Pure & Applied Mathematics Seminar

Speakers

Noah Cresp, Isabella Swadling, Matthew Delahunty

Noah Cresp: Drug Diffusion with Reversible Binding: A comparison of Analytical Methods

We extend a two-layer diffusion model for transdermal drug delivery to include reversible binding within the skin. The model describes standard (Fickian) diffusion through a drug vehicle and skin, with mobile drug in the skin able to reversibly bind to immobile binding sites, temportarily preventing its transfer to blood circulation. The resulting coupled diffusion-reaction system is analysed using the Laplace transform approach and the Unified Transform Method. We compare the resulting solution representations and discuss the additional analytical and computational challenges introduced by binding, with particular emphasis on extending these methods to more physiologically realistic drug delivery models.

Isabella Swadling: Measures for Calibration of Cohesive Bulk Materials in the Discrete Element Method

Cohesive bulk materials present a significant challenge in the modelling and design of industrial transfer systems, where flow behaviour influences performance, wear, and operating costs. Accurately characterising these materials is therefore an important component of discrete element method (DEM) calibration. This work investigates a heatmap representation of granular mound geometry as a means of quantifying changes in cohesive behaviour across a range of material conditions. By aligning and aggregating the simulated mound data from all cross-sectional viewing angles into representative mound distributions, the approach provides a statistical representation of the overall mound geometry while retaining the variability present across the full dataset. Standard deviation, area-under-the-curve, and Wasserstein distance analyses are used to examine trends in mound morphology, while linear regression is applied to assess relationships between these metrics and cohesion. The results demonstrate the potential of heatmap representations to quantify the influence of cohesion and to serve as an additional tool for the calibration and analysis of DEM simulations.

Matthew Delahunty: Inferring Particle Size Distributions from Inclined Channel Elutriation Data via Mathematical Deconvolution

Accurately measuring particle size distributions is essential for optimising mineral processing and fluidisation systems, but traditional methods are often slow and inefficient. Batch elutriation through inclined channels provides a significantly faster alternative for particle separation, though complex fluid mechanics make it difficult to directly determine feed distributions from collected samples.

In this study, we applied mathematical deconvolution and optimisation techniques to reconstruct unknown feed size distributions from batch elutriation data. Using a forward transport model with a linearly increasing fluidisation velocity, we generated synthetic elutriation datasets for both single-density and variable-density feeds. The inverse problem of deconvolving the particle size distribution from overflow bag masses is ill-posed. To resolve this, a regularised quadratic optimisation problem was formulated using zero-order spline reconstructions and non-negativity constraints.

We looked at optimising the Tikhonov regularisation parameter using L-curve analysis and direct error minimisation. The results show that constant-density feed size distributions can be reconstructed to within a relative error under 2%. Slower liquid acceleration rates improve separation clarity and accuracy, though they increase experimental runtime. Finally, we adapted this approach for two-dimensional (size-density) variable feeds using joint kernel functions. Overall, this demonstrates that inclined channel elutriation could be a practical, reliable alternative to traditional methods.

Good reduction of one-point Galois covers

In characteristic 0, one has classical tools to construct and study covers, such as the Riemann Existence Theorem. In positive characteristic p, the situation is more complicated due to the presence of wild ramification: branch points whose branching data is not described purely topologically. A useful approach in this situation is reducing a cover in characteristic 0 to obtain a new cover in characteristic p. This is particularly useful for covers with “good reduction;” loosely speaking, covers for which reduction preserves nice properties, such as smoothness. In this talk, we will learn a criterion for covers of elliptic curves in characteristic 0 branched at one point to have good reduction and what this means for covers in positive characteristic.

Faster enumeration of primes

I will discuss my recent paper on fast algorithms for finding all primes up to N (https://arxiv.org/abs/2606.22851). These are the first algorithms to achieve a speedup by a positive power of log N over the ancient sieve of Eratosthenes.

Approximation of stochastic semi-linear PDEs on the unit sphere

We study the numerical approximation of a class of semi-linear stochastic partial differential equations posed on the unit sphere. The equations considered combine diffusion, nonlinear reaction terms, and random forcing, and arise naturally in models of spatially distributed phenomena evolving on spherical domains. We construct a fully discrete numerical method by combining a Galerkin approximation based on spherical radial basis functions with a suitable time-stepping scheme. Under appropriate assumptions on the linear operator, the nonlinearity, the stochastic forcing, and the initial data, we establish stability and strong convergence of the proposed approximation. The error analysis identifies the contributions arising from the spatial discretisation, the temporal discretisation, and the truncation of the driving noise. The results extend existing numerical analyses of linear stochastic equations on the sphere to a semi-linear setting and provide a framework for approximating stochastic reaction–diffusion equations on spherical geometries.

Like Squeezing Water From a Stone; Phototrophic Subaerial Biofilms Growing on Porous Substrata

Subaerial biofilms (SABs) are microbial communities that colonize exposed surfaces, both built and natural, frequently powered by photosynthesis. Their function is tightly coupled to environmental conditions which are often harsh, particularly with respect to water availability. Even without exposure to regular rainfall, it is hypothesized that overnight radiative cooling allows SABs to utilize condensation, i.e., dew, to gather high activity water, in order to be productive during the early morning before evaporation becomes significant. Further, it is proposed that SABs are able to effectively seal substratum pores to store condensed (or precipitated) water to offset evaporation for some time. A physical theory, based on the Hertz-Knudsen equation for describing evaporation/condensation at liquid-gas interfaces, is constructed. To test the theory, the derived mathematical model of water activity in SAB-stone systems is compared to prior laboratory data of water transport in limestone samples with and without SAB growth. We show that the model is able to successfully explain differences in the data between the two cases. Implications for SAB growth in the environment are discusses.

Speakers

Aidan Ruiz De Luzuriaga, Cameron West, James Harrison

Epidemic dynamics estimated by novel mathematical models

Novel mathematical models demonstrated that durations of epidemics significantly depend on the number of hidden (asymptomatic) cases, re-infections and newborns [1-4]. Due to the hidden cases, epidemics cannot be stopped [4], but can be controlled by applying non-pharmaceutical measures (e.g., Zero-COVID strategy [5] or immediate and sufficient increase of the number of tests [6]). Despite of decrease in COVID-19 testing and reporting new infections, the numbers of new cases and deaths are still rather high in some countries [7, 8] and exceeded the endemic levels and seasonal flue mortality [9]. These facts make the development of new vaccines or the further use of existing ones very urgent. Unfortunately, existing COVID-19 vaccines cannot decrease the number of cases and deaths, but they reduce the severity of disease and case fatality risks [6-11].

1. Nesteruk, I. (2023): Endemic characteristics of SARS-CoV-2 infection. Sci Rep 13, 14841. https://doi.org/10.1038/s41598-023-41841-8
2. Nesteruk I (2025) General SIR model for visible and hidden epidemic dynamics. Front. Artif. Intell. 8:1559880. doi:10.3389/frai.2025.1559880
3. Nesteruk, I. (2025): New reproduction numbers for the visible and real epidemic dynamics. Advances in Public Health. Article ID 5469282. https://doi.org/10.1155/adph/5469282
4. Nesteruk, I. (2025) How re-infections and newborns can change the visible and hidden epidemic dynamics? Computation 2025, 13, 113.  https://doi.org/10.3390/computation13050113.
5. Nesteruk, I., Keeling, M. (2023): Population age as a key factor in the COVID-19 pandemic dynamics. Preprint. Research Square. November 30, 2023. https://doi.org/10.21203/rs.3.rs-3682693/v1
6. Nesteruk, I. (2024): Impact of vaccination and testing levels on the COVID-19 waves. J Allergy Infect Dis. 2024;5(1):44-55. https://probiologists.com/Uploads/Articles/11_638603156371379586.pdf
7. Nesteruk, I. (2024): Should we ignore SARS-CoV-2 disease? Epidemiology and Infection. 152:e57. doi:10.1017/S0950268824000487
8. Nesteruk, I. (2024): Should vaccinations against SARS-CoV-2 infection be continued? Preprint Research gate. November 6, 2024. DOI: 10.13140/RG.2.2.28947.90403
9. Nesteruk, I. (2023): Endemic characteristics of SARS-CoV-2 infection. Sci Rep 13, 14841. https://doi.org/10.1038/s41598-023-41841-8
10. Nesteruk, I. (2024): Trends of the COVID-19 dynamics in 2022 and 2023 vs. the population age, testing and vaccination levels. Front. Big Data 6:1355080. doi:10.3389/fdata.2023.1355080/li>
11. Nesteruk, I. (2025):  Impact of SARS-CoV-2 vaccinations on pandemic dynamics: Trends in cases, deaths, and fatality risks. J Allergy Infect Dis. 2025;6(1):17-24. https://www.probiologists.com/article/impact-of-sars-cov-2-vaccinations-on-pandemic-dynamics-trends-in-cases-deaths-and-fatality-risks