Term 3

Term 3 Abstracts

Week 10. Mark Lynch (Warwick MathSys CDT) - Separating compartmental behaviour for non-selfish individuals in an epidemic

How can I care about others in an epidemic? Population behaviour in an epidemic can be viewed as a “differential game”: Rational, well informed individuals seek to maximise their own utility function by modifying their behaviour. This behaviour affects the course of the epidemic in an SIR compartmental model. Typically, this analysis involves a single behavioural class, or multiple behavioural classes depending on fixed attributes such as occupation or age. Less studied are models in which individuals change their behaviour depending on which compartment they are in. Given a utility, one can solve the related game theoretic problem to derive Nash equilibrium system dynamics. When the costs in the utility only pertain to the individual, infected individuals never socially distance because modifying their behaviour cannot improve their situation. We study the case where individuals care about other members of the population. Their behaviour will then change in order to protect others from incurring the cost of being infected and/or of having to socially distance. We quantify the degree to which individuals must care about the population in order to rationally target disease eradication through social distancing.

Week 10. Sharlin Utke (Warwick Statistics CDT) - Effectiveness of Spatial Relational Reasoning in Multi-Agent Reinforcement Learning

Relational reasoning has become an important concept in machine learning and has seen notable progress in its methods like graph neural networks, which highlight the value of capturing intricate relational patterns. We explore the application of spatial relational reasoning to multi-agent reinforcement learning, a domain marked by dynamic, ever-evolving interactions between agents and their environment. While relational reasoning has shown promise in single-agent reinforcement learning, its potential in the multi-agent landscape remains largely uncharted. We aim to bridge this gap by introducing an actor-critic architecture for centralized learning and decentralized execution that uses relational graph neural networks to imbue a spatial inductive bias.

Week 9. Sotirios Stamnas (Warwick MathSys CDT) - Towards DeepFake detection using anomaly detection techniques

Recent advances in generative models, such as generative adversarial networks (GANs) and diffusion models, have enabled the creation of highly realistic fake images and videos. In particular, the generation of fake human faces, or so-called deepfakes, has become increasingly popular, with such images/videos often being used for malicious purposes. Consequently, many deepfake detection techniques have been developed to counteract this threat. The most common approach to deepfake detection involves training a binary classifier model on both real and fake images, which performs well on deepfake generation methods seen during training but struggles against unseen manipulations. To address this limitation, we propose to formulate deepfake detection as a one-class anomaly detection problem. Specifically, we introduce a differential anomaly detection framework that uses only real images during training. Preliminary results show promising generalisation performance across different manipulation methods.

Week 9. Sebastian Dooley (Warwick HetSys CDT) - Data-driven equation discovery for liquid film flows thick and thin

Partial differential equation (PDE) discovery is an exciting alternative to the standard first principles-based methodologies regularly used in mathematical modelling, particularly in regimes outside the reach of traditional approaches. This talk explores the application of PDE discovery methods to a variety of PDEs. These include introductory PDEs such as the advection, heat and advection-diffusion equations, which are complemented by looking further to the complex equation environment of liquid film flows, with the aid of direct numerical simulation data. To begin with, we focus our attention on established thin film equations, outlining important derivation aspects to build analytical understanding into the data-driven process and provide reasons for interest in data-driven methods from the thin film fluid dynamics community. Subsequently, we outline the SINDy (sparse identification of nonlinear dynamics) equation discovery method, sharing results from its application to introductory PDEs. We then gently steer the developed framework into new regimes of interest, such as thick liquid film flows, in which classical physical understanding is lacking.

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

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

Week 4. Boris Andrews (University of Oxford) - High-order time-stepping schemes for high-order systems conserving multiple high-order invariants

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

Term 1

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.