Numerical Treatment of Non-Linear singular pertubation problems

Magister Scientiae - MScThis thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.South Afric

Robust numerical methods to solve differential equations arising in cancer modeling

Philosophiae Doctor - PhDCancer 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

A fitted operator method for a model arising in vascular tumor dynamics

In this paper, we consider a model for the population kinetics of human tumor cells in vitro, differentiated by phases of the cell division cycle and length of time within each phase. Since it is not easy to isolate the effects of cancer treatment on the cell cycle of human cancer lines, during the process of radiotherapy or chemotherapy, therefore, we include the spatial effects of cells in each phase and analyse the extended model. The extended model is not easy to solve analytically, because perturbation by cancer therapy causes the flow cytometric profile to change in relation to one another. Hence, making it difficult for the resulting model to be solved analytically. Thus, in [16] it is reported that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection, reaction processes are highly dominant and the initial data are not smooth. As a result, we construct a fitted operator finite difference method (FOFDM) coupled with non-standard finite difference method (NSFDM) to solve the extended model. The FOFDM and NSFDM are analyzed for convergence and are seen that they are unconditionally stable and have the accuracy of O(Dt +(Dx)2), where Dt and Dx denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are presented

A fitted operator method for tumor cells dynamics in their micro-environment

In this paper, we consider a quasi non-linear reaction-diffusion model designed to mimic tumor cells’ proliferation and migration under the influence of their micro-environment in vitro. Since the model can be used to generate hypotheses regarding the development of drugs which confine tumor growth, then considering the composition of the model, we modify the model by incorporating realistic effects which we believe can shed more light into the original model. We do this by extending the quasi non-linear reaction-diffusion model to a system of discrete delay quasi non-linear reaction-diffusion model. Thus, we determine the steady states, provide the conditions for global stability of the steady states by using the method of upper and lower solutions and analyze the extended model for the existence of Hopf bifurcation and present the conditions for Hopf bifurcation to occur. Since it is not possible to solve the models analytically, we derive, analyze, implement a fitted operator method and present our results for the extended model. Our numerical method is analyzed for convergence and we find that is of second order accuracy. We present our numerical results for both of the models for comparison purposes

Numerical solution for a problem arising in angiogenic signalling

Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting e ects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner

Efficient numerical method for a model arising in biological stoichiometry of tumor dynamics

In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells are further modified by blood supply through the vascularization process, we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and we observe seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature

Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application

In this paper, a system of non-linear quasi-parabolic partial differential system, modeling the chemotherapy application of spatial tumor-host interaction is considered. At some certain parameters, we derive the steady state of the anti-angiogenic therapy, baseline therapy and anti-cytotoxic therapy models as well as their local stability condition. We use the method of upper and lower solutions to show that the steady states are globally stable. Since the system of non-linear quasi-parabolic partial differential cannot be solved analytically, we formulate a robust numerical scheme based on the semi-fitted finite difference operator. Analysis of the basic properties of the method shows that it is consistent, stable and convergent. Our numerical results are in agreement with our theoretical findings.https://doi.org/10.28919/cmbn/386

A fitted numerical method for a model arising in HIV-related cancer-immune system dynamics

The effect of diseases such as cancer and HIV among our societies is evident. Thus, from the mathematical point of view many models has been developed with the aim to contribute towards understanding the dynamics of diseases. Therefore, in this paper we believe by extending a system of delay differential equations (DDEs) model of HIV related cancer-immune system to a system of delay partial differential equations (DPDEs) model of HIV related cancer-immune dynamics, we can contribute toward understanding the dynamics more clearly. Thus, we analyse the extended models and use the qualitative features of the extended model to derive, analyse and implement a fitted operator finite difference method (FOFDM) and present our results. This FOFDM is analyzed for convergence and it is seen that it has has second-order accuracy. We present some numerical results for some cases of the the model to illustrate the reliability of our numerical method

Numerical solution for an extended multi-mutation and drug resistance model

Paper presented at the 5th Strathmore International Mathematics Conference (SIMC 2019), 12 - 16 August 2019, Strathmore University, Nairobi, KenyaIn this study, we extend a model that expresses intrinsic drug resistances to include SBS time required for mutation rate to take place and spatial effects of the involved cells. Furthermore, we show that the local stability condition(s) are (is) global stable. Since it is not that easy to solve the extended model analytically, we derive, analyze, implement, present a numerical solution and compare it with the solution of the original model.Department of Mathematics, University of the Western Cape, South Africa

Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application

In this paper, a system of non-linear quasi-parabolic partial differential system, modeling the chemotherapy application of spatial tumor-host interaction is considered. At some certain parameters, we derive the steady state of the anti-angiogenic therapy, baseline therapy and anti-cytotoxic therapy models as well as their local stability condition. We use the method of upper and lower solutions to show that the steady states are globally stable. Since the system of non-linear quasi-parabolic partial differential cannot be solved analytically, we formulate a robust numerical scheme based on the semi-fitted finite difference operator. Analysis of the basic properties of the method shows that it is consistent, stable and convergent. Our numerical results are in agreement with our theoretical findings