On parameter estimation in population models

Ross, Joshua V.; Taimre, Thomas; Pollett, P. K.

doi:10.1016/j.tpb.2006.08.001

Cited by 40 publications

(61 citation statements)

References 34 publications

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“…Thus, much effort has been devoted to numerical methods. Faddy and Fenlon (1999) gave a general method to compute the transition probabilities for a class of pure birth processes (extended Poisson processes); we briefly outline it, with some adaptations for pure death processes (see also Ross et al, 2006 for a more general example). Let Pr n 0 ðtÞ ¼ ðPr n 0 ;0 ðtÞ; Pr n 0 ;1 ðtÞ; .…”

Section: Matrix Exponential Methodsmentioning

confidence: 99%

“…Fortunately, a software package designed for Markov chain models, expokit (Sidje, 1998), has been implemented in MatLab. Many authors advocate its use in this context (Podlich et al, 1999;Ross et al, 2006). In most cases, computing the transition probabilities via the exponential matrix method is far more accurate and faster than computing them directly from expressions such as Eq.…”

Section: Matrix Exponential Methodsmentioning

confidence: 99%

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Stochastic modelling of prey depletion processes

Clerc

Davison

Bersier

2009

Journal of Theoretical Biology

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The modelling of prey-predator interactions is of major importance for the understanding of population dynamics. Classically, these interactions are modelled using ordinary differential equations, but this approach has the drawbacks of assuming continuous population variables and of being deterministic. We propose a general approach to stochastic modelling based on the concept of functional response for a prey depletion process with a constant number of predators. Our model could involve any kind of functional response, and permits a likelihood-based approach to statistical modelling and stable computation using matrix exponentials. To illustrate the method we use the Holling-Juliano functional response and compare the outcomes of our model with a deterministic counterpart considered by Schenk and Bacher [2002. Functional response of a generalist insect predator to one of its prey species in the field. Journal of Animal Ecology 71 (3), 524-531], who observed the depletion of Cassida rubiginosa due to its exclusive predator, Polistes dominulus. The predation was found to be Holling type III, reflecting the ability of the predator to regulate its prey. Our approach corroborates this result, but suggests that the prey depletion census should have been performed more often, and that predation features were significantly different between the two years for which data are available.

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Section: Matrix Exponential Methodsmentioning

confidence: 99%

Section: Matrix Exponential Methodsmentioning

confidence: 99%

Stochastic modelling of prey depletion processes

Clerc

Davison

Bersier

2009

Journal of Theoretical Biology

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“…Using the exponential matrix formulation, the probability of moving from state i kK1 to state i k in time t k Kt kK1 can be explicitly calculated. Any one of a range of numerical optimization techniques can then be used to find the value of q, which maximizes the likelihood (2.6) over the range of parameter space (Ross et al 2006). It should be emphasized that this method of parameter estimation uses the exact likelihood of observing the given data-assuming the model is an accurate description of disease dynamics-and also incorporates dependency between observations.…”

Section: The Sis Modelmentioning

confidence: 99%

On methods for studying stochastic disease dynamics

Keeling

Ross

2007

J. R. Soc. Interface.

174

182

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Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

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“…As before, N will usually be related to the size of the system and n/N will usually be interpreted as a population density (or vector of population densities). Condition (6) stipulates that n t changes at a rate that depends on n t only through (the density) X t = n t /N, a property shared by a wide variety of models that arise in areas as diverse as ecology [40,42,43,45], epidemiology [6,12,48], parasitology [38], chemical kinetics [4,28,35,41], telecommunications [39,46] and random graphs [14,52]. Notice that the density process (X t , t ≥ 0), being itself a Markov chain, takes values in the set E no matter what the value of N.…”

Section: Density-dependent Population Modelsmentioning

confidence: 99%

Modelling population processes with random initial conditions

Pollett

Dooley

Ross

2010

Mathematical Biosciences

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Population dynamics are almost inevitably associated with two predominant sources of variation: the first, demographic variability, a consequence of chance in progenitive and deleterious events; the second, initial state uncertainty, a consequence of partial observability and reporting delays and errors. Here we outline a general method for incorporating random initial conditions in population models where a deterministic model is sufficient to describe the dynamics of the population. Additionally, we show that for a large class of stochastic models the overall variation is the sum of variation due to random initial conditions and variation due to random dynamics, and thus we are able to quantify the variation not accounted for when random dynamics are ignored.Our results are illustrated with reference to both simulated and real data.

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On parameter estimation in population models

Cited by 40 publications

References 34 publications

Stochastic modelling of prey depletion processes

Stochastic modelling of prey depletion processes

On methods for studying stochastic disease dynamics

Modelling population processes with random initial conditions

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