Ottar N. Bjørnstad

Travelling waves and spatial hierarchies in measles epidemics.

Spatio-temporal travelling waves are striking manifestations of predator–prey and host–parasite dynamics. However, few systems are well enough documented both to detect repeated waves and to explain their interaction with spatio-temporal variations in population structure and demography. Here, we demonstrate recurrent epidemic travelling waves in an exhaustive spatio-temporal data set for measles in England and Wales. We use wavelet phase analysis, which allows for dynamical non-stationarity—a complication in interpreting spatio-temporal patterns in these and many other ecological time series. In the pre-vaccination era, conspicuous hierarchical waves of infection moved regionally from large cities to small towns; the introduction of measles vaccination restricted but did not eliminate this hierarchical contagion. A mechanistic stochastic model suggests a dynamical explanation for the waves—spread via infective ‘sparks’ from large ‘core’ cities to smaller ‘satellite’ towns. Thus, the spatial hierarchy of host population structure is a prerequisite for these infection waves.

Spatial population dynamics: analyzing patterns and processes of population synchrony

The search for mechanisms behind spatial population synchrony is currently a major issue in population ecology. Theoretical studies highlight how synchronizing mechanisms such as dispersal, regionally correlated climatic variables and mobile enemies might interact with local dynamics to produce different patterns of spatial covariance. Specialized statistical methods, applied to large-scale survey data, aid in testing the theoretical predictions with empirical estimates. Observational studies and experiments on the demography of local populations are paramount to identify the true ecological mechanisms. The recent achievements illustrate the power of combining theory, observation and/or experimentation and statistical modeling in the ecological research protocol.

Synchrony, Waves, and Spatial Hierarchies in the Spread of Influenza

Quantifying long-range dissemination of infectious diseases is a key issue in their dynamics and control. Here, we use influenza-related mortality data to analyze the between-state progression of interpandemic influenza in the United States over the past 30 years. Outbreaks show hierarchical spatial spread evidenced by higher pairwise synchrony between more populous states. Seasons with higher influenza mortality are associated with higher disease transmission and more rapid spread than are mild ones. The regional spread of infection correlates more closely with rates of movement of people to and from their workplaces (workflows) than with geographical distance. Workflows are described in turn by a gravity model, with a rapid decay of commuting up to around 100 km and a long tail of rare longer range flow. A simple epidemiological model, based on the gravity formulation, captures the observed increase of influenza spatial synchrony with transmissibility; high transmission allows influenza to spread rapidly beyond local spatial constraints.

Nonparametric spatial covariance functions: Estimation and testing

Spatial autocorrelation techniques are commonly used to describe genetic and ecological patterns. To improve statistical inference about spatial covariance, we propose a continuous nonparametric estimator of the covariance function in place of the spatial correlogram. The spline correlogram is an adaptation of a recent development in spatial statistics and is a generalization of the commonly used correlogram. We propose a bootstrap algorithm to erect a confidence envelope around the entire covariance function. The meaning of this envelope is discussed. Not all functions that can be drawn inside the envelope are candidate covariance functions, as they may not be positive semidefinite. However, covariance functions that do not fit, are not supported by the data. A direct estimate of the L0 spatial correlation length with associated confidence interval is offered and its interpretation is discussed. The spline correlogram is found to have high precision when applied to synthetic data. For illustration, the method is applied to electrophoretic data of an alpine grass (Poa alpina).

POPULATION TIME SERIES: PROCESS VARIABILITY, OBSERVATION ERRORS, MISSING VALUES, LAGS, AND HIDDEN STATES

Population sample data are complex; inference and prediction require proper accommodation of not only the nonlinear interactions that determine the expected future abundance, but also the stochasticity inherent in data and variable (often unobserved) environmental factors. Moreover, censuses may occur sporadically, and observation errors change with sample methods and effort. The state variable (usually density or abundance) may be hidden from view and known only through highly indirect observational schemes (such as public health records, hunting reports, or fossil/archeological surveys). We extend the basic state-space model for time-series analysis to accommodate these dominant sources of variability that influence population data. Using examples, we show how different types of process error and observation error, unequal sample intervals, and missing values can be accounted for within the flexible framework of Bayesian state-space models. We provide algorithms based on Gibbs sampling that can be used to obtain posterior estimates of population states and of model parameters. For models that can be linearized, results can be used for direct sampling of the posterior, including those with missing values and unequal sample intervals. For nonlinear models, we make use of Metropolis-Hastings within the Gibbs sampling framework. Examples derive from long-term census and population data. We illustrate the extension to discrete state variables with multiple stages using a Time- series Susceptible-Infected-Recovered (TSIR) model for mid 20th-century measles infec- tion in London, where birth rates are assumed known, but susceptibles and infected indi- viduals arise from imperfect reporting.

Dispersal, Environmental Correlation, and Spatial Synchrony in Population Dynamics.

Many species exhibit widespread spatial synchrony in population fluctuations. This pattern is of great ecological interest and can be a source of concern when the species is rare or endangered. Both dispersal and spatial correlations in the environment have been implicated as possible causes of this pattern, but these two factors have rarely been studied in combination. We develop a spatially structured population model, simple enough to obtain analytic solutions for the population correlation, that incorporates both dispersal and environmental correlation. We ask whether these two synchronizing factors contribute additively to the total spatial population covariance. We find that there is always an interaction between these two factors and that this interaction is small only when one or both of the environmental correlation and the dispersal rate are small. The interaction is opposite in sign to the environmental correlation; so, in the normal case of positive environmental correlation across sites, the population synchrony will be lower than predicted by simply adding the effects of dispersal and environmental correlation. We also find that population synchrony declines as the strength of population regulation increases. These results indicate that dispersal and environmental correlation need to be considered in combination as explanations for observed patterns of population synchrony.

A Geographic Gradient in Small Rodent Density Fluctuations: A Statistical Modelling Approach

The patterns of density dependence in Fennoscandian rodents are investigated statistically using a linear autoregressive scheme. Nineteen time series of microtine abundances along a latitudinal gradient in Fennoscandia from 60° N to 69° N are analysed. We provide statistical evidence that there exists a latitudinal gradient in density dependence in Fennoscandian microtines. Southern populations experience significantly stronger direct density dependence than northern populations. Delayed density dependence was significantly negative throughout the region and appeared constant across the latitudinal gradient. The populations consistently exhibit dynamics of second order throughout the region. Together, the clinal direct density dependence and constant delayed density dependence give rise to a cline in cycle period from 3 to 4.5 years. The statistical results are compared to assumptions and predictions made in previous studies on the geographic gradient in the population dynamics of these rodents. The results are in agreement with the predictions of the ‘generalist predator hypothesis’.

The dynamics of measles in sub-Saharan Africa

Although vaccination has almost eliminated measles in parts of the world, the disease remains a major killer in some high birth rate countries of the Sahel. On the basis of measles dynamics for industrialized countries, high birth rate regions should experience regular annual epidemics. Here, however, we show that measles epidemics in Niger are highly episodic, particularly in the capital Niamey. Models demonstrate that this variability arises from powerful seasonality in transmission—generating high amplitude epidemics—within the chaotic domain of deterministic dynamics. In practice, this leads to frequent stochastic fadeouts, interspersed with irregular, large epidemics. A metapopulation model illustrates how increased vaccine coverage, but still below the local elimination threshold, could lead to increasingly variable major outbreaks in highly seasonally forced contexts. Such erratic dynamics emphasize the importance both of control strategies that address build-up of susceptible individuals and efforts to mitigate the impact of large outbreaks when they occur.

Measles Metapopulation Dynamics: A Gravity Model for Epidemiological Coupling and Dynamics

Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer‐resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible‐infected‐recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.

Allee effects and pulsed invasion by the gypsy moth.

Biological invasions pose considerable threats to the world’s ecosystems and cause substantial economic losses. A prime example is the invasion of the gypsy moth in the United States, for which more than