On 'Analytical models for the patchy spread of plant disease'.

Epidemiologists are interested in using models that incorporate the effects of clustering in the spatial pattern of disease on epidemic dynamics. Bolker (1999, Bull. Math. Biol. 61, 849-874) has developed an approach to study such models based on a moment closure assumption. We show that the assumption works above a threshold initial level of disease that depends on the spatial dispersal of the pathogen. We test an alternative assumption and show that it does not have this limitation. We examine the relation between lattice and continuous-medium implementations of the approach.

Effects of dispersal mechanisms on spatio-temporal development of epidemics.

The nature of pathogen transport mechanisms strongly determines the spatial pattern of disease and, through this, the dynamics and persistence of epidemics in plant populations. Up to recently, the range of possible mechanisms or interactions assumed by epidemic models has been limited: either independent of the location of individuals (mean-field models) or restricted to local contacts (between nearest neighbours or decaying exponentially with distance). Real dispersal processes are likely to lie between these two extremes, and many are well described by long-tailed contact kernels such as power laws. We investigate the effect of different spatial dispersal mechanisms on the spatio-temporal spread of disease epidemics by simulating a stochastic Susceptible-infective model motivated by previous data analyses. Both long-term stationary behaviour (in the presence of a control or recovery process) and transient behaviour (which varies widely within and between epidemics) are examined. We demonstrate the relationship between epidemic size and disease pattern (characterized by spatial autocorrelation), and its dependence on dispersal and infectivity parameters. Special attention is given to boundary effects, which can decrease disease levels significantly relative to standard, periodic geometries in cases of long-distance dispersal. We propose and test a definition of transient duration which captures the dependence of transients on dispersal mechanisms. We outline an analytical approach that represents the behaviour of the spatially-explicit model, and use it to prove that the epidemic size is predicted exactly by the mean-field model (in the limit of an infinite system) when dispersal is sufficiently long ranged (i.e. when the power-law exponent a

Analytical methods for predicting the behaviour of population models with general spatial interactions.

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.

Mental illness and physical health in older people.

Mental illness was recorded in 17% of men and 30% of women at the first examination of a random sample of the population of north Edinburgh consisting of 215 men and 272 women aged 62-90 years. Chronic brain syndrome (CBS) was found in 6% of men and 10.7% of women. CBS severe enough to constitute clinical dementia occurred in 3% of the sample. CBS increased in prevalence with age, particularly in women. Depressive illness was found in 3.7% of men and 6.6% of women. Anxiety states were present in 7% of men and 16.5% of women. Physical health and measurements were recorded in the same random sample in respect of ischaemic heart disease, stroke, blood pressure, chronic bronchitis, respiratory function, visual acuity, audiometry, urinary incontinence and food intake. The physical findings related to CBS were poor vision, poor hearing, urinary incontinence and poor diet. Five years later 113 men and 148 women were re-examined. The annual incidence of CBS was 0.4% in men and 1.4% in women. The annual incidence of depressive illness was 0.8% in both sexes, and of anxiety neurosis was 0.3%. Mortality rates were high in subjects with CBS. All moderate/severe cases of CBS had died by the end of seven years, but 15% of the mild cases were alive after 14 years. Prevalence and incidence of CBS in the study were similar to those reported by other authors. Figures for depressive illness and anxiety neurosis tended to be higher than in other surveys.

A longitudinal study of handgrip and dementia in older people.

Handgrip was measured, using a modified sphygmomanometer cuff, in a random sample of 487 older people in Edinburgh in 1968-9. The prevalence of dementia was determined by full psychiatric examination. The error score in the Isaacs-Walkey test was also recorded. Five years later, 261 of the original sample were re-examined. In both examinations treated cross-sectionally there was a significant decrease in grip strength as age increased. However, the mean of the five-year change in individuals did not differ significantly from zero and was not related to age. Handgrip was not significantly related to dementia either at the first examination or as a predictor of new dementia at the five-year examination. Mean handgrip was significantly less at the first examination in those who subsequently died, compared with five-year survivors.

A dietary survey of older people in Edinburgh.

1. Dietary histories covering 1 week were obtained from 212 men and 263 women, aged 62--90 years, who formed a random sample of the older people living in a defined area of Edinburgh. The histories were validated by 2 d weighed-diet records for 219 of the subjects. 2. Nutrient intakes are given as mean values and standard deviations, and as percentage distributions of intake relative to the recommended intakes of nutrients. Intakes are also related to factors which might affect diet adversely. 3. Mean values obtained suggested that intakes were less than recommended intakes for more nutrients for women than for men, particularly values for energy and vitamin D. Values for percentage distributions indicated that a substantial proportion of all subjects had low intakes of ascorbic acid and vitamin D, that the energy intake of many women was low and that for most nutrients a greater proportion of men than of women had the recommended intake. Factors assocIATED With significant proportions of subjects with less than the recommended intakes were mental illness in women and deafness and severe dyspnoea in men. 4. Possible action to improve the diet of the elderly is discussed.