248 research outputs found
The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications
The statement of the mean field approximation theorem in the mean field
theory of Markov processes particularly targets the behaviour of population
processes with an unbounded number of agents. However, in most real-world
engineering applications one faces the problem of analysing middle-sized
systems in which the number of agents is bounded. In this paper we build on
previous work in this area and introduce the mean drift. We present the concept
of population processes and the conditions under which the approximation
theorems apply, and then show how the mean drift is derived through a
systematic application of the propagation of chaos. We then use the mean drift
to construct a new set of ordinary differential equations which address the
analysis of population processes with an arbitrary size
Translated Poisson approximation to equilibrium distributions of Markov population processes
The paper is concerned with the equilibrium distributions of continuous-time
density dependent Markov processes on the integers. These distributions are
known typically to be approximately normal, and the approximation error, as
measured in Kolmogorov distance, is of the smallest order that is compatible
with their having integer support. Here, an approximation in the much stronger
total variation norm is established, without any loss in the asymptotic order
of accuracy; the approximating distribution is a translated Poisson
distribution having the same variance and (almost) the same mean. Our arguments
are based on the Stein-Chen method and Dynkin's formula.Comment: 18 page
Use of approximations of Hamilton-Jacobi-Bellman inequality for solving periodic optimization problems
We show that necessary and sufficient conditions of optimality in periodic
optimization problems can be stated in terms of a solution of the corresponding
HJB inequality, the latter being equivalent to a max-min type variational
problem considered on the space of continuously differentiable functions. We
approximate the latter with a maximin problem on a finite dimensional subspace
of the space of continuously differentiable functions and show that a solution
of this problem (existing under natural controllability conditions) can be used
for construction of near optimal controls. We illustrate the construction with
a numerical example.Comment: 29 pages, 2 figure
Daphnias: from the individual based model to the large population equation
The class of deterministic 'Daphnia' models treated by Diekmann et al. (J
Math Biol 61: 277-318, 2010) has a long history going back to Nisbet and Gurney
(Theor Pop Biol 23: 114-135, 1983) and Diekmann et al. (Nieuw Archief voor
Wiskunde 4: 82-109, 1984). In this note, we formulate the individual based
models (IBM) supposedly underlying those deterministic models. The models treat
the interaction between a general size-structured consumer population
('Daphnia') and an unstructured resource ('algae'). The discrete, size and
age-structured Daphnia population changes through births and deaths of its
individuals and throught their aging and growth. The birth and death rates
depend on the sizes of the individuals and on the concentration of the algae.
The latter is supposed to be a continuous variable with a deterministic
dynamics that depends on the Daphnia population. In this model setting we prove
that when the Daphnia population is large, the stochastic differential equation
describing the IBM can be approximated by the delay equation featured in
(Diekmann et al., l.c.)
On Exceptional Times for generalized Fleming-Viot Processes with Mutations
If is a standard Fleming-Viot process with constant mutation rate
(in the infinitely many sites model) then it is well known that for each
the measure is purely atomic with infinitely many atoms. However,
Schmuland proved that there is a critical value for the mutation rate under
which almost surely there are exceptional times at which is a
finite sum of weighted Dirac masses. In the present work we discuss the
existence of such exceptional times for the generalized Fleming-Viot processes.
In the case of Beta-Fleming-Viot processes with index we
show that - irrespectively of the mutation rate and - the number of
atoms is almost surely always infinite. The proof combines a Pitman-Yor type
representation with a disintegration formula, Lamperti's transformation for
self-similar processes and covering results for Poisson point processes
Extinction times in the subcritical stochastic SIS logistic epidemic
Many real epidemics of an infectious disease are not straightforwardly super-
or sub-critical, and the understanding of epidemic models that exhibit such
complexity has been identified as a priority for theoretical work. We provide
insights into the near-critical regime by considering the stochastic SIS
logistic epidemic, a well-known birth-and-death chain used to model the spread
of an epidemic within a population of a given size . We study the behaviour
of the process as the population size tends to infinity. Our results cover
the entire subcritical regime, including the "barely subcritical" regime, where
the recovery rate exceeds the infection rate by an amount that tends to 0 as but more slowly than . We derive precise asymptotics for
the distribution of the extinction time and the total number of cases
throughout the subcritical regime, give a detailed description of the course of
the epidemic, and compare to numerical results for a range of parameter values.
We hypothesise that features of the course of the epidemic will be seen in a
wide class of other epidemic models, and we use real data to provide some
tentative and preliminary support for this theory.Comment: Revised; 34 pages; 6 figure
Bayesian inference of biochemical kinetic parameters using the linear noise approximation
Background
Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.
Results
We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.
Conclusion
The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods
Global parameter identification of stochastic reaction networks from single trajectories
We consider the problem of inferring the unknown parameters of a stochastic
biochemical network model from a single measured time-course of the
concentration of some of the involved species. Such measurements are available,
e.g., from live-cell fluorescence microscopy in image-based systems biology. In
addition, fluctuation time-courses from, e.g., fluorescence correlation
spectroscopy provide additional information about the system dynamics that can
be used to more robustly infer parameters than when considering only mean
concentrations. Estimating model parameters from a single experimental
trajectory enables single-cell measurements and quantification of cell--cell
variability. We propose a novel combination of an adaptive Monte Carlo sampler,
called Gaussian Adaptation, and efficient exact stochastic simulation
algorithms that allows parameter identification from single stochastic
trajectories. We benchmark the proposed method on a linear and a non-linear
reaction network at steady state and during transient phases. In addition, we
demonstrate that the present method also provides an ellipsoidal volume
estimate of the viable part of parameter space and is able to estimate the
physical volume of the compartment in which the observed reactions take place.Comment: Article in print as a book chapter in Springer's "Advances in Systems
Biology
Syntactic Markovian Bisimulation for Chemical Reaction Networks
In chemical reaction networks (CRNs) with stochastic semantics based on
continuous-time Markov chains (CTMCs), the typically large populations of
species cause combinatorially large state spaces. This makes the analysis very
difficult in practice and represents the major bottleneck for the applicability
of minimization techniques based, for instance, on lumpability. In this paper
we present syntactic Markovian bisimulation (SMB), a notion of bisimulation
developed in the Larsen-Skou style of probabilistic bisimulation, defined over
the structure of a CRN rather than over its underlying CTMC. SMB identifies a
lumpable partition of the CTMC state space a priori, in the sense that it is an
equivalence relation over species implying that two CTMC states are lumpable
when they are invariant with respect to the total population of species within
the same equivalence class. We develop an efficient partition-refinement
algorithm which computes the largest SMB of a CRN in polynomial time in the
number of species and reactions. We also provide an algorithm for obtaining a
quotient network from an SMB that induces the lumped CTMC directly, thus
avoiding the generation of the state space of the original CRN altogether. In
practice, we show that SMB allows significant reductions in a number of models
from the literature. Finally, we study SMB with respect to the deterministic
semantics of CRNs based on ordinary differential equations (ODEs), where each
equation gives the time-course evolution of the concentration of a species. SMB
implies forward CRN bisimulation, a recently developed behavioral notion of
equivalence for the ODE semantics, in an analogous sense: it yields a smaller
ODE system that keeps track of the sums of the solutions for equivalent
species.Comment: Extended version (with proofs), of the corresponding paper published
at KimFest 2017 (http://kimfest.cs.aau.dk/
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