The basic reproduction number of plant pathogens: matrix approaches to complex dynamics

Emerging Themes and Approaches in Plant Virus Epidemiology

Plant diseases caused by viruses share many common features with those caused by other pathogen taxa in terms of the host-pathogen interaction, but there are also distinctive features in epidemiology, most apparent where transmission is by vectors. Consequently, the host-virus-vector-environment interaction presents a continuing challenge in attempts to understand and predict the course of plant virus epidemics. Theoretical concepts, based on the underlying biology, can be expressed in mathematical models and tested through quantitative assessments of epidemics in the field; this remains a goal in understanding why plant virus epidemics occur and how they can be controlled. To this end, this review identifies recent emerging themes and approaches to fill in knowledge gaps in plant virus epidemiology. We review quantitative work on the impact of climatic fluctuations and change on plants, viruses, and vectors under different scenarios where impacts on the individual components of the plant-virus-vector interaction may vary disproportionately; there is a continuing, sometimes discordant, debate on host resistance and tolerance as plant defense mechanisms, including aspects of farmer behavior and attitudes toward disease management that may affect deployment in crops; disentangling host-virus-vector-environment interactions, as these contribute to temporal and spatial disease progress in field populations; computational techniques for estimating epidemiological parameters from field observations; and the use of optimal control analysis to assess disease control options. We end by proposing new challenges and questions in plant virus epidemiology.

Plant Virus Adaptation to New Hosts: A Multi-scale Approach

Viruses are studied at each level of biological complexity: from within-cells to ecosystems. The same basic evolutionary forces and principles operate at each level: mutation and recombination, selection, genetic drift, migration, and adaptive trade-offs. Great efforts have been put into understanding each level in great detail, hoping to predict the dynamics of viral population, prevent virus emergence, and manage their spread and virulence. Unfortunately, we are still far from this. To achieve these ambitious goals, we advocate for an integrative perspective of virus evolution. Focusing in plant viruses, we illustrate the pervasiveness of the above-mentioned principles. Beginning at the within-cell level, we describe replication modes, infection bottlenecks, and cellular contagion rates. Next, we move up to the colonization of distal tissues, discussing the fundamental role of random events. Then, we jump beyond the individual host and discuss the link between transmission mode and virulence. Finally, at the community level, we discuss properties of virus-plant infection networks. To close this review we propose the multilayer network theory, in which elements at different layers are connected and submit to their own dynamics that feed across layers, resulting in new emerging properties, as a way to integrate information from the different levels.

How the epidemiology of disease-resistant and disease-tolerant varieties affects grower behaviour

Population-scale effects of resistant or tolerant crop varieties have received little consideration from epidemiologists. When growers deploy tolerant crop, population-scale disease pressures are often unaffected. This only benefits growers using tolerant varieties, selfishly decreasing yields for others. However, resistant crop can reduce disease pressure for all. We coupled an epidemiological model with game theory to understand how this affects uptake of control. Each time a grower plants a new crop, they must decide whether to use an improved (i.e. tolerant/resistant) or unimproved variety. This decision is based on strategic-adaptive expectations in our model, with growers comparing last season's profit with an estimate of what is expected from the alternative crop. Despite the positive feedback loop promoting use of a tolerant variety whenever it is available, a mixed unimproved- and tolerant-crop equilibrium can persist. Tolerant crop can also induce bistability between a scenario in which all growers use tolerant crop and the disease-free equilibrium, where no growers do. However, due to 'free-riding' by growers of unimproved crop, resistant crop nearly always exists in a mixed equilibrium. This work highlights how growers respond to contrasting incentives caused by tolerant and resistant varieties, and the distinct effects on yields and population-scale deployment.

How large and diverse are field populations of fungal plant pathogens? The case of Zymoseptoria tritici

Pathogen populations differ in the amount of genetic diversity they contain. Populations carrying higher genetic diversity are thought to have a greater evolutionary potential than populations carrying less diversity. We used published studies to estimate the range of values associated with two critical components of genetic diversity, the number of unique pathogen genotypes and the number of spores produced during an epidemic, for the septoria tritici blotch pathogen Zymoseptoria tritici. We found that wheat fields experiencing typical levels of infection are likely to carry between 3.1 and 14.0 million pathogen genotypes per hectare and produce at least 2.1-9.9 trillion pycnidiospores per hectare. Given the experimentally derived mutation rate of 3 × 10-10 substitutions per site per cell division, we estimate that between 27 and 126 million pathogen spores carrying adaptive mutations to counteract fungicides and resistant cultivars will be produced per hectare during a growing season. This suggests that most of the adaptive mutations that have been observed in Z. tritici populations can emerge through local selection from standing genetic variation that already exists within each field. The consequences of these findings for disease management strategies are discussed.

Modeling the transmission dynamics of plant viral disease using two routes of infection, nonlinear terms and incubation delay

Plant viral diseases have devastating effects on agricultural products worldwide. In this research, a delay differential equation model has been proposed for the transmission dynamics of plant viral disease using the vector-to-plant (primary) transmission and plant-to-plant (i.e. secondary) transmissions modeled via nonlinear (saturated) terms. Also, a time delay is considered in the model due to the incubation period of the plant. Feasibility and stability analyses of the equilibria of the model have been provided based on the basic reproduction numbers. Stability changes occur through Hopf bifurcation in both the delayed and non-delayed systems. Sensitivity analysis shows the impact of a parameter on the infection. The mathematical analysis of the model and numerical examples suggested that the disease will occur if the incubation period of the plant is small. Viral disease of a plant can be controlled by maintaining the distance between plants, removing the infected plants, and increasing crop resistance towards the disease.

An epi-evolutionary model for predicting the adaptation of spore-producing pathogens to quantitative resistance in heterogeneous environments

We have modeled the evolutionary epidemiology of spore-producing plant pathogens in heterogeneous environments sown with several cultivars carrying quantitative resistances. The model explicitly tracks the infection-age structure and genetic composition of the pathogen population. Each strain is characterized by pathogenicity traits determining its infection efficiency and a time-varying sporulation curve taking into account lesion aging. We first derived a general expression of the basic reproduction number R 0 for fungal pathogens in heterogeneous environments. We show that the evolutionary attractors of the model coincide with local maxima of R 0 only if the infection efficiency is the same on all host types. We then studied the contribution of three basic resistance characteristics (the pathogenicity trait targeted, resistance effectiveness, and adaptation cost), in interaction with the deployment strategy (proportion of fields sown with a resistant cultivar), to (i) pathogen diversification at equilibrium and (ii) the shaping of transient dynamics from evolutionary and epidemiological perspectives. We show that quantitative resistance affecting only the sporulation curve will always lead to a monomorphic population, whereas dimorphism (i.e., pathogen diversification) can occur if resistance alters infection efficiency, notably with high adaptation costs and proportions of the resistant cultivar. Accordingly, the choice of the quantitative resistance genes operated by plant breeders is a driver of pathogen diversification. From an evolutionary perspective, the time to emergence of the evolutionary attractor best adapted to the resistant cultivar tends to be shorter when resistance affects infection efficiency than when it affects sporulation. Conversely, from an epidemiological perspective, epidemiological control is always greater when the resistance affects infection efficiency. This highlights the difficulty of defining deployment strategies for quantitative resistance simultaneously maximizing epidemiological and evolutionary outcomes.

Understanding biological control with entomopathogenic fungi-Insights from a stochastic pest-pathogen model

In this study, an individual-based model is proposed to investigate the effect of demographic stochasticity on biological control using entomopathogenic fungi. The model is formulated as a continuous time Markov process, which is then decomposed into a deterministic dynamics using stochastic corrections and system size expansion. The stability and bifurcation analysis shows that the system dynamic is strongly affected by the contagion rate and the basic reproduction number. However, sensitivity analysis of the extinction probability shows that the persistence of a biological control agent depends to the proportion of spores collected from insect cadavers as well as their ability to be reactivated and infect insects. When considering the migration of each species within a set of patches, the dispersion relation shows a Hopf-damped Turing mode for a threshold contagion rate. A large size population led to a spatial and temporal resonant stochasticity and also induces an amplification effect on power spectrum density.

Simulation of leaf curl disease dynamics in chili for strategic management options

Leaf curl, a whitefly-borne begomovirus disease, is the cause of frequent epidemic in chili. In the present study, transmission parameters involved in tripartite interaction are estimated to simulate disease dynamics in a population dynamics model framework. Epidemic is characterized by a rapid conversion rate of healthy host population into infectious type. Infection rate as basic reproduction number, R₀ = 13.54, has indicated a high rate of virus transmission. Equilibrium population of infectious host and viruliferous vector are observed to be sensitive to the immigration parameter. A small increase in immigration rate of viruliferous vector increased the population of both infectious host and viruliferous vector. Migrant viruliferous vectors, acquisition, and transmission rates as major parameters in the model indicate leaf curl epidemic is predominantly a vector -mediated process. Based on underlying principles of temperature influence on vector population abundance and transmission parameters, spatio-temporal pattern of disease risk predicted is noted to correspond with leaf curl distribution pattern in India. Temperature in the range of 15–35 °C plays an important role in epidemic as both vector population and virus transmission are influenced by temperature. Assessment of leaf curl dynamics would be a useful guide to crop planning and evolution of efficient management strategies.

The Epidemiology of Plant Virus Disease: Towards a New Synthesis

Epidemiology is the science of how disease develops in populations, with applications in human, animal and plant diseases. For plant diseases, epidemiology has developed as a quantitative science with the aims of describing, understanding and predicting epidemics, and intervening to mitigate their consequences in plant populations. Although the central focus of epidemiology is at the population level, it is often necessary to recognise the system hierarchies present by scaling down to the individual plant/cellular level and scaling up to the community/landscape level. This is particularly important for diseases caused by plant viruses, which in most cases are transmitted by arthropod vectors. This leads to range of virus-plant, virus-vector and vector-plant interactions giving a distinctive character to plant virus epidemiology (whilst recognising that some fungal, oomycete and bacterial pathogens are also vector-borne). These interactions have epidemiological, ecological and evolutionary consequences with implications for agronomic practices, pest and disease management, host resistance deployment, and the health of wild plant communities. Over the last two decades, there have been attempts to bring together these differing standpoints into a new synthesis, although this is more apparent for evolutionary and ecological approaches, perhaps reflecting the greater emphasis on shorter often annual time scales in epidemiological studies. It is argued here that incorporating an epidemiological perspective, specifically quantitative, into this developing synthesis will lead to new directions in plant virus research and disease management. This synthesis can serve to further consolidate and transform epidemiology as a key element in plant virus research.

Using models to provide rapid programme support for California's efforts to suppress Huanglongbing disease of citrus

We describe a series of operational questions posed during the state-wide response in California to the arrival of the invasive citrus disease Huanglongbing. The response is coordinated by an elected committee from the citrus industry and operates in collaboration with the California Department of Food and Agriculture, which gives it regulatory authority to enforce the removal of infected trees. The paper reviews how surveillance for disease and resource allocation between detection and delimitation have been addressed, based on epidemiological principles. In addition, we describe how epidemiological analyses have been used to support rule-making to enact costly but beneficial regulations and we highlight two recurring themes in the programme support work: (i) data are often insufficient for quantitative analyses of questions and (ii) modellers and decision-makers alike may be forced to accept the need to make decisions on the basis of simple or incomplete analyses that are subject to considerable uncertainty. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'. This theme issue is linked with the earlier issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'.

Plant Virus Epidemiology: Applications and Prospects for Mathematical Modeling and Analysis to Improve Understanding and Disease Control

In recent years, mathematical modeling has increasingly been used to complement experimental and observational studies of biological phenomena across different levels of organization. In this article, we consider the contribution of mathematical models developed using a wide range of techniques and uses to the study of plant virus disease epidemics. Our emphasis is on the extent to which models have contributed to answering biological questions and indeed raised questions related to the epidemiology and ecology of plant viruses and the diseases caused. In some cases, models have led to direct applications in disease control, but arguably their impact is better judged through their influence in guiding research direction and improving understanding across the characteristic spatiotemporal scales of plant virus epidemics. We restrict this article to plant virus diseases for reasons of length and to maintain focus even though we recognize that modeling has played a major and perhaps greater part in the epidemiology of other plant pathogen taxa, including vector-borne bacteria and phytoplasmas.

A mathematical model for the dynamics and MCMC analysis of tomato bacterial wilt disease

In this paper, we formulate and analyze a mathematical model to investigate the transmission dynamics of tomato bacterial wilt disease (TBWD) in Mukono district, Uganda. We derive the basic reproduction number [Formula: see text] and prove the existence of a disease-free equilibrium point which is globally stable if [Formula: see text] and an endemic equilibrium which exists if [Formula: see text]. Model parameters are estimated using the Markov Chain Monte Carlo (MCMC) methods and robustness tested. The model parameters were observed to be identifiable. Numerical simulations show that soil solarization and sensitization of farmers can help to eliminate the disease in Uganda. A modified tomato bacterial wilt model with control terms is formulated.

Quantitative trait locus mapping reveals complex genetic architecture of quantitative virulence in the wheat pathogen Zymoseptoria tritici

We conducted a comprehensive analysis of virulence in the fungal wheat pathogen Zymoseptoria tritici using quantitative trait locus (QTL) mapping. High-throughput phenotyping based on automated image analysis allowed the measurement of pathogen virulence on a scale and with a precision that was not previously possible. Across two mapping populations encompassing more than 520 progeny, 540 710 pycnidia were counted and their sizes and grey values were measured. A significant correlation was found between pycnidia size and both spore size and number. Precise measurements of percentage leaf area covered by lesions provided a quantitative measure of host damage. Combining these large and accurate phenotypic datasets with a dense panel of restriction site-associated DNA sequencing (RADseq) genetic markers enabled us to genetically dissect pathogen virulence into components related to host damage and those related to pathogen reproduction. We showed that different components of virulence can be under separate genetic control. Large- and small-effect QTLs were identified for all traits, with some QTLs specific to mapping populations, cultivars and traits and other QTLs shared among traits within the same mapping population. We associated the presence of four accessory chromosomes with small, but significant, increases in several virulence traits, providing the first evidence for a meaningful function associated with accessory chromosomes in this organism. A large-effect QTL involved in host specialization was identified on chromosome 7, leading to the identification of candidate genes having a large effect on virulence.

The basic reproduction number of vector-borne plant virus epidemics

The basic reproduction number R₀ is a key parameter in plant disease epidemiology, which largely determines whether or not an epidemic will occur in a plant population. The next generation matrix approach to deriving and calculating the basic reproduction number of a plant virus epidemic is described. The approach is illustrated through a series of examples of increasing complexity, ranging from the simplest case of one vector transmitting one virus to a single host, to the case of multiple vectors, to combined horizontal (vector) and vertical (seed) transmission, and where vector control using insecticides is practised. The importance of parameters representing host and vector population dynamics and their interaction in the absence of disease is stressed, and the constraints these place on the calculation of the basic reproduction number. Finally, mention is made of further elaborations to the approach that could prove useful in plant virus epidemiology.

Invasiveness of plant pathogens depends on the spatial scale of host distribution

Plant diseases often cause serious yield losses in agriculture. A pathogen's invasiveness can be quantified by the basic reproductive number, R₀. Since pathogen transmission between host plants depends on the spatial separation between them, R₀ is strongly influenced by the spatial scale of the host distribution. We present a proof of principle of a novel approach to estimate the basic reproductivenumber, R₀, of plant pathogens as a function of the size of a field planted with crops and its aspect ratio. This general approach is based on a spatially explicit population dynamical model. The basic reproductive number was found to increase with the field size at small field sizes and to saturate to a constant value at large field sizes. It reaches amaximum in square fields and decreases as the field becomes elongated. This pattern appears to be quite general: it holds for dispersal kernels that decrease exponentially or faster, as well as for fat-tailed dispersal kernels that decrease slower than exponential (i.e., power-law kernels). We used this approach to estimate R₀ in wheat stripe rust(an important disease caused by Puccinia striiformis), where we inferred both the transmission rates and the dispersal kernels from the measurements of disease gradients. For the two largest datasets, we estimated R₀ of P. striiformis in the limit of large fields to be of the order of 30. We found that the spatial extent over which R₀ changes strongly is quite fine-scaled (about 30 m of the linear extension of the field). Our results indicate that in order to optimize the spatial scale of deployment of fungicides or host resistances, the adjustments should be made at a fine spatial scale. We also demonstrated how the knowledge of the spatial dependence of R₀ can improve recommendations with regard to fungicide treatment.

The effect of time delay in plant--pathogen interactions with host demography

Botanical epidemic models are very important tools to study invasion, persistence and control of diseases. It is well known that limitations arise from considering constant infection rates. We replace this hypothesis in the framework of delay differential equations by proposing a delayed epidemic model for plant--pathogen interactions with host demography. Sufficient conditions for the global stability of the pathogen-free equilibrium and the permanence of the system are among the results obtained through qualitative analysis. We also show that the delay can cause stability switches of the coexistence equilibrium. In the undelayed case, we prove that the onset of oscillations may occur through Hopf bifurcation.

Mixtures as a fungicide resistance management tactic

We have reviewed the experimental and modeling evidence on the use of mixtures of fungicides of differing modes of action as a resistance management tactic. The evidence supports the following conclusions. 1. Adding a mixing partner to a fungicide that is at-risk of resistance (without lowering the dose of the at-risk fungicide) reduces the rate of selection for fungicide resistance. This holds for the use of mixing partner fungicides that have either multi-site or single-site modes of action. The resulting predicted increase in the effective life of the at-risk fungicide can be large enough to be of practical relevance. The more effective the mixing partner (due to inherent activity and/or dose), the larger the reduction in selection and the larger the increase in effective life of the at-risk fungicide. 2. Adding a mixing partner while lowering the dose of the at-risk fungicide reduces the selection for fungicide resistance, without compromising effective disease control. The very few studies existing suggest that the reduction in selection is more sensitive to lowering the dose of the at-risk fungicide than to increasing the dose of the mixing partner. 3. Although there are very few studies, the existing evidence suggests that mixing two at-risk fungicides is also a useful resistance management tactic. The aspects that have received too little attention to draw generic conclusions about the effectiveness of fungicide mixtures as resistance management strategies are as follows: (i) the relative effect of the dose of the two mixing partners on selection for fungicide resistance, (ii) the effect of mixing on the effective life of a fungicide (the time from introduction of the fungicide mode of action to the time point where the fungicide can no longer maintain effective disease control), (iii) polygenically determined resistance, (iv) mixtures of two at-risk fungicides, (v) the emergence phase of resistance evolution and the effects of mixtures during this phase, and (vi) monocyclic diseases and nonfoliar diseases. The lack of studies on these aspects of mixture use of fungicides should be a warning against overinterpreting the findings in this review.

A generic risk-based surveying method for invading plant pathogens

Invasive plant pathogens are increasing with international trade and travel, with damaging environmental and economic consequences. Recent examples include tree diseases such as sudden oak death in the Western United States and ash dieback in Europe. To control an invading pathogen it is crucial that newly infected sites are quickly detected so that measures can be implemented to control the epidemic. However, since sampling resources are often limited, not all locations can be inspected and locations must be prioritized for surveying. Existing approaches to achieve this are often species specific and rely on detailed data collection and parameterization, which is difficult, especially when new arrivals are unanticipated. Consequently regulatory sampling responses are often ad hoc and developed without due consideration of epidemiology, leading to the suboptimal deployment of expensive sampling resources. We introduce a flexible risk-based sampling method that is pathogen generic and enables available information to be utilized to develop epidemiologically informed sampling programs for virtually any biologically relevant plant pathogen. By targeting risk we aim to inform sampling schemes that identify high-impact locations that can be subsequently treated in order to reduce inoculum in the landscape. This "damage limitation" is often the initial management objective following the first discovery of a new invader. Risk at each location is determined by the product of the basic reproductive number (R0), as a measure of local epidemic size, and the probability of infection. We illustrate how the risk estimates can be used to prioritize a survey by weighting a random sample so that the highest-risk locations have the highest probability of selection. We demonstrate and test the method using a high-quality spatially and temporally resolved data set on Huanglongbing disease (HLB) in Florida, USA. We show that even when available epidemiological information is relatively minimal, the method has strong predictive value and can result in highly effective targeted surveying plans.

Variation and selection of quantitative traits in plant pathogens

The first section presents the quantitative traits of pathogenicity that are most commonly measured by plant pathologists, how the expression of those traits is influenced by environmental factors, and why the traits must be taken into account for understanding pathogen evolution in agricultural systems. Particular attention is given to the shared genetic control of these traits by the host and the pathogen. Next, the review discusses how quantitative traits account for epidemic development and how they can be related to pathogen fitness. The main constraints that influence the evolution of quantitative traits in pathogen populations are detailed. Finally, possible directions for research on the management of pathogen virulence (as defined by evolutionists) and host quantitative resistance are presented. The review evaluates how the theoretical corpus developed by epidemiologists and evolutionists may apply to plant pathogens in the context of agriculture. The review also analyzes theoretical papers and compares the modeling hypotheses to the biological characteristics of plant pathogens.

The failure of R0

The basic reproductive ratio, R(0), is one of the fundamental concepts in mathematical biology. It is a threshold parameter, intended to quantify the spread of disease by estimating the average number of secondary infections in a wholly susceptible population, giving an indication of the invasion strength of an epidemic: if R(0) < 1, the disease dies out, whereas if R(0) > 1, the disease persists. R(0) has been widely used as a measure of disease strength to estimate the effectiveness of control measures and to form the backbone of disease-management policy. However, in almost every aspect that matters, R(0) is flawed. Diseases can persist with R(0) < 1, while diseases with R(0) > 1 can die out. We show that the same model of malaria gives many different values of R(0), depending on the method used, with the sole common property that they have a threshold at 1. We also survey estimated values of R(0) for a variety of diseases, and examine some of the alternatives that have been proposed. If R(0) is to be used, it must be accompanied by caveats about the method of calculation, underlying model assumptions and evidence that it is actually a threshold. Otherwise, the concept is meaningless.

Transmission of insect-vectored pathogens: effects of vector fitness as a function of infectivity status

The transmission of insect-vectored pathogens is dependent on the population dynamics of the vector. Epidemiological models typically assume that birth and death rates of pathogen-free and inoculative vectors are equal, an assumption that is not true for all pathosystems. Here, a series of simple and general epidemiological models were used to explore how assumptions about birth and death rates of vectors based on their infectivity status influence disease incidence. With fixed death rate of pathogen-free vectors, increasing the death rate of inoculative vectors reduced vector density, the proportion of vectors that were inoculative, and the proportion of hosts infected. This effect was mediated by acquisition rate. Specifically, increasing the acquisition rate increased the proportion of vectors that were inoculative, thereby increasing the proportion of the vector population that experienced the increased death rate. With fixed birth rate of pathogen-free vectors, variation in birth rate of inoculative vectors had little influence on disease incidence provided that the birth rate of pathogen-free vectors was much greater than their death rate. However, when the birth rate of pathogen-free vectors was only slightly greater than their death rate, large increases in the birth rate of inoculative vectors increased total vector density and disease incidence. The results indicate that assumptions about birth and death rates of vectors based on infectivity status can have important effects on the vector population that in turn affects disease incidence.

Models of fungicide resistance dynamics

We describe two classes of models used for fungicide and antibiotic resistance dynamics. One class assumes that the density of the pathogen (or severity of the disease caused by the pathogen) has no feedback effects on the rate at which new infections arise. The second class does not make this assumption. A quantitative relationship between these two classes is derived. We then discuss the two sets of assumptions made in the literature about initial conditions: either both the fungicide-sensitive strain and the -resistant strain are initially at low density, or the sensitive strain is resident at nonlow density and the resistant strain is initially at low density. We show that models of fungicide resistance dynamics with and without density-dependent feedback give contrasting predictions on the effects of pathogen life-cycle parameters and the effects of the fungicide (dose, frequency, use of mixtures, spatial usage restrictions) on the evolution, invasion, and spread of fungicide resistance. We further show that the evaluation of a resistance management strategy requires a very precise definition of what constitutes a good strategy.

Epidemiological models for invasion and persistence of pathogens

Motivated by questions such as "Why do some diseases take off, while others die out?" and "How can we optimize the deployment of control methods," we introduce simple epidemiological concepts for the invasion and persistence of plant pathogens. An overarching modeling framework is then presented that can be used to analyze disease invasion and persistence at a range of scales from the microscopic to the regional. Criteria for invasion and persistence are introduced, initially for simple models of epidemics, and then for models with greater biological realism. Some ways in which epidemiological models are used to identify optimal strategies for the control of disease are discussed. Particular attention is given to the spatial structure of host populations and to the role of chance events in determining invasion and persistence of plant pathogens. Finally, three brief case studies are used to illustrate the practical applications of epidemiological theory to understand invasion and persistence of plant pathogens. These comprise long-term predictions for the persistence and control of Dutch elm disease; identification of methods to manage the spread of rhizomania on sugar beet in the U.K. by matching the scale of control with the spatial and temporal scales of the disease; and analysis of evolutionary change in virus control to identify risks of inadvertent selection for damaging virus strains.