Mathematical Science http://hdl.handle.net/10530/903 Mathematics 2022-01-22T02:42:05Z Mathematical modeling for optimal control of breast cancer http://hdl.handle.net/10530/1846 Mathematical modeling for optimal control of breast cancer Oke, Segun Isaac Breast cancer, which often occurs in the inner lining of milk ducts, is the deadliest and most common form of invasive cancer among females according to a 2017 report of the World Health Organization. The purpose of this study was to develop a four compartmental mathematical model using a system of nonlinear Ordinary Di erential Equations (ODEs) which investigates the impact of anti-cancer drugs, ketogenic-diets and immune boosters on the dynamics of breast cancer. The study focused on the dynamical interaction of normal and tumor cells as well as the invasion of tumor cells during the metastasis stage of breast cancer. The systems of ODEs were analytically solved for the equilibria. Using the next generation matrix method, a threshold quantity called the treatment in- duced invasion reproduction number (R i ) was computed. Center manifold theory was used to investigate the possibility of the bifurcation analysis of R i being greater than unity. Using a suitable Lyapunov functions, the global stability of the tumor-free equilibrium was achieved in conjuction with LaSalle's invariance principle. Uncertainty and sensitivity analyses were performed on R i using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coe cient (PRCC). R i was used as the response function while investigating the most signi cant parameters (such as: 1, 2, 1, d, and 1 ) that a ects disease progression and cell invasion. Optimal control theory was applied using the Pontryagins' Maximum Principle to investigate optimal strategies for controlling and eliminating tumor cells using time dependent controls such as u1(t) (anti-cancer drugs) and u2(t) (ketogenic diets). Numerical simulation results using a set of parameter values were provided to validate the analytical results. It was found that the tumor-free equilibrium points for ix breast cancer was locally asymptotically stable when the associated invasion reproduction number was less than unity and that it was otherwise unstable. The tumor-free equilibrium was found to be globally asymptotically stable if (Ri) < 1 . Sensitivity analysis showed that the natural death rate of normal cells has the most positive sensitivity index. However, increasing the death rate as a control measure is unreasonable biologically. The level of ketogenic diet rate was found to be most negatively sensitive to Ri. Therefore, the formulated model showed that reduction of the invasion reproduction number (R i ) below unity can be achieved by maintaining the level of ketogenic diet and by reducing tumor progression rate. It was shown from this study that the breast cancer model exhibited backward bifurcation with bifurcation parameter 1 which implies that the reduction R i below unity alone is not su cient to eradicate tumor cells from the body system while in the case of forward bifurcation, the reduction of R i above unity is su cient to eradicate tumor cells from the body system . The incremental cost-e ectiveness analysis of control strategies adapted in treating breast cancer has shown that the integration of ketogenic diet and anti-cancer drugs as intervention strategy is the most cost-e ective in ghting tumor cells. x 2019-01-01T00:00:00Z Some investigation into causal viscous cosmological models with varieble Lambda http://hdl.handle.net/10530/904 Some investigation into causal viscous cosmological models with varieble Lambda Nkosi, Zakhele Thomas In this thesis we investigate the evolution of viscous Friedmann-Lemaitre-Robertson-Walker models with a variable cosmological term. The numerical solutions are obtained and represented graphically. Each graph depends on the value of gamma. The present values of the deceleration parameter and density parameter were obtained by using Eckart theory and the truncated theory and given in the tabular form. Power law solutions for the Hubble parameter are shown to exist and we give the values of all the other cosmological variables. The behaviour of the temperature depends on the initial conditions. Furthermore, the equations are also transformed into a plane autonomous system by using dimensionless variables and a dimensionless equation of state, and the qualitative behaviour of the system is investigated. The sets of equilibrium points are determined and their behaviour discussed. The exact value of the Hubble parameter, cosmological term, bulk viscosity pressure and the energy density are obtained and discussed. Thesis submitted to the Faculty of Science and Agriculture in fulfillment for the degree, Doctor of Philosophy, in the Department of Mathematical Sciences, at the University of Zululand, South Africa, 2001. 2001-01-01T00:00:00Z The energy-momentum problem in general relativity http://hdl.handle.net/10530/815 The energy-momentum problem in general relativity Xulu, Sibusiso S. Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energ3r and momentum in general relativity. Attempts aimed at finding a quantity for describing distribution of energy-momentum due to matter, non-gravitational and gravitational fields only resulted in various energy-momentum complexes (these are nontensorial under general coordinate transformations) whose physical meaning have been questioned. The problems associated with energy-momentum complexes re¬sulted in some researchers even abandoning the concept of energy-momentum localization in favor of the alternative concept of quasi-localization. However, quasi-local masses have their inadequacies, while the remarkable work of Virbhadra and some others, and recent results of Cooperstock and Chang et ai have revived an interest in various energy-momentum complexes. Hence in this work we use energy-momentum complexes to obtain the energy dis¬tributions in various space-times. We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifshitz, Papapetrou, Weinberg, and Moller to investigate energy distributions in var¬ious space-times. It is shown that several of these energy-momentum com¬plexes give the same and acceptable results for a given space-time. This shows the importance of these energy-momentum complexes. Our results agree with Virbhadra's conclusion that the Einstein's energy-momentum complex is still the best tool for obtaining energy distribution in a given space-time. The Cooperstock hypothesis (that energy and momentum in a curved space-time are confined to the the regions of non-vanishing energy-momentum of matter and the non-gravitational field) is also supported. Thesis presented for the Degree of Doctor of Philosophy in Applied Mathematics, Department of Mathematical Sciences at the University of Zululand, 2002. 2002-01-01T00:00:00Z