arXiv subjects

Thomas E. Woolley

Publications and source records attributed to Thomas E. Woolley.

At least 19 recordsLinked to original sources

arXiv2026

Patterns in Time and Space from a Single Morphogen via Nonlinear Layering

Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This $N$-layer model is analysed via a thin-domain limit, which reduces to an $N$-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.

nlin.PS↗

arXiv2025

Reducing Size Bias in Epidemic Network Modelling

Epidemiological models help policymakers mitigate disease spread by predicting transmission metrics based on disease dynamics and contact networks. Calibrating these models requires representative network sampling. We investigate the Random Walk (RW) and Metropolis-Hastings Random Walk (MHRW) algorithms for three network types: Erdős-Rényi (ER), Small-world (SW), and Scale-free (SF). Disease transmission is simulated using a stochastic susceptible-infected-recovered (SIR) framework. For ER and SW networks, RW overestimates infected individuals and secondary infections by $25\%$ due to size bias, favouring highly connected nodes. MHRW, though more computationally intensive, reduces size bias and provides more representative samples. For time-to-infection, both algorithms provide representative estimates. However, neither algorithm samples SF networks representatively, exhibiting significant variability. Furthermore, removing duplicate sample nodes reduces MHRW's accuracy across three network types. We apply both algorithms to a cattle movement network of 46,512 farms combining ER, SW, and SF features. RW overestimates infected farms by about $100\%$ and secondary infections by over $900\%$, reflecting significant size bias, while MHRW estimates align within $1\%$ of the cattle network values. RW underestimates time-to-infection by about $40\%$, while MHRW overestimates it by $10\%$. Accuracy, again, deteriorates when duplicates nodes are removed. Our findings guide algorithm selection and intervention strategies based on network structure and disease severity; RW's conservative estimates suit high-mortality, fast-spreading epidemics, while MHRW enables more precise interventions for slower epidemics.

q-bio.PE↗

arXiv2024

3D Topological Modeling and Multi-Agent Movement Simulation for Viral Infection Risk Analysis

In this paper, a method to study how the design of indoor spaces and people's movement within them affect disease spread is proposed by integrating computer-aided modeling, multi-agent movement simulation, and airborne viral transmission modeling. Topologicpy spatial design and analysis software is used to model indoor environments, connect spaces, and construct a navigation graph. Pathways for agents, each with unique characteristics such as walking speed, infection status, and activities, are computed using this graph. Agents follow a schedule of events with specific locations and times. The software calculates "time-to-leave" based on walking speed and event start times, and agents are moved along the shortest path within the navigation graph, accurately considering obstacles, doorways, and walls. Precise distance calculations between agents are enabled by this setup. Viral aerosol concentration is then computed and visualized using a reaction-diffusion equation, and each agent's infection risk is determined with an extension of the Wells-Riley ansatz. Infection risk simulations are improved by this spatio-temporal and topological approach, incorporating realistic human behavior and spatial dynamics. The resulting software is designed as a rapid decision-support tool for policymakers, facility managers, stakeholders, architects, and engineers to mitigate disease spread in existing buildings and inform the design of new ones. The software's effectiveness is demonstrated through a comparative analysis of cellular and open commercial office plan layouts.

cs.MA↗

arXiv2024

VIRIS: Simulating indoor airborne transmission combining architectural design and people movement

A Viral Infection Risk Indoor Simulator (VIRIS) has been developed to quickly assess and compare mitigations for airborne disease spread. This agent-based simulator combines people movement in an indoor space, viral transmission modelling and detailed architectural design, and it is powered by topologicpy, an open-source Python library. VIRIS generates very fast predictions of the viral concentration and the spatiotemporal infection risk for individuals as they move through a given space. The simulator is validated with data from a courtroom superspreader event. A sensitivity study for unknown parameter values is also performed. We compare several non-pharmaceutical interventions (NPIs) issued in UK government guidance, for two indoor settings: a care home and a supermarket. Additionally, we have developed the user-friendly VIRIS web app that allows quick exploration of diverse scenarios of interest and visualisation, allowing policymakers, architects and space managers to easily design or assess infection risk in an indoor space.

cs.CY↗

arXiv2022

Differential Dynamic Microscopy can be applied to Differential Interference Contrast images despite shadowing effects

During In Vitro Fertilisation (IVF), combining time-lapse contrast microscopy with suitable image processing techniques could facilitate non-invasive classification of oocyte health, driving oocyte selection to maximise success rates. One image processing technique, Differential Dynamic Microscopy (DDM), has been used in a variety of microscopy settings including bright field, dark field and fluorescence. However, in some applications, effects stemming from the choice of microscopy may invalidate underlying assumptions of DDM. Here, we study the DDM analysis of differential interference contrast (DIC) microscopy movies. DIC exhibits a characteristic shadowing effect that gives the illusion of a 3D appearance, which we show causes deformation of the output DDM matrix. We present a mathematical description of this deformation, and conclude that, when studying isotropic motion, no account of the DIC shadow needs to be considered during DDM analysis. We validate our conclusions with simulated particle data, and with DIC images of colloidal dispersions. Although this conclusion does not generally extend to anisotropic motion, we demonstrate that for directed advection and diffusion behaviour, parameter fitting invariance still holds. These results validate the current practice of applying DDM to DIC, and are a foundation for further exploration of DDM in other phase-contrast image datasets.

cond-mat.stat-mech↗

arXiv2022

Modelling polarity-driven laminar patterns in bilayer tissues with mixed signalling mechanisms

Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localised signalling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e. contact-based or diffusive signalling, which generates both a spatial and temporal dependence on cell behaviours. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyse the role of signalling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layer-wise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for existence, uniqueness and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue.

math.DS↗

arXiv2021

Coupling Bayesian theory and static acoustic detector data to model bat motion and locate roosts

We propose a novel approach for modelling bat motion dynamics and use it to predict roost locations using data from static acoustic detectors. Specifically, radio tracking studies of Greater Horseshoe bats demonstrate that bat movement can be split into two phases: dispersion and return. Dispersion is easily understood and can be modelled as simple random motion. The return phase is much more complex, as it requires intelligent directed motion and results in all agents returning home in a stereotypical manner. Critically, combining reaction-diffusion theory and domain shrinking we deterministically and stochastically model a ``leap-frogging'' motion, which fits favourably with the observed tracking data.

q-bio.QM↗

arXiv2021

Algebraic framework for determining laminar pattern bifurcations by lateral-inhibition in 2D and 3D bilayer geometries

Fine-grain patterns produced by juxtacrine signalling, have been studied using static monolayers as cellular domains. Unfortunately, analytical results are restricted to a few cells due to the algebraic complexity of nonlinear dynamical systems. Motivated by concentric patterning of Notch expression observed in the mammary gland, we combine concepts from graph and control theory to represent cellular connectivity. The resulting theoretical framework allows us to exploit the symmetry of multicellular bilayer structures in 2D and 3D, thereby deriving analytical conditions that drive the dynamical system to form laminar patterns consistent with the formulation of cell polarity. Critically, the conditions are independent of the precise dynamical details, thus the framework allows for the utmost generality in understanding the influence of cellular geometry on patterning in lateral-inhibition systems. Applying the analytic conditions to mammary organoids suggests that intense cell signalling polarity is required for the maintenance of stratified cell-types within a static bilayer using a lateral-inhibition mechanism. Furthermore, by employing 2D and 3D cell-based models, we highlight that the cellular polarity conditions derived from static domains have the capacity to generate laminar patterning in dynamic environments. However, they are insufficient for the maintenance of patterning when subjected to substantial morphological perturbations. In agreement with the mathematical implications of strict signalling polarity induced on the cells, we propose an adhesion dependent Notch-Delta biological process which has the potential to initiate bilayer stratification in a developing mammary organoid.

physics.bio-ph↗

arXiv2021

A stochastic framework for atomistic fracture

We present a stochastic modeling framework for atomistic propagation of a Mode I surface crack, with atoms interacting according to the Lennard-Jones interatomic potential at zero temperature. Specifically, we invoke the Cauchy-Born rule and the maximum entropy principle to infer probability distributions for the parameters of the interatomic potential. We then study how uncertainties in the parameters propagate to the quantities of interest relevant to crack propagation, namely, the critical stress intensity factor and the lattice trapping range. For our numerical investigation, we rely on an automated version of the so-called numerical-continuation enhanced flexible boundary (NCFlex) algorithm.

cond-mat.mtrl-sci↗

arXiv2020

Turing Patterning in Stratified Domains

Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarisation. We develop a modelling framework for bi-layer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally-relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

nlin.PS↗

arXiv2020

Likely cavitation and radial motion of stochastic elastic spheres

The cavitation of solid elastic spheres is a classical problem of continuum mechanics. Here, we study this problem within the context of "stochastic elasticity" where the constitutive parameters are characterised by probability density functions. We consider homogeneous spheres of stochastic neo-Hookean material, composites with two concentric stochastic neo-Hookean phases, and inhomogeneous spheres of locally neo-Hookean material with a radially varying parameter. In all cases, we show that the material at the centre determines the critical load at which a spherical cavity forms there. However, while under dead-load traction, a supercritical bifurcation, with stable cavitation, is obtained in a static sphere of stochastic neo-Hookean material, for the composite and radially inhomogeneous spheres, a subcritical bifurcation, with snap cavitation, is also possible. For the dynamic spheres, oscillatory motions are produced under suitable dead-load traction, such that a cavity forms and expands to a maximum radius, then collapses again to zero periodically, but not under impulse traction. Under a surface impulse, a subcritical bifurcation is found in a static sphere of stochastic neo-Hookean material and also in an inhomogeneous sphere, whereas in composite spheres, supercritical bifurcations can occur as well. Given the non-deterministic material parameters, the results can be characterised in terms of probability distributions.

physics.class-ph↗

arXiv2020

Likely cavitation and radial motion of stochastic elastic spheres 2: Impulse driven

Cavitation in solids can be caused by tensile dead-load traction or impulse traction. The two different types of boundary conditions lead to different static and dynamic solutions. In addition, if the material is stochastic, i.e., the model parameters are represented by probability distributions, the expected behaviour is more complicated to describe. Here, following the first instalment of this work, we examine the static and dynamic cavitation of a stochastic material under a uniform tensile impulse traction in different spherical geometries. We find that the critical load at which a cavity forms at the centre of the sphere is the same as for the homogeneous sphere composed entirely of the material found at its centre, while the post-cavitation radial motion is non-oscillatory. However, there are some important differences in the nonlinear elastic responses. Specifically, subcritical bifurcation, with snap cavitation, is obtained in a static sphere of stochastic neo-Hookean material and also in a radially inhomogeneous sphere, whereas for composite spheres, a supercritical bifurcation, with stable cavitation, is possible as well. Given the non-deterministic material parameters, the results are characterised by probability distributions.

physics.class-ph↗

arXiv2019

From One Pattern into Another: Analysis of Turing Patterns in Heterogeneous Domains via WKBJ

Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.

nlin.PS↗

arXiv2019

Likely cavitation in stochastic elasticity

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as `likely cavitation'. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.

cond-mat.soft↗

arXiv2019

Likely oscillatory motions of stochastic hyperelastic solids

Stochastic homogeneous hyperelastic solids are characterised by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney-Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterised. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as `likely oscillatory motions'.

physics.class-ph↗

arXiv2019

Likely equilibria of stochastic hyperelastic spherical shells and tubes

In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limit-point, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.

cond-mat.soft↗

arXiv2019

Likely chirality of stochastic anisotropic hyperelastic tubes

When an elastic tube reinforced with helical fibres is inflated, its ends rotate. In large deformations, the amount and chirality of rotation is highly non-trivial, as it depends on the choice of strain-energy density and the arrangements of the fibres. For anisotropic hyperelastic tubes where the material parameters are single-valued constants, the problem has been satisfactorily addressed. However, in many systems, the material parameters are not precisely known, and it is therefore more appropriate to treat them as random variables. The problem is then to understand chirality in a probabilistic framework. Here, we develop a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially-independent random variables defined by probability density functions. The tube is subjected to uniform dead loading consisting of internal pressure, axial tension and torque. Assuming that the tube wall is thin and that the resulting deformation is the combined inflation, extension and torsion from the reference circular cylindrical configuration to a deformed circular cylindrical state, we derive the probabilities of radial expansion or contraction, and of right-handed or left-handed torsion. We refer to these stochastic behaviours as `likely inflation' and `likely chirality', respectively.

physics.class-ph↗

arXiv2019

Likely equilibria of the stochastic Rivlin cube

The problem of the Rivlin cube is to determine the stability of all homogeneous equilibria of an isotropic incompressible hyperelastic body under equitriaxial dead loads. Here, we consider the stochastic version of this problem where the elastic parameters are random variables following standard probability laws. Uncertainties in these parameters may arise, for example, from inherent data variation between different batches of homogeneous samples, or from different experimental tests. As for the deterministic elastic problem, we consider the following questions: what are the likely equilibria and how does their stability depend on the material constitutive law? In addition, for the stochastic model, the problem is to derive the probability distribution of deformations, given the variability of the parameters.

math-ph↗