Data on the county level distributions of E. chaffeensis and A. phagocytophilum were available from previous research on their serostatus in Odocoileus virginianus (white-tailed deer) populations 2627. This surveillance approach was based on immunofluorescent antibody (IFA) tests performed on serum samples from white-tailed deer, and its efficacy has been confirmed by comparisons with polymerase chain reaction assays and culture isolations. E. chaffeensis and A. phagocytophilum were each sampled from 567 white-tailed deer populations distributed across 18 states. Serological data for each population were georeferenced by county. E. chaffeensis and A. phagocytophilum were classified as present in counties where one or more deer had antibodies reactive to the pathogen (Figure 1).

A descriptive analysis was carried out to quantify differences in the spatial patterns of these pathogens. Indicator semivariograms 28 were computed to characterize spatial autocorrelation for E. chaffeensis and A. phagocytophilum. The spatial location of each county was represented by its centroid, and presence (1) or absence (0) of the pathogen in each county was used as the indicator variable.

Environmental variables characterizing climate, land cover, and host populations within each county were obtained from a variety of sources. Climate variables were computed from 1-km Daymet grids which summarized monthly minimum temperature, maximum temperature, and precipitation over the period 1980–97 29. Monthly relative humidity was computed using estimates of ambient and saturation vapor pressure derived from monthly minimum and maximum temperatures 30. Land-cover variables were derived from the National Land Cover Dataset, which was created using 30-m resolution Landsat imagery collected in 1992 31. These data were used to compute the proportion of each county covered by forests, which included evergreen, deciduous and mixed forest as well as forested wetlands. The spatial pattern of forest cover within each county was characterized using a fragmentation index, which quantified the frequency of edges between forest and human land-use pixels (e.g. urban, agriculture) relative to the frequency of adjacent forested pixels 32. Deer density data from 1999 were obtained as a paper map from the Quality Deer Management Association (Watkinsville, GA, USA). Deer density was mapped as an index with five levels: (1) deer absent, rare or urban with unknown population; (2) < 15 deer/km² ; (3) 15–30 deer/km² ; (4) 30–45 deer/km²; and (5) > 45 deer/km². The map was digitized, georeferenced and converted to a 1-km grid. Deer density was summarized for each county as the density index that characterized the majority of the county.

A set of predictor variables was previously chosen for each pathogen through a multi-model comparison exercise 18, and these variables were used to develop the environmental models considered in this study (Figure 2). July-September mean maximum monthly temperature, March-June mean monthly humidity, annual precipitation, and percent forested land cover were used as environmental predictors for both E. chaffeensis and A. phagocytophilum. In addition, the fragmentation index was used as a predictor variable for E. chaffeensis, and deer density was used as a predictor variable for A. phagocytophilum.

Geographic zones were previously identified to characterize spatial heterogeneity in the influences of environmental variables on the distributions of E. chaffeensis and A. phagocytophilum. The zones were created via k-means clustering of the results of a geographically weighted regression (GWR) analysis of pathogen distributions, as documented in a previous study 18. GWR produces local estimates of regression coefficients for each sample location 16. Each cluster thus delineates an area in which pathogen-environment relationships are relatively homogeneous, but distinctive from the other clusters. This method identified a set of four geographic zones for each pathogen (Figure 3). Although the underlying GWR models and the resulting geographic zones were different for E. chaffeensis and A. phagocytophilum, both pathogens exhibited a general shift from climatic constraints in the southeastern U.S. to land cover and deer density constraints in the south-central U.S. 18.

We used a hierarchical Bayesian modelling approach to fit statistical models of pathogen presence/absence at the county level. We chose this technique because it allowed us to examine environmental correlates, spatial autocorrelation, and spatial heterogeneity in a consistent statistical framework. The binary response variable, Y_i, denoting pathogen presence/absence in each county was assumed to follow a Bernoulli distribution Y_i ~ Bernoulli(p_i)

where p_i was the probability of pathogen presence in county i, hereafter referred to as the endemicity probability 26. The probability of pathogen presence was in turn modelled as a function of predictor variables characterizing local environmental characteristics and spatial association with neighbouring counties. The following set of five alternative models was considered.

(1) The global environmental model predicted p_i as a function of co-located environmental variables. In the global model, a single parameter was fitted to quantify the influence of each environmental variable across the entire study area.

logit ( p i ) = log ⁡ [ p i / ( 1 − p i ) ] = b 0 + ∑ j = 1 v b j x i j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeiBaWMaee4Ba8Maee4zaCMaeeyAaKMaeeiDaqNaeiikaGIaemiCaa3aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkcqGH9aqpcyGGSbaBcqGGVbWBcqGGNbWzdaWadaqaaiabdchaWnaaBaaaleaacqWGPbqAaeqaaOGaei4la8IaeiikaGIaeGymaeJaeyOeI0IaemiCaa3aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaiaawUfacaGLDbaacqGH9aqpcqWGIbGydaWgaaWcbaGaeGimaadabeaakiabgUcaRmaaqahabaGaemOyai2aaSbaaSqaaiabdQgaQbqabaGccqWG4baEdaWgaaWcbaGaemyAaKMaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWG2bGDa0GaeyyeIuoaaaa@5BC4@

where j indexed v explanatory variables, b₀ was the intercept, b_j were the parameters, and x_ij were the environmental variables.

(2) The local environmental model also predicted p_i as a function of co-located environmental variables. To account for spatial heterogeneity, multiple parameters were fitted for each environmental variable to account for spatial heterogeneity in environmental effects across geographic zones (Figure 3).

logit ( p i ) = b 00 + ∑ j = 1 v b j 0 x i j + ∑ k = 1 s − 1 b 0 k z k + ∑ j = 1 v ∑ k = 1 s − 1 b j k z k x i j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7939@

where k indexed geographic zones, s _{was the number of geographic zones, b00 was the intercept for the baseline zone, bj0 were the parameters for the baseline zone, zk were indicator variables for the s-1 other zones, b0k were the deviations of the intercept in zone k from b00, and bjk were the deviations of the parameter for environmental variable j in zone k from bj0. Zone 1 was used as the baseline zone in all models (Figure 3).}

(3) The spatial autoregressive model predicted p_i as a function of endemicity in neighbouring counties

logit(p_i) = b₀ + ρ_i

where ρ_i was a spatial random effect that was modelled as a conditionally autoregressive (CAR) process. These random effects adjusted the endemicity probability up or down depending on the values of ρ_i in surrounding counties 33.

(4) The global environmental-autoregressive model was a combination of models (1) and (3).

logit ( p i ) = b 0 + ∑ j = 1 v b j x i j + ρ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeiBaWMaee4Ba8Maee4zaCMaeeyAaKMaeeiDaqNaeiikaGIaemiCaa3aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkcqGH9aqpcqWGIbGydaWgaaWcbaGaeGimaadabeaakiabgUcaRmaaqahabaGaemOyai2aaSbaaSqaaiabdQgaQbqabaGccqWG4baEdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabgUcaRiabeg8aYnaaBaaaleaacqWGPbqAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdAha2bqdcqGHris5aaaa@4E78@

(5) The local environmental-autoregressive model was a combination of models (2) and (3).

logit ( p i ) = b 00 + ∑ j = 1 v b j 0 x i j + ∑ k = 1 s − 1 b 0 k z k + ∑ j = 1 v ∑ k = 1 s − 1 b j k z k x i j + ρ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7D6C@

Models were fitted via Markov Chain Monte Carlo (MCMC) simulation using WinBUGS software 34. Vague prior distributions for the environmental parameters were specified as b_j ~ b_jk ~ N(0, 10⁶). The spatial random effect was modelled as a conditional autoregressive (CAR) process in which the distribution of each spatial effect had a Gaussian distribution centred on the mean of the neighbouring values.

ρ i | ρ i ≠ j ~ N ( ∑ i ≠ j w i j ρ i ∑ i ≠ j w i j , σ ρ 2 ∑ i ≠ j w i j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@662C@

where w_ij were the neighbourhood weights and σ²_ρ was a hyperparameter specifying the prior variance of the spatial random effects. The w_ij were specified based on a queen's adjacency rule, in which counties sharing a common boundary were considered neighbours. In spatial Bayesian models, a hyperprior for 1/σ²_ρ is commonly specified as a gamma distribution such as Γ(0.001, 0.001) or Γ(0.5, 0.0005) 35. However, in the present application these specifications led to difficulties with MCMC convergence. Instead, we specified a truncated normal hyperprior for σ²_ρ, which has been suggested as one alternative to the gamma distribution 36. We used a moderately informative specification of σ²_ρ ~ N(0, 10), truncated at zero so that σ²_ρ was always positive. Sensitivity analyses using alternative specifications of σ²_ρ ~ N(0, 5) and σ²_ρ ~ N(0, 20) yielded similar parameter estimates and prediction accuracies, demonstrating that our results were robust to changes in the specification of σ²_ρ. Flat priors were used for the intercepts b₀ and b₀₀.

The data used to fit the models and generate predictions included Y_i values for the counties with serology data, along with x and z values for all of the counties in the study area. Initial values were specified for all model parameters, including the coefficients for each environmental variable and the spatial random effects for each county. The posterior values of these parameters were updated during each step of the MCMC algorithm, and the parameters were then used to compute values of p_i for all counties. Convergence of the MCMC algorithm was evaluated through visual examination of the trace plots and through Brooks-Rubin-Gelman diagnostic plots 37. Based on these evaluations, a burn in of 20,000 steps was sufficient to achieve convergence for all models, and the posterior parameters values were sampled at 20,000 additional steps. The endemicity probability for each county was computed as the mean of p_i across the 20,000 steps.

Cross-validation was used to compare model performance at predicting pathogen presence in unsampled counties. The 567 counties with serological data were randomly split into four subsets of approximately equal size, and four WinBUGS runs were carried out for each model. In each of these runs, one of the four subsets was set aside for model evaluation, and the remaining three subsets were used to fit the model.

The predictive capabilities of the models were evaluated by computing the area under the receiver operating characteristic curve (AUC) for each model. The receiver operating characteristic curve describes relationship between the true positive rate and the false positive rate using a range of thresholds to classify pathogen presence and absence based on p_i 38. The AUC statistic can be interpreted as the probability that a randomly selected county where the pathogen is present will have a higher p_i value than a randomly selected county where the pathogen is absent. We also selected an optimal classification threshold for each model by computing classification accuracy (percent of counties correctly classified) for a range of thresholds and choosing the threshold with the highest accuracy. Sensitivity (percent of positive counties correctly classified) and specificity (percent of negative counties correctly classified) were also computed using this optimal threshold.

Maps of the predicted distributions of each pathogen were generated by plotting the spatial distribution of predicted p_i values for each of the five models. To generate these maps, models were fitted using pathogen data from all 567 counties with serology data to utilize all the available information and generate the best possible spatial predictions. Pathogen presence/absence data from the serology database were overlaid on the predicted endemicity probabilities to visually assess spatial patterns of prediction accuracy for the various models