# Mathematical Modelling of Optimal Strategies for Improving Industrial Productive Population in the Presence of Perverse Diseases Pandemic

on August 10, 2021

Mathematical Modelling of Optimal Strategies for Improving Industrial Productive Population in the Presence of Perverse Diseases Pandemic.

### ABSTRACT

In this thesis, we investigate certain key aspects of mathematical modelling to explain the epidemiology of HIV/AIDS, Tuberculosis, Hepatitis B, Tumour, diabetes and stroke at the workplace and assess the potential benefits of proposed control strategies.

The compartmental epidemiological modelling approach was used in the formulation of the models on HIV/AIDS, tuberculosis (TB), Hepatitis B (HBV), Tumour and Diabetes pandemic.

In each of the cases, the dynamics of the disease was studied according to the various compartments based on the transmission dynamics of the disease. The resulting model in each of the diseases was a system of nonlinear ordinary differential equations.

The solutions of the various models were obtained using the ODE45 module in MATLAB software built based on Runge-Kutta 4th Order method and the results were plotted on graphs.

The model on stroke was formulated using a fluid dynamics approach where the geometry of the arteries of the employee(s) was used in determining the flow patterns of blood most especially in an occluded internal carotid artery.

The resulting model here is a partial differential equation which was solved using the Galerkindiscretisation scheme implemented by the finite element method in MATLAB and the results plotted on graphs.

In the case of HIV/AIDS, a combination of intervention strategies including prevention, Education/enlightenment, and HAART treatment was studied showing a great potential to control HIV transmission in the workplace and indirectly improving the productivity of the labour force population and also the availability of good labour force.

In the TB model, the two strategies employed, optimal education strategy and chemoprophylaxis clearly showed that both controls reduced/minimised the infected workforce population. In HBV, after introducing therapy, the viral load decreased after 10 days.

In addition, the number of free virions at the final time tf= 100 (days) in the case with control is less than that without control thereby increasing the efficiency of drug therapy in inhibiting viral production.

In tumour disease, the models described how DCs and NK cells of workers, as the innate immune system, and CD8 + T cells, as the specific immune system, affect the growth of the tumour cell population in the body of workers.

In the diabetes model, without control, the workforce population is lower than that with control. The workforce population increased progressively as the control increases.

As the stenotic height increased, the diameter of the arteries reduced leading to occlusion thereby lowering the blood flow velocity with high blood pressure leading to stroke. The equilibrium analysis showed that the models were globally and asymptotically stable at both the disease-free and endemic states.

The optimal control measure was established alongside the various strategies for the controls which showed prodigious improvement in the workforce population on the application of the controls.

Title Page i
Declaration ii
Certification iii
Dedication iv
Acknowledgement v
Abstract x
List of Figures xvi
List of Symbols (Parameters and Variables) xvii
List of Tables xxvii

Chapter One: Introduction

1.1 Background of Study 1
1.2Statement of the Problem 4
1.3Aims and Objective of Study 9
1.4 Justification of Study 9
1.5 Scope of Study 10

Chapter Two: Literature Review

2.1 HIV/AIDS Pandemic 11
2.2 Tuberculosis (TB) Pandemic 14
2.3 Hepatitis B Virus 21
2.4 The Human and Economic Burden of Stroke 25
2.5 Mathematical Models of Tumour 27
2.6 Mathematical Models and Data Used in Diabetology 38
2.7 Ill-Health and its Economic Consequences 40
2.8 Human Capital, Health and Productivity 43
2.9 Sensitivity Analysis 49
2.10 Guidelines for Improving the Labour Productivity 50

Chapter Three: Methodology

3.0 Introduction 52
3.1 Physiognomies of Workers 52
3.2 Tuberculosis 64
3.2.1 Patterns of TB Infection 65
3.3 Hepatitis B 68
3.3.1 Geographical Distribution of Hepatitis B 68
3.3.2 Transmission of Hepatitis B 69
3.3.3 Symptoms of Hepatitis B 69
3.3.4 People at Risk for Chronic Hepatitis Disease 69
3.3.5 Diagnosis of Hepatitis B 70
3.3.6 Treatment of Hepatitis B 70
3.3.7 Prevention of Hepatitis B 71
3. 4 Diabetes Mellitus 72
3.4.1 Types of Diabetes Mellitus 73
3.4.2 Symptoms of Diabetes 76
3.4.3 Complications Caused by Diabetes 77
3.5 Stroke (Cerebrovascular Accident) 79
3.5.2 Diagnosis of Stroke 85
3.5.3 Treatments of Stroke 85
3.5.4 Prevention of Stroke 88
3.6 Tumour Growth 89
3.6.1 Benign tumours 89
3.6.2 Precancerous conditions 89
3.6.3 Malignant tumours 90
3.6.4 How tumours and cancers are named 90
3.6.6 Prognosis and Survival from Cancer 91
3.7 Theorems Governing the Dissertation 92
3.7.1 Theorem 93
3.7.2 Gronwall’s Inequality 93
3.7.3 Well-Posed Problem 95
3.6.1 Definition 95
3.7.4 Hartman-Grobman Theorem 96

Chapter Four: Model Construction

4.0 Introduction 99
4.1 Formulation of the Various Models 99
4.1 Formulation of HIV/AIDS Model 99
4.1.1 Assumptions of the Model on HIV/AIDS 99
4.1.2 Model Variables and Parameters 99
4.1.3 Model Flow Diagram/Compartmental Analysis 100
4.1.4 Mathematical Model for HIV/AIDS 100
4.1.5 Formulation of the Optimal Control Problems 102
4.1.6 Disease-Free Equilibrium (DFE) and Endemic Equilibrium 104
4.1.7 Formulation of Optimal Control Problem 105
4.2 Formulation of Tuberculosis (TB) Model 107
4.2.1 Assumptions of the Model on Tuberculosis 107
4.2.2 Variables for Tuberculosis (TB) Model 107
4.2.3 Parameters of Tuberculosis (TB) Model 108
4.2.4 Model Flow Diagram for the TB Model 108
4.2.5 Tuberculosis Model 109
4.2.6 Modelling the Optimal Control Problem for TB 110
4.2.7 Existence of an Optimal Control Solution 112
4.2.8 Characterization of Optimal Controls 112
4.3 Formulation of the Model for Optimal Control of Hepatitis B 115
4.3.1 Assumptions of the Model for Optimal Control of Hepatitis B 115
4.3.2 Variables for Hepatitis B Virus Model 116
4.3.3 Parameters of Hepatitis B Model 116
4.3.4 Mathematical Model for Hepatitis B 116
4.3.5 The Optimal Control Problems 117
4.4 Formulation of the Model on Tumour/Cancer 119
4.4.1 Assumptions of the Model on Tumour/Cancer 120
4.4.2 Model Variables for Tumour Growth 120
4.4.3 Parameters of Tumour Growth Model 120
4.4.4 Mathematical Model for Tumour/Cancer 121
4.4.5 Non –Dimensionalisation 121
4.4.6 Steady State and Stability Analysis 122
4.5 Formulation of the Diabetes Model 124
4.5.1 Assumptions of the Diabetes Model 124
4.5.2 Variables of the Diabetes Model 124
4.5.3 Parameters of Diabetes Model 124
4.5.4 Model Flow Diagram for Diabetes 125
4.5.5 Mathematical Model for Diabetes 125
4.5.6 The Optimal Control: Existence and Characterization 126
4.5.6.1 Existence and Positivity of Solutions 126
4.5.6.2 Characterization of the Optimal Control 128
4.6 Formulation of Model on Cardiovascular Accident/High Blood Pressure 129
4.6.1 Assumptions of the Model on Cardiovascular Accident/High Blood Pressure 129
4.6.2 Variables and Parameters of the Model on Cardiovascular Accident 131
4.6.3 Mathematical Model for Cardiovascular Accident/High Blood Pressure 131
4.6.5 Governing Equations 137
4.5.7 Estimating wave generation in the base of the brain 140
4.7 Solution of the Various Models 145

Chapter Five: Results, Discussions, Summary/Conclusion and Recommendations

5.0 Introduction 148
5.1 Results/Discussion 148
5.1.1 Results/Discussion on workforce Productivity in the presence of HIV/AIDS 148
5.1.2 Results/Discussion on Workforce Productivity in the presence of Tuberculosis 161
5.1.3 Results/Discussion on Workforce Productivity in the presence of Hepatitis 182
5.1.4 Results/Discussion on Workforce Productivity in the presence of Tumour 186
5.1.5 Results/Discussion on Workforce Productivity in the presence of Diabetes 195
5.1.6 Results/Discussion on Workforce Productivity in the presence of Stroke 202
5.2 Summary 207
5.3 Conclusion 207
5.4 Contribution to Knowledge 210
References 211

### INTRODUCTION

It is obvious that the welfare of individuals, the growth of enterprises and the development of the national economies are largely dependent on their comparative productivity.

There exist differences among the various countries of the world based on political ideologies, economic systems or some such reasons but all unanimously recognize the importance of the improvement in the productivity levels.

Productivity is a ratio between the output of the wealth produced and the input of resources used in the process of any economic activity (Rao, 2009).

The input creativity can yield a greater amount of output through conversion efficiency and here lies the importance of improvement in productivity levels. The concept of productivity of course, with some degree of confusion, has remained a continuous and challenging area of study.

The changes in the productivity levels greatly influence a wide range of human, economic and social considerations, such as higher standard of living, rapid economic growth, improvement in the balance of payments, control of inflation culture of the nation etc.

Productivity in a most simple way may be defined as the ratio of output to input. It is expressed as under: P = O I where P = Productivity, O = Output, I = Input.

According to Oxford Illustrated Dictionary 2nd Edition (2003), productivity is defined as efficiency in industrial production to be measured by some relationship of outputs to inputs.

The Encyclopedia Britannica (2009) defined productivity according to economics as the ratio of what is produced to what is required to produce it.

Usually, this ratio is in the form of an average expressing the total output of some category of goods divided by the total input of say, labour and raw materials.

In principle, any input can be used in the denominator of the productivity ratio. Thus one can speak of productivity of land, labour, capital or sub-categories of any of these factors of production.

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