SA Graduate Modelling Camp Problems

Presenters:

Professor Gideon Ngwa, Applied Mathematical and Computer Assisted Modelling Unit, Department of Mathematics, University of Buea, Cameroon.

Professor Calistus Ngonghala, Department of Mathematics, University of Florida, USA

Problem Statement

Mosquito-borne diseases remain a critical global health challenge, causing over a million cases and hundreds of thousands of deaths annually, particularly in sub-Saharan Africa. These diseases impose significant economic costs, with billions of dollars lost each year due to treatment, prevention, and productivity losses. Mosquito-borne diseases are transmitted through the bite of female mosquitoes, which acquire the pathogen from an infected human and then pass it on to others. These mosquitoes exhibit specific behaviours, including questing for blood by seeking out human habitats, and resting in or near these habitats after feeding to digest the blood and develop eggs. Mosquitoes often return to breeding sites to lay their eggs, completing the reproductive or gonotrophic cycle. This behaviour allows mosquitoes to effectively transmit diseases like mosquito-borne disease between humans.

Insecticide-treated nets (ITNs) are a cornerstone of mosquito-borne disease control, acting as both a physical barrier and a means to kill or repel mosquitoes through the insecticide coating. By reducing mosquito bites and lowering mosquito populations, ITNs significantly reduce mosquito-borne disease transmission. Additionally, breeding site removal, which targets mosquito habitats, further decreases mosquito numbers and transmission potential. Together, these interventions are crucial in reducing mosquito-borne disease's burden.

The goal of this project is to develop and apply a mathematical model to understand the combined impact of ITNs and breeding site removal on long-term Mosquito-borne disease control. The model will aim to identify the minimum intervention level needed to achieve a decline in mosquito populations, specifically targeting breeding site, questing, and resting mosquitoes.

Research Question. What is the minimum level of combined ITN use and breeding site removal required to achieve a sustainable reduction in mosquito populations and long-term mosquito-borne disease control?

References

[1] Ngwa G A. On the population dynamics of the malaria vector, Bulletin of Mathematical Biology, 68(2006) pp 2161-2189.

[2] Ngwa G A, Wankah T T, Fomboh-Nforba M Y, Ngonghala C N, Teboh-Ewungkem M I. On a repro-ductive stage-structured model for the population dynamics of the malaria vector, Bulletin of Mathema-tical Biology, 76 (2014) pp 2476-2516.

[3] Ngonghala C N, Del Valle S Y, Zhao R, Mohammed-Awel J. Quantifying the impact of decay in bed-net efficacy on malaria transmission, Journal of Theoretical Biology, 363 (2014) pp 247-261.

[4] Ngonghala C N, Mohammed-Awel J, Zhao R, Prosper O. Interplay between insecticide-treated bed-netsand mosquito demography: implications for malaria control, Journal of Theoretical Biology, 397 (2016)pp 179-192.

[5] Ngwa G A, Teboh-Ewungkem M I, Dumont Y, Quifki R, Banasiak J. On a three-stage structured modelfor the dynamics of malaria transmission with human treatment, adult vector demographics and oneaquatic stage, Journal of Theoretical Biology, 481 (2019) pp 202-222.

[6] Ghakanyuy B M, Teboh-Ewungkem M I, Schneider K A, Ngw G A. Investigating the impact of multiplefeeding attempts on mosquito dynamics via mathematical models, Mathematical Biosciences,350 (2022) 108832.

[7] Ngwa G A, Teboh-Ewungkem M I, Njongwe J A. Continuous-time predator-prey-like systems used toinvestigate the question: Can human consciousness help eliminate temporary mosquito breeding sites fromaround human habitats?, Mathematics and Computers in Simulation, 206 (2023) pp 437 469.

[8] Keegan L T, Dushoff J, Population-level effects of clinical immunity to malaria. BMC infectious diseases, 13 (2013) pp 1-11.

[9] Aron J L. Dynamics of acquired immunity boosted by exposure to infection, Mathematical Biosciences, 64(1983) pp 249-259.

[10] Fowkes F, McGready R, Cross N, Hommel M, Simpson J, Elliott S, Richards J, Lackovic K,Viladpai- Nguen J, Narum D, Tsuboi D, Anders R, Nosten F, Beeson J. New insights into acquisition,boosting, and longevity of immunity to malaria in pregnant women, The Journal of Infectious Diseases, 206 (2012) 1612-21.

[11] Langhorne J, Ndungu F M, Sponaas A-M, Marsh K. Immunity to malaria: more questions thananswers, Nature Immunology, 9 (2008) pp 725-732.

Supporting Material

First-day Presentation

MISG 2025 Modelling Camp Problem 1 First-day presentation

Report-back Presentation

MISG 2025 Modelling Camp Problem 1 Report-back presentation

Presenter: Professor Neville Fowkes, School of Mathematics and Applied Mathematics, University of Western Australia

Problem Statement

Penguins need sardines and other sea food  to survive,  and fishermen need sardines `to live'.  They compete (unfairly).  The South African Little Penguin is critically endangered. Governments   can  regulate   the fishermen’s  catch in various ways  and penguins just do what they naturally do.  

• Explore the possible outcomes if various regulations are introduced.
• Check out available information about the African Penguin in this context.
• One way of improving the outcome is to exclude fishing in regions near penguin colonies. How much area should be `reserved' to ensure survival?

This problem relates to the MISG penguin problem which is concerned with the effect of climate change on the survivability of the penguin. The penguin population is critically endangered even without climate change with many  factors playing a role (predators, tourists, oil spill,...), however sardine biomass is thought to be the most critical factor. I  am  not  an expert and there has been much work on the problem so  we all will need to inform ourselves.

 For a start we might extend/modify  the standard predator/prey model to deal with   two competing (sardine)   predators. This may enable us to understand what  parameter combinations are important. 

Students should be able to handle ODE solution methods analytically and numerically using  any of the available packages.   We will introduce students to predator/prey and species competition modelling if necessary.

References

Supporting Material

First-day Presentation

MISG 2025 Modelling Camp Problem 2 First-day Presentation

Report-back Presentation

MISG 2025 Graduate Modelling Camp Problem 2 Report Back Presentation

Presenter:

Professor Graeme Hocking, School  of Mathematics and Statistics, Murdoch University, Perth, Australia.

Problem Statement

Th onset of the effects of climate change have motivated the development of wind farms throughout the world. Many of these are in the sea and now many more are being built on land.  The design of wind turbines is to be optimised so that they generate the maximum amount of energy in the most efficient manner.  Barotrauma is damage caused to organs by variations in pressure.

In planning for a wind farm, the developers must consider the environmental impact of their construction and the ongoing effect on the environment and seek approval from the appropriate Government Authorities. There is a lot of evidence now that these wind turbines can have a significant impact on birds through possible collisions with the blades and this has been the focus of many studies.  However, there are other environmental factors that have not been studied, one of which is the impact of the wind turbines on bats.

There is evidence that although it is unlikely that bats fly high enough to be struck by the turbines, there do appear to be many casualties due to their co-location.  It is theorised that this may be due to the low pressure developed by the so-called wing-tip vortices that are shed from the downstream edge of the turbine blades.  The blades of the turbine are aerodynamic objects much like aircraft wings and are designed to generate sideways lift causing the rotor to spin most effectively.  Bats have weak lungs and if they are “struck” by a low pressure region from the blades of the turbine, it is possible that their lung tissue may rupture.

In this project, you would consider the basics of wind turbine blade design and the aerodynamics associated with this in order to examine the flow of air over the blades and how that can form into a vortex sheet that rolls into a spiral.   This spiral can create a very low pressure patch that may impact on bats.  There are some general theories for these aerodynamic features that do not rely on complex simulations and so these should be considered.  If time permits, design of the turbine blades to minimise the intensity of the “wing-tip” vortices might be considered.

References

Supporting Material

 First-day Presentation

MISG 2025 Modelling Camp Problem 3 First-day presentation

Report-back Presentation

Graduate Modelling Camp Problem 3 Report-back presentation - Barotrauma in bats

Presenter: Matthews M. Sejeso, University of the Witwatersrand, Johannesburg

Problem Statement

Data clustering involves grouping entities in a dataset into clusters such that items within the same cluster exhibit greater similarity to one another than to those in other clusters. A detailed review of clustering techniques can be found in [1]. Frequently, datasets consist of points in a Euclidean space, and the clustering task is simplified to identifying hidden groups among these vectors. Many methods utilize distance metrics as a measure of similarity [1]. However, these conventional metrics lose reliability in high-dimensional space, because the data points are sparsely located. It is shown in [2] that the distance between any two high dimensional points becomes equal as the dimension n → ∞. As a result, most clustering algorithms which perform reasonably well in lower dimensions, fail in high dimensions.

To address this, numerous algorithms [3] have been proposed for handling high-dimensional data. In many real-world scenarios, data points are not evenly distributed across the high-dimensional space but instead are concentrated along lower-dimensional structures [4]. For example, images of a face captured under varying lighting conditions are known to reside in a 9-dimensional subspace, despite being represented by high-resolution pixel data [5]. Principal Component Analysis (PCA) [6] is a widely used technique to identify such low-dimensional linear subspaces within high-dimensional datasets. However, when datasets contain multiple categories, the assumption that all points lie in a single subspace becomes invalid. For instance, images of several faces under different lighting conditions would span a union of multiple 9-dimensional subspaces. Subspace clustering addresses this challenge by grouping data points such that each cluster corresponds to a distinct, low-dimensional subspace [7].

Subspace clustering techniques can be categorized into four major types [7]: (i) algebraic, (ii) iterative, (iii) statistical, and (iv) spectral clustering-based methods. Algebraic approaches assume noise-free data that perfectly adheres to the subspace union structure, though some extensions accommodate moderate noise levels. Iterative methods alternate between assigning data points to clusters and estimating subspaces for those clusters. Statistical approaches rely on generative models for the data. This discussion focuses on spectral clustering, which has gained significant attention in recent years [8]. Unlike traditional clustering methods, spectral clustering employs the eigenstructure of a similarity graph to uncover latent patterns in the data. Despite its effectiveness, applying spectral clustering to practical datasets presents challenges, including similarity graph construction, hyperparameter selection, computational scalability, and handling noise or outliers.

Constructing an effective similarity graph is critical to spectral clustering's success, yet it remains a complex task. The similarity measure must balance capturing meaningful relationships between points while avoiding spurious connections caused by noise or sparse data. Additionally, spectral clustering relies on computationally expensive eigenvalue decomposition, which limits its application to large-scale datasets. The selection of hyperparameters, such as the kernel bandwidth or number of neighbours, further complicates the process, often requiring extensive experimentation and domain knowledge.

The graduate modelling camp will focus on exploring, understanding, and addressing these challenges in spectral clustering. Participants will work to improve spectral clustering's robustness and efficiency using the fact that subspaces are sparse in high-dimensional data. They will also investigate techniques to optimize similarity graph construction, reduce computational complexity, and enhance performance on large-scale datasets.

References:

1. Xu and D. C. Wunsch II, “Survey of clustering algorithms,” IEEE Trans. Neural Networks, vol. 16, no. 3, pp. 645–678, 2005.
2. Beyer, J. Goldstein, R. Ramakrishnan, and U. Shaft, “When is “nearest neighbor” meaningful?” in Int. Conf. Database Theory. Springer, 1999, pp. 217–235.
3. Parsons, E. Haque, and H. Liu, “Subspace clustering for high dimensional data: a review,” ACM SIGKDD Explorations Newsletter, vol. 6, no. 1, pp. 90–105, 2004.
4. S. Cherkassky and F. Mulier, Learning from Data: Concepts, Theory, and Methods, 1^st ed. New York, NY, USA: John Wiley & Sons, Inc., 1998.
5. Basri and D. W. Jacobs, “Lambertian reflectance and linear subspaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 2, pp. 218–233, 2003.
6. Jolliffe, Principal component analysis (Springer Series in Statistics). Berlin, Germany: Springer, 2002.
7. Vidal, “Subspace clustering,” IEEE Signal Processing Magazine, vol. 28, no. 2, pp. 52–68, 2011.
8. L Ulrike, “A tutorial on spectral clustering”, Statistics and computing, 17, pp.395-416, 2007.

Supporting Material

First-day Presentation

MISG 2025 Modelling Camp Problem 4 First-day presentation

Report-back Presentation

MISG 2025 Modelling Camp Problem 4 Report-back presentation

Presenter:  Professor David P Mason, School of Computer Science and Applied Mathematics, University of the Witwatersrand.

Problem Statement

The generation of lift is important in the aircraft industry, wind turbines and in nature with birds and insects.

The generation of lift by a thin aerofoil will be considered. Circulation which is important in lift generation is first discussed. The Blasius Theorem for the net force per unit length on a body and the Kutta-Joukowski Lift Theorem which relates the lift to the circulation are then outlined. Vortex flow, vortex filaments, vortex sheets and the strength of the vortex sheet and the connection to circulation will then be reviewed. An important condition is the Kutta condition that for a given aerofoil and given angle of attack the value of the circulation is such that the fluid  flow leaves the trailing edge smoothly. The aerofoil equation which is a singular integral equation for the strength of the vortex sheet is derived and solved using real variable theory subject to the Kutta condition. The integral of the vortex strength along the aerofoil gives the circulation which is related to the lift through the Kutta-Joukowski Theorem.

The Blasius Theorem and the Kutta-Joukowski Lift Theorem are derived using complex analysis. The approach is self- contained and the necessary complex variable theory is explained for students who have not done complex analysis. Alternatively students who have not done complex analysis can accept the results without derivation.

Aerofoil theory applies to a fixed aerofoil. We will investigate the twist in a wind turbine blade to accommodate blade rotation. We will also investigate the lift mechanism in the flight of a mosquito. We will try and understand the physical mechanisms although we may not be able to develop a mathematical model.

References

Supporting Material

First-day Presentation

Graduate Modelling Camp Problem 5 First-day Presentation

Report-back Presentation

MISG 2025 Graduate Modelling Camp Problem 5 Report-back Presentation

Presenter: Dr Thama Duba and Dr Erick Mubai, School of Computer Science and Applied Mathematics, University of the Witwatersrand

Problem Statement:

What is a tsunami?

A tsunami is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake, which may be caused by an underwater earthquake, volcano and underwater landslides or any tectonic shift in the ocean.

Tsunamis in history

The 2004 Indian Ocean tsunami is the deadliest in history, caused by 9.1 earthquake on the Richter scale on the coast of Indonesia, Sumatra. This occurred on 26 December 2004. Tsunami waves moved at a speed of 700km/h and they were as high as 40.5 metres. They destroyed about 15 countries and caused deaths of about 280,000 people.

The March 2011 Japan earthquake, which also measured at 9.1 on the Richter scale, was not as destructive, however, due to the Tsunami signals already established. In this case the community had 8 minutes to evacuate after the first signal from an earthquake was observed. This tsunami destroyed the nuclear plant in Japan.

Research into tsunami

Studies into tsunami separate the tsunami mechanism into three stages:

1. Generation and its evolution near the source,
2. Propagation of waves in open ocean, and
3. Inundation in the shallow water and on the shore.

Most common model used to describe a tsunami is the linear, shallow water, long wave model.

Nonlinear equations are important at the shore where most damages occur. Most common model used is the Korteweg-de Vries (KdV) equation whose solutions are localised structures called solitons.

Depending on the depth of the ocean, the wave may be deadly as the speed of these water is given by c=\sqrt{gH} where g is the acceleration according to gravity and H is the depth of the ocean floor.

Earlier models do not include the rotation of the Earth in the model. However, geophysical fluids are characterised by rotation and stratification. Since the ocean and atmospheres have the Rossby number which is small, studies involving oceans must take rotation into consideration. Many studies that take rotation into consideration use the geophysical KdV equation (gKdV) which adds the Coriolis force into the system.

Many simulated studies use the forced KdV equation to include the source term.

The main issue in the study of tsunami is that a wave propagates in both the deep water and the shallow waters, hence the discussions and the variations in the modelling. In some cases, due to varying topography, some studies assume flat topography and others take into consideration a varying topography.

The good news is that NOT all tsunamis are catastrophic. In December of 2017, a tsunami on the coast of Durban was recorded, but it was not deadly.

Task:

1. You are to find the best mathematical model for the tsunami propagation considering the debates in literature showing the strengths and the pitfalls of each of the models.
2. Make arguments about the models at each of the 3 stages considered in the study of a tsunami, that is, how a tsunami forms, propagates and breaks down onshore and where possible create your own mathematical model, find the solution and simulate the propagation.

References

1. Tsunami: Race against Time https://youtu.be/4V_xX3UDHb8
2. Modeling study of tsunami wave propagation. M Y Regina and E S Mohamed. https://doi.org/10.1007/s13762-022-04484-2
3. On the relevance of soliton theory to tsunami modelling. Adrian Constantin https://doi.org/10.1016/j.wavemoti.2009.05.002
4. On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves. Adrian Constantin and Robyn Stanley Johnson. Journal of Nonlinear Mathematical Physics, volume 15, no 2, 2008, 58-73
5. Solitons: An introduction. P G Drazin and R S Johnson. Cambridge University Press, 1989

First-day Presentation

MISG 2025 Modelling Camp Problem 6 First-day presentation

Report-back Presentation

MISG 2025 Graduate Modelling Camp Problem 6 Report-back presentation