dissertation work is dedicated to my beloved Brother, Mr. Akande Abdul Azeez.

iii

ACKNOWLEDGMENT

All praises and adoration are due to Almighty Allah, the creator of heaven and earth.

The peace and blessing of Him (Allah) be with the Prophet Muhammad Bin Abdullahi,

his household, companions and those who follow and who will follow him until the day of

resurrection.

I next acknowledge the effort, tolerance and immeasurable guidance of my father in Is-

lam, a father, a supervisor, the Head of department of Mathematics, Professor M.O.

Ibrahim , towards the successful completion of programme. I appreciated Allah in him,

and pray to Allah to continue to provide all his need. In addition, it will be a sign of

ungrateful not to acknowledge the effort of my lecturers in the department most especially,

Dr. (Mrs.) C.N. Ejieji and the entire staff of the department.

My sincere gratitude goes to my parents, Alhaji S. Akande and Hajia H.O. A. Akande

for their unquantified and infinite care, support and advice since the day i was born with-

out tiredness. I pray to Allah to reward you and help in taking care of you. You are role

model to emulate from, no one like you.

Furthermore, and in distinct, I can not forget my brother like a father, a family relative

like the Messenger of Allah (Prophet Muhammad s.a.w.), Mr. Akande Abdul Azeez. After

Allah s.w.a. and the Rossulullahi s.a.w., my immediate parent, he is next, he is a man of

understanding and made me to discover my potential, he never tired of me not even for a

nano-second. I am grateful. In addition, my sincere gratitude goes to the entire member of

my family, start from the head to toe, among are: Mr. Luqman Akande, Mr. Abdul wasi

Akande, Mrs. Aminat Motunrayo Akande Olosan, Mr. Abdul Hakeem Akande, B.Khadijat

Jumai(a.k.a. Jumong female), Fasilat Akande (Mrs.), Akande Olayiwola Abubakar among

others.

Last but one, I am also using this medium to express my gratitude to friends among

are: Abdullahi Ibrahim, Khalid Soddiq, Biala Toyeeb, BOSAKEP members, families, rel-

atives and well-wisher, many but few to mention.

Jazakumullahu Khaeran.

iv

ABSTRACT

This dissertation proposes two deterministic mathematical models of the prevention

of Mother-To-Child tuberculosis. Disease dynamics taking into consideration passive im-

munization, treatment of exposed individuals at the latent period and infectious adult Tu-

berculosis treatment. We developed two compartmental models M-S-I-R (Passive Immune

Infants, Susceptible, Infectious and Removed) and M-S-E-I-R (Passive Immune Infants,

Susceptible, Exposed, Infectious and Removed). The dynamics of the compartments were

described by a system of ordinary differential equations. The equations were solved alge-

braically and analysed for stability. It was established that the disease free equilibrium

states for each model are stable when the number of susceptible individuals produced is

less than natural death rate. It was also established that the endemic equilibrium states

for each model are stable using Bellman and Cooke’s theorem.

v

Contents

TITLE PAGE i

CERTIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 GENERAL INTRODUCTION 1

1.1 ABOUT TUBERCULOSIS (TB) . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 TB DIAGNOSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 TREATMENT FOR TUBERCULOSIS . . . . . . . . . . . . . . . . . . . . 2

1.4 TUBERCULOSIS FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 AIM AND OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 5

1.7 BACKGROUND OF THE STUDY . . . . . . . . . . . . . . . . . . . . . . 6

1.8 SIGNIFICANCE OF THE STUDY . . . . . . . . . . . . . . . . . . . . . . 7

1.9 STATIC AND DYNAMIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.10 DETERMINISTIC AND PROBABILISTIC

(STOCHASTIC): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.11 SCOPE AND LIMITATION OF THE RESEARCH . . . . . . . . . . . . . 8

1.12 DISCRETE AND CONTINUOUS . . . . . . . . . . . . . . . . . . . . . . . 9

1.13 The Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.14 STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

vi

1.15 DEFINITION OF TERMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.16 DEFINITION OF PARAMETER . . . . . . . . . . . . . . . . . . . . . . . 12

1.16.1 Review Model presented by Enagi and Ibrahim (2011) . . . . . . . 12

1.16.2 MODIFIED MODEL PARAMETER . . . . . . . . . . . . . . . . . 13

2 LITERATURE REVIEW 14

2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 LITERATURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 RESEARCH METHODOLOGY 17

3.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The model of Enagi and Ibrahim (2011) . . . . . . . . . . . . . . . . . . . 18

3.2.1 MODEL DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 MODEL FLOW DIAGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 MODEL EQUATIONS OF ENAGI AND IBRAHIM (2011) . . . . 21

3.3.2 EQUILIBRIUM STATE OF ENAGI AND IBRAHIM MODEL . . . 21

3.4 MODIFIED MODEL EQUATION . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 EQUILIBRIUM STATE OF THE MODIFIED MODEL . . . . . . 23

4 ANALYSIS OF STABILITY 25

4.0.2 STABILITY ANALYSIS OF THE MODEL EQUATIONS . . . . . 25

4.1 The stability analysis of ENAGI and IBRAHIM model . . . . . . . . . . . 25

4.1.1 Stability of the Zero Equilibrium State . . . . . . . . . . . . . . . . 26

4.1.2 Stability of the Non-Zero equilibrium Using Bellman and Cooke’s

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 BELLMAN AND COOKE’S THEOREM . . . . . . . . . . . . . . . . . . . 26

4.3 The stability analysis of the modified model . . . . . . . . . . . . . . . . . 29

4.3.1 Stability of the Zero Equilibrium State . . . . . . . . . . . . . . . . 29

4.3.2 Stability of the Non-Zero Equilibrium State . . . . . . . . . . . . . 30

5 DISCUSSION CONCLUSION AND RECOMMENDATION 35

5.1 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii

5.2 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 RECOMMENDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

Chapter 1

GENERAL INTRODUCTION

1.1 ABOUT TUBERCULOSIS (TB)

TB is an infections disease caused by bacteria whose scientific name is Mycobacterium

TB. it was first isolated in 1882 by a German physician named Robert Koch who received

the noble prize for this discovery. TB most commonly affects the lungs but also can involve

almost any organ of the body. Many years ago, this disease was referred to as “Consump-

tion” because without effective treatment, these patients often would waste away. Today,

of course, TB usually can be treated successfully with antibiotics (George 2007; WHO

2010; Patel 2011).

There is also a group of organism referred to as a typical tuberculosis. these involve

other types of bacterium family. Often, these organisms do not cause disease and are

referred to as “colonizers” because they simply live alongside with other bacteria in our

bodies without causing damage. At times, these bacteria can cause an infections that is

sometimes clinically like typical TB. When these a typical mycobacteria cause infection,

they are often very difficult to cure. Often drug therapy for these organisms must be ad-

ministered for one and a half to two years and requires multiple medications.George (2007).

1

1.2 TB DIAGNOSE

TB can be diagnosed in several different ways, including chest X-rays, analysis of

sputum and skin test. sometimes, the chest X-rays can reveal evidence of active TB pneu-

monia. other times, the X-Rays may show scaring (Fibrosis) or hardening (Calcification)

in the lungs, suggesting that the TB is contained and inactive. Examination of sputum

on a slide (smear) under the microscope can show the presence of the TB like family,

including atypical mycobacteria, stain positive with special dyes and are referred to as

Acid-Fast-Bacteria (AFB), George (2007).

A new technology, Light Emitting-Diode Fluorescence Microscopy (LED-FM), a type

of smear microscopy, is more sensitive than the standard Ziehl-Nelsen AFB stain. This

test is faster to perform and again may help identify patients in need of therapy quicker.

George (2007).

Several types of skin tests are used to screen for TB-infection. these so-called tuberculin

skin tests include the time test and the Mantoux test, also known as the PPD (purified

Protein Derivative) test. Research has also revealed that the Tb skin test cannot deter-

mine whether the disease is active or not. this determination requires the chest X-rays

and / or sputum analysis (smear and culture) in the laboratory. The organism can take

up to six weeks to grow in culture in the microbiology lab. A special test to diagnose Tb

called the PCR (Polymerase Chain Reaction) detects the genetic material of the bacteria.

This test is extremely sensitive (it detects minute amounts of the bacteria) and specific

(it detects only tb bacteria). One can usually get results from the PCR test within a few

days (George 2007).

1.3 TREATMENT FOR TUBERCULOSIS

A person with a positive skin test, a normal chest x-ray and no symptoms most likely

has only a few tb germs in an inactive state and is not contagious. Nevertheless, treatment

with an antibiotic may be recommended for this person prevent the tb from turning into

an active infection. The antibiotic used for this purpose is called ISONAIZID (INH). If

taken for six to twelve months, it will prevent the tb from becoming active in the future.

2

In fact, if a person with a positive skin test does not take INH, there is a 5-10% lifelong

risk that the TB will become active (Cohen et al 2004; George, 2007).Taking Isoniazid can

be inadvisable (contraindicated) during pregnancy or for those suffering from alcoholism

or liver disease. Also, Isonaizid can have side effects, the side effects occur infrequently,

but a rash can develop, and the individual can feel tired or irritable. Liver damage from

Isonaizid is a rare occurrence and typically reverses once the drug is stopped. Very rarely,

however, especially in older people, the liver damage(INH hepatitis) can even be fatal.

Another side effect of INH is a decrease sensation in the extremities referred to as a pe-

ripheral neuropathy. This can be avoided by taking vitamin B6 (pyridoxine), and this is

often prescribed drug with INH (George, 2007).

Tb is treated with a combination of medication along with Isonaizid, Rifampin (Ri-

fadin), ethambutol (myambutol) and pyrazinamide are the drugs commonly used to treat

active Tb in conjunction with Isonaizid (INH). Four drugs are often taken for the first

two months of therapy to help kill any potentially resistant strains of bacteria. Then the

number is usually reduced to two drugs for the remainder of the treatment based on drug

sensitivity testing. Streptonysin, a drug that is given by injection, may be used as well,

particularly when the disease is extensive and/or the patients do not take their oral medi-

cations reliably (termed “poor compliance”). Treatment usually lasts for many months and

sometimes for years. Successful treatment of TB is dependent largely on the compliance of

the patient. (Melisa, 2011). It is thus important that alternative strategies are sought, one

of them being the use of mathematical models to give insights into what intervent could

be used to control the transmission of mother to child tuberculosis in endemic areas. The

use of Bacillus Calmette Gverin (BCG) is a vaccine given throughout many parts of the

world (Melisa, 2011).

1.4 TUBERCULOSIS FACTS

1. Tuberculosis (TB) is an infection primarily in the lungs (a pneumonia), caused by

bacteria called Mycobacterium TB. It is spread usually from person to person by

breathing infected air during close contact.

3

2. TB can remain in an inactive (dominant) state for years without causing symptoms

or spreading to other people.

3. When the immune system of a patient with dominant TB is weakened, the Tb can

become active (reactive) and cause infection in the lungs or other parts of the body.

4. The risk factors of acquiring Tb include close-contact situations, alcohol and IV drug

abuse, and certain diseases (for example, diabetes, cancer, and HIV) and occupations

(for example, health-care workers).

5. The most common symptoms and signs of Tb are fatigue, fever, weight loss, coughing

and night sweats.

6. The diagnosis of TB involves skin tests, Chest X-rays, sputum analysis (smear and

culture), and PCR tests to detect the genetic material of the causative bacteria.

7. Inactive TB may be treated with an antibiotic, Isoniazid (INH), to prevent the Tb

infection from becoming active

8. Active TB is treated, usually successfully, with INH in combination with one or more

of several drugs, including rifampin (Rifadin), ethambutol (Myambutol), Pryrazi-

namide and streptomycin.

9. Drug-resistant TB is a serious, as yet unsolved, public-health problem, especially

in South-East, Asia, Africa and in prison populations. Poor patient compliance,

lack of detection of resistant strains and unavailable therapy are key reasons for the

development of drug-resistant Tb.

10. The occurrence of HIV has been responsible for an increased frequency of tuber-

culosis. Control of HIV in the future, however, should substantially decrease the

frequency of TB (Melissa 2011).

1.5 AIM AND OBJECTIVES

The aim of this research is to investigate the stability analysis of tuberculosis on pre-

vention of Mother-To-Child transmission of TB and an adult expose to TB using Bacillus

4

Calmette Gverin vaccine: A deterministic Modeling Approach.

Objectives include:

1. To review a vaccination model developed by Enagi and Ibrahim (2011),

2. To modify and extend the reviewed model for the vaccination

3. To find the diseases free equilibrium of the modified model

4. To find the endemic equilibrium of the modified model

5. To carry out a stability analysis of the modified model

6. perform simulations on the model to determine the effects of various variables in the

model

7. To make appropriate recommendations

1.6 STATEMENT OF THE PROBLEM

Globally, TB is ranked the seventh most important cause of premature mortality and

disability and is projected to remain among the 10 leading causes of disease burden even

in the year 2020 (ARFH, 2013). Nigeria remains one of the high burden countries (HBC’s)

according to the WHO Global TB report for 2012 (ARFH, 2013). Sub-optimal access to

tuberculosis services remains a major bottleneck to the low level of awareness about TB

at the population level. Also, Misleading myths and social stigma affect the demand for

TB services at the grassroots in spite of the fact that the services are free, hence the said

reality that people continue to die from a curable disease.(ARFH,2013)

Tuberculosis thrives in the context of poverty i.e. in addition to its impact on an

individuals ability to work and even earn a living, the costs of seeking accurate diagnosis

and treatment can be considerable for low-income household. TB patients face substantial

cost before diagnosis in that, they often consult several public and private providers before

and in the process of being diagnosed. adherence to treatment of at least six months

by the poor, rural and marginalized patients (such as asylum dwellers, migrants and the

homeless) remains a challenge. In light of all these, it has become pertinent to engage

5

the communities as an important stakeholder in the control of tuberculosis, particularly to

refer persons suspected of TB, provide treatment support and to trace patients who have

defaulted from completing the treatment regimen. (ARFH, 2012).

However, various studies including mathematical modelling of HIV, TB and its control

have been conducted by many researchers, some of them are: (colditz et al, 1995; Blower et

al, 1996; Blower et al, 1995; Colijn, 2006; Luju Liu, 2010; Egbetade, 2012) among others.

And research still continued to in order to see total eradication of the disease.

1.7 BACKGROUND OF THE STUDY

When the incidence of an infection starts to increase resulting to infant death, people

start to look at how best to combat the out break. Launching nationwide vaccination

campaigns (or even vaccinating a small group of a population) can be a costly and time

consuming endeavour, so any tool that will enable the campaign to become directed or to

predict the outcome is highly valuable. Therefore, with the employment of mathematical

model, hopefully the problem will be solved (Abdulrazak et al 2012).

Mathematical model, is the description of a system using mathematical concepts and

language. The process of developing a mathematical model is termed mathematical

modeling. Mathematical models are used not only in the natural sciences (such as Physics,

biology, earth science, metrology) and engineering disciplines (e.g. computer science) but

also in the social sciences (such as economics, psychology, sociology and political science);

physicists, engineers, statisticians, operations research analysts and economists use math-

ematical models most extensively. A models may help to explain a system and to study

the effects of different components, and to make predictions about behaviour (wikipedia).

Mathematical models can take many forms, including but not limited to dynamical sys-

tems, statistical models, differential equations, or game theoretic models. These and other

types of models can overlap, with a given model involving a variety of abstract structures

(Keeling 2008).

A mathematical model uses the language of mathematics to produce a more refined and

precise description of the system. In epidemiology, models allows us to translate between

behaviour at various scales, or extrapolate from a known set of conditions to another. As

6

such, models allow us to predict the population-level epidemic dynamic from an individual-

level knowledge of epidemiological factors, the long-term behaviour from the early invasion

dynamics, or the impact of vaccination of the spread of infection (Keeling, 2008).

Formulating a model for a particular problem is a trade-off between three important

and often conflicting elements: accuracy, transparency, and flexibility. Accuracy, in the

sense that, the ability to reproduce the observed data and reliably predict future dynamics,

is clearly vital, but whether a qualitative or quantitative fit is necessary depends on the

details of the problem. Qualitative is used to gain insight to dynamic infection while

quantitative is sufficiently used to advise on future control policies. Transparency comes

from being able to understand (either analytical or more often numerically) how the various

model components influence the dynamics and interact. Flexibility measures the ease with

which the model can be adopted to new situations; this is vital if the model is to evaluate

control policies or predict future disease levels in an ever-changing environment (Keeling,

2008).

1.8 SIGNIFICANCE OF THE STUDY

The primary reason for the study of infectious disease is to improve, control and ul-

timately eradicate the infection from the population. Model can be powerful tools in this

approach, giving the chance to optimize the limited resources. Several forms of control

measure exist, all operate by reducing the average amount of transmission between infec-

tious and susceptible individuals, which control strategy or mixture of strategies is used

will depend on the disease, the host and the scale of the epidemic. The practice of vacci-

nation began with Edward Jenner in 1796 who developed vaccines against smallpox-which

remains the only disease to date that has been eradicated worldwide.(Abdulrazak, et al.

2012)

Vaccine acts by stimulating a host immune response, such that immunized individuals

are protected against infection. In 1988, the World Health Organization (WHO) resolved

to use similar campaigns strategy to eradicate polio worldwide by 2005. This is still on-

going work although much progress has been made to date (Pegman, 2008; Abdulrazak,

et all. 2012). It may be on the noted outbreak of infectious diseases cause mortality of

7

millions of people as well as expenditure of enormous amount of money in health care and

disease control. It is therefore, essential that adequate attention must be paid to stop the

spread of such disease by taking necessary control measures, vaccination is an important

control measure to reduce the spread of such diseases (Naresh, et al. 2008; Abdulrazak, et

al. 2012). There is believe that with the ongoing research and progress made on Tuber-

culosis vaccine, the model can be applied and yield results that will go a long way in the

control and eradication of tuberculosis.

1.9 STATIC AND DYNAMIC

A dynamic model accounts for time-dependent changes in the state of the system,

while a static (or steady-state) model calculates the system in equilibrium, and thus is

time-invariant. Dynamic models typically are represented by differential equations (Keeling

2008).

1.10 DETERMINISTIC AND PROBABILISTIC

(STOCHASTIC):

A deterministic model is one in which every set of variable states is uniquely de-

termined by parameters in the model and by sets of previous states of these variables.

Therefore, deterministic models perform thesame way for a given set of initial conditions.

Conversely, in a Stochastic model, randomness is present, and variable states are not de-

scribed by unique values, but rather by probability distributions (Keeling 2008).

1.11 SCOPE AND LIMITATION OF THE RESEARCH

The scope of this research work is to determining what situations the model is to give

stability of tuberculosis, we modify the MSIR model by (Enagi and Ibrahim, 2012) into

MSEIR model. The classes or compartments are assumed to interact simultaneously with

each other.

8

It may be unrealistic to expect any model to deliver a quantitative and qualitative

prediction of a disease dynamics. A model is only useful if it represents a simplification,

indicating which element of the process being analyzed is important. However, the better

the fit to reality the more likely the predictions are to be accurate. In the case of tuber-

culosis, the need of accurate models requires the simulation of many different processes.

The requirement for a good fit to field data has thus committed us to developing a model

with many different components and parameters. At present, some of the process we have

modeled are ill-understood that cannot be captured by statistical measures of imprecision.

The model developed so far do not take to account of ethnic group or geographical region

with their response to infection. This model do not capture all epidemiological phenomena

that are relevant to immunity to tuberculosis. In endemic areas, chronic asymptomatic

infection appears to play important role in effective clinical immunity (Smith T. et al,

1999).

This research has limitation of age, infant to adult of fifteen (15) years old but it is

subject to further research on the effect of using BCG vaccine on adults older than fifteen

(15) years.

1.12 DISCRETE AND CONTINUOUS

Discrete models are characterized with discrete time step and formed as difference

equation while continuous models are characterized with continuous time and are formu-

lated as differential equations. We note that despite the differences in these model, they

can all be used to study similar scenarios and give results in the same range. In this re-

search work, we introduce the concept of deterministic models with continuous time step.

Consider time to be the independent variable and Ui, i=1(1)n as the dependent variables

for a particular conceptual, then the system of differential equations can be formulated as:(dUi

dt

)=(Fi(Ui(t)

)), i = 1, 2, ...n (1.1)

such a system is considered to be autonomous since it does not depend on the inde-

pendent variable. Although system (1.1) seems to consider first order derivatives, higher

orders can also be used to describe biological phenomena. However, we restrict our study

9

to first order derivatives. In order to capture the biological picture of the real world, all

initial conditions must be non-negative.

Using the fundamental theorem of existence and uniqueness of initial valued problems, the

solution to (1.1) exists and is unique if it is continuous and differentiable (Perko, 2000).

For examination of system (1.1), we determine the equilibrium points by setting the right

hand side to zero, then compute the Jacobian Matrix evaluated at those equilibrium points

(Keeling 2008)(Abdulrazak et al 2012).

Suppose U∗ = (U∗1, U∗2, · · ·U∗n) is any arbitrary equilibrium point of (1.1) so that

F (U∗) = 0, then the Jacobian Matrix evaluated at U∗ is given by:

J =

∂F1(U∗)

∂U1

∂F2(U∗)

∂U2

, · · · ∂Fn(U∗)

∂Un∂F2(U

∗)

∂U1

∂F2(U∗)

∂U2

, · · · ∂Fn(U∗)

∂Un∂F3(U

∗)

∂U1

∂F2(U∗)

∂U2

, · · · ∂Fn(U∗)

∂Un...

∂Fn(U∗)

∂U1

∂Fn(U∗)

∂U2

, · · · ∂Fn(U∗)

∂Un

(1.2)

When all the eigenvalues of J have negative real parts, then, (U1(t), U2(t), · · ·Un(t))→ U∗

as t → +∞ and the equilibrium point U∗ is said to be locally asymptotically stable.

This concept is widely applied to dynamical systems describing the dynamics of infectious

diseases to predict the extinction or persistence of an infection in a given population

(Keeling 2008).

1.13 The Equilibrium State

The inclusion of host demographic dynamics may permit a disease to persist in a

population in the long term. one of the most useful ways of thinking about what may

happen eventually is to explore when the system is at equilibrium (i.e. by setting all the

ODE to zero) and work out values of the variables. The ODE that has constant solution

is called a critical point of the equation.(Keeling 2008).

10

1.14 STABILITY

The restriction on parameter values of equilibra to be biologically meaningful, there

is need to know how likely the values are observed, in mathematical terms, this call for

a”Stability Analysis”.(Keeling, 2008). Hence, Stability means the zero growth rate with

constant numbers of births and deaths over a period

1.15 DEFINITION OF TERMS

* Compartmental Model: A model whose members of a host population are assigned to

compartments on the basis of their infection status or other attributes, and changes

in the size of compartments are described as a dynamic system.

* Contact Rate: The rate at which individuals in a host population interact in a way

that could potentially transmit an infection.

* Endemic: The constant presence of a disease or infectious agent within a given

geographical area or population group.

* Epidemic Equilibrium: a time invariant state with the infection present in the host

population.

* Epidemic: An event during which an infection sweeps through a population

* Exposed: The member of the host population is classified as exposed if they have

been infected but are not infectious. The exposed compartment is a collection of

such individuals

* A member of the host population is classified as infective if they have been infected

and are infectious. The infective compartment is a collection of such individuals

* The number of members of the host population that become infected in a given

period of time. Often referred to as incidence rate, which is the incidence per unit

time

11

* Population Density: The number of individuals of the host population in a given

geographical area per unit area.

* Prevalence: The population of the host population that is infected

* Recovered: A member of the host population is classified as removed if they are

unable to take part in the transmission of infection, either because they are no longer

infectious and have vaccinated or have gone back to the susceptible compartment.

* Susceptible: or vulnerability - A member of the host population is classified as sus-

ceptible if they are not infected and are capable of being infected. The Susceptible

compartment is the collection of such individuals.

* Latent Infection: Persistence of an infectious agent within the host without symp-

toms.

* Latent Period: Either the interval from initiation of the disease to clinical emergence

of disease/disease detection or the interval between initiation of exposure to disease

or detection of disease.

1.16 DEFINITION OF PARAMETER

1.16.1 Review Model presented by Enagi and Ibrahim (2011)

θ is Proportion of new births given BCG vaccines at birth to protect against infection

θρ is the proportion of incoming individuals immunized against infection

µ represents Natural death

αrepresents the Rate of vaccine efficacy

(1− θ)ρ represent Population of individual not immunized against infection

β represents Instantaneous incidence rate of infection

γ stands for Rate of successful cure of infections TB patients

δ represents Death rate caused as a result of chronic TB infection.

12

1.16.2 MODIFIED MODEL PARAMETER

“a” stands for Rate of vaccine efficacy

α represents Proportion of new births given BCG vaccines at birth to protect against

infection

θα represent the proportion of incoming individuals immunized against infection

µ stands Natural death

“b” is the Instantaneous incidence rate of infection

“c” is Rate of fast progression to infection

d represents Recovery/Remove rate

“f” represents Rate of slow progression to infection

“e” represents Death rate caused as a result of chronic TB infection.

“g” stands for Recovery/Removed rate returned to increase population

13

Chapter 2

LITERATURE REVIEW

2.1 INTRODUCTION

Tuberculosis is a leading cause of infectious mortality (Colijn et al, 2006; Kalu A.U.

2012). Despite the infection agent that causes tuberculosis having be discovered in 1882,

many aspect of natural history and transmission of dynamics of TB are still not fully

understood. This is reflected in the differences in the structures of mathematical models of

TB, which in turn produce differences in the predicted impacts of interactions. Gaining a

greater understanding of TB transmission dynamics requires further empirical laboratory

and field work, mathematical modeling and interaction between them. Modeling can be

used to quantify uncertainty due to different gaps in our knowledge to help identify research

priorities. Fortunately, the present moment is an exciting time for TB epidemiology, with

rapid progress being made in applying new mathematical modeling techniques, new tools

for TB diagnosis and generic analysis and a growing interest in developing more effective

public-health interventions. (White, et al. 2010; Colijn, et al 2006)

2.2 LITERATURES

The first mathematical model of TB was presented by Waaler et al., (1962) (Colijn et

al, 2006). Since this period, the dynamics of tuberculosis epidemic has been a subject of

rigorous research among many researcher and have influenced policy; the spread of HIV

14

and emergence of drug-resistant TB strains motivate the use of mathematical models to-

day (Egbetade S.A. et al, 2012; Blower et al, 1995; Cohen et al, 2007; Colijn et al, 2006;

Murray et al, 2006). Among of the mathematical models include: the stochastic model,

the deterministic (compartmental) model such as:the SIR, SIS, SIRS, SEIS, SEIR, MSIR,

MSEIR, and the MSEIRS models. (Where S - Susceptible class; I - Infective class; M -

passively immune infants; E - Exposed class; and R - Removed or Recovered class) etc

(Kalu A.U., et al., 2012).

Following this, there were several numerical studies, primarily focusing on cost-effectiveness

of different interventions (Brogger, 1967; Revelle et al., 1969). Revelle et al., (1969) used a

model with one progression rate and various latent classes representing different treatment

and control strategies, and argued that vaccination was cost-effective in countries with

high TB burdens. Waaler continued his work in Waaler (1968a), Waaler (1968b), Waaler

and Piot (1969), Waaler (1970) and Waaler and Piot (1970). After the 1970s little work on

models of tuberculosis appeared in the literature until the mid-1990s (Colijn et al, 2006).

In 1995, Blower et al. presented two differential equation models of TB, a simpler model

SIR and a more detailed one. Both are SEIR-type models; the detailed model has both

infectious and non-infectious active TB as well as recovery. In their second model, Blower

et al. (1995) use a similar approach but specify two active TB classes (one infectious and

one noninfectious), a recovered class (with entry cI for cure), and relapse into active class.

Roberto M.A. and Pablo V.N. developed a model for the dynamic interaction between

macrophage T cell and mycobacterium tuberculosis based on six populations. This model

incorporates logistic and Michaelis Menten kinetics for reproduction rates. They tested

the model by using some parameter values from the available literature together with the

stability condition for the infection’s free state. When the parameter values were set to

satisfy that condition, the simulation as predicted by hypothesis of R¡1 converged to the

infections free equilibrium point. On the other hand, when the parameters values violated

that condition, the simulation converged to an equilibrium point different to the infection’s

free equilibrium point.

(Colijn et al, 2006) presented two models: a spatial stochastic individual-based model

and a set of delay differential equations encapsulating the same biological assumptions.

They compared two different assumptions about partial immunity and explore the effect

15

of preventative treatments. They further discussed sharp threshold behaviour and asymp-

totical dynamics are determined by a parameter R0 that is, R0 < 1, the disease free

equilibrium is (usually globally) asymptotically stable, and when R0 > 1, there exist a

unique endemic equilibrium, which is also (usually globally) stable.

Mahboobeh et al, (2011), introduce a model for some diseases which have temporary

immunity. It means after recovery, there is immunity but it is not permanent. In the

model, the people are divided into some groups: susceptible, infective, immune and deatd

people with attention being paid to people who are born or die because of any reasons

except of the disease and was proved to be unstable.

16

Chapter 3

RESEARCH METHODOLOGY

In this chapter, we adopt the model by (Enagi and Ibrahim 2011) to describe a deter-

ministic modeling approach on Mother-To-Child transmission of tuberculosis using Bacil-

lus Camette Guerin vaccine. The aim is to get a clear understanding of the mathematical

model on the effect of the vaccine in the transmission of tuberculosis. This model will later

be extended and modified to include the deterministic of tuberculosis.

3.1 Basic Assumptions

The epidemiological features of tuberculosis lead to the following assumptions about

the transmission of the disease.

• That the population is heterogeneous. That is, the individuals that make up the

population can be grouped into different compartments or groups according to their

epidemiological state: four (4) and five (5) compartments MSIR and MSEIR

• That the population size in a compartment is differentiable with respect to time

and that the epidemic process is deterministic. In other words, that the changes in

population of a compartment can be calculated using only history to develop the

model.

• Birth and death occurs at constant rate

17

• The population is a mixing in a homogeneous manner i.e. everyone has equal chances

of contacting the disease.

• That a proportion of the population of newborns is immunized against TB infection

through vaccination.

• Expiration of duration of vaccine efficacy at constant rate

• That the infection does not confer immunity to the cured and recovered individuals

and so they go back to the susceptible class at a given rate.

• That people in each compartment have equal natural death rate

• Recovery occurs at a constant rate

• That all newborns are previously uninfected by TB and therefore join either the

immunized compartment or the susceptible compartment depending on whether they

are vaccinated or not.

• That there are no immigrants and emigrants. The only way of entry into the popu-

lation is through new born babies and the only way of exit is through death from

natural causes or death from TB-related causes.

3.2 The model of Enagi and Ibrahim (2011)

3.2.1 MODEL DESCRIPTION

Enagi and Ibrahim 2011 model was reviewed, analysis and later extended by including

some important factors. The population is partitioned into four compartments. A propor-

tion θ of new births were given BCG vaccines at birth to protect them against infection.

The Immunized compartment changes due to the coming in of the immunized children

into the population where we assumed that a proportion θρ of the incoming individuals

are immunized against infection, the Susceptible population increases due to the coming in

of new births not immunized against infection into the population at the rate (1−θ)ρ, this

compartment reduces due to expiration of duration of vaccine efficacy at the rate αand

18

also by natural death at the rate µ and infection with an incident rate of infection β.

In the same way the population dynamic of the Infectious class grows with the instan-

taneous incidence rate of infection β resulting from contacts of members of Susceptible

class with Infectious class. This class also reduces by natural death rate µ, successful cure

of infectious TB patients at the rate γ and death caused as result of chronic TB infection

at the rate δ.

Lastly, the dynamics of the Recovered class increases with successful cure of infectious

TB patients at the rate γ and decreases by natural death rate µ.

19

3.3 MODEL FLOW DIAGRAM

(a) REVIEWED MODEL DIAGRAM OF ENAGI AND IBRAHIM (2011)

(b) MODIFIED MODEL DIAGRAM FLOW DIAGRAM.jpg

20

3.3.1 MODEL EQUATIONS OF ENAGI AND IBRAHIM (2011)

dM

dt= θα− (a+ µ)M (3.1)

dS

dt= (1− θ)ρ+ αM − βSI − µS (3.2)

dI

dt= βSI − (γ + µ+ δ)I (3.3)

dR

dt= γI − µR (3.4)

3.3.2 EQUILIBRIUM STATE OF ENAGI AND IBRAHIM MODEL

At equilibrium state,dM

dt=dS

dt=dI

dt=dR

dt= 0

The Disease-Free-Equilibrium (DFE) state:

from equation (3.1),

M =θρ

α + µ(3.5)

from equation (3.3),dI

dt=(βS − (γ + µ+ δ)

)I = 0

⇒ I = 0, and (βS − (γ + µ+ δ)) = 0

S =γ + µ+ δ

β(3.6)

from equation (3.4), γI − µR = 0

⇒ R = 0

forI = 0, and M in equation (3.5), substitute in equation (3.2)

⇒ (1− θ)ρ+ (θρ

α + µ)− µS

Therefore,

S =αρ+ (1− θ)µρ

µ(α + µ)(3.7)

Hence, the DFE state is

P0 =

{θρ

α + µ,αρ+ (1− θ)µρ

µ(α + µ), 0, 0

}(3.8)

21

The endemic equilibrium state: substitute equation (3.5) and (3.6) in (3.2) and in solv-

ing to have

I =(α + µ)

[ρβ − µ(γ + µ+ δ)

]− βθµρ

(α + µ)β(γ + µ+ δ)(3.9)

substitute equation (3.9) in (3.4)

Therefore,

R =γ[(α + µ)

[ρβ − µ(γ + µ+ δ)

]− βθµρ

]µ(α + µ)β(γ + µ+ δ)

(3.10)

Hence, the endemic equilibrium state is given as

P ∗ =

{θρ

α+ µ,γ + µ+ δ

β,(α+ µ)[ρβ − µ(γ + µ+ δ)]− βθµρ

(α+ µ)β(γ + µ+ δ),γ[(α+ µ)

[ρβ − µ(γ + µ+ δ)

]− βθµρ

]µ(α+ µ)β(γ + µ+ δ)

}(3.11)

3.4 MODIFIED MODEL EQUATION

dM

dt= θα− (a+ µ)M (3.12)

dS

dt= aM + (1− θ)α− βSI − µS + gR (3.13)

dE

dt= bSI − (c+ f + µ)E (3.14)

dI

dt= (c+ f)E − (d+ e+ µ)I (3.15)

dR

dt= dI − (g + µ)R (3.16)

22

3.4.1 EQUILIBRIUM STATE OF THE MODIFIED MODEL

At equilibrium the model equations becomes:

θα− (a+ µ)M = 0 (3.17)

aM + (1− θ)α− βSI − µS + gR = 0 (3.18)

bSI − (c+ f + µ)E = 0 (3.19)

(c+ f)E − (d+ e+ µ)I = 0 (3.20)

dI − (g + µ)R = 0 (3.21)

At Disease-Free-Equilibrium (DFE), to have from equation (3.17),

M =θρ

a+ µ(3.22)

from equation (3.18),

S =(a+ (1− θ)µ)α

µ(a+ µ)(3.23)

E = 0 (3.24)

I = 0 (3.25)

R = 0 (3.26)

Therefore, the DFE is

P0 = { θρ

a+ µ,(a+ (1− θ)µ)α

µ(a+ µ), 0, 0, 0} (3.27)

Now, the endemic equilibrium of the modified model, we use equation (3.17) - (3.21)

from equation (3.19)

E =bIS

c+ f + µ(3.28)

from equation (3.20)

E =(d+ e+ µ)

c+ f(3.29)

equating equation (3.28) and (3.29) to have

S =(c+ f + µ)(d+ e+ µ)I

b(c+ f)(3.30)

23

from equation (3.21)

R =dI

g + µ(3.31)

substitute (3.17),(3.30) and (3.31) in (3.18), we obtain

I =(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg(3.32)

E =(d+ e+ µ)

(c+ f)

{(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg

}(3.33)

R =d

(g + µ)

{(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg

}(3.34)

24

Chapter 4

ANALYSIS OF STABILITY

4.0.2 STABILITY ANALYSIS OF THE MODEL EQUATIONS

Having established the equilibrium states, we then investigate the stability of the

equilibrium states. To obtain this, we examine the behaviour of the models near the

equilibrium states.

4.1 The stability analysis of ENAGI and IBRAHIM

model

θα− (a+ µ)M = 0 (4.1)

(1− θ)ρ+ αM − βSI − µS = 0 (4.2)

βSI − (γ + µ+ δ)I = 0 (4.3)

γI − µR = 0 (4.4)

J =

−(α + µ) 0 0 0

α −(βI + µ) 0 0

0 βI −(γ + µ+ δ − βS) 0

0 0 γ −µ

25

The Characteristic equation for the Jacobi’s matrix

|J − λI| = 0 =

∣∣∣∣∣∣∣∣∣∣∣

−(α + µ+ λ) 0 0 0

α −(βI + µ+ λ) 0 0

0 βI −(γ + µ+ δ − βS + λ) 0

0 0 γ −(µ+ λ)

∣∣∣∣∣∣∣∣∣∣∣|J − λI| = (α + µ+ λ){−(βI + µ+ λ)[−(µ+ λ)(γ − µ− δ + βS − λ)]}

4.1.1 Stability of the Zero Equilibrium State

At the DFE state:

λ1 = −α− µλ2 = λ3 = −µλ4 =

αβρ+ (1− θ)βµρµ(α + µ)

− (γ + µ+ δ)

It can be seen that that λ1, λ2, λ3 are all negative, meaning, Tuberculosis will be completely

eradicate at this state and for the λ4, need to be negative

⇒ αβρ+ (1− θ)βµρµ(α + µ)

< (γ + µ+ δ)

whereαβρ+ (1− θ)βµρ

µ(α + µ)is said to be number of latent infections produced and (γ+µ+δ)

is the total removal rate from the infections class.

4.1.2 Stability of the Non-Zero equilibrium Using Bellman and

Cooke’s Theorem

4.2 BELLMAN AND COOKE’S THEOREM

Let H(z) = P (z, ex)where P (z, w) is a polynomial in with principal term.

Suppose, H(iy), y ∈ <, is separated into its real and imaginary parts:

H(iy) = F (y) + iG(y)

if the zeros of H(y)have negative real parts then the zeros of F (y) and G(y) are real, simple

and alternate and

F (0)G′(0)− F ′(0)G(0) > 0,∀y ∈ < (4.5)

26

Conversely, all zeros of H(z) will be in the left half plane provided that either of the

following condition is satisfied:

1. All the zeros of F (y) and G(y) are real, simple and alternate and the inequality (4.5)

is satisfied at least for one y

2. All zeros of F (y) are real and for each zero, the relation (4.5) is satisfied

3. All zeros of G(y) are real and for each zero, the relation (4.5) is satisfied

The above fundamental theorem establishes the analysis of the stability of character-

istic equation as stated by (Jack, 1997) and (Momoh, 2011).

Applying Bellman ad Cooke’s Theorem for endemic equilibrium state

|J − λI| = 0 = λ4 + (4µ− βS + γ + δ + βI + α)λ3 + (−3βSµ+ 3γµ+ 6µ2 + 3δµ+ 3µβI

+3αµ− βSβI − βSα + γβI + γα + βδI + δα + βIα)λ2 + (3γµ2 + 3δµ2+

3µ2βI + 3αµ2 − 2µβSβI − 2µβSα + 2µγβI + 2µγα + 2µδα + 2µβαI

−βSβIα + γβαI + δβαI − 3βµ2S + 4µ3)λ+ µ(−βS + γ + µ+ δ)(βI + µ)(α + µ)

H(iq) = 0 = (iq)4 + (4µ− βS + γ + δ + βI + α)(iq)3 + (−3βSµ+ 3γµ+ 6µ2 + 3δµ+ 3µβI

+3αµ− βSβI − βSα + γβI + γα + βδI + δα + βIα)(iq)2 + (3γµ2 + 3δµ2+

3µ2βI + 3αµ2 − 2µβSβI − 2µβSα + 2µγβI + 2µγα + 2µδα + 2µβαI

−βSβIα + γβαI + δβαI − 3βµ2S + 4µ3)(iq) + µ(−βS + γ + µ+ δ)(βI + µ)(α + µ)

H(iq) = F (q) + iG(q)

where

F (q) = q4 − (3βSµ+ 3γµ+ 6µ2 + 3δµ+ 3µβI + 3αµ− βSβI − βSα + γβI + γα + βδI+

δα + βIα)(q)2 + µ(−βS + γ + µ+ δ)(βI + µ)(α + µ)

F (0) = µ(−βS + γ + µ+ δ)(βI + µ)(α + µ)

F ′(q)= 4q3 + 2(3βSµ+ 3γµ+ 6µ2 + 3δµ+ 3µβI + 3αµ− βSβI − βSα + γβI + γα+

βδI + δα + βIα)q

F ′(0) = 0

27

G(q) = (4µ− βS + γ + δ + βI + α)(q)3 + (3γµ2 + 3δµ2 + 3µ2βI + 3αµ2 − 2µβSβI−2µβSα + 2µγβI + 2µγα + 2µδα + 2µβαI − βSβIα + γβαI + δβαI − 3βµ2S + 4µ3)(q)

G(0) = 0

G′(0) = (3γµ2 + 3δµ2 + 3µ2βI + 3αµ2 − 2µβSβI − 2µβSα + 2µγβI + 2µγα+

2µδα + 2µβαI − βSβIα + γβαI + δβαI − 3βµ2S + 4µ3)

Therefore, J = F (0)G′(0) > 0

Substitute S,Iand R in the above expression to obtain value for J

α β γ δ ρ θ µ J REMARK

0.2 .1 0.0 .1 0.9 0.5 0.0 0.000000000 Threshold

0.2 0.2 0.1 0.1 0.9 0.5 0.1 0.000000000 Threshold

0.2 0.3 0.1 0.2 0.9 0.5 0.1 −1.557796869x10−13 Unstable

0.2 0.4 0.3 0.2 0.9 0.5 0.1 0.000000000 Threshold

0.2 0.5 0.4 0.2 0.9 0.5 0.1 0.000000000 Threshold

0.2 0.6 0.5 0.2 0.9 0.5 0.1 −6.218437552x10−14 Unstable

0.2 0.7 0.5 0.2 0.9 0.5 0.1 2.993554720x10−14 Stable

0.2 0.8 0.6 0.2 0.9 0.5 0.1 0.000000000 Threshold

0.2 0.9 0.7 0.0 0.9 0.5 0.1 0.000000000 Threshold

0.2 1.0 0.8 0.0 0.9 0.5 0.1 0.000000000 Threshold

28

4.3 The stability analysis of the modified model

θα− (a+ µ)M = 0 (4.6)

aM + (1− θ)α− βSI − µS + gR = 0 (4.7)

bSI − (c+ f + µ)E = 0 (4.8)

(c+ f)E − (d+ e+ µ)I = 0 (4.9)

dI − (g + µ)R = 0 (4.10)

The Jacobian Matrix

J =

−(a+ µ) 0 0 0 0

a −(bI + µ) 0 −bS g

0 bI −(c+ f + µ) bS 0

0 0 (c+ f) −(d+ e+ µ) 0

0 0 0 d −(g + µ)

The Characteristic equation for the Jacobi’s matrix

|J − λI| =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−(a+ µ+ λ) 0 0 0 0

a −(bI + µ+ λ) 0 −bS g

0 bI −(c+ f + µ+ λ) bS 0

0 0 (c+ f) −(d+ e+ µ+ λ) 0

0 0 0 d −(g + µ+ λ)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣= (a+ µ+ λ)

{−(bI(c+ f)

(−Sb(g + µ+ λ) + dg

)+ Sb(c+ f)(g + µ+ λ)(Ib+ µ+ λ)

− (g + µ+ λ)(c+ d+ µ+ λ)(d+ e+ µ+ λ)(Ib+ µ+ λ)}

= 0

4.3.1 Stability of the Zero Equilibrium State

At zero equilibrium state (M,S,I,R)=

{θρ

α + µ,αρ+ (1− θ)µρ

µ(α + µ), 0, 0

}λ1 = −(a+ µ)

λ2 = −(g + µ)

λ3 = −µFrom the determinant matrix, we see that the first three eigen values all have negative

29

real parts. We now establish the necessary and sufficient conditions for the remaining two

eigenvalues to have negative real part. The remaining two eigenvalues of the equation will

have negative real part if and only if Trace (A) ¡0 and det. A¿0 i.e. The Routh-Hurwitz

theorem states that the equilibrium state will be asymptotically stable if and only if the

eigenvalues of the characteristics equation of Det. A have negative real part.

⇒ (c+ f + µ+ λ)(d+ e+ µ+ λ)− bS(c+ f) > 0

⇒ (c+ f + µ+ λ)(d+ e+ µ) > bS(c+ f) (4.11)

This means that the necessary and sufficient for DFE of this model to be asymptotically

stable, is that the product of total contraction and total breakdown of latent infection class

given by bS(c+f) must be less than the total removal rate from both infectious and latent

class given by (c+ f + µ)(d+ e+ µ).

4.3.2 Stability of the Non-Zero Equilibrium State

Applying Bellman ad Cooke’s Theorem for endemic equilibrium state

|J − λI| = 0 = λ5 + (Ib+ a+ c+ 2d+ e+ g + 5µ)λ4 + (Iab+ Ibc+ 2Ibd+ Ibe+ Ibg + 4Ibµ

−Sbc− Sbf + ac+ 2ad+ ae+ ag + 4aµ+ cd+ ce+ cg + 4cµ+ d2 + de+ 2dg+

8dµ+ eg + 4eµ+ 4gµ+ 10µ2)λ3 + (Iabc+ 2Iabd+ Iabe+ Iabg + 3Iabµ+ Ibcd+

Ibce+ Ibcg + 3Ibcµ+ Ibd2 + Ibde+ 2Ibdg + 6Ibdµ+ Ibeg + 3Ibeµ+ 3Ibgµ+

6Ibµ2 − Sabc− Sabf − Sbcg − 3Sbcµ− Sbfg − 3Sbfµ+ acd+ ace+ acg + 3acµ+

ad2 + ade+ 2adg + 6adµ+ aeg + 3aeµ+ 3agµ+ 6aµ2 + cdg + 3cdµ+ ceg + 3ceµ+

3cgµ+ 6cµ2 + deg + 3deµ+ 6dgµ+ 12dµ2 + 3egµ+ 6eµ2 + gd2 + 6gµ2 + 10µ3+

3d2µ)λ2 + (Iabcd+ Iabce+ Iabcg + 2Iabcµ+ Iabde+ 2Iabdg + 4Iabdµ+ Iabeg+

+2Iabeµ+ 2Iabgµ+ Iabd2 + 3Iabµ2 + 2Ibcdµ+ Ibceg + 2Ibceµ+ 2Ibcgµ+ 3Ibcµ2

+Ibdeg + 2Ibdeµ+ Ibdfg + 4Ibdgµ+ 6Ibdµ2 + 2Ibegµ+ 3Ibeµ2 + Ibgd2 + 3Ibgµ2+

4Ibµ3 + 2Ibd2 − Sabcg − 2Sabcµ− Sabfg − 2Sabfµ− 2Sbcgµ− 3Sbcµ2 − 2Sbfgµ−3Sbfµ2 + acdg + 2acdµ+ aceg + 2aceµ+ 2acgµ+ 3acµ2 + adeg + 2adeµ+ 4adgµ+

6adµ2 + 2aegµ+ 3aeµ2 + agd2 + 3agµ2 + 4aµ3 + 2ad2µ+ 2cdgµ+ 3cdµ2 + 2cegµ+

3ceµ2 + 3cgµ2 + 4cµ3 + 2degµ+ 3deµ2 + 6dgµ2 + 8dµ3 + 3egµ2 + 4eµ3 + 4gµ3+

2gd2µ+ 3d2µ2 + 5µ4)λ+ (Iabcdµ+ Iabceg + Iabceµ+ Iabcgµ+ Iabcµ2 + Iabdeg+

30

Iabdeµ− Iabdfg + 2Iabdgµ+ 2Iabdµ2 + Iabegµ+ Iabeµ2 + Iabgd2 + Iabgµ2+

Iabµ3 + Iabd2µ+ Ibcdµ2 + Ibcegµ+ Ibceµ2 + Ibcgµ2 + Ibcµ3 + Ibd2µ2 + Ibdegµ+

Ibdeµ2 + Ibdfgµ+ 2Idgµ2 + 2Ibdµ3 + Ibegµ2 + Ibeµ3 + Ibgµ3 + Ibgd2 + Ibµ4−Sabcgµ− Sabcµ2 − Sabfgµ− Sabfµ2 − Sbcgµ2 − Sbcµ3 − Sbfgµ2 − Sbfµ3+

acdgµ+ acdµ2 + acegµ+ aceµ2 + acgµ2 + aceµ3 + ad2µ2 + adegµ+ adeµ2 + 2adgµ2

+2adµ3 + aegµ2 + aeµ3 + agµ3 + agd2µ+ aµ4 + cdgµ2 + cdµ2 + cegµ2 + ceµ2 + ceµ3+

cgµ3 + cµ4 + d2µ3 + degµ2 + deµ3 + 2dgµ3 + 2dµ4 + egµ3 + eµ4 + gd2 + gµ4 + µ5)

H(iy) =(iy)5 + (Ib+ a+ c+ 2d+ e+ g + 5µ)(iy)4 + (Iab+ Ibc+ 2Ibd+ Ibe+ Ibg + 4Ibµ− Sbc−Sbf + ac+ 2ad+ ae+ ag + 4aµ+ cd+ ce+ cg + 4cµ+ d2 + de+ 2dg + 8dµ+ eg + 4eµ+

4gµ+ 10µ2)(iy)3 + (Iabc+ 2Iabd+ Iabe+ Iabg + 3Iabµ+ Ibcd+ Ibce+ Ibcg + 3Ibcµ+

Ibd2 + Ibde+ 2Ibdg + 6Ibdµ+ Ibeg + 3Ibeµ+ 3Ibgµ+ 6Ibµ2 − Sabc− Sabf − Sbcg−3Sbcµ− Sbfg − 3Sbfµ+ acd+ ace+ acg + 3acµ+ ad2 + ade+ 2adg + 6adµ+ aeg+

3aeµ+ 3agµ+ 6aµ2 + cdg + 3cdµ+ ceg + 3ceµ+ 3cgµ+ 6cµ2 + deg + 3deµ+ 6dgµ+

12dµ2 + 3egµ+ 6eµ2 + gd2 + 6gµ2 + 10µ3 + 3d2µ)(iy)2 + (Iabcd+ Iabce+ Iabcg+

2Iabcµ+ Iabde+ 2Iabdg + 4Iabdµ+ Iabeg + 2Iabeµ+ 2Iabgµ+ Iabd2 + 3Iabµ2+

2Ibcdµ+ Ibceg + 2Ibceµ+ 2Ibcgµ+ 3Ibcµ2 + Ibdeg + 2Ibdeµ+ Ibdfg + 4Ibdgµ+

6Ibdµ2 + 2Ibegµ+ 3Ibeµ2 + Ibgd2 + 3Ibgµ2 + 4Ibµ3 + 2Ibd2 − Sabcg − 2Sabcµ− Sabfg−2Sabfµ− 2Sbcgµ− 3Sbcµ2 − 2Sbfgµ− 3Sbfµ2 + acdg + 2acdµ+ aceg + 2aceµ+ 2acgµ+

3acµ2 + adeg + 2adeµ+ 4adgµ+ 6adµ2 + 2aegµ+ 3aeµ2 + agd2 + 3agµ2 + 4aµ3 + 2ad2µ+

2cdgµ+ 3cdµ2 + 2cegµ+ 3ceµ2 + 3cgµ2 + 4cµ3 + 2degµ+ 3deµ2 + 6dgµ2 + 8dµ3 + 3egµ2+

4eµ3 + 4gµ3 + 2gd2µ+ 3d2µ2 + 5µ4)(iy) + (Iabcdµ+ Iabceg + Iabceµ+ Iabcgµ+ Iabcµ2+

Iabdeg + Iabdeµ− Iabdfg + 2Iabdgµ+ 2Iabdµ2 + Iabegµ+ Iabeµ2 + Iabgd2 + Iabgµ2+

Iabµ3 + Iabd2µ+ Ibcdµ2 + Ibcegµ+ Ibceµ2 + Ibcgµ2 + Ibcµ3 + Ibd2µ2 + Ibdegµ+

Ibdeµ2 + Ibdfgµ+ 2Idgµ2 + 2Ibdµ3 + Ibegµ2 + Ibeµ3 + Ibgµ3 + Ibgd2 + Ibµ4 − Sabcgµ−Sabcµ2 − Sabfgµ− Sabfµ2 − Sbcgµ2 − Sbcµ3 − Sbfgµ2 − Sbfµ3 + acdgµ+ acdµ2 + acegµ+

aceµ2 + acgµ2 + aceµ3 + ad2µ2 + adegµ+ adeµ2 + 2adgµ2 + 2adµ3 + aegµ2 + aeµ3 + agµ3+

agd2µ+ aµ4 + cdgµ2 + cdµ2 + cegµ2 + ceµ2 + ceµ3 + cgµ3 + cµ4 + d2µ3 + degµ2 + deµ3+

2dgµ3 + 2dµ4 + egµ3 + eµ4 + gd2 + gµ4 + µ5)

F (q)= (Ib+ a+ c+ 2d+ e+ g + 5µ)(q)4 + (Iabc+ 2Iabd+ Iabe+ Iabg + 3Iabµ+ Ibcd+ Ibce+

Ibcg + 3Ibcµ+ Ibd2 + Ibde+ 2Ibdg + 6Ibdµ+ Ibeg + 3Ibeµ+ 3Ibgµ+ 6Ibµ2 − Sabc− Sabf−

31

Sbcg − 3Sbcµ− Sbfg − 3Sbfµ+ acd+ ace+ acg + 3acµ+ ad2 + ade+ 2adg + 6adµ+ aeg+

3aeµ+ 3agµ+ 6aµ2 + cdg + 3cdµ+ ceg + 3ceµ+ 3cgµ+ 6cµ2 + deg + 3deµ+ 6dgµ+ 12dµ2+

3egµ+ 6eµ2 + gd2 + 6gµ2 + 10µ3 + 3d2µ)(q)2 + (Iabcdµ+ Iabceg + Iabceµ+ Iabcgµ+ Iabcµ2+

Iabdeg + Iabdeµ− Iabdfg + 2Iabdgµ+ 2Iabdµ2 + Iabegµ+ Iabeµ2 + Iabgd2 + Iabgµ2 + Iabµ3

+Iabd2µ+ Ibcdµ2 + Ibcegµ+ Ibceµ2 + Ibcgµ2 + Ibcµ3 + Ibd2µ2 + Ibdegµ+ Ibdeµ2 + Ibdfgµ

+2Idgµ2 + 2Ibdµ3 + Ibegµ2 + Ibeµ3 + Ibgµ3 + Ibgd2 + Ibµ4 − Sabcgµ− Sabcµ2 − Sabfgµ−Sabfµ2 − Sbcgµ2 − Sbcµ3 − Sbfgµ2 − Sbfµ3 + acdgµ+ acdµ2 + acegµ+ aceµ2 + acgµ2+

aceµ3 + ad2µ2 + adegµ+ adeµ2 + 2adgµ2 + 2adµ3 + aegµ2 + aeµ3 + agµ3 + agd2µ+ aµ4+

cdgµ2 + cdµ2 + cegµ2 + ceµ2 + ceµ3 + cgµ3 + cµ4 + d2µ3 + degµ2 + deµ3 + 2dgµ3 + 2dµ4+

egµ3 + eµ4 + gd2 + gµ4 + µ5)

G(q)= (iq)5 + (Iab+ Ibc+ 2Ibd+ Ibe+ Ibg + 4Ibµ− Sbc− Sbf + ac+ 2ad+ ae+ ag + 4aµ+

cd+ ce+ cg + 4cµ+ d2 + de+ 2dg + 8dµ+ eg + 4eµ+ 4gµ+ 10µ2)(iq)3 + (Iabcd+ Iabce+

Iabcg + 2Iabcµ+ Iabde+ 2Iabdg + 4Iabdµ+ Iabeg + 2Iabeµ+ 2Iabgµ+ Iabd2 + 3Iabµ2+

2Ibcdµ+ Ibceg + 2Ibceµ+ 2Ibcgµ+ 3Ibcµ2 + Ibdeg + 2Ibdeµ+ Ibdfg + 4Ibdgµ+ 6Ibdµ2+

2Ibegµ+ 3Ibeµ2 + Ibgd2 + 3Ibgµ2 + 4Ibµ3 + 2Ibd2 − Sabcg − 2Sabcµ− Sabfg − 2Sabfµ−2Sbcgµ− 3Sbcµ2 − 2Sbfgµ− 3Sbfµ2 + acdg + 2acdµ+ aceg + 2aceµ+ 2acgµ+ 3acµ2+

adeg + 2adeµ+ 4adgµ+ 6adµ2 + 2aegµ+ 3aeµ2 + agd2 + 3agµ2 + 4aµ3 + 2ad2µ+

2cdgµ+ 3cdµ2 + 2cegµ+ 3ceµ2 + 3cgµ2 + 4cµ3 + 2degµ+ 3deµ2 + 6dgµ2 + 8dµ3+

3egµ2 + 4eµ3 + 4gµ3 + 2gd2µ+ 3d2µ2 + 5µ4)(iq)

F (0)= (Iabcdµ+ Iabceg + Iabceµ+ Iabcgµ+ Iabcµ2 + Iabdeg + Iabdeµ− Iabdfg + 2Iabdgµ+

2Iabdµ2 + Iabegµ+ Iabeµ2 + Iabgd2 + Iabgµ2 + Iabµ3 + Iabd2µ+ Ibcdµ2 + Ibcegµ+

Ibceµ2 + Ibcgµ2 + Ibcµ3 + Ibd2µ2 + Ibdegµ+ Ibdeµ2 + Ibdfgµ+ 2Idgµ2 + 2Ibdµ3+

Ibegµ2 + Ibeµ3 + Ibgµ3 + Ibgd2 + Ibµ4 − Sabcgµ− Sabcµ2 − Sabfgµ− Sabfµ2 − Sbcgµ2−Sbcµ3 − Sbfgµ2 − Sbfµ3 + acdgµ+ acdµ2 + acegµ+ aceµ2 + acgµ2 + aceµ3 + ad2µ2+

adegµ+ adeµ2 + 2adgµ2 + 2adµ3 + aegµ2 + aeµ3 + agµ3 + agd2µ+ aµ4 + cdgµ2 + cdµ2+

cegµ2 + ceµ2 + ceµ3 + cgµ3 + cµ4 + d2µ3 + degµ2 + deµ3 + 2dgµ3 + 2dµ4 + egµ3 + eµ4+

gd2 + gµ4 + µ5)

G(0) = 0

F ′(q) = 4(Ib+ a+ c+ 2d+ e+ g + 5µ)(q)3 + 2(Iabc+ 2Iabd+ Iabe+ Iabg + 3Iabµ+ Ibcd+

32

Ibce+ Ibcg + 3Ibcµ+ Ibd2 + Ibde+ 2Ibdg + 6Ibdµ+ Ibeg + 3Ibeµ+ 3Ibgµ+ 6Ibµ2−Sabc− Sabf − Sbcg − 3Sbcµ− Sbfg − 3Sbfµ+ acd+ ace+ acg + 3acµ+ ad2 + ade+

2adg + 6adµ+ aeg + 3aeµ+ 3agµ+ 6aµ2 + cdg + 3cdµ+ ceg + 3ceµ+ 3cgµ+

6cµ2 + deg + 3deµ+ 6dgµ+ 12dµ2 + 3egµ+ 6eµ2 + gd2 + 6gµ2 + 10µ3 + 3d2µ)(q)

F ′(0) = 0

G′(q)= 5(iq)4 + 3(Iab+ Ibc+ 2Ibd+ Ibe+ Ibg + 4Ibµ− Sbc− Sbf + ac+ 2ad+ ae+ ag + 4aµ+

cd+ ce+ cg + 4cµ+ d2 + de+ 2dg + 8dµ+ eg + 4eµ+ 4gµ+ 10µ2)(iq)2 + (Iabcd+ Iabce+

Iabcg + 2Iabcµ+ Iabde+ 2Iabdg + 4Iabdµ+ Iabeg + 2Iabeµ+ 2Iabgµ+ Iabd2 + 3Iabµ2+

2Ibcdµ+ Ibceg + 2Ibceµ+ 2Ibcgµ+ 3Ibcµ2 + Ibdeg + 2Ibdeµ+ Ibdfg + 4Ibdgµ+ 6Ibdµ2+

2Ibegµ+ 3Ibeµ2 + Ibgd2 + 3Ibgµ2 + 4Ibµ3 + 2Ibd2 − Sabcg − 2Sabcµ− Sabfg − 2Sabfµ−2Sbcgµ− 3Sbcµ2 − 2Sbfgµ− 3Sbfµ2 + acdg + 2acdµ+ aceg + 2aceµ+ 2acgµ+ 3acµ2+

adeg + 2adeµ+ 4adgµ+ 6adµ2 + 2aegµ+ 3aeµ2 + agd2 + 3agµ2 + 4aµ3 + 2ad2µ+ 2cdgµ+

3cdµ2 + 2cegµ+ 3ceµ2 + 3cgµ2 + 4cµ3 + 2degµ+ 3deµ2 + 6dgµ2 + 8dµ3 + 3egµ2 + 4eµ3+

4gµ3 + 2gd2µ+ 3d2µ2 + 5µ4)

G′(0) = (Iabcd+ Iabce+ Iabcg + 2Iabcµ+ Iabde+ 2Iabdg + 4Iabdµ+ Iabeg + 2Iabeµ+ 2Iabgµ+

Iabd2 + 3Iabµ2 + 2Ibcdµ+ Ibceg + 2Ibceµ+ 2Ibcgµ+ 3Ibcµ2 + Ibdeg + 2Ibdeµ+ Ibdfg+

4Ibdgµ+ 6Ibdµ2 + 2Ibegµ+ 3Ibeµ2 + Ibgd2 + 3Ibgµ2 + 4Ibµ3 + 2Ibd2 − Sabcg − 2Sabcµ−Sabfg − 2Sabfµ− 2Sbcgµ− 3Sbcµ2 − 2Sbfgµ− 3Sbfµ2 + acdg + 2acdµ+ aceg + 2aceµ+

2acgµ+ 3acµ2 + adeg + 2adeµ+ 4adgµ+ 6adµ2 + 2aegµ+ 3aeµ2 + agd2 + 3agµ2 + 4aµ3+

2ad2µ+ 2cdgµ+ 3cdµ2 + 2cegµ+ 3ceµ2 + 3cgµ2 + 4cµ3 + 2degµ+ 3deµ2 + 6dgµ2 + 8dµ3+

3egµ2 + 4eµ3 + 4gµ3 + 2gd2µ+ 3d2µ2 + 5µ4)

Substituting these into the condition stated in the Bellman and Cooke’s Theorem, we have

J = F (0)G′(0) > 0

Hence, the condition holds for J > 0

S =(c+ f + µ)(d+ e+ µ)I

b(c+ f)

33

I =(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg

E =(d+ e+ µ)

(c+ f)

{(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg

}

R =d

(g + µ)

{(a+ µ)(c+ f)(g + µ)αb− (c+ f)(g + µ)bαθµ− (a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)µ

b(a+ µ)(g + µ)(c+ f + µ)(d+ e+ µ)− (a+ µ)(c+ f)bdg

}

substitute these in (4.11) and then computed with Mapple software, we have the following result

in table below:

a b c d e f g α µ θ J REMARK

0.2 0.1 0.1 0.1 0.1 0.05 0.1 0.9 0.0 0.5 0.000002187000000 Stable

0.2 0.2 0.2 0.1 0.1 0.10 0.0 0.9 0.1 0.5 3.097500000x10−7 Stable

0.2 0.3 0.3 0.1 0.2 0.15 0.0 0.9 0.1 0.5 −3.86370092210−7 Unstable

0.2 0.4 0.4 0.3 0.2 0.20 0.2 0.9 0.1 0.5 0.0001103040000 Stable

0.2 0.5 0.5 0.4 0.2 0.25 0.3 0.9 0.1 0.5 0.0006216124297 Stable

0.2 0.6 0.6 0.5 0.2 0.30 0.4 0.9 0.1 0.5 0.002427660300 Stable

0.2 0.7 0.7 0.5 0.2 0.35 0.4 0.9 0.1 0.5 0.003613638150 Stable

0.2 0.8 0.8 0.6 0.2 0.40 0.5 0.9 0.1 0.5 0.01027592198 Stable

0.2 0.9 0.9 0.7 0.0 0.45 0.6 0.9 0.1 0.5 0.04130270245 Stable

0.2 1.0 1.0 0.8 0.0 0.50 0.7 0.9 0.1 0.5 0.09568518464 Stable

34

Chapter 5

DISCUSSION CONCLUSION AND

RECOMMENDATION

5.1 DISCUSSION

In this dissertation, stability analysis of tuberculosis has been investigated on the effect of BCG

to the prevention of Mother-To-Child tuberculosis using mathematical model presented by Enagi

and Ibrahim (2011). It was started by reviewing the model to understand the effect of the BCG

on tuberculosis infection dynamics.

Bellman and Cooke’s Theorem technique was used to analysis this model to test for the

stability. For the the Enagi and Ibrahim model to be stable, it was shown that at DFE, the first

three eigenvalues are negative and for the fourth eigenvalues to be stable requires to be negative,

this will be achieved by having

αβρ+ (1− θ)βµρµ(α+ µ)

< (γ + µ+ δ)

whereαβρ+ (1− θ)βµρ

µ(α+ µ)is said to be number of latent infections produced and (γ+µ+ δ) is the

total removal rate from the infections class.

Furthermore, in the modified model, research equally give the first three (3) eigenvalues to

be negative and for the last two to be, we have

(c+ f + µ+ λ)(d+ e+ µ) > bS(c+ f)

This means that the necessary and sufficient condition for DFE of the modified model to be

asymptotically stable, is that the product of total contraction and total breakdown of latent

35

infection class given by bS(c + f) must be less than the total removal rate from both infectious

and latent class given by (c+ f + µ)(d+ e+ µ).

5.2 CONCLUSION

In this seminar work, we presented two deterministic mathematical models, (MSIR) and (MSEIR)for

stability analysis of tuberculosis: The effect of BCG vaccine in preventing mother to child trans-

mission of tuberculosis and from chapter 4, we have successfully proved that the disease free

equilibrium state is stable for the first three (3) eigenvalues being negative and for the last to

be, the total contraction and total breakdown of latent infection class must be less than the total

removal rate from both infectious and latent class.

However, it is important to note that for the population to be sustained, the recovery rate

from infectious class “d” must be greater or equal to the natural death and the death rate due

to infection combined with natural death else the population will tend towards extinction.

The model proposed in this paper can used in interactive workshops with health planners and

other stakeholders in the analysis of TB control so that participants could gain a better under-

standing of how BCG vaccines could be used to control the disease. In addition, the model can

be applied to simulate data of a community or specific country over a given time frame in order

to estimate the number of Mother-To-Child TB infections so that prevention and intervention

strategies could be properly designed.

5.3 RECOMMENDATION

Globally, Tuberculosis is ranked the seventh most important cause of premature mortality and

disability and is projected to remain among the ten (10) leading causes of disease burden even

forecast up to the year 2020 and Nigeria is counted to be among ARFH (2013). Furthermore,

TB also thrives in the context of poverty, the costs of seeking accurate diagnosis and treatment

can be considerable for low-income household. TB patients face substantial cost before diagnosis

in that they often consult several public and private providers before and in the process of being

diagnosed. Hence, in view to these, the following are my recommendation:

1. The Government, non-governmental organizations and stakeholders should help in creating

awareness because prevention is better and cure

36

2. Provision should be made for the diagnose and treatment of TB because for the economy

of any country to be stable, it involves manpower

3. TB patient should be refers to necessary place for proper treatment to avoid spread of

disease

4. TB patients should stick to their medication for at least six (6) month for effective cure.

5. Individuals should shun unprescribed drug usage, this contributed to active tuberculosis

6. Use of antibiotic can also be use to cure non-clinical tuberculosis (See Treatment of TB in

Chapter 1)

37

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