24 research outputs found
The pseudo-compartment method for coupling PDE and compartment-based models of diffusion
Spatial reaction-diffusion models have been employed to describe many
emergent phenomena in biological systems. The modelling technique most commonly
adopted in the literature implements systems of partial differential equations
(PDEs), which assumes there are sufficient densities of particles that a
continuum approximation is valid. However, due to recent advances in
computational power, the simulation, and therefore postulation, of
computationally intensive individual-based models has become a popular way to
investigate the effects of noise in reaction-diffusion systems in which regions
of low copy numbers exist.
The stochastic models with which we shall be concerned in this manuscript are
referred to as `compartment-based'. These models are characterised by a
discretisation of the computational domain into a grid/lattice of
`compartments'. Within each compartment particles are assumed to be well-mixed
and are permitted to react with other particles within their compartment or to
transfer between neighbouring compartments.
We develop two hybrid algorithms in which a PDE is coupled to a
compartment-based model. Rather than attempting to balance average fluxes, our
algorithms answer a more fundamental question: `how are individual particles
transported between the vastly different model descriptions?' First, we present
an algorithm derived by carefully re-defining the continuous PDE concentration
as a probability distribution. Whilst this first algorithm shows strong
convergence to analytic solutions of test problems, it can be cumbersome to
simulate. Our second algorithm is a simplified and more efficient
implementation of the first, it is derived in the continuum limit over the PDE
region alone. We test our hybrid methods for functionality and accuracy in a
variety of different scenarios by comparing the averaged simulations to
analytic solutions of PDEs for mean concentrations.Comment: MAIN - 24 pages, 10 figures, 1 supplementary file - 3 pages, 2
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First passage time density for the Ehrenfest model
We derive an explicit expression for the probability density of the first passage time to state 0 for the Ehrenfest diffusion model in continuous time
Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics
Two algorithms that combine Brownian dynamics (BD) simulations with
mean-field partial differential equations (PDEs) are presented. This
PDE-assisted Brownian dynamics (PBD) methodology provides exact particle
tracking data in parts of the domain, whilst making use of a mean-field
reaction-diffusion PDE description elsewhere. The first PBD algorithm couples
BD simulations with PDEs by randomly creating new particles close to the
interface which partitions the domain and by reincorporating particles into the
continuum PDE-description when they cross the interface. The second PBD
algorithm introduces an overlap region, where both descriptions exist in
parallel. It is shown that to accurately compute variances using the PBD
simulation requires the overlap region. Advantages of both PBD approaches are
discussed and illustrative numerical examples are presented.Comment: submitted to SIAM Journal on Applied Mathematic
Turing pattern or system heterogeneity? A numerical continuation approach to assessing the role of Turing instabilities in heterogeneous reaction-diffusion systems
Turing patterns in reaction-diffusion (RD) systems have classically been
studied only in RD systems which do not explicitly depend on independent
variables such as space. In practise, many systems for which Turing patterning
is important are not homogeneous with ideal boundary conditions. In
heterogeneous systems with stable steady states, the steady states are also
necessarily heterogeneous which is problematic for applying the classical
analysis. Whilst there has been some work done to extend Turing analysis to
some heterogeneous systems, for many systems it is still difficult to determine
if a stable patterned state is driven purely by system heterogeneity or if a
Turing instability is playing a role. In this work, we try to define a
framework which uses numerical continuation to map heterogeneous RD systems
onto a sensible nearby homogeneous system. This framework may be used for
discussing the role of Turing instabilities in establishing patterns in
heterogeneous RD systems. We study the Schnakenberg and Gierer-Meinhardt models
with spatially heterogeneous production as test problems. It is shown that for
sufficiently large system heterogeneity (large amplitude spatial variations in
morphogen production) it is possible that Turing-patterned and base states
become coincident and therefore impossible to distinguish. Other exotic
behaviour is also shown to be possible. We also study a novel scenario in which
morphogen is produced locally at levels that could support Turing patterning
but on intervals/patches which are on the scale of classical critical domain
lengths. Without classical domain boundaries, Turing patterns are allowed to
bleed through; an effect noted by other authors. In this case, this phenomena
effectively changes the critical domain length. Indeed, we even note that this
phenomena may also effectively couple local patches together and drive
instability in this way.Comment: 10 figure
Optimal Interruption of P. vivax Malaria Transmission Using Mass Drug Administration
Plasmodium vivax is the most geographically widespread malaria-causing parasite resulting in significant associated global morbidity and mortality. One of the factors driving this widespread phenomenon is the ability of the parasites to remain dormant in the liver. Known as ‘hypnozoites’, they reside in the liver following an initial exposure, before activating later to cause further infections, referred to as ‘relapses’. As around 79–96% of infections are attributed to relapses from activating hypnozoites, we expect it will be highly impactful to apply treatment to target the hypnozoite reservoir (i.e. the collection of dormant parasites) to eliminate P. vivax. Treatment with radical cure, for example tafenoquine or primaquine, to target the hypnozoite reservoir is a potential tool to control and/or eliminate P. vivax. We have developed a deterministic multiscale mathematical model as a system of integro-differential equations that captures the complex dynamics of P. vivax hypnozoites and the effect of hypnozoite relapse on disease transmission. Here, we use our multiscale model to study the anticipated effect of radical cure treatment administered via a mass drug administration (MDA) program. We implement multiple rounds of MDA with a fixed interval between rounds, starting from different steady-state disease prevalences. We then construct an optimisation model with three different objective functions motivated on a public health basis to obtain the optimal MDA interval. We also incorporate mosquito seasonality in our model to study its effect on the optimal treatment regime. We find that the effect of MDA interventions is temporary and depends on the pre-intervention disease prevalence (and choice of model parameters) as well as the number of MDA rounds under consideration. The optimal interval between MDA rounds also depends on the objective (combinations of expected intervention outcomes). We find radical cure alone may not be enough to lead to P. vivax elimination under our mathematical model (and choice of model parameters) since the prevalence of infection eventually returns to pre-MDA levels