Accurate and efficient discretizations for stochastic models providing near agent-based spatial resolution at low computational cost

Understanding how cells proliferate, migrate and die in various environments is essential in determining how organisms develop and repair themselves. Continuum mathematical models, such as the logistic equation and the Fisher–Kolmogorov equation, can describe the global characteristics observed in commonly used cell biology assays, such as proliferation and scratch assays. However, these continuum models do not account for single-cell-level mechanics observed in high-throughput experiments. Mathematical modelling frameworks that represent individual cells, often called agent-based models, can successfully describe key single-cell-level features of these assays but are computationally infeasible when dealing with large populations. In this work, we propose an agent-based model with crowding effects that is computationally efficient and matches the logistic and Fisher–Kolmogorov equations in parameter regimes relevant to proliferation and scratch assays, respectively. This stochastic agent-based model allows multiple agents to be contained within compartments on an underlying lattice, thereby reducing the computational storage compared to existing agent-based models that allow one agent per site only. We propose a systematic method to determine a suitable compartment size. Implementing this compartment-based model with this compartment size provides a balance between computational storage, local resolution of agent behaviour and agreement with classical continuum descriptions

Asymptotic analysis of a multiphase drying model motivated by coffee bean roasting

Recent modelling of coffee bean roasting suggests that in the early stages of roasting, within each coffee bean, there are two emergent regions: a dried outer region and a saturated interior region. The two regions are separated by a transition layer (or, drying front). In this paper, we consider the asymptotic analysis of a recent multiphase model in order to gain a better understanding of its salient features. The model consists of a PDE system governing the thermal, moisture, and gas pressure profiles throughout the interior of the bean. By obtaining asymptotic expansions for these quantities in relevant limits of the physical parameters, we are able to determine the qualitative behaviour of the outer and interior regions, as well as the dynamics of the drying front. Although a number of simplifications and scalings are used, we take care not to discard aspects of the model which are fundamental to the roasting process. Indeed, we find that for all of the asymptotic limits considered, our approximate solutions faithfully reproduce the qualitative features evident from numerical simulations of the full model. From these asymptotic results, we have a better qualitative understanding of the drying front (which is hard to resolve precisely in numerical simulations), and hence, of the various mechanisms at play including heating, evaporation, and pressure changes. This qualitative understanding of solutions to the multiphase model is essential when creating more involved models that incorporate chemical reactions and solid mechanics effects

Modelling structural deformations in a roasting coffee bean

Macroscale deformations in a roasting coffee bean are important mechanisms in determining flavour development, moisture loss, and consistency of the bean. In this paper, we model the stresses and strains in the cellulose structure of a roasting coffee bean via temperature-dependent poroviscoelastic constitutive equations. This model accounts for the deformations that are created and controlled by the moisture content, temperature, and gas pressure inside of the roasting coffee bean. The model combines previously derived multiphase heat and mass transfer models for roasting coffee beans with these poroviscoelastic equations, to determine when and where macroscale deformations of the cellular matrix are likely to occur. By exploiting reasonable asymptotic reductions of the poroviscoelastic equations, we find that a large surge of stress is produced in the interior of a coffee bean. We determine that this build-up of stress is due to the viscoelastic interior of the bean being contained by a rigid elastic exterior and unable to expand. Our theoretical results suggest directions for possible improvement in standard industrial coffee roasting techniques, which may allow the macroscale deformations of the cellular matrix to be controlled and thereby improve properties such as flavour, moisture loss, and consistency of the final product

Understanding viral video dynamics through an epidemic modelling approach

Motivated by the hypothesis that the spread of viral videos is analogous to the spread of a disease epidemic, we formulate a novel susceptible–exposed–infected–recovered–susceptible (SEIRS) delay differential equation epidemic model to describe the popularity evolution of viral videos. Our models incorporate time-delay, in order to accurately describe the virtual contact process between individuals and the temporary immunity of individuals to videos after they have grown tired of watching them. We validate our models by fitting model parameters to viewing data from YouTube music videos, in order to demonstrate that the model solutions accurately reproduce real behaviour seen in this data. We use an SEIR model to describe the initial growth and decline of daily views, and an SEIRS model to describe the long term behaviour of the popularity of music videos. We also analyse the decay rates in the daily views of videos, determining whether they follow a power law or exponential distribution. Although we focus on viral videos, the modelling approach may be used to understand dynamics emergent from other areas of science which aim to describe consumer behaviour

Understanding viral video dynamics through an epidemic modelling approach

Motivated by the hypothesis that the spread of viral videos is analogous to the spread of a disease epidemic, we formulate a novel susceptible–exposed–infected–recovered–susceptible (SEIRS) delay differential equation epidemic model to describe the popularity evolution of viral videos. Our models incorporate time-delay, in order to accurately describe the virtual contact process between individuals and the temporary immunity of individuals to videos after they have grown tired of watching them. We validate our models by fitting model parameters to viewing data from YouTube music videos, in order to demonstrate that the model solutions accurately reproduce real behaviour seen in this data. We use an SEIR model to describe the initial growth and decline of daily views, and an SEIRS model to describe the long term behaviour of the popularity of music videos. We also analyse the decay rates in the daily views of videos, determining whether they follow a power law or exponential distribution. Although we focus on viral videos, the modelling approach may be used to understand dynamics emergent from other areas of science which aim to describe consumer behaviour

Emergent structures in reaction-advection-diffusion systems on a sphere

We demonstrate novel effects due to the addition of advection into a two-species reaction-diffusion system on the sphere. We find that advection introduces emergent behavior due to an interplay of the traditional Turing patterning mechanisms with the compact geometry of the sphere. Unidirectional advection within the Turing space of the reaction-diffusion system causes patterns to be generated at one point of the sphere, and transported to the antipodal point where they are destroyed. We illustrate these effects numerically, and deduce conditions for Turing instabilities on local projections to understand the mechanisms behind these behaviors. We compare this behavior to planar advection which is shown to only transport patterns across the domain. Analogous transport results seem to hold for the sphere under azimuthal transport or away from the antipodal points in unidirectional flow regimes

Emergent structures in reaction-advection-diffusion systems on a sphere

We demonstrate novel effects due to the addition of advection into a two-species reaction-diffusion system on the sphere. We find that advection introduces emergent behavior due to an interplay of the traditional Turing patterning mechanisms with the compact geometry of the sphere. Unidirectional advection within the Turing space of the reaction-diffusion system causes patterns to be generated at one point of the sphere, and transported to the antipodal point where they are destroyed. We illustrate these effects numerically, and deduce conditions for Turing instabilities on local projections to understand the mechanisms behind these behaviors. We compare this behavior to planar advection which is shown to only transport patterns across the domain. Analogous transport results seem to hold for the sphere under azimuthal transport or away from the antipodal points in unidirectional flow regimes

Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays

We examine the role of the travel time of a predator along a spatial network on predator-prey population interactions, where the predator is able to partially or fully sustain itself on a resource subsidy. The impact of access to food resources on the stability and behaviour of the predator-prey-subsidy system is investigated, with a primary focus on how incorporating travel time changes the dynamics. The population interactions are modelled by a system of delay differential equations, where travel time is incorporated as discrete delay in the network diffusion term in order to model time taken to migrate between spatial regions. The model is motivated by the Arctic ecosystem, where the Arctic fox consumes both hunted lemming and scavenged seal carcass. The fox travels out on sea ice, in addition to quadrennially migrating over substantial distances. We model the spatial predator-prey-subsidy dynamics through a “stepping-stone” approach. We find that a temporal delay alone does not push species into extinction, but rather may stabilize or destabilize coexistence equilibria. We are able to show that delay can stabilize quasi-periodic or chaotic dynamics, and conclude that the incorporation of dispersal delay has a regularizing effect on dynamics, suggesting that dispersal delay can be proposed as a solution to the paradox of enrichment

A heat and mass transfer study of coffee bean roasting

Understanding heat, moisture and mass transport during the roasting of a coffee bean is essential to identifying how the colour and flavours are produced. This paper first considers a slightly simplified version of an existing heat and moisture transport model proposed by Fabbri et al. [Numerical modeling of heat and mass transfer during coffee roasting process. Journal of Food Engineering 105 (2011) 264-269], and we show that this model can be fitted well to data for the moisture content of a coffee bean but has some stability issues and lacks some important physical mechanisms. Building on these ideas, a new model is derived from conservation equations. This model is then simplified; in particular, issues of CO2 production are neglected as there is currently insufficient experimental data to fit parameters. This new model is fitted to the same experimental data as presented by Fabbri et al. The new model predicts significantly different internal structure and behaviour of the moisture than the existing model, while both show qualitatively similar average behaviour. This is due to the fact that our model tracks local, rather than bulk, quantities. One benefit to this new model is that it accurately predicts the existence of a sharp drying front, which partitions the bean into an outer dry region and an inner moist region. A detailed comparison of the two models is provided, in order to cast light on the relative importance of various heat and mass transfer mechanisms inherent in coffee bean roastin

A heat and mass transfer study of coffee bean roasting

Understanding heat, moisture and mass transport during the roasting of a coffee bean is essential to identifying how the colour and flavours are produced. This paper first considers a slightly simplified version of an existing heat and moisture transport model proposed by Fabbri et al. [Numerical modeling of heat and mass transfer during coffee roasting process. Journal of Food Engineering 105 (2011) 264-269], and we show that this model can be fitted well to data for the moisture content of a coffee bean but has some stability issues and lacks some important physical mechanisms. Building on these ideas, a new model is derived from conservation equations. This model is then simplified; in particular, issues of CO2 production are neglected as there is currently insufficient experimental data to fit parameters. This new model is fitted to the same experimental data as presented by Fabbri et al. The new model predicts significantly different internal structure and behaviour of the moisture than the existing model, while both show qualitatively similar average behaviour. This is due to the fact that our model tracks local, rather than bulk, quantities. One benefit to this new model is that it accurately predicts the existence of a sharp drying front, which partitions the bean into an outer dry region and an inner moist region. A detailed comparison of the two models is provided, in order to cast light on the relative importance of various heat and mass transfer mechanisms inherent in coffee bean roastin