Department of Mathematical and Computational Sciences

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Open Access
q- Enumeration of permutations avoiding adjacent patterns
(2009-09) Takalani, Ntendeni Annah; Tshifhumulo, T. A.
Open Access
Mathematical modelling of transmission and control of malaria
(2012-12-19) Mulaudzi, Matodzi Stanley; Garira, Winston
Malaria starts with plasmodium sporozoites infection of the host liver, where development into blood stage parasites occurs. A number of deterministic models are developed in this thesis. The release of modified mosquitoes aims to displace gradually the wild (natural) mosquito from the habitat. We discuss the suitability of this technique when applied to pre-domestically adapted plasmodium falciparum mosquitoes which are transmissor of malaria disease. The dynamics of interaction of sporozoites, liver cells, merozoites and red blood cells which cause the symptoms and pathology of the disease is comprehensively studied. We then show how variability of host-parasite immunity is incorporated in the model which are constructed to include liver and blood compartments by subdividing the host population into various mutually exclusive compartments. The increase in eggs, larval and pupal stages of mosquitoes increase the vector mosquito population and transmission of the disease, hence the suggestion that immature and adult mosquitoes be controlled extensively. The models which are in the form of nonlinear ordinary differential equations are rigorously analysed using ex tensively analytic and numerical techniques to determine important epidemiological thresholds, stability of the steady states and the persistence of infection in the respective populations. Conclusions are made based on the results obtained from the analysis of the models of malaria that have been developed
Open Access
Analysis of a boundary value problem for a system on non-homogeneous ordinary differential equations (ODE), with variable coefficients
(2015-01-16) Makhabane, Paul Suunyboy; Hlomuka, V. J.; Garira, W.
In this study we present a condition for the existence and uniqueness of the solution y(x) for a system of nonhomogeneous linear first order Ordinary Differential Equations (ODE). The existence and uniqueness of the solution of y(x) was confirmed through the Picard Lindelof Theorem. We then study the stability of matrix A(x) using its spectrum, moreover, A(x) is symmetric. This is a pre-condition for the application of Lefschetz direct stability method. We then modify the given Lefschetz system (Meyer, 1964) to suit the problem at hand. The direct method requires the construction of a suitable Lyapunov function; not easy for a time-independent (non-dynamic) problem. For a time-dependent problem the energy thereof becomes a suitable candidate for a Lyapunov function. For a non-dynamic problem it is harder to construct a Lyapunov function as there are no rules for that purpose. In our study we modified the Lefschetz system for the direct stability method and applied it to confirm the Lefschetz stability criterion using the modified systems of linear first order ODEs with variable coefficients. The Lefschetz method afforded us the construction of a credible Lyapunov function which enabled us to confirm the stability of the null solution to our problem. From our modified Lefschetz direct stability system, we solved the Makhabane / Hlomuka equation (5) for B(x) (7) which we later confirmed as both symmetric and positive definite.
Open Access
Factors associated with maternal mortality in South Africa (2003-2008)
(2015-03-02) Mukondeleli, Livhuwani Ellen; Amey, A. K. A.; Kyei, K. A.
Maternal death is the death of woman during pregnancy or within 42 days of termination of pregnancy from any cause related to or aggravated by the pregnancy or its management but not from accidental or external causes. Health complications during pregnancy and child birth are a major challenge worldwide. Statistics South African reports mortality data for South Africa classified according to various demographic characteristics and causes. The causes are classified using ICD10. Using the Statistics South Africa data, this study focused on the death of a women aged between 15 to 49 over the period of 2003 to 2008. The purpose of this study is to identify the level, trends, leading causes and factors associated with death due to maternal related causes among women aged 15-49 in South Africa and to explore the relationship between the socio-demographic and clinical factors on one hand with maternal deaths over the period 2003 to 2008. Logistic regression and log linear analysis were used to explore the relationships. The results show that the maternal mortality ratio of South Africa increased from 114 per 100 000 live births in 2003 to 195 per 100 000 live births in 2008. Free State province had the lowest maternal mortality ratio and the Western Cape province had the highest ratio. Northern Cape had the same levels of maternal mortality as Western Cape in 2003 - 2004, but increased at a faster rate afterwads. By 2008, the rate of maternal mortality in Northern Cape was about twice that of Western Cape. The top five causes of maternal death were eclampsia (015), puerperal sepsis (072), postpartum hemorrhage (085), maternal infectious and parasitic diseases classifiable elsewhere but complicating pregnancy, childbirth and the puerperium (098), other maternal diseases classifiable elsewhere but complicating pregnancy, childbirth and the puerperium (099). There is a significant association of maternal mortality with socio demographic variables (Marital Status, Age, Death Province, Place of Death) and other illnesses (HIV, TB, pneumonia and diarrhoea). There was no significant association between maternal deaths and educational level. When four or five causes of death were listed on the death notification form, maternal mortality is not directly associated with HIV. The link to HIV is only through the opportunistic diseases associated with HIV. When only two or three causes are listed, direct association exist.
Open Access
Existence and Uniqueness of a solution to a flow problem about a Rotating Obstacle at low Reynolds number
(2015-05) Nyathi, Freeman; Moyo, S.
The flow is described by the Navicr-Stokes equations in the domain n C R3. The open-bounded domain is assumed to have a cone property. The rotation of a 3- dimensional symmetrical impermeable cylindrical rigid body in the fluid is studied. The model is constructed in a manner that the equations describe a system in a frame attached to the obstacle.The system of the governing equations is constructed on the basis of conservation of angular momentum of the rigid body and the conservation of linear momentum of the fluid. When the conservation of angular momentum is taken into account, a dynamic boundary condition is considered. The uniqueness of this unknown velocity vector field is confirmed by using the so called energy method. In this study we chose the incompressible viscous Navier-Stokes flow and thus the fluid density does not change through out the flow.
Open Access
A stochastic programming framework for financial intermediaries liquidity in South Africa
(2015-05) Chagwiza, Wilbert; Garira, W.; Moyo, S.
We provide a thorough overview of the current liquidity problem, dis cussing what are considered to be state-of-the-art approaches in both in dustry and in academia, and clearly establish our motivation to depart from current liquidity management standards. We noted that during 2007 to 2009 financial crisis different countries throughout the whole world faced serious challenges that ranged from chronic liquidity problems, deep-rooted risk management deficiencies and poor corporate governance practices. The liq uidity crisis, especially in the USA, was caused by low real interest rates stimulating an asset price bubble fuelled by new financial products that were not stress-tested and that failed in the downturn. Large depository institu tions face the question of how to optimally utilise funds and manage liquid ity problems. The existing liquidity models are not standardised, and do not take into consideration the complexity and the nature of the bank. As evi denced by the late 2000 financial crisis, it is evident that nowadays there is not a metric that seems to be completely adequate to prevent liquidity crises. Our contribution is fourfold. First we investigate appropriate scenario gener ation methods and perform a rigorous investigation on how to generate mul tistage scenario trees. Further, we investigate the inclusion of all qualifying liquid asset instruments into the portfolio optimisation. We propose a novel multistage stochastic programming methodology for liquid asset control. Thus we define how to construct and solve stochastic programming models for liquidity needs-driven sub-portfolios. Our approach is based on scenario trees and makes no assumption on the distributions of random variables. Fi nally, we investigate the inclusion of liquidity needs-driven strategies which are core liquidity, cash cushion, operational cash and discretionary liquidity into the overall liquidity portfolio. Stochastic programming is a multi-faceted problem, and even the most fo cused treatment necessarily incorporates techniques from a wide range of disciplines. Through the development of fully coherent models and a suffi ciently robust solution methodology, we provide a thorough overview of the problem at hand, discussing approaches in both industry and in academia. We introduce practical and theoretical advances that are to our best knowl edge unexplored in the current literature, and document the usefulness of these avenues through a systematic series of increasingly complex applica tions and experiments. The data used is from the South African Reserve Bank (SARB) and International Monetary Fund (IMF) from January 1988 to May 2014. Most corporations such as commercial banks generally hold a unitary liquidity portfolio and not necessarily segmented to focus on bank specific cash and liquidity needs. Banks do have different cash needs at different times. Financial intermediaries to manage liquidity efficiently and effectively, segmenting the optimal liquid asset portfolio is the best manage ment strategy. We construct different stochastic programming models based on decision making under risk and provision of powerful paradigm for de cision making under uncertainty. The stochastic programming models with recourse are clear, easy to implement, very efficient and provide the optimal solution according to future possible set of scenarios. In defining and con structing stochastic programming models with recourse, sensitivity analysis should be carried out to increase the decision maker's understanding of the problem and to show the effect of different assumptions. Constructing an SP model with more stages provides better approximations compared to a single-period model. We found that the use of the proposed models through segmenting the unitary liquidity portfolio improves the management of liq uidity compared to current techniques which are based on simulation, ex perience and trial-and-error. Thus high quality liquidity buffers insure the bank against any adverse liquidity risk. We found that liquidity buffer is best modelled as a stochastic process than deterministic. High net cash flow enhance effective and efficient liquidity management. Finally, excess re serves improve country monetary policy and broaden the scope of central bank lending programmes to address conditions in credit markets but on the other hand, may lead banks to loosen standards by weakening lending crite ria in an attempt to increase returns. In designing the sub-portfolios, liquidity determinants such as liquid asset returns, loan returns, payment flow, interest rates and gross income are un certain, and to avoid liquidity problems caused by these variables, we need to include randomness on them. To effectively and efficiently manage liq uidity, a bank may be required to follow certain key liquidity principles. The first is that the bank needs to understand and categorise its cash needs into at least four liquidity sub-portfolios as explained in the research. In this study, we strongly recommend that the banks' management should design the following sub-portfolios; liquidity buffer, cash cushion, operational cash and discretionary liquidity. The bank should clearly maintain the holdings of high quality liquid assets that can provide reliable reserves under all con ditions. In addition, strict and relatively comprehensive disclosure practices in relation to liquidity risk management objectives should be submitted to central banks. Finally, there should be improvement on funding markets and public confidence by broadening the scope of bank guarantees to ensure future financial stability. We need to emphasise the point that bank liquid ity is restricted by capital adequacy, required reserves, liability and deposit insurance defined by the central bank. Research can therefore be done on models that can be used to efficiently and accurately forecast future cash outflow taking into consideration the behavioural cash flows and estimating the discretionary and excess reserves. Further research can also be done on comparing the benefits of the calculated strategies in the stochastic programming context to more traditional methods.
Open Access
Stochastic modelling of HIV/AIDS epidemiology with TB co-infection drug reaction in South Africa
(2015-07-16) Shoko, Claris; Garira, W; Amey, A. K. A; Bessong, P. O
The study explores the stochastic approach to multi-state modeling of HIV dynamic evolution and identification of the model that best describes HIV progression on individuals under ART. The effects of TB co-infection, as well as the patients' development of adverse reaction to drugs to the transition rates are also examined. The study uses a cohort analysis of the surveillance data for HIV-infected patients under antiretroviral (ART) from the Wellness Clinic in Bela Bela, South Africa. The survey was conducted between 2005 and 2009 and a follow up was done after every 6 months. The method par titions the HIV infection period into five CD4-cell count intervals followed by the end points, that is, death and withdrawal from study. The analysis is based on transition probabilities, transition rates (hazards), mean sojourn times, and time to absorption. The effects of the covariates, namely sex, age, TB co-infection, drug reaction, body mass index (BMI), baseline viral load (VLBL) and the CD4+ cell count baseline on enrollment, on transition in tensities for each model are also analysed. The likelihood ratio test is used to compare the fitted models, and the test shows that the time inhomogeneous model describes the data better than the time homogeneous models. The results show that the rates of immune recovery are generally higher than the rates of immune deterioration. The patients who developed TB during treat ment have higher rates of immune deterioration than recovery. Having TB as the initial marker of AIDS has higher contributory effects to the deaths from all the stages except from the AIDS defining stage. Reaction to drugs was the leading cause of transition from a CD4+ cell count 2: 750 to a CD4+ cell count between 500 and 750.
Open Access
Modelling flood heights of the Limpopo River at Beitbridge Border Post using extreme value distributions
(2016) Kajambeu, Robert; Sigauke, Caston; Bere, Alphonce
Haulage trucks and cross border traders cross through Beitbridge border post from landlocked countries such as Zimbabwe and Zambia for the sake of trading. Because of global warming, South Africa has lately been experiencing extreme weather patterns in the form of very high temperatures and heavy rainfall. Evidently, in 2013 tra c could not cross the Limpopo River because water was owing above the bridge. For planning, its important to predict the likelihood of such events occurring in future. Extreme value models o er one way in which this can be achieved. This study identi es suitable distributions to model the annual maximum heights of Limpopo river at Beitbridge border post. Maximum likelihood method and the Bayesian approach are used for parameter estimation. The r -largest order statistics was also used in this dissertation. For goodness of t, the probability and quantile- quantile plots are used. Finally return levels are calculated from these distributions. The dissertation has revealed that the 100 year return level is 6.759 metres using the maximum likelihood and Bayesian approaches to estimate parameters. Empirical results show that the Fr echet class of distributions ts well the ood heights data at Beitbridge border post. The dissertation contributes positively by informing stakeholders about the socio- economic impacts that are brought by extreme flood heights for Limpopo river at Beitbridge border post
Open Access
A class of efficient iterative solvers for the steady state incompressible fluid flow : a unified approach
(2016-02-01) Muzhinji, Kizito;
Open Access
A mathematical modelling frame-work for immuno-epidemiology of Guinea worm infection
(2016-02-12) Netshikweta, Rendani; Garira, W.; Moyo, S.
Open Access
Mathematical modelling of fungal contamination of citrus produce along the pre-harvest supply chain
(2016-05) Muleya, Nqobile; Garira, W.; Mchau, G. R. A.
See the attached abstract below
Open Access
Mathematical modelling of Cholera Immunology
(2016-05) Maphiri, Azwindini Delinah; Garira, W.; Musie, E.
See the attached abstract below
Open Access
Profit risk models for South African banking sector
(2016-05) Antwi, Albert; Kyei, K. A.; Manda, D. C.; Gyamfi, E. N.
See the attached abstract below
Open Access
Determination of factors contributing towards women's unemployment in the Capricorn and Sekhukhune districts in the Limpopo Province
(2017-09-18) Maboko, Tumisho; Kyei, K. A.
See the attached abstract below
Open Access
Modeling and Forecasting Ghana's Inflation Rate Under Threshold Models
(2017-09-18) Antwi, Emmanuel; Kyei, K. A.; Gyampi, E. N
Over the years researchers have been modeling inflation rate in Ghana using linear models such as Autoregressive Integrated Moving Average (ARIMA), Autoregressive Moving Average (ARMA) and Moving Average (MA). Empirical research however, has shown that financial data, such as inflation rate, does not follow linear patterns. This study seeks to model and forecast inflation in Ghana using nonlinear models and to establish the existence of nonlinear patterns in the monthly rates of inflation between the period January 1981 to August 2016 as obtained from Ghana Statistical Service. Nonlinearity tests were conducted using Keenan and Tsay tests, and based on the results, we rejected the null hypothesis of linearity of monthly rates of inflation. The Augmented Dickey-Fuller (ADF) was performed to test for the presence of stationarity. The test rejected the null Hypothesis of unit root at 5% significant level, and hence we can conclude that the rate of inflation was stationary over the period under consideration. The data were transformed by taking the logarithms to follow nornal distribution, which is a desirable characteristic feature in most time series. Monthly rates of inflation were modeled using threshold models and their fitness and forecasting performance were compared with Autoregressive (AR ) models. Two Threshold models: Self-Exciting Threshold Autoregressive (SETAR) and Logistic Smooth Threshold Autoregressive (LSTAR) models, and two linear models: AR(1) and AR(2), were employed and fitted to the data. The Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) were used to assess each of the fitted models such that the model with the minimum value of AIC and BIC, was judged the best model. Additionally, the fitted models were compared according to their forecasting performance using a criterion called mean absolute percentage error (MAPE). The model with the minimum MAPE emerged as the best forecast model and then the model was used to forecast monthly inflation rates for the year 2017. The rationale for choosing this type of model is contingent on the behaviour of the time-series data. Also with the history of inflation modeling and forecasting, nonlinear models have proven to perform better than linear models. The study found that the SETAR and LSTAR models fit the data best. The simple AR models however, out-performed the nonlinear models in terms of forecasting. Lastly, looking at the upward trend of the out-sample forecasts, it can be predicted that Ghana would experience double digit inflation in 2017. This would have several impacts on many aspects of the economy and could erode the economic gains i made in the year 2016. Our study has important policy implications for the Central Bank of Ghana which can use the data to put in place coherent monetary and fiscal policies that would put the anticipated increase in inflation under control.
Open Access
Hybrid multi-scale mathematical modelling of malaria infection transmission
(2017-09-18) Vele, Khathutshelo; Garira, W.; Moyo, S.
See the attached abstract below
Open Access
Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction
(De Gruyter, 2017-11-27) Shateyi, Stanford; Marewo, Gerald T.
We numerically investigate a mixed convection model for a magnetohydrodynamic (MHD) Jeffery fluid flowing over an exponentially stretching sheet. The influence of thermal radiation and chemical reaction is also considered in this study. The governing non-linear coupled partial differential equations are reduced to a set of coupled non-linear ordinary differential equations by using similarity functions. This new set of ordinary differential equations are solved numerically using the Spectral Quasi-Linearization Method. A parametric study of physical parameters involved in this study is carried out and displayed in tabular and graphical forms. It is observed that the velocity is enhanced with increasing values of the Deborah number, buoyancy and thermal radiation parameters. Furthermore, the temperature and species concentration are decreasing functions of the Deborah number. The skin friction coefficient increases with increasing values of the magnetic parameter and relaxation time. Heat and mass transfer rates increase with increasing values of the Deborah number and buoyancy parameters.
Open Access
Alternative methods for solving nonlinear two-point boundary value problems
(2018-03-18) Ghomanjani, Fateme; Shateyi, Stanford
In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary di erential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.
Open Access
Modelling average maximum daily temperature using r largest order statistics: An application to South African data
(OASIS, 2018-05-02) Nemukula, M. M.; Sigauke, Caston
Natural hazards (events that may cause actual disasters) are established in the literature as major causes of various massive and destructive problems worldwide. The occurrences of earthquakes, floods and heat waves affect millions of people through several impacts. These include cases of hospitalisation, loss of lives and economic challenges. The focus of this study was on the risk reduction of the disasters that occur because of extremely high temperatures and heat waves. Modelling average maximum daily temperature (AMDT) guards against the disaster risk and may also help countries towards preparing for extreme heat. This study discusses the use of the r largest order statistics approach of extreme value theory towards modelling AMDT over the period of 11 years, that is, 2000–2010. A generalised extreme value distribution for r largest order statistics is fitted to the annual maxima. This is performed in an effort to study the behaviour of the r largest order statistics. The method of maximum likelihood is used in estimating the target parameters and the frequency of occurrences of the hottest days is assessed. The study presents a case study of South Africa in which the data for the non-winter season (September–April of each year) are used. The meteorological data used are the AMDT that are collected by the South African Weather Service and provided by Eskom. The estimation of the shape parameter reveals evidence of a Weibull class as an appropriate distribution for modelling AMDT in South Africa. The extreme quantiles for specified return periods are estimated using the quantile function and the best model is chosen through the use of the deviance statistic with the support of the graphical diagnostic tools. The Entropy Difference Test (EDT) is used as a specification test for diagnosing the fit of the models to the data.
Open Access
Unsteady hydromagnetic chemically reacting mixed convection MHD flow over a permeable stretching sheet embedded in a porous medium with thermal radiation and heat source/sink
(2018-05-18) Machaba, Mashudu Innocent; Shateyi, Stanford; Netshiozwi, N. J.
The unsteady hydromagnetic chemically reacting mixed convection MHD ow over a permeable stretching sheet embedded in a porous medium with thermal radiation and heat source/sink is investigated numerically. The original partial di erential equations are converted into ordinary di erential equations by using similarity transformation. The governing non-linear partial di erential equations of Momentum, Energy, and Concentration are considered in this study. The e ects of various physical parameters on the velocity, temperature, and species concentration have been discussed. The parameters include the Prandtl number (Pr), Magnetic parameter (M), the Schmidt number (Sc), Unsteady parameter (A), buoyancy forces ratio parameter (N), Chemical reaction (K), Radiation parameter (Nr), Eckert number (Ec), local heat source/sink parameter (Q) and buoyancy parameter due to temperature ( ). The coe cient of Skin friction and Heat transfer are investigated. The coupled non-linear partial di erential equations governing the ow eld have been solved numerically using the Spectral Relaxation Method (SRM). The results that are obtained in this study are then presented in tabular forms and on graphs and the observations are discussed.