Representation of space and time

The model simulates X. fastidiosa outbreaks with discrete annual timesteps and over a 200 x 200 m discrete grid covering the infected region of Puglia, southern Italy (195,491 grid cells; Fig 2). This grid cell size approximates the median size of olive groves used in our previous modelling of local disease dynamics [10], while reducing the computational demands of a higher resolution grid. The number of olive trees in each grid cell (N) was estimated from its proportion cover of olive orchards (Ω, see Fig A in S1 Appendix), derived from the InnovaPuglia Spa 2011 land use map (Uso del Suolo – 2011; https://dati.puglia.it/ckan/dataset/uso-del-suolo-2011-uds), and the median olive tree density of 91 ha^-1 in the olive orchard plots used in our previous study [10]. Based on this, the model assumed a grid cell with 100% olive cover would contain 364 olive trees (i.e., N = 364Ω).

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Fig 2. Maps showing the locations of Xylella fastidiosa inspections and laboratory-confirmed positive detections carried out in the modelled region and in each model year (May 1^st to April 30^th of the named years), which were used to determine the locations of inspections in the model simulations.

Each inspection covered a 100 x 100 m area, meaning up to four inspections per year in each model grid cell (200 x 200 m). The presumed approximate location of introduction is also shown. The base map is reproduced from the GADM Global administrative areas dataset, under Creative Commons Attribution-ShareAlike 2.0 (https://gadm.org/license.html).

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Model years used May 1^st as their start date, corresponding to the approximate start of the adult vector flight period [3]. All simulations initiated the epidemic at the approximate ‘ground zero’ area where tree decline was first detected (Fig 2). However, the year of X. fastidiosa introduction (Y₀) was varied as a parameter to estimate with the model.

Epidemiological model

Within infected grid cells, X. fastidiosa transmission and disease progression was simulated with a stochastic and spatial version of an existing discrete-time compartmental model [10]. The structure and parameters of the existing model were estimated to best explain 2–3 year snapshots of disease progression in 17 olive groves [10], justifying the current model compartments and informative priors for several key parameters.

Model compartments are the uninfected susceptible trees (S) and three classes of infected trees: asymptomatic (I_A, assumed to be non-infective to vectors, so equivalent to the more standard ‘exposed’ or ‘pre-symptomatic’ compartments), symptomatic (I_S, assumed to be infectious, so equivalent to the standard ‘infected’ compartment) and desiccated (I_D, most foliage is scorched but assumed to still be infectious at a lower level through basal regrowth of green leaves). All trees that become infected progress over time from I_A to I_S to I_D (Fig 1). The transmission model was based on our previous modelling [10], which concluded that asymptomatic trees do not drive transmission but could not rule out the possibility of transmission from desiccated trees, which could occur from any remaining live foliage or basal resprouting (suckers).

During each annual time step of simulations, individual trees progress through these compartments via Bernoulli trials using transition probabilities according to the following equations. The probability of each individual uninfected tree in a grid cell i becoming infected in year t () depends on the current density of infective trees (i.e., symptomatic or desiccated) in the same grid cell and surrounding areas (Fig 1) formulated as:

Parameter β is the contact rate for infected olive trees with symptoms and b_D is the relative infectivity of desiccated trees. Since the original model analysis found similar support for a model in which desiccated trees were not infective (equivalent to SEIR) and a model where they were fully infective (equivalent to SEI) [10], here we included parameter b_D as a proportion, reducing the infectiveness of desiccated trees relative to symptomatic trees. To represent infection from multiple locations, j indexes the summation over all grid cells in the landscape (including i). The function K represents pathogen dispersal by scaling a reduction in transmission probability with increasing distance from cell j to cell i (see below for further details).

Based on our previous modelling [10], the infected asymptomatic trees are assumed to develop symptoms at a constant rate, while desiccation of symptomatic trees is assumed to occur at a constant rate after an initial delay period of τ years, representing a minimum time for desiccation. Annual probabilities of individual trees changing from asymptomatic to symptomatic () and symptomatic to desiccated () are:

Parameter T_A is the mean length of the asymptomatic period (years). Variable y is the number of years the tree has been symptomatic, τ is the desiccation delay parameter (minimum years for desiccation) and T_D is the mean time to desiccation after this delay period. We set τ = 3 years in all simulations since in our previous modelling there was very low support for models with other values of τ [10].

The dispersal function K(i,j) represents a decline in X. fastidiosa transmission with increasing distance from source j to destination i (d_i,j) which is implicitly linked to vector dispersal [34,39]. We formulated K as a mixture of short- and long-distance functions [34], to represent local diffusive-like movements of vectors as well as rarer and less predictable jumps across large distances that can spread pathogens rapidly to new regions [40,41]. Short distance diffusive movement was modelled with a Gaussian function and long-distance dispersal (LDD) with an exponential function:

Parameters m_short and m_long determine the spatial scales of distance decay at both scales. Parameter L is the proportion of LDD, which was fixed to a single low number (L = 10^-6). L was fixed to avoid confounding the fitting of L with that of the scale parameters m_short and m_long. We chose this value by “trial and error” to give realistic patchy spread patterns over a wide range of other parameter values.

A further function M(i,j) was included to represent different LDD mechanisms in a computationally minimal way that did not add extra parameters to estimate. M(i,j) was formulated to yield a proportion that modifies the amount of LDD from cell j to cell i, approximating three alternative scenarios:

1. A “basic” scenario with isotropic LDD in which M(i,j) = 1.
2. A “wind” scenario with anisotropic LDD following prevailing winds in the region, to represent wind-assisted vector flights [32]. We analysed wind directions in the region to derive a function for M(i,j) giving a value of 1 when the direction from j to i was aligned to the prevailing wind (approximately southeast), and proportionately lower values when the direction from j to i was in a less frequent wind direction (Figs B and C in S1 Appendix).
3. A “road” scenario in which dispersing vectors preferentially landed near to major roads as a result of hitchhiking on stationary vehicles in or near olive orchards and then being deposited along the roadside as the vehicle travels [33]. We modelled M(i,j) as declining from 1 towards 0 with increased distance of the destination grid cell from a major road (as motorways, state highways and provincial roads; see Fig D in S1 Appendix) [33]. We modelled M(i,j) as declining from 1 towards 0 with increased distance of the destination grid cell from a major road scaled using the short-distance kernel (Fig D in S1 Appendix).

Containment strategy

Disease surveillance was modelled so that we could compare the spread observed in the database of the official monitoring programme (http://www.emergenzaxylella.it/portal/portale_gestione_agricoltura) with spread patterns produced by the spatial epidemiological model. This was important since spatial variation in the surveillance locations (Fig 2) has a major influence on the observed spread, as opposed to unbiased random surveillance locations. Additionally, tree felling after positive inspections [13] may have slowed the spread of the pathogen, necessitating inclusion of these measures in the spread model.

The surveillance model was a simplified representation of the actual regional containment strategy [12,13]. Real surveillance involves visual inspections by professional surveyors of the Regional Phytosanitary Service [13], followed by sampling of trees and laboratory testing for the disease. If any samples test positive for X. fastidiosa this triggers felling of all olive trees within a 100 m radius [12,13].

Inspection locations from 2013 to the end of April 2020 (the latest complete year at the time of the study) were derived from records of testing individual geolocated olive trees in the monitoring programme database (http://www.emergenzaxylella.it/portal/portale_gestione_agricoltura). A small number of outlying isolated records were removed from the data as we suspected they may be ‘incidental’ inspections following reported incidental sightings of disease symptoms rather than part of the systematic survey. To define inspections for the model, individual tree GPS coordinates in the database from the same model year were assigned to the 100 x 100 m grid of the surveillance program. As such up to four inspections could occur per year in each 200 x 200 m model grid cell (Fig 2).

In total, the data contained 495,626 inspections of which 2,329 were positive for X. fastidiosa (0.47%). The surveillance intensity and spatial pattern changed year on year (Fig 2) as the outbreak progressed and demarcated areas were updated. Initial surveillance was an area-wide low intensity monitoring to delimit the extent of the outbreak (2013/14). Surveillance then focussed on the most northerly part of the infection front (2014/15) and then targeted northerly clusters of infection with the aim of eradicating them (2015/16). Latterly, surveillance focused on containment and buffer zones spanning and extending beyond the moving infection front (2016/17, onwards) with the aim of containing the spread [13].

During each simulated visual inspection by professional surveyors of the Regional Phytosanitary Service, we assumed the probability of detecting visible symptoms increased with the proportion of trees that were symptomatic. As such, the probability of visual detection in an inspection of cell i in year t was formulated as

where is the proportion of trees with symptoms. v is a free parameter scaling the inefficiency in visual detection (i.e., small values of v increase the probability of detecting rare symptoms).

Following visual inspection, olive leaves are sampled for laboratory testing. The number of trees sampled per inspection depended on whether symptoms had been visually detected and was drawn randomly from the empirical distribution of numbers of trees tested per inspection in the monitoring database (mean of 3.2 trees when symptoms were seen and 1.9 trees when no symptoms were seen). When symptoms were seen, we assumed only infected symptomatic trees were selected for testing. When symptoms were not found, we assumed only randomly-selected asymptomatic trees were testing so the proportion infected is . We assumed that some sampled leaves from an infected tree might contain no (or undetectably low) levels of X. fastidiosa, so introduced a parameter u for the probability of sampling a leaf with detectable X. fastidiosa load from an infected tree. Finally, we assumed laboratory testing would have a probability of false negative results Z, which differs for symptomatic or asymptomatic trees, as testing procedures depend on symptom status and symptomatic trees will likely have higher bacterial load and therefore lower false negatives.

Therefore, the probability of each sample from cell i returning a positive result in year t () was formulated as:

The false negative rates (Z) were estimated using Latent Class Analysis on the monitoring data as Z_symp = 0.0512 and Z_asymp = 0.0614 for symptomatic and asymptomatic trees, respectively (Section S1.3 in S1 Appendix). False positives were not included in the model as this analysis found negligible rates of false positives.

Using these probabilities, Bernoulli trials were used to simulate whether each sampled tree was confirmed as X. fastidiosa positive. If any tree returned a laboratory-confirmed positive that inspection was deemed positive. Following a positive inspection, all trees in the 200 x 200 m grid cell were felled, approximating the 100 m felling radius used in the actual containment strategy [12,13].

Approximate Bayesian computation

Approximate Bayesian Computation (ABC) using rejection sampling [37,42] was used for model selection among the three long-distance dispersal scenarios and then for parameter estimation of the chosen dispersal scenario. ABC is based on matching summary statistics from data to those produced by the model rather than using likelihoods [43,44]. It was used since we considered it intractable to produce a likelihood function for the observed spread data derived from the monitoring scheme. Furthermore, ABC is well suited for parameter estimation for complex stochastic models as large numbers of model runs can be run in parallel on high-performance computing clusters [42].

To produce reliable results from ABC, a range of different summary statistics relevant to the pattern of interest should be used [43]. From the monitoring database, we chose the following summary statistics to capture the pattern of spread for each year between 2013 and 2020: number of positive inspections; spread distance (99^th percentile distance from introduction point for positive inspections); clustering of positive inspections (local Moran’s I on positive grid cells, weighted using a Gaussian moving window with 200 m standard deviation); and association of positive inspections with major roads (median distance of positive inspections to a major road, which could indicate an effect of roads on the spread [33]).

In addition, we obtained independent estimates of the size of the new area with severe damage in each year from 2010 to 2017 from a previous remote sensing study [45]. In that study, remote sensing was able to detect large olive orchards (>12.5 ha) with high levels of desiccation. X. fastidiosa progression in olives trees is monitored on a visual severity scale from 0 (unaffected) to 5 (canopy entirely desiccated) [46,47]. In the remote sensing it was estimated that a mean severity score above ~2.9 was detectable [45]. To estimate an equivalent area of severe damage from the model, we first identified the model grid cells corresponding to large olive orchards, and then estimated their mean disease severity score. The modelled severity score of individual trees increased in yearly increments for symptomatic trees, reaching a maximum of 5 for desiccated or felled trees. This aligns with the definition of the severity scores and recorded within cell disease progression [10]. The new area of large orchards with mean severity >2.9 was then calculated for each year of the simulation.

ABC rejection sampling was performed using these summary statistics [43,44]. First, 10⁵ simulations were performed for each long-distance dispersal scenario (basic, wind and road) with parameter values drawn randomly from the prior distributions in Table 1.

For each simulation, a standardised distance ρ between the observed summary statistics (m_obs) and the summary statistics calculated from simulated outbreaks (m_sim) was calculated as

where k indexes over all the summary statistics and σ_sim,k is the standard deviation of summary statistic k among all the simulations. Those simulations with ρ less than a certain acceptance threshold (detailed below) are accepted as the best matches to the observed data and their parameter values form the posterior distribution estimates [43,44].

The relative support for the more complex dispersal scenarios (wind and road) was evaluated by calculating Bayes Factors compared to the simpler basic scenario (Section S1.4 in S1 Appendix). Having selected the single dispersal scenario with the greatest support from Bayes Factors, posterior parameters were estimated for the chosen dispersal model. To improve the estimation, a further 4x10⁵ simulations were performed for this model, giving a total of 5x10⁵ simulations on which rejection sampling was implemented. An acceptance threshold of 0.0002 was used giving a posterior sample of 100 simulations [42].

The accuracy of the parameter estimation was assessed using a leave-one-out cross validation (LOO CV) procedure [42,43]. For this, a randomly selected simulation was used as the data for parameter estimation (i.e., its simulated summary statistics were used as m_obs), rejection sampling was performed with the remaining simulations and the estimated parameter values were compared to the true values used for the focal simulation. We ran LOO CV 10,000 times with randomly chosen focal simulations, to calculate both the RMSE for the parameter estimates and the proportion of parameter estimates whose 95% credible interval contained the true values.

Containment modelling

Finally, the calibrated model was used to simulate spread under the current containment strategy, where trees are felled after positive detections, and for a “no containment” scenario with no felling performed. One simulation was run for each of the 100 posterior samples from the ABC, and the results were averaged to estimate pathogen spread for both scenarios in terms of the total number of infected trees, number of felled trees and spread distance (quantified as the 99^th percentile distance of infected trees from the origin point).

Comparison of dispersal scenarios

The Bayes Factor analysis (Section S.2.1 in S1 Appendix) found lower support for the highly simplified ‘road’ long-distance dispersal model than the ‘basic’ model with isotropic long-distance dispersal. The ‘wind’ long-distance dispersal model was slightly better supported than the ‘basic’ model, but not sufficiently so to select it over the conceptually-simpler basic model. Therefore, the basic model was used in the remainder of this study.

Parameter estimation for the basic dispersal model

Leave-one-out cross validation (LOO CV) of the rejection sampling on the basic model showed that ABC parameter estimation was reliable for the two dispersal parameters and introduction year, shown by a 1:1 relationship between the parameter value and its estimate (Fig 3, panels for m_short, m_long and Y₀). The LOO CV showed ABC was not effective at estimating the three disease dynamics parameters with highly informative priors (β, T_A, and T_D), but this was not a concern as we already had good estimates of these from our previous study [10] and used informative priors. The infectivity of desiccated trees (b_D) and two inspection parameters (v and u) were also estimated poorly, although for u the LOO CV yielded accurate estimates in about half of the cases (Fig 3). These parameters had uninformative priors capturing a wide range of plausible values, and so even though we could not estimate them, the posterior simulations captured a wide range of uncertainty in their values.

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Fig 3. Accuracy of parameter estimation estimated with 10,000 leave-one-out cross validations (LOO CV) of the Approximate Bayesian Computation rejection sampling and acceptance threshold of 2x10^-4.

Background shading shows the number of cross-validations. Where the highest densities of values align to the grey 1:1 lines, this indicates that parameter estimation is reliable. See Table 1 for full parameter explanations: β = transmission rate; T_A =asymptomatic period (years); T_D = desiccation period (years); b_D = infectiveness of desiccated trees; m_short = short-range dispersal distance (km); m_long = long-distance dispersal distance (km); Y₀ = Introduction year; v = visual inspection inefficiency; u = leaf sampling probability.

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Posterior distributions of the three identifiable parameters are shown in Fig 4. The median estimates for the short- and long-distance dispersal ranges were 0.317 km (95% CI [0.146, 0.606]) and 6.376 km (95% CI [4.990, 10.881]), respectively. The median estimate of the introduction year Y₀ was 2003 (95% CI [2000, 2009]), although the posterior was somewhat bimodal.

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Fig 4. Posterior estimates for the three identifiable parameters in the basic model shown as kernel density plots with vertical lines showing the medians (solid lines) and 95% credible intervals (dashed lines).

Grey background histograms show the uniform prior distributions.

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Posterior simulations

Using these parameter estimates, posterior predictive checks indicated a good correspondence between observed and simulated summary statistics (Fig F in S1 Appendix). The main exception was that the model over-predicted the number of positive inspections in 2017/18. The average modelled spread pattern produced from these parameters is shown in Fig 5, noting that individual simulations produce a patchier and more stochastic spread pattern than the smoother average of many simulations shown in Fig 5 (see Fig G in S1 Appendix for an example). This was especially so at the invasion front, where new infection foci are seeded and grow.

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Fig 5. Average model Xylella fastidiosa spread, shown as the mean number of infected trees (I_A+ I_S + I_D) per grid cell at the end of each modelled year in single simulations with each of the 100 posterior parameter values.

Simulations started at the estimated introduction year for each posterior parameter, but only 2008/9 onwards is shown as there was little earlier spread. For scale, the grey background grid represents 20x20 km divisions. The base map is reproduced from the GADM Global administrative areas dataset, under Creative Commons Attribution-ShareAlike 2.0 (https://gadm.org/license.html).

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Management scenario simulations

Simulations in which no felling occurred were compared to the default model of felling after positive inspections (Fig 6). Linear regression of the spread distance for infected trees against year gives an estimated spread rate of 5.7 km y^-1 (95% CI [5.4, 5.9]) between 2013 and 2020 when felling was applied. This compares to 7.2 km y^-1 (95% CI [6.9, 7.5]) without felling. This reduction in modelled X. fastidiosa spread rate appears to have been driven by a large amount of felling in the model in the year 2017/18, when there were high numbers of positive inspections and felling (Figs 6 and Fig F in S1 Appendix). Further, jointly considering the spread rate of both the infected and felled trees (7.6 km y^-1, 95% CI [7.3, 7.9]) there was similar spread to the simulation with no felling. Therefore, modelled felling overall had only a small influence on simulated pathogen spread, in terms of distance from the origin, and reductions in trees lost to infection were largely countered by losses from felling (Fig 6).

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Fig 6. Modelled effect of containment measures on spread of Xylella fastidiosa in 100 simulations from the posterior model parameters.

Panels show the mean (a) trees infected and felled and (b) 99^th percentile spread distance. Also shown is a “none” scenario in which no felling is applied after positive infection detections. Ribbons are bootstrapped 95% confidence intervals for the means.

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