Mathematical modelling of the tumour treatment with chemotherapy
- Authors: Nyaweni, Fundile Sindy
- Date: 2024-04
- Subjects: Mathematical models , Mathematics , Chemotherapy
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10948/64289 , vital:73672
- Description: In this study, a cancer disease model incorporating predator-prey dynamics between immune cells and tumor cells, as well as competition between tumor cells and normal cells, is formulated in an attempt to understand the interaction dynamics that govern the complex interplay within the tumor microenvironment. Mathematical analysis is employed to derive conditions for the boundedness of solutions and to discuss disease thresholds such as the basic reproduction number, R0. Equilibrium points are identified, and stability conditions for the system are derived. Sensitivity analysis is used to assess uncertainties in the parameters of the model, specifically their impact on R0. Numerical simulations are conducted, utilizing the Runge-Kutta scheme to solve the model. Furthermore, optimal control (OC) techniques are applied to examine the role of chemotherapy in enhancing tumor cell elimination and minimizing adverse effects on immune cells and normal cells. The improved model, incorporating normal cells, anticancer drugs, and immune cells, allows for a more comprehensive analysis of the dynamic interactions within the biological system. The optimal control is determined using the forward-backward sweep numerical method. The numerical simulation is carried out in MATLAB to validate and complement the analytical findings. The results indicate that a low concentration of the drug leads to a prolonged period for tumor clearance. Conversely, a higher drug concentration results in quicker tumor clearance, albeit with adverse effects on normal and effector cells. The OC identifies when to stop the treatment once the tumor clears. Furthermore, OC facilitates the regeneration of normal and immune cells beyond treatment, in contrast to a control scenario without optimization. The findings highlight the importance of early detection and careful chemotherapy dosage selection for effective and personalized cancer treatment strategies. , Thesis (MSc) -- Faculty of Science, School Computer Science, Mathematics, Physics and Statistics, 2024
- Full Text:
- Date Issued: 2024-04
- Authors: Nyaweni, Fundile Sindy
- Date: 2024-04
- Subjects: Mathematical models , Mathematics , Chemotherapy
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10948/64289 , vital:73672
- Description: In this study, a cancer disease model incorporating predator-prey dynamics between immune cells and tumor cells, as well as competition between tumor cells and normal cells, is formulated in an attempt to understand the interaction dynamics that govern the complex interplay within the tumor microenvironment. Mathematical analysis is employed to derive conditions for the boundedness of solutions and to discuss disease thresholds such as the basic reproduction number, R0. Equilibrium points are identified, and stability conditions for the system are derived. Sensitivity analysis is used to assess uncertainties in the parameters of the model, specifically their impact on R0. Numerical simulations are conducted, utilizing the Runge-Kutta scheme to solve the model. Furthermore, optimal control (OC) techniques are applied to examine the role of chemotherapy in enhancing tumor cell elimination and minimizing adverse effects on immune cells and normal cells. The improved model, incorporating normal cells, anticancer drugs, and immune cells, allows for a more comprehensive analysis of the dynamic interactions within the biological system. The optimal control is determined using the forward-backward sweep numerical method. The numerical simulation is carried out in MATLAB to validate and complement the analytical findings. The results indicate that a low concentration of the drug leads to a prolonged period for tumor clearance. Conversely, a higher drug concentration results in quicker tumor clearance, albeit with adverse effects on normal and effector cells. The OC identifies when to stop the treatment once the tumor clears. Furthermore, OC facilitates the regeneration of normal and immune cells beyond treatment, in contrast to a control scenario without optimization. The findings highlight the importance of early detection and careful chemotherapy dosage selection for effective and personalized cancer treatment strategies. , Thesis (MSc) -- Faculty of Science, School Computer Science, Mathematics, Physics and Statistics, 2024
- Full Text:
- Date Issued: 2024-04
Evaluation of the effectiveness of small aperture network telescopes as IBR data sources
- Authors: Chindipha, Stones Dalitso
- Date: 2023-03-31
- Subjects: Computer networks Monitoring , Computer networks Security measures , Computer bootstrapping , Time-series analysis , Regression analysis , Mathematical models
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/366264 , vital:65849 , DOI https://doi.org/10.21504/10962/366264
- Description: The use of network telescopes to collect unsolicited network traffic by monitoring unallocated address space has been in existence for over two decades. Past research has shown that there is a lot of activity happening in this unallocated space that needs monitoring as it carries threat intelligence data that has proven to be very useful in the security field. Prior to the emergence of the Internet of Things (IoT), commercialisation of IP addresses and widespread of mobile devices, there was a large pool of IPv4 addresses and thus reserving IPv4 addresses to be used for monitoring unsolicited activities going in the unallocated space was not a problem. Now, preservation of such IPv4 addresses just for monitoring is increasingly difficult as there is not enough free addresses in the IPv4 address space to be used for just monitoring. This is the case because such monitoring is seen as a ’non-productive’ use of the IP addresses. This research addresses the problem brought forth by this IPv4 address space exhaustion in relation to Internet Background Radiation (IBR) monitoring. In order to address the research questions, this research developed four mathematical models: Absolute Mean Accuracy Percentage Score (AMAPS), Symmetric Absolute Mean Accuracy Percentage Score (SAMAPS), Standardised Mean Absolute Error (SMAE), and Standardised Mean Absolute Scaled Error (SMASE). These models are used to evaluate the research objectives and quantify the variations that exist between different samples. The sample sizes represent different lens sizes of the telescopes. The study has brought to light a time series plot that shows the expected proportion of unique source IP addresses collected over time. The study also imputed data using the smaller /24 IPv4 net-block subnets to regenerate the missing data points using bootstrapping to create confidence intervals (CI). The findings from the simulated data supports the findings computed from the models. The CI offers a boost to decision making. Through a series of experiments with monthly and quarterly datasets, the study proposed a 95% - 99% confidence level to be used. It was known that large network telescopes collect more threat intelligence data than small-sized network telescopes, however, no study, to the best of our knowledge, has ever quantified such a knowledge gap. With the findings from the study, small-sized network telescope users can now use their network telescopes with full knowledge of gap that exists in the data collected between different network telescopes. , Thesis (PhD) -- Faculty of Science, Computer Science, 2023
- Full Text:
- Date Issued: 2023-03-31
- Authors: Chindipha, Stones Dalitso
- Date: 2023-03-31
- Subjects: Computer networks Monitoring , Computer networks Security measures , Computer bootstrapping , Time-series analysis , Regression analysis , Mathematical models
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/366264 , vital:65849 , DOI https://doi.org/10.21504/10962/366264
- Description: The use of network telescopes to collect unsolicited network traffic by monitoring unallocated address space has been in existence for over two decades. Past research has shown that there is a lot of activity happening in this unallocated space that needs monitoring as it carries threat intelligence data that has proven to be very useful in the security field. Prior to the emergence of the Internet of Things (IoT), commercialisation of IP addresses and widespread of mobile devices, there was a large pool of IPv4 addresses and thus reserving IPv4 addresses to be used for monitoring unsolicited activities going in the unallocated space was not a problem. Now, preservation of such IPv4 addresses just for monitoring is increasingly difficult as there is not enough free addresses in the IPv4 address space to be used for just monitoring. This is the case because such monitoring is seen as a ’non-productive’ use of the IP addresses. This research addresses the problem brought forth by this IPv4 address space exhaustion in relation to Internet Background Radiation (IBR) monitoring. In order to address the research questions, this research developed four mathematical models: Absolute Mean Accuracy Percentage Score (AMAPS), Symmetric Absolute Mean Accuracy Percentage Score (SAMAPS), Standardised Mean Absolute Error (SMAE), and Standardised Mean Absolute Scaled Error (SMASE). These models are used to evaluate the research objectives and quantify the variations that exist between different samples. The sample sizes represent different lens sizes of the telescopes. The study has brought to light a time series plot that shows the expected proportion of unique source IP addresses collected over time. The study also imputed data using the smaller /24 IPv4 net-block subnets to regenerate the missing data points using bootstrapping to create confidence intervals (CI). The findings from the simulated data supports the findings computed from the models. The CI offers a boost to decision making. Through a series of experiments with monthly and quarterly datasets, the study proposed a 95% - 99% confidence level to be used. It was known that large network telescopes collect more threat intelligence data than small-sized network telescopes, however, no study, to the best of our knowledge, has ever quantified such a knowledge gap. With the findings from the study, small-sized network telescope users can now use their network telescopes with full knowledge of gap that exists in the data collected between different network telescopes. , Thesis (PhD) -- Faculty of Science, Computer Science, 2023
- Full Text:
- Date Issued: 2023-03-31
Phase Plane Analysis of Linear Systems in Dynamic Mathematical Models
- Authors: Marange, Simukai Daniel
- Date: 2019
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14715 , vital:40064
- Description: A plethora of dynamic mathematical models exist and to understand and master all of them would be a gargantuan task. The author had, nonetheless, attempted to outline some of the methods used to analyse linear systems in modeling. Systems techniques are fundamental to current research in molecular cell-biology. The systems-approach stands in stark contrast to the historically, reductionist paradigm of molecular biology. Field work can be very dangerous. The main purpose of this study was to come up with the best analysis that would be used without going to the real field and thus saving time, money and risks associated with remote field localities. This research showed that the best analysis depends on the nature of the objectives intended to be solved by the model. Phase plane analysis on linear systems assisted in gaining deeper knowledge on the characteristics of such systems. This work analysed some dynamic models looking at phase planes, bifurcation, sensitivity and stability. The research provided a qualitative analysis of the processes not a numerical analysis.
- Full Text:
- Date Issued: 2019
- Authors: Marange, Simukai Daniel
- Date: 2019
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14715 , vital:40064
- Description: A plethora of dynamic mathematical models exist and to understand and master all of them would be a gargantuan task. The author had, nonetheless, attempted to outline some of the methods used to analyse linear systems in modeling. Systems techniques are fundamental to current research in molecular cell-biology. The systems-approach stands in stark contrast to the historically, reductionist paradigm of molecular biology. Field work can be very dangerous. The main purpose of this study was to come up with the best analysis that would be used without going to the real field and thus saving time, money and risks associated with remote field localities. This research showed that the best analysis depends on the nature of the objectives intended to be solved by the model. Phase plane analysis on linear systems assisted in gaining deeper knowledge on the characteristics of such systems. This work analysed some dynamic models looking at phase planes, bifurcation, sensitivity and stability. The research provided a qualitative analysis of the processes not a numerical analysis.
- Full Text:
- Date Issued: 2019
A derivation of the the black-scholes equation using martingales
- Authors: Nyarko , Ebenezer Narh
- Date: 2018
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14572 , vital:40016
- Description: This work focuses on the application of stochastic differential equations, with martingales, in finance. The emphasis is on the derivation of the Black-Scholes model for the valuation of options. A theoretical framework in stochastic analysis, together with Itô calculus (Kiyoshi Itô), is explored. The Girsanov Theorem is applied in order to transform a modelled stochastic equation based, on predetermined stock and bond prices, into equivalent martingale measures. A replication strategy is then adopted to solve the two equations analytically, by finding the natural logarithm of the expectation of the solution to the stochastic models. We finally compute the resulting solution based on a standard, normal distribution to get the desired outcome of the Black-Scholes model.
- Full Text:
- Date Issued: 2018
- Authors: Nyarko , Ebenezer Narh
- Date: 2018
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14572 , vital:40016
- Description: This work focuses on the application of stochastic differential equations, with martingales, in finance. The emphasis is on the derivation of the Black-Scholes model for the valuation of options. A theoretical framework in stochastic analysis, together with Itô calculus (Kiyoshi Itô), is explored. The Girsanov Theorem is applied in order to transform a modelled stochastic equation based, on predetermined stock and bond prices, into equivalent martingale measures. A replication strategy is then adopted to solve the two equations analytically, by finding the natural logarithm of the expectation of the solution to the stochastic models. We finally compute the resulting solution based on a standard, normal distribution to get the desired outcome of the Black-Scholes model.
- Full Text:
- Date Issued: 2018
User experience metrics for Dr Math
- Authors: Ngaye, Zonke
- Date: 2012
- Subjects: Mathematics -- Data processing , Mathematical models , Mathematics -- Study and teaching , Numerical analysis
- Language: English
- Type: Thesis , Masters , MTech
- Identifier: vital:9799 , http://hdl.handle.net/10948/d1012036 , Mathematics -- Data processing , Mathematical models , Mathematics -- Study and teaching , Numerical analysis
- Description: The purpose of this research study is to propose guidelines for providing a positive user experience for pupils using Dr Math®. User experience was found to have a positive impact on the acceptance and adoption of a product. Thus the proposed guidelines contribute in maximizing the adoption and acceptance of Dr Math® among pupils. This study begins with an introductory chapter that describes the problem that forms the basis for this research. The chapter defines the objectives that this study is intended to achieve in order to accomplish its ultimate goal. The methodology followed to conduct this research study as well as its scope are also defined here. The results from a preliminary survey revealed that despite its potential accessibility, Dr Math® has a low adoption rate. However, when compared to other mobile learning (m-learning) applications for mathematics learning, Dr Math® is more popular. Thus Dr Math® was selected as a case for study. Chapter 2 of this study provides a detailed description of Dr Math® as a local mobile application for mathematics learning. It was found that the affordability and accessibility of Dr Math® did not necessarily imply a high adoption rate. There are various possible barriers to its low adoption. User experience (UX), which is the focus of this study, is one of them. Thus, a subsequent chapter deals with UX. Chapter 3 discusses UX, its scope, components and definition and places particular emphasis on its significance in the success of any product. The chapter also highlights the characteristics of a positive UX and the importance of designing for this outcome. In Chapter 4, a discussion and justification of the methodology used to conduct this research is discussed. This study primarily employs a qualitative inductive approach within an interpretivism paradigm. An exploratory single case study was used to obtain an in-depth analysis of the case. Data was collected using Dr Math® log files as a documentary source. Gathered data was then analysed and organized into themes and categories using qualitative content analysis as outlined in Chapter 5. Also the findings obtained from the results, which are mainly the factors that were found to have an impact on the user interaction with Dr Math®, are presented here. The identified factors served as a basis from which the guidelines presented in Chapter 6 were developed. Chapter 7 presents the conclusions and recommendations of the research. From both theoretical and empirical work, it was concluded that Dr Math® has the potential to improve mathematics learning in South Africa. Its adoption rate, however, is not satisfying: hence, the investigation of the factors impacting on the user interaction with Dr Math®, from which the proposed guidelines are based.
- Full Text:
- Date Issued: 2012
- Authors: Ngaye, Zonke
- Date: 2012
- Subjects: Mathematics -- Data processing , Mathematical models , Mathematics -- Study and teaching , Numerical analysis
- Language: English
- Type: Thesis , Masters , MTech
- Identifier: vital:9799 , http://hdl.handle.net/10948/d1012036 , Mathematics -- Data processing , Mathematical models , Mathematics -- Study and teaching , Numerical analysis
- Description: The purpose of this research study is to propose guidelines for providing a positive user experience for pupils using Dr Math®. User experience was found to have a positive impact on the acceptance and adoption of a product. Thus the proposed guidelines contribute in maximizing the adoption and acceptance of Dr Math® among pupils. This study begins with an introductory chapter that describes the problem that forms the basis for this research. The chapter defines the objectives that this study is intended to achieve in order to accomplish its ultimate goal. The methodology followed to conduct this research study as well as its scope are also defined here. The results from a preliminary survey revealed that despite its potential accessibility, Dr Math® has a low adoption rate. However, when compared to other mobile learning (m-learning) applications for mathematics learning, Dr Math® is more popular. Thus Dr Math® was selected as a case for study. Chapter 2 of this study provides a detailed description of Dr Math® as a local mobile application for mathematics learning. It was found that the affordability and accessibility of Dr Math® did not necessarily imply a high adoption rate. There are various possible barriers to its low adoption. User experience (UX), which is the focus of this study, is one of them. Thus, a subsequent chapter deals with UX. Chapter 3 discusses UX, its scope, components and definition and places particular emphasis on its significance in the success of any product. The chapter also highlights the characteristics of a positive UX and the importance of designing for this outcome. In Chapter 4, a discussion and justification of the methodology used to conduct this research is discussed. This study primarily employs a qualitative inductive approach within an interpretivism paradigm. An exploratory single case study was used to obtain an in-depth analysis of the case. Data was collected using Dr Math® log files as a documentary source. Gathered data was then analysed and organized into themes and categories using qualitative content analysis as outlined in Chapter 5. Also the findings obtained from the results, which are mainly the factors that were found to have an impact on the user interaction with Dr Math®, are presented here. The identified factors served as a basis from which the guidelines presented in Chapter 6 were developed. Chapter 7 presents the conclusions and recommendations of the research. From both theoretical and empirical work, it was concluded that Dr Math® has the potential to improve mathematics learning in South Africa. Its adoption rate, however, is not satisfying: hence, the investigation of the factors impacting on the user interaction with Dr Math®, from which the proposed guidelines are based.
- Full Text:
- Date Issued: 2012
- «
- ‹
- 1
- ›
- »