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Statistics, Probability, Analysis and Applied Mathematics (SPAAM)

SPAAM Seminar Series 2023/24

The Statistics, Probability, Analysis and Applied Mathematics (SPAAM) seminar series will take place on Thursdays between 3-4pm in room B3.02 and virtually on the SPAAM Microsoft Teams ChannelLink opens in a new window during term time. It will host a variety of talks from PhD students involved in applied mathematics research at Warwick and invited guests from other institutions (see below for the schedule and talk abstracts).

The seminars will usually host two speakers (unless otherwise stated) with each talk taking around 15-20 minutes with 5-10 minutes of questions afterwards. Speakers and committee members will hang around for some time after the talks for social tea/coffee and further questions.

This seminar series is hosted by the Warwick SIAM-IMA Student ChapterLink opens in a new window. Please do contact one of the committee if you would like to join and be added to the MS Teams channel. Note that these talks may be recorded so do join with audio and video off if you don't wish to feature!

If you missed the seminar, head over to our Youtube channelLink opens in a new window to find the recorded talks!

If you would like to give a talk this academic year, please contact:

Andrew Nugent (a.nugent@warwick.ac.uk) or

Oscar Holroyd (o.holroyd@warwick.ac.uk)

and we will find you a slot!

Find out more about the
Warwick SIAM-IMA Student Chapter

Term 3

Date

Talk 1

Talk 2

25th April 2024 (Week 1)

Andrew Nugent

(Steering opinion dynamics through control of social networks)

Luisa Fernanda Estrada Plata

(Learning a Social Network by Influencing Opinions)

2nd May 2024 (Week 2)

Social Event

9th May 2024 (Week 3)

SIAM Careers Event (Auto Trader)

16th May 2024 (Week 4)

Alex Kaye

(Quantifying epidemic risks: A practical approach for seasonal pathogens)

Phurinut Srisawad

(Exploration of the Best Option under Changing Environments)

23rd May 2024 (Week 5)

Emily Claughton

(Understanding linguistic dynamics in agent-based communal networks)

Hanyang Wang

(Respecting the limit: Bayesian optimization with a bound on the optimal value)

30th May 2024 (Week 6)

Yi Ting Loo

(In-silico image-based modelling of morphogen diffusion)

TBC

6th June 2024 (Week 7)

Social Event

13th June 2024 (Week 8)

No event

20th June 2024 (Week 9)

Sotirios Stamnas

Seb Dooley

27th June 2024 (Week 10)

Mark Lynch Sharlin Utke

Term 3 Abstracts

Week 6. Yi Ting Loo (Warwick MathSys CDT) - In-silico image-based modelling of morphogen diffusion

Morphogens are intercellular signalling molecules providing spatial information to cells in developing tissues to coordinate cell fate decisions. The spatial information is encoded within long-ranged concentration gradients of the morphogen. Experimental measurements of morphogen diffusivity vary significantly depending on experimental approach. Such differences have been used to argue against diffusion as a viable mechanism of morphogen gradient formation. Using particle modelling on realistic zebrafish brain images, we demonstrate that accounting for the local tissue architecture in concert with including receptor binding is sufficient to explain a range of biological observations. This demonstrates that (hindered) diffusion-driven transport is a viable mechanism of gradient formation of morphogens.

Week 5. Emily Claughton (University of Nottingham) - Understanding linguistic dynamics in agent-based communal networks

Linguistic variation has been well studied on an individual basis with several accepted mechanisms as to why individuals change the way that they speak and the vocabulary they use. However, the answer to the same question of a larger population remains largely unclear. Lack of data on individual dialects and the scale of populations has traditionally rendered this problem difficult to solve, issues that mathematical modelling is ideally suited to deal with. Representing a community of speakers as an interconnected network, we demonstrate that agent-based network modelling can help us understand linguistic change in networks and allows us to investigate it as a function of network parameters.

Week 5. Hanyang Wang (Warwick MathSys CDT) - Respecting the limit: Bayesian optimization with a bound on the optimal value

In many real-world optimization problems, we have prior information about what values are achievable under the objective function. In this talk, we will discuss the scenario that we have either exact knowledge of the minimum value or a, possibly inexact, lower bound on its value. We propose bound-aware Bayesian optimization (BABO), a Bayesian optimization method that uses a new surrogate model and acquisition function to utilize such prior information.

Week 4. Alex Kaye (Warwick MathSys CDT) - Quantifying epidemic risks: A practical approach for seasonal pathogens

What does it mean for an outbreak to be "major", and what is the probability of this happening? This question is well-studied when the assumption is made that there are no changes in transmission after the introduction of an initial infectious case. However, this assumption is often untrue. For example, the transmission of the flu follows a clear seasonal pattern. Here, we introduce two new metrics that can help evaluate the probability that a major outbreak will occur when there is seasonal transmission present. We will motivate this study by applying this idea to the diseases dengue and chikungunya, whose seasonality is driven by yearly changes in local mosquito populations.

Week 4. Nut Srisawad (Warwick MathSys CDT) - Exploration of the Best Option under Changing Environments.

The problem of identifying the best option from stochastic samples can be applied in various fields, for example, A/B testing to choose the best web designs for customers, clinical trials to investigate the best treatment for patients, etc. In a non-stationary world, this problem becomes more challenging since the best option may change depending on the environment. One raises a question: “Can we learn the best option across environmental change?” When an environment shares the same influence on every option, it is possible to find the uniquely best option by learning such influence. In this talk, I will present the exploitation of learning about environmental change to design an efficient adaptive sampling policy and a prediction of the best option.

Week 1. Andrew Nugent (Warwick MathSys CDT) - Steering opinion dynamics through control of social networks

We propose a new control problem for opinion dynamics on evolving networks. The controls modify the strength of connections in the network, rather than influencing opinions directly, with the overall goal of steering the population towards a target opinion. This requires that the social network remains sufficiently connected, the population does not break into separate opinion clusters, and that the target opinion remains accessible. We present several results on the existence of controls and discuss methods for finding them.

Week 1. Luisa Fernanda Estrada Plata (Warwick MathSys CDT) - Learning a Social Network by Influencing Opinions

We study a campaigner who wants to learn the structure of a social network by observing the underlying diffusion process and intervening on it. Using synchronous majoritarian updates on binary opinions as the underlying dynamics, we offer upper bounds on the campaigner's budget for learning any network with certainty, considering both observation and intervention resources, and further improving them for the case of clique networks. Additionally, we investigate the learning progress of the campaigner when her budget falls below these upper bounds. For such cases, we design a greedy campaigning strategy aimed at optimising the campaigner's information gain at each opinion diffusion step.

Term 2

Date

Talk 1

Talk 2

11th January 2024 (Week 1)

Pheobe Asplin
(An introduction to health economic modelling in the context of infectious diseases)

18th January 2024 (Week 2)

Olayinka Ajayi

(Finding order in near-disorder: Position Encoding for Graphs)

25th January 2024 (Week 3)

Social Event!

1st February 2024 (Week 4)

Patricia Lamirande

(Mean first time and its application in ocular drug delivery)

Boris Andrews

(High-order time-stepping schemes for high-order systems conserving multiple high-order invariants)

8th February 2024 (Week 5)

Nathan Doyle

(How should lockdown be introduced? Devising cost-effective strategies for novel outbreaks amid vaccine uncertainty)

Rachel Seibel

(Unifying human infectious disease models and real-time awareness of population- and subpopulation-level intervention effectiveness)

15th February 2024 (Week 6)

Andres Trujillo Miniguano

(A nonlocal PDE-constrained optimisation model for containment of infectious diseases)

22nd February 2024 (Week 7)

Ed Brambley

(How to write a paper)

29th February 2024 (Week 8)

Social Event!

7th March 2024 (Week 9)

Ryan Teo

(Exploring de Bruijn graph representations of environmental metagenomes)

Byron Tzamarias

(An Optimal Control Theory formulation that is based on maximizing the life expectancy of cancer chemotherapy patients)

14th March 2024 (Week 10)

Yueting Han Elliot Vincent

Term 2 Abstracts

Week 1. Phoebe Asplin (Warwick MathSys CDT) - An introduction to health economic modelling in the context of infectious diseases

How can we measure the cost of an outbreak? Or how effective an intervention strategy is? How can we determine which strategy is most effective beyond simply looking at case numbers? How can we say when an intervention is not worth doing despite it saving lives? In this seminar, we will explore the answers to these questions and discuss the techniques most commonly used in health economic modelling for infectious diseases.

Week 2. Olayinka Ajayi (Warwick MathSys CDT) - Finding order in near-disorder: Position Encoding for Graphs

This talk is about sharing possible ideas to order the nodes of some random graph. Position encodings was popularised courtesy of the transformer model for NLP to account for the order of words in a sentence. But unlike sentences (sequences) that inherently come with an order, general graphs do not have an inherent order (if we ignore grids, cycles and path graphs). In this talk, we would look at how the position encodings in the transformer architecture is derived from graphs, and how this derivation extends to general large scale network.

Week 4. Patricia Lamirande (University of Oxford) - Mean first time and its application in ocular drug delivery
Wet age-related macular degeneration is a progressive disease that leads to severe visual impairment. Standard of care treatment involves drug injections into the eye, determining the yet unmet medical need of reducing injection frequency. This motivates modelling efforts to understand and help select prospective drugs with longer retention times. To this end, we developed a mean first passage time (MFPT) modelling framework, to investigate the scaling relationships of ocular pharmacokinetics in humans and animal species and to inform drug development.
The MFPT describes how long it takes, on average, for a random walker to reach a given target, and is a valuable method to quantify the efficacy of diffusion transport. In this work, we derived a partial differential equation system that describes the MFPT of a particle in a 3D finite domain, bounded by reflective and semi-permeable conditions, modelling the diffusion of a drug injected into the eye. We applied our model to quantify the influence of anatomical and physiological parameters of the eye on the kinetics of protein therapeutics, to better relate retention times between different species. We also investigated the impact of the injection location and of the variability in human eyes on drug elimination.
Week 4. Boris Andrews (University of Oxford) - High-order time-stepping schemes for high-order systems conserving multiple high-order invariants
Time-dependent ODEs and PDEs typically feature physically meaningful invariants, such as the energies in Hamiltonian ODEs or the ideal Navier–Stokes PDEs. Naive numerical schemes for such systems will generally not conserve these invariants; this infers undesirable, unphysical behaviour.
Modified Runge–Kutta methods have been the typical approach for constructing conservative time-stepping schemes. Distinct difficulties have persisted, however, in certain cases where:
- One seeks a scheme with order in time ≥ 2
- The invariants have order ≥ 2
- The system has order ≥ 1
In this talk, we propose a novel framework for constructing conservative schemes for potentially multiple invariants. The approach relies on a finite-element-in-time reformulation, alongside the systematic introduction of auxiliary variables; this allows us to circumvent these difficulties.
We conclude with various numerical demonstrations.
Week 5. Nathan Doyle (Warwick MathSys CDT) - How should lockdown be introduced? Devising cost-effective strategies for novel outbreaks amid vaccine uncertainty

During an infectious disease outbreak, public health policy makers are tasked with strategically implementing control interventions whilst weighing competing objectives. To provide a quantitative framework that can be used to guide these decisions, it is helpful to devise a clear and specific objective function that can be evaluated to determine the optimal outbreak response. In this study, we have developed a mathematical model to simulate outbreaks of a novel emerging pathogen for which non-pharmaceutical interventions (NPIs) are imposed or removed based on thresholds for hospital occupancy. These thresholds are set at different levels to define four unique control strategies. We illustrate that the optimal intervention strategy is contingent on the choice of objective function. Specifically, the optimal strategy depends on the extent to which policy makers prioritise reducing unmitigated health costs due to infection over control-associated costs. Motivated by the scenario early in the COVID-19 pandemic, we incorporate the development of a vaccine and demonstrate how uncertainty in future vaccination availability and coverage (and/or effectiveness) can influence the optimal control strategy to adopt at the outbreak's onset. These analyses highlight the benefits of policy makers being explicit about the precise objectives of introducing interventions.

Week 5. Rachel Seibel (Warwick MathSys CDT) - Unifying human infectious disease models and real-time awareness of population- and subpopulation-level intervention effectiveness

Background. During infectious disease outbreaks, humans often base their decision to adhere to an intervention strategy on their personal opinion towards the intervention, perceived risk of infection and intervention effectiveness. However, due to data limitations and inference challenges, infectious disease models usually omit variables that may impact an individual's decision to get vaccinated and their awareness of the intervention's effectiveness of disease control within their social contacts as well as the overall population.

Methods. We constructed a compartmental, deterministic Susceptible-Exposed-Infectious-Recovered (SEIR) disease model that includes a behavioural function with parameters influencing intervention uptake. The behavioural function accounted for an initial subpopulation opinion towards an intervention, their outbreak information sensitivity and the extent they are swayed by the real-time intervention effectiveness information (at a subpopulation- and population-level). Applying the model to vaccination uptake and three human pathogens - pandemic influenza, SARS-CoV-2 and Ebola virus - we explored through model simulation how these intervention adherence decision parameters and behavioural heterogeneity in the population impacted epidemiological outcomes.

Results. From our model simulations we found that differences in preference towards outbreak information were pathogen-specific. Therefore, in some pathogen systems, outbreak information types at different outbreak stages may be more informative to an information-sensitive population and lead to less severe epidemic outcomes. In both behaviourally-homogeneous and behaviourally-heterogeneous populations, pandemic influenza showed patterns distinct from SARS-CoV-2 and Ebola for cumulative epidemiological metrics of interest. Furthermore, there was notable sensitivity in outbreak size under different assumptions regarding the population split in behavioural traits. Outbreak information preference was sensitive to vaccine efficacy, which demonstrates the importance of considering human behaviour during outbreaks in the context of the perceived effectiveness of the intervention.

Implications. Incorporating behavioural functions that modify infection control intervention adherence into epidemiological models can aid our understanding of adherence dynamics during outbreaks. Ultimately, by parameterising models with what we know about human behaviour towards vaccination (and other infection control interventions) adherence, such models can help assist decision makers during outbreaks. Such progress will be particularly important for emerging infectious diseases when there is initially little information on the disease dynamics and intervention effectiveness.

Week 6. Andres Trujillo Miniguano (University of Edinburgh) -A nonlocal PDE-constrained optimisation model for containment of infectious diseases

Nonpharmaceutical interventions have proven crucial in the containment and prevention of Covid-19 outbreaks. In particular public health policy makers have to assess the effects of strategies such as social distancing and isolation to avoid exceeding social and economical costs. In this work, we study an optimal control approach for parameter selection applied to a dynamical density functional theory model. This is applied in particular to a spatially-dependent SIRD model where social distancing and isolation of infected persons are explicitly taken into account. Special attention is paid when the strength of these measures is considered as a function of time and their effect on the overall infected compartment. A first order optimality system is presented, and numerical simulations are presented using a spectral-Newton method. This work could potentially provide some mathematical insights into the management of disease outbreaks.

Week 9. Ryan Teo (University of Birmingham) - Exploring de Bruijn graph representations of environmental metagenomes

In the face of the evolving challenges posed by emerging or novel pathogens, advanced analytical techniques for microbial surveillance are essential. This includes our ability to sequence and characterise microbial communities in the environment, contained in a metagenome, which complements traditional syndromic surveillance systems. Unfortunately, environmental metagenomes are highly complex and variable, and regular analysis methods for smaller scale microbiome studies do not necessarily scale as well.

For this project, I explore the utility of using de Bruijn Graphs (DBGs) to represent environmental metagenomes and analyse them without having to classify DNA into Operational Taxonomic Units. DBGs are directed graphs consisting of nodes which represent DNA sequences of length k, which share an edge if they share an overlap of k-1. Representing complex environmental metagenome data as graphs opens up the possibility of graph based analysis that may yield deeper insights into microbial community dynamics. Preliminary results, based on the publicly available datasets from longitudinal wastewater sampling, reveal promising insights into the use of graphs to understand microbial diversity and community evolution over time. However, this application still requires further validation to establish meaningful connections between quantitative graph properties and biological signals. If successful, the utilisation of DBGs in metagenomic analysis may lay the foundation for characterising a baseline for environmental metagenomes and detecting emerging pathogens.
Week 9. Byron Tzamarias (Warwick MathSys CDT) - An Optimal Control Theory formulation that is based on maximizing the life expectancy of cancer chemotherapy patients
We propose a novel optimal control theory (OCT) formulation that evaluates the efficacy of cancer-chemotherapy treatments in terms of the expected lifetime of patients and all possible cancer futures/outcomes. Outcomes include tumor clearance, tumor relapse as well as death due to the treatment. A key advantage of the proposed formulation is that all parameters have direct biological interpretations and hence could be potentially used in the development of personalized treatments, provided that individual risk can be determined. Our model is based on a deterministic OCT framework, however stochastic effects such as cancer-cell lineage die-out have been accounted on. In this talk we present the formulation in a general/theoretical setting. Individual characteristics, such as the age of the patient and the susceptibility of the patient to drug-induced side effects, are considered. We employ a direct optimal control method and find that optimal solutions are Bang-Bang and have either no switch or a single switch giving solutions that are (i) continuous treatment at maximum tolerated dose, (ii) no treatment or (iii) treat-and-stop solutions, treating at the maximum tolerated dose (MTD) and stopping drug administration before the time horizon is reached. For the latter, the treatment time (and the optimal solutions) are independent of the time horizon given that drug treatment has stopped. Optimizing over the time horizon, treatments are either no treatment, i.e. patients are untreatable since there is no benefit under treatment, or MTD for a specified time. Patients thus split into an untreatable class and a treatable class, with patient demographics, tumor size, tumor response and drug toxicity determining a patient's benefit under treatment and their class. Future directions will be discussed such as applications of the OCT formulation to study acute myeloid leukemia (AML) chemotherapy treatments.
Week 10. Yueting Han (Warwick MathSys CDT) - Modelling and Predicting Online Vaccination Views Using Bow-tie Structure -- network analysis, machine learning & mechanistic simulation

Social media has become increasingly important in shaping public vaccination views, especially since the COVID-19 outbreak. In the realm of online social networks, this paper explores a more nuanced division of roles each user plays in information flow, going beyond the “creator-receiver” dynamics through the lens of “bow-tie structure”.The dataset we work on describes the information exchange among anti-vaccination, pro-vaccination, and neutral Facebook pages, covering the period before and during the initial stage of COVID-19. In our research, we consistently observe statistically significant bow-tie structures with different dominant components for each vaccination group over time. We further investigate changes in opinions over time, as measured by fan count variations, using agent-based simulations and machine learning models. Across both methods, accounting for bow-tie decomposition better reflects information flow differences among vaccination groups and improves our opinion dynamics prediction results. The modelling frameworks we consider can be applied to any multi-stance temporal network and could form a basis for exploring opinion dynamics using bow-tie structure in a wide range of applications.
Week 10. Elliot Vincent (Warwick MathSys CDT) - Modelling the emergence of finch trichomonosis in the UK

Emerging infectious diseases (EIDs) can have severe and unprecedented impacts on wildlife populations, especially when exacerbated by human-driven factors. Since 2005, British finch populations have undergone drastic declines as a result of an outbreak of the disease trichomonosis. In this work I have constructed a mathematical model to describe the biological system, and used this model in combination with real data to gain insight into properties of the outbreak, such as transmission dynamics and disease progression.

Term 1

Date

Talk 1

Talk 2

19th October 2023 (Week 3)

Oscar Holroyd

(Linear Quadratic Regulation Control for Falling Liquid Films)

Mark Lynch

(Nash Neural Networks: Inferring Utilities from Optimal Behaviours in Epidemics)

26th October 2023 (Week 4)

Peter Lewin-Jones

(Computational Modelling of Drop-Drop Collisions in the Presence of Gas Microfilms: When Do Drops Bounce?)

2nd November 2023 (Week 5)

Social Event

9th November 2023 (Week 6)

Niall Rodgers

(Strong Connectivity and Influence In Real Directed Networks)

Nathan Coombs

(Colloidal Deposits from Evaporating Sessile Droplets)

16th November 2023 (Week 7)

Margherita Botticelli

(How does ECM stiffness affect spheroid growth?)

Andrew Nugent

(Scaling in Opinion Dynamics)

23rd November 2023 (Week 8)

Blaine Van Rensburg

(Adaptive Dynamics of Diverging Fitness Optima)

Freddie Jensen

(Nonlinear acoustics in a general 3D duct)

30th November 2023 (Week 9)

Christmas Social!
7th December 2023 (Week 10)

Jack Buckingham

(Exploration vs Exploitation: The Art of Acquisition Functions in Bayesian Optimisation)

Term 1 Abstracts

Week 3. Oscar Holroyd (Warwick HetSys CDT) - Linear Quadratic Regulation Control for Falling Liquid Films

We propose a new framework based on linear-quadratic regulation (LQR) for stabilising falling liquid films via injecting and removing fluid from the base at discrete locations. Our methodology bridges the gap between the reduced-order models accessible to the LQR controls and the full, nonlinear Navier-Stokes system describing the fluid flow. We find that not only is this technique successful, but that it works far beyond the anticipated range of validity of the reduced order models. The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems, and is also generalisable to other forms of actuation.

Week 3. Mark Lynch (Warwick MathSys CDT) - Nash Neural Networks: Inferring Utilities from Optimal Behaviours in Epidemics

We consider rational individuals socially distancing in an epidemic as a “differential game”, where different interacting individuals are each seeking to simultaneously maximise their own utility function by modifying their behaviour. Given a specific form of utility, one can solve the related constrained optimal control problem to derive optimal system dynamics that result in the maximal utilities for each individual. Machine Learning techniques can be used to solve the inverse problem, that of inferring some unknown utility function that is being optimised by given system dynamics. We seek to derive an ambitious machine learning framework that is able to infer this hidden utility assuming no knowledge of the form of this function. The main issue to address is how to perform the learning of such a function using solely measurable data (the state of the system at any given time), that is, without knowledge of the hidden variables required to define the underlying constrained optimization problem (i.e., the Lagrange multipliers). This talk presents variations of the forward problem, as well as discusses some possible methods for solving the inverse problem.

Week 4. Peter Lewin-Jones (Warwick HetSys CDT) - Computational Modelling of Drop-Drop Collisions in the Presence of Gas Microfilms: When Do Drops Bounce?

Collisions and impacts of drops are critical to numerous processes, including raindrop formation, inkjet printing, food manufacturing and spray cooling. We will see that with increasing speed, drop collisions undergo multiple transitions: from merging to bouncing and then back to merging, which were recently discovered to be surprisingly sensitive to the radius of the drops as well as the ambient gas pressure. To provide new insight into the physical mechanisms involved and as an important predictive tool, we have developed a novel, open-source computational model for the collision and impact of drops, using the finite element package oomph-lib. This uses a lubrication framework for the gas film, incorporating micro and nano-scale effects into an interfacial flow. Our simulations show strong agreement with experiments of impacts and collisions, but can also go beyond the regimes considered experimentally. We will show how our model enables us to explore the parameter space, probe different regimes of contact and gas film behaviour, with the aim of predicting the minimum film height and the critical impact speed for contact to occur. Beyond this, we can extend with novel lubrication models to consider Leidenfrost conditions and impacts of drops onto liquid films.

Week 6. Niall Rodgers (University of Birmingham) - Strong Connectivity and Influence In Real Directed Networks

I present recent work on the structure and dynamics of directed networks. Using the technique of Trophic Analysis which can be used to measure the hierarchical ordering and global directionality of a directed network. Firstly, we tackle the problem of predicting strong connectivity in directed networks. In many real, directed networks, the strongly connected component of nodes which are mutually reachable is small. This does not fit with current theory, based on random graphs. We find that strong connectivity depends crucially on the extent to which the network has an overall direction. Using percolation theory, we find the critical point separating weakly and strongly connected regimes, and confirm our results on many real-world networks, including ecological, neural, trade and social networks. We also show how Trophic Analysis can be used to study network influence by examining how the presence of a hierarchical ordering impacts structural and dynamic interpretations of network influence. And if time permits, I will highlight in-progress work on how network hierarchy can be induced in fitness based growing network models.

Week 6. Nathan Coombs (Warwick Maths CDT) - Colloidal Deposits from Evaporating Sessile Droplets

The coffee ring effect (CRE) refers to the accumulation of solute particles near the contact line of an evaporating sessile droplet and arises due to evaporation-induced capillary flows. Suppression of the CRE is desirable in many industrial applications which utilize colloidal deposition from an evaporating liquid, notably inkjet printing. It is therefore important that the influence of experimentally accessible physical parameters (ambient temperature, humidity, particle size/shape etc.) on the deposit morphology are well understood.Of critical importance in CRE modelling is the inclusion of particle “jamming”: when solute reaches a threshold volume fraction (approximately 64% for mono-disperse spherical particles), a transition towards a porous solid is observed. Jammed particles have a semi-crystalline structure and can exhibit both ordered and disordered phases depending on the local advection speed. Since jammed solute is incompressible, it also influences the shape of the drop’s surface, ultimately leading to a reversal in surface curvature and meniscus touchdown at the late stages of evaporation.Existing CRE models that include jamming are limited in scope to pre-touchdown dynamics and so are not able to describe the drying process in full. In this talk I will introduce a modelling framework that remedies this issue. Though much of the focus will be on axisymmetric drops, the model can be easily generalised to arbitrary drop shapes, allowing us to explore the influence of contact line curvature on the local CRE intensity.Time permitting, I will also look at the role of particle assembly at the drop surface and how it can be exploited to attenuate the CRE.

Week 7. Andrew Nugent (Warwick MathSys CDT) - Scaling in Opinion Dynamics

There is a rich literature on microscopic models for opinion dynamics; most of them fall into one of two categories - agent-based models or differential equation models. Agent-based models more closely mirror real life interactions: randomly chosen agents meet in pairs and may or may not change their opinions at that specific time. By constant, in differential equation models, individuals can interact constantly with the entire population and continually update their opinions. In this talk I describe how differential equation models can be obtained from agent-based models by simultaneously rescaling time and the distance by which agents update their opinions. Not only does this provide a rigorous justification of differential equation models, it also provides a route through which choices in these two modelling approaches can be compared.

Week 7. Margherita Botticelli (University of Birmingham) - How does ECM stiffness affect spheroid growth?

When a tumour develops in a primary organ, the cancer cells can invade and migrate collectively. This can lead to metastasis, which is the primary cause of death in cancer patients. Collective cell migration is strongly influenced by how cells interact with each other and with the surrounding environment, which includes the extracellular matrix (ECM). Mathematical models can be used to study and predict the dynamics of collective cell migration. A type of model commonly used in cell migration is a hybrid discrete-continuous model, which couples a discrete agent-based model for the cells with a continuum model for the microenvironment. The model we want to build is based on the underlying biology of in vitro experiments in 3D, with the aim to identify how the stiffness of the extracellular matrix affects the growth of a spheroid of cancer cells in 3D. We are developing the model in PhysiCell, an open-source agent-based modelling platform which implements an off-lattice, centre-based model for the cells, together with a PhysiCell ECM extension to model the matrix.

Week 8. Blaine Van Rensburg (University of Birmingham) - Adaptive Dynamics of Diverging Fitness Optima

We analyse a non-local parabolic integro-differential equation modelling the evolutionary dynamics of a phenotypically structured population in a changing environment. Such models arise in the study of species adapting to climate change, or cancer adapting to therapy. Our results concern the long-time behaviour, in the small mutation limit, of the model. The main novelty of our work is that the time- and trait-dependent per capita growth rate is characterised by having multiple (locally) optimal traits which shift at possibly different velocities. Our results imply that in populations undergoing competition in temporally changing environments, both the true optimal fitness and the required rate of adaptation for each of the diverging optimal traits contribute to the eventual dominance of one trait.

Week 8. Freddie Jensen (Warwick Maths CDT) - Nonlinear acoustics in a general 3D duct

The aim of this project is to reproduce and build upon the work of McTavish, J. P. (2019) in modelling nonlinear sound propagation in 3D waveguides, with an application to the harmonic series of brass instruments. Time-harmonic perturbations about a state of rest are considered up to second order (weak nonlinearity). The governing equations for weakly nonlinear sound are then projected onto a basis of straight, cylindrical duct modes before the consideration of the duct outlet physics. Eventually the impedance along the duct is calculated for various geometries and endpoint conditions; this can be informative as to the harmonic series of a brass instrument, for example.

Week 10. Jack Buckingham (Warwick MathSys CDT) - Exploration vs Exploitation: The Art of Acquisition Functions in Bayesian Optimisation

Have you ever spent ages tuning hyperparameters in an ML model? Or wondered how to find the best parameters for your fluids simulation? Or perhaps you’re familiar with the words ‘Bayesian’ and ‘optimisation’, and would love to know what the fuss is when you put them together! In this introductory seminar, we will cover how Gaussian processes and acquisition functions can be used to efficiently solve expensive (in some sense) optimisation problems. Like you, I also get bamboozled by too many equations during talks, so the focus will be on the intuition behind the ideas. It’s my aim that you’ll be able to follow along comfortably without any prior exposure to Bayesian optimisation.