Philosophiae Doctor - PhD (Mathematics)
http://hdl.handle.net/11394/3210
Thu, 22 Oct 2020 04:47:46 GMT2020-10-22T04:47:46ZMathematical modeling of TB disease dynamics in a crowded population.
http://hdl.handle.net/11394/7357
Mathematical modeling of TB disease dynamics in a crowded population.
Maku Vyambwera, Sibaliwe
Tuberculosis is a bacterial infection which is a major cause of death worldwide. TB is a
curable disease, however the bacterium can become resistant to the first line treatment
against the disease. This leads to a disease called drug resistant TB that is difficult
and expensive to treat. It is well-known that TB disease thrives in communities in overcrowded
environments with poor ventilation, weak nutrition, inadequate or inaccessible
medical care, etc, such as in some prisons or some refugee camps. In particular, the World
Health Organization discovered that a number of prisoners come from socio-economic disadvantaged
population where the burden of TB disease may be already high and access
to medical care may be limited. In this dissertation we propose compartmental models of
systems of differential equations to describe the population dynamics of TB disease under
conditions of crowding. Such models can be used to make quantitative projections of TB
prevalence and to measure the effect of interventions. Indeed we apply these models to
specific regions and for specific purposes. The models are more widely applicable, however
in this dissertation we calibrate and apply the models to prison populations.
Philosophiae Doctor - PhD
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11394/73572020-01-01T00:00:00ZEfficient Variable Mesh Techniques to solve Interior Layer Problems
http://hdl.handle.net/11394/7324
Efficient Variable Mesh Techniques to solve Interior Layer Problems
Mbayi, Charles K.
Singularly perturbed problems have been studied extensively over the past
few years from different perspectives. The recent research has focussed on the
problems whose solutions possess interior layers. These interior layers appear
in the interior of the domain, location of which is difficult to determine a-priori
and hence making it difficult to investigate these problems analytically. This
explains the need for approximation methods to gain some insight into the behaviour
of the solution of such problems. Keeping this in mind, in this thesis
we would like to explore a special class of numerical methods, namely, fitted
finite difference methods to determine reliable solutions. As far as the fitted
finite difference methods are concerned, they are grouped into two categories:
fitted mesh finite difference methods (FMFDMs) and the fitted operator finite
difference methods (FOFDMs). The aim of this thesis is to focus on the
former. To this end, we note that FMFDMs have extensively been used for
singularly perturbed two-point boundary value problems (TPBVPs) whose
solutions possess boundary layers. However, they are not fully explored for
problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions
possess interior layers and to improve accuracy of these approximation
methods via methods like Richardson extrapolation. Then we extend these
two ideas to solve such singularly perturbed TPBVPs with variable diffusion
coefficients. The overall approach is further extended to parabolic singularly
perturbed problems having constant as well as variable diffusion coefficients.
Philosophiae Doctor - PhD
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11394/73242020-01-01T00:00:00ZHigh Accuracy Fitted Operator Methods for Solving Interior Layer Problems
http://hdl.handle.net/11394/7320
High Accuracy Fitted Operator Methods for Solving Interior Layer Problems
Sayi, Mbani T
Fitted operator finite difference methods (FOFDMs) for singularly perturbed
problems have been explored for the last three decades. The construction of
these numerical schemes is based on introducing a fitting factor along with the
diffusion coefficient or by using principles of the non-standard finite difference
methods. The FOFDMs based on the latter idea, are easy to construct and they
are extendible to solve partial differential equations (PDEs) and their systems.
Noting this flexible feature of the FOFDMs, this thesis deals with extension
of these methods to solve interior layer problems, something that was still outstanding.
The idea is then extended to solve singularly perturbed time-dependent
PDEs whose solutions possess interior layers. The second aspect of this work is
to improve accuracy of these approximation methods via methods like Richardson
extrapolation. Having met these three objectives, we then extended our
approach to solve singularly perturbed two-point boundary value problems with
variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses
followed by extensive numerical simulations supporting theoretical findings
are presented where necessary.
Philosophiae Doctor - PhD
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11394/73202020-01-01T00:00:00ZRobust numerical methods to solve differential equations arising in cancer modeling
http://hdl.handle.net/11394/7250
Robust numerical methods to solve differential equations arising in cancer modeling
Shikongo, Albert
Cancer is a complex disease that involves a sequence of gene-environment interactions
in a progressive process that cannot occur without dysfunction in multiple systems.
From a mathematical point of view, the sequence of gene-environment interactions often
leads to mathematical models which are hard to solve analytically. Therefore, this
thesis focuses on the design and implementation of reliable numerical methods for nonlinear,
first order delay differential equations, second order non-linear time-dependent
parabolic partial (integro) differential problems and optimal control problems arising
in cancer modeling. The development of cancer modeling is necessitated by the lack of
reliable numerical methods, to solve the models arising in the dynamics of this dreadful
disease. Our focus is on chemotherapy, biological stoichometry, double infections,
micro-environment, vascular and angiogenic signalling dynamics. Therefore, because
the existing standard numerical methods fail to capture the solution due to the behaviors
of the underlying dynamics. Analysis of the qualitative features of the models with
mathematical tools gives clear qualitative descriptions of the dynamics of models which
gives a deeper insight of the problems. Hence, enabling us to derive robust numerical
methods to solve such models.
Philosophiae Doctor - PhD
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11394/72502020-01-01T00:00:00Z