arXiv subjects

Christian A. Yates

Publications and source records attributed to Christian A. Yates.

At least 19 recordsLinked to original sources

arXiv2026

Equation Learning for multiscale models of infectious diseases

Tuberculosis (TB) is an airborne disease caused by the pathogen Mycobacterium tuberculosis. In 2023, according to the World Health Organization, it ''probably'' replaced COVID-19 as the leading cause of death from an infectious agent globally; in the nineteenth century, one in seven of all humans deaths were as a result of tuberculosis. More than 10 million people are diagnosed with TB every year. The majority of cases in adults occur in males (62.5% of all global adult cases in 2023, compared to 37.5% in females). The main reasons for males suffering from a higher burden of global TB cases, compared to females, is likely to be a combination of within-host factors, such as differences in immune response, and population-scale factors, such as likelihood of completing treatment. To investigate the impact different scales have in determining this higher TB burden in males, we have developed a gender/sex-stratified multiscale framework. We have learnt ordinary differential equations (ODEs) to capture the average output of an agent-based within-host model, and used the resulting equations to describe the within-host scales of the multiscale framework. We evolve the population demographics at the between-host scale using ODEs, and link the scales with stochastic coupling functions. We have considered counterfactual scenarios to elucidate the impact of sex and gender on the infectious disease dynamics of TB. This paper is intended to provide a proof-of-concept for the development and implementation of the presented multiscale framework.

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arXiv2026

Multi-stage volume exclusion models for cell proliferation

Cell proliferation and cell movement are fundamentally stochastic processes which lead to variability in the growth and spatial structure of cell populations in many biological settings, such as cell invasion, wound healing, and tumour growth. We develop stochastic, on-lattice agent-based models (ABMs) which incorporate volume exclusion, random movement, and multi-stage representations of the cell cycle. The multi-stage framework enables a more realistic representation of true cell cycle time distributions. We also introduce a novel form of myopic behaviour, in which cells sense their local environment when attempting to proliferate. For each ABM, we derive a corresponding continuum partial differential equation (PDE) description under the mean-field approximation. Using numerical simulations, we investigate how different proliferation mechanisms influence population-level dynamics in both the discrete and continuum models. In particular, we consider biologically relevant contexts of growth-to-confluence assays (using uniform initial conditions) and travelling wave behaviour associated with cell invasion. We examine how the PDE solutions compare with the behaviour of the corresponding ABMs averaged over many realisations.

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arXiv2026

A model of tuberculosis progression using CompuCell3D

Tuberculosis (TB) is an airborne disease caused by the bacterium Mycobacterium tuberculosis (M. tb). Prior to the COVID-19 pandemic, TB was the leading cause of death from an infectious agent globally. However, most people exposed to M. tb do not develop active TB and go on to display symptoms. Instead, in the majority of cases, the bacteria are contained within a granuloma (an aggregation of immune cells) without being eliminated; this is called latent TB. The spatial organisation of the bacteria and immune cells is important in determining whether an individual exposed to M. tb will develop latent or active TB. In this paper, we present a multi-cell, multiscale model of TB progression to investigate the importance of the spatial organisation. This is a novel TB within-host dynamics modelling framework, having been developed using CompuCell3D (CC3D), an open-source computer software used for simulating cellular biological processes both within and between cells. We used this model to compare the generated results with those from a previously developed within-host infectious disease model. We found that, although the results of our CC3D model mostly agree qualitatively with those from the previously developed model, there are quantitative differences. Additionally, we conducted a robustness analysis of key model parameters from the CC3D model to determine their importance to the CC3D model output, using a methodology specifically designed for agent-based models. The model output appears to be robust in response to perturbations in parameters controlling chemotactic movement, but less so in response to perturbations in parameters controlling persistence of movement in cells, cell adhesion and volume constraints. This work compares our CC3D model of TB progression with another agent-based modelling approach to the same problem.

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arXiv2026

An agent-based modelling approach to investigate the impact of gender on tuberculosis transmission in Uganda

Tuberculosis (TB) is an airborne disease caused by the pathogen Mycobacterium tuberculosis. In 2023, it returned to being the leading cause of death from an infectious agent globally, replacing COVID-19; in the nineteenth century, one in seven of all humans died of tuberculosis. More than 10 million people are diagnosed with TB every year. The majority of cases in adults occur in males (62.5% of all global adult cases in 2023, compared to 37.5% in females). The main reasons for males suffering from a higher burden of global TB cases, compared to females, may be in large part due to population-scale factors, such as employment type, the quantity and type of social contacts they make, and their health-seeking behaviours (e.g. differences in diagnostic and treatment delays between genders). To investigate which population-scale factors are most important in determining this higher TB burden in males, we have developed an age- and gender-stratified, spatially heterogeneous epidemiological agent-based model. We have focused specifically on Kampala, the capital of Uganda, which is a high-burden TB country. We considered counterfactual scenarios to elucidate the impact of gender on the epidemiology of TB. Setting disease progression parameters equal between the genders leads to a reduction in both male-to-female case ratio and total case numbers.

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arXiv2025

The Spatial Regime Conversion Method

We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction-diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction-diffusion master equation (RDME) representation in regions of low concentration and continuum partial differential equations (PDEs) where concentrations are high, dynamically switching based on local thresholds. This enables efficient and accurate simulation of systems in which stochasticity plays a key role but is not required uniformly across the domain. We specify the full mathematical formulation of the SRCM, including conversion reactions, hybrid kinetic rules, and consistent numerical updates. The method is validated across several one-dimensional test systems, including simple diffusion from a region of high concentration, the formation of a morphogen gradient, and the propagation of FKPP travelling waves. Results show that the SRCM captures key stochastic features while offering substantial gains in computational efficiency over fully stochastic models.

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arXiv2024

Modelling adhesion in stochastic and mean-field models of cell migration

Adhesion between cells plays an important role in many biological processes such as tissue morphogenesis and homeostasis, wound healing and cancer cell metastasis. From a mathematical perspective, adhesion between multiple cell types has been previously analysed using discrete and continuum models including the Cellular Potts models and partial differential equations (PDEs). While these models can represent certain biological situations well, Cellular Potts models can be computationally expensive and continuum models only capture the macroscopic behaviour of a population of cells, ignoring stochasticity and the discrete nature of cell dynamics. Cellular automaton models allow us to address these problems and can be used for a wide variety of biological systems. In this paper, we consider a cellular automaton approach and develop an on-lattice agent-based model (ABM) for cell migration and adhesion in a population composed of two cell types. By deriving and comparing the corresponding PDEs to the ABM, we demonstrate that cell aggregation and cell sorting are not possible in the PDE model. Therefore, we propose a set of stochastic mean equations (SMEs) which better capture the behaviour of the ABM in one and two dimensions.

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arXiv2023

Swapping in lattice-based cell migration models

Cell migration is frequently modelled using on-lattice agent-based models (ABMs) that employ the excluded volume interaction. However, cells are also capable of exhibiting more complex cell-cell interactions, such as adhesion, repulsion, pulling, pushing and swapping. Although the first four of these have already been incorporated into mathematical models for cell migration, swapping has not been well studied in this context. In this paper, we develop an ABM for cell movement in which an active agent can 'swap' its position with another agent in its neighbourhood with a given swapping probability. We consider a two-species system for which we derive the corresponding macroscopic model and compare it with the average behaviour of the ABM. We see good agreement between the ABM and the macroscopic density. We also analyse the movement of agents at an individual level in the single-species as well as two-species scenarios to quantify the effects of swapping on an agent's motility.

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arXiv2023

Stochastic drift in discrete waves of non-locally interacting-particles

In this paper, we investigate a generalised model of $N$ particles undergoing second-order non-local interactions on a lattice. Our results have applications across many research areas, including the modelling of migration, information dynamics and Muller's ratchet -- the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as $(\log N)^{-2}$ and $(\log N)^{-3}$, respectively -- much slower than the usual $N^{-1/2}$ factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

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arXiv2022

An equivalence framework for an age-structured multi-stage representation of the cell cycle

We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multi-stage Markov process for approximating cell cycle time distributions. We further demonstrate that the resulting mean behaviour is equivalent, over large timescales, to the classical McKendrick-von Foerster integro-partial differential equation. We conclude by extending this framework to a spatial context, facilitating the modelling of travelling wave phenomena and cell-mediated pattern formation. More generally, this methodology may be extended to myriad reaction-diffusion processes for which the age of individuals is relevant to the dynamics.

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arXiv2022

Critical weaknesses in shielding strategies for COVID-19

The COVID-19 pandemic, caused by the coronavirus SARS-CoV-2, has led to a wide range of non-pharmaceutical interventions being implemented around the world to curb transmission. However, the economic and social costs of some of these measures, especially lockdowns, has been high. An alternative and widely discussed public health strategy for the COVID-19 pandemic would have been to 'shield' those most vulnerable to COVID-19 (minimising their contacts with others), while allowing infection to spread among lower risk individuals with the aim of reaching herd immunity. Here we retrospectively explore the effectiveness of this strategy using a stochastic SEIR framework, showing that even under the unrealistic assumption of perfect shielding, hospitals would have been rapidly overwhelmed with many avoidable deaths among lower risk individuals. Crucially, even a small (20%) reduction in the effectiveness of shielding would have likely led to a large increase (>150%) in the number of deaths compared to perfect shielding. Our findings demonstrate that shielding the vulnerable while allowing infections to spread among the wider population would not have been a viable public health strategy for COVID-19 and is unlikely to be effective for future pandemics.

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arXiv2020

Incorporating domain growth into hybrid methods for reaction-diffusion systems

Reaction--diffusion mechanism are a robust paradigm that can be used to represent many biological and physical phenomena over multiple spatial scales. Applications include intracellular dynamics, the migration of cells and the patterns formed by vegetation in semi-arid landscapes. Moreover, domain growth is an important process for embryonic growth and wound healing. There are many numerical modelling frameworks capable of simulating such systems on growing domains, however each of these may be well suited to different spatial scales and particle numbers. Recently, spatially extended hybrid methods on static domains have been produced in order to bridge the gap between these different modelling paradigms in order to represent multiscale phenomena. However, such methods have not been developed with domain growth in mind. In this paper, we develop three hybrid methods on growing domains, extending three of the prominent static domain hybrid methods. We also provide detailed algorithms to allow others to employ them. We demonstrate that the methods are able to accurately model three representative reaction-diffusion systems accurately and without bias.

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arXiv2020

The blending region hybrid framework for the simulation of stochastic reaction-diffusion processes

The simulation of stochastic reaction-diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction-diffusion at distinct spatial scales, we allow them to overlap in a "blending region". Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary "blending functions" which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction-diffusion scenarios.

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arXiv2020

Efficiently simulating discrete-state models with binary decision trees

Stochastic simulation algorithms (SSAs) are widely used to numerically investigate the properties of stochastic, discrete-state models. The Gillespie Direct Method is the pre-eminent SSA, and is widely used to generate sample paths of so-called agent-based or individual-based models. However, the simplicity of the Gillespie Direct Method often renders it impractical where large-scale models are to be analysed in detail. In this work, we carefully modify the Gillespie Direct Method so that it uses a customised binary decision tree to trace out sample paths of the model of interest. We show that a decision tree can be constructed to exploit the specific features of the chosen model. Specifically, the events that underpin the model are placed in carefully-chosen leaves of the decision tree in order to minimise the work required to keep the tree up-to-date. The computational efficencies that we realise can provide the apparatus necessary for the investigation of large-scale, discrete-state models that would otherwise be intractable. Two case studies are presented to demonstrate the efficiency of the method.

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arXiv2019

Unbiased on-lattice domain growth

Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously well-used method in order to incorporate domain growth into on-lattice reaction-diffusion models causes a build up of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here, we present a new method which addresses this unphysical build up of particles at the boundaries, and demonstrate that it is accurate even for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.

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arXiv2019

Pulling in models of cell migration

There are numerous scenarios in which populations of cells migrate in crowded environments. Typical examples include wound healing, cancer growth and embryo development. In these crowded environments cells are able to interact with each other in a variety of ways. These include excluded volume interactions, adhesion, repulsion, cell signalling, pushing and pulling. One popular way to understand the behaviour of a group of interacting cells is through an agent-based model (ABM). A typical aim of modellers using such represtations is to elucidate how the microscopic interactions at the cell-level impact on the macroscopic behaviour of the population. The complex cell-cell interactions listed above have also been incorporated into such models; all apart from cell-cell pulling. In this paper we consider this under-represented cell-cell interaction, in which an active cell is able to `pull' a nearby neighbour as it moves. We incorporate a variety of potential cell-cell pulling mechanisms into on- and off-lattice agent-based volume exclusion models of cell movement. For each of these agent-based models we derive a continuum partial differential equation which describes the evolution of the cells at a population-level. We study the agreement between the ABMs and the continuum, population-based models, and compare and contrast a range of ABMs (accounting for the different pulling mechanisms) with each other. We find generally good agreement between the ABMs and the corresponding continuum models that worsens as the agent-based models become more complex. Interestingly, we observe that the partial differential equations that we derive differ significantly, depending on whether they were derived from on- or off-lattice ABMs of pulling. This hints that it is important employ the appropriate ABM when representing pulling cell-cell interactions.

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arXiv2018

Robustly simulating biochemical reaction kinetics using multi-level Monte Carlo approaches

In this work, we consider the problem of estimating summary statistics to characterise biochemical reaction networks of interest. Such networks are often described using the framework of the Chemical Master Equation (CME). For physically-realistic models, the CME is widely considered to be analytically intractable. A variety of Monte Carlo algorithms have therefore been developed to explore the dynamics of such networks empirically. Amongst them is the multi-level method, which uses estimates from multiple ensembles of sample paths of different accuracies to estimate a summary statistic of interest. {In this work, we develop the multi-level method in two directions: (1) to increase the robustness, reliability and performance of the multi-level method, we implement an improved variance reduction method for generating the sample paths of each ensemble; and (2) to improve computational performance, we demonstrate the successful use of a different mechanism for choosing which ensembles should be included in the multi-level algorithm.

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arXiv2018

Stochastic and deterministic modelling of cell migration

Mathematical models are vital interpretive and predictive tools used to assist in the understanding of cell migration. There are typically two approaches to modelling cell migration: either micro-scale, discrete or macro-scale, continuum. The discrete approach, using agent-based models (ABMs), is typically stochastic and accounts for properties at the cell-scale. Conversely, the continuum approach, in which cell density is often modelled as a system of deterministic partial differential equations (PDEs), provides a global description of the migration at the population level. Deterministic models have the advantage that they are generally more amenable to mathematical analysis. They can lead to significant insights for situations in which the system comprises a large number of cells, at which point simulating a stochastic ABM becomes computationally expensive. However, finding an appropriate continuum model to describe the collective behaviour of a system of individual cells can be a difficult task. Deterministic models are often specified on a phenomenological basis, which reduces their predictive power. Stochastic ABMs have advantages over their deterministic continuum counterparts. In particular, ABMs can represent individual-level behaviours (such as cell proliferation and cell-cell interaction) appropriately and are amenable to direct parameterisation using experimental data. It is essential, therefore, to establish direct connections between stochastic micro-scale behaviours and deterministic macro-scale dynamics. In this Chapter we describe how, in some situations, these two distinct modelling approaches can be unified into a discrete-continuum equivalence framework. We provide an overview of some of the more recent advances in this field and we point out some of the relevant questions that remain unanswered.

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arXiv2018

The invasion speed of cell migration models with realistic cell cycle time distributions

Cell proliferation is typically incorporated into stochastic mathematical models of cell migration by assuming that cell divisions occur after an exponentially distributed waiting time. Experimental observations, however, show that this assumption is often far from the real cell cycle time distribution (CCTD). Recent studies have suggested an alternative approach to modelling cell proliferation based on a multi-stage representation of the CCTD. In order to validate and parametrise these models, it is important to connect them to experimentally measurable quantities. In this paper we investigate the connection between the CCTD and the speed of the collective invasion. We first state a result for a general CCTD, which allows the computation of the invasion speed using the Laplace transform of the CCTD. We use this to deduce the range of speeds for the general case. We then focus on the more realistic case of multi-stage models, using both a stochastic agent-based model and a set of reaction-diffusion equations for the cells' average density. By studying the corresponding travelling wave solutions, we obtain an analytical expression for the speed of invasion for a general N-stage model with identical transition rates, in which case the resulting cell cycle times are Erlang distributed. We show that, for a general N-stage model, the Erlang distribution and the exponential distribution lead to the minimum and maximum invasion speed, respectively. This result allows us to determine the range of possible invasion speeds in terms of the average proliferation time for any multi-stage model.

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