New data and tools for integrating discrete and continuous population modeling strategies.

Realistic population models have interactions between individuals. Such interactions cause populations to behave as systems with nonlinear dynamics. Much population data analysis is done using linear models assuming no interactions between individuals. Such analyses miss strong influences on population behavior and can lead to serious errors--especially for infectious diseases. To promote more effective population system analyses, we present a flexible and intuitive modeling framework for infection transmission systems. This framework will help population scientists gain insight into population dynamics, develop theory about population processes, better analyze and interpret population data, design more powerful and informative studies, and better inform policy decisions. Our framework uses a hierarchy of infection transmission system models. Four levels are presented here: deterministic compartmental models using ordinary differential equations (DE); stochastic compartmental (SC) models that relax assumptions about population size and include stochastic effects; individual event history models (IEH) that relax the SC compartmental structure assumptions by allowing each individual to be unique. IEH models also track each individual's history, and thus, allow the simulation of field studies. Finally, dynamic network (DNW) models relax the assumption of the previous models that contacts between individuals are instantaneous events that do not affect subsequent contacts. Eventually it should be possible to transit between these model forms at the click of a mouse. An example is presented dealing with Cryptosporidium. It illustrates how transiting model forms helps assess water contamination effects, evaluate control options, and design studies of infection transmission systems using nucleotide sequences of infectious agents.

Analyzing sensitivity to model form assumptions of infection transmission system models.

PURPOSE: Transmission system models make restrictive assumptions that might distort the conclusions of model analyses. We propose methods to progressively relax the following assumptions of classical deterministic compartmental models: 1) that the population has an effectively infinite size 2) that contact is instantaneous with no duration, 3) that mixing in this large population is instantaneously thorough after contact.METHODS: Analyses of contact patterns between high and low risk groups on gonorrhea transmission were performed. Initial models were similar to those analyzed by Hethcote and Yorke with compartments corresponding to sets of individuals. The instantaneous contact assumption in these models was relaxed by using continuous deterministic pairing models in the style of models presented by Dietz and Hadelar. That model makes restrictive assumptions about concurrent contacts, population sizes, and instantaneously random mixing. To relax these assumptions, we simulated our GERMS model of discrete individuals forming pairings and transmitting infection in continuous time.RESULTS: Relaxing the instantaneous contact assumption demonstrated a progressively decreased effect of mixing between high and low risk groups as the duration of contact was increased. The GERMS model simulations were shown to effectively reproduce pairing model behavior given the same restrictive assumptions as the pairing model. Further GERMS model analysis then demonstrated that concurrency assumptions alter the effects of contact rates between risk groups in ways that are dependent upon contact parameters. Finally GERMS models were used to structure mixing into four local areas. This affected the dynamics of reaching equilibrium but not the equilibrium value.CONCLUSIONS: Assessing the effects of assumptions in continuous compartmental models of transmission systems is feasible and important.

A k nearest neighbour test for space-time interaction.

This paper describes a k nearest neighbour statistic sensitive to the pattern of cases expected of space-time clusters of health events. The Knox and Mantel tests are frequently used for space-time clustering but have two disadvantages. First, the selection of critical space-time distances for the Knox test and of a data transformation for the Mantel test is subjective. Second, the Mantel statistic is the sum of the products of space and time distances, is linear in form, and is not sensitive to non-linear associations between small space and time distances expected of contagious processes. The k nearest neighbour statistic is the number of case pairs that are k nearest neighbours in both space and time, and is evaluated under the null hypothesis of independent space and time nearest neighbour relationships. The test was applied to simulated and real data and compared to the Knox and Mantel tests using statistical power comparisons. The k nearest neighbour test proved sensitive to the space-time interaction pattern expected of disease clusters, does not require parameters (such as critical distances) to be estimated from the data, and may be used to test hypotheses about the spatial and temporal scale of the cluster process. The method addresses significant weaknesses in existing space-time cluster tests and should prove useful in the quantification and evaluation of clusters of human health events. Additional research is needed to further document the power of the test under different cluster processes.

The analysis of disease clusters, Part II: Introduction to techniques.

Public health professionals often are asked to investigate apparent clusters of human health events or "disease clusters." A cluster is an excess of cases in space (a geographic cluster), in time (a temporal cluster), or in both space and time. This is the second part of an introductory-level review of the analysis of disease clusters for physicians and health professionals concerned with infection surveillance in hospitals. It reviews the status of the field with the hope of expanding the use of cluster analysis methods for the routine surveillance of infectious diseases in the hospital environment.

The analysis of disease clusters, Part I: State of the art.

Public health professionals often are asked to investigate apparent clusters of human health events, or "disease clusters." A cluster is an excess of cases in space (a geographic cluster), in time (a temporal cluster), or in both space and time. This is part I of an introductory-level review of the analysis of disease clusters for physicians and health professionals concerned with infection surveillance in hospitals. It reviews the status of the field with the hope of expanding the use of cluster analysis methods for the routine surveillance of infectious disease in the hospital environment.

Disease cluster statistics for imprecise space-time locations.

Health professionals are investigating an increasing number of possible disease clusters, and statistical tests play an important role in cluster description and analysis. Existing cluster statistics assume precise data, when in reality health events are often imprecise (for example, place of residence is known only to the census district or zip code) and uncertain (for example, 'I first became ill sometime in 1985'). This incompatibility--precise methods used to analyse imprecise data--is largely ignored, resulting in test statistics of unknown accuracy. Most cluster statistics can be written as the cross-product of two matrices where one matrix reflects nearest-neighbour, distance or adjacency relationships and the second matrix is health related (for example, case-control identities). This paper explores a general approach to clustering, which incorporates uncertainty regarding space-time locations into these nearest neighbour, distance or adjacency relationships. Because the approach is general it can be used with almost all existing cluster tests, and, because it accounts for imprecise location data, it is suited to the 'real-world' nature of disease cluster investigations.

Realistic power simulations compare point- and area-based disease cluster tests.

One can roughly divide disease cluster tests into area-based (using regional data) and point-based (using exact locations). We have compared the power of two area-based methods (Moran's I and I* (pop), a new method) to that of two point-based methods (the Cuzick-Edwards test and Grimson's test), using three realistic simulations of disease (fox rabies in England, childhood leukaemia in North Humberside, England, and Lyme disease in Georgia). The naive belief that point-based methods should be better is not supported: for the complex data simulated here, I* (pop) and the Cuzick-Edwards test had higher power than Grimson's method or Moran's I. I* (pop) capitalizes on high inter-region variability, while Moran's I cannot.

Disease models implicit in statistical tests of disease clustering.

State and local health departments investigate an increasing number of cluster allegations, for which the selection of appropriate statistical methods is an important problem. Many of the methods for the spatial analysis of health data assume, either implicitly or explicitly, some model of disease occurrence, and comparisons of methods can be difficult when their underlying disease models differ. We review some of the issues involved in the statistical analysis of spatial disease patterns and describe several methods recently proposed to detect areas of increased disease rates. The disease models upon which the methods are based are explicitly described, and they provide a useful basis for comparing alternative clustering methods.

The map comparison problem: tests for the overlap of geographic boundaries.

The quality of environmental studies is often compromised by the use of statistics, such as correlation and regression for example, which presuppose a statistical model, linear or otherwise, between two variables. When investigating hypotheses about relationships among geographically distributed variables, an alternative approach is to measure the amount of boundary overlap. Boundaries are geographic zones of rapid change in the intensity of a variable, and are often of scientific interest in their own right. Examples of boundaries include ecotones, genetic hybrid zones, pollution plumes, and the front of the wave of advance of an epidemic. Boundary overlap describes zones where boundaries from two or more variables coincide, and are useful for evaluating epidemiologic hypotheses relating health to environmental exposures. This paper proposes four statistics of boundary overlap, and explores their performance using simulation models and real data describing ozone concentrations and hospital admissions for respiratory conditions. The statistics are found sensitive to different aspects of boundary overlap, and provide an additional diagnostic tool in the analysis of geographically distributed variables. Overlap statistics are expected to come into increasing use as the installed base of geographic information systems increases.

Cuzick and Edwards' test when exact locations are unknown.

Cuzick and Edwards (JR Stat Soc [B] 1990;52:73-104) have proposed a case-control test to detect spatial clustering. The test statistic is the sum, over all cases, of the number of each case's k nearest neighbors that also are cases. Their approach is attractive in that it accounts for geographic variation in population density and because it allows one to account for confounders, both known and unknown, through the judicious selection of controls. However, the test assumes case locations are known exactly, when, in practice, case locations are usually approximated by the centers of areas such as census tracts and zip code zones. In such situations, "ties" arise when cases and controls are assigned to the same area, and the loss of information precludes calculation of the test statistic. The author's approach enumerates the ways in which the ties may be resolved to obtain upper and lower bounds on the exact, unobserved, test statistic. The null hypothesis of no clustering is rejected when the upper and lower bounds are significant, and it is accepted when they are not significant. Judgment is withheld when the upper bound is significant but the lower bound is not significant. This approach allows Cuzick and Edwards' test to be used with inexact locations typical of most cluster investigations.

Criteria for the evaluation of alternative environmental monitoring variables: Theory and an application using winter flounder (Pleuronectes americanus) and Dover sole (Microstomus pacificus).

The design of environmental monitoring programs is frequently hampered by a lack of objective, quantitative criteria for evaluating alternative monitoring variables. In this paper we describe two such criteria, which we call samples required - the number of samples required to detect a given change in value - and information imparted - the amount of environmental information revealed by the monitoring variable. We then use these criteria to evaluate fin erosion in winter flounder (Pleuronectes americanus) and Dover sole (Microstomus pacificus) as marine environmental monitoring variables. Two methods for determining the samples required use contaminated and reference areas to estimate the sample statistics of a hypothetical impacted population. The first method is based on the overall difference in the proportions of diseased fish in the reference and hypothetical populations. The second treats the proportion of diseased fish in individual trawls as the variate and determines the samples required based on the mean and variance of the reference and contaminated populations. We use both methods to predict the number of trawls needed to detect an increase of 200% in fin erosion in the reference population. The first method had greater statistical power but assumes spatially homogeneous populations. The second method accounts for environmental patchiness. For Dover sole it predicted 1661 trawls would be needed to detect the 200% increase. An estuarine winter flounder population would require 74 trawls, and an oceanic winter flounder population would require 142.5 trawls. It appears that fin erosion in winter flounder may be a useful indicator of environmental contamination, but several stipulations apply. Migration may inflate the number of diseased fish observed in the reference population, and a more detailed etiology of the disease is required, including an understanding of what contaminants are responsible for manifestation of the disease.

Synthetic cancer variables and the construction and testing of synthetic risk maps.

Cancer cluster investigations are usually univariate in nature; they focus on a particular cancer, such as leukaemia, and attempt to determine whether excess risk is associated with a suspected cancer-causing agent. Although several causes of death (such as leukaemia, lymphoma, Hodgkin's) may be considered, the approach is univariate because the causes of death are analysed sequentially and independently of one another. This approach is consistent with a one-cause one-effect model. Rarely, however, is the action of a carcinogen manifested at only one body site, and correlations among causes of death are the norm rather than the exception. A multiple effects model is therefore appropriate, and the multivariate nature of cancer mortality data should be exploited when exploring geographic pattern in cancer risks. This paper describes such an approach. We construct maps based on a principal components analysis of cancer mortality rates from different geographic areas. The resulting principal components are called synthetic cancer variables (SCVs), and maps of the SCV scores are synthetic risk maps (SRMs). These maps quantify geographic variation in cancer risk at several body sites simultaneously, and may be analysed for (1) spatial structure and (2) geographic association with potential risk factors. As an example, we use synthetic risk maps to determine whether high-risk counties in Illinois cluster near nuclear facilities. Much work remains to be done, but synthetic cancer risk maps appear to be a useful tool for quantifying geographic pattern and multivariate structure in cancer mortality.

Genetic relationships of European populations reflect their ethnohistorical affinities.

From 420 records of ethnic locations and movements since 2000 B.C., we computed vectors describing the proportions which peoples of the various European language families contributed to the gene pools within 85 land-based 5 x 5-degree quadrats in Europe. Using these language family vectors, we computed ethnohistorical affinities as arc distances between all pairs of the 85 quadrats. These affinities are significantly correlated with genetic distances based on 26 genetic systems, even when geographic distances, a common causative factor, are held constant. Thus, the ethnohistorical distances explain a significant amount of the genetic variation observed in modern populations. Randomizations of the records by chronology result in loss of significance for the observed partial correlation between genetics and ethnohistory, when geography is held constant. However, a randomization of records by location only results in reduced significance. Thus, while the historical sequence of the movements does not seem to matter in Europe, their geographic locations do. We discuss the implications of these findings.

Diversity of some gene frequencies in European and Asian populations. V. Steep multilocus clines.

Regions of abrupt genetic change, which result from either rapid spatial change of selective pressures or limited admixture, were investigated in Europe and Asia on the basis of eight red cell markers typed in 960 samples. Two methods were employed, one based on genetic distances and one on evaluation of the first derivative of the surfaces representing allele-frequency variation. Genetic divergence tends to be maximal between populations that are separated by physical factors (mountain ranges and seas) but also separated by cultural barriers (different language affiliation). This suggests that mating isolation, rather than adaptive response to environmental change, accounts for spatially abrupt genetic change at the loci studied and that cultural differences associated with language contribute to isolating populations. Although selection may have determined two wide allele-frequency gradients, the genetic structure of European and Asian populations seems primarily to reflect isolation by distance when investigated on a small scale and migration patterns (or absence of migration) when investigated on a larger scale.

Spatial autocorrelation analysis of migration and selection.

We test various assumptions necessary for the interpretation of spatial autocorrelation analysis of gene frequency surfaces, using simulations of Wright's isolation-by-distance model with migration or selection superimposed. Increasing neighborhood size enhances spatial autocorrelation, which is reduced again for the largest neighborhood sizes. Spatial correlograms are independent of the mean gene frequency of the surface. Migration affects surfaces and correlograms when immigrant gene frequency differentials are substantial. Multiple directions of migration are reflected in the correlograms. Selection gradients yield clinal correlograms; other selection patterns are less clearly reflected in their correlograms. Sequential migration from different directions and at different gene frequencies can be disaggregated into component migration vectors by means of principal components analysis. This encourages analysis by such methods of gene frequency surfaces in nature. The empirical results of these findings lend support to the inference structure developed earlier for spatial autocorrelation analysis.