Using spatially explicit surveillance models to provide confidence in the eradication of an invasive ant

Effective detection plays an important role in the surveillance and management of invasive species. Invasive ants are very difficult to eradicate and are prone to imperfect detection because of their small size and cryptic nature. Here we demonstrate the use of spatially explicit surveillance models to estimate the probability that Argentine ants (Linepithema humile) have been eradicated from an offshore island site, given their absence across four surveys and three surveillance methods, conducted since ant control was applied. The probability of eradication increased sharply as each survey was conducted. Using all surveys and surveillance methods combined, the overall median probability of eradication of Argentine ants was 0.96. There was a high level of confidence in this result, with a high Credible Interval Value of 0.87. Our results demonstrate the value of spatially explicit surveillance models for the likelihood of eradication of Argentine ants. We argue that such models are vital to give confidence in eradication programs, especially from highly valued conservation areas such as offshore islands.


Results and Discussion
Probability of eradication. The probability of eradication increased sharply as each survey was conducted, reflecting the cumulative surveillance effort and system sensitivity with no ants detected ( Table 1). The overall median probability of eradication of Argentine ants from all survey and surveillance methods using parameter set 2 was 0.957 (Table 1). There was a high level of confidence in this result, with a high Credible Interval Value of 0.87 (i.e. 87% of the probability of eradication estimates were greater than the threshold value of 0.9). The overall sensitivity of the surveillance system was relatively insensitive to the parameter set used, although parameter set 2, with different decay curves for different search methods, gave greater overall sensitivity (Table 1). In addition, the estimates of the probability of eradication were insensitive to the prior distributions. The parameter set1 with highly informative priors produced a posterior probability of eradication of 0.943, but all other trials were above 0.95. Using an uninformed prior (i.e. a Pert with a range 0-1) that is skewed low (value = 0.25) requires sizable surveillance data in order to achieve a posterior > 0.95. Given that our results meet or exceed this target, there is substantial reason to be confident that Argentine ants have been eradicated.

Eradication of ants.
Invasive ants are generally regarded as being very difficult to eradicate 19,20 . Eradication is often possible from small scale plots, or when a species is discovered early in its establishment phase in a new area. Larger-scale eradication of invasive ants, whilst possible, remains much more demanding 19,21 . Currently one of the major challenges for Argentine ant eradication programmes is finding the location of small 'survivor' nests after the initial control operation has occurred. Such nests remain restricted to a very small area (< few metres), and thus a large proportion of time and resources are spent on finding and controlling only 1% of ants.
Recent work on formulating requirements for the eradication success of invasive ants includes a number of factors, such as the importance of operational requirements (e.g. lines of authority, sufficient resources for the task) and on early intervention 19 , which currently gives a far greater chance of success for invertebrate eradication efforts 20 . However, improving detectability at low densities is also one of the most critical factors to increase the success of ant eradication 19 . Typically 'eradication success' is declared after a certain time when the pest has not been found, for example, two years post-treatment has been the minimum standard used for most published ant 22 and vertebrate eradications 23 . However, the lack of detection after just waiting a certain time period does not prove a pest is absent. A pest may still be present in an area somewhere but remains undetected through poor surveillance coverage and/or intensity, or purely by chance. That is why it is critical to interpret negative surveillance data (non-detections) in light of the surveillance system sensitivity.
Proof of eradication modelling. Modelling the probability of eradication is a valuable technique, providing a level of confidence in the overall result and justification for the allocation of resources required to meet a set probability level. This type of modelling is also very useful in understanding the resources needed to achieve certain confidence levels. For example, modelling will help avoid prematurely declaring success due to insufficient survey effort or, conversely, avoid wasting resources on surveys when the pest has been eradicated from an area. Modelling may also identify spatial gaps in surveys, where search effort has been too low, allowing for subsequent surveys to be directed to search these areas. This ensures there are no refuges available for a pest population, and ultimately gives increased confidence of the overall probability of eradication.

Conclusions
Here, we demonstrated a technique that estimated the confidence in the eradication of an invasive ant based on defining the relationship between surveillance effort and the probability of detecting the target organism, a technique previously used in eradications of vertebrates 23 and wildlife disease 24 . In general, this modelling technique provides useful information about the probability of eradication of a pest species, and thus gives confidence of eradication at very low densities. Such modelling could be applied to many populations of Argentine ants, and assist in the eradication of this pest from many locations around the world.  We used several 'paths' to cover the entire Schoolhouse bay area. These paths were recorded with GPS and used for all three surveillance methods ( Table 2; Fig. 1). Areas covered by houses were excluded from the area of inference because no internal searching of houses has been carried out by us (but informally by residents). Data was analysed with a spatially explicit surveillance model, as described below. The model estimated the sampling sensitivity for each survey and updated the overall probability of eradication with each survey.
Proof of eradication modeling. We used a spatially explicit surveillance model of Anderson et al. 24 , initially developed for disease eradication in vertebrates, to estimate the probability that Argentine ants had been eradicated from the Schoolhouse Bay area given negative surveillance results since control was applied. The model estimated the overall sensitivity of the surveillance (i.e. the probability of detecting Argentine ants given they were present) using a 1-m grid-cell resolution across the modelled landscape. Following a survey where ants were not detected, the sensitivity of the surveillance method was used to update the probability of ant eradication using Bayes theorem.
Under the framework from the aforementioned model, there are two key parameters for estimating the probability of detection in a given grid cell 24 . The first was the maximum probability of detecting an ant or its nest (g 0 ) if the surveillance device was placed directly on top of it. The second was a spatial decay parameter (σ), which modifies how the probability of detection declined with distance from the surveillance device. Consequently, a device can detect ants or nests that were in the same or neighboring grid cells.
The baited vials were spaced 2 m apart along the path. The g 0 and σ parameters for the baited vials were derived by fitting a half-normal function to the curve shown in Fig. 2 of Stringer et al. 6 , which described the probability of detecting foragers from a small nest of the fire ant (Solenopsis invicta) using baited vials placed out for 1-2 hours at different distances from the nest. The fitted values were: g 0 = 0.548 and σ = 1.331 (Fig. 2). Although the above parameters were developed for fire ants, they are biologically relevant to Argentine ants, as when ant colonies (no matter what species) are very small, they have a very small foraging area, and thus their detection (by baits, visual searching, etc.) is harder. We developed two model sets. Because human visual searchers and sniffer dogs move continuously along a path, we divided the path up into regularly spaced 'search points' . In the first instance (parameter set1) we assumed the same g 0 and σ values as those described by Stringer et al. 6 and 2-m spacing between search points for all three surveillance methods. Secondly (parameter set2), we attempted to better represent the detection  Table 2. Details of the surveillance methods utilised along different paths (see Fig. 2) and survey periods. '×' surveillance was undertaken.
range of visual searching and sniffer dogs, for which information was specifically available for Argentine ants 25,26 .
We modified values from Stringer et al. 6 , restricting visual searching to a 1-m decay and increasing the decay of the sniffer dog to 4 m. We also changed the search-point spacing to give an equal detection probability along the path bearing (distance = 0) as a result of the overlapping detection kernels. For the visual searchers in parameter set2 we used g 0 = 0.733, σ = 0.4 and a search-point spacing of 1 m. For the sniffer dogs in parameter set2 we used g 0 = 0.750, σ = 1.65 and a search-point spacing of 2 m (Fig. 2). This was equivalent to a per-cell probability of detection 0.75 and 0.9 of detecting an ant nest if the search point of person or dog, respectively, was in the same cell as the ant. In both parameter sets σ and g 0 parameter values were allocated a standard deviation equivalent to a 10% Coefficient of Variation around the mean value to account for the uncertainty in model parameters. A relatively pessimistic 'prior' probability of eradication being successful was specified using a Pert distribution with a most likely value of 0.25 (range 0-0.75). The 'prior' is our estimated probability that ants were absent from the island at the time of the first post-treatment surveys. We suggest that the value of 0.25 is appropriate given the uncertainty in the control operation. We assessed the sensitivity of the estimated probability of eradication to the prior distribution. Retaining a most likely prior value of 0.25, analysis for both parameter sets were repeated using a highly informative distribution (range 0-0.4) and an uninformative distribution (0-1). The design prevalence or the minimum number of 1-m 2 grid cells within the Schoolhouse Bay Area likely to be infested if ants were still present was set to one (i.e. one surviving colony).
The annual probability of re-introduction of ants into Schoolhouse Bay was assumed to be very low (because of ongoing education of the public and surveillance in the area), and described by a Pert distribution with a most likely value of 0.01 (range 0-0.012). Point locations of the baited vials used in this study were derived by spacing 500 vials along the Schoolhouse Bay search paths 27 using the ArcGIS 10.2.1 'Construct Points' function. The same tool was used to assign equally spaced search points for the visual searchers and sniffer dog.