Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework

Infection, Genetics and Evolution xxx (2014) xxx–xxx

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Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework Julien Papaïx a,b,⇑, Katarzyna Adamczyk-Chauvat b, Annie Bouvier b, Kiên Kiêu b, Suzanne Touzeau b, Christian Lannou a, Hervé Monod b a b

INRA, UR 1290 BIOGER-CPP, 78850 Thiverval Grignon, France INRA, UR 341 Mathématiques et Informatique Appliquées, 78350 Jouy-en-Josas, France

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Landscape epidemiology Landscape pattern indices Metapopulation model Model analysis Sensitivity analysis Tessellation models

a b s t r a c t Modelling processes that occur at the landscape scale is gaining more and more attention from theoretical ecologists to agricultural managers. Most of the approaches found in the literature lack applicability for managers or, on the opposite, lack a sound theoretical basis. Based on the metapopulation concept, we propose here a modelling approach for landscape epidemiology that takes advantage of theoretical results developed in the metapopulation context while considering realistic landscapes structures. A landscape simulator makes it possible to represent both the field pattern and the spatial distribution of crops. The pathogen population dynamics are then described through a matrix population model both stage- and space-structured. In addition to a classical invasion analysis we present a stochastic simulation experiment and provide a complete framework for performing a sensitivity analysis integrating the landscape as an input factor. We illustrate our approach using an example to evaluate whether the agricultural landscape composition and structure may prevent and mitigate the development of an epidemic. Although designed for a fungal foliar disease, our modelling approach is easily adaptable to other organisms. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Integrating the principles governing ecosystems into crop protection approaches is a powerful way to meet the challenge of agriculture regarding productivity improvement, while decreasing its environmental impact (Tilman, 1999). In particular, this will require shifting the scale at which crop protection strategies are designed from the field to the agricultural landscape (Plantegenest et al., 2007; Thrall et al., 2011). The concepts of landscape ecology – which relates to interactions between environment spatial patterns and ecological processes (Burel and Baudry, 2004; Turner, 2005) – will be essential for building a new and extended paradigm for crop protection. In this article, we present a modelling approach for landscape epidemiology that explicitly considers spatial structures typical of agricultural landscapes and provides a general framework for its analysis. The term ‘Landscape’ is understood here sensu Turner and Gardner (1991) and Forman (1995), i.e., as a spatially and/or temporally heterogeneous area at any scale relevant to the ecological process or organism under investigation. ⇑ Corresponding author. Address: Centre d’Ecologie Fonctionnelle et Evolutive, UMR 5175, Campus du CNRS, 1919, route de Mende, 34293 Montpellier cedex 5, France. Tel.: +33 (0)4 67 61 32 17. E-mail address: [email protected] (J. Papaïx).

The modelling of processes that occur at the landscape scale is gaining more and more attention from theoretical ecologists as well as agricultural managers. Among the diversity of existing models, a continuum of complexity can be identified. First, theoretical ecologists often describe space as a simple continuum or as a discrete lattice, sometimes as a 1D-environment. On the opposite, landscape ecologists tend to consider the landscape physical structure with a very high level of details. While the first approach lacks applicability for managers, the second one lacks a sound theoretical basis (Seppelt et al., 2009). Several authors have emphasized the need for agro-ecological models specifically designed to cope with the particularities of agricultural landscapes (e.g., Dalgaard et al., 2003). Indeed, the main difference between crops and wild vegetation is that the population dynamics and evolutionary trajectories of wild systems are not under direct human control (even though almost all plant communities have been influenced, directly or indirectly, by human activities (Kareiva et al., 2007)). In contrast, the type and density of plants that are grown in a cropping area and, to some extent, the crop environment itself are directly controlled by man. This has direct consequences on pathogen dynamics and evolution, as exemplified by Johnson (1961). The metapopulation structure (Levins, 1969) is particularly adapted to the representation of agricultural landscapes. It

http://dx.doi.org/10.1016/j.meegid.2014.01.022 1567-1348/Ó 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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requires, however, the integration of the specificities of such landscapes due to man-guided processes (e.g., the geometry of landscape structures as fields, roads. . .). In this modelling context the landscape is considered as a set of habitat patches linked by dispersal and immersed in an unsuitable environment with regard to the organism under investigation. A major interest of using metapopulation models is that a strong theoretical background is available, allowing to conciliate a rather flexible and realistic description of the landscape structures, easily accepted by agricultural managers, with the possibility of analytical developments, as promoted by theoretical ecologists (Hanski, 1998). In this article we present a modelling framework that benefits from theoretical results developed in the metapopulation context, while integrating a detailed representation of the man-guided landscape structures. This framework is flexible and is referred to as the Ddal (disease dynamics in agricultural landscapes, Fig. 1) framework (Papaïx, 2011). To illustrate the description of Ddal, a case study on the potential of landscape diversification for reducing the disease risk in crops is presented. It will be taken as an example all along the article. Then the two main components of Ddal are described. The first is a landscape simulator (Section 3) allowing the representation of both the physical landscape structure (i.e., the field pattern) and the qualitative landscape structure (i.e., the spatial distribution of host types). The second component concerns the population dynamics, described through dispersal (Section 4.1) and local life cycle (Section 4.2). In addition to a classical invasion analysis we present a stochastic simulation experiment and the framework is completed by a sensitivity analysis approach integrating the landscape as an input factor.

the mixture effectiveness in controlling the disease (Mundt and Brophy, 1988; Zhu et al., 2000). More generally, host diversification at the landscape scale could be of great interest for reducing epidemic severity in agricultural systems (Priestley and Bayles, 1980; Skelsey et al., 2005, 2010; Mundt et al., 2011). Up to now, however, most studies on crop resistance diversification were designed at the field scale and still few theoretical investigations on host diversification at the landscape scale are available in the phytopathological literature (Plantegenest et al., 2007). To illustrate the Ddal modelling framework, we used a case study of how the agricultural landscape composition and structure may prevent and mitigate the development of an epidemic. We assumed that the landscape is composed of two host types, one with a complete resistance to the disease and the other susceptible. We considered the spread of a single pathogen type which population remains genetically homogeneous over time. The model was designed for simulating polycyclic foliar diseases, such as rusts, but the metapopulation structure of the model makes it adaptable to other pathogen life-styles and even other organisms. We assessed the influence of the main characteristics of the landscape spatial heterogeneity (see Section 3.1. for a definition of ‘‘spatial heterogeneity’’) on the pathogen capacity to successfully invade the landscape and, when pathogen invasion was successful, on the global disease severity by varying (i) the field pattern characteristics, giving priority to the average field size; (ii) the landscape composition, measured by the proportion of resistant vs. susceptible hosts; (iii) the spatial allocation of host types to the fields, in order to reflect the contrast between aggregated or mixed host types distributions; (iv) the dispersal range of the pathogen.

2. A case study

3. Landscape representation

It is widely recognized that more sustainable cropping systems with respect to plant disease will require increased genetic diversity for resistance factors in space and time (Pautasso et al., 2005; Keesing et al., 2006). Mixtures of resistant and susceptible plants within the same field (variety mixtures) have been proved to slow down epidemics due to dilution effects (Mundt, 2002). In addition, increasing the size of the land area occupied by the host mixture from the field to the landscape has a positive impact on

3.1. Quantification of spatial heterogeneity Li and Reynolds (1995) based their definition of spatial heterogeneity on two components: the factor of interest, either continuous (e.g., temperature) or discrete (e.g., habitat types), and its variability in space. The spatial variability can be quantified through non-spatial components, the composition of the environment (e.g., proportion of a particular crop in an agricultural

Fig. 1. Schematic representation of the Ddal modelling framework.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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landscape), and spatial components, the configuration of the environment (Li and Reynolds, 1995; Gustafson, 1998). Several descriptors, called Landscape Pattern Indices (LPIs), were developed to quantify each of these components of spatial heterogeneity (O’Neill, 1988; Riitters et al., 1995; He et al., 2000; Ahlqvist and Shortridge, 2010). Much bibliography has been dedicated to devise new LPIs and to avoid some pitfalls due to the high correlation between LPIs and the fact that they quantify a complex combination of several heterogeneity components (Tischendorf, 2001; Li and Wu, 2004; Peng et al., 2010). LPIs do not define a unique landscape but they make it possible to decompose the landscape variability into a measurable and controllable part through quantitative variables, and a residual variability. Thus, the main features of a landscape can be varied in order to better study their influence on the agro-ecological processes. The following two sections describe a landscape simulator for modelling agricultural landscapes in this way. 3.2. Simulating the landscape physical structure The physical structure of an agricultural landscape (e.g., field pattern, roads, remnant wildlands...) is highly constrained by geometrical features of human origin. In this context, a vectorial approach is well adapted to describe spatial structures. In Ddal, the landscape physical structure is assumed to be exclusively due to the field pattern. Several methods are currently available for simulating a field pattern as a polygonal meshing of a plane (Okabe et al., 1992). The Voronoi tessellation, based on point processes, allows controlling the average and variability of field surfaces (e.g., Le Ber et al., 2006; Adamczyk et al., 2007; Le Ber et al., 2009). The Voronoi tessellation method produces convex polygons with a high number of vertices, which is not adapted to field patterns of agricultural landscapes. Here, we propose to use the T-tessellation simulation algorithm developed by Kiêu et al. (2013). These authors proposed a stochastic model that can be considered as a completely random distribution on the T-tessellation space and is closely related to a previous model introduced by Arak et al. (1993). Kiêu et al. (2013) improved the model by considering Gibbsian extensions based on an energy function so as to make it possible to control key LPIs of the field pattern (Fig. 2a–c). The simulation algorithm is based on the Metropolis–Hastings–Green principle that makes it possible to generate several field patterns with the same LPIs distribution (Fig. 2d–g). Based on this approach, the tessellation algorithm allows us to control the field surfaces average and variability as well as the square-like form of the fields. The first two LPIs are biologically important because they control the level of landscape fragmentation and the frequency of fields with extreme (small or high) surface, respectively. The latter LPI allows a more realistic landscape representation by avoiding triangular fields. 3.3. Distributing habitat types across the landscape The quality of the pathogen local habitat can depend on meteorological conditions, agricultural practises and crop species or varieties (host types in the following). For the sake of simplicity, the habitat quality is assumed here to be entirely determined by the host type and its spatial variability. In Ddal, habitat quality is described by three LPIs: the number of host types, the proportions in surface that they cover and their level of spatial aggregation (Fig. 3a–c). The aggregation level of host type k is described by the mean proportion, over the landscape, of neighbour fields that share the same host type:

P

i;hðiÞ¼k AIk ¼ P

tik

i;hðiÞ¼k N i

;

ð1Þ

3

where h(i) is the host type grown in the field i, tik is the number of fields that share the host type k in the neighbourhood of field i and Ni is the total number of fields in the neighbourhood of field i. In the computation of AIk, two fields are considered as neighbours if they shared a common edge. The host types are then allocated to the fields by the mean of a simulated annealing algorithm (Kirkpatrick et al., 1983). This is a stochastic algorithm that performs constrained optimisation and makes it possible to generate pseudorandom variety allocation replicates while controlling both their proportion and aggregation level (Fig. 3d–f). 3.4. Landscape generation for the case study Landscapes were generated according to three nested steps. First, the simulator described in Section 3.2. was used to generate the field patterns with two different average field sizes. The whole landscape consisted of a square whose surface was set to one, by convention. The fields surface average and variability were controlled and equalled 2.0 ⁄ 102 ± 4.7 ⁄ 104 and 6.5 ⁄ 103 ± 8.0 ⁄ 105 in terms of landscape unit square (Supporting Information, Figure A.1). As a result, five field patterns composed of around 50 fields (respectively, 52, 49, 51, 48 and 52) and five composed of around 155 fields (respectively, 155, 154, 152, 153, 156) were constructed. On each of these field patterns, the two host types were allocated to the fields controlling the proportion of the resistant host (10%, 30%, 50%, 70% and 90% – Supporting Information, Figure A.2) and their aggregation level (low for a mixed landscape, medium for a mosaic landscape, high for a aggregated landscape – Supporting Information, Figure A.3). Finally, each landscape structure (a field pattern by a crop proportion by a crop spatial aggregation) was replicated twice leading to 10 field patterns by 5 crop proportions by 3 spatial allocations by 2 replicates = 300 landscapes.

4. Population dynamics model 4.1. Dispersal 4.1.1. Dispersal scale Most of air-borne plant pathogens disperse at a smaller level than the whole field within a pathogen generation (Lannou et al., 2008; Frezal et al., 2009). Thus intra-field population dynamics require the spatial decomposition of fields into smaller units called patches. The patch is then the smallest spatial unit of the system: the pathogen population is supposed to be perfectly mixed at the patch level and it disperses among patches. As in metapopulation models, habitat conditions are homogeneous within a patch. Thus in our case, the host type present in a patch is determined by the field to which the patch belongs. In Ddal, the patches are generated as the intersections between the polygons of the field pattern and the polygons associated with a regular rectangular or triangular grid. The patch size has an influence on disease spread. In general, unless there is a natural patch size in the problem under study, we suggest testing different patch sizes to determine a value such that a finer grid would not alter the epidemic spread, given an average field size. 4.1.2. Dispersal function and dispersal rates In the case of fungal pathogens, the large majority of the spores are deposited relatively close to the source (Lannou et al., 2008; Frezal et al., 2009). Such dispersal is well represented by an individual dispersal function (IDF). An IDF is defined as the probability density of the deposit position, (x, y), of a spore emitted from a

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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Fig. 2. Simulated field patterns with controlled LPIs. (a) The number of fields is controlled (number of fields = 53, surface variability = 1.6  103); (b) the number of fields and their surface variability are controlled (number of fields = 51, surface variability = 1.8  104); (c) the three LPIs (number, surface variability and square-like form of the fields) are controlled (number of fields = 52, surface variability = 5.2  105). (d–g) examples of field patterns having the same LPIs. (d) number of fields = 49 and surface variability = 6.2  105; (e) number of fields = 51 and surface variability = 5.3  105; (f) number of fields = 48 and surface variability = 4.2  105; (g) number of fields = 52 and surface variability = 6.9  105.

Fig. 3. Examples of simulated landscape structures with two host types (light (50%) and dark (50%) grey) dispatched among 155-field landscapes. From (a) to (c): the spatial aggregation is increasing ((a) AI1 = 0.56, (b) AI1 = 0.63, (c) AI1 = 0.71) but the field pattern remains unchanged. (d–f) Landscape replicates (different field patterns with the same level of spatial aggregation – (d) AI1 = 0.64, (e) AI1 = 0.68, (f) AI1 = 0.63).

punctual source in (0, 0). Note that this implicitly introduces continuous space as a third and finest scale in the model. Let g(z, z0 ) denote the individual dispersal function between spatial points z and z0 . The dispersal rates between two patches could be done by cumulating the total amount of spores produced

by the source polygon and distribute these spores over a target polygon. Instead, and to better account for the geometry between neighbour patches, the integration of g(, ) is performed between pairs of points that belong to the area Ai and Aj of patches i and j, respectively, according to the formula:

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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R R mij ¼

Ai

Aj

0

gðz; z0 Þdz dz Ai

ð2Þ

:

The dispersal rates mij are computed using the CaliFloPP algorithm (Bouvier et al., 2009). This algorithm computes the integral of a point-wise dispersal function between any pair of source and target polygons. The CaliFloPP algorithm uses algorithmic geometry tools in order to decompose, triangulate and intercept polygons, then numerical integration tools in order to compute the flow of spores. 4.2. Matrix population model Matrix population models (Caswell, 2001) have proved their efficiency by including the population internal structures either due to different life-cycle stages (age-structured population) or to space (metapopulation). The matrix structure of the model makes it adaptable to other pathogen life-cycles and other organisms but we take as an example here a classical stage-structured model in plant epidemiology (Madden et al., 2008): the HealthyLatent-Infectious-Removed (HLIR, also called SEIR) model. The main terms and parameters are described in Table 1. The model describes the evolution of the number of plants in each patch and in each of the following states: healthy (uninfected – H), latent (L), infectious (I) and removed (R). The carrying capacity of patch i, Ki, is fixed and proportional to the surface of patch i. In patch i, a healthy plant becomes infected with the probability pðxÞ  ehðiÞ , where p(x) is the probability for a healthy plant to be contaminated by a spore, and eh(i) is the infection efficiency. The function pðÞ is an increasing function of x = Hi/Ki, with p(0) = 0 and p(1) = 1. We assume here that susceptible plants become less easily contaminated when the local disease level increases by using the following sigmoid function for pðÞ: Table 1 Definition of the main parameters, variables and functions. Notation

Definition

Landscape description Number of fields Ai Surface of field i

Square-like form of fields Spatial aggregation of host type k Proportion of host types

– (Landscape unit)2 (Landscape unit)4 – – –

Individual dispersal function Dispersal rate from patch i to patch j Mean dispersal distance

– – –

Field surface variability

AIk Dispersal g() mij m0

Units

Pathogen life cycle H Healthy hosts L Latent hosts I Infectious hosts R Removed hosts Ki Carrying capacity of patch i p() Contamination function ek Infection efficacy on the host type k s Latency period T Infectious period r Spore production Model analysis R0 Basic reproductive number based on the next generation matrix 0 Basic reproductive number based on the heuristic R0 of Park et al. (2001) Y Observed pathogen invasion during simulations HS Integrated healthy surface –: the parameter, variable or function is dimensionless.

– – – – – – – Days Days –

pðxÞ ¼ 1 

5

expð5:33x3 Þ  expð5:33Þ ; 1  expð5:33Þ

giving an inflection point for x  0.5 equal to p(0.5) = 0.5. The use of a sigmoid function instead of a linear function implies that the contamination of a susceptible plant is easier when the proportion of susceptible plants is higher than 50%, and harder when the proportion of susceptible plants is lower than 50%. Spores that do not succeed in contaminating a plant are removed. When infected, the plant remains latent during s days before producing spores (the latent period). After T days of sporulation, the plant is removed. An infectious plant produces r spores per day. A spore remains in patch i with the probability mii and disperse to patch j with the probability mij. These probabilities are computed by the dispersal formula (Eq. (2)). The deterministic version of the model can be synthesised, for patch i (i = 1, 2, . . .), by the following system of equations: 8 N   X > > dHi > ¼ ehðiÞ p HK i r mij Ij > > dt i > > j¼1 > > > > N   X < dLi ¼ ehðiÞ p HK ii r mij Ij  1s Li : dt > > j¼1 > > > > dIi 1 1 > > dt ¼ s Li  T Ii > > > : dRi 1 ¼ T Ii dt

ð3Þ

4.3. Application to the case study A solid body of literature suggests that air-borne pathogen dispersal follows an inverse power function of dispersal (e.g., Frantzen and van den Bosch, 2000; Sackett and Mundt, 2005). Accordingly, the spore density emitted from a given source point z and arriving at a given reception point z0 can be given by:

gðz; z0 Þ ¼

ða  2Þða  1Þ 2

2pb

 a kz  z0 k 1þ ; b

where kz  z0 k is the Euclidean distance between z and z0 , b > 0 is a scale parameter and a > 2 determines the weight of the dispersal tail. The mean dispersal distance (mean distance travelled by a spore) is defined only when a > 3 and equals m0 = 2b/(a  3). While the weight of the dispersal tail was fixed to a = 3.4 the mean dispersal distance, expressed in proportion of the landscape size, varied in m0 e {0.025, 0.1, 0.25} by varying parameter b. Epidemics were also simulated over 1000 days using a discretetime and stochastic version of the model described in Eq. (3) (see also Fig. 4 and Appendix A). The number of spores produced by an infectious plant during one day was set to r = 2. Infected plants were supposed to remain latent during s = 5 days before producing spores during T = 10 days. On the susceptible host SH, the infection efficiency was fixed to eSH = 0.1. The resistant host RH was assumed to be completely resistant by setting the infection efficiency at eRH = 0. Epidemics were initialised by one infectious plant in one SH patch chosen at random. Model stochasticity was taken into account by simulating two epidemics with two different starting points on each of the 300 landscape structures. Note that more than two model replicates would be necessary for a refined study of the effects due to model stochasticity in a specific landscape structure. However, two replicates by 300 landscape structures (and 3 dispersal distances) provide a lot of degrees of freedom to estimate the average variability due to model stochasticity.

–

5. Population growth rate and invasion threshold – – –

5.1. General considerations In epidemiology, the basic reproductive number, R0, is classically used for predicting the occurrence of epidemics (Gilligan,

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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2008; Madden et al., 2008). R0 gives the expected number of new infections originating from a single infected host during the average duration of the infectious period. In a determinist model, the basic reproductive number must be higher than 1 for the epidemics to occur. Classically, the basic reproductive number is computed using the next generation matrix (van den Driessche and Watmough, 2002; Fulford et al., 2002). Such invasion criteria are based on the rarity hypothesis that assumes that the pathogen population is initially composed of an infinitesimal individual equally distributed among patches. Thus, this method takes into account the spatial structure of the host population but not the spatial distribution among patches of the pathogen population at initialisation (when rare). The next generation matrix gives then the number of plants in each life-cycle stage and for each patch at time t from the same quantities at time t  1. With this method, only the dynamics of the infected states have to be considered to compute the basic reproductive number. They can be decomposed into new infections (F) and transfers between infectious states (V). The next generation matrix is then given by FV 1, where F is the Jacobian matrix of F and V is the Jacobian matrix of V, evaluated at the disease-free equilibrium (DFE). The basic reproductive number is then defined as the spectral radius, i.e., the highest eigenvalue in magnitude, of FV 1, R0 = q(FV 1). If R0 < 1, then DFE is locally stable, whereas if R0 > 1 the DFE is unstable and the pathogen can spread. The rarity hypothesis is difficult to characterise in the context of spatially structured populations: the few individuals that are potentially at the origin of an epidemic can be dispatched in several patches or together within the same patch. An alternative is to take into account the spatial location of the initial inoculums as the heuristic proposed by Park et al. (2001) – see Massol et al. (2009) for another criterion.

5.2. Application to the case study 5.2.1. Computation of the basic reproductive number We used the two approaches described in Section 5.1 to compute the basic reproductive number. We first computed R0 based on the next generation matrix. We show in Appendix B that R0 is equal to the dominant eigenvalue of a matrix whose elements in

line i and column i are Treh(i)mji. It is computed numerically according to the landscape structure. Additionally to R0 based on the next generation matrix, we computed R00 following Park et al. (2001). In the model described by Eq. (3), a single infectious plant in patch i potentially gives birth to Treh(i)mii new infectious plants in the same patch and to Treh(i)mij potential new infectious plants in patch j. Thus, the number of new infections produced by a single infectious plant in patch i P can be calculated as R00 ¼ Tr j ehðjÞ mij . 5.2.2. Experimental design and output variables Regarding the landscape, the LPIs of interest were: (i) the average field size (2 values); (ii) the proportion of resistant vs. susceptible hosts (5 values); (iii) the spatial allocation of host types to the fields (3 values). In addition, we varied, (iv) the dispersal range of the pathogen (3 values) which is directly connected to the sensitivity of the pathogen to the spatial heterogeneity. As described in Section 3.4., for each of these 90 conditions, 20 replicates were available through 5 field pattern replicates, 2 variety allocation replicates and 2 model replicates. For each simulation we numerically computed R0 and R00 . In addition we run the stochastic version of the model described in the Appendix A of Supporting Information. The capacity of the pathogen to invade successfully the host population in simulation s was then characterised by three binary variables: whether or not R0(s) > 1, whether or not R00 ðsÞ > 1, and whether or not the pathogen actually invaded the host population in the simulation (Y(s) = 1 if pathogen invasion was successful in simulation s and Y(s) = 0 otherwise). We considered that the pathogen successfully invaded the host population when, at the end of the simulation, more than 5% of the global susceptible host population was diseased. From Y we estimated the probability for the pathogen to invade as a function of the independent variables using the following Generalized Linear Mixed model (GLMm): 8 YðsÞjPobsðsÞ  BernðPobsðsÞÞ > > > > > > < logitðPobsðsÞÞ ¼ a0 þ a1;AGGðsÞ þ a2;AGGðsÞ FRAGðsÞ þ a3;AGGðsÞ DISPðsÞ > > > > > > :

þa3;AGGðsÞ PROPðsÞ

;

þa4;AGGðsÞ FRAGðsÞDISPðsÞ þ a5;AGGðsÞ FRAGðsÞPROPðsÞ þa6;AGGðsÞ DISPðsÞPROPðsÞ þ a7;AGGðsÞ FRAGðsÞDISPðsÞPROPðsÞ ð4Þ

Fig. 4. Example of an epidemic simulated with a stochastic version of the model described by Eq. (3). Landscape structure (a): white, susceptible host; grey: resistant host; red dot, spatial position of inoculum. Spatial spread of the pathogen population (b–f): grey scale, proportion of diseased plants (latent, infectious and removed plants – the darker the grey, the higher the proportion). The global dynamics of the pathogen population on the susceptible host are displayed in (g). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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where AGG is the three-level factor indicating the crop allocation strategy (mixed, mosaic or clustered, see Section 3.4), FRAG is the scaled average field surface, DISP is the scaled pathogen mean dispersal distance and PROP is the scaled proportion of the resistant host in the landscape. 5.2.3. Effect of landscape structure on pathogen invasion There were significantly more successful invasions in clustered than in mosaic than in mixed landscapes (Table 2, Fig. 5a–c). Decreases in the average field surface (more fragmented landscapes) did not affect much the pathogen invasion. Increases in the mean dispersal distance of the pathogen and increases in RH proportion led to a decrease in the probability of successful invasion by the pathogen. Indeed, under these conditions, the probability for a spore to land in a field containing the resistant host was higher, which increased dilution effects. For example, mixtures at the field scale are generally found less effective to control diseases caused by splash-dispersed pathogen (e.g., wheat septoria) with a steeper dispersal gradient than wind dispersed pathogen (e.g., wheat rusts or powdery mildew – Mundt, 2002). In addition interactions were consistent with increased dilution effects. Thus the mean dispersal distance and the spatial allocation of RH were found to interact positively so that the effect of mixing the varieties was higher when the pathogen dispersal ability was higher (Table 2 and compare Fig. 5a vs. b vs. c). A similar result was obtained regarding the interaction between host spatial allocation and RH proportion (Table 2 and compare the three curves of Fig. 5a–c). The mean dispersal distance of the pathogen was found to interact negatively with RH proportion. This implies that the negative effect on invasion obtained by increasing RH proportion was greater when the mean dispersal distance of the pathogen was longer (Table 2 and compare Fig. 5a vs. b vs. c). Due to demographic stochasticity present in the simulation model, the estimated probability of successful invasion by the pathogen was globally lower than predicted by the analytical criteria R0 and R00 (compare the top vs. the middle and bottom rows of

Fig. 5). R0 predicted pathogen invasion in all cases except for the mixed landscapes with high RH proportion and pathogen dispersal. Conversely, the general trends were comparable between the estimated probability of invasion and the R00 predictions (compare the top vs. the bottom rows of Fig. 5). This illustrates the importance to take into account the spatial structure of the pathogen population at inoculation because the local environment may prevent the epidemic spread. 6. Sensitivity analysis Classically, local sensitivity analysis (or elasticity analysis) is based on the computation of the model derivatives with respect to the model inputs of interest (Caswell, 2001). However, this derivative-based approach is unwarranted when the inputs vary in large intervals or in discrete sets, especially when the model is highly non-linear. In the case of the model presented here, sensitivity analysis can rather be based on the global approach promoted by Saltelli et al. (2008), adapted to include the landscape features among the factors of interest. Below we first present a global sensitivity analysis, taking simultaneously into account the influence of several input factors, then we make a focus on the system sensitivity to the landscape variables. 6.1. Global sensitivity analysis Each input factor is given on a discrete or continuous domain of variation, called the uncertainty domain. Several methods are then available to quantify the influence of the factors (Monod et al., 2006; Iooss, 2011) but we expose the variance-based methods, based on a decomposition of the model’s output variance (Sobol, 1990; Saltelli et al., 2000). When the input factors vary independently in their uncertainty domains, the Sobol decomposition leads to a unique decomposition of the output variance:

V G ¼ V 1 þ    þ V n þ V 1;2 þ    þ V n1;n þ    þ V 1;;n ;

Table 2 Estimated effects of each input variable on the probability to observe pathogen invasion (see Eq. 4).

Intercept Mosaic Clustered Mixed:FRAG Mosaic:FRAG Clustered:FRAG Mixed:DISP Mosaic:DISP Clustered:DISP Mixed:PROP Mosaic:PROP Clustered:PROP Mixed:FRAG:DISP Mosaic:FRAG:DISP Clustered:FRAG:DISP Mixed:FRAG:PROP Mosaic:FRAG:PROP Clustered:FRAG:PROP Mixed:DISP:PROP Mosaic:DISP:PROP Clustered:DISP:PROP Mixed:FRAG:DISP:PROP Mosaic:FRAG:DISP:PROP Clustered:FRAG:DISP:PROP

Estimate

Standard error

Z-value

Pr(>|Z|)

1.38814 0.98705 1.44153 0.44728 0.11604 0.14853 1.0885 0.22764 0.19840 1.04029 0.39221 0.01437 0.32221 0.03354 0.04713 0.27055 0.05196 0.09052 0.93403 0.28734 0.05676 0.20043 0.02858 0.01387

0.16075 0.18267 0.18066 0.16079 0.08680 0.08247 0.17284 0.08899 0.08269 0.14864 0.08827 0.08256 0.17289 0.08901 0.08272 0.14868 0.08830 0.08258 0.15606 0.09110 0.08282 0.15611 0.09113 0.08285

8.636 5.403 7.979 2.782 1.337 1.801 6.298 2.558 2.399 6.999 4.443 0.174 1.864 0.377 0.570 1.820 0.588 1.096 5.985 3.154 0.685 1.284 0.314 0.167

<2e16 (***) 6.54e08 (***) 1.47e15 (***) 0.00541 (**) 0.18128 0.07171 (.) 3.01e10 (***) 0.01052 (*) 0.01643 (*) 2.58e12 (***) 8.87e06 (***) 0.86178 0.06236 (.) 0.70633 0.56885 0.06880 (.) 0.55623 0.27300 2.16e09 (***) 0.00161 (**) 0.49312 0.19916 0.75383 0.86704

Signif. codes: (.) 0.1. 0.05. ** 0.01. *** 0.001. *

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where, VG is the global variance, Vi is the first order effect (or main effect) of input factor i, Vi,j is the interaction of order one between factors i and j, and the rest of the decomposition consists of higher order interactions. When the model is stochastic, an additional term must be added to this decomposition to consider the output variance due to stochasticity (when the input factors are fixed).

The Sobol’s sensitivity indices are calculated as the ratios between the variance terms and the global variance. For example, SIi = Vi/VG denotes the first order sensitivity index of factor i and SIi,j = Vi,j/VG denotes the sensitivity index of the interaction of factors i and j. Note that the sensitivity indices belong to the (0, 1) interval, that they sum to one, and that a high index indicates a

Fig. 5. Estimated probability of successful invasion (a–c) and predicted successful invasions ((d–f), R0; (g–i), R00 ) as a function of RH proportion (x-axes), RH spatial aggregation (grey scale – light grey: mixed allocation; dark grey: mosaic allocation; black: clustered allocation) and mean dispersal distance (a and g: m0 = 0.025; b and h: m0 = 0.1; c and i: m0 = 0.25). Solid lines: response curves, dashed lines: 95% bootstrap confidence intervals. Bootstrap confidence envelopes were constructed based on 1000 samplings with replacement among simulations. These results are for 155-field landscapes and based on the GLMm described in Eq. 4.

Fig. 6. Schematic representation of a sensitivity analysis with a landscape as an input factor. Landscape descriptors (LPIs) make it possible to define a set of landscapes which are used as an input of the model. The sensitivity of model outputs to landscape descriptors and other parameters could then be done through variance decomposition.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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high influence on the model output. The total sensitivity index of factor i is defined as the sum of all first order and interaction sensitivity indices that involve factor i. It also belongs to (0, 1) and gives a global measure of the influence of factor i on the system. In practise, the sensitivity indices cannot be calculated exactly and they must be estimated based on the model simulations. This requires to perform a computer experiment by (i) defining the input factors and their uncertainty domains, (ii) choosing the most appropriate method for designing the experimental plan (Fang et al., 2006) and the number of simulations to run, (iii) drawing a sample for each input factor within the input domain according to the experimental plan, (iv) running the simulations, (v) calculating the estimated sensitivity indices. We propose to apply the meta-modelling technique developed by Sudret (2008), which allows to take into account a large number of levels/values per variable, to estimate the sensitivity indices. 6.2. Landscape as a complex input factor Regarding sensitivity analysis, landscape is a complex input factor that requires specific attention. Problems and possible solutions associated with complex input factors are discussed by Iooss and Ribatet (2009). Agricultural landscape structure and composition are integrated explicitly in the sensitivity analyses performed in Viaud et al. (2008), Lavigne et al. (2008) and Colbach et al. (2009). However these studies cope with deterministic models and with factorial designs that involve a small number of levels per factor. Here we adapt their method to stochastic models. In Section 3.1., the landscapes are described by Landscape Pattern Indices (LPIs). As stressed by Li and Wu (2004), LPIs do not define a unique landscape but make it possible to decompose the landscape variability into a part that can be measured and controlled through quantitative variables and a part that is considered as residual variability. Two main questions then arise when studying the influence of the landscape structure on an ecological process: how much is the process influenced by the LPIs of interest? And how robust are the results with regard to the landscape residual variability? The LPIs of interest are input factors that have to be considered at the sampling stage of the sensitivity analysis (Fig. 6). Each set of values of these LPIs determines a ‘patron’ and each generated landscape is then a random realisation around this patron. The sensitivity analysis is based on the following decomposition:

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the relative healthy surface was defined as the proportion of healthy plants in the global susceptible host population, and then integrated over time to obtain HS. HS is a proxy of disease impact on the host (van den Bosch and Gilligan, 2003). We then studied the sensitivity of R0, R00 , and HS to the input variables (spatial aggregation, average field surface, pathogen mean dispersal distance and proportion of the resistant host in the landscape) using the framework described in Section 6.1 and a meta-model consisting of Legendre polynomials of degree 3 (Sudret, 2008).

6.3.2. Sensitivity of epidemics severity to the landscape structure Fig. 7 shows the result of the sensitivity of the basic reproductive numbers (R0 and R00 ) and the integrated healthy surface (HS) to the input variables. The sensitivity indices were consistent for the three output variables: the larger part of the variance was explained by RH proportion and spatial aggregation. Generally, habitat loss (i.e., increase in RH proportion) has a strong negative effect on population dynamics and biodiversity. On the contrary, habitat fragmentation (i.e., decrease in aggregation level and increase in the number of fields) has a weaker effect (Fahrig, 2003). Our results are consistent with this literature, since the RH proportion had the greatest effect on R0, R00 and HS. The mean dispersal distance was also found to play an important role. While the four input variables explained well the variability of R0, the residual variance explained approximately 30% of the R00 variance. The high residual variance is consistent with differences observed between predictions of pathogen invasion by R0 and by R00 in Section 5.2.3, potentially reflecting the influence of local spatial structures on R00 as it takes into account the spatial structure of the pathogen population at inoculation. This was also the case for

1. The LPIs of interest are included in the main sampling design among the other input factors; 2. a set of landscape replicates is generated for each LPI combination of the sampling design; 3. for each LPI combination defined in the sampling design and each landscape replicate, several simulations are performed. This hierarchical structure allows us to compute the sensitivity indices associated with all input factors, including the LPIs. It also makes it possible to assess the output variability that can be attributed to landscape residual variability (through landscape replicates) and to other sources of model stochasticity (through model replicates). 6.3. Application to the case study 6.3.1. Experimental design and outputs The experimental design is described in Section 5.2.2. For each simulation, the spreading capacity of the pathogen population was characterised by R0 and R00 . In addition, for the simulations in which the pathogen was able to invade the host population we computed the integrated host healthy surface (HS). At a given time,

Fig. 7. Total sensitivity indices of R0 (a), R00 (b) and HS (c) to the four input variables (RH proportion, RH aggregation level, mean dispersal distance and number of fields) and the residuals.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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HS but to a lesser extent. In addition HS was found more influenced than R0 and R00 by the spatial aggregation of host types. 7. Discussion The definition of the agro-metapopulation by Bousset and Chèvre (2013) was a first step in adjusting the theoretical concept of metapopulation (Levins, 1969) to the specificities of the agro-ecosystems. Here we propose the modelling framework Ddal to extend and implement practically this concept (Fig. 1). The metapopulation structure of Ddal makes it possible to consider a high level of details of the landscape structure and the specificities of the organism under study while keeping a sound theoretical basis. We illustrated how such continuum can be maintained on an example evaluating whether the agricultural landscape composition and structure may prevent and mitigate the development of an epidemic caused by an air-borne plant pathogen. The first step of the model consisted in defining the environmental structures. The agricultural context had naturally led to consider space as both continuous and structured by 2D-objects, the fields. Field patterns were obtained by using a T-tessellation algorithm that makes it possible to control the size, number and shape of fields (Kiêu et al., 2013). This approach was chosen in order to obtain field patterns sharing common characteristics (LPIs) but all different from each other. As we explained, such landscape replicates are of prime interest for testing the robustness of results or to study the variability of a model output between a set of landscapes. Then the spatial variability of habitat quality was described by three LPIs: the number of host types, the proportion in surface that they cover and their level of spatial aggregation. This approach is simple and purely spatial but advanced softwares can be used as an alternative to simulate crop allocation on an agricultural landscape and their rotation over several years. For example, Le Ber et al. (2006) developed the CarrotAge model for analysing spatio-temporal databases and studying the cropping patterns on a territory. More recently, Castellazzi et al. (2010) provided the LandSFACTS software, which makes it possible to simulate crop allocation strategies under specific spatio-temporal constraints. Most spatially explicit models for population dynamics consider migration as a reaction–diffusion model (e.g., Débarre et al., 2009; Sapoukhina et al., 2010). Another more realistic way for modelling dispersal at the landscape scale is the use of an individual dispersal function (IDF). IDFs are classified in three categories. Empirical models are made of a parametric probability density function. Quasi-mechanistic models take into account the major atmospheric mechanisms. Mechanistic models describe the physical movement of a particle in the atmosphere with parameters that have a physical meaning. We considered here empirical models because they are open to the empirical/experimental estimation of parameters, which often is not the case for mechanistic models. In addition, they involve less parameters and take less time to compute. Then, we used a particular algorithm, CaliFloPP (Bouvier et al., 2009), in order to compute patch-to-patch dispersal rates. The CaliFloPP algorithm performs the integration of an individual dispersal function between any pair of polygons. Respectively to raster approaches it insures non biased dispersal rates for three reasons. First, raster methods suppose that dispersal occurs from barycentre to barycentre. However, due to the high non-linearity for short distances of the IDF, dispersal rates can be biased. Second, by integrating the IDF, the geometrical shape of polygons is taken into account, which is not the case otherwise. Third, if fields are not contiguous but separated by a gap or a road, this is automatically taken into account. Note that anisotropies in the dispersal function (Soubeyrand et al., 2007), for example due to a preferential winds or wind gusts, can be handled in the computation of dispersal rates.

The computation of patch-to-patch dispersal rates allows us to describe the pathogen population dynamics as a matrix population model both stage (life-cycle states) and space (the patches) structured. The flexibility and the huge amount of theoretical results available for matrix population models open the way for the exploration of landscape management strategies, at least in part, both through analytical and simulation based analysis, insuring thus a more solid theoretical basement for complex simulation models. The possibility to compare theoretical to simulation based criteria was illustrated using the classical basic reproductive number (R0). We assessed the sensitivity of two basic reproductive numbers to the landscape structures. We found that computing R0 by considering the spatial structure of the pathogen population at inoculation markedly changed the prediction of successful pathogen invasion. This result was also found in the simulation study showing that the spatial structure of the pathogen population at inoculation is crucial for predicting pathogen invasion at the landscape scale because the local environment may prevent the epidemic spread. We illustrated the Ddal modelling framework with a simple demographic model. However, when resistant crop varieties are introduced in agrosystems, they usually show a decreasing resistance level over time, due to the rapid adaptation of pathogens (Stukenbrock and McDonald, 2008). Recently, Papaïx et al. (2013) developed an adaptive dynamics model to study phenotypic changes in an evolving organism inhabiting a spatially explicit metapopulation. Due to the metapopulation structure, such tools could be easily transferred to the Ddal framework to study the role of landscape structures in driving the adaptation of pathogen population to crops. The issue of data-based inference must also be considered. The modelling framework exposed here could help to find the most influential features in the landscape on the population dynamics and spread in order to guide further statistical investigation or set up experiments (Fortin et al., 2003). In addition, the increasing development of statistical inference when likelihood is not available (Beaumont, 2010; Hartig et al., 2011) offers an interesting framework for parameter estimation at the landscape scale. In plant pathology, the scale at which control strategy is defined has proved crucial for its effectiveness (Gilligan et al., 2007). More generally, the effect of spatial scale is an important issue in landscape ecology (Rickelfs, 1987). The meta-ecosystem concept (Loreau et al., 2003) appears interesting to operate the transition into a regional scale model. A meta-ecosystem is defined as ‘a set of ecosystems connected by spatial flows of energy, materials and organisms across ecosystem boundaries’ (Loreau et al., 2003). In this context, the transition to larger scales would be done by considering a set of landscapes interconnected via spatial flows. This hierarchical concept could appear useful to identify the role of spatial heterogeneity at different scales on pathogen invasion at the regional scale (Melbourne et al., 2007). Ecological management of agricultural landscapes relies on the understanding of complex biotic interactions which are likely to be highly variable and difficult to predict. However, the evaluation of management practises based on a better perception of the agroecosystem functioning need for predictive tools (Shennan, 2008; Thrall et al., 2011). On one hand, the consideration of complex environmental structures, the ecological and evolutionary processes – including interactions between agricultural practises, crops, pests, natural enemies and wild plant communities – occurring in agro-ecological landscapes (Burdon and Thrall, 2008) make unavoidable complex modelling approaches based on numerical investigation and simulation methods (Evans et al. 2013). On the other hand, it is necessary to ensure coherence with analytical solutions available in the case of simpler systems to avoid pitfalls and to keep safe the dialogue between theoretical ecologists and environmental practitioners. The modelling approach developed

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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here attempts to provide a unified framework to understand and predict large scale processes in agricultural landscapes. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.meegid.2014.01. 022. References Adamczyk, K., Angevin, F., Colbach, N., Lavigne, C., Le Ber, F., Mari, J.-F., 2007. Genexp, un logiciel simulateur de paysages agricoles pour l’étude de la diffusion de transgènes. Géomatique 17, 469–487. Ahlqvist, O., Shortridge, A., 2010. Spatial and semantic dimensions of landscape heterogeneity. Landscape Ecol. 25, 573–590. Arak, T., Clifford, P., Surgailis, D., 1993. Point-based polygonal models for random graphs. Adv. Appl. Probab. 25, 348–372. Beaumont, M.A., 2010. Approximate Bayesian computation in evolution and ecology. Annu. Rev. Ecol. Syst. 41, 379–406. Bousset, L., Chèvre, A.-M., 2013. Stable epidemic control in crops based on evolutionary principles: Adjusting the metapopulation concept to agroecosystems. Agric. Ecosyst. Environ. 165, 118–129. Bouvier, A., Kiêu, K., Adamczyk, K., Monod, H., 2009. Computation of the integrated flow of particles between polygons. Environ. Modell. Softw. 24, 843–849. Burdon, J.J., Thrall, P.H., 2008. Pathogen evolution across the agro-ecological interface. Implications for disease management. Evol. Appl. 1, 57–65. Burel, F., Baudry, J., 2004. Landscape Ecology: Concepts, Methods and Applications. Science Publishers Inc., Plymouth (UK). Castellazzi, M.S., Matthews, J., Angevin, F., Sausse, C., Wood, G.A., Burgess, P.J., Brown, I., Conrad, K.F., Perry, J.N., 2010. Simulation scenarios of spatio-temporal arrangement of crops at the landscape scale. Environ. Modell. Softw. 25, 1881– 1889. Caswell, H., 2001. Matrix Population Models. Sinauer Associates Inc. Publishers, Sunderland (USA). Colbach, N., Monod, H., Lavigne, C., 2009. A simulation study of the medium-term effects of field patterns on cross-pollination rates in oilseed rape (Brassica napus L.). Ecol. Model. 220, 662–672. Dalgaard, T., Hutchings, N.J., Porter, J.R., 2003. Agroecology, scaling and interdisciplinarity. Agric. Ecosyst. Environ. 100, 39–51. Débarre, F., Lenormand, T., Gandon, S., 2009. Evolutionary epidemiology of drugresistance in space. PLoS Comput. Biol. 5 (4), e1000337. http://dx.doi.org/ 10.1371/journal.pcbi.1000337. Evans, M., Grimm, V., Johst, K., Knuuttila, T., de Langhe, R., Lessells, C.M., Merz, M., O’Malley, M.A., Orzack, S.H., Weisberg, M., Wilkinson, D.J., Wolkenhauer, O., Benton, T.G., 2013. Do simple models lead to generality in ecology? Trends Ecol. Evol. 28, 578–583. Fahrig, L., 2003. Effects of habitat fragmentation on biodiversity. Annu. Rev. Ecol. Evol. Syst. 34, 487–515. Fang, K.-T., Li, R., Sudjianto, A., 2006. Design and Modeling for Computer Experiments. Chapman & Hall/CRC. Forman, R.T.T., 1995. Some general principles of landscape and regional ecology. Landscape Ecol. 10, 133–142. Fortin, M.J., Boots, B., Csillag, F., Remmel, T.K., 2003. On the role of spatial stochastic models in understanding landscape indices in ecology. Oikos 102, 203–212. Frantzen, J., van den Bosch, F., 2000. Spread of organisms: can travelling and dispersive waves be distinguished? Basic Appl. Ecol. 1, 83–91. Frezal, L., Robert, C., Bancal, M.O., Lannou, C., 2009. Local dispersal of Puccinia triticina and wheat canopy structure. Phytopathology 99, 1216–1224. Fulford, G.R., Roberts, M.G., Heesterbeek, J.A.P., 2002. The metapopulation dynamics of an infectious disease: tuberculosis in possums. Theor. Popul. Biol. 61, 15–29. Gilligan, C.A., 2008. Sustainable agriculture and plant diseases: an epidemiological perspective. Philos. Trans. R. Soc. Lond. B. 363, 741–759. Gilligan, C.A., Truscott, J.E., Stacey, A.J., 2007. Impact of scale on the effectiveness of disease control strategies for epidemics with cryptic infection in a dynamical landscape: an example for a crop disease. J. R. Soc. Interface 4, 925–934. Gustafson, E.J., 1998. Quantifying landscape spatial pattern: what is the state of the art? Ecosystems 1, 143–156. Hanski, I., 1998. Metapopulation dynamics. Nature 396, 41–49. Hartig, F., Calabrese, J.M., Reineking, B., Wiegand, T., Huth, A., 2011. Statistical inference for stochastic simulation models – theory and application. Ecol. Lett. 14, 816–827. He, H.S., DeZonia, B.E., Mladenoff, D.J., 2000. An aggregation index (AI) to quantify spatial patterns of landscapes. Landscape Ecol. 15, 591–601. Iooss, B., 2011. Revue sur l’analyse de sensibilité globale de modèles numériques. J. de la Société Française de Statistique 152, 3–25. Iooss, B., Ribatet, M., 2009. Global sensitivity analysis of computer models with functional inputs. Reliab. Eng. Syst. Saf. 94, 1194–1204. Johnson, T., 1961. Man-guided evolution in plant rusts. Science 10, 357–362. Kareiva, P.M., Watts, S., McDonald, R., Boucher, T., 2007. Domesticated nature: shaping landscapes and ecosystems for human welfare. Science 316, 1866– 1869.

11

Keesing, F., Holt, R.D., Ostfeld, R.S., 2006. Effects of species diversity on disease risk. Ecol. Lett. 9, 485–498. Kiêu, K., Adamczyk-Chauvat, K., Monod, H., Stoica, R.S., 2013. A completely random T-tessellation model and Gibbsian extensions. Spat. Stat. 6, 118–138. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by simulated annealing. Science 220, 671–680. Lannou, C., Soubeyrand, S., Frezal, L., Chadoeuf, J., 2008. Autoinfection in wheat leaf rust epidemics. New Phytol. 177, 1001–1011. Lavigne, C., Klein, E.K., Mari, J.-F., Le Ber, F., Adamczyk, K., Monod, H., Angevin, F., 2008. How do genetically modified (GM) crops contribute to background levels of GM pollen in an agricultural landscape? J. Appl. Ecol. 45, 1104–1113. Le Ber, F., Benoît, M., Schott, C., Mari, J.-F., Mignolet, C., 2006. Studying crop sequences with CarrotAge, a HMM-based data mining software. Ecol. Modell. 191, 170–185. Le Ber, F., Lavigne, C., Adamczyk, K., Angevin, F., Colbach, N., Mari, J.-F., Monod, H., 2009. Neutral modelling of agricultural landscapes by tessellation methods – application for gene flow simulation. Ecol. Modell. 220, 3536–3545. Levins, R., 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240. Li, H., Reynolds, J.F., 1995. On definition and quantification heterogeneity. Oikos 73, 280–284. Li, H., Wu, J., 2004. Use and misuse of landscape indices. Landscape Ecol. 19, 389–399. Loreau, M., Mouquet, N., Holt, R.D., 2003. Meta-ecosystems: a theoretical framework for a spatial ecosystem ecology. Ecol. Lett. 6, 673–679. Madden, L.V., Hughes, G., van den Bosch, F., 2008. The Study of Plant Disease Epidemics. APS Press, St. Paul (USA). Massol, F., Calcagno, V., Massol, J., 2009. The metapopulation fitness criterion: proof and perspectives. Theor. Popul. Biol. 75, 183–200. Melbourne, B.A., Cornell, H.V., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A.L., Hall, R.J., Harrison, S., Hastings, A., Holland, M., Holyoak, M., Lambrinos, J., Moore, K., Yokomizo, H., 2007. Invasion in a heterogeneous world: resistance, coexistence or hostile takeover? Ecol. Lett. 10, 77–94. Monod, H., Naud, C., Makowski, D., 2006. Uncertainty and sensitivity analysis for crop models. In: Makowski, D., Wallach, D., Jones, J.W. (Eds.), Working with Dynamic Crop Models: Evaluation, Analysis, Parameterization, and Applications. Elsevier, pp. 55–100, chapter 4. Mundt, C.C., 2002. Use of multiline cultivars and cultivar mixtures for disease management. Annu. Rev. Phytopathol. 40, 381–410. Mundt, C.C., Brophy, L.S., 1988. Influence of number of host genotype units on the effectiveness of host mixtures for disease control: a modeling approach. Phytopathology 78, 1087–1094. Mundt, C.C., Sackett, K.E., Wallace, L.D., 2011. Landscape heterogeneity and disease spread: experimental approaches with a plant pathogen. Ecol. Appl. 21, 321– 328. Okabe, A., Boots, B., Sugihara, K., 1992. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley. O’Neill, R.V., 1988. Indices of landscape pattern. Landscape Ecol. 1, 153–162. Papaïx, J., 2011. Landscape Structure and Epidemic Risk, a Demo-Genetic Approach (Ph.D. thesis). AgroParisTech – ABIES. Papaïx, J., David, O., Lannou, C., Monod, H., 2013. Dynamics of adaptation in spatially heterogeneous metapopulations. PLoS One 8, e54697. Park, A.W., Gubbins, S., Gilligan, C.A., 2001. Invasion and persistence of plant parasites in a spatially structured host population. Oikos 94, 162–174. Pautasso, M., Holdenrieder, O., Stenlid, J., 2005. Susceptibility to fungal pathogens of forests differing in tree diversity. Ecol. Stud. 176, 263–289. Peng, J., Wang, Y., Zhang, Y., Wu, J., Li, W., Li, Y., 2010. Evaluating the effectiveness of landscape metrics in quantifying spatial patterns. Ecol. Indic. 10, 217–223. Plantegenest, M., Le May, C., Fabre, F., 2007. Landscape epidemiology of plant diseases. J. R. Soc. Interface 4, 963–972. Priestley, R.H., Bayles, R.A., 1980. Varietal diversification as a means of reducing the spread of cereal disease in the United Kingdom. J. Nat. Inst. Agric. Bot. 15, 205– 214. Rickelfs, R.E., 1987. Community diversity: relative roles of local and regional processes. Science 235, 167–171. Riitters, K.H., O’Neill, R.V., Hunsaker, C.T., Wickham, J.D., Yankee, D.H., Timmins, S.P., Jones, K.B., Jackson, B.L., 1995. A factor analysis of landscape pattern and structure metrics. Landscape Ecol. 10, 23–39. Sackett, K.E., Mundt, C.C., 2005. The effects of dispersal gradient and pathogen life cycle components on epidemic velocity in computer simulations. Phytopathology 95, 992–1000. Saltelli, A., Tarantola, S., Campolongo, F., 2000. Sensitivity analysis as an ingredient of modeling. Stat. Sci. 15, 377–395. Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global sensitivity analysis. The primer. Wiley, Chichester (UK). Sapoukhina, N., Tyutyunov, Y., Sache, I., Arditi, R., 2010. Spatially mixed crops to control the stratified dispersal of airborne fungal diseases. Ecol. Modell. 221, 2793–2800. Seppelt, R., Muller, F., Schroder, B., Volk, M., 2009. Challenges of simulating complex environmental systems at the landscape scale: a controversial dialogue between two cups of espresso. Ecol. Modell. 220, 3481–3489. Shennan, C., 2008. Biotic interactions, ecological knowledge and agriculture. Philos. Trans. R. Soc. Lond. B. 363, 717–739. Skelsey, P., Rossing, W.A.H., Kessel, G.J.T., Powell, J., van der Werf, W., 2005. Influence of host diversity on development of epidemics: an evaluation and elaboration of mixture theory. Phytopathology 95, 328–338.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022

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J. Papaïx et al. / Infection, Genetics and Evolution xxx (2014) xxx–xxx

Skelsey, P., Rossing, W.A.H., Kessel, G.J.T., van der Werf, W., 2010. Invasion of Phytophthora infestans at the landscape level: how do spatial scale and weather modulate the consequences of spatial heterogeneity in host resistance? Phytopathology 100, 1146–1161. Sobol, I.M., 1990. Sensitivity estimates for non linear mathematical models. Matematicheskoe Modelirovanie 2, 112–118 (in Russian). English translation in: Mathematical Modeling and Computational Experiment, 1993, 1, 407–414.. Soubeyrand, S., Enjalbert, J., Sanchez, A., Sache, I., 2007. Anisotropy, in density and in distance, of the dispersal of yellow rust of wheat: experiments in large field plots and estimation. Phytopathology 97, 1315–1324. Stukenbrock, E.H., McDonald, B.A., 2008. The origins of plant pathogens in agroecosystems. Annu. Rev. Phytopathol. 46, 75–100. Sudret, B., 2008. Global sensitivity analysis using polynomials chaos expansions. Reliab. Eng. Syst. Saf. 93, 964–979. Thrall, P.H., Oakeshott, J.G., Fitt, G., Southerton, S., Burdon, J.J., Sheppard, A., Russell, R.J., Zalucki, M., Heino, M., Denison, R.F., 2011. Evolution in agriculture: the application of evolutionary approaches to the management of biotic interactions in agro-ecosystems. Evol. Appl. 4, 200–215.

Tilman, D., 1999. Global environmental impacts of agricultural expansion: the need for sustainable and efficient practices. Proc. Natl. Acad. Sci. USA 96, 5995–6000. Tischendorf, L., 2001. Can landscape indices predict ecological processes consistently? Landscape Ecol. 16, 235–254. Turner, M.G., 2005. Landscape ecology: what is the state of the science? Annu. Rev. Ecol. Evol. Syst. 36, 319–344. Turner, M.G., Gardner, R.H., 1991. Quantitative Methods in Landscape Ecology: An Introduction. Springer-Verlag, New York (USA). van den Bosch, F., Gilligan, C.A., 2003. Measures of durability of resistance. Phytopathology 93, 616–625. van den Driessche, P., Watmough, J., 2002. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48. Viaud, V., Monod, H., Lavigne, C., Angevin, F., Adamczyk, K., 2008. Spatial sensitivity of maize gene-flow to landscape pattern: a simulation approach. Landscape Ecol. 23, 1067–1079. Zhu, Y., Chen, H., Fan, J., Wang, Y., Li, Y., Chen, J., Fan, J.X., Yang, S., Hu, L., Leung, H., Mew, T.W., Teng, P.S., Wang, Z., Mundt, C.C., 2000. Genetic diversity and disease control in rice. Nature 406, 718–722.

Please cite this article in press as: Papaïx, J., et al. Pathogen population dynamics in agricultural landscapes: The Ddal modelling framework. Infect. Genet. Evol. (2014), http://dx.doi.org/10.1016/j.meegid.2014.01.022