The study period was from 1 January 1995 to 31 December 2003. During this period Denmark was divided into 16 counties, each of which was generally served by only one clinical microbiological laboratory. In order to avoid a possible laboratory bias we choose to restrict our study to only one county and thereby one laboratory. We choose the county of Funen which has natural geographic borders; it consists of one main island in addition to several smaller islands (Figure 1). The population of Funen County ranged from 468,099 in 1995 to 473,852 in 2003 10. During the study period Funen County was served by the clinical microbiological laboratory at Statens Serum Institut. All fecal samples submitted for analysis for gastrointestinal pathogens – both from GPs and hospitals – from Funen County were referred to this laboratory.

Data from the Statens Serum Institut laboratory systems were extracted as described elsewhere 11. The samples were identified and geocoded to exact address using the CPR (Central Person Registration) number, which is a ten digit unique identifier given to all individuals with permanent residency in Denmark 12. Two datasets were established. One consisted of all individuals who had had one or more stool samples submitted for microbiological analysis, from a physician from Funen County regardless of the result of the laboratory analysis (n = 50,049), in the following referred to as diarrhea cases. The second dataset consisted of all individuals with a positive Campylobacter diagnosis, henceforth referred to as Campylobacter cases (n = 2,984). If more than one sample (diarrhea cases) or positive sample (Campylobacter cases) existed for the same individual, only the first within a 12 month period was retained. In addition to Campylobacter and diarrhea cases, the background population of Funen County was similarly geocoded to exact address. We searched for clustering of the Campylobacter cases and subsequently performed an identical cluster detection on the diarrhea cases to visually check if clustering co-occurred spatio-temporally for the two datasets, because this could point to a potential sampling bias when searching for Campylobacter clusters. We return to this problem in the discussion section.

The dataset had a spatial extent of 4,900 km² and we aggregated all data in a grid with a resolution of 5 by 5 km per cell resulting in 209 cells of 25 km² each. We arrived pragmatically at this spatial resolution. Using 1 by 1 km cells resulted in a large number of cells neither containing any population nor any Campylobacter cases, while 10 by 10 km cells were too large, evening out local variations in Campylobacter incidences and thereby hiding potential point sources in small, local areas. While we are aware of the modifiable area unit problem described by Openshaw 13, we have not quantitatively analyzed the impact of changing size and shape of the cells (the units of analysis) on our results and acknowledge this as a potential shortcoming of this study.

Combining the 25 km² cells with a one year temporal resolution of the nine year period covered (1995–2003) resulted in a three-dimensional data frame, where the unit of analysis is a space-time cell with the dimensions 25 km² * 1 year, henceforth referred to as a feature. In total, the data frame was composed of 1,881 features (209 * 9). The space-time continuum was thus divided into discrete units where the third spatial dimension, elevation, was substituted for time. In the following, we will refer to all features within a given year as a 'layer' and to all features from the same geographical location as a 'stack'. Figure 2 shows the data frame with the study area (corresponding to a layer) demarked in green and a nine year 'stack' of data for the 25 km² area covering Odense, the major city on Funen.

Spatial- and spatio-temporal cluster analyses can be performed in various software applications, of which SaTScan (the Spatial and Space-Time Scan Statistics)14 is probably the most widely used within public health and ArcGIS from ESRI is the dominant commercial application. Analyzing the distribution of disease incidences in time and space is associated with a large number of methodological challenges, of which we list a few here: First, in order to calculate incidences, a population denominator must be specified. As described above, we used a unit of analysis of 5 by 5 km times 1 year for aggregation of cases and population respectively, but there are large analytical impacts of deciding on how to combine spatial and temporal dimensions. Second, when calculating Campylobacter incidences, features with few cases and a small population will result in disproportionally high incidences 151617. Third, defining spatial associations between the phenomena under study is not straightforward and clusters could be searched for in cubic, cylindrical, conical or other shaped windows 18 with various levels of distance decay.

Having no hands-on experience with SaTScan we decided to construct our own tool using the software Netlogo 19 for the present purpose. This approach had the advantages that we had a priori experience with Netlogo, which has build-in 3D data visualization options, that we could control the data flow through the model and that we had perfect knowledge of all data processing steps.

We assumed that foodborne Campylobacter infections are independent events occurring randomly in space and time while Campylobacter cases associated with an environmental factor are grounded in space expressing spatio-temporal autocorrelation. This phenomenon has been put forward by Tobler 20 as "Everything is related to everything else, but near things are more related than distant things" and is sometimes referred to as Tobler's Law, or the first law of Geography. To detect campylobacteriosis associated with other factors than food we therefore searched for features where the incidence of Campylobacter infections deviated from the mean incidence, and where the incidence of the neighboring features (in time and space) also deviated from the mean incidence. To do so, we used the algorithm for computing local spatial autocorrelation developed by Luc Anselin 92122:

I i = ( x i − x ¯ ) ∑ i ( x i − x ¯ ) 2 ⋅ ∑ j w i j ( x j − x ¯ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@541C@

Where I_i is the local Moran's Index computed for each feature i, x_i is the incidence for feature i, x¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebaaaa@2D66@ is the mean incidence for the entire dataset, x_j is the incidence for the neighbor and w_ij is a weighting function for all pairs of i and j.

I_i can be either positive or negative. A) If the incidence in feature i (the target feature) is higher than the mean incidence, and the mean incidence of the neighboring features is also above the mean incidence, I_i will be positive. B) If both the incidence of the target feature and the mean incidence of the neighbors are below the mean incidence, I_i will also be positive, because it is the product of the negative deviations from the mean. In both cases, A and B, there is a spatial cluster of similar values and positive spatial autocorrelation. C) If the incidence of the target feature is higher than the mean incidence and the mean incidence of the neighboring features is lower than the mean incidence, or D), vice versa, I_i will be negative indicating spatial dissimilarity, or negative spatial autocorrelation. Situations A and B are sometimes referred to as high/high and low/low clusters, while C and D are called high/low and low/high, respectively.

In the study period there was a secular trend that, if not corrected for, in itself would affect the analyses. The input data for the analyses shows two similar "inverted U"-shaped trends with initial increases in the number of Campylobacter cases and also, although to a lesser extend, in the number of diarrhea cases followed by a decrease in numbers toward the end of the study period (Figure 3). Uncorrected, this temporal trend could lead to identification of clusters in the middle of the study period when the incidences peak. As we considered this peak a global phenomenon within the study area we therefore de-trended the data during the analysis by replacing x¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebaaaa@2D66@ of Equation 1 with the mean incidence for a given year (x⌢it MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbambadaWgaaWcbaGaemyAaKMaemiDaqhabeaaaaa@3060@ and x⌢jt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbambadaWgaaWcbaGaemOAaOMaemiDaqhabeaaaaa@3062@) (Equation 2).

In our analyses we expanded the neighbor weighting function, w_ij, to take spatial as well as temporal neighbors into account. We used a cubic Moore neighborhood definition of 26 neighbors but restrained the neighborhood to only include features with Campylobacter cases. The implications on the analyses of excluding zero-activity features are 1) we avoid detection of zero-activity (low/low) clusters. If not excluded, a target feature and its neighbors which don't have any Campylobacter cases will yield negative values for Z_it and Z_jt (Equation 3) and result in a positive I_it value indicating a cluster. 2) We also reduce detection of spatial dissimilarity (low/high and high/low clusters) to only detect dissimilarities between Campylobacter positive features.

In features with a small population, a single Campylobacter case will result in a disproportionally high incidence which would likely bias the analysis. To correct for this potential bias, we replaced s² for each feature with a variance estimate that took the feature's population into account and calculated the deviation of the feature's incidence from the mean incidence of the given year, divided by the square root of the variance estimate:

Z i t = x i t − x ^ i t ( 1 p i t − 1 p ⋅ t ) 2 ⋅ p i t ⋅ ( x ^ i t ) + ( 1 p ⋅ t ) 2 ⋅ ( p ⋅ t − p i t ) ⋅ x ^ i t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@78CC@

where p_it is the population of the ith feature at time t, p_·t is the sum of the population of features at time t and x⌢it MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbambadaWgaaWcbaGaemyAaKMaemiDaqhabeaaaaa@3060@ is the expected incidence for the ith feature at time t, calculated as the mean incidence for time t.

The resulting equation for calculating local Moran's I for a given feature to a given time thus became:

I i t = Z i t ⋅ ∑ j w i j ( Z j t ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemysaK0aaSbaaSqaaiabdMgaPjabdsha0bqabaGccqGH9aqpcqWGAbGwdaWgaaWcbaGaemyAaKMaemiDaqhabeaakiabgwSixpaaqafabaGaem4DaC3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGcdaqadaqaaiabdQfaAnaaBaaaleaacqWGQbGAcqWG0baDaeqaaaGccaGLOaGaayzkaaaaleaacqWGQbGAaeqaniabggHiLdaaaa@457E@

To estimate a statistical significance of I_it we calculated the expected I_i, E(I_it) given no spatial autocorrelation:

E ( I i t ) = − ∑ j w i j ( n − 1 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrau0aaeWaaeaacqWGjbqsdaWgaaWcbaGaemyAaKMaemiDaqhabeaaaOGaayjkaiaawMcaaiabg2da9KqbaoaalaaabaGaeyOeI0YaaabuaeaacqWG3bWDdaWgaaqaaiabdMgaPjabdQgaQbqabaaabaGaemOAaOgabeGaeyyeIuoaaeaadaqadaqaaiabd6gaUjabgkHiTiabigdaXaGaayjkaiaawMcaaaaaaaa@41F0@

where n is the count of features in the data frame. This was subtracted from the calculated I_it, and divided by the standard deviation of the features:

Z ( I i t ) = I i t − E ( I i t ) var ⁡ ( I i t ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aaeWaaeaacqWGjbqsdaWgaaWcbaGaemyAaKMaemiDaqhabeaaaOGaayjkaiaawMcaaiabg2da9KqbaoaalaaabaGaemysaK0aaSbaaeaacqWGPbqAcqWG0baDaeqaaiabgkHiTiabdweafjabcIcaOiabdMeajnaaBaaabaGaemyAaKMaemiDaqhabeaacqGGPaqkaeaadaGcaaqaaiGbcAha2jabcggaHjabckhaYnaabmaabaGaemysaK0aaSbaaeaacqWGPbqAcqWG0baDaeqaaaGaayjkaiaawMcaaaqabaaaaaaa@4A39@

Throughout the analyses we used a Z-score of 2.76 (99% confidence level) and to further sort out 'misclassified' features identified as significant clusters, we passed the features through two logical tests: 1) to avoid cluster features which could be caused by transient outbreaks, cluster features should have at least one other cluster feature in its stack and, 2) to avoid 'coldspot' clusters, the incidence of the cluster feature should be larger than the mean incidence of the 3 by 3 stack surrounding the feature. Finally, the strength of our approach was tested in a permutation analysis: Using a Poisson distribution, we generated 10,000 permutations of the spatio-temporal distribution of cases in the study area based on the population of each feature, and restrained the expected number of simulated cases to match the total number of observed cases. We recorded the number of accepted cluster features for each permutation, and used the simulated data to test if the observed number of clusters differed significantly from what could occur by random.

Data analyses and visualizations were performed in Netlogo, a software capable of performing 3D visualization 19. While originally being developed for the programming of agent-based models, the cellular automata facilities of this software also allow for cell operations such as calculations on spatial relations.