Ethics statement

The decision and implementation of euthanasia of animals was conducted by county authorities, before samples were received for the current study (i.e., we had no role in this). For the current study, the State of Colorado Department of Natural Resources issued annual Scientific Collection Licenses (14SALV2060, 15SALV2060) in order for us to receive a subset of dead carcasses for testing.

Surveillance data

The study area included 8448 km² (88 x 96 km; Fig 1) of Larimer, Boulder and Weld counties, Colorado, USA. Skunk samples were obtained by reports made by public to local health departments, which decided whether to submit a skunk for rabies testing to one of two diagnostic laboratories in the state. The decision and implementation of euthanasia of animals was conducted by county authorities, before samples were received for the current study. For the current study, the State of Colorado Department of Natural Resources issued annual Scientific Collection Licenses (14SALV2060, 15SALV2060) in order for us to receive a subset of dead carcasses for testing. There was no vaccination campaign (trap-vaccinate-release or oral rabies vaccination) targeting wildlife during the epizootic, but there were alerts through local public media and leash laws in effect.

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Fig 1. Map of study area.

Each dot represents the location where a dead skunk was reported and subsequently tested for rabies. Grid cells show the scale at which the study area was gridded (8 x 8 km sites, i = 1, …, M).

https://doi.org/10.1371/journal.pntd.0005822.g001

Strange-acting or dead mammals found on the landscape with no reported human or domestic animal contact were not typically tested by the local health departments—especially later in the study period when the perceived risk of rabies infection was much lower and funding for testing had decreased. Because of the ongoing epizootic, these samples were referred for testing by United States Department of Agriculture, Animal and Plant Health Inspection Service, Wildlife Services, National Wildlife Research Center (NWRC); hereafter referred to as ‘enhanced surveillance’. Enhanced surveillance accounted for 0%, 1% and 47% of surveillance data in 2012, 2013 and 2014 respectively (Table SM1.1 in S1 Text). Enhanced surveillance samples were identified through the public health surveillance system by the local health departments, but with carcass referral to NWRC for rabies testing. Carcasses were referred to NWRC when the perceived risk to public health was low (i.e., due to a lack of contact with humans or pets). Because the rabies epizootic was waning in 2014, the high proportion of enhanced surveillance in 2014 relative to 2012 and 2013 did not contribute much extra information (except for improving uncertainty levels) using the occupancy model. Thus, typical surveillance systems could likely make the same type of inferences, but would show greater uncertainty as sampling decreased. An address matching-geocoding technique, using ArcGIS 10.3 (Environmental Systems Research Institute, Redlands, CA, USA), was used to convert street addresses of the case reports to UTM (Universal Transverse Mercator) coordinates.

Before 2012, carnivore rabies had not been detected in the study area for several decades. From 2012–2014 there were 246 skunk reports and 379 non-skunk terrestrial animal reports tested for rabies (Fig 1, Table SM1.2 in S1 Text), with a total of 139 rabies-positive skunks (raw prevalence ~ 57%, Table SM1.2 in S1 Text), and 21 rabies-positive cases from the 379 non-skunk terrestrial species tested (5.5% positive; Table SM1.2 in S1 Text) including: raccoons (6), bison (2), foxes (6) cats (2), horses (2), cows (2), coyotes (1). All non-skunk terrestrial animals were excluded from the analysis because they comprised such a small proportion of the samples and because surveillance efforts may have differed in some of these hosts—especially raccoons which are higher density and more peridomestic, and were experiencing an ongoing epizootic of canine distemper virus. Human population data from 2013 [24] were organized at the grid-cell scale (8 x 8km) to be included in the occupancy model as a potential factor contributing to reporting rate.

Occupancy model formulation

We used a discrete-time, dynamic, patch-occupancy model to quantify the occurrence probability of skunk rabies in space and time. In a typical occupancy framework, there is the partially observable occupancy process and an observation process that is conditional on the occupancy process. For this dynamic model the occupancy process describes the probability of rabies being present by site (grid cells) and time (months). The ecological process of rabies occurrence changed over grid cells and months due to the local component processes of colonization, local extinction, and persistence. Within the latent ecological process, we estimated important parameters describing the initial colonization dynamics (i.e., spatial invasion) of skunk rabies, specifically: direction of invasion, distance to nearest infected grid cell, and density of infection in the local neighborhood. Conditional on the latent ecological process of occupancy, we modeled the observation process as the probability that an animal sampled has rabies, given that rabies is present (prevalence given rabies occupancy). Within the observation process, we considered the effects of human population size on the proportion of skunks that were rabies positive. Thus, our approach allowed us to quantify parameters underlying the spatio-temporal dynamics of geographic spread while accounting for imperfect detection. For the occupancy model, we assumed the landscape was homogenous because our genetic model suggested that spread rates were homogenous across the landscape and we did not have enough degrees of freedom to incorporate more parameters in our occupancy model.

We used a monthly time step to scale with the incubation period for rabies virus in skunks [25,26]. We divided the study area (much of Larimer, Weld and Boulder Counties in Northern Colorado, USA) into 8 x 8 km grid cells (“patches” in occupancy modeling terms; Fig 1). This spatial scale was much larger than a skunk home range size [27,28] suggesting that local transmission (i.e., due to skunk movements alone) should be primarily within grid cells or nearest-neighbor grid cells. Also, at this spatial scale, there were multiple samples collected within grids (Fig 1) at a given time step (mean number of samples in grids with samples: 3.6 sampled skunks, range 1–19 sampled skunks), which is necessary for estimating grid-cell level prevalence and informing the observation process, but at finer spatial scales multiple samples per grid cell and time step were rare. Thus, because of the sparse sampling intensity in our study area, the 8 x 8 km grid cell size was the smallest size we could choose for approximating the skunk home range (~ 2 x 2 km; [25, 26]) and hence skunk-to-skunk transmission distances.

Data on the presence of rabies in dead skunks (sampling unit = 1 skunk) were collected repeatedly in grid cells i = 1, …, M during each month t = 1, …, T (spanning January 2012 through December 2014). We modeled the true occupancy status z_{i, t} conditioned on the previous time step z_{i, t-1}, as a latent Bernoulli variable defined by parameter Ψ_{i, t} (occupancy probability), where z_{i, t} = 0 indicates that grid cell i is not occupied (no rabies in skunks) in time step t, and z_{i, t} = 1 indicates that grid cell i is occupied (at least one skunk with rabies) in time step t.

(1)

We assumed the initial occupancy state for each grid cell (z_{i, 1}) was also a Bernoulli random variable described by occupancy probability Ψ_{i, 1}, which had a Uniform prior distribution (Ψ_{i, 1}~Unif(0, 1)). We allowed occupancy probability in grid cell i at time t (Ψ_{i, t}) to be determined by three types of local dynamic processes: initial colonization (γ_i,t), recolonization (ζ_i,t) and persistence (ϕ_i,t), which were conditional on the latent occupancy status in the previous time step (z_{i, t-1}).

(2)

We used parameter A_{i, t} to distinguish an initial colonization event from subsequent colonization events. Although it is more typical to consider only occupancy and extinction dynamics in occupancy models, we further distinguished initial colonization from recolonization (as in [29]) because we were most interested in factors (described below) driving spatial invasion into new sites. We quantified the effects of multiple different factors (direction, distance to nearest infected neighbor, neighborhood infection density (Eqs 3–8) on initial colonization probability (γ_i,t-1), and treated the other processes as inherent contributors to occupancy probability (Ψ_i,t) without quantifying their potential drivers (i.e. we used a global parameter for persistence- ϕ_it ~ Beta(α_ϕ, β_ϕ) and recolonization—ζ_it ~ Beta(α_ζ, β_ζ). For γ_i,t-1, an intercept term (β_o) absorbed non-spatial effects (Eq 3). Direction (north-south (N) or east-west (E)) was modeled as increasing integers (1 representing the southernmost grids and 12 representing the northernmost grids for N; and 1 representing the westernmost grids and 11 representing the easternmost grids for E; Eq 4, β_k, where k = 1, …, K for the number of initial colonization effects considered). East-west direction was represented similarly but using columns in the grid. We modeled an interaction between direction and a trend in time (T) by multiplying the direction covariate data by the time step (N_i · T or E_i · T). We calculated distance to the nearest infected grid cell by taking the minimum distance for all pairwise distances between the centroid of target grid cell i* in time t and all other grid cell centroids (i) in time t-1 (d_ii*). We estimated the relationship of initial colonization probability and distance using an exponential decay function which included a parameter describing the decay rate of initial colonization probability (α) with distance and a scaling parameter relative to maximum initial colonization probability (β_k, Eq 5). We estimated the local neighborhood infection density for grid cell i* in time t using the occupancy prevalence in all immediate grid-cell neighbors (queen’s neighbors; j = 1, …, J; note j’s are a special subset of i—restricted to the closest neighbors of i*) in time step t-1 and estimated its effect with parameter β_k (Eq 6). We incorporated a seasonal effect as a factor where one level represented spring/summer (Feb.–Aug.) and the second level represented fall/winter (Sept.–Jan.) (Eq 7). We chose these time frames because the literature and surveillance data suggest peaks of rabies in spring and summer following arousal and mating activities, and subsequent peaks in fall and winter associated with dispersal and contact involving susceptible young of the year [16, 25]. We modelled the infection density (Eq 6) and distance effects (Eq 5) separately because we were interested in the separate effects of distance versus infection intensity. Together these effects describe the spatial kernel for local transmission. Additionally, we modelled a weighted spatial kernel (Eq 8) which accounted for the density of neighbors with infections but discounted the impact an infected cells at further distances from the grid of interest (d_ii* represents the distance between grid cell ‘i' and the grid cell of interest ‘i*’).

logit(γ_i,t-1) = (3) (4) (5) (6) (7) (8)

We considered these effects (Eqs 4–8) separately and additively (up to three effects including the Eq 3) in our model selection procedure (described below). We did not present the fullest model because parameter estimates became inconsistent when more than three additional effects were in the model. Persistence and recolonization were important processes in the occupancy dynamics, thus we modeled them explicitly using Beta prior distributions with shape and scale parameters (α_ϕ = 1, β_ϕ = 1, α_ζ = 1, β_ζ = 1). We did not include covariates that could potentially drive persistence and recolonization because we were only interested in the process of spatial spread (i.e., initial colonization). Prior distributions for all β_k parameters were: β_k ~ Norm(0, 1), except for the scaling parameter for the exponential decay model (Eq 5) which was modeled as a Gamma(5, 1).

For the observation layer of our model, we represented the number of skunks that were positive for rabies, y, in grid cell i at time t when rabies was present as the observed data using a binomial distribution where p was the estimated prevalence of rabies in dead skunks (given occupancy) and the number of trials (R_{i, t}) were the observed number of samples collected in grid cell i at time t.

(9)

To account for variation in rabies detection due to variation in human population size, we examined prevalence as a linear function of human population size (N).

(10)

We only investigated this effect in the intercept-only model (Eq 3) and the two-effect model we were most interested in (Table SM2.1 in S1 Text for list of all models that were fit). The general model specification is given in section SM2.1 in S1 Text.

To calculate the posterior distribution for the parameters of interest, we fit models using a Markov chain Monte Carlo (MCMC) algorithm with a Gibbs sampler including Metropolis-Hastings steps [30] custom written in program R [31]. Posterior estimates for the data are based on 50,000 iterations of the MCMC algorithm with the first 5,000 iterations discarded as burn-in. Convergence and mixing were assessed graphically. Example MCMC chains and posterior distributions are given in SM2.1 in S1 Text. Convergence of the best predictive model was additionally assessed using the Gelman-Rubin statistic [30]. The joint and conditional distributions are specified in SM2.3 in S1 Text. We evaluated the ability of our models to recover parameters by simulating data from known parameter values and estimating the parameter values using our fitting framework (Model Validation Method in SM5 in S1 Text).

Model selection and evaluation

We used a combination of Watanabe Akaike Information Criterion (WAIC) [32, 33], Area Under the receiver operator Curve (AUC) [34], and leave-one-out cross validation (looCV) [31] to compare the importance of different effects on γ and p and assess goodness-of-fit (presented in Table 1, schematic of work flow shown in Fig. SM3.1). WAIC is a model selection criterion based on the posterior predictive distribution, and was used to compare fits of models. WAIC was not used to compare models with human population modelled on prevalence (p) because the data were different and thus the WAIC values would not be comparable. AUC is a measure of how well variation is explained by the model—in our case, the ability to distinguish a presence from an absence—and was used to assess how well a particular model explained the data (i.e. a measure of goodness of fit as in [35]). LooCV is a measure of the model’s ability to predict out of sample, and thus was used to compare predictive ability among models. When comparing predictive ability between models we used both AUC and looCV.

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Table 1. Model selection and fit statistics for within- and out-of-sample predictions.

https://doi.org/10.1371/journal.pntd.0005822.t001

We calculated WAIC and AUC (Section SM4 in S1 Text) for fits to the full data as well as the looCV predictions. WAIC is preferable to DIC (Deviance Information Criterion—another Bayesian method for model selection) for model selection using hierarchical models [32] because it considers the posterior predictive distribution explicitly and penalizes for complexity of model structure, not just the number of parameters. Similar to DIC, lower values of WAIC indicate a better model of the data. However, in contrast to DIC [36], there is no standard quantitative difference between WAIC values from alternative models that indicates a significant difference between them (i.e., the only criterion is that lower is better). We presented two AUC scores: 1) AUC1 measured predictive ability of p using predicted y’s from posterior values of z and observed y’s, 2) AUC2 measured predictive ability of Ψ using the posterior values of z and the observed y’s transformed to binary data. For the out-of-sample predictions, we conducted looCV for each point in the data (using all other points as the training data) and presented means of AUC1 and AUC2 as measures of predictive ability.

Out-of-sample prediction

To further evaluate the predictive ability of our approach we predicted occupancy status over space and time using only parameters estimated from the “best predictive model” (i.e., considering out-of-sample statistics for AUC1, AUC2 and looCV) and the sample size data. First, we fit the best model to the full data and then predicted all y_{i, t} from the posterior predictive distribution [30]. Each predicted y_{i, t} depended on the predicted y_{i, t-1}, rather than the data, but the actual data for sample size (R_{i, t}) were used in the prediction in order to scale y_{i, t} predictions appropriately.

Rates of spatial spread from occupancy model

Using the model with the best out-of-sample looCV score, we predicted grid cell occupancy over time (z_{i, t}; rabies presence -1, or absence -0), and estimated the rate of southerly spread of the predictions using regression. First, we obtained the z_{i, t} values from each MCMC iteration (minus the burn in) for grids across time. Each grid cell corresponded to a distance in kilometers from the southernmost border of the study area (e.g., 1, …, 88 km; “North value”). For z_{i, t} = 1 values, we ran a simple linear regression where the independent values were the months in time and the dependent values were the north values. The slope of this regression represents the monthly rate of southerly spread. We then converted this to an annual rate of southerly spread by multiplying the monthly rate by 12. We calculated the annual variance using the delta method [37].

Genetic sequencing

To verify spatial patterns indicated by our occupancy model, we sequenced the whole or partial glycoprotein (G, 1575 base pairs [bp]) and the non-coding region between the glycoprotein and polymerase genes (GL, 560 bp) for 53 viruses collected in Colorado between August 2012 and December 2014 (Table SM1.3 in S1 Text). We added sequence data from 20 viruses collected through May 2015 (27.4% of the total genetic dataset) to increase the precision of parameters of the molecular evolutionary models. We confirmed that these additional sequences comprised an extension of the same epizootic though preliminary phylogenetic analysis and the absence of changes in the inferred rate of spread in 2015 which could have biased our overall spread rate. Rabies virus RNA was extracted from brainstem tissue samples of rabies positive skunks using Trizol reagent following the manufacturer’s protocol. The conversion of RNA to cDNA (RT) and primary PCR amplification was accomplished using a Superscript III One-step RT-PCR system with Platinum Taq DNA Polymerase (Invitrogen), targeting a 2,135bp region including the full glycoprotein gene with previously published primers [18,38]. PCR products were visualized by UV-light on a 2% agarose gel with ethidium bromide, and cleaned using ExoSap-IT (Affymetrix) following the manufacturer’s protocol. Sequencing reactions were performed using Big Dye Terminator v.3.1 (Applied Biosystems), with flanking and internal primers as previously described [18,38]. Sequencing products were cleaned using Sephadex G-50 columns (GE Healthcare) and run on a 3130 or 3500 analyzer (Applied Biosystems). Forward and reverse sequences were aligned in Sequencher v.5.2.4 (Gene Codes Corporation), and ambiguities were resolved visually. Alignment of full or partial glycoprotein sequences was performed using BioEdit v.7.2.0.

Phylogeographic analysis of viral spread

Continuous phylogeographic analysis was conducted in BEAST v. 1.8.4 [39]. Briefly, this method estimates the phylogenetic history connecting samples and conducts ancestral state reconstruction of the latitudes and longitudes of inferred nodes using time and location-annotated sequence data. Phylogenetic uncertainty is accommodated by summarizing estimates across the posterior distribution of trees from a Bayesian search. Analyses in TempEst (software that analyzes correlations between the temporal and genetic distances between sequences) showed evidence of a molecular clock signal in the coding (G) and non-coding (GL) regions of the RABV genome, but faster evolution in the G-L region compared to G [40]. Our BEAST analysis therefore modeled a single tree topology (given that the samples represent the same underlying epidemic history) using different molecular clock and substitution models to allow for differences in evolutionary rates in different parts of the RABV genome. Preliminary BEAST runs using the lognormal relaxed molecular clock indicated little variation in rates among branches, indicating the use of the strict molecular clock for both partitions, which was supported by equivocal differences in Bayes Factors (BF) between models assuming strict or relaxed molecular clocks (BF = 1.5 in favor of strict clocks) [41]. Final runs used customized substitution models for the following data partitions: G codon positions 1+2 = TPM1uf; G codon position 3 = TIM1+G, GL = TPM1uf+I, as suggested by AIC in jModeltest2 [42]. Among the 3 random walk phylogeographic models tested (homogenous, Cauchy and gamma), marginal likelihoods estimated by the stepping stone method were highest in the homogenous model (-3574.86) followed by the gamma (-3577.27) and Cauchy (-3594.05) models, suggesting relatively little variation in spread rates among branches or little power to detect such variation. To account for the possibility of among branch variation, we present the results of the gamma model, but note that parameter estimates were nearly identical in the similarly supported homogeneous rate model. All analyses used the Bayesian skyline model of demographic growth. MCMC chains were run for 100 million generations which generated effective samples sizes >200 after removal of burn-in.

We used the Seraphim package of R to calculate viral diffusion rates from 500 randomly selected (post burn-in) trees from the posterior distribution of the BEAST analysis [43]. We estimated viral spread rates as (i) the “mean branch velocity” calculated as the average velocity across the branches of each tree, averaged across all trees and (ii) the weighted “diffusion rate”, obtained by summarizing distances and times spanning each tree and taking the average of that value across all trees (i.e., sum distances/sum time lengths over entire tree). To calculate the transmission distance per viral generation (i.e., per infected animal), we divided the distance traversed along each branch by the expected number of infections along that branch, assuming a generation time of 30 days, which corresponds to the incubation period of rabies virus in skunks [25,26]. However, the substitution rates that we estimated (G: 4.1x10^-4 substitutions per site per year (95% highest posterior density (HPD) = 2.3–6.05x10^-4); GL: 8.7x10^-4 [HPD₉₅ = 4.1–13.4 x10^-4]), imply that some transmission events would be expected to occur without detectable evolution in the partial genomes that we sequenced. This led to branch lengths that were less than the assumed generation time of RABV, which caused an upward bias in inferred transmission distances per infection. We corrected for this bias by forcing all branches to have a minimum generation time of 1 infection.

Model selection

The probability of initial colonization (γ) was best explained by the minimum distance to infections, infection density in the local neighborhood, season, and spatial kernel (see AUC2 of Models 1–7; Table 1). Similarly, the best two-effect models included pairs of these effects (distance + neighborhood, distance + season, kernel + season; Models 9, 11, 15; Table 1). Accounting for direction (i.e., N∙T or E∙T) provided more minor improvements to quantifying initial colonization probability (AUC2 Table 1). The inclusion of human population size on p (rabies prevalence in occupied grid cells) significantly improved predictive power of the model (see AUC1 and looCV in Table 1, Models 1b, 9b, 11b, 16b) because it explained some of the prevalence variation due to differences in the number of samples reported by humans. We considered Model 11b to be the best predictive model because it had the lowest looCV, and we used this model for prediction. However, in order to study the effects of the most significant single predictors (neighborhood, season and distance), we fit Model 16 to quantify the relative importance of these factors (results presented in Fig 2).

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Fig 2. Effects of covariates on initial colonization probability.

Predicted γ as a function of different factors using Model 16 Transmission distance = 3.9 km 95% CI (1.4, 11.3) (neighborhood+distance+season, Table 1). A) Decay of initial colonization probability with distance to nearest infected grid cell in km. B) Proportion of local neighborhood (“queen’s neighbors”) infected. C) Season factor with two levels. Median values of initial colonization probability are indicated by the horizontal black line. D) Relative contribution of different effects (see section SM5 in S1 Text for calculations). Shading in A and B indicates 95% credible intervals.

https://doi.org/10.1371/journal.pntd.0005822.g002

Effects on initial colonization probability

We modeled the effect of distance to nearest infection using an exponential decay function. Using the estimated decay rate parameter, we calculated the distance at which initial colonization probability decayed to half its maximum (referred to as “transmission distance”) using the asymptotic limit as the minimum (see section SM54 in S1 Text for calculation). We interpreted the transmission distance as a measure of skunk-to-skunk transmission distance. Transmission distance was 3.9 km (95% credible intervals (CI₉₅): 1.4, 11.3; Fig 2A). Similarly, using the genetic data independently, the mean distance per viral generation (another proxy for skunk-to-skunk transmission distance) was estimated to be 2.3 km (95% Highest Posterior Density (HPD₉₅) = 0.04–5.7; Fig 3C).

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Fig 3. Phylogeographic inference of viral spatial dynamics.

A) Spatio-temporal projection of the maximum clade credibility tree from the phylogeographic analysis. Transparent polygons indicate the 80% HPD of infected area through time. Grey shading on the left indicates landscape that is above 1829 m. Grey lines indicate major highways. Blue lines indicate all waterways, including very minor and ephemeral ones. Stars indicate the centers of major cities (as in Fig 1). B) Spatial expansion of RABV in skunks, displayed as the distance from the inferred epizootic origin in the phylogeographic analysis. Grey shading indicates the HPD₉₅. The solid black line is the median distance. C) The distance traversed by each branch in the phylogenetic tree divided by the expected number of infections along that branch, assuming a generation time of 30 days. The histogram shows distances calculated from the 144 branches of 500 randomly-selected trees from the posterior distribution of the phylogeographic analysis. Grey bars are the inner HPD₉₅.

https://doi.org/10.1371/journal.pntd.0005822.g003

Initial colonization probability increased exponentially with local neighborhood infection density, especially after 0.2 (i.e., > 1grid cell occupied; Fig 2B, Model 16, Table 1). Initial colonization probability was substantially higher during the spring/summer season relative to the fall/winter season (Fig 2C, Model 16). Of the 76% of variation in occupancy probability that was explained by the three-factor model (AUC2, Model 16, Table 1), 57% of initial colonization probability was explained by local effects (distance or neighborhood), 41% was explained by season and 2% was explained by other factors (which could include translocation or other unknown factors; Fig 2D). Although direction explained less variation in initial occupancy probability relative to other factors (Table 1, Models 3 and 4 versus Models 2, 5, 6, and 7), there appeared to be a significant increase in initial colonization probability in the southerly direction with time during the study period indicating southerly spread of the disease (Fig SR1.1 in S1 Figures). However, east-west movement over time did not show significant visual trends (Fig SR1.2 in S1 Figures).

Prediction

The “best” predictive model we examined (Model 11b, Table 1) performed very well at predicting rabies cases in space and time using parameters trained on different data than the data being predicted (Table 1, out-of-sample prediction AUC1 & AUC2). The model also performed very well at out-of-sample prediction using only the initial conditions, sample size data and parameters fit to the full data (Fig SR2.1 in S1 Figures). Average occupancy probability across the study area was low over time (Fig 4, top row, mean: 0.034, CI₉₅: 0.006–0.24) but prevalence in sampled dead skunks in occupied grid cells was high (Table 2, mean: 0.91, CI₉₅: 0.86, 0.95). Occupancy probability was predicted to decrease substantially in the north and increase in the south during the time course of the study period (Fig 4, Figs SR3.1 & SR3.2 in S1 Figures). Also, earlier during the time course occupied patches were larger and more contiguous whereas later occupied patches became more fragmented and isolated (Fig 4). The overall rate of southerly spread estimated by the occupancy model was predicted to be 21.1 km/year (CI₉₅: 16.7, 25.5, Fig 5). This was similar to the phylogeographic diffusion rate assuming either gamma 21.8 [HPD₉₅: 14.9–29.0] or homogenous rate 22.6 [HPD₉₅: 15.3–29.7] models. Mean branch velocities from the phylogeographic models were slightly higher but had overlapping HPD₉₅s with the genetic and occupancy-based models (gamma: 28.4 [HPD₉₅: 19.6–39.8] or homogenous 29.8 [HPD₉₅: 20.8–39.8]).

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Fig 4. Rabies occupancy probability over time.

Top. Occupancy probability for all sites over time (blue line, shading: 95% credible intervals) using Model 11b, Table 1). Grey bars: number of samples collected. Middle. Occupancy probability in space and time. Bottom. Coefficient of variation for occupancy probabilities over space and time. Middle and bottom: Data were aggregated over a 6-month time frame (corresponding to the X-axis labels of the top row plot). Black triangles represent city locations (Fort Collins, Greeley, Boulder and Longmont); sizes scaled to the human population size of the cities. Topographical divide from Fig 1, which indicates land above 1829 m (6000 feet), is indicated by the black line.

https://doi.org/10.1371/journal.pntd.0005822.g004

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Table 2. Parameter estimates from the best predictive model of occupancy probability (Model 11b, Table 1) and the phylogeographic model.

https://doi.org/10.1371/journal.pntd.0005822.t002

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Fig 5. Rates of spatial spread.

Model 11b was used to predict spatial spread of rabies. Points are the predicted distance in km of new colonizations from the southern-most grid-cell row. Size of the points corresponds to numbers of new colonizations at a given distance from the southern-most row. The slope (red line) gives a monthly rate of southerly spread (21.1 km/yr). Shaded area gives the 95% confidence intervals of the slope. The 6-month intervals shown on the X-axis correspond to those in Fig 4.

https://doi.org/10.1371/journal.pntd.0005822.g005