ABSTRACT
This study proposes and analyzes a non-linear mathematical model for the dynamics
of HIV/AIDS with treatment and vertical transmission. The equilibrium points of the
model system are found and their stability is investigated.
The model exhibits two equilibria namely, the disease-free and the endemic
equilibrium. It is found that if the basic reproduction number R0 1, the disease-free
equilibrium is always locally asymptotically stable and in such a case the endemic
equilibrium does not exist. If 0 R 1, a unique equilibrium exist which locally
asymptotically stable and becomes globally asymptotically stable under certain
conditions showing that the disease becomes endemic due to vertical transmission.
By using stability theory and computer simulation, it is shown that by using
treatment measures (ARVs) and by controlling the rate of vertical transmission, the
spread of the disease can reduced significantly and also the equilibrium values of
infective, pre-AIDS and AIDS population can be maintained at desired levels.
A numerical study of the model is also used to investigate the influence of certain
key parameters on the spread of the disease.
TABLE OF CONTENTS
Certification...............................................................................................................i
Declaration and Copyright ........................................................................................ii
Acknowledgements................................................................................................. iii
Dedication ...............................................................................................................iv
Abstract ....................................................................................................................v
Table of Contents.....................................................................................................vi
List of Figures .......................................................................................................viii
List of Tables............................................................................................................x
CHAPTER ONE: INTRODUCTION....................................................................1
1.1 General Introduction.....................................................................................1
1.2 Statement of the Problem..............................................................................5
1.3 General Objectives .......................................................................................6
1.4 Specific Objectives.......................................................................................6
1.5 Significant of the Study ................................................................................6
CHAPTER TWO: LITERATURE REVIEW.......................................................8
2.1 Research Hypothesis ..................................................................................10
2.2 Methodology..............................................................................................11
CHAPTER THREE: THE MODEL DESCRIPION AND ANALYSIS .............12
3.0 Historical Background................................................................................12
3.1 Model Formulation.....................................................................................15
3.2 Positivity of Solutions ................................................................................21
3.3 Stability Analysis of the Model...................................................................25
3.4 Equilibrium Points of the Model.................................................................25
3.5 Computation of the Basic Reproduction Number, 0 R .................................26
3.6 Local Stability of the Disease Free Equilibrium..........................................28
3.7 The Endemic Equilibrium and Local Stability ............................................31
3.8 Existence of Forward Bifurcation ...............................................................36
3.9 Global Stability of the Endemic Equilibrium..............................................43
CHAPTER FOUR: THE NUMERICAL SIMULATIONS OF THEMODEL...49
4.1 Model Simulation.......................................................................................49
CHAPTER FIVE: DISCUSSION, CONCLUTION AND FUTUREWORK.....70
5.1 Discussion..................................................................................................70
5.2 Conclusions................................................................................................74
5.3 Future Work...............................................................................................75
REFFERENCES...................................................................................................76