ABSTRACT: Infectious disease has become a source of fear and superstition since the first ages of human civilization. In this study, we consider the Discrete SIR model for disease transmission to explain the use of this model and also show significant explanation as regard the model. We discuss the mathematics behind the model and various tools for judging effectiveness of policies and control methods. The model has two equilibrium states, namely the disease free equilibrium and the endemic equilibrium state. The stability of each equilibrium state is discussed and the endemic equilibrium has been found to be stable while that of the disease free equilibrium was unstable. The basic reproduction number was computed and gotten to be 1.2. The disease was found to persist with Ro > 1, whenever the natural death rate is reduced, the death rate caused by the disease and the transmission rate is increased but the disease dies out with Ro < 1, whenever the transmission rate and the birth rate is reduced and recovery rate is increased. The data used in the study was extracted from a journal stated in chapter 4 and were analysed using the one-way sensitivity analysis, and also used matlab 2013 to run respectively simulations for the change in each parameter extracted. The results of the sensitivity analysis showed that the birth rate, the transmission rate reduced and increasing the recovery rate will make the disease die out of the population.
TABLE OF CONTENTS:
Title page
Undertaking
Certification
Dedication
Acknowledgement
Table of content
Abstract
CHAPTER 1: INTRODUCTION 1.1 Background of the Study
1 1.1.1 Infectious Disease………………………………………………..4 1.1.1.1 Notes on Infectious Disease……………………………………4 1.1.1.2 Modelling of Infectious Disease………………………………6
1.2 Significance of Study…...….……………………………………………………8 1.3 Objectives of Study…………………………………………………………….8 1.4 Scope of Study………………………………………………………………..9
1.5 Definition of Terms………………..……….…………………………………9
1.6 Organization of Study……....……………...………………………………......11
CHAPTER 2: LITERATURE REVIEW 2.1 Review of various SIR Models…………………………………………………12
2.2 Difference Equation….………………......…………………………………….17 2.2.1 Classification of Difference Equation…………………………………17
2.3 Description of some Deterministic Discrete Time Epidemic Models…………18
2.3.1 The SI Model …..………………………………………………... 19
2.3.1.1 The Reproduction Number of an SI Model……… 20
2.3.2 The SI Model …..………………………………………………... 21
2.3.3 The SEIR Model…………………………………………...22
2.4 Linear and Nonlinear Models……………….………………………23
2.5 Equilibrium States……………………………………………..24
2.6 Uncertainty and Sensitivity analysis in Modelling………………………
CHAPTER 3: METHODOLOGY 3.1 Introduction……..…………………………………………………….. 27
3.2 Preliminaries………….……………………………………………..……28
3.3 Model Formulation……….……………………………………………………28
3.3.1 Model Assumptions……………………………………………………..28
3.3.2 Description of the Discrete SIR Model………………………………….29
3.4 Model Equations……………………………………………………………….31
3.5Equilibrium Points………………………..……………………………………36
3.5.1Disease-Free Equilibrium Point…………………………………………...37 3.5.2Endemic Equilibrium Point………………………………………………...38
3.6 Stability Analysis of the Equilibrium Points………………………...................40 3.6.1 Stability Analysis of the Disease-free Equilibrium……………………….41
3.6.2 Stability Analysis of the EndemicEquilibrium………………..........43
3.7 The Basic Reproduction Number …………..……………………………...46
CHAPTER 4: ANALYSIS AND NUMERICAL SIMULATION 4.0 Introduction…..………………….……………………………………………..49
4.1 Equilibrium Points………………………………………………..50
4.2 Stability Analysis…………………………………….……..51
4.2.1 Stability Analysis of the Disease-Free Equilibrium Point……………51
4.2.2 Stability analysis of the Endemic Equilibrium Point………………....53
4.3 Sensitivity Analysis………………...………………………………………….55
4.3.1 Sensitivity Analysis using the Basic Reproduction Number………….55
4.3.2 Sensitivity Analysis of Disease transmission by Simulation……56
4.4 Summary..................................................................................................62
CHAPTER FIVE: SUMMARY AND CONCLUSION 5.0 Introduction……….……………………………………………………………66
5.1 Summary and Conclusion……………………………………………………...67 References………………………………………………………68