Browsing by Author "Witbooi, Peter J."
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Item The fibre of a pinch map in a model category(Springer Verlag, 2013) Hardie, Keith A.; Witbooi, Peter J.In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X ∪ CA → X ∪ CA/ X from the mapping cone on a cofibration A → X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.Item Generalizing the Hilton–Mislin genus group(Elsevier, 2001) Witbooi, Peter J.For any group H, let H be the set of all isomorphism classes of groups K such that K H . For a finitely generated group H having finite commu- Ž .tator subgroup H, H , we define a group structure on H in terms of embed- dings of K into H, for groups K of which the isomorphism classes belong to Ž . H . If H is nilpotent, then the group we obtain coincides with the genus group Ž .GG H defined by Hilton and Mislin. We obtain some new results on Hilton Mislin genus groups as well as generalizations of known results.Item A model for control of HIV/AIDS with parental care(World Scientific Publishing, 2013) Abiodun, Gbenga Jacob; Marcus, Nizar; Okosun, Kazeem Oare; Witbooi, Peter J.In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.Item A model of HIV/AIDS population dynamics including ARV treatment and pre-exposure prophylaxis(SpringerOpen, 2018) Nsuami, Mozart U.; Witbooi, Peter J.Antiretroviral treatment (ART) and oral pre-exposure prophylaxis (PrEP) have recently been used efficiently in management of HIV infection. Pre-exposure prophylaxis consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals. We propose a new model for the transmission of HIV/AIDS including ART and PrEP. Our model can be used to test the effects of ART and of the uptake of PrEP in a given population, as we demonstrate through simulations. The model can also be used to estimate future projections of HIV prevalence. We prove global stability of the disease-free equilibrium. We also prove global stability of the endemic equilibrium for the most general case of the model, i.e., which allows for PrEP individuals to default. We include insightful simulations based on recently published South-African data.Item A model of population dynamics of TB in a prison system and application to South Africa(BioMed Central, 2017) Witbooi, Peter J.; Vyambwera, Sibaliwe MakuBACKGROUND: Tuberculosis (TB) continues to spread in South African prisons in particular, as prisons are over-capacitated and have poor ventilation. The awaiting trial detainees are not screened on admission and are at high risk of getting infected with TB. RESULTS: We propose a compartmental model to describe the population dynamics of TB disease in prisons. Our model considers the inflow of susceptible, exposed and TB infectives into the prison population. Removal of individuals out of the prison population can be either by death or by being released from prison, as compared to a general population in which removal is only by death. We describe conditions, including non-inflow of infectives into the prison, which will ensure that TB can be eradicated from the prison population. The model is calibrated for the South African prison system, by using data in existing literature. The model can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Illustrative simulations in this regard are presented. The model can be used for other prison populations too, if data is available to calculate the model parameters. CONCLUSIONS: Various simulations generated with our model serve to illustrate how it can be utilized in making future projections of the levels of prevalence of TB, and to quantify the effect of interventions such as screening, treatment or reduction of transmission parameter values through improved living conditions for inmates. This makes it particularly useful as there are various targets set by the World Health Organization and by governments, for reduction of TB prevalence and ultimately its eradication. Towards eradication of TB from a prison system, the theorem on global stability of the disease-free state is a useful indicator.Item Modeling the dynamics of an epidemic under vaccination in two interacting populations(Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, KailashMathematical modeling of the numerical evolution of infectious diseases has become an important tool for disease control and eradication when possible. Much work has been done on the problem of how a given population is affected by another population when there is mutual interaction. The mere presence of migrant people poses a challenge to whatever health systems are in place in a particular region. Such epidemiological phenomena have been studied extensively, described by mathematical models with suggestions for intervention strategies. The epidemiological effect of migration within the population itself was modeled for sleeping sickness in a paper 1 by Chalvet-Monfray et al. In the case of malaria, there is for instance a study 2 by Tumwiine et al. on the effect of migrating people on a fixed population. The latter two diseases are vector borne. Diseases that propagate without a vector spread perhaps more easily when introduced into a new region. Various studies of models with immigration of infectives have been undertaken for tuberculosis, see for instance 3 by Zhou et al., or the work 4 of Jia et al., and for HIV, see the paper 5 of Naresh et al.Item Multiplication of Crowns(Instituto de Matematica de Bahia Blanca (INMABB-CONICET), 2013) Witbooi, Peter J.It is known that the only nite topological spaces that are H-spaces are the discrete spaces. For a nite poset which is weakly equivalent to an H-space, a generalized multiplication may be found after suitable sub-division. In this paper we construct minimal models of the k-fold generalised multiplications of circles in the category of relational structures, including poset models. In particular, we obtain higher dimensional analogues of a cer-tain ternary multiplication of crownsItem An optimal portfolio and capital management strategy for basel III compliant commercial banks(Hindawi Publishing Corporation, 2014) Muller, Grant E.; Witbooi, Peter J.We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security, a marketable security, and a loan and we regard the interest rate in the market as being stochastic. We find the investment strategy that maximizes an expected utility of the bank’s asset portfolio at a future date. This entails obtaining formulas for the optimal amounts of bank capital invested in different assets. Based on the optimal investment strategy, we derive a model for the Capital Adequacy Ratio (CAR), which the Basel Committee on Banking Supervision (BCBS) introduced as a measure against banks’ susceptibility to failure. Furthermore, we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level. We derive a formula for the bank’s asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenarios. Under the optimal investment strategy, the CAR is above the minimum prescribed level. The value of the asset portfolio is improved if the CAR is at its (constant) minimum value.Item An SEIR model with infected immigrants and recovered emigrants(Springer Science and Business Media Deutschland GmbH, 2021) Witbooi, Peter J.We present a deterministic SEIR model of the said form. The population in point can be considered as consisting of a local population together with a migrant subpopulation. The migrants come into the local population for a short stay. In particular, the model allows for a constant inflow of individuals into different classes and a constant outflow of individuals from the R-class. The system of ordinary differential equations has positive solutions and the infected classes remain above specified threshold levels. The equilibrium points are shown to be asymptotically stable. The utility of the model is demonstrated by way of an application to measles. © 2021, The Author(s).Item An SEIRS epidemic model with stochastic transmission(Springer, 2017) Witbooi, Peter J.For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant R? is below unity. Here R?Item Stability of an SEIR epidemic model with indepenent stochastic perturbations(Elsevier, 2013) Witbooi, Peter J.For an epidemic model of the type mentioned, we prove a theorem on almost sure exponential stability of the disease-free equilibrium. For small values of the diffusion parameter, σ, we describe the stability of the disease free equilibrium point in terms of an appropriate analogue, Rσ , of the basic reproduction number R0 of the deterministic special case. Whenever σ > 0 then Rσ < R0. For small values of σ, the stability theorem guarantees almost sure exponential stability whenever Rσ < 1. We also discuss the effect of increasing σ.Item Stochastic Volatility Models for Contingent Claim Pricing and Hedging(University of the Western Cape, 2008) Manzini, Muzi Charles; Witbooi, Peter J.; Faculty of ScienceThe present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.Item Tuberculosis in Ethiopia: Optimal intervention strategies and cost-effectiveness analysis(MDPI, 2022) Mengistu, Ashenafi Kelemu; Witbooi, Peter J.This paper searches for optimal strategies for the minimization of the number of high-risk latent and active tuberculosis (TB) infectious individuals using real data from Ethiopia. Optimal control theory is harnessed for investigation and analysis of the optimal combination of interventions for controlling the transmission of TB using distancing, case finding, and case holding as controls. We calculate and compare the incremental cost-effectiveness ratio (ICER) for each of the strategies to determine the most effective combination of interventions for curbing the spread of the disease. Our findings suggest that, for optimal cost-effective management of the TB disease, the government of Ethiopia must focus more on prevention strategies such as isolation of infectious people, early TB patient detection, treatment, and educational programs. The optimal strategy is quantified through simulation.