# Bayesian inference for spatio-temporal stochastic transmission of plant disease in the presence of roguing: A case study to characterise the dispersal of Flavescence dorée

Hola K Adrakey, Gavin J Gibson, Sandrine Eveillard, Sylvie Malembic-Maher, Frederic Fabre

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## To cite this version:

Hola K Adrakey, Gavin J Gibson, Sandrine Eveillard, Sylvie Malembic-Maher, Frederic Fabre. Bayesian inference for spatio-temporal stochastic transmission of plant disease in the presence of roguing: A case study to characterise the dispersal of Flavescence dorée. PLoS Computational Biology, 2023, 19 (9), pp.e1011399. 10.1371/journal.pcbi.1011399. hal-04216679

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RESEARCH ARTICLE

# Bayesian inference for spatio-temporal stochastic transmission of plant disease in the presence of roguing: A case study to characterise the dispersal of Flavescence dorée

Hola K. Adrakey ¹, Gavin J. Gibson¹,², Sandrine Eveillard¹,³, Sylvie Malembic-Maher¹,³,
Frederic Fabre¹,¹*

1 UMR SAVE, INRAE, Bordeaux Sciences Agro, Villenave d'Ornon, France, 2 Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, United Kingdom, 3 UMR BFP, INRAE, Univ. Bordeaux, Villenave d'Ornon, France

* frederic.fabre@inrae.fr

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## Abstract

Estimating the distance at which pathogens disperse from one season to the next is crucial for designing efficient control strategies for invasive plant pathogens and a major milestone in the reduction of pesticide use in agriculture. However, we still lack such estimates for many diseases, especially for insect-vectored pathogens, such as Flavescence dorée (FD). FD is a quarantine disease threatening European vineyards. Its management is based on mandatory insecticide treatments and the removal of infected plants identified during annual surveys. This paper introduces a general statistical framework to model the epidemiological dynamics of FD in a mechanistic manner that can take into account missing hosts in surveyed fields (resulting from infected plant removals). We parameterized the model using Markov chain Monte Carlo (MCMC) and data augmentation from surveillance data gathered in Bordeaux vineyards. The data mainly consist of two snapshot maps of the infectious status of all the plants in three adjacent fields during two consecutive years. We demonstrate that heavy-tailed dispersal kernels best fit the spread of FD and that on average, 50% (resp. 80%) of new infection occurs within 10.5 m (resp. 22.2 m) of the source plant. These values are in agreement with estimates of the flying capacity of *Scaphoideus titanus*, the leafhopper vector of FD, reported in the literature using mark-capture techniques. Simulations of simple removal scenarios using the fitted model suggest that cryptic infection hampered FD management. Future efforts should explore whether strategies relying on reactive host removal can improve FD management.

## Author summary

The dispersal of pathogen propagules is an important feature of spatial epidemiology that has a major impact on the prevalence and distribution of disease in a population. In

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## OPEN ACCESS

**Citation:** Adrakey HK, Gibson GJ, Eveillard S, Malembic-Maher S, Fabre F (2023) Bayesian inference for spatio-temporal stochastic transmission of plant disease in the presence of roguing: A case study to characterise the dispersal of Flavescence dorée. PLoS Comput Biol 19(9): e1011399. https://doi.org/10.1371/journal.pcbi.1011399

**Editor:** Alex Perkins, University of Notre Dame, UNITED STATES

**Received:** January 27, 2023

**Accepted:** July 28, 2023

**Published:** September 1, 2023

**Copyright:** © 2023 Adrakey et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability Statement:** The authors confirm that all data underlying the findings are fully available without restriction. The FD dataset, MCMC codes (written in C language) and scripts to generate the figures and tables (written in R language) are available in INRAE Dataverse at https://doi.org/10.57745/YXOEHX.

**Funding:** This study received financial support from Plan National Déperissément du Vignoble
under research contracts RISCA and Co-Act2. H.A. was funded by RISCA and Co-Act2 through a postdoctoral fellowship. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests:** The authors have declared that no competing interests exist.

agriculture, properly characterising the dispersal of emerging diseases is of great impor-
tance in designing science-based control strategies that allow pesticide use to be reduced.
Although field epidemiological surveys can provide informative data, they are by nature
rare while resulting from the interactions between disease spread and the undergoing sur-
veillance and control. Here, we take advantage of a general statistical framework to model
the epidemiological dynamics of Flavescence dorée (FD), a quarantine disease threatening
European vineyards, in a mechanistic manner that can take into account missing hosts in
surveyed fields (resulting from infected plant removals). We parameterized the model
with a Bayesian approach using mainly two snapshot maps of the infectious status of all
plants in three adjacent fields during two consecutive years. We demonstrate that on aver-
age, 50% (resp. 80%) of new FD infection occurs within 10.5 m (resp. 22.2 m) of the source
plant. Although FD mainly spreads locally from one year to the next, our results also indi-
cate frequent long-distance dispersal events, a feature crucial to consider when designing
control strategies.

**Introduction**

The management of emerging plant pathogens is of foremost importance in the effort to reduce pesticide use in agriculture, particularly since the rate of introduction of alien species is higher than ever due to increasing globalisation [1]. The design of efficient management strategies has much to gain from the recent advances in epidemiological modelling. In particular, spatially explicit stochastic models are able to simulate realistic patterns of epidemic spread and test the effectiveness of a range of management strategies (for a review, see [2] and, more recently, [3–8]). However, for this approach to be feasible, the underlying model parameters must be estimated.

The main issue encountered when fitting spatially explicit mechanistic models is the type
and nature of the available data. First, epidemiological surveys most often provide information
on epidemic dynamics in the presence of control, which must accordingly be explicitly
accounted for in models and fitting procedures [9, 10]. Second, there is a trade-off between the
number of fields that can be studied and the number of seasons over which they can be
observed. The data collected most often contain only partial spatial and/or temporal informa-
tion on the location of infected individuals. Moreover, the timing of true infection events is
typically not observed. The first symptom expression in an individual plant (the most easily
accessible variable during field surveys) can take place a long time after that plant becomes
infected and even infectious. The latter situation results from the presence of cryptic infection,
during which an infectious host is asymptomatic [2, 11]. All of this context involves the exis-
tence of missing data. Data augmentation, a technique first introduced to plant disease epide-
miology by [12], enables tractability of model likelihoods and so, in turn, facilitates rigorous
Bayesian inference for mechanistic models. Within this framework, unobserved events, typi-
cally the timing of infection events, are treated as additional unknown parameters, and the
inference is based on Markov chain Monte Carlo (MCMC) methods. These methods have had
great success in analysing plant epidemic data over several years [3, 9, 10, 13–18].

Here, Markov chain Monte Carlo (MCMC) was used to fit a spatially explicit stochastic
model of Flavescence dorée (FD) spread. FD is a quarantine disease threatening European
vineyards. FD is one of the most damaging diseases in European vineyards. It is caused by the
FD phytoplasma (taxonomic subgroups 16SrV-C and 16SrV-D), a small bacterium that multi-
plies in the phloem sap of infected plants. FD phytoplasma is transmitted from grapevine to
grapevine by the leafhopper vector *Scaphoideus titanus* Ball [19]. Typical symptoms are leaf yellowing or reddening, downward rolling, incomplete lignification of canes, abortion of flowers, and grape wilting. FD phytoplasma, which is endemic to European alders, emerged as a disease in South West France in the 1950s, following the accidental introduction of the amplexophagous vector *S. titanus* from North America [20]. Since then, the disease has spread throughout European vineyards [21]. Due to the severe economic consequences of the disease, FD phytoplasma has been classified as a quarantine organism in Europe since 1993. There is currently no means of curing plants of FD phytoplasma. The current regulation of the disease relies mainly on four mandatory measures: (i) the planting of disease-free material, (ii) the application of insecticides to kill the vector, (iii) the establishment of annual vineyard surveys for monitoring plant infection, and (iv) the removal of infected plants.

The epidemiology of FD in vineyards is driven primarily by the spatio-temporal dynamics of its univoltine vector *S. titanus*. The eggs hatch in May, and there are then five nymphal instars before the first adults appear, usually in July. The adults live from one to two months [22, 23]. The fertilised females can lay eggs from around 10 days after their emergence till the end of summer [22]. Considering as starting point a plant infected during year *t*, the FD epidemic cycle occurs as follows. The vine plant usually becomes infectious from the spring of year *t* + 1 [24–26] as a result, during fall and winter, of its own dormancy stage and of the concomitant diapause of the eggs of the vector. Phytoplasmas are acquired passively in year *t* + 1, through feeding on infected plants, from the first larval stage onwards hatched in May. Winged adults mainly, but also last instar larvae, can transmit the phytoplasma to healthy plants, from late spring to late summer [27, 28]. While some non-specific FD symptoms such as delayed bud break may appear in early spring of year *t* + 1, the set of specific symptoms mostly appear throughout the summer [22, 26]. In the Bordeaux vineyard, the probability of detecting the set of typical FD symptoms is substantially increased after the third week of August [29]. In the vineyard, the epidemiology of FD is also influenced by vine cultivar effects. Although no genetic resistance to FD is known, vine cultivars differ in their susceptibility to (i) phytoplasma transmission by *S. titanus* and (ii) phytoplasma multiplication and diffusion in the plant [30–32]. For example, Merlot and Cabernet Sauvignon, the two main cultivars in the Bordeaux area, display contrasting susceptibilities. Merlot is characterised by a lower proportion of infected shoots and reduced phytoplasma titers in the leaves as compared to the highly susceptible cultivar Cabernet Sauvignon [31]. These features affect FD epidemiology, as phytoplasma acquisition efficiency by *S. titanus* increases with its titers in the plant [27, 33]. Direct cultivar effects on the feeding behaviour of *S. titanus* have also been shown to explain some cultivars' higher susceptibility to FD [34].

A feature of FD epidemiology that is poorly understood is its spatial dispersal kernel. Dispersal kernels represent the statistical distribution of the location of the infected hosts after inoculum dispersal from a focal plant source [35, 36]. Dispersal kernels play a crucial role in the design of control strategies for emerging pathogens that rely on reactive host removal. They consist of removing plants within a particular distance of locations identified as containing the infection. The rationale is to remove (or treat) locations that are likely to be infected but not yet showing symptoms [11, 36–38]. Here, recent georeferenced inspection and eradiation data supplied by extensive surveillance at the landscape scale (representing an area of 28 hectares) was used to fit a mechanistic and spatio-temporal model of FD spread from one season to the next. The model builds on the work of [14] regarding the spread of Huanglongbing through citrus groves. It provides insights into the dispersal kernel that underlies FD spread. For coherent integration of information from different observations gathered at plant and field scales, we worked within a Bayesian framework. A key feature of the approach considered here is the use of data augmentation techniques to account for the missing information in the
data related to plants removed before the initial survey date in order to construct the full trajec-
tory of the epidemic. Accounting for this information enables us to use MCMC, a Bayesian
computational method to sample from the posterior distribution of the model parameters.

**Materials and methods**

Data

**A plant-by-plant survey of FD symptoms in three fields.** The study area is situated in Faleyras (S1(A) Fig), a district near Bordeaux in South West France, where three plots labelled F₁, F₂ and F₃ (S1(B) Fig) were monitored. These plots belong to the same owner. The plot F₁ was planted in 2002, F₂ in 1976 and F₃ in 1998. They are located in an open vineyard landscape at an altitude of 60-70 m with some woods and meadows (S1(A) Fig). The three plots are not separated by any hedgerow. The initial state of the fields before the FD incursion is shown in S1(B) Fig. In that initial state, there were 2259, 677 and 3120 plants distributed in F₁, F₂ and F₃ respectively, with two different cultivars, 2354 Cabernet Sauvignon (CS) and 3702 Merlot (M), the most widely grown cultivars in the region (S1(B) Fig). Of these cultivars, 2259 CS were planted in F₁, 677 M in F₂ and 3025 M and 95 CS in F₃.

The three fields were extensively monitored on October 11, 2018, and on September 23, 2019, following the methodology described in [31]. The three fields were mapped by distinguishing the non-symptomatic, the symptomatic and the missing (due to removal) plants. Over the two studies performed by the same team (this one and [31]), in fields with high FD prevalence and sampling done at the end of summer, 98% of symptomatic Merlot plants (n = 207) and 99% of symptomatic Cabernet Sauvignon plants (n = 202) were positive for FD. Similarly, 97% of non-symptomatic Merlot plants sampled (n = 88) and 98% of non-symptomatic Cabernet Sauvignon plants sampled (n = 96) were negative in real-time PCR tests. Genotyping of FD phytoplasma was performed by nested PCR followed by sequencing of the secY-map gene [39] on 4 samples of Merlot (in F₂ and F₃) and 4 samples of Cabernet Sauvignon (in F₁ and F₃) from 2018 and 2019. All samples were infected by the M54 genotype (Map-FD2) which is predominant in outbreaks from Europe and France [20].

For each symptomatic plant, symptom severity was recorded. A severity score from 1 to 4 was assigned depending on the proportion of symptomatic branches on the stock: 1) 0-25%, 2) 26-50%, 3) 51-75% and 4) >75%. The symptomatic plants were marked, and the wine grower was asked to remove them during the winter following the inspection. The growers may have also removed some non-symptomatic plants for other agronomic reasons. The two snapshots of the health status of the fields in October 2018 and September 2019 are shown in Fig 1A and 1B respectively. In 2018, 713 symptomatic plants including 668 CS and 45 M were detected, while 411 were missing (352 CS and 59 M). Field F₁ was entirely removed during winter 2018 in compliance with FD management rules because its prevalence (i.e. the percentage of diseased plant in the field) exceeded 20%. In 2019, 638 new symptomatic plants including 46 CS and 592 M were recorded on F₂ and F₃. During this same year, 50 additional missing plants, arising from the removal of asymptomatic plants, were identified in F₂ and F₃.

**Disease prevalence in the extended area of the focal fields.** The three fields, F₁, F₂ and F₃, are located in a larger region where FD surveys are performed by a professional organisation known as GDON (Groupements de Défense contre les Organismes Nuisibles) des Bordeaux. The district of Faleyras was surveyed in 2014. The survey is conducted as follows. Inspectors walk around all the fields in the area in search of symptomatic plants. Once symptomatic plants are located in a given field, the inspector collects symptomatic leaves from one to five plants and pools them into a single sample. The detection of the FD phytoplasmas in samples collected is performed by accredited laboratories using a real-time PCR test [40]. Each
<figure><img src="image_5.png" /><figcaption>**Fig 1. Individual status of the plants in the three vineyard fields F₁, F₂ and F₃ considered.** The field F₁ was planted with 2259 Cabernet Sauvignon, the field F₂ with 677 Merlot, and the field F₃ with 3025 Merlot and 95 Cabernet Sauvignon in two rows surrounded by a dotted line. The status of the plants is i) asymptomatic, ii) symptomatic or iii) removed. A: Spatial distribution of 713 FD-symptomatic plants including 668 CS and 45 M detected (red) in 2018 (October 11) as well as the 411 (352 CS and 59 M) holes showing plant removals before 2018. B: Spatial distribution of 638 FD-symptomatic plants including 46 CS and 592 M detected in 2019 (September 23), 2011 plants removed in 2018 (in grey) and the same 411 (352 CS and 59 M) missing plants indicating removal before 2018. Note that real-time PCR tests indicate that more than 98% of symptomatic plants are infected with FD, while more than 97% of asymptomatic plants are FD negative. Maps were drawn by authors using own data set released in the dataverse at https://doi.org/10.57745/YXOEHX.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1011399.g001

field is then classified as infected or not. If the field is infected, the inspectors count its total number of symptomatic plants. The field will be monitored in subsequent years. The infection status of the fields monitored by the GDON des Bordeaux in 2014 within 300 m of the three focal fields is shown in Fig 2A. Note that the fields in the neighbouring district of Romagne (the northeastern part of the map) were not inspected. From 2015 to 2017, the survey continued in the fields found to be infected in 2014 and spread to nearby fields after 2017 (Fig 2B, 2C, 2D and 2E). Note that GDON inspectors did not record the precise localisation of symptomatic plants. This information is only available in F₁, F₂ and F₃ for 2018 and 2019 following a dedicated survey performed by INRAE staff.

As shown in Fig 2A, FD was first detected in the area in 2014, most prevalently in field F₁ but also in three other fields. FD has since spread to more fields. The symptoms in the fields F₂ and F₃ were first reported by GDON in 2018 during the second round of inspection of F₂ and F₃ since 2014. Accordingly, their year of initial infection is unknown. In agreement with the set of mandatory measures to control FD, one or two insecticide treatments were recommended from 2015 to 2018 in the area corresponding to the districts of Faleyras and Romagne.

## Epidemiological models

A mechanistic compartmental SIR model with a discrete time step of one year was considered to describe the infectious status of each plant from 2014 to 2019 within the two focal fields F₂ and F₃. Three statuses were considered: Susceptible (noted S, i.e. healthy plant), Infectious (noted I, i.e. infected plant able to transmit infection) and Removed (noted R). The annual discrete time step assumed fits the univoltine cycle of *S. titanus* (i.e. one generation per year) and the existence of a dormancy phase (both for the plants and the eggs of the vector). This
<figure><img src="image_6.png" /><figcaption>Fig 2. Status of fields within 300 m around the targeted fields from 2014 to 2018. The status of fields is either i) not inspected, ii) not infected by FD (if symptomatic plants were sampled but the PCR test was negative) or iii) infected by FD. For the latter category, the values inside each infected correspond to the number of symptomatic plants recorded and subsequently removed. The area with the three focal fields, F₁, F₂ and F₃, is circled in dotted line. Maps were drawn by authors using own data set released in the dataverse at https://doi.org/10.57745/YXOEHX.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1011399.g002

dormancy phase, which results in infected plants in year t becoming infectious in year t + 1 (see Introduction), explained why an exposed compartment (host infected, but not yet able to transmit infection) was not considered here. Moreover, all infectious plants in year t + 1 were assumed to be removed in year t + 2 in compliance with the mandatory measures applied in France to manage FD (see below). Note that, as this transition is deterministic, the model corresponds formally to a SI model.

**Modelling the infection pressure.** Let first introduce the probability $P_i(t)$ that a susceptible plant i is infected in year t (and therefore becomes infectious in year t + 1). This probability is defined as

$$P_i(t) = 1 - \exp(-\phi_i(t)). \qquad (1)$$

where

$$\phi_i(t) = \epsilon(t) + \beta \sum_{j \in I_{t-1}} c_j K(r_{ij}; \alpha) \mathbb{1}_{j \in F_1} + \beta \sum_{j \in I_{t-1}} c_j K(r_{ij}; \alpha) \mathbb{1}_{j \in F_2 \cup F_3} \qquad (2)$$

Eq 2 is the infection pressure on host i at year t. Its first two terms together represent the infection pressure exerted on plants located in fields F₂ and F₃. The first term represents the instantaneous infection pressure exerted by all the fields other than F₂ and F₃ at time t = 2014, 2015, 2016, 2017, and by all the fields other than F₁, F₂ and F₃ in 2018 as an infection snapshot was available for F₁ in 2018. It corresponds to a primary infection rate that varies with time in proportion to the total number of infectious plants in the neighbouring fields (except for the plants in field F₁ in 2018). Specifically, $\epsilon(t) = \epsilon n_{t-1}$, where $n_{t-1}$ is the number of symptomatic
<table><caption>**Table 1.** Total number of FD-symptomatic plants detected in a radius of 300 m around the focal fields F₂ and F₃. The values of *n*<sub>t</sub> include the number of symptomatic plants in all the neighbouring fields including the field F₁ from 2014 to 2017 but excluding F₁ in 2018. These symptomatic plants are assumed to be infectious in the model.</caption><thead><tr><th>t</th><th>2014</th><th>2015</th><th>2016</th><th>2017</th><th>2018</th></tr></thead><tbody><tr><th>n<sub>t</sub></th><td>11</td><td>24</td><td>29</td><td>21</td><td>47</td></tr></tbody></table>

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plants at year *t* − 1 within a radius 300 m measured from the centre of the two focal fields F₂ and F₃ (Fig 2 and Table 1). The second term represents the external force of infection exerted by the infectious plants in field F₁ in 2018 on the plants in the fields F₂ and F₃ in 2019. It incorporates a distance-dependent kernel K(r<sub>ij</sub>; α) shared with the third term, which represents the secondary infections taking place within the fields F₂ and F₃. In the second and third terms, the sum is over infectious individuals *j* (i.e. individuals infected during year *t* − 1). These terms also include a parameter *c*<sub>*j*</sub> that accounts for the differences in infectivity between cultivars. Differences in infectivity were assumed to be related to the proportion of symptomatic branches on the stock. Accordingly, it was estimated independently using the severity score ranging from 1 to 4 that was described previously. Of the 668 CS plants recorded in 2018, 328 scored 1, 126 scored 2, 87 scored 3 and 127 scored 4, leading to a mean severity score value of 2.02. Of the 637 M plants recorded in 2018 and 2019, 437 scored 1, 147 scored 2, 38 scored 3 and 15 scored 4, leading to a mean severity score value of 1.42. Accordingly, *c*<sub>*j*</sub> was set to 1.42 (≈ 2.02/1.42) if the cultivar of the source plant is Cabernet Sauvignon and to 1 if the cultivar is Merlot.

The dispersal kernel *K* is a non-negative function also known as the infection kernel. It characterises the spatial dispersal process with a parameter *α*. Specifically, it characterises the relationship between infective challenge and the relative positions of infected and susceptible plants. A number of candidate models were proposed for *K*:

$$ \text{Exponential :} \quad K(r; \alpha) = \frac{1}{2\pi\alpha^2} \exp\left(-\frac{r}{\alpha}\right) \quad \text{with } \mu = 2\alpha \quad (3) $$

$$ \text{Gaussian :} \quad K(r; \alpha) = \frac{1}{\pi\alpha^2} \exp\left(-\frac{r^2}{\alpha^2}\right) \quad \text{with } \mu = \alpha\sqrt{\pi}/2 \quad (4) $$

$$ \text{Power law :} \quad K(r; \alpha) = \frac{(b-2)(b-1)}{2\pi\alpha^2} \left(1 + \frac{r}{\alpha}\right)^{-b}, \quad \text{with } \mu = \frac{2\alpha}{b-3} \frac{2\pi}{(b-2)(b-1)} \quad (5) $$

for $b = 4, 6, 8$

where *r* is the Euclidean distance (measured in m) between a given pair of plants. The mean of the dispersal kernel is given by *μ* and corresponds to the mean distance to the first infection in a totally susceptible population. The Gaussian and exponential kernels correspond to thin-tailed kernels, while the power law kernel corresponds to thick-tailed kernels. Kernels such as power law result in a rapid and long dispersal ahead of the source [41, 42]. Note that these 1-dimensional kernels are isotropic and can represent different spatial patterns of epidemic dynamics. They also allow these different spatial patterns to be compared directly.

Overall, 20 models were fitted. They represented all combinations of two hypotheses regarding the external source *ϵ*(*t*) (1: constant, by setting to *ϵ*(*t*) = *ϵ* in Eq 2, or 2: variable between years) with two hypotheses regarding a cultivar infectivity effect (1: presence, by setting *c*<sub>*y*</sub> = 1.42 for a Cabernet Sauvignon source plant and *c*<sub>*y*</sub> = 1 for a Merlot source plant, or 2:
<table><caption>**Table 2. Specification of the 20 models considered.** They include 2 × 2 × 5 combinations of hypotheses regarding the primary infection rate (column External source), the presence of a cultivar effect (column Cultivar effect) and the shape of the dispersal kernels (column Kernel). The numbering of equations (columns Eqn) corresponds to the equations detailed in Text A in S1 Text for the infection pressure and in the main text for the kernels.</caption><thead><tr><th rowspan="2">Label</th><th colspan="3">Infection pressure (φ(x; t))</th><th colspan="2">Kernel K(r; α)</th></tr><tr><th>External source</th><th>Cultivar effect</th><th>Eqn</th><th>Name</th><th>Eqn</th></tr></thead><tbody><tr><td>Model 0</td><td rowspan="5">Constant</td><td rowspan="5">No</td><td rowspan="5">S1</td><td>Gaussian</td><td>4</td></tr><tr><td>Model 1</td><td>Exponential</td><td>3</td></tr><tr><td>Model 2</td><td>Power law, b = 4</td><td>5</td></tr><tr><td>Model 3</td><td>Power law, b = 6</td><td>5</td></tr><tr><td>Model 4</td><td>Power law, b = 8</td><td>5</td></tr><tr><td>Model 5</td><td rowspan="5">Variable</td><td rowspan="5">No</td><td rowspan="5">S2</td><td>Gaussian</td><td>4</td></tr><tr><td>Model 6</td><td>Exponential</td><td>3</td></tr><tr><td>Model 7</td><td>Power law, b = 4</td><td>5</td></tr><tr><td>Model 8</td><td>Power law, b = 6</td><td>5</td></tr><tr><td>Model 9</td><td>Power law, b = 8</td><td>5</td></tr><tr><td>Model 10</td><td rowspan="5">Constant</td><td rowspan="5">Yes</td><td rowspan="5">S3</td><td>Gaussian</td><td>4</td></tr><tr><td>Model 11</td><td>Exponential</td><td>3</td></tr><tr><td>Model 12</td><td>Power law, b = 4</td><td>5</td></tr><tr><td>Model 13</td><td>Power law, b = 6</td><td>5</td></tr><tr><td>Model 14</td><td>Power law, b = 8</td><td>5</td></tr><tr><td>Model 15</td><td rowspan="5">Variable</td><td rowspan="5">Yes</td><td rowspan="5">S4</td><td>Gaussian</td><td>4</td></tr><tr><td>Model 16</td><td>Exponential</td><td>3</td></tr><tr><td>Model 17</td><td>Power law, b = 4</td><td>5</td></tr><tr><td>Model 18</td><td>Power law, b = 6</td><td>5</td></tr><tr><td>Model 19</td><td>Power law, b = 8</td><td>5</td></tr></tbody></table>

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absence, by setting $c_y = 1$ regardless of the cultivar of the source plant) and five hypotheses regarding the shape dispersal kernel. The 20 models considered are summarised in Table 2, and the corresponding equations for the infection pressure are detailed in Text A in S1 Text. Finally, the parameters to be estimated are summarised in Table 3.

**Modelling the initial infection year of the fields F₂ and F₃.** The fields F₂ and F₃ were not inspected from 2015 to 2017 (Fig 2). Let $t_0$ denote the year the inoculum was first transmitted to either of these fields. The unobserved levels of initial infection of the fields F₂ and F₃ (i.e. number and position of infected plants at $t_0$) are treated as additional model parameters.

<table><caption>**Table 3. Definition of model parameters and their mean posterior values along with their 95% credible interval obtained from the MCMC algorithm using Model 18.** The credible interval for $t_0$ is ignored, as there was a negligible posterior weight on years other than 2016.</caption><thead><tr><th>Process</th><th>Symbol</th><th>Description</th><th>Estimated value</th><th>Unit<fnref n="1" /></th></tr></thead><tbody><tr><td>Dispersal</td><td>α</td><td>Scale parameter</td><td>22.9 [20.1–26.5]</td><td>m</td></tr><tr><td rowspan="3">Infection</td><td>ε</td><td>Primary infection rate</td><td>10<sup>-4</sup>[10<sup>-4</sup>–2.10<sup>-4</sup>]</td><td>year<sup>-1</sup></td></tr><tr><td>β</td><td>Secondary infection rate</td><td>106.1[94.5–118.4]</td><td>m<sup>2</sup>.year<sup>-1</sup></td></tr><tr><td>c<sub>y</sub></td><td>Cultivar's infectivity<sup>2</sup></td><td>c<sub>y</sub> = 1 or c<sub>y</sub> = 1.42</td><td>na</td></tr><tr><td>Removal</td><td>q</td><td>Probability of removal for reasons other than FD</td><td>0.004 [0.004–0.005]</td><td>na</td></tr><tr><td>Initial infection date in F<sub>2</sub> ∪ F<sub>3</sub></td><td>t<sub>0</sub></td><td>Year of first infection</td><td>2016</td><td>na</td></tr></tbody></table>

https://doi.org/10.1371/journal.pcbi.1011399.t003

<footnote n="1">¹ na when not applicable or for dimensionless parameters.</footnote>

<footnote n="2">² Values estimated independently using data published by [31].</footnote>
**Modelling removal process.** The set of specific FD symptoms mostly appear through-out the summer and are best identified between mid-August and mid-October in the Bordeaux vineyard [29]. Symptomatic plants were assumed to be removed during the winter following their identification. This succession during the same year of the periods where a plant is first infectious, then expressed the typical set of FD symptoms, next identified as such and finally removed led us to consider thereafter that the FD survey currently implemented in the Bordeaux area is characterised by the existence of “cryptic” infection from late spring/early summer until optimal detection of symptoms in late summer/early autumn. It corresponds sensu [2] to a period of time where infectious hosts go undetected while providing a source of inoculum for spread. Finally, we assumed that any susceptible individual could be removed in a given year for some agronomic reason unrelated to FD infection. The probability *q* of this event was considered as an additional model parameter.

## Model inference

**Likelihood.** The data y record the status of plants at *n* discrete observation times in [$t_0, t_{max}$], along with information on removed plants. In addition, let denote the set of removed hosts with $\mathcal{R}$, the corresponding set of reasons hosts are removed with $R = (r_1, r_2, \dots, r_{|\mathcal{R}|})$ and the corresponding set of removal times with $\underline{t}^R = (t_1^R, t_2^R, \dots, t_{|\mathcal{R}|}^R)$. It is worth noting that $|\mathcal{R}|$ represents the size of the removed hosts in the population. For simplicity, $r_j$ is set to 0 if the host *j* was removed due to causes other than FD, or to 1 if the host was removed due to FD infection. The set of reasons hosts are removed *R* comprises those with known reason $R_k$ and those for which the reason for removal is unknown $R_u$. In other words, $R = R_k \cup R_u$. Finally, let us denote the set of the model parameters with $\theta = (\alpha, \epsilon, \beta, q, t_0)$. Then, taking the reasons hosts are removed and their corresponding times as known, the complete data likelihood function is given as

$$L(\theta, R, \underline{t}^R; y) = \prod_{j \in \mathcal{R}} \left[ \left( \prod_{i=1}^{t_j^R-1} \exp(-\phi_j(i))(1-q) \right) \left( (1 - \exp(-\phi_j(t_j^R)))(1-q) \right) \mathbb{1}_{\{r_j=1\}} + \exp(-\phi_j(t_j^R)) q \mathbb{1}_{\{r_j=0\}} \right] \times \prod_{j \in \mathcal{R}} \prod_{i=1}^{t_{max}} \exp(-\phi_j(i))(1-q) \quad (6)$$

The first two lines of Eq 6 represent the contribution to the likelihood arising from the removal events. Two classes of removed hosts are considered: those removed due to the disease ($r_j = 1$) and those removed due to causes other than FD ($r_j = 0$). The term $(1 - q)$ ensures that symptomatic plants are not removed due to any other causes. The third line represents the contribution to the likelihood of plants that escaped the removal with probability $1 - q$ and also escaped infection.

**Bayesian inference.** The likelihood $L(\theta, R, \underline{t}^R; y)$ is not straightforward to compute, as the locations, dates and natures of infections between 2014 and 2017 were not observed. However, this problem can be overcome using the data-augmented approach by treating the missing plants in 2018 as an additional unknown “parameter” and investigating the joint posterior distribution $P(\alpha, \epsilon, \beta, q, R_u, \underline{t}^R|y)$ [3, 13–15, 17, 43]. The algorithm involves proposing accepting changes to the times and nature of removals before 2018 and provides posterior distribution of infection sources as a by-product. This is done using methods which are now standard in computational epidemiology, such as reversible-jump [44, 45] MCMC techniques. Details of algorithms are given in Text B in S1 Text.
## Model checking and model comparison

The MCMC algorithm described in Text B in S1 Text resulted in a joint posterior distribution of both the model parameters and the time and nature of missing plants during the first visit to fields F₂ and F₃ in 2018. These outputs can be used both to perform posterior model checking and to select the subset of candidate models that best explains the data. In the context of a spatio-temporal transmission model, natural model selection criteria such as the deviance information criterion (DIC) suffer from important limitations because of their sensitivity and complexity [46]. Following work by [12, 14, 15, 17] in a closed context, model selection was performed by comparing (i) the posterior predictive distribution of the counts of symptomatic and removed plants and (ii) the posterior predictive distribution of the spatial structure of the epidemic to the observed ones.

**Count of symptomatic and removed plants in 2018 and 2019.** Samples were randomly drawn from the joint posterior distribution of the model parameters to simulate the epidemic process, assuming that the first infection(s) occurred in 2013 on the field F₁. It is worth recalling that the initial infection on F₂ ∪ F₃ was estimated and therefore also sampled from the joint posterior distribution. The counts of symptomatic and removed plants observed in 2018 and 2019 were compared to those of the simulated epidemics.

**Spatial autocorrelation.** The spatial structure of the simulated epidemics described above was compared to those of the actual observations. Specifically, Moran's I index and Ripley's L index [47–49] were considered as our spatial summary statistics.

The region was divided into 5 × 10 square sub-regions, each containing numbers of plants ranging from 1 to 19 (with median value 16), and count of the number of symptomatic sites for the time point used for the computation of Moran's I

$$I = \frac{n \sum_{i=1}^{n} \sum_{j=1}^{n} w_{ij} (y_i - \bar{y})(y_j - \bar{y})}{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{ij} (y_i - \bar{y})^2} \quad (7)$$

where $y_i = 1$ if plant i is symptomatic before the observation time and $y_i = 0$ otherwise, and $\bar{y}$ is the mean of the observation $y_i$ over n sub-regions. Here, $w_{ij}$ represents a distance-dependent weight between sub-regions i and j. There are many ways to define $w_{ij}$, and the distance weights between plant i and j was used here. The index was computed by using the ape package [50] available in the statistical software R.

Similarly, the Ripley's L function to characterise clustering/dispersion of point patterns at multiple distances was defined as

$$L(r) = \sqrt{K(r)/\pi} \quad (6)$$

where K(r) is the Ripley K function defined as

$$K(r) = \frac{A}{n(n-1)} \sum_{i,j} \mathbb{1}_{d_{ij} \le r}.$$

A represents the area of the window, n is the number of data points, and the sum is taken over all ordered pairs of points i and j in the window. $d_{ij}$ is the distance between the two points, and $\mathbb{1}_{d_{ij}}$ is the indicator that equals 1 if the distance is less than or equal to r. This spatial index is computed without requiring any aggregation of the points and therefore gives a more complete description than statistics that utilise aggregation. The index was computed by using the spatstat package available in the statistical software R [51].
## Effect of removal efficiency on FD spread

For further insight into how FD spreads, FD epidemics were simulated in a single large square field of 1 km² with 167 334 plants. Vines were assumed to be of the same cultivar and planted to mimic the practices in the Bordeaux vineyard, with spacing along rows of 2 m and spacing between rows of 3 m. Disease progress was simulated over 10-year time spans, starting with a single initial infection located at the centre of the field. FD spread only through secondary infections (i.e. no primary infection) according to the rate $\beta$ and to the kernel scale $\alpha$ estimated from the data. Realisations of the epidemic process were generated by sampling sets of parameter ($\alpha, \beta$) from the joint posterior distribution of Model 18. Note that, with these assumptions, the maximum distance FD can spread from the initial source is $(1000\sqrt{2})/2 \approx 707$ m.

Removal due to causes other than FD was ignored ($q = 0$), but the assumption that symptomatic plants will be systematically removed one year after their infection was relaxed by introducing a parameter $\tau$ that represents the efficiency of the removal process. More precisely, the parameter $\tau$ corresponds to the probability of removing during the autumn/winter of year $t+1$, that is after the infectiousness period which extends from late spring to late summer, a plant infected during year $t$. Viewed another way, $1 - \tau$ corresponds to the proportion of infectious plants that escape removal each year, as a result of either flawed identification of symptomatic plants or flawed plant removal. Removed plants are replaced immediately. The algorithm is described in Text C in S1 Text. For each removal efficiency ranging from 0 (no removal) to 100% (full removal) by steps of 20%, 500 realisations of the epidemic process were run. For each run, the observations consisted of the annual snapshots of infected hosts in the 1 km² square field considered. Specifically, the distances between the newly infected plants and the initial foci were computed for each year. The median distance, 95% quantile distance and maximum distance between the newly infected plants and the initial source plant were characterized based on the 500 realisations performed. Note that the secondary infection rate $\beta$ used in the simulation is representative of the average conditions of FD spread in the area and period considered for parameter estimation. In particular, it depends on (and represents) the local population dynamics of *S. titanus* in a context where one to two annual insecticide applications are recommended (but not necessarily done, or done in suboptimal conditions).

## Results

### Model selection and parameter estimation

The 20 models (Table 2) were fitted to the data using reversible-jump MCMC with data augmentation (Text B in S1 Text and Table A in S2 Text). The MCMC algorithm was iterated for 200 000 iterations, discarding the first 50 000 iterations (the burn-in period) to ensure the convergence to the stationary distribution is reached, with uninformative exponential priors to obtain a joint posterior distribution density for the parameters $\epsilon, \beta, \alpha, q$ and $t_0$. Trace plots demonstrated that there is no evidence of non-convergence across all models.

We first discuss those parameters whose posterior distributions did not differ greatly with the model fitted. First, the posterior distribution of the starting time of the epidemic revealed that 2016 was the most likely year of introduction of the inoculum into the fields $F_2 \cup F_3$ (Fig A in S2 Text). This introduction probably occurred from field $F_1$, as infection in this field was first spotted in 2014. Second, the 95% credible interval for the probability $q$ of a plant being removed in any given year due to causes other than FD was consistently estimated to be [0.004, 0.005]. Third, the posterior distribution of $\epsilon$ (rate of primary infection from surrounding fields) depended on the assumption made (constant rate or varying with the number of symptomatic plants in the neighbourhood fields, Table 1). Nevertheless, this estimation
<figure><img src="image_7.png" /><figcaption>Fig 3. Comparison of the 20 models using the posterior predictive distribution of the counts of observed symptomatic and removed plants in 2018. Each panel corresponds to a model. The dotted line is the actual observed count, and the density of counts is obtained from 1 000 simulations. The colors correspond to the symptomatic (red) and the removed (grey) plants. The 20 models (described in Table 2) differ according to their dispersal kernel (by column) and formulation of infection pressure (by row).</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1011399.g003

revealed that the number of initial infections arising from primary sources is likely to be 4 for all models.

In contrast, the posterior distribution of other parameters differed significantly with the fitted models. As expected without normalizing the kernel, the infection rate $\beta$ and the kernel shape $\alpha$ were highly correlated (Table A in S2 Text). Normalizing the kernel by following the approach developed in [52, 53] would have reduced this correlation. The model outputs were used to select the best model(s) by comparing the dynamics and the spatial pattern of simulated epidemics with those of the actual observation. To do so, 1 000 epidemics were simulated for each model using 1 000 samples from their joint posterior distributions.

The first model selection criteria compared the simulated counts of symptomatic and removed plants to observed values in 2018 and 2019 (Fig 3 and S2 Fig). For clarity of these figures and similar ones, the labelling has been chosen so that each line shows the results of different parameterization of the infection pressure and each column the results of different dispersal kernels. The actual counts lay within the distribution of simulated counts only for
<figure><img src="image_8.png" /><figcaption>**Fig 4.** Comparison of the 20 models using the posterior predictive distribution of Ripley's L function. The Ripley's L function determines the clustering/dispersion of point data over a range of distances. In each panel, corresponding to a model, the red line represents the actual measure of Ripley's L function for the observed data, and the grey area indicates the 95% marginal credible region from the simulated epidemic. The 20 models (described in Table 2) differ in their dispersal kernel (column) and formulation of the infection pressure (row).</figcaption></figure>

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Model 18. Models relying on Gaussian and exponential thin-tail kernels did not seem to offer an appropriate fit. Observed counts lay in the good range of the posterior count distributions provided by Models 3, 4, 8 and 9, except for the counts of symptomatic plants in 2018.

The second model selection criterion compared the spatial structures of the simulated epidemics to those of the actual observation using the indices of Moran's I (S3 Fig) and Ripley's L (Fig 4). While the actual value of Moran's I index lay in a good range of the posterior distribution for all models, models with thin-tail kernels were completely ruled out when considering Ripley's L index. Power law kernel models fitted reasonably well for this criterion, especially for the exponent parameters *b* = 4 and *b* = 6.

To summarise, four of the 20 models considered (Model 3, Model 4, Model 8 and Model 18) displayed the best fit for both observed counts and observed spatial structures. Of these, Model 18 performed slightly better. Interestingly, similar to Model 18, Models 3 and 8 rely on a power law kernel with *b* = 6. They differed by the hypothesis underlying the infection pressure, with Model 8 utilising a variable primary infection rate (but no cultivar effect) and Model 18 assuming both a variable primary infection rate and a cultivar effect, Cabernet Sauvignon being a better source of infection than the cultivar Merlot. In the following, we will focus on the results obtained from these four models, particularly Model 18.
Dispersal kernels and FD transmission distance

A key epidemiological feature obtained in this study was the dispersal kernel of FD. It repre-
sents the rate of disease transmission between a pair of hosts at a given distance. The estimated
spatial kernels are shown in Fig 5 for the 4 selected models. The posterior distribution of the
kernel means μ showed little variation. This was evidenced by the 95% credible intervals lying
within [9, 18] m (Fig 5A). In particular, the posterior mean of μ estimated from the best model
(Model 18) was 13 m. It was comparable to the 15 m obtained from Model 3 and Model 8 (that
only differ by the formulation of the primary infection rate). Model 4 resulted in a slightly
lower estimate of 10 m compared to the three others. Besides the posterior distribution of the
kernel mean, the probability of transmission was consistently observed to vanish beyond 20 m
for the 4 models considered in a uniformly distributed host landscape (Fig 5B).

Running simulations without removal in the 1 km² square field considered enable to locate
infected hosts after a single cropping season and thus estimate the posterior predictive distri-
bution of the transmission distance, i.e. the mean distance of hosts infected from a given
source in a totally susceptible population (Fig 5C). The posterior median (resp. mean) of the

<figure><img src="image_9.png" /><figcaption>**Fig 5.** Posterior distribution of the spatial transmission under the four best models. The set of best models is Models 3, 4, 8 and 18, the last being the best one. A: Posterior distribution of the kernel mean, the mean distance to the first infection in a totally susceptible population. B: Probability density of the distance between an infectious plant and the first secondary infection that it causes assuming a uniformly distributed population of susceptible hosts when the scale parameter α is set to its posterior mean. C: Posterior predictive distribution of the mean transmission distance obtained by simulating epidemic realisations using 500 samples from the posterior distribution of parameters estimated from Model 18. D: Corresponding posterior predictive distribution of the cumulative distribution function (CDF) of the distance between the initial source plant and new infections in a single epidemic season (i.e. one year). Error bars represent 95% credible intervals of the CDF. The dotted lines show that on average, 50%, 80% and 95% of the new infections occur within 10.5, 22.2 and 55.2 m of the source, respectively.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1011399.g005
transmission distance was 15.01 m (resp. 16.9 m), with the 95% credible interval estimated to be [5.67, 33.59] m. The corresponding posterior predictive distribution of the cumulative distribution function of the distance of the newly infected plants after a single annual epidemic cycle is shown in Fig 5D. On average, 50%, 80% and 95% of the new infection occurred within 10.5, 22.2 and 55.2 m of the source, respectively.

## Effect of removal efficiency on FD spread

The simulations also allowed us to estimate how removal efficiency affects the distribution of the distance of the newly infected plants over 10 annual epidemic cycles (Fig 6A, 6B and 6C). The median distance remained in the same range regardless of the level of removal efficiency during the first 5 years (Fig 6A). Then, they substantially increased for low removal efficiencies ($\tau \le 20\%$) while remaining in the same range for higher removal efficiencies ($\tau \ge 60\%$). Ten seasons after epidemic onset, infected plants were at a median distance of 273 m from the source without removal. This distance was reduced to 212 m for low removal efficiencies ($\tau = 20\%$) while remaining close to 140 m for removal efficiencies $\ge 60\%$. The higher level of removal efficiencies was particularly important for reducing the upper quantiles of the distance between newly infected plants of the initial source (Fig 6B and 6C). The maximum distance without removal is 604 m, meaning that FD infection almost reached the border of the square field of $1 \text{ km}^2$ considered, as the maximum distance FD can spread from the initial source is 707 m. With removal, the maximum distance decreased almost linearly to 443 m with a removal efficiency of 80%, while remaining at a close value with full removal. Importantly, a full removal strategy slowed down but did not stop FD spread. This is mainly due to the presence of a delay between plant infectiousness and optimal symptom detection, followed by removal. For example, ten seasons after epidemic onset, 166 m separated the maximum distance of localisation of newly infected plants between the no removal and full removal strategies. Besides these effects on the distributions of the distance of the new infections, increasing removal efficiencies strongly reduces the number of newly infected plants in each cropping season (Fig 6D), and thus, epidemic size. Ten years after the epidemic onset, this reduction amounted to nearly 85% between the no removal and the full removal strategy.

## Discussion

While the estimation of dispersal parameters of plant pathogens using spatio-temporal stochastic mechanistic models has been an active research area for more than 20 years [12], information is still lacking for many diseases, especially for insect-vectored pathogens [36]. In this study, data incorporating two consecutive annual snapshots indicating the symptomatic status of thousands of host plants in three close vineyard plots were used to fit a spatially explicit model representing the epidemiology of FD, a major quarantine disease damaging European vineyards. We assumed that symptom expressions reveal the infected status of plants to FD, based on quantitative real-time PCR tests indicating that more than 98% of symptomatic (resp. more than 97% of asymptomatic) plants noted by the team in charge of the survey were positive (resp. negative) for the phytoplasma. The corresponding high sensitivity and specificity of FD detection are strongly supported for the sampling period in autumn. Indeed, the visual identification of FD symptoms is much easier in autumn [29], as this period is the ideal time for the expression of certain specific symptoms, such as grape shrivelling and the non-lignification of canes [26]. Based on this premise, model inference relies on Markov chain Monte Carlo methods with data augmentation and allows extraction of new insights into the spread of FD. In particular, as far as we know, this study provides the first characterisation of the dispersal kernel of FD. The inference was challenging in our case study because the observed data
<figure><img src="image_10.png" /><figcaption>Fig 6. Posterior predictive distribution of the distance between newly infected plants and the initial FD source and posterior predictive distribution of numbers of new infections as a function of the number of seasons after epidemic onset and removal efficiency. Distance and new infections are obtained by simulating epidemics in a square field of 1 km² with 167334 vine plants. Epidemics, initiated by a single infected plant at the centre of the field, spread as a result of secondary infections and of the removal efficiency of FD-infectious plants. A: Posterior distribution of the median distance during 10 cropping seasons as a function of removal efficiency (6 levels, from no to full removal). B: Same as A for the 95% quantile of the distance. C: Same as A for the maximum distance. The dashed line indicates the maximum distance FD can spread in the field considered. D: Posterior distributions of the mean number of newly infected plants occurring each year during 10 cropping seasons as a function of 6 levels of removal efficiency.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1011399.g006

mainly consist of only two snapshots taken in two consecutive years, a situation similar to the
data analysed by [12]. Such situations can be frequent for quarantine disease. Many detailed
studies of plant epidemics have benefitted from more frequent spatio-temporal observations
(e.g. 4 in [14], 6 in [16] and ≥ 15 in [10, 17, 18]). The inference framework, which reconstructs
the infection dynamics in the area of interest, notably relies on the presence of missing plants
(i.e. vine plants removed) before the year of observation of the first snapshot (2018) in the two
target fields. The data augmentation methodology used here makes it feasible to infer the
causes of these observed removals by considering the probability *q* that a vine plant is removed
due to agronomic causes other than FD. Including this parameter allows the spread of FD dur-
ing the first years after its initial incursion in the area to be slowed in a manner reflecting the
situation in the field. Our results strongly suggest that the most likely year of the first infection
of fields *F*₂ and *F*₃ is 2016 (Fig A in S2 Text).

Dispersal kernels are the key to understanding the spatial transmission of pathogens. They
are firstly characterised by their scale parameter. The mean of the posterior distribution of the
kernel mean μ obtained from the best model is 13 m (Fig 5A), and the values estimated with
the 4 best models range from 10 to 15 m (Fig 5A). In addition, the predictions-based epidemic
simulations indicate that the mean (resp. the median) of the posterior distribution of the mean
transmission distance is 16.9 m (resp. 15.02 m) (Fig 5C). It also indicates that 50% (resp. 80%)
of the new infections are likely to be found within 10.5 m (resp. 22.2 m) of the infectious source
plant (Fig 5D). Accordingly, FD mainly spreads locally from one year to the next. Interestingly,
these estimations are in broad agreement with the space-time point pattern analysis of an FD
epidemic (surveyed from 2011 to 2015) in a field located in Italy [54]. These authors found
that the dominant feature was an aggregated cluster of symptomatic plants in moving windows
of 8 × 8 plants (equivalent to a square of 11 m/side) to 12 × 12 plants (equivalent to a square of
17 m/side). They also estimated that about 50% of newly symptomatic plants (on year *t*) were
found within 3 m of plants already symptomatic (on year *t* − 1) while discussing how this fig-
ure was somewhat blurred by their statistical approaches, which could not necessarily disen-
tangle primary from secondary infections. Our estimations are also in agreement with
observations on the fly activity of *S. titanus*, the vector of FD [55]. These authors have shown
using mark-recapture techniques that 80% of *S. titanus* adults disperse over distances of less
than 30 m along the whole season, a value in line with the aggregated spatial distribution of
this monophagous leafhopper previously observed [56–58]. Some authors suggest that the cre-
puscular flight activity of *S. titanus* adults makes them likely to rely on an active wandering
movement rather than on passive wind-borne transport [59]. More broadly, few dispersal ker-
nels are known among insect-vectored pathogens. Huanglongbing, a major disease of citrus
worldwide, is mainly transmitted by the Asian citrus psyllid, *Diaphorina citri*. The estimation
of its dispersal kernel also indicates a short mean transmission distance of 5 m for 5-year-old
trees and 10 m for 18-year-old trees [14]. By contrast, the mean transmission distance of plum
pox virus by *Aphis gosypii* amounts to 92.8 m [10], a higher figure possibly linked to the better
dispersal capacity of winged aphids that rely on wind. More broadly, it should be noted that
the mean dispersal distance of fungal plant pathogens propagated by airborne spores is often
much higher (ranging from 213 to 2560 m) in the four pathosystems listed by [36].

Dispersal kernels can be further defined by their shape, which in particular informs the “fatness” of their tails. Tail fatness can be used to categorise kernels in a binary fashion [41]. When at a relatively large distance the shape of the tail decreases less slowly than exponential distribution, or equally slowly, a kernel is termed “short-tailed” or “thin-tailed” [60]. In contrast, if the probability of dispersal decreases more slowly than an exponential distribution at long distances from the source, kernels are termed “long-tailed” or “fat-tailed”. Long-distance dispersal events are more frequent than with an exponential kernel that shares the same mean dispersal distance. Here, the annual spread of FD is best described by a fat-tailed power law with exponent parameter *b* = 6, whereas the “thin-tailed” kernels considered (Gaussian and exponential kernels) were clearly discarded by model selection. This result is in line with the maximum distances occasionally observed for *S. titanus* by [14] (330 m) and [58] (600 m). The small-scale dispersion ability of *S. titanus* does not exclude the larger-scale passive dissemination of infectious *S. titanus* over longer distances by the wind, as is the case for many insects [61]. A
consequence of fat tails is the patchiness of the epidemics that can arise. Rather than having a
wave that propagates from an initial focus, you get patterns that appear to have multiple foci
[12, 62] and in some occasions accelerating epidemic waves [63]. These features are crucial to
consider when designing control strategies for emerging plant disease [36].

Given the importance of properly characterising the speed of propagation of the front of
the spread of an emerging disease, it would be interesting to consider other fat-tailed dispersal
kernels, such as exponential power kernels. Indeed, fat-tailed kernels can be further distin-
guished depending on whether they are “regularly varying” (e.g. power-law kernels) or “rap-
idly varying” (e.g. exponential power kernels) [60]. Mathematically, this implies that power
law kernels decrease even more slowly than any exponential power function. Biologically, this
means that fat-tailed exponential power kernels display rarer long-distance dispersal events
than power-law kernels. This affects the dynamics of an epidemic. Indeed, e.g. with weak Allee
effects, subtle interactions between tail fatness (rapidly versus regularly varying) and per capita
growth rate of the epidemic near zero determine whether the spread is accelerating [64, 65].
However, our observations are mainly collected in three fields within a bounding box of 232 m
by 202 m (i.e. over 4.7 *ha*). It is well known that data confined to relatively small spatial scales
can blur the precise estimates of the form of dispersal at large distances, and in particular the
shape of the kernel’s tail [66]. The selected dispersal at long distances depends on both the ker-
nel considered and the distances over which the dataset is collected [67].

The inference framework used considered a non-spatialized primary infection rate term, $\epsilon(t)$, modelling the introduction of FD in the two targeted fields from a set of remote fields. Recent work by [14, 17] also incorporates seasonality into the primary or external rate of infection in modelling plant epidemics, but assumes the rate to be homogeneous. Here, we utilise the data on the disease dynamic in adjacent fields by building the primary infection rate as a function of the density of symptomatic plants in those adjacent fields. This could be further refined by considering the primary infection rate as a spatially decaying source of inoculum as, in our case study, the spatial polygons of the remote fields are known (Fig 2). Thus, a further improvement of the inference framework could consist in using the same dispersal kernel for handling both the data collected at plant scale (in the fields $F_1, F_2$ and $F_3$) and the data aggregated at field scale (number of symptomatic plants recorded in remote fields). Regarding this, the method developed in [68] is of interest, as it allows the quantification of the flow of particles over a heterogeneous area by integrating a pointwise dispersal function over source and target polygons. A simpler approach could also be used, e.g. through connectivity matrices between fields. However, the latter approach neglects field geometry and can be expected to bias connectivity estimates, such as when field shapes and sizes are disparate. Furthermore, given the evidence that the FD epidemic spreads more rapidly along than across rows ([54]), anisotropic kernels, in which propagules can disperse differently depending on the direction (e.g. [69]), are worth considering. A different aspect, in light of [10], that could be interesting to take into account in the inference framework is the sensitivity and specificity of FD detection (as detailed in the materials and methods section), given that FD detection in the focal fields was assumed to be errorless. Finally, no records of vector population densities were available for the focal fields. Undoubtedly, incorporating such data, typically throughout a secondary infection rate depending annually on sticky traps data, would have improved model fit.

Despite its simplicity, the spatially explicit SIR model developed and fitted here captures the
main features of FD epidemiology in vineyard fields: (i) a single annual infection cycle result-
ing from the univoltine life cycles of *S. titanus*, (ii) a dormancy phase in the life cycle of the
plants and the vector that results in infected plants in year *t* becoming infectious in spring of
year *t* + 1 and (iii) the existence of a delay between plant infectiousness from spring of year
*t* + 1 and plant removal the next autumn/winter. This time gap results from practical
considerations for implementing FD management in the field: the beginning of the survey in August as the set of specific symptoms mostly appear throughout the summer, molecular verification of diagnosis, and finally plant removal. The simulations consider a basic scenario, all other factors impacting FD epidemiology being equal: (i) FD is spreading in an isolated and large field (100 ha) after its introduction in a single infected plant, (ii) plants infected during the year *t* are removed with an efficiency *τ* infected plants during autumn/winter of year *t* + 1 and (iii) removed plants are replanted immediately. Such removal strategy mimics the current uprooting practice in France, with the exception of the rule that states that fields must be entirely removed as soon as their FD prevalences exceed 20%. This rule would be important for modelling FD control strategies in agricultural landscapes, which is out of the scope of this study. Overall, in the context of a FD survey as currently carried out in the area, simulations suggest that even a fully efficient uprooting strategy (*τ* = 1) is unable to avoid FD spread (Fig 6D). It illustrates, as already emphasised in several pathosystems, that the control of emerging disease is the most difficult when it is hampered by invisible cryptic infection [11, 16]. In our case study, “cryptic” infections are the consequence of taking into account practical constraints in defining the modalities of FD management in the field. It imposes a time gap between the infectious period and the period of optimal detection of symptoms (and an even larger gap until effective plant removal). The major interest of mandatory insecticide treatments directed against nymphs and adults of the vector is precisely to reduce the risk of FD phytoplasma acquisition/transmission during this time gap.

In a context of a necessary reduction of insecticide treatments, previous results should lead us to reconsider the component of the global FD control strategy which is identifying and removing infected plants. Currently, only symptomatic plants are removed in France. In the presence of “cryptic” infection, strategies relying on reactive host removal are of particular interest. They consist in removing plants, regardless of their infectious status (i.e. both asymptomatic and symptomatic plants), within a particular distance of infection foci. The rationale is to remove (and/or treat) locations that are likely to be infected without yet showing symptoms [11, 37, 38]. In the case of FD, the plants should be removed during the winter of year *t*, before their infectiousness. Note that, in line with these reasons, insecticide treatments against *S. titanus* are already advised in a buffer of 500 m around infected plots. Another strategy could consist in cutting the vine twigs as soon as the very first symptoms appear on them, all along the summer, as suggested by [33] and as practiced in some vineyards in Italy. It must be followed by a complete removal in winter as imposed by quarantine measures. The model developed in this work is a first step in redesigning FD control strategies. Future research should take advantage of the (*S*)usceptible-(*E*)xposed-(*C*)ryptic-(*D*)etectable-(*I*)nfected-(*R*)emoved framework proposed by [11] to optimise control strategies for invasive plant disease. Research should also benefit from one of the few attempts to model the long-term epidemiology of FD [70]. Using a mean-field model approximation (i.e. ignoring the transmission dispersal kernel), these authors explore the role of hotbeds (defined as vine-growing areas such as abandoned vineyards and woods containing wild grapevines) and of insecticide spray on FD dynamics. In keeping with these and our approaches, future works should consider the full spectrum of existing control measures (e.g. healthier planting material, removal of hotbeds, improving vector control with insecticides but also vibrational disturbances and “push-pull” strategies, see [22] for a review) when we only consider here the uprooting of infected plants. Finally, following [7], this approach should account for economic criteria that balance benefits (generated by the cultivation of healthy/asymptomatic plants) against costs (particularly due to FD control: surveillance, plant removal, replanting and insecticide treatments).
**Supporting information**

**S1 Text.** A: Equations used to model the infection pressure. B: Description of the MCMC algorithm. C: Simulation algorithm.
(PDF)

**S2 Text.** Parameters' estimates for all models.
(PDF)

**S1 Fig.** Map of the three vineyard fields F₁, F₂ and F₃ in their landscape.
(PDF)

**S2 Fig.** Comparison of the 20 models using the counts of observed symptomatic and removed plants in 2019.
(PDF)

**S3 Fig.** Comparison of the 20 models using the spatial correlation index of Moran.
(PDF)

**Acknowledgments**

The authors wish to thank the people who sampled or provided data: Sophie Bentejac, Morgane Le Goff and Charlotte Labit (GDON des Bordeaux); Dominique Vergnes (FREDON Aquitaine); and Gontran Casella (Syndicat des Bordeaux et Bordeaux Supérieur). We also thank Sophie Bentejac and Charlotte Labit for their critical and careful reading of the manuscript.

**Author Contributions**

**Conceptualization:** Hola K. Adrakey, Gavin J. Gibson, Frederic Fabre.

**Formal analysis:** Hola K. Adrakey, Gavin J. Gibson.

**Funding acquisition:** Sylvie Malembic-Maher, Frederic Fabre.

**Investigation:** Hola K. Adrakey, Sandrine Eveillard, Sylvie Malembic-Maher, Frederic Fabre.

**Visualization:** Hola K. Adrakey, Frederic Fabre.

**Writing – original draft:** Hola K. Adrakey, Frederic Fabre.

**Writing – review & editing:** Hola K. Adrakey, Gavin J. Gibson, Sandrine Eveillard, Sylvie Malembic-Maher, Frederic Fabre.

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