Department of Mathematical Sciences

Investigating grade eleven learners’ mathematical modelling competencies in algebraic problem solving

Thu, 01 Jan 2026 00:00:00 GMT

Investigating grade eleven learners’ mathematical modelling competencies in algebraic problem solving Dlamini, Reuben The National Curriculum Statement (NCS), implemented in 2003, marked the formal introduction of mathematical modelling into the South African curriculum. This represented a significant shift, as mathematical modelling had previously been largely absent from the curriculum. This study investigated the mathematical modelling proficiency of Grade 11 learners from three high schools in the Pongola Circuit, KwaZulu-Natal Province, with a specific focus on determining their competency in solving non-routine problems. The study was motivated by the observation that many teachers did not receive adequate training in mathematical modelling during their pre-service education. Existing research indicates that inadequately trained teachers can negatively affect learners’ academic performance. Furthermore, the teacher is widely regarded as the most critical agent in the successful implementation of instructional reforms at the classroom level (Shepherd, 2019; Theophile et al., 2020). It was therefore necessary to examine learners’ competencies in mathematical modelling. Learners’ competencies were assessed using the five-step modelling process proposed by Kaiser and Stender (2013), which includes: (1) understanding the problem, (2) formulating a mathematical model, (3) solving the model, (4) interpreting the results, and (5) validating the results. For a learner to be considered competent in mathematical modelling, all five stages of the process had to be correctly executed. A total of 75 Grade 11 learners from three purposively selected schools participated in the study. Qualitative data were collected through document analysis. The data analysis process involved familiarisation with learners’ written responses, followed by thematic analysis to interpret meaning, identify patterns, and generate insights related to the research questions. The findings revealed that all learners demonstrated incomplete competency in mathematical modelling. Specifically: • None of the learners successfully completed all five stages of the modelling process in any of the four test questions. • Learners did not make or attempt to make assumptions, which are essential in solving real-life problems. - 6 - • Variables were used without clear definitions, and final solutions were often expressed in terms of unidentified variables. This indicated a lack of interpretation of results within the context of the real-world problems. Interpretation involves translating mathematical outcomes back into meaningful real-life conclusions. • Learners did not verify or validate their solutions. Validation is critical for assessing the accuracy and appropriateness of both the mathematical model and its results in relation to the real-world context. The findings suggest that Mathematics teachers should be encouraged to adopt a mathematical modelling approach in teaching and learning. It can be inferred that learners had limited or no exposure to mathematical modelling, as key processes—such as defining variables, making assumptions, interpreting results, and validating solutions—were consistently omitted. According to the Curriculum and Assessment Policy Statement (CAPS), mathematical modelling should serve as a central focus of the Mathematics curriculum (Department of Basic Education [DBE], 2011). Therefore, the Department of Education has a responsibility to ensure the effective implementation of mathematical modelling, emphasising the integration of real-life contexts across all aspects of the curriculum.

Compartmental model for the spread of infectious disease with hererogenous population: a case study of COVID-19 and Lassa fever

Thu, 01 Oct 2015 00:00:00 GMT

Compartmental model for the spread of infectious disease with hererogenous population: a case study of COVID-19 and Lassa fever Oluwagunwa, Abiodun Peter This study investigates the transmission dynamics of COVID-19 and Lassa fever, with particular attention to the risks and implications of co-infection. By dividing the human population into epidemiological compartments, the models capture the real course of disease spread in a structured and realistic way. The mathematical correctness of the models was validated by proving positivity, boundedness, and reliability of solutions for public health interpretation. For COVID-19, the basic SYR framework was analysed to obtain the reproduction number RY , and the model was further extended to an SEAIHR structure to include exposed, asymptomatic, infectious, and hospitalised individuals. For Lassa fever, a deterministic compartmental model was developed and subjected to stability analysis. Numerical simulations were carried out for both diseases to assess intervention strategies and transmission behaviour. The COVID–19 analysis revealed that the disease can be eliminated when RY < 1, but once RY > 1, infection becomes persistent. The extended SEAIHR model also produced the overall reproduction number R0, showing how reductions in contact rates, timely detection, effective hospital care, and faster recovery can substantially suppress transmission. For Lassa fever, the disease-free equilibrium remained stable only when the reproduction number was kept below one. Simulations highlighted the significant influence of asymptomatic carriers and showed that no single intervention— whether treatment, health education, or rodent control—can fully control the disease on its own. Instead, the most meaningful reduction in cases occurred when human-focused measures were combined with strong rodent control, leading to the elimination of infectious rodents by the 35th day. Together, these results emphasise the urgency of studying co-infections in regions such as West Africa, where both diseases circulate at the same time. Incorporating optimal control theory provides a systematic and cost-effective framework for coordinating interventions across multiple pathways of transmission. This study therefore deepens our understanding of the dynamics of both COVID– 19 and Lassa fever and offers practical insights for improving interventions, epivdemic preparedness, and public health responses. The centre-manifold analysis further shows that both models experience a forward transcritical bifurcation at R0 = 1: once transmission exceeds this threshold, a stable endemic state emerges. This reinforces a critical message—maintaining transmission below the threshold is essential for preventing long-term persistence of either disease.

Lie group analysis and conserved vectors of multidimensional nonlinear Partial differential equations

Mon, 01 Dec 2025 00:00:00 GMT

Lie group analysis and conserved vectors of multidimensional nonlinear Partial differential equations Sebogodi, Motshidisi Charity This thesis studies the applications of Lie group analysis and conserved vectors to multi-dimensional nonlinear partial differential equations. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are the generalized Chaffee-Infante equation in (3+1) dimensions, the (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev- Petviashvili equation and a generalized (2+1)-dimensional generalized Korteweg-de Vries equation. The generalized Chaffee-Infante equation with power-law nonlinearity in (3+1) dimensions is analyzed. Ansatz methods are utilized to provide topological and nontopological soliton solutions of this equation. Soliton solutions to nonlinear evolution equations have several practical applications in many areas of mathematical physics such as plasma physics and diffusion process. It will be shown that for certain values of the parameters, the power-law nonlinearity Chaffee-Infante equation permits soliton solutions. The requirements and restrictions for existence of soliton solutions are mentioned. Conservation laws are derived for the aforementioned equation. The Lie symmetry method investigates a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev-Petviashvili equation. Symmetry reduction is performed and group invariant solutions are obtained. Furthermore, the multiplier method is employed to derive conservation laws for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev-Petviashvili equation. Finally a generalized nonlinear (2+1)-dimensional equation is investigated from the view point of symmetry analysis in conjunction with ansatz methods and multipleexp function method. Moreover, conservation laws based on the multiplier approach will be investigated. Abstract and text in English

Analysis of a mathematical model of malaria transmission

Tue, 12 Dec 2023 00:00:00 GMT

Analysis of a mathematical model of malaria transmission Gizachew Tirite Gellow In this thesis, we analyze a mathematical model for the spread of malaria that consists of ten components. The human host population is divided into two main categories: semiimmune, which included all individuals who were immune to malaria, and non-immune, which included all individuals who were not. However, we further categorized semiimmune people into vulnerable, exposed, infectious, and recovered; non-immune people into vulnerable, exposed, and infectious; and the mosquito population into three classes: susceptible, exposed, and infected. We compute an explicit formula for the reproductive number, which depends on the weight of transmission from non-immune people to mosquitoes and from mosquitoes to non-immune humans, as well as the weight of transmission from semi-immune humans to mosquitoes and from mosquitoes to semi-immune humans. As a result, the square root of the sum of the squares of these weights for the two contact kinds represents the reproductive number for the entire population. The DFE point is GAS if R0 ≤ 1, indicating that malaria dies away, and stable if R0 > 1, indicating that malaria persists in the population. The model outcome confirms that the disease-free equilibrium is asymptotically stable when the reproductive number less than one and unstable when the reproductive number greater than one, and we discuss the possibility of a control for malaria transmission throughout a definite sub-group such as non-immune, semi-immune, or mosquitoes.