Mathematical Modelling of Optimal Strategies for Improving Industrial Productive Population in the Presence of Perverse Diseases Pandemic
In this theses, 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 ODE45 module in MATLAB software built based on Runge-Kutta 4th Order method and the results plotted on graphs. The model on stroke was formulated using 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 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 virons 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 work force population is lower than that with control. The work force population increased progressively as the controlincreases. 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 with the various strategies for the controls which showed prodigious improvement in the workforce population on the application of the controls.
TABLE OF CONTENTS
Title Page i
List of Figures xvi
List of Symbols (Parameters and Variables) xvii
List of Tables xxvii
Table of Contents xxiii
Chapter One : Introduction
1.1 Background of the Study 1
1.2Statement of the Problem 4
1.3Aims and Objective of the Study 9
1.4 Justification of the Study 9
1.5 Scope of the 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
3.6.5 How cancer spreads 91
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
220.127.116.11 Existence and Positivity of Solutions 126
18.104.22.168 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
5.0 Introduction 148
5.1 Results/Discussion 148
5.1.1 Results/Discussion onWorkforce Productivity in the presence of
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
1.1 Background of Study
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 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 balance of payments, control of inflation culture of the nation etc. The productivity in a most simple way may be defined as ratio of output to input. It is expressed as under:
P = O 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.
It should be noted that, the concept of productivity is so closely attached to the labour input that the term productivity is almost used as the synonym of productivity of labour. International Labour Organization (ILO) (2007) defined productivity as, the ratio between the volume of output as measured by production indices and corresponding volume of labour input as measured by employment indices. The most important reasons why labour is used as the commonest factor in measuring productivity are: It is easy and precise to measure the units of labour inputs as compared to other inputs like materials, capital etc. Productivity can be very easily measured in terms of output per man, output per man hour or output per unit of labour time, Labour input is universally applied to all types of plants processes, productions and it has become a common practice to link wages, with the productivity.
Productivity is a technique of extracting greater output from the inherent input creativity of various resources through the conversion efficiency. The conversion efficiency which changes the level of productivity is largely affected by numerous factors. All these factors affect the level of productivity either individually or jointly. Some important factors are classified as: Technological; Managerial; Financial; Natural; Sociological, and Government.
The technological advancement always strives to achieve the increase of production with minimum costs and efforts, which always result into increased productivity for example, application of mechanized power, automation etc. Progressive and imaginative managerial skills always tap greater output of the human and nonhuman resources. Delegation of authority, true recognition of human factor, imaginative judgment results into increased productivity and contented labour force. The availability of financial resources enables the organization to spend moneys for the research and development, employment of professional executives, adaptation of latest technology, provision of amenities, effective stock piling and material control. All these factors directly affect the level of productivity. The low level of productivity and poor industrial growth of the underdeveloped countries is due to poor capital
formation constraints on the financial resources and good quality manpower management.
The natural resources like geographical, physical and climatic conditions directly affect the level of productivity. The effect of these factors is confined to certain type of industries and the possibility of bringing them within control as in humidification in textile industry, quality thickness and depth of the mineral resources; climate effect on the labour efficiency etc.
The generic characteristics and racial quality has a great impact on the productivity of the labour. Productivity is also affected by the attitude of the workers towards the work and the approach of the management towards the workforce and the provision of working conditions.The Government policy regarding financial incentives, taxation policy, tariff policy, industrial licensing labour laws etc also affect the productivity.Provision of concessional loans for modernization, tax incentives for the expenditures on research and development etc help increase
the level of productivity.
There are still many other ways of improving productivity. In construction, productivity is usually taken to mean labour productivity, that is, units of work placed or produced per man hour. The inverse of labour productivity, man-hours per unit (unit rate), is also commonly used.
Productivity is the ratio of output to all or some of the resources used to produce that output. Output can be homogenous or heterogeneous. Resources comprise: labour, capital, energy, raw materials, etc.
Horner and Talhouni (2008) stated “A popular concept in the USA, and increasingly in the UK, is the concept of earned hours. It relies on the establishment of a set of standard outputs or “norms” for each unit operation. Thus, a number of earned hours are associated with each unit of work completed.” “Productivity may then be seen also as the ratio of earned to actual hours. The problem with this concept is in establishing reliable “norms”, for setting standards. It also depends on the method used to measure productivity, and on the extent to which account is taken of all the factors which affect it.
Few discussions of the potential economic impact of the global AIDS epidemic fail to observe that HIV/AIDS, unlike most other infectious diseases, strikes working-age adults during what should be their most productive working years. The mortality component of this loss is clear: lives lost to AIDS cannot contribute to economic growth. The morbidity component, however, has rarely been addressed. Although it is generally accepted that the morbidity associated with
1.2 Statement of the Study
The identification and evaluation of factors affecting labour productivity have become a critical issue facing project managers for a long time in order to increase industrial productivity.
Understanding critical factors affecting productivity are both positive and negative and can be used to prepare a strategy to reduce inefficiencies and to improve the effectiveness of industrial performance.
Knowledge and understanding of the various factors affecting labour productivity is needed to determine the focus of the necessary steps in an effort to reduce project cost overrun and project completion delay, thereby increasing productivity and overall project performance. Based on the study and survey, factors affecting industrial productivity have been identified and are grouped into 15 categories according to their characteristics, namely: design factors, execution plan factors, material factors, equipment factors, labour factors, health and safety factors, supervision factors, working time factors, project factors, quality factors, financial factors, leadership and coordination factors, organization factors, owner/consultant factors, and external factors.
There are also factors that specifically affect the small and medium companies/establishments. The researcher has presented among many others to include: Lack of material, Labour strikes, delay in arrival of materials, financial difficulties of the owner, unclear instruction to labourer and high absenteeism of labours, bad weather (e.g. rain, heat, etc.), non-discipline labour and use of alcohol and drugs, no supervision method, design changes, repairs and repetition of work and bad resources management, bad supervisors, absenteeism and far away from location of material
storage, and bad leadership.
On the other hand there are some top factors that affect large companies which the researcher has presented among many others to include: unclear instruction to labourer, delay in arrival of materials, lack of material and financial difficulties of the owner, no definite schedule, low supervisor’s capability, supervision method, lack of equipment, and high absenteeism of labours, supervisors absenteeism, frequent damage of equipments, and labour strikes, design changes, incomplete drawing and inspection delay, poor communication in site and inaccurate design.
Considering all these factors, we have also seen that it is still very evident that these factors affect industrial productivity both negatively and positively. Those factors affecting productivity positively would have to be adhered to and still given a new lease of life while those affecting productivity negatively would have to be optimized for an improved industrial productivity.
In the course of looking into the factors, the barriers to productivity were detected and presented thus: The countries’ economy has become increasingly more dynamic and complex. As a result, economic measurement and analysis, particularly relating to productivity, have become more difficult and complicated. The main problem involves properly defining units of measurement, evaluating qualitative changes and obtaining reliable data for both inputs and outputs. This process is further complicated by the need to price – deflate this data in order to evaluate changes in productivity in real terms. Measurement of inputs is problematic. Variations in the rate of input utilization are at best partially picked up in data series. In particular, the rate of capital equipment utilization, i.e. the measurement of machine hours, is rarely accomplished. Labour input, if measured by hours actually worked, is better suited to reflect the changing rate of manpower utilization, but remains an imperfect measure. The increasing prominence of the service sector within the national economy has generated increased mis-measurement of labour hours. Information technology may aggravate this measurement error by allowing increased work flexibility and longer effective workdays that are not properly captured by the official statistics.
It is therefore evident that to optimize productivity, the aforementioned factors have to be critically looked into and optimized. It is not out of place that the human factor has a very important role to play in the areas of apportioning work, monitoring/supervising and also in the provision of skilled labour. The health of the individuals cannot be undermined in this course.
Going on the present report from World Health Organization (2010), Global Summary of the AIDS Epidemic – 2010 has shown that; Number of People Living with HIVTotal: 34.0 million [31.6million–35.2million], Adults: 30.1 million [28.4million–31.5million], Women: 16.8 million [15.8million–17.6million], Children: 3.4million [3.0million–3.8million], Number of People Newly Infected, Total: 2.7 million [2.4million–2.9million], Adults: 2.3 million [2.1million–2.5million], Children: 390 000 [340 000–450 000]. AIDS Deaths Total: 1.8 million [1.6million– 1.9million], Adults: 1.5million [1.4million–1.6million], and Children: 250000 [220 000–290 000].
In a similar scenario, the Sub-Saharan Africa HIV and AIDS Statistics – 2010 has shown the Number of People Living with HIV Total – 22.9 million [21.6 million – 24.1 million], Children – 3.1 million [2.8 million – 3.5 million]. Number of people newly infected, total – 1.9 million [1.7 million – 2.1 million], Children – 350 000 [300 000 – 410 000], adult prevalence (15-49 years) [4.7% – 5.2%] and death due to AIDS has a Total of 1.2 million [1.1 million – 1.4 million], children – 230 000 [200 000 – 260 000]. Over 7 000 new HIV infections a day about 97% are in
low and middle income countries, about 6 000 are in adults aged 15 years and older, of whom: almost 48% are women, about 42% are young people (15-24). In 2002, it was estimated that globally, 40 million people were infected with HIV/AIDS and more than 70% of this disease burden was borne in sub-Saharan Africa where HIV/AIDS is now the number one overall cause of death (Guinness et al. 2003). Besides the human cost, HIV/AIDS is having profound effects on productive employment and economic development in Africa, and hence its inability to cope with the pandemic (Bloom et al. 2006). The spread of HIV/AIDS is being aided by stigma and discrimination which keep the disease underground and discourage persons from being tested and seeking treatment. The total number of people tested for HIV globally remains unacceptably low with an estimated 90% of people who are HIV infected worldwide unaware of their status (Waxman et al. 2007).
Despite the magnitude of the problem, little rigorous empirical research on the micro level effects of HIV/AIDS at the firm and industry levels has been published in the literature on African development. Unpublished case studies from Kenya and Botswana in 1994 found widely varying impacts, with costs ranging from less than 1% of profits to nearly 9% (Rosen et al. 2004).
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