Mathematical Modelling of Optimal Strategies for Improving Industrial Productive Population in the Presence of Perverse Diseases Pandemic
Chapter One
Aims and Objective of theStudy
The main objective of the study is specifically to determine
- and formulate mathematical models that will optimize industrial productivity in the presence of HIV/AIDS, Tubercolosis, Stroke (cardiovascular accident), diabetes, tumour and hepatitisB,
- the impact of these diseases on industrial productive workforcepopulation,
- if productive infected employees/employers are less infectious than the non-productive infected,
- labour force recruitments strategies in the presence of perverse diseasespandemic,
- the stability of the disease-free and endemic equilibrium points with the aim of improving industrial productivity,
- and establish the optimal strategies for disease control in the
CHAPTER TWO
LITERATURE REVIEW
The HIV epidemic which has caused millions of human deaths worldwide is a serious public health problem and therefore calling upon all stakeholders in HIV prevention and control to use appropriate intervention strategies useful in curbing down the epidemic.HIV epidemic control strategies through optimal control techniques have attracted many researchers since the beginning of the disease as presented here;
Yusuf et al. (2010) formulated the strategy for controlling the spread of the pandemic as an optimal control problem. Results indicated that increased change in sexual habits and providing ARV treatment at the pre-AIDS stage, reduce both the incidence and prevalence much faster than starting treatment after progression into AIDS.
Bihara et al. (2011), considered an HIV model with immune response, with the aim of exhibiting the basic reproduction number (R0), proving the global stability of the virus free equilibrium and studying the stability of the endemic equilibrium in a special case, and then apply optimal control to the model to maximize the healthy cells and minimize the cost of treatments. Result shows that if Ro>1, there exist a unique endemic equilibrium and if Ro<1, the virus free equilibrium is globally asymptotically stable. In their study, it was noted that increasing the effectiveness of control has a positive effect on uninfected CD4+T-cells and negative effect on viral load.
Shabani et al. (2011), examined the combined effects of screening and variable in flow of infective immigrants on the spread of HIV/AIDS in a population of varying size. They proposed and analyzed this qualitatively using the stability theory of differential equations as nonlinear deterministic mathematical model. Results show that as the reproductive number R0>1, the rate of inflow of infective immigrants increases leading to persistence of the disease in the population; However the presence of screening greatly reduces the spread of HIV/AIDS.
Mlyashmbi et al. (2011) investigated a mathematical model of infectious diseases by immigrants in a vaccinated and temporary immune protected population. They analyzed the stability of the system for the existence of the disease free and endemic equilibrium points and their result shows that the disease free equilibrium point is asymptotically stable when Ro<1 and unstable when Ro>1. It is also shown that the increase of immigrants in a population tends to lower the first dose-vaccination coverage, hence the disease become endemic in the population. Numerical simulations of the model showed that in the absence of the immigrants the disease can be eradicated in a population with a single dose vaccination only.
Kibonaet al. (2011) studied a non-linear mathematical model for HIV epidemic that spreads in a variable population size through both horizontal and vertical transmission in the presence of HIV infected immigrants. Results show that if the basic reproduction number Ro<1, the disease-free equilibrium is always locally asymptotically stable and in such a case the endemic equilibrium does not exist. If Ro>1, a unique endemic equilibrium exists, which is locally asymptotically stable under certain conditions. This shows that the disease becomes endemic due to constant immigration of both HIV infected and non-infected individuals into the community.
Amsalework and Ejigu (2008),discussed an efficient numerical method (based on optimal control) to identify the best treatment strategy in order to maximize the healthy cells by using chemotherapies with minimum side effects. Results from numerical simulations of the model shows that the amount of chemotherapy in the body should be controlled by the control variables. It also compared the CD4+T – cells count before and after using these treatments and the result showed that the immune system is strong when the treatments with controls are used.
Musekwa et al. (2009), proposed a model for the heterogeneous transmission of HIV/AIDS in the presence of disease carriers, to assess the role of screening, as an intervention program that can slow the epidemic. Result shows that treatment in the presence of screening reduces the spread of the disease as compared to treatment without blood screening. Blood screening helps to counsel and educate carriers on safe sexual interaction methods and encourage them to seek treatment since there is a lot of fear of the side effects of using ARV therapy or HAART among people living with HIV/AIDS. Result also shows that screening alone cannot be used as a comprehensive HIV fighting strategy. Multiple strategies should therefore be emphasized in the fight against HIV/AIDS. In this study three control strategies will be considered; control rate of infective immigrant inflow through screening and treatment; control of contact rate through treatment and intensive public awareness campaign and control due to behavioural change through intensive public awareness campaign.
CHAPTER THREE
METHODOLOGY
The methodology of model building aimed at improving industrial productivity in presence of perverse diseases pandemic has been presented in this chapter of the thesis. The physiognomies of workers in presence of diseases are presented alongside with the background information on the epidemiology of each of the disease under consideration in this Disertation. The diseases among many others presented here include HIV/AIDS, Tuberculosis , Hepatitis B, Cancer/Tumour, Diabetes, Cerebrovascular Accident (Stroke)//High Blood Pressure.
Physiognomies of Workers
In Presence of HIV/AIDS The Susceptible
The susceptible class here is being referred to as a individuals that are free from the disease. These individuals are bound to exhibit the following characters:
- The Person displays no symptoms, and looks and feels very
- Some susceptible individuals may engage in substance
- Susceptible group can become infected through sex with infected partners or through contact with infected material like blood or\
CHAPTER FOUR
MODEL FORMULATION AND SOLUTIONS
Formulation of the Various Models
In this section of the work, we are interested in the formulation of the models on HIV/AIDS, TB, HBV, Tumour, Diabetes and Stroke.
CHAPTER FIVE
RESULTS/DISCUSSION, SUMMARY AND CONCLUSION
Introduction
This very chapter of this thesis presented the results of the study with the aid of graphs and as a matter of different diseases under consideration, the researcher deemed it fit to present the results based on the individual diseases.
Results/Discussion
The discussion is done alongside with the graphs such that the discussion on the results presented on each of the graphs is presented immediately just below the graph.
Results/Discussion on Workforce Productivity in the presenceof HIV/AIDS
We explore a simple model with screening, prevention, education, and treatment as control measures to study the effects of control practices and transmission of HIV. We use various combinations of the four controls, three controls and two controls at a time. This is done under the following scenarios to compare numerical results.
- Strategy A: Using Prevention ( u1 ) and education/monitoring ( u2 ) without treatment of infectives ( u3 =0 ).
- Strategy B: Using screening ( u1 ) and HAART treatment of infectives ( u3 ) without education/monitoringcontrol ( u2 = 0 ) and no prevention ( u3 = 0 ).
- Strategy C: Prevention ( u1 ) and without education/monitoring ( u2 =0 ) and no HAART treatment of infectives ( u3 = 0 ).
- Strategy D: Using education/monitoring ( u2 ) and prevention of infectives ( u3 =0 ).
Summary
In this thesis, we considered the theoretical analysis of compartmental HIV/AIDS, TB, Hepatites B, Tumour, and Diabetes models alongside with models governed by fluid dynamics to account for the cases of stroke in the workplace. The work was motivated by the possibility that mathematical modelling could improve our understanding of the dynamics of these diseases, particularly the impact of infection on an employee’s productivity at the workplace.
Conclusion and Recommendation
In the case of HIV/AIDS, we developed relatively simple but reliable models which confirmed our expectations: that a combination of intervention strategies including Prevention (condom use, protected sex, use of screened blood transfussion, avoidance of unsterilized sharp objects usage), Education/enlightenment, screening and HAART treatment has the potential to control HIV transmission at a community level, indirectly also improving the productivity of a labour force and also the availability of good labour force.
In the TB model, we employed two strategies which are optimal education strategy, i.e. the effort on education which allows workers to go to the hospital, to be diagnosed early be treated from the disease. This treatment control measure is the chemoprophlaxis.
We were able to seethat the burden of diabetes in the work place can be reduced at three levels which included controlling:
- the number of people evolving from prediabetes (workers) to diabetes without complication (β1(1 −u(t))),
- the number of diabetic patients developing complications (β2(1−u(t)))and
- the number of people evolving directly from pre-diabetes to diabetes with omplications due to delayed diagnosis ( β3(1−u(t))).
We extended this analysis to recruitment and employment policies regarding wellness of staff. This was done by adapting principles from fields such as dynamical systems, control theory and economics utilising data available in literature. The end result is, in our opinion, a good framework for integrating available data with a toolkit of mathematical methods which can be continuously improved in accordance with the ever-changing picture of these diseases pandemic.
Initially, we were motivated by the work of Okosunet al (2012) which enabled us undertake investigations on the impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives by incorporating condom use, screening of unaware infectives and treatment of the infected. We observed that unawareness by infectives has a great cost positive impact on the community.
Furthermore, we showed that interventions result in more people being in the productive classes and less people in the non-productive and AIDS classes. The study equally showed that optimal control theory, numerical simulations and cost effectiveness analysis can be used to make meaningful decisions on choosing appropriate combinations of interventions to manage epidemics.
We derived and analyzed the various models for the transmission of HIV/AIDS, TB, Hepatites B, Tumour, Diabetes, and Stroke diseases to examine the recruitment effect of the following: non-productive susceptible individuals, non-productive infected individuals, productive susceptible individuals and productive infected individuals into an organizational workforce population.
We established the stability analysis of the equilibrium and found that the models for HIV/AIDS, TB, Hepatites B, Tumour, and Diabetes exhibit backward and Hopf bifurcation under different scenarios respectively. Furthermore, we also performed optimal control analysis of the models by using Pontryagin’s Maximum Principle to derive and analyze the conditions for optimal control of the disease with effective screening, prevention, HAART treatment regime and enlightenment of non-productive susceptible and infectious individuals.
The results suggest that the effective recruitment strategy must include screening and prevention for the susceptible and treatment of the infected towards reducing the nonproductivity of employees thereby ensuring productivity. However, deliberate budgetary provisions need to be put in place so that enlightenment/monitoring of employees may be included observing as it has a significant impact in reducing the non-productivity of employees (labour force) in the presence of HIV/AIDS.
We conclude that the successful screening of unaware infectives has a significant impact in reducing the endemicity of HIV/AIDS. This may be as a result of awareness by infectives who also took necessary precautionary measures not to spread the disease. Control programs that follow these strategies can effectively reduce the spread of HIV/AIDS in a population.
We deduced from this thesis the transition from one disease state to another is not instantaneous, it is worth considering time delay in the models for the first five diseases formulated during this study as part of future work. The importance of taking epidemiological modelling from the realm of the purely theoretical and applying it to real world situations cannot be overstated. Thus the future work will, therefore, extend to other problems related to dynamics of all these diseases, for instance substance abuse and Ebola, common problems faced by communities. Although this may limit the range of suitable mathematical tools, it is envisaged that even fairly simple models based on limited available data could lead to valuable insights. We would then have to cherry- pick the most useful mathematical tools and apply them to the most complete data in these diseases. This may entail having to conduct surveys wherein it may become necessary to incorporate parameter estimation techniques.
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