Interconnected population processes

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.

Stepping stone models for population growth

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours. An approximate solution is obtained for the probability structure of the population size, and exact results are derived for the process when immigration is introduced into a single colony from outside the system.

Spatio-temporal stochastic modelling of Clostridium difficile

Clostridium difficile-associated diarrhoea (CDAD) occurs sporadically or in small discrete outbreaks. Stochastic models may help to inform hospital infection control strategies. Bayesian framework using data augmentation and Markov chain Monte Carlo methods were applied to a spatio-temporal model of CDAD. Model simulations were validated against 17 months of observed data from two 30-bedded medical wards for the elderly. Simulating the halving of transmission rates of C. difficile from other patients and the environment reduced CDAD cases by 15%. Doubling the rate at which patients become susceptible increased predicted CDAD incidence by 63%. By contrast, doubling environmental load made hardly any difference, increasing CDAD incidence by only 3%. Simulation of different interventions indicates that for the same effect size, reducing patient susceptibility to infection is more effective in reducing the number of CDAD cases than lowering transmission rates.

Applying the saddlepoint approximation to bivariate stochastic processes

The problem of moment closure is central to the study of multitype stochastic population dynamics since equations for moments up to a given order will generally involve higher-order moments. To obtain a Normal approximation, the standard approach is to replace third- and higher-order moments by zero, which may be severely restrictive on the structure of the p.d.f. The purpose of this paper is therefore to extend the univariate truncated saddlepoint procedure to multivariate scenarios. This has several key advantages: no distributional assumptions are required; it works regardless of the moment order deemed appropriate; and, we obtain an algebraic form for the associated p.d.f. irrespective of whether or not we have complete knowledge of the cumulants. The latter is especially important, since no families of distributions currently exist which embrace all cumulants up to any given order. In general the algorithm converges swiftly to the required p.d.f.; analysis of a severe test case illustrates its current operational limit.

Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.

Stochastic effects in a model of nematode infection in ruminants

We illustrate the importance of stochastic effects in population models of biological systems and demonstrate a number of analytic and simulation-based approaches that can usefully be applied to such models. In so doing, we compare the stochastic approach to the more usual deterministic one. The model studied represents the gastrointestinal infection of ruminants by nematodes when the hosts maintain a fixed density. The incorporation of a feedback mechanism, which accounts for the immune response of the infected animals, results in a highly nonlinear model; similar forms of nonlinearity are a feature of many plausible models in population biology. In the absence of an analytic solution to the full stochastic model we explore a number of approximations and compare them to simulations of the full stochastic process. We explore three modes of behaviour of the system. In the endemic regime the stochastic system fluctuates widely around the non-zero fixed points of the deterministic model. In the managed regime, where the system is subject to external periodic perturbation, stochastic effects are negligible. Finally, we find that in a regime in which the deterministic model predicts the long-term persistence of oscillations the stochastic model shows that extinction can occur. Of the approximation procedures we consider, the Normal approximation to the full stochastic process is the most generally applicable, and it is also the most accurate in the light of simulation results. Local linearization provides reasonably accurate prediction of the variance-covariance structure, and a transfer function approach allows calculation of the time-lagged auto- and cross-correlations in the endemic regime. Linearization of the stochastic updates themselves results in poor prediction of the population variances.

Chaos in biometry

Although the study of chaotic and periodic phenomena began as recently as the 1960s, its subsequent development during the past few years has been extremely rapid in terms of both theory and practical application. The purpose of this paper is therefore to present an overview which will enable researchers with little prior knowledge to assess the relevance and potential application of nonlinear systems to problems in medicine and biology. Deterministic dynamic behaviour is examined through discrete logistic-type equations; stochastic behaviour is studied by superimposing an appropriate birth-death structure. Analysis of a variety of insect data sets shows that periodic and chaotic structures do indeed feature in natural populations; the classic Nicholson's blowfly data are viewed from both stochastic limit-cycle and deterministic chaos standpoints. Determination of the attractor dimension can be an invaluable aid to the understanding of biological and medical phenomena, and convincing examples include phase-space comparisons between healthy and sick humans for both EEG and ECG records.

Tracer studies to locate the site of platinum ions within filamentous and inhibited cells of Escherichia coli

The distribution of platinum ions within Escherichia coli after the induction of filaments with cis-Pt(NH(3))(2)Cl(4), and after growth inhibition by PtCl(6) (2-), has been determined with radioactive metal compounds ((191)Pt, with a half-life of approximately 3 days) by the simple chemical procedure of Roberts et al. In the filamentous cells, the platinum metal is associated with metabolic intermediates, nucleic acids, and cytoplasmic proteins; whereas, in inhibited cells, the platinum is combined only with the cytoplasmic protein. Similar experiments with gram-positive cells of Bacillus cereus and Staphyloccus aureus, which show no filamentous growth in the presence of cis-Pt(NH(3))(2)Cl(4), reveal that the metal complex does penetrate the cell wall and subsequently becomes bound predominantly by metabolic intermediates.

Platinum-induced filamentous growth in Escherichia coli

Certain group VIIIB transition metal compounds were found to inhibit cell division in Escherichia coli, causing marked filamentous growth. Gram-negative bacilli were the most sensitive to this effect, whereas gram-positive bacilli responded only at near-toxic levels of the metal. None of the cocci tested showed any apparent effect. Cytokinesis (cross-septation) can be initiated by removal or decrease of platinum, but not by treatment with pantoyl lactone, divalent cations, or a temperature of 42 C.