[Show abstract][Hide abstract] ABSTRACT: We consider a model for an epidemic in a population that occupies
geographically distinct locations. The disease is spread within subpopulations
by contacts between infective and susceptible individuals, and is spread
between subpopulations by the migration of infected individuals. We show how
susceptible individuals can act collectively to limit the spread of disease
during the initial phase of an epidemic, by specifying the distribution that
minimises the growth rate of the epidemic when the infectives are migrating so
as to maximise the growth rate. We also give an explicit strategy that
minimises the basic reproduction number, which is also shown be optimal in
terms of the probability of extinction and total size of the epidemic.
[Show abstract][Hide abstract] ABSTRACT: We study a variant of Hanski's incidence function model that accounts for the
evolution over time of landscape characteristics which affect the persistence
of local populations. In particular, we allow the probability of local
extinction to evolve according to a Markov chain. This covers the widely
studied case where patches are classified as being either suitable or
unsuitable for occupancy. Threshold conditions for persistence of the
population are obtained using an approximating deterministic model that is
realized in the limit as the number of patches becomes large.
[Show abstract][Hide abstract] ABSTRACT: Hanski’s incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.
Journal of Applied Probability 06/2014; 51(2). DOI:10.1239/jap/1402578626 · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the relationship between certain stochastic and
deterministic versions of Hanski's incidence function model and the spatially
realistic Levins model. We show that the stochastic version can be well
approximated in a certain sense by the deterministic version when the number of
habitat patches is large, provided that the presence or absence of individuals
in a given patch is influenced by a large number of other patches. Explicit
bounds on the deviation between the stochastic and deterministic models are
Journal of Mathematical Biology 01/2014; DOI:10.1007/s00285-015-0865-4 · 1.85 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We construct a stochastic patch occupancy metapopulation model that incorporates variation in habitat quality and an Allee-like effect. Using some basic results from stochastic ordering, we investigate the effect of habitat degradation on the persistence of the metapopulation. In particular, we show that for a metapopulation with Allee-like effect habitat degradation can cause a dramatic decrease in the level of persistence while in the absence of an Allee-like effect this decrease is more gradual.
[Show abstract][Hide abstract] ABSTRACT: We consider a Markov chain model similar to the stochastic logistic model except that it allows for variation amongst individuals in the population. We prove that as the population size grows, the process can be approximated by a deterministic process. The equilibrium points of the limiting process and their stability are determined. Applications to modelling epidemics and metapopulations are discussed.
[Show abstract][Hide abstract] ABSTRACT: Metapopulation models have been used to better understand the conditions necessary for the persistence of the metapopulation. In this paper, we study a stochastic patch occupancy model that incorporates variation in quality and connectivity of the habitat patches. Two important assumptions are imposed in our analysis. Firstly, the distance between patches has a special form. This amounts to assuming that migrating individuals follow certain pathways. Secondly, the area of the habitat patches is assumed to scale with the number of patches in the metapopulation. Under these assumptions, a deterministic limit is obtained as the number of patches goes to infinity. Using the deterministic limiting process, a condition for persistence of the metapopulation is derived.
[Show abstract][Hide abstract] ABSTRACT: Stochastic patch occupancy models (SPOMs) are a class of discrete time Markov chains used to model the presence/absence of a population in a collection of habitat patches. This class of model is popular with ecologists due to its ability to incorporate important factors of the habitat patch network such as connectivity and distance between patches as well as heterogeneity in patch characteristics. We present an asymptotic examination of a simple type of SPOM called the mainland-island model. In this model a single patch called the mainland is connected to a large number of smaller patches called islands and each island is only connected to the mainland. We discuss the limiting behaviour of the SPOM as the number of islands increases and the size of the islands decrease relative to the mainland. We demonstrate that a variety of limiting behaviours is possible depending on the scaling of the island size and on the heterogeneity of habitat quality.
[Show abstract][Hide abstract] ABSTRACT: To model a fault that can be caused by more than one source, a mixture of conditional Gaussian transitions is proposed. The conditional means are modelled by recurrent neural networks. An expectation-maximization (EM) algorithm is used to estimate model parameters. By grouping known types of faults it is possible to form a bank of different fault models.
Control & Automation (MED), 2011 19th Mediterranean Conference on; 01/2011
[Show abstract][Hide abstract] ABSTRACT: We propose a model for the presence/absence of a population in a collection of habitat patches. This model assumes that colonisation and extinction of the patches occur as distinct phases. Importantly, the local extinction probabilities are allowed to vary between patches. This permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number of habitat patches increases to ∞. This is done in the case where the number of patches and the initial number of occupied patches increase at the same rate, and for the case where the initial number of occupied patches remains fixed.
Advances in Applied Probability 12/2010; 42(4). DOI:10.1239/aap/1293113156 · 0.71 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A new approximate Bayesian computation (ABC) algorithm is proposed specifically designed for models involving quantile distributions.
The proposed algorithm compares favourably with two other ABC algorithms when applied to examples involving quantile distributions.
KeywordsApproximate Bayesian computation-Likelihood-free inference-Markov chain Monte Carlo-Quantile distributions-Quantile regression
Statistics and Computing 11/2010; 22(6):1-9. DOI:10.1007/s11222-010-9209-9 · 1.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider the problem of estimating the parameters of a Markov queuing model from discrete time observations of the queue length. The proposed approach is an application of the martingale estimating function methodology which has been used extensively in mathematical finance. A small simulation study sug-gests that the estimator performs well, even for moderate sample size, and that it is an improvement over the Gaussian diffusion based, approximate maximum likelihood estimator.