Post on 24-Feb-2023
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
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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