Willem Proesmans
email: willem.proesmans@inrae.fr

Matthias Albrecht

Anna Gajda

Peter Neumann

Robert J Paxton

Maryline Pioz

Christine Polzin

Oliver Schweiger

Josef Settele

Hajnalka Szentgyörgyi

Hans-Hermann Thulke

Adam J Vanbergen
email: adam.vanbergen@inrae.fr

Pathways for Novel Epidemiology: Plant-Pollinator-Pathogen Networks and Global Change

Multiple global change pressures, and their interplay, cause plant-pollinator extinctions and modify species assemblages and interactions. This may alter the risks of pathogen host shifts, intra-or interspecific pathogen spread, and emergence of novel population or community epidemics. Flowers are hubs for pathogen transmission. Consequently, the structure of plant-pollinator interaction networks may be pivotal in pathogen host shifts and modulating disease dynamics. Traits of plants, pollinators, and pathogens may also govern the interspecific spread of pathogens. Pathogen spillover-spillback between managed and wild pollinators risks driving the evolution of virulence and community epidemics. Understanding this interplay between host-pathogen dynamics and global change will be crucial to predicting impacts on pollinators and pollination underpinning ecosystems and human wellbeing.

Pollinator Disease in the Anthropocene

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, including spillover between wild and managed bee species. Given the importance of pollinators and pollination [1], it is vital to understand how anthropogenic pressures such as land use change [13], chemical stressors [14], altered habitat resources [15], Highlights Flower sharing amongst pollinator species represents a conduit for interspecific insect pathogen transmission. Plant-pollinator network structure and species traits shape pathogen dynamics in the pollinator community. Global change (climate change, invasive species, agricultural intensification, and urbanisation) can modulate species interactions, host susceptibility, and pathogen virulence, thereby creating novel epidemiological risks. Multiple global change effects can interact in synergistic or antagonistic ways, creating additional risks and complicating predictions of pathogen dynamics and evolution. Flower-mediated insect pathogen transmission provides a model framework to understand the interplay of interspecific pathogen dynamics under global change.

climate change [START_REF] Cornelissen | Global warming promotes biological invasion of a honey bee pest[END_REF], and alien species [START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF] modulate pollinator-pathogen dynamics. Such pressures may raise pollinator disease risk by altering the dominance, distribution, and interaction of primary or secondary host species and asymptomatic pathogen vectors. Understanding this interplay between host-pathogen dynamics and global change will be crucial to predicting impacts on pollinator populations, communities, and pollination functions underpinning ecosystems and human wellbeing (Figure 1).

Disease and the Plant-Pollinator-Pathogen Network

The distribution, diversity, and abundance of floral resources in a landscape are key to concentrating or diluting interspecific pollinator interactions and potential pathogen transfer. Pollinator species are obligate flower visitors, foraging and feeding on pollen and/or nectar for at least part of their life cycle, and therefore both compete for and share floral resources in time and space [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF]. This makes flowers hotspots of co-occurrence, direct contact, or indirect interaction (mediated by pollen or nectar) between pollinator species in a habitat or landscape [START_REF] Durrer | Shared use of flowers leads to horizontal pathogen transmission[END_REF][START_REF] Mcart | Arranging the bouquet of disease: floral traits and the transmission of plant and animal pathogens[END_REF][START_REF] Graystock | Dominant bee species and floral abundance drive parasite temporal dynamics in plantpollinator communities[END_REF][START_REF] Graystock | Parasites in bloom: flowers aid dispersal and transmission of pollinator parasites within and between bee species[END_REF]. determined by the species pool (A). Pathogen transmission through shared flowers is mediated by intrinsic species traits and plant-pollinator interaction network structure, with potential network rewiring following differential disease impacts on pollinator species (B). Environmental pressures (climate change, invasive species, agricultural intensification, and urbanisation, see Figure 2) affect pathogen dynamics directly by introducing new species differing in disease resistance or vectoring capacity (C). Environmental pressures may also rewire the network through changes in plant-pollinator species composition, abundance, behaviour, and phenology (D) or introduce novel pathogens and/or alter infectivity, virulence, and susceptibility in pollinator populations (E), unbroken blue lines show established pathogenicity, broken blue lines show hypothetical vectoring or pathogenicity. The Western honey bee Apis mellifera (red circle) is the most widespread managed pollinator species and plays a dominant role in plant-pollinator network structure (F). A. mellifera may act as a superspreader (H) exacerbated through intra-colony co-infection dynamics (G), as in the case of Deformed wing virus (DWV) which became more prevalent and virulent through association with Varroa destructor parasitism in managed hives (G). Honey bees can thus introduce pathogens into the wild pollinator community (spillover), or contract them from co-occurring wild pollinators (spillback) (H), which may drive further evolution of virulence (G, H). Icons sourced from: https://thenounproject.com/CC BY-NC-ND 2.0: (lluisa Iborra, Yu Luck, Les vieux garçons, Gan Khoon Lay, Blair Adams, Nareerat Jaikaew, and Prosymbols) and https://www.shutterstock.com (a free trial). Accordingly, flowers can serve as hubs of pathogen transmission within and between species and pathogens may 'travel' through the network of plant-pollinator interactions, potentially crossing species boundaries and infecting new hosts [START_REF] Graystock | Parasites in bloom: flowers aid dispersal and transmission of pollinator parasites within and between bee species[END_REF]. Studying this tripartite plant-pollinatorpathogen network can therefore help to identify potential flower-mediated transmission routes and elucidate how anthropogenic disruptions to network structure may alter or create novel pathogen dynamics.

Plant-Pollinator Traits Governing Pathogen Transmission

Pollinator species traits determine their susceptibility to different environmental pressures and their potential to transmit pathogens [START_REF] Williams | Ecological and life-history traits predict bee species responses to environmental disturbances[END_REF]. Species' diet breadth and preference may modulate the probability of encountering pathogens in nectar/pollen, on infected hosts, or deposited on flowers. Relatively specialised pollinators that forage on few plant taxa may have a lower exposure risk than species with broader floral diets. Less specialised species function as generalist pollinators in interaction networks, being highly connected to multiple forage plant species and thus indirectly to other pollinators [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF]. Consequently, generalists may have a greater probability of pathogen exposure or infection (novel or otherwise) and accordingly vector multiple pathogens across the network. Generalist species, however, may differ in foraging preferences and individual foragers of a generalist species may select for specific subsets of forage plant species, both of which may mitigate the risk of flower-mediated pathogen spread through a community [START_REF] Ellner | Individual specialization and multihost epidemics: disease spread in plant-pollinator networks[END_REF]. Moreover, generalist pollinators may dilute the risk of transmission per plant species [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF] and an emergent community epidemic by spreading their flower visits over multiple plant species.

Sociality is a key trait affecting disease dynamics. Eusocial pollinators (Apis, Bombus, Meliponini) play a disproportionately large role due to their life history, cooperative brood care, and overlapping generations in densely populated colonies exacerbating intra-colony disease spread [START_REF] Graystock | Dominant bee species and floral abundance drive parasite temporal dynamics in plantpollinator communities[END_REF]. Along with their generalised floral diets and long flying season bridging the phenology of other pollinator species, eusocial bees likely elevate the probability of interspecific contact and pathogen spillover. However, eusocial species may also mitigate the risk of increased pathogen burden through social immunity (e.g., behaviours to combat disease such as removing diseased larvae and dead workers from the colony [START_REF] Meunier | Social immunity and the evolution of group living in insects[END_REF]). Elucidating how gradients in foraging traits (e.g., pollen uptake and deposition rates, pollen transportation mode), colony sizes, and trait plasticity (e.g., facultative sociality in Halictidae) among taxa affect pathogen transmission, remains largely unexplored and is an important future direction for pollinator eco-epidemiology.

Plant traits may also modulate pollinator disease risk, with the potential for both facilitation and inhibition of pathogen transfer [START_REF] Mcart | Arranging the bouquet of disease: floral traits and the transmission of plant and animal pathogens[END_REF]. Floral architecture, by constraining access to pollen and nectar to pollinator species with the requisite morphological adaptations, filters insect visitor interactions and contacts [START_REF] Mcart | Arranging the bouquet of disease: floral traits and the transmission of plant and animal pathogens[END_REF]. Plants with structurally simple, large floral displays (e.g., Apiaceae, Rosaceae) attract numerous pollinator species and may serve as hubs facilitating interspecific pathogen transmission [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF]. Additionally, plants with a more central position in a plant-pollinator network often contain a high pathogen load [START_REF] Piot | Network centrality as an indicator for pollinator parasite transmission via flowers[END_REF]. More specialised plants with complex flower structures requiring intimate and sustained contact with a specialised pollinator may limit transmission to a particular species or functional group [START_REF] Mcart | Arranging the bouquet of disease: floral traits and the transmission of plant and animal pathogens[END_REF]. Alternatively, other floral characteristics, such as organic volatiles produced by plants to attract mutualists, may be toxic to certain pathogens [START_REF] Hammerbacher | Roles of plant volatiles in defence against microbial pathogens and microbial exploitation of volatiles[END_REF]. Nectar and pollen rewards may contain phytochemicals that inhibit pathogens when consumed by the host (e.g., reducing Crithidia sp. infection in bumble bees [START_REF] Giacomini | Medicinal value of sunflower pollen against bee pathogens[END_REF]), potentially reducing pathogen loads and community spread. Combined with limited pathogen resistance to environmental degradation (e.g., temperature, UV-radiation), such plant traits may limit pathogen survival on or within floral resources and subsequent pollinator infections [START_REF] Adler | Disease where you dine: plant species and floral traits associated with pathogen transmission in bumble bees[END_REF][START_REF] Figueroa | Bee pathogen transmission dynamics : deposition , persistence, and acquisition on flowers[END_REF].

Host-Pathogen Interactions and other Biota Affecting Disease

The interaction between the physiology of a pathogen and alternative hosts also has implications for epidemiological dynamics at population and community levels. Certain pollinator individuals or species may be immune to a particular pathogen or function as asymptomatic vectors, whereas others suffer from disease, leading to physiological stress, altered behaviour, and reduced fitness or death [START_REF] Manley | Contrasting impacts of a novel specialist vector on multihost viral pathogen epidemiology in wild and managed bees[END_REF]. Additionally, pathogen infections may differ in their effects according to the host insect nutritional status, influence of other stressors, and environmental context [START_REF] Brown | Strong context-dependent virulence in a host-parasite system: reconciling genetic evidence with theory[END_REF].

While pathogen research in pollinators is heavily biased towards honey bees and bumble bees, recent studies point to broader pathogen host ranges, such as Crithidia bombi infections in multiple bee families [START_REF] Ngor | Cross-infectivity of honey and bumble beeassociated parasites across three bee families[END_REF]. Pathogen strains or taxa can behave as specialists or generalists in terms of host range, virulence, and infectivity, affecting their fitness and ability to cross species boundaries [START_REF] Manley | Contrasting impacts of a novel specialist vector on multihost viral pathogen epidemiology in wild and managed bees[END_REF]. Single-stranded RNA viruses tend to pose the highest risk of co-infecting multiple species due to their high mutation rate, short generation time, and ability to cross phylogenetic boundaries [START_REF] Mcmahon | A sting in the spit: widespread cross-infection of multiple RNA viruses across wild and managed bees[END_REF][START_REF] Dobelmann | Genetic strain diversity of multi-host RNA viruses that infect a wide range of pollinators and associates is shaped by geographic origins[END_REF][START_REF] Dalmon | Possible spillover of pathogens between bee communities foraging on the same floral resource[END_REF]. Multiple co-infections of pathogens and parasites (RNA viruses, microsporidia, Acari) may further alter dynamics. For example, colonies of Apis mellifera infested with Varroa destructor mites can show a higher viral load and virulence of the RNA virus Deformed wing virus (DWV) [START_REF] Wilfert | Deformed wing virus is a recent global epidemic in honeybees driven by Varroa mites[END_REF]. Alternatively, pathogen co-infections may be asymmetric with intra-host competition inhibiting infection and disease manifestation [START_REF] Doublet | Within-host competition among the honey bees pathogens Nosema ceranae and Deformed wing virus is asymmetric and to the disadvantage of the virus[END_REF].

Pathogens may also alter pollinator host behaviour to increase transmission potential. For example, bumble bees infected with trypanosomatid parasites defecate more frequently on flowers thus spreading the pathogen [START_REF] Figueroa | Bee pathogen transmission dynamics : deposition , persistence, and acquisition on flowers[END_REF] and fungal pathogens may alter floral scent to lure pollinators [START_REF] Cellini | Pathogen-induced changes in floral scent may increase honeybee-mediated dispersal of Erwinia amylovora[END_REF]. Certain pathogens (e.g., Nosema spp.) increase their infectivity by forming cysts or spores that promote environmental persistence and probability of encountering susceptible hosts [START_REF] Evison | The biology and prevalence of fungal diseases in managed and wild bees[END_REF].

Beyond pathogens, other biota (e.g., gut microbiota) interacting with the host in mutualistic, commensal, or antagonistic ways, may modify pathogen dynamics. Lactobacilli inadvertently collected with pollen during flower visits may improve bee resistance to pathogens [START_REF] Mcfrederick | Environment or kin: whence do bees obtain acidophilic bacteria?[END_REF]. Arbuscular mycorrhizal fungi infecting plant roots can interact with soil fertility to alter nectar chemistry (e.g., alkaloids) in ways that influence pathogen prevalence [START_REF] Davis | From plant fungi to bee parasites: mycorrhizae and soil nutrients shape floral chemistry and bee pathogens[END_REF]. Interactions between virus-vectoring and non-vector herbivores mediated by changes in plant secondary chemicals (e.g., reduction of flavonoids) can indirectly influence plant virus transmission [START_REF] Su | A non-vector herbivore indirectly increases the transmission of a vector-borne virus by reducing plant chemical defences[END_REF]. It remains unknown whether similar complex trophic interactions might affect pathogens infecting pollinators.

Architecture of Plant-Pollinator Networks Governs Potential Pathogen Dynamics

Plant-pollinator networks describe the community of interacting species [START_REF] Nielsen | Ecological networks, nestedness, and sampling effort[END_REF][START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF]. The pool of species traits in a habitat or landscape, pollinator behaviour, metabolic and dietary needs, and the phenological synchrony of potential mutualisms dictate the architecture of a network [START_REF] Vázquez | Uniting pattern and process in plant-animal mutualistic networks: a review[END_REF] (Figure 1A). Plant-pollinator networks are typically asymmetrical in form (Box 1). Specialist pollinators tend to forage on a limited number of plant taxa (e.g., within a single family) that represent a subset of the plants visited by generalist species leading to a nested structure of the community network [START_REF] Nielsen | Ecological networks, nestedness, and sampling effort[END_REF][START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF]. Plant-pollinator networks can also consist of weakly interlinked subsets of species modules that comprise sets of strongly interacting species [START_REF] Olesen | The modularity of pollination networks[END_REF]. This modularity can arise from convergent trait sets and mutualist coevolution, caused for example by inter-and intraspecific competition among generalist pollinators, driving flexible foraging at the individual level and concentrating visitation on a subset of the available floral species [START_REF] Spiesman | Flexible foraging shapes the topology of plant-pollinator interaction networks[END_REF].

Stochastic or environmentally driven variation in this fundamental process of community assembly (Figure 1A,C,D) results in different network structures (Box 1) that provide variable transmission routes for pathogen propagation through the community (Figure 1B). Highly connected networks with a large diversity or number of interactions may elevate interspecific contact and potential pathogen transmission [START_REF] Strona | The intrinsic vulnerability of networks to epidemics[END_REF], or decrease pathogen prevalence due to a dilution effect [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF], the latter of which may explain lower disease prevalence in diverse pollinator communities [START_REF] Fearon | Pollinator community species richness dilutes prevalence of multiple viruses within multiple host species[END_REF]. Highly nested networks may result in increased pathogen persistence [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF], with generalist species serving as hubs of pathogen spread to specialist species. Increased modularity should buffer against risk of flower-mediated pathogen spillover and spread across entire communities, by compartmentalising pathogens and disease outbreaks within modules [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF][START_REF] Gilarranz | Effects of network modularity on the spread of perturbation impact in experimental metapopulations[END_REF]. As a corollary, pollinator species within a module may be at increased risk of pathogen spillover from strongly interacting species (e.g., honey bees and bumble bees) with high pathogen loads [START_REF] Gilarranz | Effects of network modularity on the spread of perturbation impact in experimental metapopulations[END_REF].

Plant-pollinator-pathogen network structure will not be static, however, but dynamically shift with intra-and inter-annual temporal shifts in community composition and interactions [START_REF] Miele | Core-periphery dynamics in a plantpollinator network[END_REF] (Box 2). Foraging plasticity of many pollinator species can rewire interaction networks [START_REF] Trøjelsgaard | Geographical variation in mutualistic networks: similarity, turnover, and partner fidelity[END_REF][START_REF] Caradonna | Interaction rewiring and the rapid turnover of plant-pollinator networks[END_REF] that may affect host-pathogen exchange and dynamics [START_REF] Graystock | Dominant bee species and floral abundance drive parasite temporal dynamics in plantpollinator communities[END_REF][START_REF] Chacoff | Interaction frequency, network position, and the temporal persistence of interactions in a plantpollinator network[END_REF]. Even in modular networks, generalist individuals may switch floral fidelity during their lifespan potentially enhancing pathogen spread between network modules [START_REF] Ellner | Individual specialization and multihost epidemics: disease spread in plant-pollinator networks[END_REF]. Longer-term species turnover and species population dynamics can also rewire interaction networks through species extinctions, range extensions, and uptake of vacant niches [START_REF] Schleuning | Ecological networks are more sensitive to plant than to animal extinction under climate change[END_REF][START_REF] Ramos-Jiliberto | Topological plasticity increases robustness of mutualistic networks[END_REF]. While network rewiring lends stability to plant-pollinator networks [START_REF] Ramos-Jiliberto | Topological plasticity increases robustness of mutualistic networks[END_REF], it offers a mechanism by which novel interactions may lead to pathogen shifts and microevolution. The relative contribution and interrelationships of different network properties [START_REF] Fortuna | Nestedness versus modularity in ecological networks: two sides of the same coin?[END_REF], and the precise effect on host-pathogen dynamics, remain key questions in pollinator epidemiology (Box 2). Recent advances in predicting the structure and dynamics of plant-pollinator interactions by considering trait-matching of species [START_REF] Valdovinos | Mutualistic networks: moving closer to a predictive theory[END_REF][START_REF] Peralta | Trait matching and phenological overlap increase the spatio-temporal stability and functionality of plant-pollinator interactions[END_REF][START_REF] Pichler | Machine learning algorithms to infer trait-matching and predict species interactions in ecological networks[END_REF] promise a route towards a more mechanistic understanding of pathogen transmission in plant-pollinator networks [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF] (Box 2).

Anthropogenic Environmental Change

Aside from their separate impacts on pollinators and flora [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF][START_REF] Vanbergen | and the Insect Pollinators Initiative (2013) Threats to an ecosystem service: pressures on pollinators[END_REF], anthropogenic environmental pressures are likely to alter plant-pollinator-pathogen dynamics [START_REF] Redhead | Potential landscape-scale pollinator networks across Great Britain: structure, stability, and influence of agricultural land cover[END_REF], directly or via their synergistic or antagonistic effects (Figure 2). These global change pressures and their interlinked effects are ultimately caused by common drivers (e.g., human population growth, resource consumption) making it very difficult to disentangle and rank their relative importance [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF]. Impacts of pressures may occur through apparent competitive and facilitative interactions [START_REF] Schleuning | Ecological networks are more sensitive to plant than to animal extinction under climate change[END_REF][START_REF] Magrach | Honeybee spillover reshuffles pollinator diets and affects plant reproductive success[END_REF][START_REF] Simmons | Motifs in bipartite ecological networks: uncovering indirect interactions[END_REF] driven by altered abundances, diversity, or multispecies interactions in plant-pollinator-pathogen networks. We

Box 1. Key Network Metrics affecting Pathogen Transmission

Plant-pollinator interactions can be displayed as bipartite networks, in which species are nodes and interactions connecting lines. Connectance, nestedness, and modularity (Figure I) are arguably the most important metrics predicting pathogen transmission throughout a network [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF]. While often used for unweighted (binary) networks, indices exist for these metrics that can characterise weighted networks (i.e., where the interactions are weighted by their frequency), which is the most common and most accurate way to display plant-pollinator networks.

Connectance is the proportion of realised ecological interactions to potential ecological interactions. Hence, a plant-pollinator network containing five pollinator species and ten plant species, with ten realised interactions, has a connectance of 0.20, since 50 interactions are theoretically possible.

Modularity refers to the division of a network in separate modules, which have a high within-module connectance, but few interactions between modules. As for nestedness, multiple indices exist to express the level of modularity. Newman's modularity (Q), for example, is calculated as the fraction of interactions within each module minus the fraction of expected interactions in a random network with the same connectance [115].

Nestedness in bipartite networks is described as the tendency for specialist species to interact with generalist species, resulting in specialist species' interactions being a subset of generalist species' interactions [116]. There are several nestedness indices, of which nestedness metric based on overlap and decreasing fill (NODF) is considered the most robust [117].

expect that pathogen spread within a pollinator population or interspecific spillover may be reduced or exacerbated by: (i) invasions by novel species or subspecies [START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF] (Figure 1C); (ii) the extirpation of pre-existing populations or species [START_REF] Gómez-Ruiz | Climate change, range shifts, and the disruption of a pollinator-plant complex[END_REF] (Figure 1C); (iii) modification of the relative abundance of species and dominant interactions [START_REF] Morrison | Distinct responses of antagonistic and mutualistic networks to agricultural intensification[END_REF] (Figure 1D); (iv) disruption of species' phenology [START_REF] Gérard | Global warming and plant-pollinator mismatches[END_REF][START_REF] Fisogni | Urbanization drives an early spring for plants but not for pollinators[END_REF] (Figure 1D); (v) shifting species distributions at landscape, regional, or global levels [START_REF] Gómez-Ruiz | Climate change, range shifts, and the disruption of a pollinator-plant complex[END_REF]; or (vi) changing species physiology, altering disease resistance, infectivity, or virulence [START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF][START_REF] Laughton | Responses to a warming world: integrating life history, immune investment, and pathogen resistance in a model insect species[END_REF] (Figure 1E). While scientific and policy awareness of pressures on wild pollinators has increased and led to regulations at various levels protecting pollinators and their habitats, less regulatory of 12 pollinator species (yellow nodes) and 14 plant species (green nodes). Possible routes of disease spread, starting at pollinator species 1 are displayed by indicating path length between nodes (1, dark red; 2, light red; 3, orange; 4, yellow; 5, dark blue; 6, light blue; 7, light green; 8, green; 9, dark green). High connectance and nestedness may increase disease spread, while high modularity may keep disease spread limited to the affected module. In the network with high modularity, the three modules are indicated by horizontal lines. In the highly nested network, specialist pollinators visit a subset of plants visited by generalists. Abbreviations: NODF, nestedness metric based on overlap and decreasing fill; Q, Newman's modularity.

Trends in Ecology & Evolution

OPEN ACCESS attention has been devoted to wild pollinator diseases and their interplay with managed bees and environmental change (Table S1 in the supplemental information online).

Climate Change

Climate change has the potential to cause changes in pollinator disease risk. Elevated temperatures may increase pollinator resistance or susceptibility to pathogens according to physiological constraints (e.g., thermal stress) and changes in immune response [START_REF] Laughton | Responses to a warming world: integrating life history, immune investment, and pathogen resistance in a model insect species[END_REF][START_REF] Ferguson | Paradoxical acclimation responses in the thermal performance of insect immunity[END_REF][START_REF] Zaragoza-Trello | Interactions among global change pressures act in a non-additive way on bumblebee individuals and colonies[END_REF]. Local climatic changes may increase niche suitability for pathogens, facilitating range extensions and enhancing virulence or conversely, lower virulence and abundance of pathogens or their vectors [START_REF] Natsopoulou | Interspecific competition in honeybee intracellular gut parasites is asymmetric and favours the spread of an emerging infectious disease[END_REF][START_REF] Giacobino | Varroa destructor and viruses association in honey bee colonies under different climatic conditions[END_REF] (Figure 1E). Similarly, as with human diseases [START_REF] Ogden | Climate change and vector-borne diseases of public health significance[END_REF], pathogen vectoring by range expanding Box 2. Mechanistic Process-Oriented Modelling of Wild Pollinator Disease Empirically detecting disease events in wild pollinator-plant-pathogen systems is challenging because of the tangled network of direct and indirect interactions between pathogens, plants, and pollinators. Agent-based spatially explicit models may address this complexity [118]. Mechanistic models have previously been used to upscale from individual life cycle, resource use, and behaviours, to population level impacts of stressors and diseases in managed bee colonies [119,120].

Forecasting the course and impact of disease in multispecies pollinator systems is another order of complexity [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF]. Recent use of the susceptible-infectious-susceptible (SIS) epidemiological cohort framework illustrated how disease persistence was promoted by network nestedness and specialisation, while pollinator foraging selectivity, specialised mutualist pairs, and generalist flowers increased bee contact and disease spread [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF]. Whilst addressing the key assumption of disease modelling of individuals mixing uniformly and randomly, this study overlooked how contacts can emerge from individual behaviour and life cycle or other trait variation among species. Integrating interaction networks [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF]116] with dynamic pathogen infestation states creates temporally variable contact structures [121]. This may elucidate whether pollinator traits (e.g., diet breadth, feeding rates) or environmental heterogeneity (e.g., resource accessibility, clustering) limit or facilitate pathogen persistence or transmission [START_REF] Ellner | Individual specialization and multihost epidemics: disease spread in plant-pollinator networks[END_REF]. Equally, they may identify methodological caveats to empirical detection of pathogens (e.g., sampling effort or observer's collection window), which can improve monitoring protocols and target experiments. pollinators can introduce novel infections to native pollinator communities. Climate-induced plant extinctions may also affect flower-mediated pathogen transmission by shifting community structure to be dominated by plant species with different potential to act as hubs of pathogen transmission [START_REF] Adler | Flowering plant composition shapes pathogen infection intensity and reproduction in bumble bee colonies[END_REF] (Figure 1D). Additionally, pollinator forage plants may be extirpated or produce less flowers or lower-quality floral rewards under changed climatic circumstances [START_REF] Walter | Dryness, wetness, and temporary flooding reduce floral resources of plant communities with adverse consequences for pollinator attraction[END_REF], causing nutritional stress and higher susceptibility of hosts [START_REF] Pike | On the diverse and opposing effects of nutrition on pathogen virulence[END_REF].

Invasive species

Agric. intensificaƟon

UrbanisaƟon

Climate change

Managed pollinators [START_REF] Natsopoulou | Interspecific competition in honeybee intracellular gut parasites is asymmetric and favours the spread of an emerging infectious disease[END_REF][START_REF] Giacobino | Varroa destructor and viruses association in honey bee colonies under different climatic conditions[END_REF] [65]

[ [START_REF] Cornelissen | Global warming promotes biological invasion of a honey bee pest[END_REF][START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF] [ [START_REF] Laurino | Vespa velutina: an alien driver of honey bee colony losses[END_REF][START_REF] Graystock | Do managed bees drive parasite spread and emergence in wild bees?[END_REF] [80]

[110]

[106] Phenological or spatial mismatch of species interactions [START_REF] Gérard | Global warming and plant-pollinator mismatches[END_REF][START_REF] Schleuning | Trait-based assessments of climate-change impacts on interacting species[END_REF] due to life-history differences in tracking climate changes, or climate-induced species extirpations, or range shifts, may alter host activity and species relations [START_REF] Gómez-Ruiz | Climate change, range shifts, and the disruption of a pollinator-plant complex[END_REF][START_REF] Schweiger | Climate change can cause spatial mismatch of trophically interacting species[END_REF]. Such mismatches, extinctions, and subsequent rewiring of networks [START_REF] Schleuning | Ecological networks are more sensitive to plant than to animal extinction under climate change[END_REF][START_REF] Schleuning | Trait-based assessments of climate-change impacts on interacting species[END_REF], may change the prevalence of pathogens or parasites in pollinator communities and pathways to infection [START_REF] Farzan | Experimental shifts in phenology affect fitness, foraging, and parasitism in a native solitary bee[END_REF] (Figure 1D).

Non-random processes such as trait matching will modulate the impact of climate change on plant-pollinator-pathogen communities [START_REF] Schleuning | Trait-based assessments of climate-change impacts on interacting species[END_REF]. Overall, rarer species with narrow climate niches, high specialisation, low dispersal abilities, and restricted geographic distribution are likely to suffer most. Climate impacts may thus inflate network-level generality of interactions [START_REF] Schleuning | Ecological networks are more sensitive to plant than to animal extinction under climate change[END_REF], possibly increasing pathogen spread, although this may be mitigated should specialist interactions be sufficiently robust to disturbance [START_REF] Peralta | Trait matching and phenological overlap increase the spatio-temporal stability and functionality of plant-pollinator interactions[END_REF]. Climate change has been shown to increase nestedness and decrease modularity of plant-pollinator networks [START_REF] Takemoto | Human impacts and climate change influence nestedness and modularity in foodweb and mutualistic networks[END_REF], indicating potential for increased pathogen prevalence [START_REF] Truitt | Trait-based modeling of multi-host pathogen transmission: plant-pollinator networks[END_REF] (Box 1).

Introduced Species

Invasive Alien Species Anthropogenic translocations of species often lead to alien species invasions, which have had severe consequences for plant-pollinator populations and communities [START_REF] Vanbergen | and the Insect Pollinators Initiative (2013) Threats to an ecosystem service: pressures on pollinators[END_REF][START_REF] Vanbergen | Risks to pollinators and pollination from invasive alien species[END_REF][START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF]. Species invasions and the novel contact opportunities they create present risks of pathogen host shifts, coinfections, and the evolution of virulence with detrimental effects on species lacking immunity or resistance [START_REF] Arbetman | Global decline of bumblebees is phylogenetically structured and inversely related to species range size and pathogen incidence[END_REF]. Alien pollinators can directly compete with native species via interference and resource competition, or impact them indirectly by introducing novel pathogens [START_REF] Aizen | Invasive mutualists erode native pollination webs[END_REF] or enhancing pathogen transmission [START_REF] Morales | Rapid ecological replacement of a native bumble bee by invasive species[END_REF]. Potentially invasive pollinators (e.g., Bombus terrestris), often introduced deliberately or accidentally via the bee trade into regions outside of their native distribution range, are usually highly competitive generalist foragers with an extensive flight period that profoundly disrupt recipient communities [START_REF] Aizen | Invasive mutualists erode native pollination webs[END_REF][START_REF] Valdovinos | Species traits and network structure predict the success and impacts of pollinator invasions[END_REF]. Dominant invasive species may push native species to shift onto and compete for suboptimal and less abundant forage plants, potentially elevating nutritional stress and pathogen susceptibility [START_REF] Magrach | Honeybee spillover reshuffles pollinator diets and affects plant reproductive success[END_REF].

Invasive plant species often provide copious floral resources dominating pollinator diets and rearranging interactions to disrupt the modular structure and increase the generalism and nestedness of the plant-pollinator network [START_REF] Hernández-Castellano | A new native plant in the neighborhood: effects on plant-pollinator networks, pollination, and plant reproductive success[END_REF][START_REF] Albrecht | Consequences of plant invasions on compartmentalization and species' roles in plant-pollinator networks[END_REF]. Consequently, invasive plant flowers can serve as potential hubs for pathogen transfer by concentrating pollinator activity and diversity [START_REF] Majewska | Urbanization predicts infection risk by a protozoan parasite in non-migratory populations of monarch butterflies from the southern coastal U.S. and Hawaii[END_REF] and reducing network modularity to facilitate pathogen spread throughout the community. Similarly, non-insect pollinated invasive plant species can concentrate pollinator activity and contact by suppressing native vegetation and limiting forage plant availability [START_REF] Hachuy-Filho | Changes in plant community structure and decrease in floral resource availability lead to a high temporal β-diversity of plant-bee interactions[END_REF]. Invasive plants may also increase pollinator susceptibility to pathogens by supplying copious yet suboptimal pollen or nectar lacking essential nutrients or containing secondary compounds toxic to certain pollinator species [START_REF] Tiedeken | Nectar chemistry modulates the impact of an invasive plant on native pollinators[END_REF]. In the longer term, invasive plants can lead to differential reproduction of native plant species due to facilitation or competition for insect pollination [START_REF] Albrecht | Pollinator-mediated impacts of alien invasive plants on the pollination of native plants: the role of spatial scale and distinct behaviour among pollinator guilds[END_REF], potentially reconfiguring plant-pollinator interactions to produce a novel epidemiology (Figure 1D,E).

Invasions by other functional groups may also profoundly affect pollinator-pathogen dynamics.

Invasive non-pollinating insects may affect pathogen dynamics by directly vectoring pathogens to shared plant species [START_REF] Dobelmann | Genetic strain diversity of multi-host RNA viruses that infect a wide range of pollinators and associates is shaped by geographic origins[END_REF], exerting novel predation pressure upon pollinator populations [START_REF] Laurino | Vespa velutina: an alien driver of honey bee colony losses[END_REF] or reorganising the plant community through novel herbivores [START_REF] Glaum | Functional reduction in pollination through herbivore-induced pollinator limitation and its potential in mutualist communities[END_REF]. Such changes to food webs may affect pathogen circulation and interspecific processes (facilitation, competition) governing disease dynamics.

Managed Bee Species

Within their original and translocated ranges, honey bees and some bumble bees, stingless bees, and solitary bees, are managed for hive products and/or crop pollination services, with A. mellifera the most widespread and abundant managed species [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF]. Apart from the risk of emerging disease in previously unaffected regions due to transport of managed bees (particularly A. mellifera, but also e.g., Bombus terrestris and Bombus impatiens) over large distances [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF][START_REF] Vanbergen | Risks to pollinators and pollination from invasive alien species[END_REF][START_REF] Aizen | Invasive mutualists erode native pollination webs[END_REF][START_REF] Graystock | Do managed bees drive parasite spread and emergence in wild bees?[END_REF], bee management presents additional epidemiological risks. Decreases in managed A. mellifera genetic diversity, caused by selective breeding [START_REF] Espregueira Themudo | Declining genetic diversity of European honeybees along the twentieth century[END_REF], may weaken A. mellifera immune responses [START_REF] Youngsteadt | Urbanization increases pathogen pressure on feral and managed honey bees[END_REF] increasing population pathogen levels. Selective pollinator breeding could increase pathogen resistance or tolerance [START_REF] Guichard | Advances and perspectives in selecting resistance traits against the parasitic mite Varroa destructor in honey bees[END_REF], the latter potentially exacerbating pathogen spillover. Managed A. mellifera are kept at higher colony densities than occur naturally, which together with their supergeneralism positions them as a central hub or a 'superspreader' reservoir host in plant-pollinator networks (Figure 1F-H) [START_REF] Stein | Super-spreaders in infectious diseases[END_REF].

One consequence of the global translocation of A. mellifera, for pollination services and hive products was parasite and pathogen transfer (e.g., V. destructor, Nosema ceranae) from the original Asian host species, Apis cerana [START_REF] Vanbergen | Risks to pollinators and pollination from invasive alien species[END_REF][START_REF] Higes | How natural infection by Nosema ceranae causes honeybee colony collapse[END_REF]. Parasitism of A. mellifera by V. destructor exacerbates transmission and increases prevalence of several RNA viruses including DWV infecting the honey bee [START_REF] Mcmahon | A sting in the spit: widespread cross-infection of multiple RNA viruses across wild and managed bees[END_REF][START_REF] Wilfert | Deformed wing virus is a recent global epidemic in honeybees driven by Varroa mites[END_REF] (Figure 1G), hence potentially spreading pathogens to these susceptible alternative hosts during foraging [START_REF] Pike | On the diverse and opposing effects of nutrition on pathogen virulence[END_REF][START_REF] Manley | Knock-on community impacts of a novel vector: spillover of emerging DWV-B from Varroainfested honeybees to wild bumblebees[END_REF][START_REF] Purkiss | Pathogen spillover from Apis mellifera to a stingless bee[END_REF] (Figure 1H), while driving nutritional stress in wild pollinators by outcompeting them for scarce food. Weakening of the host's immune system by combined parasite and pathogen infections may induce rapid microevolutionary changes that increase virulence [START_REF] Mcmahon | A sting in the spit: widespread cross-infection of multiple RNA viruses across wild and managed bees[END_REF][START_REF] Manley | Knock-on community impacts of a novel vector: spillover of emerging DWV-B from Varroainfested honeybees to wild bumblebees[END_REF]. Moreover, this may be a dynamic process with multiple spillover and spillback events iteratively increasing the risk of pathogen exchanges and development of novel community epidemiology (Figure 1G,H). Such host shifts highlight the risks of species invasions for creating novel species interactions and epidemiology for populations and communities [START_REF] Vanbergen | Risks to pollinators and pollination from invasive alien species[END_REF].

Conventional Agricultural Intensification

Conventional intensive agriculture is a globally important pressure on pollinators and pollination [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF][START_REF] Vanbergen | and the Insect Pollinators Initiative (2013) Threats to an ecosystem service: pressures on pollinators[END_REF], that may modulate pathogen transmission or pollinator susceptibility. Generalist pollinators able to adapt to the cropped environment are usually less affected than more specialised species by landscape simplification and management intensification [START_REF] Ekroos | High land-use intensity in grasslands constrains wild bee species richness in Europe[END_REF]. Use of herbicides, fertiliser, and destruction of seminatural elements reduce richness and abundance of floral resources providing nectar and pollen [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF], which can lead to network generalism [100]. Consequently, plant-pollinator networks in intensively farmed areas tend to be generalised with low modularity and nestedness and high connectance [START_REF] Redhead | Potential landscape-scale pollinator networks across Great Britain: structure, stability, and influence of agricultural land cover[END_REF][START_REF] Morrison | Distinct responses of antagonistic and mutualistic networks to agricultural intensification[END_REF]. Such a network architecture can either limit or propagate community-wide pathogen transmission (Box 1).

Intensively managed agricultural landscapes often hold transient or low pollinator population densities [START_REF] Figueroa | Landscape simplification shapes pathogen prevalence in plant-pollinator networks[END_REF] that may reduce pathogen spread [101]. Conversely, the risk of pathogen exposure could increase if pollinator interactions become concentrated on the few remaining natural floral resource patches in intensive agricultural landscapes [START_REF] Potts | Safeguarding pollinators and their values to human well-being[END_REF][START_REF] Hachuy-Filho | Changes in plant community structure and decrease in floral resource availability lead to a high temporal β-diversity of plant-bee interactions[END_REF] (Figure 1D). Mass-flowering crops provide huge, transient pulses of pollen and nectar food resources that attract and provide nutrition supporting the fitness of large numbers of diverse pollinators [START_REF] Vanbergen | and the Insect Pollinators Initiative (2013) Threats to an ecosystem service: pressures on pollinators[END_REF]102]. Domination of pollinator interactions by mass-flowering crops could create infection hubs by concentrating the pollinator community, reducing pathogen transmission by spreading pollinator activity over a large continuous area of floral resource [START_REF] Graystock | Dominant bee species and floral abundance drive parasite temporal dynamics in plantpollinator communities[END_REF] (Figure 1D).

Mass flowering crops present additional potential risks of pesticide exposure that may also weaken pollinator immune responses, leading to increased pathogen load and virulence [START_REF] Fine | An inert pesticide adjuvant synergizes viral pathogenicity and mortality in honey bee larvae[END_REF]. Recently, the fungicide chlorothalonil has been associated with high parasite loads in bumble bees and resulting

Urbanisation

Urban environments are increasing in extent worldwide. They have higher ambient temperature and levels of pollution (e.g., pesticides, heavy metals, and airborne particulates), large areas of impervious surfaces, and few semi-natural areas, but provide pollinator habitats in anthropic green spaces and gardens supporting a richness of native and alien plant species [107]. Multiple factors may affect pollinator-pathogen dynamics in urban areas where high pathogen loads have been detected in honey bees and bumble bees [START_REF] Youngsteadt | Urbanization increases pathogen pressure on feral and managed honey bees[END_REF]108]. Similar to climatic influences at a continental or global scale, higher urban temperatures may drive the evolution of virulence in pathogens [START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF] or vector activity [START_REF] Ogden | Climate change and vector-borne diseases of public health significance[END_REF], creating local patterns of pathogen prevalence (Figure 1E). Urban heating may also drive phenological mismatch between plants and bees, due to differential adaptation to the novel microclimate [START_REF] Fisogni | Urbanization drives an early spring for plants but not for pollinators[END_REF]. Urban heat islands coupled to a proliferation of non-native horticultural and weed plant species, possessing intrinsic differences in flowering phenology, can extend the pollinator foraging season in urban areas [START_REF] Schweiger | Multiple stressors on biotic interactions: how climate change and alien species interact to affect pollination[END_REF]. This lengthened foraging season has potential impacts on pathogen transmission risk especially as levels of V. destructor and associated viruses in honey bees tend to peak later in season, exposing late-season active species to higher levels of pathogens [109]. Patchily distributed floral resources in urban landscapes increase pollinator interspecific contacts at shared flowers, which together with growing popularity of urban beekeeping (Figure 1G,H) may elevate pathogen transmission and co-infections [START_REF] Majewska | Urbanization predicts infection risk by a protozoan parasite in non-migratory populations of monarch butterflies from the southern coastal U.S. and Hawaii[END_REF]108,110].

Overall, urbanisation favours generalist species [111] and very high levels of urbanisation reduce pollinator diversity [112] (Figure 1E). Urban environments thus generally produce a network containing fewer, generalised pollinator species, with high connectance and low evenness and robustness [113,114]. This network architecture may promote pathogen transmission and infections in concert with other pressures of urban settings.

Concluding Remarks

Flower-mediated pathogen transmission is driven by a complex interplay between species traits, plant-pollinator network architecture, and anthropogenic drivers, often interacting in synergistic or antagonistic ways. While recent studies have shed light on several individual facets, many questions remain, especially on the complex interactions between the various aspects modulating disease transmission and on the exact mechanisms that drive host shifts (see Outstanding Questions). As anthropogenic environmental change continues and intensifies the risks of novel plant-pollinator-pathogen interactions and epidemics are likely to rise. Given the complex and highly context dependent outcomes, future research needs to cover a wide range of landscapes, climates, and local circumstances, to derive a robust overview on how pathogen dynamics will respond to anthropogenic pressures at different spatiotemporal scales. The interplay of global change pressures may reorganise pollinator assemblages and species interactions in ways that elevate the risks of emergent pathogens and population, or community epidemics, an outcome that can undermine policies (Table S1 in the supplemental information online) to safeguard pollinators and pollination services.

Figure 1 .

 1 Figure 1. Potential Impacts of Anthropogenic Environmental Pressures and Bee Management on Flower-Mediated Pathogen Transmission between Pollinator Species. Plant-pollinator community interactions are

Figure I .

 I Figure I. Example Unweighted (for Simplicity) Networks Created by Randomly Compiling Plant-Pollinator Networks at a Predefined Connectance (C = 0.15 in Network with Low Connectance; C = 0.40 in Network with High Connectance; C ≈ 0.20 in Other Networks). Networks are composed

  Mechanistic process-oriented modelling of wild pollinator disease in support of planning experiments and data exploration requires knowledge of emergent trait-based life cycles and foraging activities. Sampling for construction of tripartite networks will provide individual (plant-pollinator-pathogen) data (occurrence, abundance, and contact), which can be translated to individual-oriented model characteristics for experimentation. Potential components for modelling of pollinator disease tailored to experimentation/observations with literature examples: Pollinator Population Dynamics [13,119,120] -Static populations -Population-level reproduction rate -Explicit population turnover by individual nesting and reproduction Landscape Heterogeneity, Dynamic Landscapes, Explicit Forage Environment [20,122-124] -Homogeneous landscapes -Spatial habitat distribution using, for example, species distribution modelling (SDM) -Landscape maps by fitness indices using resource needs and reachability -Standardised and systematic landscape generation algorithms Foraging and Visitation Behaviour, Pollen Collection [20,125,126] -Diffusive foraging and kernel models -Explicit traits and behavioural trade-offs -Temporally varying individual-level preferences of food or habitat -Emergent foraging behavioural responses by exploring spatial explicit structures Virtual Insect Sampling: Simulated Data and Models Mimicking Real Species in Landscapes -Model sampling methods as empirically tested -Resample transect path in real landscapes

Figure 2 .

 2 Figure 2. Interactive Effects between Environmental Pressures. Green arrows indicate synergistic effects, orange arrows complex, trait, and context-dependent effects. Arrow points in the direction of the effect. References mentioning the respective interactions are displayed in square brackets [16,17,65,69,70,80,85,89,91,106,110]. Concurring environmental drivers may reinforce each other, as is the case with climate change exacerbating the effects of urban heating, urbanisation leading to a higher pressure of invasive species, and higher density of managed pollinators, or intensive agriculture increasing transport and demand for managed pollinators. Conversely, most interactions between global change drivers have contextdependent effects on pathogen transmission, as is the case with climate change, which has far-reaching effects on agricultural practices and modulates potential for invasive species depending on species traits, or invasive plants, pollinators, and pathogens interacting in complex ways with managed pollinator behaviour, physiology, and epidemiology. Photographs by Dan Chapman, ©NERC, (WT-EN) Bulliver CC-BY-SA 1.0, Piscisgate CC BY-SA 4.0, and Arnold Paul CC BY-SA 3.0.

  declines and range contractions[103]. Additionally, the paucity of floral resources in intensive agricultural landscapes may mean pollinators are unable to optimise their nutrient intake [104] and suffer nutrient deficits that compromise immune function and pathogen resistance[105]. Future changes in agricultural practices driven by climate change [106] may significantly reshape the agricultural landscape, possibly with large impacts on plant-pollinator-pathogen dynamics.

Connectance Trends Trends in in Ecology Ecology & Evolution Evolution

						Low													High
	1	2	3	4	5	6	7	8	9		10	11	12		1	2	3	4	5	6	7	8	9		10	11	12
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Connectance = 0.17											Connectance = 0.39
	NODF = 18.3												NODF = 43.7
	Q = 0.57													Q = 0.29
	1	2	3	4	5	6	7	8	9		10	11	12		1	2	3	4	5	6	7	8	9		10	11	12
	Modularity
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Connectance = 0.26												Connectance = 0.20
	NODF = 30.7													NODF = 14.3
	Q = 0.36													Q = 0.60
	1	2	3	4	5	6	7	8	9	10	11	12		1	2	3	4	5	6	7	8	9	10	11	12
	Nestedness
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	1	2	3	4	5	6	7	8	9	10	11	12	13	14
	Connectance = 0.21											Connectance = 0.23
	NODF = 8.7													NODF = 52.3
	Q = 0.49													Q = 0.40

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Acknowledgements

We acknowledge the Biodiversa project VOODOO (Viral eco-evolutionary dynamics of wild and domestic pollinators under global change www.voodoo-project.eu) and its funders (France: ANR-19-EBI3-0006; Poland: NCN UMO-2019/32/Z/NZ8/00006;

• What is the role of dominant species (e.g., honey bees) in driving pathogen community dynamics?

• How is pathogen fitness affected by novel host and abiotic environments? Does evolutionary change precede or follow from a host shift and is host switching accompanied by a loss of fitness in the original reservoir host?

• Do pollinator hosts evolve increased tolerance or resistance to pathogen infection?

Network Ecology: Transmission

• How strongly does pollinator network structure predict pathogen transmission within and between hosts?

• How dynamic is the flower-pollinatorpathogen network within and between years?

Declaration of Interests

No interests are declared.

Supplemental Information

Supplemental information associated with this article can be found online https://doi.org/10.1016/j.tree.2021.03.006. • How do agri-environment measures (e.g., wildflower strips) affect pollinator health and disease?

Trends in Ecology & Evolution

Pollinators and Global Change

• How do beekeeping practices affect pathogen spillover between managed and wild pollinators, and how can it be reduced?

• How does the degree of urbanisation (town to city) affect the structure of plant-pollinator-pathogen networks?

• Can we experimentally determine how invasive plant species shift the community structure and pollinator behaviour to drive declines in co-flowering native species and changes in pathogen and parasite loads?

• How do synergistic and antagonistic effects of global change drivers affect pathogen dynamics in different pollinator hosts?

• What are the effects of policy actions (agricultural, climate, biodiversity conservation, and pollinator conservation) on pathogen dynamics?


Denis Nikitin

Carlos Canudas-De-Wit

Paolo Frasca

Boundary Control for Stabilization of Large-Scale Networks through the Continuation Method

In this work we study a continuation method which transforms spatially distributed ODE systems into PDEs that respect the spatial structure of the original ODE systems. Such PDE description can be used not only for analysis but also for a continuous control design which, being discretized back, results in a nontrivial control law for the original ODE system. In this paper we focus on the continuation for linear systems, including multidimensional inhomogeneous systems and in particular linear networks, showing that such systems can be transformed into general second-order parabolic PDEs. The method is applied to the stabilization of a chain of coupled semiconductor lasers. We obtain a PDE model of this system, design a backstepping-based boundary control to stabilize the obtained PDE and then translate the control policy back to the original laser chain, effectively stabilizing it.

I. INTRODUCTION

Large-scale networks are often used to describe physical systems, such as urban traffic [START_REF] Bianchin | Gramian-based optimization for the analysis and control of traffic networks[END_REF], brain activity [START_REF] Van Veen | Localization of brain electrical activity via linearly constrained minimum variance spatial filtering[END_REF], power networks [START_REF] Dorfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF], robotics formations [START_REF] Olfati-Saber | Flocking for multi-agent dynamic systems: Algorithms and theory[END_REF] or epidemic spreading. Entities in these systems have a predefined position in the real-world space, representing individual nodes in the network.

Due to increasing complexity of such systems, methods of simplification of the model are of a great importance. Mostly, the methods of network analysis "forget" information about positions, relying only on interaction topology. Among examples of methods of this type, there is model reduction, in particular clustering of the original network depending on the topology [START_REF] Aoki | Control of large-scale dynamic systems by aggregation[END_REF], [START_REF] Niazi | Structure-based clustering algorithm for model reduction of largescale network systems[END_REF] or reduction towards the average state [START_REF] Niazi | Scale-free estimation of the average state in large-scale systems[END_REF], [START_REF] Nikitin | Control of average and deviation in large-scale linear networks[END_REF].

Some of methods of network analysis represent the original system using continuous models. When every agent in the network is coupled to all others, the method called population density approach [START_REF] Nykamp | A population density approach that facilitates large-scale modeling of neural networks: Analysis and an application to orientation tuning[END_REF], [START_REF] Grabert | Projection operator techniques in nonequilibrium statistical mechanics[END_REF] is used to track the whole probability distribution over all agents' states in the network. This method is mostly used to model large biological neural networks. More sophisticated topologies can be dealt with a recently emerged theory of graphons which studies graph limits, i.e. structural properties that the graph possesses if the number of nodes tends to infinity while preserving interaction topology, see [START_REF] Glasscock | What is... a graphon[END_REF]. Using graphons it is possible to describe any dense graph as a linear operator in continuum space [START_REF] Lovász | Large networks and graph limits[END_REF]. This method was further used to control large-scale linear networks [START_REF] Gao | Graphon control of large-scale networks of linear systems[END_REF] and to study sensitivity of epidemic networks [START_REF] Vizuete | Graphon-based sensitivity analysis of SIS epidemics[END_REF].

*This work was supported by the funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 694209) 1 Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-Lab, Grenoble, France Contrary to previous works, we recently developed a method [START_REF] Nikitin | A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip[END_REF] of continuous model transformation which can utilize the intrinsic information of agents' positions. Our idea is to replace the original spatially distributed ODE system by a continuous PDE whose state and space variables preserve the state and space variables of the original system. We name this method as a continuation, since it is exactly opposite to the discretization procedure.

In this paper we focus on the linear network analysis. It is widely known [START_REF] Frihauf | Leader-enabled deployment onto planar curves: A PDE-based approach[END_REF], [START_REF] Jafarizadeh | Weighted average consensus-based optimization of advection-diffusion systems[END_REF] that the Laplacian consensus networks are closely related to the diffusion PDEs. Performing continuation of a general linear spatially-distributed network, we show how the network dynamics can be written as a linear second-order PDE with space-dependent coefficients. In particular we show that several additional properties, such as absence of self-loops, regularity or undirected topology of the network, can simplify the resulting PDE.

The continuation method allows to recover a PDE which describes the same physical system as the original ODE network. The obtained model can be then used for analysis and control. It is possible to design a control for the continuous PDE model which, being discretized back, results in a control law for the original ODE system: the design framework is illustrated in Fig. 1. We demonstrate this continuation framework by designing an explicit control for a chain of coupled semiconductor lasers, suppressing undesirable oscillations.

One could argue: why do we want to use a PDE instead of ODEs, if PDEs are generally considered to be harder to analyze and to control? The answer is that a suitable use of PDEs can lead to explicit and scalable algorithms. Indeed, the centralized computation of feedback control gains for a large-scale linear system with N agents requires at least O(N) operations by methods such as ODE-based backstepping High-average-power laser-diode 41kW array, composed of 28 silicon monolithic microchannels (SiMMs) each consisting of thousands of diodes. Image from Lawrence Livermore National Laboratory, https://lasers.llnl.gov/science/photon-science/ highpowered-lasers/hapl, licensed under CC BY-NC-SA 4.0.

[18] and at least O(N 3 ) operations by methods like LQR, which require solving a Riccati matrix equation. On the contrary, in the next section we will give an example of such situation where the continuation helps to design a control with gains computed in O(1) operations. In particular, in case of unstable 1-dimensional PDE we can use the result of [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF], where the general second-order linear space-dependent system is stabilized to zero state using backstepping control, based on [START_REF]Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations[END_REF]. The case of two controlled boundaries of 1dimensional system is covered in [START_REF] Vazquez | Bilateral boundary control of onedimensional first-and second-order PDEs using infinite-dimensional backstepping[END_REF].

a) Outline: The rest of this paper is organized as follows. Section 2 contains our motivating example, the stabilization of a chain of laser diods: we describe the mathematical model, define the stabilization problem, and solve it by applying the continuation method and the backstepping method. The subsequent sections develop the continuation method. More precisely, Sections 3 and 4 contain a concise but self-contained account of the continuation theory that we developed in [START_REF] Nikitin | A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip[END_REF]: Section 3 regards systems without boundaries and Section 4 regards systems with boundary conditions. Then, Section 5 contains novel continuation results that are tailored to linear networks and that permit the continuation of laser chain model. Section 6 contains some concluding remarks.

II. SYNCHRONIZATION OF A LASER CHAIN

Our motivating example will be the problem of stabilization of a chain of coupled semiconductor lasers. Coupled laser systems are important for high-precision power transmission applications such as welding, laser surgery or fusion research as well as many others [START_REF] Saxena | Amplitude death: The emergence of stationarity in coupled nonlinear systems[END_REF]. Recently, large arrays of semiconductor laser diodes were shown to be more powerand cost-efficient compared to single crystal lasers due to lower electrical resistance and optical load [START_REF] Pietrzak | Heading to 1 kW levels with laser bars of high-efficiency and emission wavelength around 880 nm and 940 nm[END_REF]. A typical array of coupled lasers is depicted in Fig. 2.

In [START_REF] Carr | Negative-coupling resonances in pump-coupled lasers[END_REF] it was shown that coupling of several Class-B lasers [START_REF] Arecchi | Deterministic chaos in laser with injected signal[END_REF] can lead to a resonance effect, greatly increasing intensity comparing to the uncoupled laser system. However, such a system is prone to instabilities: electrical fields of lasers start to oscillate around the operating point, destroying resonance effect. It was further shown in [START_REF] Carr | Negative-coupling resonances in pump-coupled lasers[END_REF] that these oscillations, up to the first order, are described by coupled Stuart-Landau oscillators.

Stuart-Landau oscillators are prototypical models for Andronov-Hopf bifurcation, and apart from laser applications they are used to describe many oscillatory systems such as electronic oscillators [START_REF] Bergner | Remote synchronization in star networks[END_REF] or biological neural networks [START_REF] Aoyagi | Network of neural oscillators for retrieving phase information[END_REF]. Usually in laser analysis Stuart-Landau model describes electrical field of one laser, thus the oscillating behaviour is the desired one. However we base our analysis on [START_REF] Carr | Negative-coupling resonances in pump-coupled lasers[END_REF], where Stuart-Landau model is used to describe deviation from the synchronized steady state: thus, oscillations should be suppressed. Let the deviation of one laser be z ∈ C, then one Stuart-Landau oscillator is described by the evolution equation

ż = Γ + iΩ -η|z| 2 z, (1)
where Γ > 0 is an excitation gain, Ω is a natural frequency and η ∈ C is a nonlinear damping coefficient. For Γ < 0 the system has one stable equilibrium point z = 0, while for Γ > 0 zero equilibrium point is unstable, and system has a stable limit cycle with frequency Ω and with amplitude |z| = Γ/η. The authors of [START_REF] Saxena | Amplitude death: The emergence of stationarity in coupled nonlinear systems[END_REF] proposed to design laser hardware having in mind an effect called amplitude death to suppress laser electrical field's undesirable oscillations and thus prevent loss of efficiency. This effect appears when many inhomogeneous oscillators are strongly coupled, thus making their limit cycles unstable and the zero fixed point stable. Contrarily to this approach of hardware-designed amplitude death, we propose to use an active feedback stabilization from one boundary to suppress oscillations. We consider here a chain of N + 2 coupled Stuart-Landau oscillators. Let the position of i-th oscillator for i ∈ {0, .., N + 1} be x i = i∆x with ∆x = 1/(N +1) being distance between two neighbours, thus x 0 = 0 and x N+1 = 1. The state of i-th oscillator is z i ∈ C. We assume that the oscillators on the boundaries are directly controllable, namely the left boundary oscillator has fixed zero state z 0 = 0 and the state of the right boundary oscillator is a control variable z N+1 := u. We also assume that the coupling of lasers is realized by an overlapping of their evanescent fields [START_REF] Winful | Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers[END_REF], thus the evolution equation of i-th oscillator depends on the nearest neighbours' states with gains a i,i-1 and a i,i+1 . Since it is a conservative force, a i,i+1 = a i+1,i , thus the network is undirected. The system is given by żi

= (µ -η|z i | 2 )z i + a i,i-1 (z i-1 -z i ) + a i,i+1 (z i+1 -z i ), z 0 = 0, z N+1 = u, (2)
with µ = Γ + iΩ. In general, coupling a i,i+1 can be spacedependent. We assume that it is monotone, for example as in case of an increasing electrical permeability of the medium along the laser chain. Therefore, to approximate monotone dependencies we restrict ourselves to a class of coupling gains

a i,i+1 ≈ α(x i -β ) 2 with α > 0, β ∈ R \ [0, 1]. Note
that this class includes also homogeneous couplings in case β → ±∞ and α → 0 such that αβ 2 ≡ const.

A. Continuation process and boundary control design

Since Γ > 0, system (2) has unstable zero equilibrium. Our goal is to design a feedback control law z N+1 = u(z) such that zero solution is stabilized, thus suppressing oscillations. Linearizing system (2) around zero and assuming

|z i | is small, we get żi = µz i + a i,i-1 (z i-1 -z i ) + a i,i+1 (z i+1 -z i ), z 0 = 0, z N+1 = u. (3)
In case of thousands of coupled laser diodes, the implementation of traditional control algorithms for the system (3) would require a lot of computational power. Instead, we can perform continuation to the system (3), the process which will be explained in following sections. This process leads to the following PDE:

∂ z(x,t) ∂t = µz(x) + ∂ ∂ x α∆x 2 (x -β ) 2 ∂ z(x,t) ∂ x (4)
for x ∈ (0, 1) and with boundary conditions z(0,t) = 0 and z(1,t) = u.

Although system (4) is formulated in complex domain, one can use backstepping method from [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF] to stabilize it. Indeed, the stabilizing controller is given by

u := 1 0 k(x)z(x,t)dx, (5)
and the kernel is found by formula (44) from [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF]:

k(x) = - x (µ + c) α|β | (1 -β ) 3/2 (x -β ) 5/2 × × I 1 (µ + c)( ȳ2 -x2 )/(αβ 2 ) (µ + c)( ȳ2 -x2 )/(αβ 2 ) , (6)
where c > 0 is an adjustable gain, I 1 (s) is the modified Bessel function of order one, x = -β log(1x/β ) and ȳ = -β log(1 -1/β ).

B. Control discretization and numerical simulation

Finding a control law for the original ODE system (2) can be easily done by performing a numerical integration of (5) using the trapezoidal rule

u := ∆x N ∑ i=1 k(x i )z i , (7)
Each control gain k(x i ) can be computed directly in O(1) operations.

We validated the obtained control law by numerical simulation of system (2) with N = 30 coupled oscillators. We took Γ = 5 for excitation gain, Ω = 4 for natural frequency and ν = 10 for damping, thus a steady state magnitude of an uncoupled oscillator would be |z| = 1/ √ 2 ≈ 0.7071. Further, we took α = 5 and β = -10 as parameters for the coupling coefficients a i,i+1 . Due to the coupling, steady state magnitudes of the network (2) diminish, which can be seen on the graph in Fig. 3(a) depicting simulation of the uncontrolled system (2) with u = 0. However, the system still oscillates. The oscillations can be suppressed by applying control [START_REF] Niazi | Scale-free estimation of the average state in large-scale systems[END_REF] with kernel [START_REF] Niazi | Structure-based clustering algorithm for model reduction of largescale network systems[END_REF], where we took c = 10. Successful suppression is depicted in Fig. 3(b).

III. CONTINUATION METHOD FOR LINEAR SPACE-INDEPENDENT SYSTEMS

We start presenting the continuation method by considering the simplest class of systems for which the transformation of ODE into PDE can be performed, namely linear ODE systems corresponding to the dynamics of states of nodes, which are aligned on the 1D line in space and depend only on some fixed set of their neighbours. Let the node i have a state ρ i ∈ R and a geographical position x i ∈ R such that for every i the distance between two consecutive nodes in space is constant, x i+1x i = ∆x (the assumption of ∆x being constant will be relaxed later on). Then the systems of our interest take the form ρi

= N ∑ j=1 a j ρ i+s j , (8)
that is ρi linearly depends only on N neighbouring nodes shifted by s j ∈ Z for j ∈ {1..N}, and a j ∈ R are the system gains, see Fig. 4. This type of systems belongs to the class of linear spatially invariant systems [START_REF] Bamieh | Distributed control of spatially invariant systems[END_REF], which is a natural class for distributed control.

A. Discretization

The discretization of PDEs is usually performed by a finite difference method, where the partial derivatives are approximated by finite differences. For example, in the case of Transport ODE,

∂ ρ ∂ x ≈ 1 ∆x (ρ i+1 -ρ i ) .
This approximation is valid when ∆x is small. Indeed, assuming that the solution to PDE is given by a smooth function ρ(x) and using Taylor series, we can write

ρ i+1 = ρ(x i+1 ) = ρ(x i ) + ∂ ρ ∂ x ∆x + ∂ 2 ρ ∂ x 2 ∆x 2 2 + ..., (9)
where all partial derivatives are calculated in x i . Thus, subtracting ρ i and dividing by ∆x, we get

∂ ρ ∂ x = 1 ∆x (ρ i+1 -ρ i ) - ∂ 2 ρ ∂ x 2 ∆x 2 -..., (10)
which means that the residual belongs to the class O(∆x) of all functions which go to zero at least as fast as ∆x. Thus, taking ∆x sufficiently small, one can ensure the arbitrary accuracy of the approximation, provided all the partial derivatives are bounded. Accuracy can be further increased by taking different points where the function is sampled, called stencil points. For example, writing

ρ i-1 = ρ(x i-1 ) = ρ(x i ) - ∂ ρ ∂ x ∆x + ∂ 2 ρ ∂ x 2 ∆x 2 2 -..., (11)
ρ i ρ i+s 2 ρ i+s 1 ρ i+s j a 2 a 1 a j x x i-1 x i x i+1
x i+s j ∆x ∆x subtracting [START_REF] Glasscock | What is... a graphon[END_REF] from ( 9) and dividing by 2∆x, we get

∂ ρ ∂ x = 1 2∆x (ρ i+1 -ρ i-1 ) - ∂ 3 ρ ∂ x 3 ∆x 2 6 + .... (12)
Thus, using stencil points {i -1, i + 1} to approximate the first-order derivative in the point i the obtained residual belongs to the class O(∆x 2 ), which means that this discretization of the Transport PDE is accurate to the second order.

B. Continuation

Essentially the same process can be applied to the equation ( 8) to get the PDE version. For every term in the summation in [START_REF] Nikitin | Control of average and deviation in large-scale linear networks[END_REF] we can write

ρ i+s j = ρ(x i+s j ) = ρ(x i ) + ∂ ρ ∂ x ∆xs j + ∂ 2 ρ ∂ x 2 ∆x 2 s 2 j 2 + ... (13)
Thus, assume we state the problem of finding the PDE approximation of (8) in form

N ∑ j=1 a j ρ i+s j ≈ d ∑ k=0 c k ∆x k k! ∂ k ρ ∂ x k , ( 14
)
where d is the highest order of derivative (order of continuation) we want to use. Note that zero is also included in the right summation, since the function itself can be used in the resulting PDE. Then the vector of unknown coefficients c can be found by inserting ( 13) into ( 14):

c k = N ∑ j=1 a j s k j ∀k ∈ {0, .., d}. (15)
Once ( 15) is solved, we write the PDE approximation to (8):

∂ ρ ∂t = d ∑ k=0 c k ∆x k k! ∂ k ρ ∂ x k . ( 16
)
Remark 1 (Multidimensional systems). The method can be easily generalized to include more classes of systems, such as systems having several spatial dimensions. Multidimensionality can be accounted for by assuming that a position of a node ρ i is described by x i ∈ R n and taking multidimensional Taylor expansion at [START_REF] Gao | Graphon control of large-scale networks of linear systems[END_REF].

Remark 2 (Inhomogeneous systems). Space-dependent systems can also be continualized using the same method. Indeed, choosing a unique d for all agents, we can perform continuation at positions x i of all agents, obtaining possibly different coefficients c k (x i ) for k ∈ {0, .., d}. We can then interpret them as sampled values of some continuous functions c k (x) and approximate it by any method for continuous approximation, for example interpolation or least squares approximation.

C. Accuracy of continuation

Procedures of discretization and continuation look very similar from the algebraic point of view, however they are qualitatively different in the way how the problem is formulated and how we should interpret their results. Indeed, the discretization step ∆x is usually an adjustable parameter which can be set by a system engineer arbitrarily small to satisfy the desired performance. Thus the notion of accuracy of a discretization is used to describe how fast the solution of the discretized equation tends to the solution of the original equation when ∆x tends to zero.

Instead, when the original system is given by the ODE, the nodes have fixed locations, thus ∆x is a true constant representing properties of an underlying physical system and it cannot be changed by an engineer. However, the higher order of continuation is taken, the better the original ODE operator ( 8) is approximated by the PDE [START_REF] Frihauf | Leader-enabled deployment onto planar curves: A PDE-based approach[END_REF]. This intuition is supported by the following theorem, proven in [START_REF] Nikitin | A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip[END_REF]: Theorem 1. The spectrum of the PDE (16) converges to the spectrum of the original ODE (8) pointwise as d → ∞.

The convergence of spectrums is not uniform, therefore for any d at high enough frequencies the approximation will be not accurate. This effect can lead to a loss of stability properties of the original system. In particular, one of corollaries in [START_REF] Nikitin | A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip[END_REF] shows that if the order of continuation is a multiple of four, d = 4m, and in ( 16) the last gain c d > 0, the PDE ( 16) becomes unstable even if the original system (8) was stable, thus an artificial instability is introduced. In the same way if d = 4m + 2 and if c d < 0, ( 16) is also unstable. Finally, another corollary in [START_REF] Nikitin | A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip[END_REF] shows that odd terms in ( 16) don't change stability properties since they act only on imaginary plane.

Taking high orders of continuation in ( 16) gives a highorder PDE which is difficult to work with. Thus it makes sense to take the smallest order which provides preservation of stability properties of [START_REF] Nikitin | Control of average and deviation in large-scale linear networks[END_REF]. Having in mind the discussion above, in this paper we will use d = 2, restricting our analysis only to the systems for which c 2 > 0 such that instability is not artificially introduced.

IV. BOUNDARY CONDITIONS

We now consider the problem of defining appropriate boundary conditions for the PDE approximation [START_REF] Frihauf | Leader-enabled deployment onto planar curves: A PDE-based approach[END_REF]. Let us look at the Heat PDE:

∂ ρ ∂t = ∂ 2 ρ ∂ x 2 . (17)
Imagine that this equation is defined on an interval x ∈ [0, +∞), that is there is a boundary in the point x = 0. There are two types of boundary conditions (or BC) which can be supplied to provide a well-posed boundary value problem. For example for some a ∈ R,

1) Dirichlet BC: ρ(0) = a, 2) Neumann BC: ∂ ρ/∂ x (0) = a. (18)
There can also exist a linear combination of these boundary conditions, called Robin BC.

If the Heat Equation ( 17) is discretized in stencil points {i -1, i, i + 1}, the result is

ρi = 1 ∆x 2 (ρ i-1 -2ρ i + ρ i+1 ) . (19)
Assume now that there exists i 0 = 1 such that x i 0 -1 = 0. Depending on the type of boundary conditions, the equation for the state ρ 1 can be obtained by the discretization of a boundary value problem ( 17)-( 18) in two ways:

1) Dirichlet BC: ρ1 = (a -2ρ 1 + ρ 2 ) /∆x 2 , 2) Neumann BC: ρ1 = (ρ 2 -ρ 1 ) /∆x 2 -a/∆x. (20)
Now imagine the system ( 17) is obtained by the continuation process from the system [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF]. We can notice that states of [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF] are governed by the same dynamics except for the boundary state ρ 1 . The question is how to recover the boundary conditions [START_REF] Kanellakopoulos | Systematic design of adaptive controllers for feedback linearizable systems[END_REF] for the PDE from the dynamics of ρ 1 in [START_REF]Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations[END_REF].

This indeed can be done if one assumes that there exists a "ghost cell" ρ 0 such that it has no dynamics, but is algebraically connected with adjacent states. With a proper definition of ρ 0 the equation for ρ1 can be represented in the same way as for other states [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF] and thus has the same continuation [START_REF] Jafarizadeh | Weighted average consensus-based optimization of advection-diffusion systems[END_REF]. For example, algebraic equations for ρ 0 representing ( 19)-( 20) are 1) Dirichlet BC: ρ 0 = a, 2) Neumann BC:

ρ 0 = ρ 1 -a∆x. (21)
The ghost cell ρ 0 = a for the case of Dirichlet BC is depicted in Fig. 5. Notice that equations ( 21) can be directly continualized, obtaining [START_REF] Kanellakopoulos | Systematic design of adaptive controllers for feedback linearizable systems[END_REF]. This procedure can be generalized to any ODE system: once the states near boundaries change their dynamics with respect to the general governing equation, this change can be represented by "ghost cells" with algebraic dependences on the "real" states. Continualizing these algebraic equations leads to the boundary conditions for the obtained PDE.

V. GENERAL LINEAR NETWORK

In this section we move on to the analysis of a network system. Assume the system is given by a linear model, with ρ i ∈ R being the state of i-th agent, and x i ∈ R n being its spatial position. We can assume that every agent i is influenced by its neighbourhood N i and also has its own dynamics:

ρi = a ii ρ i + ∑ j∈N i a i j ρ j . (22)
Note that ( 22) is a space-dependent multidimensional generalization of [START_REF] Nikitin | Control of average and deviation in large-scale linear networks[END_REF]. Also we assume that ghost cells are added to the system [START_REF] Saxena | Amplitude death: The emergence of stationarity in coupled nonlinear systems[END_REF] to ensure boundary conditions as in Section IV. A particular choice would be to have a set of boundary nodes, placed on the boundaries of the domain, with either fixed or controlled states.

Based on Theorem 1 and on the discussion in Section III-C, we choose the order of continuation d = 2 to study transportation and diffusion properties of large-scale network.

Therefore, the continuation of the state ρ j at the point x i can be performed in the following way:

ρ j = ρ(x j ) ≈ ρ(x i ) + (x j -x i ) T • ∇ρ+ + 1 2 (x j -x i ) T ∂ 2 ρ ∂ x 2 (x j -x i ), (23)
or using the property of trace:

ρ j = ρ(x j ) ≈ ρ(x i ) + (x j -x i ) T • ∇ρ+ + 1 2 Tr (x j -x i )(x j -x i ) T ∂ 2 ρ ∂ x 2 , (24)
which leads to the PDE, which can be written at agents' positions as

∂ ρ ∂t = a ii + ∑ j∈N i a i j ρ + ∑ j∈N i a i j (x j -x i ) T • ∇ρ+ + Tr 1 2 ∑ j∈N i a i j (x j -x i )(x j -x i ) T ∂ 2 ρ ∂ x 2 . ( 25
) Define λ (x) ∈ R, b(x) ∈ R n and ε(x) ∈ R n×n such that λ (x i ) ≈ a ii + ∑ j∈N i a i j , b(x i ) ≈ ∑ j∈N i a i j (x j -x i ) , ε(x i ) ≈ 1 2 ∑ j∈N i a i j (x j -x i )(x j -x i ) T , (26)
thus these functions are found by a continuous approximation of coefficients of [START_REF] Arecchi | Deterministic chaos in laser with injected signal[END_REF]. With the help of these functions we finally formulate the main continuation result:

Theorem 2. The continuation of a linear network [START_REF] Saxena | Amplitude death: The emergence of stationarity in coupled nonlinear systems[END_REF] is given by

∂ ρ ∂t = λ (x)ρ + b(x) T • ∇ρ + Tr ε(x) ∂ 2 ρ ∂ x 2 , (27)
where λ (x), b(x) and ε(x) are given by (26).

Remark 3. Note that if a i j > 0 then the matrix inside of the trace is positive-semidefinite, which means that under suitable affine transformation of local coordinates the secondorder term can be represented as a stable Laplacian diffusion. This corresponds to c 2 > 0 in [START_REF] Frihauf | Leader-enabled deployment onto planar curves: A PDE-based approach[END_REF], required in Section III-C.

It is possible to derive several important corollaries for different classes of networks:

Corollary 1 (Laplacian network). If the original system [START_REF] Saxena | Amplitude death: The emergence of stationarity in coupled nonlinear systems[END_REF] depends only on the differences of states

ρi = ∑ j∈N i a i j (ρ j -ρ i ), then (27) has λ (x) ≡ 0.
Proof. This property corresponds to the fact that the network has no self-loops. For the Laplacian network a ii = -∑ j∈N i a i j , thus by ( 26) λ (x) ≡ 0.

Corollary 2 (Symmetric network). If the original system is symmetric, that is for every j ∈ N i there exists such j ∈ N i that x jx i = -(x jx i ) and a i j = a i j , then b(x) ≡ 0.

Proof. Straightforward by [START_REF] Bergner | Remote synchronization in star networks[END_REF].

Corollary 3 (Undirected regular network). If a i j = a ji for all i, j and if for every j ∈ N i there exists such j ∈ N i that x j -x i = -(x j -x i ), then (27) can be represented in the form

∂ ρ ∂t = λ (x)ρ + ∇ • ε(x) ∂ ρ ∂ x , (28)
Proof. Indeed, by taking the derivative we see that

∇ • ε(x) ∂ ρ ∂ x = (∇ • ε(x)) • ∇ρ + Tr ε(x) ∂ 2 ρ ∂ x 2 , ( 29
)
and it remains to prove that b T = ∇ • ε. Since a i j = a ji , we can assume that there exists some continuous function α(x, n) dependent on the coordinate x and the direction n even with respect to the direction such that

a i j = α x i + x j 2 , x j -x i ||x j -x i || = α x i + x j 2 , x i -x j ||x i -x j || = a ji
Denote n j = (x ix j )/||x ix j ||. Further, define y = x i to be the point where the function α is investigated, thus α ((x i + x j )/2, n j ) = α (y + (x jx i )/2, n j ). We can now take the Taylor expansion of this function with respect to the coordinate:

α y + x j -x i 2
, n j ≈ α(y, n j ) + 1 2 ∇α(y, n j ) • (x jx i ).

Inserting this expansion into the definition of b(x) we obtain b(y) T = ∑ j∈N i α(y, n j )(x j -

x i ) T + + 1 2 ∑ j∈N i ∇α(y, n j ) • (x j -x i )(x j -x i ) T = = 1 2 ∑ j∈N i ∇α(y, n j ) • (x j -x i )(x j -x i ) T , (30)
since the first sum vanishes because for every j there exists j such that (x jx i ) = -(x jx i ) and α(y, n j ) = α(y, n j ). Now, analyzing ε(x), we get

ε(y) = 1 2 ∑ j∈N i α(y, n j )(x j -x i )(x j -x i ) T + + 1 4 ∑ j∈N i (x j -x i )∇α(y, n j )(x j -x i )(x j -x i ) T = = 1 2 ∑ j∈N i α(y, n j )(x j -x i )(x j -x i ) T , (31)
where the second sum vanishes by the same reasons. Now it is clear that taking the divergence of (31) with respect to y one ends up with (30), which finishes the proof.

VI. CONCLUSION

In this paper we used a recently developed continuation method to transform a general linear spatially-distributed network in a single second-order linear PDE with spacedependent coefficients. We showed how the properties of the original network can affect the resulting PDE. Further we demonstrated the effectiveness of such approach by stabilizing a network of semiconductor lasers using a continuous PDE model and an explicit PDE-based backstepping boundary control.

Fig. 1 .

 1 Fig. 1. Proposed framework for control design based on the continuation method and a continuous representation of the system.

Fig. 2 .

 2 Fig. 2.

Fig. 3 .

 3 Fig. 3. Numerical simulation of system (2) with N = 30 oscillators with parameters µ = 5 + 4i, η = 10, α = 5 and β = -10. The absolute values of all states |z i (t)| for i ∈ {1..N} are depicted. Inset (a): uncontrolled system, u = 0. Inset (b): controlled system with controller (7) with c = 10.

Fig. 4 .

 4 Fig.[START_REF] Olfati-Saber | Flocking for multi-agent dynamic systems: Algorithms and theory[END_REF]. System of nodes aligned in 1D line with dynamics given by (8) with s 1 = -1 and s 2 = 1.

Fig. 5 .

 5 Fig.5. Boundary of the system[START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF] with Dirichlet boundary condition[START_REF]Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations[END_REF], represented by a ghost cell ρ 0 = a.

ACKNOWLEDGMENT

The Scale-FreeBack project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N 694209).