DIMACS Workshop on Models/Methodological Problems of Botanical Epidemiology

March 16 - 18, 2009
DIMACS Center, CoRE Building, Rutgers University

Chris Gilligan, Cambridge, cag1@cus.cam.ac.uk
Presented under the auspices of the Special Focus on Computational and Mathematical Epidemiology.


Don Aylor, Plant Pathology and Ecology, The Connecticut Agricultural Experiment Station, New Haven, CT

Title: Modeling disease spread by aerially dispersed pathogens in a patchy landscape

Anticipation of a new disease introduction in an area, management of a periodically introduced pathogen, or optimal application of disease control measures for an indigenous pathogen all depend on a quantitative understanding of pathogen spread and host and pathogen development over a wide range spatial and temporal scales. Disease spread can be modeled in terms of basic biological and physical parameters, viz., latent period, infectious period, basic infection rate, dispersal distance and survival time scales, and host and pathogen spatial distribution and their phenology, including over wintering potential. A mean waiting time for new infections to appear on discrete patches of host plants a certain distance from a focus of disease can be defined in terms of these basic parameters. We intend to illustrate how this waiting time can be used to help establish guidelines for minimizing application of fungicides and for minimizing the risk of pesticide resistance, while maintaining acceptable yield. We examine the following questions: 1) Once disease or the pathogen is detected locally, can a buffer zone around the new focus be identified and protected without the necessity of spraying the whole field? 2) When can small fields separated by a given distance be treated as separate management units? 3) For hosts distributed on the regional or landscape scale, can we define waiting times that allow us to forgo or delay control measures in a neighboring region? These ideas will be illustrated using apple scab, potato late blight, tobacco blue mold, and stem rust of wheat as examples.

Inez Demon, Department of Biomathematics and Bioinformatics, Rothamsted Research, West Common, Harpenden, Hertfordshire, AL5 2JQ, United Kingdom

Title: Sampling for detecting and mapping invasive plant pathogens.

Invasive pathogens are known to cause mayor damage to the environments they invade. Effective disease management can depend on early detection and the ability to accurately estimate the spatial distribution of plant disease. In plant pathology the main focus has been on sampling to estimate disease incidence and in this area much progress has been made. Here, we focus on sampling with the aims of (i) detecting an invading plant pathogen and (ii) estimating the geographic distribution of plant diseases from a sample. To that end, we introduce the concept of spatial optimisation. The method allows for the incorporation of epidemiological processes and the optimum allocation of sampling points to obtain the largest detection probability. We show the benefits of the method to plant pathology and in the instance of detection sampling we compare its performance with that of three other sampling methods. We show: (i) that the optimisation method can easily be applied for disease detection, (ii) that the method, which incorporates aspects of the disease and the host distribution accurately maps the disease distribution, (iii) that it outperforms the conventional sampling methods and (iv) that combining it with a spatially explicit epidemic model, we can develop optimum sample schemes, i.e. optimum locations to sample that maximise the probability of detecting an invasive pathogen.

Alain Franc, Dept Ecology of Forests, Grasslands and Water, INRA-Bordeaux, France and Nathalie Peyrard

Title: How does the geometry of migration routes of pathogens impact an epidemics?

Epidemics have been modelled for decades within several tools as contact process, or mean field models such as SIS or SIR models. Contact process is intractable, even on a line. We will analyse the impact of the graph topology on the outcome of an epidemic. Therefore, we select three classical ways for characterizing a graph: degree distribution, diameter, and clustering index. We show how to derive refined mean field approximations, pair approximation from these quantities, how to go slightly beyond, and how to translate the approximations into quantification of an epidemic prevalence, or threshold for outbreak. We finally show how this is related to many other research areas where similar questions arise. We show that graph topology impacts the outcome of an epidemic, and how. It then can be an extended phenotype of a pathogen, and subject of selection pressures. We show how vectorization of a disease, i.e. selecting an active agent for selecting routes towards hosts, is an evolutionary advantage for the pathogen. An analogy is suggested with vectorization of pollination, dominant in Tropical Rainforests, common in temperate grasslands, as in both processes, the topology of the transmission routes impacts the success of the individuals, the transmission of a pathogen or a gene.

Michael Jeger, Imperial College

Title: Network models for the spread of plant disease in trade pathways

Network models have been developed for the spread of plant pathogens through the movement of diseased plants in the nursery trade. The model structure is based on small-size directed graphs of different topologies with two simple parameters: the probabilities of disease transmission between nodes and disease persistence at nodes. Within this framework it is shown how invasion criteria and epidemic final size depend on network properties including connectance, clustering, and in-out correlations at nodes for the different topologies. The analysis suggests that control interventions based on inspection and eradication regimes ideally require some knowledge of the underlying trade network, which in the UK at least is imperfectly understood for reasons of commercial sensitivity. The modelling has been linked to the incidence and spread of Phytophthora ramorum on hardy nursery stock in both UK nurseries and the semi-natural environment.

Michael Jeger, Imperial College

Title: Do pathogens regulate host population dynamics? A model exploration for the biennial Tragopogon pratensis affected by a systemic, sterilising rust Puccinia hysterium

Tragopogon pratensis is a biennial plant found in grassland communities. It is commonly infected by Puccinia hysterium an autecious demicyclic (lacking a repeating spore stage). First year seedlings are infected by aeciospores/teliospores produced on second year flowering plants. The rust goes systemic between seasons. Model equations are based on a discrete time SEIR model appropriate for the biological system. Invasion criteria and (implicit) final size expressions are derived for different scenarios, including allowance for disease-induced mortality and a short-lived seed bank. Model outputs for realistic parameter values include population crashes of the host (and pathogen), cycling of healthy and diseased hosts, and steady state host populations with the pathogen going extinct. A second-order recurrence relationship is derived which when compared with long-term data sets suggests that host regulation is dependent on disease-induced mortality and is not dependent on host density.

Michael Jeger, Imperial College

Title: How to model transmission for vectored plant virus diseases

Plant virus transmission by vectors has been modelled in several ways, commonly using SEIR-type models and more rarely linking these to vector population dynamics explicitly. In the latter case, although useful qualitative results have been obtained, the extended number of parameters involved makes detailed analysis difficult. A further complication arises when within-plant dynamics determining virus titre are linked to the population models. A variety of ways of incorporating plant virus titre has been explored by considering alternatives to common assumptions on transmission, with no more than three parameters involved to ensure model parsimony. Different outcomes result in terms of the number of parameters in the invasion criteria and final epidemic sizes derived. In one case a closed form solution is obtained relating time to a function of mean virus titre per infected plant, that in principle could be fitted to field data.

Chris Mundt, Dept Botany & Plant Pathology, Oregon State University

Title: Aerial dispersal, invasive epidemics, and scaling relationships: empirical evidence

Our overall approach has been to derive experimental data from wheat plots inoculated with stripe rust, use these data to develop simple principles of epidemic spread, and then test these principles against observational data at a continental scale. As a foundation, we use a simple model that assumes inverse square dispersal and that the initial position of the epidemic front is proportional to the temporal rate of disease increase. The model predicts that a plot of epidemic velocity versus distance will have a slope of 1/2 regardless of temporal rate, but that it will take an epidemic with a lower rate a longer time to reach any given distance as compared to one with a faster rate. The position of the epidemic front increases exponentially as 2 raised to the exponent of time, with low rate epidemics having a lower intercept than high rate epidemics. Experimental data for wheat stripe rust in monoculture and mixture stands of wheat are fit well by this model. Due to scale invariance of the power law, we were able to scale-up to describe continental-scale spread of wind-dispersed plant pathogens, as well as to animal/human pathogens spread by migratory birds (West Nile virus and avian bird flu H5N1). Our next step is to describe effects of landscape heterogeneity on epidemic spread. Focus size and host diversity have a large influence on epidemic spread, and can be described with simple scaling relationships. In contrast, host population size had no effect on epidemic spread, likely because autoinfection rates are substantially higher than previously thought. Experimental and theoretical studies of the "grain" of diversity are underway, and the effect of the grain of diversity is likely to be a function of the ratio of size of genotype unit to size of initial disease focus. Landscape heterogeneity variables are now being evaluated for the continental-scale spread of soybean rust in North America.

Stephen Parnell, Rothamsted Research

Title: The influence of landscape pattern on the eradication of an invading plant pathogen

Recent increases in international trade and travel have led to a significant rise in the number of introductions of exotic plant pathogens. The potential destructiveness of these pathogens combined with a lack of effective disease management measures means that their eradication is often sought. Eradication typically involves the elimination of inoculum via the removal of symptomatic hosts and their neighbours. This works by targeting local cryptic infections which escape detection during standard disease surveys. However, although effective, this process can involve the removal of large numbers of hosts and thus substantial cost. Recent epidemiological modelling studies have demonstrated that optimal strategies can be design which minimise the adverse impacts of eradication. The design of optimal eradication strategies depends critically on the spatial pattern of epidemic increase. This is strongly influenced by the spatial pattern of susceptible hosts in the landscape. In this paper we demonstrate how landscape pattern influences the eradication of an invading pathogen and why it should be a key consideration in the design of optimal eradication policy. We use a spatially explicit stochastic model to simulate an epidemic and eradication process in a range of simulated host landscapes. We consider landscapes which differ in density and aggregation of hosts and show how this influences the optimal eradication strategy and also the total costs associated with it.

Michael Shaw, University of Reading, UK

Title: Coupling models to weather, and weather to climate: can we yet do it?

This is an intentionally open-ended paper. The standard predictive modelling framework is to take a description of the instantaneous processes controlling change in a system and turn them into a differential or difference equation model; we may bend things a bit to make them fit conveniently to, for example, an SEIR model, already departing from the physics paradigm out of which this dynamical modelling arises. To produce a model like this which relates disease to weather and ultimately climate we must couple the parameters to environmental variables in a meaningful way. The obstacle has often been seen as lying in acquisition of adequate data or the suitability of fitting methods; but I will argue that there are much more fundamental problems in relating experimental or observational data to model parameters which are, and that understanding these better might make the statistical problems more transparent and guide epidemiological experimenters more effectively. The problems arise from three directions. (a) We almost never have a full, exact, process description ? and if we did, it would probably be on too small a scale to allow its use in a model (compare the problems in using quantum mechanics ab initio in protein structure calculation). This means that we are using empirical relations between the outcomes of processes (eg infection) and environmental variables; but there are near-infinite degrees of freedom in the environmental variables, so our relationships ? inevitably used in an extrapolative form ? are based on exploring a very limited volume of the external parameter space, which might justify interpolation. (b) Model structures generally (and I would argue, correctly) use a number of abstractions which mean that model parameters are not simply related to the observables in experiments. (c) When we turn to climate change issues, the obvious approach is to use climate predictions to generate weather ensembles and then use these to generate epidemic ensembles. But unless the epidemiological model is based on a full process description (incorporating changes in plant defences and microbial relations due to changes in nutrient and water relations) parameter values unrelated to weather will have changed ? and we are extremely likely to be extrapolating environmental relations outside the range for which we have experimental evidence. I would argue that we need to look for new ideas on how to formulate our weather-based process models, and new ideas on how to solve the dimensionality problem in parameter estimation. If I knew what these ideas were, I would suggest them!

Aurélien Tellier, Section of Evolutionary Biology, Biocenter, University of Munich, 82152 Planegg-Martinsried, Germany and James K.M. Brown, Department of Disease and Stress Biology, John Innes Centre, Colney, Norwich, NR4 7UH, UK

Title: Maintenance of polymorphism in host-parasite interactions: role of ecological, epidemiological and genetic factors

Allelic diversity is widespread at host loci involved in parasite recognition such as those controlling the major histocompatibility complex in vertebrates or gene-for-gene (GFG) relationships in plants. It has been proposed that spatial sub-division of populations may promote stable polymorphism, maintaining diversity in matching host and parasite loci, in a model known as the Geographical Mosaic of Coevolution. The theory of this process, however, has involved diverse and generally complex theories invoking cost of resistance and virulence, as well as a wide range of genetic, ecological and epidemiological parameters in addition to restricted migration between demes. Here, we unify and simplify existing theory by describing a simple condition required to maintain polymorphism in a matching pair of host and parasite loci. The condition is that there must be negative direct frequency-dependent selection (FDS), implying that the selection coefficient for a given allele decreases as its frequency increases. We show here that this condition is fulfilled if there is migration of the host, the parasite or both between demes of a meta-population with different environments, such that the fitness cost of parasite virulence or host resistance, or the cost to a host of being diseased differs between demes. We also summarize the various factors that have been found to generate negative direct FDS such as host seed banks, host perenniality, and polycyclic parasites.

Femke van den Berg and F. van den Bosch, Rothamsted Research, and Chris Gilligan, University of Cambridge

Title: The Evolutionary Ecology of Plant Pathogen

The evolutionary responses of plant pathogens to the use of fungicides and/or resistant crops have been intensively studied. The selection pressures imposed on the evolution of a pathogen's life-cycle characteristics by other disease management strategies or changes in the environment have been studied much less frequently. Such adaptations can, however, have a major effect on epidemic dynamics and therewith host performance. This presentation aims to give an overview of theoretical and experimental approaches presently undertaken in a BBSRC-INRA collaboration.

A short overview of one of the approaches used in theoretical evolutionary studies will be followed by a variety of examples of biological questions that can be answered using these approaches and examples of challenges met when studying the evolutionary ecology of plant pathogens.

One of the major challenges in evolutionary ecology of plant pathogens is designing suitable experiments to detect trade-off relationships between pathogen life-cycle dynamics. Using two examples I will show that despite these challenges progress is under way.

The next example shows the effect of a period of host absence between two consecutive growing seasons on the evolution of a pathogen's reproductive capacity. It will be shown that longer periods of host absence seem to select for a higher reproductive capacity in airborne pathogens, yet a lower reproductive capacity in soil-borne pathogens.

An example on virus titre evolution shows that different management strategies can also select for both less or more aggressive pathogen strains. More specifically it will be shown that the use of virus free material though in vitro propagation may select virus strains with a larger within cell multiplication rate, whereas roguing tends to select virus strains with a smaller multiplication rate.

Data from the Broadbalk experiment at Rothamsted Research show evolutionary bi-stability patterns in the ratio of horizontal to vertical pathogen transmission in Phaeosphaeria nodorum. It will be shown that the pattern found from the data analyses can be explained by using a simple model of the evolutionary ecology of this pathogen.

The presentation will finish with a brief overview of the key future aims in the study of the evolutionary ecology of plant pathogens.

Wopke van der Werf and Pete Skelsey, Wageningen University, the Netherlands

Title: Multi-scale modeling of potato late blight epidemics.

Phytopthora infestans, causal agent of potato late blight, causes multi-billion dollar losses annually in global production of potatoes and other Solanaceous crops. Although the spatial dimension of potato late blight epidemiology is widely acknowledged by researchers who study plant disease, the corresponding theoretical framework is underdeveloped. This is because spatial increase of potato late blight disease is the product of a complex interplay between meteorological driving forces, management practices, and the spatial characteristics of crops and landscapes.

Field-scale disease dynamics were quantified through the development and validation of a spatial-temporal model of the potato late blight pathosystem. The model was used to compare disease dynamics in pragmatic layouts for variety mixtures. A sensitivity analysis of the model resulted in a heuristic for distinguishing epidemics driven by lesion expansion from those driven by lesion propagation. The model was also used to study the vulnerability of potato crops to disease invasion from an initial influx of sporangia, in order to determine the feasibility of spore dispersal modeling as a risk assessment tool. Regional-scale inoculum dynamics were quantified through the development and validation of a numerical and a fully analytical atmospheric spore dispersion and deposition model.

Two modeling tools for management were developed. In the first instance, a full complement of aerobiological models (release of spores from sporangiophores, escape from the canopy, numerical dispersion model and survival during transportation) were combined with an existing decision support system (DSS) to develop a novel concept for inclusion of regional aerobiological modeling in disease risk warning forecasts for an aerially transmitted plant pathogen. A field trial with the new system showed adequate disease control while the number of chemical treatments was reduced in with the full complement of aerobiological models (including the analytical dispersion model) and a landscape generator to create a multi-scale simulator of the late blight pathosystem. The simulator was used to evacuate various landscape designs for the suppression of invasions of resistance breaking pathogen strains. Simulation results showed that the large capacity of P. infestans for long-distance transport of viable inoculum nullified the effectiveness of spatial barriers to disease spread at scales up to several kilometres. The most successful landscape designs for suppressing disease spread were those that increased the level of genetic diversity in host populations, and/or the degree of mixing of host genotypes.

Many of the modeling tools and concepts developed in this research are extendable to other pathosystems characterized by airborne inoculum.

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Document last modified on February 25, 2009.