Rational design of a survey protocol for avocado sunblotch viroid in commercial orchards to demonstrate pest freedom

Avocado sunblotch viroid (ASBVd) is a subcellular pathogen of avocado that reduces yield from a tree, diminishes the appearance of the fruit by causing unsightly scarring and impedes trade because of quarantine conditions that are imposed to prevent spread of the pathogen via seed-borne inoculum. For countries where ASBVd is officially reported, permission to export fruit to another country may only be granted if an orchard can be demonstrated to be a pest free production site. The survey requirements to demonstrate pest freedom are usually defined in export protocols that have been mutually agreed upon by the trading partners. In this paper, we introduce a flexible statistical protocol for use in optimizing sampling strategies to establish pest free status from ASBVd in avocado orchards. The protocol, which is supported by an interactive app, integrates statistical considerations of multistage sampling of trees in orchards with a RT-qPCR assay allowing for detection of infection in pooled samples of leaves taken from multiple trees. While this study was motivated by a need to design a survey protocol for ASBVd, the theoretical framework and the accompanying app have broader applicability to a range of plant pathogens in which hierarchical sampling of a target population is coupled with pooling of material prior to diagnosis.

Wheat rust epidemics damage Ethiopian wheat production: A decade of field disease surveillance reveals national-scale trends in past outbreaks

Wheat rusts are the key biological constraint to wheat production in Ethiopia-one of Africa's largest wheat producing countries. The fungal diseases cause economic losses and threaten livelihoods of smallholder farmers. While it is known that wheat rust epidemics have occurred in Ethiopia, to date no systematic long-term analysis of past outbreaks has been available. We present results from one of the most comprehensive surveillance campaigns of wheat rusts in Africa. More than 13,000 fields have been surveyed during the last 13 years. Using a combination of spatial data-analysis and visualization, statistical tools, and empirical modelling, we identify trends in the distribution of wheat stem rust (Sr), stripe rust (Yr) and leaf rust (Lr). Results show very high infection levels (mean incidence for Yr: 44%; Sr: 34%; Lr: 18%). These recurrent rust outbreaks lead to substantial economic losses, which we estimate to be of the order of 10s of millions of US-D annually. On the widely adopted wheat variety, Digalu, there is a marked increase in disease prevalence following the incursion of new rust races into Ethiopia, which indicates a pronounced boom-and-bust cycle of major gene resistance. Using spatial analyses, we identify hotspots of disease risk for all three rusts, show a linear correlation between altitude and disease prevalence, and find a pronounced north-south trend in stem rust prevalence. Temporal analyses show a sigmoidal increase in disease levels during the wheat season and strong inter-annual variations. While a simple logistic curve performs satisfactorily in predicting stem rust in some years, it cannot account for the complex outbreak patterns in other years and fails to predict the occurrence of stripe and leaf rust. The empirical insights into wheat rust epidemiology in Ethiopia presented here provide a basis for improving future surveillance and to inform the development of mechanistic models to predict disease spread.

Will an outbreak exceed available resources for control? Estimating the risk from invading pathogens using practical definitions of a severe epidemic

Forecasting whether or not initial reports of disease will be followed by a severe epidemic is an important component of disease management. Standard epidemic risk estimates involve assuming that infections occur according to a branching process and correspond to the probability that the outbreak persists beyond the initial stochastic phase. However, an alternative assessment is to predict whether or not initial cases will lead to a severe epidemic in which available control resources are exceeded. We show how this risk can be estimated by considering three practically relevant potential definitions of a severe epidemic; namely, an outbreak in which: (i) a large number of hosts are infected simultaneously; (ii) a large total number of infections occur; and (iii) the pathogen remains in the population for a long period. We show that the probability of a severe epidemic under these definitions often coincides with the standard branching process estimate for the major epidemic probability. However, these practically relevant risk assessments can also be different from the major epidemic probability, as well as from each other. This holds in different epidemiological systems, highlighting that careful consideration of how to classify a severe epidemic is vital for accurate epidemic risk quantification.

Applying optimal control theory to complex epidemiological models to inform real-world disease management

Mathematical models provide a rational basis to inform how, where and when to control disease. Assuming an accurate spatially explicit simulation model can be fitted to spread data, it is straightforward to use it to test the performance of a range of management strategies. However, the typical complexity of simulation models and the vast set of possible controls mean that only a small subset of all possible strategies can ever be tested. An alternative approach-optimal control theory-allows the best control to be identified unambiguously. However, the complexity of the underpinning mathematics means that disease models used to identify this optimum must be very simple. We highlight two frameworks for bridging the gap between detailed epidemic simulations and optimal control theory: open-loop and model predictive control. Both these frameworks approximate a simulation model with a simpler model more amenable to mathematical analysis. Using an illustrative example model, we show the benefits of using feedback control, in which the approximation and control are updated as the epidemic progresses. Our work illustrates a new methodology to allow the insights of optimal control theory to inform practical disease management strategies, with the potential for application to diseases of humans, animals and plants. 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'.

Applying optimal control theory to complex epidemiological models to inform real-world disease management

Mathematical models provide a rational basis to inform how, where and when to control disease. Assuming an accurate spatially explicit simulation model can be fitted to spread data, it is straightforward to use it to test the performance of a range of management strategies. However, the typical complexity of simulation models and the vast set of possible controls mean that only a small subset of all possible strategies can ever be tested. An alternative approach-optimal control theory-allows the best control to be identified unambiguously. However, the complexity of the underpinning mathematics means that disease models used to identify this optimum must be very simple. We highlight two frameworks for bridging the gap between detailed epidemic simulations and optimal control theory: open-loop and model predictive control. Both these frameworks approximate a simulation model with a simpler model more amenable to mathematical analysis. Using an illustrative example model, we show the benefits of using feedback control, in which the approximation and control are updated as the epidemic progresses. Our work illustrates a new methodology to allow the insights of optimal control theory to inform practical disease management strategies, with the potential for application to diseases of humans, animals and plants. 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'.

Quantifying airborne dispersal routes of pathogens over continents to safeguard global wheat supply

Infectious crop diseases spreading over large agricultural areas pose a threat to food security. Aggressive strains of the obligate pathogenic fungus Puccinia graminis f.sp. tritici (Pgt), causing the crop disease wheat stem rust, have been detected in East Africa and the Middle East, where they lead to substantial economic losses and threaten livelihoods of farmers. The majority of commercially grown wheat cultivars worldwide are susceptible to these emerging strains, which pose a risk to global wheat production, because the fungal spores transmitting the disease can be wind-dispersed over regions and even continents ^1-11 . Targeted surveillance and control requires knowledge about airborne dispersal of pathogens, but the complex nature of long-distance dispersal poses significant challenges for quantitative research ^12-14 . We combine international field surveys, global meteorological data, a Lagrangian dispersion model and high-performance computational resources to simulate a set of disease outbreak scenarios, tracing billions of stochastic trajectories of fungal spores over dynamically changing host and environmental landscapes for more than a decade. This provides the first quantitative assessment of spore transmission frequencies and amounts amongst all wheat producing countries in Southern/East Africa, the Middle East and Central/South Asia. We identify zones of high air-borne connectivity that geographically correspond with previously postulated wheat rust epidemiological zones (characterized by endemic disease and free movement of inoculum) ^10,15 , and regions with genetic similarities in related pathogen populations ^16,17 . We quantify the circumstances (routes, timing, outbreak sizes) under which virulent pathogen strains such as 'Ug99' ^5,6 pose a threat from long-distance dispersal out of East Africa to the large wheat producing areas in Pakistan and India. Long-term mean spore dispersal trends (predominant direction, frequencies, amounts) are summarized for all countries in the domain (Supplementary Data). Our mechanistic modelling framework can be applied to other geographic areas, adapted for other pathogens and used to provide risk assessments in real-time ³ .

Large-Scale Atmospheric Dispersal Simulations Identify Likely Airborne Incursion Routes of Wheat Stem Rust Into Ethiopia

In recent years, severe wheat stem rust epidemics hit Ethiopia, sub-Saharan Africa's largest wheat-producing country. These were caused by race TKTTF (Digalu race) of the pathogen Puccinia graminis f. sp. tritici, which, in Ethiopia, was first detected at the beginning of August 2012. We use the incursion of this new pathogen race as a case study to determine likely airborne origins of fungal spores on regional and continental scales by means of a Lagrangian particle dispersion model (LPDM). Two different techniques, LPDM simulations forward and backward in time, are compared. The effects of release altitudes in time-backward simulations and P. graminis f. sp. tritici urediniospore viability functions in time-forward simulations are analyzed. Results suggest Yemen as the most likely origin but, also, point to other possible sources in the Middle East and the East African Rift Valley. This is plausible in light of available field surveys and phylogenetic data on TKTTF isolates from Ethiopia and other countries. Independent of the case involving TKTTF, we assess long-term dispersal trends (>10 years) to obtain quantitative estimates of the risk of exotic P. graminis f. sp. tritici spore transport (of any race) into Ethiopia for different 'what-if' scenarios of disease outbreaks in potential source countries in different months of the wheat season.

What a Difference a Stochastic Process Makes: Epidemiological-Based Real Options Models of Optimal Treatment of Disease

The real options approach has been used within environmental economics to investigate the impact of uncertainty on the optimal timing of control measures to minimise the impacts of invasive species, including pests and diseases. Previous studies typically model the growth in infected area using geometric Brownian motion (GBM). The advantage of this simple approach is that it allows for closed form solutions. However, such a process does not capture the mechanisms underlying the spread of infection. In particular the GBM assumption does not respect the natural upper boundary of the system, which is determined by the maximum size of the host species, nor the deceleration in the rate of infection as this boundary is approached. We show how the stochastic process describing the growth in infected area can be derived from the characteristics of the spread of infection. If the model used does not appropriately capture uncertainty in infection dynamics, then the excessive delay before treatment implies that the full value of the option to treat is not realised. Indeed, when uncertainty is high or the disease is fast spreading, ignoring the mechanisms of infection spread can lead to control never being deployed. Thus the results presented here have important implications for the way in which the real options approach is applied to determine optimal timing of disease control given uncertainty in future disease progression.

The consequence of tree pests and diseases for ecosystem services

Trees and forests provide a wide variety of ecosystem services in addition to timber, food, and other provisioning services. New approaches to pest and disease management are needed that take into account these multiple services and the different stakeholders they benefit, as well as the likelihood of greater threats in the future resulting from globalization and climate change. These considerations will affect priorities for both basic and applied research and how trade and phytosanitary regulations are formulated.

Percolation-based risk index for pathogen invasion: application to soilborne disease in propagation systems

Propagation systems for seedling growth play a major role in agriculture, and in notable cases (such as organic systems), are under constant threat from soil and seedborne fungal plant pathogens such as Rhizoctonia solani or Pythium spp. Yet, to date little is known that links the risk of disease invasion to the host density, which is an agronomic characteristic that can be readily controlled. We introduce here, for the first time in an agronomic system, a percolation framework to analyze the link. We set up an experiment to study the spread of the ubiquitous fungus R. solani in replicated propagation systems with different planting densities, and fit a percolation-based epidemiological model to the data using Bayesian inference methods. The estimated probability of pathogen transmission between infected and susceptible plants is used to calculate the risk of invasion. By comparing the transmission probability and the risk values obtained for different planting densities, we are able to give evidence of a nonlinear relationship between disease invasion and the inter-plant spacing, hence to demonstrate the existence of a spatial threshold for epidemic invasion. The implications and potential use of our methods for the evaluation of disease control strategies are discussed.

Prominent effect of soil network heterogeneity on microbial invasion

Using a network representation for real soil samples and mathematical models for microbial spread, we show that the structural heterogeneity of the soil habitat may have a very significant influence on the size of microbial invasions of the soil pore space. In particular, neglecting the soil structural heterogeneity may lead to a substantial underestimation of microbial invasion. Such effects are explained in terms of a crucial interplay between heterogeneity in microbial spread and heterogeneity in the topology of soil networks. The main influence of network topology on invasion is linked to the existence of long channels in soil networks that may act as bridges for transmission of microorganisms between distant parts of soil.

Time-dependent infectivity and flexible latent and infectious periods in compartmental models of plant disease

Compartmental models have become the dominant theoretical paradigm in mechanistic modeling of plant disease and offer well-known advantages in terms of analytic tractability, ease of simulation, and extensibility. However, underlying assumptions of constant rates of infection and of exponentially distributed latent and infectious periods are difficult to justify. Although alternative approaches, including van der Plank's seminal discrete time model and models based on the integro-differential formulation of Kermack and McKendrick's model, have been suggested for plant disease and relax these unrealistic assumptions, they are challenging to implement and to analyze. Here, we propose an extension to the susceptible, exposed, infected, and removed (SEIR) compartmental model, splitting the latent and infection compartments and thereby allowing time-varying infection rates and more realistic distributions of latent and infectious periods to be represented. Although the model is, in fact, more general, we specifically target plant disease by demonstrating how it can represent both the van der Plank model and the most commonly used variant of the Kermack and McKendrick (K & M) model (in which the infectivity response is delay Gamma distributed). We show how our reformulation retains the numeric and analytic tractability of SEIR models, and how it can be used to replicate earlier analyses of the van der Plank and K & M models. Our reformulation has the advantage of using elementary mathematical techniques, making implementation easier for the nonspecialist. We show a practical implication of these results for disease control. By taking advantage of the easy extensibility characteristic of compartmental models, we also investigate the effects of including additional biological realism. As an example, we show how the more realistic infection responses we consider interact with host demography and lead to divergent invasion thresholds when compared with the "standard" SEIR model. An ever-increasing number of analyses purportedly extract more biologically realistic invasion thresholds by adding additional biological detail to the SEIR model framework; we contend that our results demonstrate that extending a model that has such a simplistic representation of the infection dynamics may not, in fact, lead to more accurate results. Therefore, we suggest that modelers should carefully consider the underlying assumptions of the simplest compartmental models in their future work.

Spatial sampling to detect an invasive pathogen outside of an eradication zone

Invasive pathogens are known to cause major damage to the environments they invade. Effective control of such invasive pathogens depends on early detection. In this paper we focus on sampling with the aim of detecting an invasive pathogen. To that end, we introduce the concept of optimized spatial sampling, using spatial simulated annealing, to plant pathology. It has been mathematically proven (15) that this optimization method converges to the optimum allocation of sampling points that give the largest detection probability. We show the benefits of the method to plant pathology by (i) first illustrating that optimized spatial sampling can easily be applied for disease detection, and then we show that (ii) combining it with a spatially explicit epidemic model, we can develop optimum sample schemes, i.e., optimum locations to sample that maximize the probability of detecting an invasive pathogen. This method is then used as baseline against which other sampling methods can be tested for their accuracy. For the specific example case of this paper, we test (i) random sampling, (ii) stratified sampling as well as (iii) sampling based on the output of the simulation model (using the most frequently infected hosts as sample points), and (iv) sampling the hosts closest to the outbreak point.

Epidemics in networks of spatially correlated three-dimensional root-branching structures

Using digitized images of the three-dimensional, branching structures for root systems of bean seedlings, together with analytical and numerical methods that map a common susceptible-infected-recovered ('SIR') epidemiological model onto the bond percolation problem, we show how the spatially correlated branching structures of plant roots affect transmission efficiencies, and hence the invasion criterion, for a soil-borne pathogen as it spreads through ensembles of morphologically complex hosts. We conclude that the inherent heterogeneities in transmissibilities arising from correlations in the degrees of overlap between neighbouring plants render a population of root systems less susceptible to epidemic invasion than a corresponding homogeneous system. Several components of morphological complexity are analysed that contribute to disorder and heterogeneities in the transmissibility of infection. Anisotropy in root shape is shown to increase resilience to epidemic invasion, while increasing the degree of branching enhances the spread of epidemics in the population of roots. Some extension of the methods for other epidemiological systems are discussed.

Periodicity in host availability does not account for evolutionary branching as observed in many plant pathogens: an application to Gaeumannomyces graminis var. tritici

Periodicity in host availability is common in agricultural systems. Although it is known to have profound effects on plant pathogen abundance, the evolutionary consequences of periodicity for the pathogen population have not previously been analyzed. An epidemiological model incorporating periodic absence of the host crop is combined with the theory of adaptive dynamics to determine whether or not seasonality in host presence plays a role in the occurrence of evolutionary branching, leading to coexisting yet genetically distinct pathogen phenotypes. The study is motivated and illustrated by the specific example of take-all disease of wheat, caused by the pathogen Gaeumannomyces graminis var. tritici, for which two coexisting but genetically distinct types and a trade-off related to seasonality in host presence have been identified. Numerical simulations are used to show that a trade-off between the pathogen transmission rate and the survival of the pathogen between cropping seasons cannot account for the evolutionary branching observed in many pathogens. Model elaborations show that this conclusion holds for a broad range of putative mechanisms. Although the analysis is motivated and illustrated by the specific example of take-all of wheat, the results apply to a broad range of pathogens.

The effect of landscape pattern on the optimal eradication zone of an invading epidemic

A number of high profile eradication attempts on plant pathogens have recently been attempted in response to the increasing number of introductions of economically significant nonnative pathogen species. Eradication programs involve the removal of a large proportion of a host population and can thus lead to significant social and economic costs. In this paper we use a spatially explicit stochastic model to simulate an invading pathogen and show that it is possible to identify an optimal control radius, i.e., one that minimizes the total number of hosts removed during an eradication campaign that is effective in eradicating the pathogen. However, by simulating the epidemic and eradication processes in multiple landscapes, we demonstrate that the optimal radius depends critically on landscape pattern (i.e., the spatial configuration of hosts within the landscape). In particular, we find that the optimal radius, and also the number of host removals associated with it, increases with both the level of aggregation and the density of hosts in the landscape. The result is of practical significance and demonstrates that the location of an invading epidemic should be a key consideration in the design of future eradication strategies.

Optimal strategies for the eradication of asiatic citrus canker in heterogeneous host landscapes

ABSTRACT The eradication of nonnative plant pathogens is a key challenge in plant disease epidemiology. Asiatic citrus canker is an economically significant disease of citrus caused by the bacterial plant pathogen Xanthomonas citri subsp. citri. The pathogen is a major exotic disease problem in many citrus producing areas of the world including the United States, Brazil, and Australia. Various eradication attempts have been made on the disease but have been associated with significant social and economic costs due to the necessary removal of large numbers of host trees. In this paper, a spatially explicit stochastic simulation model of Asiatic citrus canker is introduced that describes an epidemic of the disease in a heterogeneous host landscape. We show that an optimum eradication strategy can be determined that minimizes the adverse costs associated with eradication. In particular, we show how the optimum strategy and its total cost depend on the topological arrangement of the host landscape. We discuss the implications of the results for invading plant disease epidemics in general and for historical and future eradication attempts on Asiatic citrus canker.

Epidemiological analysis of take-all decline in winter wheat

Take-all dynamics within crops differing in cropping history (the number of previous consecutive wheat crops) were analyzed using an epidemiological model to determine the processes affected during take-all decline. The model includes terms for primary infection, secondary infection, inoculum decay, and root growth. The average rates of root production did not vary with cropping history. The force of primary infection increased from a low level in 1st wheat crops, to a maximum in 2nd to 4th wheat crops, and then to intermediate levels thereafter. The force of secondary infection was low but increased steadily during the season in first wheat crops, was delayed but rose and fell sharply in 2nd to 4th wheat crops, and for 5th and 7th wheat crops returned to similar dynamics as that for 1st wheat crops. Chemical seed treatment with silthiofam had no consistent effect on the take-all decline process. We conjecture that these results are consistent with (i) low levels of particulate inoculum prior to the first wheat crop leading to low levels of primary infection, low levels of secondary infection, and little disease suppression; (ii) net amplification of inoculum during the first wheat crop and intercrop period; (iii) increased levels of primary and secondary infection in subsequent crops, but higher levels of disease suppression; and (iv) an equilibrium between the pathogen and antagonist populations by the 5th wheat, reflected by lower overall rates of primary infection, secondary infection, disease suppression and hence, disease severity.

Constructing the effect of alternative intervention strategies on historic epidemics

Data from historical epidemics provide a vital and sometimes under-used resource from which to devise strategies for future control of disease. Previous methods for retrospective analysis of epidemics, in which alternative interventions are compared, do not make full use of the information; by using only partial information on the historical trajectory, augmentation of control may lead to predictions of a paradoxical increase in disease. Here we introduce a novel statistical approach that takes full account of the available information in constructing the effect of alternative intervention strategies in historic epidemics. The key to the method lies in identifying a suitable mapping between the historic and notional outbreaks, under alternative control strategies. We do this by using the Sellke construction as a latent process linking epidemics. We illustrate the application of the method with two examples. First, using temporal data for the common human cold, we show the improvement under the new method in the precision of predictions for different control strategies. Second, we show the generality of the method for retrospective analysis of epidemics by applying it to a spatially extended arboreal epidemic in which we demonstrate the relative effectiveness of host culling strategies that differ in frequency and spatial extent. Some of the inferential and philosophical issues that arise are discussed along with the scope of potential application of the new method.

A fungal growth model fitted to carbon-limited dynamics of Rhizoctonia solani

Here, a quasi-steady-state approximation was used to simplify a mathematical model for fungal growth in carbon-limiting systems, and this was fitted to growth dynamics of the soil-borne plant pathogen and saprotroph Rhizoctonia solani. The model identified a criterion for invasion into carbon-limited environments with two characteristics driving fungal growth, namely the carbon decomposition rate and a measure of carbon use efficiency. The dynamics of fungal spread through a population of sites with either low (0.0074 mg) or high (0.016 mg) carbon content were well described by the simplified model with faster colonization for the carbon-rich environment. Rhizoctonia solani responded to a lower carbon availability by increasing the carbon use efficiency and the carbon decomposition rate following colonization. The results are discussed in relation to fungal invasion thresholds in terms of carbon nutrition.

Parameter estimation and prediction for the course of a single epidemic outbreak of a plant disease

Many epidemics of plant diseases are characterized by large variability among individual outbreaks. However, individual epidemics often follow a well-defined trajectory which is much more predictable in the short term than the ensemble (collection) of potential epidemics. In this paper, we introduce a modelling framework that allows us to deal with individual replicated outbreaks, based upon a Bayesian hierarchical analysis. Information about 'similar' replicate epidemics can be incorporated into a hierarchical model, allowing both ensemble and individual parameters to be estimated. The model is used to analyse the data from a replicated experiment involving spread of Rhizoctonia solani on radish in the presence or absence of a biocontrol agent, Trichoderma viride. The rate of primary (soil-to-plant) infection is found to be the most variable factor determining the final size of epidemics. Breakdown of biological control in some replicates results in high levels of primary infection and increased variability. The model can be used to predict new outbreaks of disease based upon knowledge from a 'library' of previous epidemics and partial information about the current outbreak. We show that forecasting improves significantly with knowledge about the history of a particular epidemic, whereas the precision of hindcasting to identify the past course of the epidemic is largely independent of detailed knowledge of the epidemic trajectory. The results have important consequences for parameter estimation, inference and prediction for emerging epidemic outbreaks.

Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops; a theoretical assessment

Viral diseases are a key constraint in the production of staple food crops in lesser developed countries. New and improved disease control methods are developed and implemented without consideration of the selective pressure they impose on the virus. In this paper, we analyse the evolution of within-plant virus titre as a response to the implementation of a range of disease control methods. We show that the development of new and improved disease control methods for viral diseases of vegetatively propagated staple food crops ought to take the evolutionary responses of the virus into consideration. Not doing so leads to a risk of failure, which can result in considerable economic losses and increased poverty. Specifically in vitro propagation, diagnostics and breeding methods carry a risk of failure due to the selection for virus strains that build up a high within-plant virus titre. For vegetatively propagated crops, sanitation by roguing has a low risk of failure owing to its combination of selecting for low virus titre strains as well as increasing healthy crop density.

Large-scale fungicide spray heterogeneity and the regional spread of resistant pathogen strains

ABSTRACT Most models for the spread of fungicide resistance in plant pathogens are focused on within-field dynamics, yet regional invasion depends upon the interactions between field populations. Here, we use a spatially implicit metapopulation model to describe the dynamics of regional spread, in which subpopulations correspond to single fields. We show that the criterion for the regional invasion of pathogens between fields differs from that for invasion within fields. That is, the ability of a fungicide-resistant strain of a pathogen to invade a field population does not necessarily imply an ability to spread through many fields at the regional scale. This depends upon an interaction between the fraction of fields that is sprayed and the reproductive capacity of the pathogen. This result is of practical significance and indicates that resistance management strategies which currently target within-field processes, such as the use of mixtures and alternations of fungicides, may be more effective if between-field processes also were targeted; for example, through the restricted deployment of fungicides over large areas. We also show that the fraction of disease-free fields is maximized when the proportion of fields that is sprayed is just below the threshold for invasion of the resistant strain.

An Epidemiological Analysis of the Role of Disease-Induced Root Growth in the Differential Response of Two Cultivars of Winter Wheat to Infection by Gaeumannomyces graminis var. tritici

ABSTRACT Epidemiological modeling combined with parameter estimation of experimental data was used to examine differences in the contribution of disease-induced root production to the spread of take-all on plants of two representative yet contrasting cultivars of winter wheat, Ghengis and Savannah. A mechanistic model, including terms for primary infection, secondary infection, inoculum decay, and intrinsic and disease-induced root growth, was fitted to data describing changes in the numbers of infected and susceptible roots over time at a low or high density of inoculum. Disease progress curves were characterized by consecutive phases of primary and secondary infection. No differences in root growth were detected between cultivars in the absence of disease and root production continued for the duration of the experiment. However, significant differences in disease-induced root production were detected between Savannah and Genghis. In the presence of disease, root production for both cultivars was characterized by stimulation when few roots were infected and inhibition when many roots were infected. At low inoculum density, the transition from stimulation to inhibition occurred when an average of 5.0 and 9.0 roots were infected for Genghis and Savannah, respectively. At high inoculum density, the transition from stimulation to inhibition occurred when an average of 4.5 and 6.7 roots were infected for Genghis and Savannah, respectively. Differences in the rates of primary and secondary infection between Savannah and Genghis also were detected. At a low inoculum density, Genghis was marginally more resistant to secondary infection whereas, at a high density of inoculum, Savannah was marginally more resistant to primary infection. The combined effects of differences in disease-induced root growth and differences in the rates of primary and secondary infection meant that the period of stimulated root production was extended by 7 and 15 days for Savannah at a low and high inoculum density, respectively. The contribution of this form of epidemiological modeling to the better management of take-all is discussed.

The role of initial inoculum on epidemic dynamics

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.

Extinction of epidemics in lattice models with quenched disorder

The extinction of the contact process for epidemics in lattice models with quenched disorder is analyzed in the limit of small density of infected sites. It is shown that the problem in such a regime can be mapped to the quantum-mechanical one characterized by the Anderson Hamiltonian for an electron in a random lattice. It is demonstrated both analytically (self-consistent mean field) and numerically (by direct diagonalization of the Hamiltonian and by means of cellular automata simulations) that disorder enhances the contact process, given the mean values of random parameters are not influenced by disorder.

Small-scale fungicide spray heterogeneity and the coexistence of resistant and sensitive pathogen strains

ABSTRACT Empirical evidence indicates that fungicide-resistant and sensitive strains can coexist for prolonged periods. Coexistence has important practical implications, for example, for the posttreatment recovery of sensitivity and consequently the life expectancy of fungicide products. Despite this, the factors influencing coexistence remain relatively unexplored. Ecological studies have shown that environmental heterogeneity can facilitate the coexistence of different species and subspecific groups. Here we use a simple differential equation model and show that fungicide spray heterogeneity per se is not sufficient for coexistence but that the outcome depends crucially on the competitive relationship between resistant and sensitive strains. The model incorporates the competition between resistant and sensitive pathogen strains for a limited supply of susceptible host tissue on a crop which has received an incomplete coverage of fungicide. We use a combination of invasibility analysis and model simulations to explore the conditions under which coexistence can occur. We further show that the maximum density of healthy host tissue isrealized when resistant and sensitive pathogen strains coexist. A set of key influencing parameters are identified and analyzed, and the consequences of the results for disease and resistance management are discussed.

Epidemiology and chemical control of take-all on seminal and adventitious roots of wheat

ABSTRACT Epidemiological modeling is used to examine the effect of silthiofam seed treatment on field epidemics of take-all in winter wheat. A simple compartmental model, including terms for primary infection, secondary infection, root production, and decay of inoculum, was fitted to data describing change in the number of diseased and susceptible roots per plant over thermal time obtained from replicated field trials. This produced a composite curve describing change in the proportion of diseased roots over time that increased monotonically to an initial plateau and then increased exponentially thereafter. The shape of this curve was consistent with consecutive phases of primary and secondary infection. The seed treatment reduced the proportion of diseased roots throughout both phases of the epidemic. However, analysis with the model detected a significant reduction in the rate of primary, but not secondary, infection. The potential for silthiofam to affect secondary infection from diseased seminal or adventitious roots was examined in further detail by extending the compartmental model and fitting to change in the number of diseased and susceptible seminal or adventitious roots. Rates of secondary infection from either source of infected roots were not affected. Seed treatment controlled primary infection of seminal roots from particulate inoculum but not secondary infection from either seminal or adventitious roots. The reduction in disease for silthiofam-treated plants observed following the secondary infection phase of the epidemic was not due to long-term activity of the chemical but to the manifestation of disease control early in the epidemic.

Transmission rates and adaptive evolution of pathogens in sympatric heterogeneous plant populations

Diversification in agricultural cropping patterns is widely practised to delay the build-up of virulent races that can overcome host resistance in pathogen populations. This can lead to balanced polymorphism, but the long-term consequences of this strategy for the evolution of crop pathogen populations are still unclear. The widespread occurrence of sibling species and reproductively isolated sub-species among fungal and oomycete plant pathogens suggests that evolutionary divergence is common. This paper develops a mathematical model of host-pathogen interactions using a simple framework of two hosts to analyse the influences of sympatric host heterogeneity on the long-term evolutionary behaviour of plant pathogens. Using adaptive dynamics, which assumes that sequential mutations induce small changes in pathogen fitness, we show that evolutionary outcomes strongly depend on the shape of the trade-off curve between pathogen transmission on sympatric hosts. In particular, we determine the conditions under which the evolutionary branching of a monomorphic into a dimorphic population occurs, as well as the conditions that lead to the evolution of specialist (single host range) or generalist (multiple host range) pathogen populations.

Controlling disease spread on networks with incomplete knowledge

Models for control of highly infectious diseases on local, small-world, and scale-free networks are considered, with only partial information accessible about the status of individuals and their connections. We consider a case when individuals can be infectious before showing symptoms and thus before detection. For small to moderately severe incidence of infection with a small number of nonlocal links, it is possible to control disease spread by using purely local methods applied in a neighborhood centered around a detected infectious individual. There exists an optimal radius for such a control neighborhood leading to the lowest severity of the epidemic in terms of economic costs associated with disease and treatment. The efficiency of a local control strategy is very sensitive to the choice of the radius. Below the optimal radius, the local strategy is unsuccessful; the disease spreads throughout the system, necessitating treatment of the whole population. At the other extreme, a strategy involving a neighborhood that is too large controls the disease but is wasteful of resources. It is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated.

On ?Analytical models for the patchy spread of plant disease?

Epidemiologists are interested in using models that incorporate the effects of clustering in the spatial pattern of disease on epidemic dynamics. Bolker (1999, Bull. Math. Biol. 61, 849-874) has developed an approach to study such models based on a moment closure assumption. We show that the assumption works above a threshold initial level of disease that depends on the spatial dispersal of the pathogen. We test an alternative assumption and show that it does not have this limitation. We examine the relation between lattice and continuous-medium implementations of the approach.

Bayesian analysis of botanical epidemics using stochastic compartmental models

A stochastic model for an epidemic, incorporating susceptible, latent, and infectious states, is developed. The model represents primary and secondary infection rates and a time-varying host susceptibility with applications to a wide range of epidemiological systems. A Markov chain Monte Carlo algorithm is presented that allows the model to be fitted to experimental observations within a Bayesian framework. The approach allows the uncertainty in unobserved aspects of the process to be represented in the parameter posterior densities. The methods are applied to experimental observations of damping-off of radish (Raphanus sativus) caused by the fungal pathogen Rhizoctonia solani, in the presence and absence of the antagonistic fungus Trichoderma viride, a biological control agent that has previously been shown to affect the rate of primary infection by using a maximum-likelihood estimate for a simpler model with no allowance for a latent period. Using the Bayesian analysis, we are able to estimate the latent period from population data, even when there is uncertainty in discriminating infectious from latently infected individuals in data collection. We also show that the inference that T. viride can control primary, but not secondary, infection is robust to inclusion of the latent period in the model, although the absolute values of the parameters change. Some refinements and potential difficulties with the Bayesian approach in this context, when prior information on parameters is lacking, are discussed along with broader applications of the methods to a wide range of epidemiological systems.

Modeling and analysis of disease-induced host growth in the epidemiology of take-all

ABSTRACT Epidemiological modeling, together with parameter estimation to experimental data, was used to examine the contribution of disease-induced root growth to the spread of take-all in wheat. Production of roots from plants grown in the absence of disease was compared with production of those grown in the presence of disease and the precise form of diseaseinduced growth was examined by fitting a mechanistic model to data describing change in the number of infected and susceptible roots over time from a low and a high density of inoculum. During the early phase of the epidemic, diseased plants produced more roots than their noninfected counterparts. However, as the epidemic progressed, the rate of root production for infected plants slowed so that by the end of the epidemic, and depending on inoculum density, infected plants had fewer roots than uninfected plants. The dynamical change in the numbers of infected and susceptible roots over time could only be explained by the mechanistic model when allowance was made for disease-induced root growth. Analysis of the effect of disease-induced root production on the spread of disease using the model suggests that additional roots produced early in the epidemic serve only to reduce the proportion of diseased roots. However, as the epidemic switches from primary to secondary infection, these roots perform an active role in the transmission of disease. Some consequence of disease-induced root growth for field epidemics is discussed.

Inferring the dynamics of a spatial epidemic from time-series data

Spatial interactions are key determinants in the dynamics of many epidemiological and ecological systems; therefore it is important to use spatio-temporal models to estimate essential parameters. However, spatially-explicit data sets are rarely available; moreover, fitting spatially-explicit models to such data can be technically demanding and computationally intensive. Thus non-spatial models are often used to estimate parameters from temporal data. We introduce a method for fitting models to temporal data in order to estimate parameters which characterise spatial epidemics. The method uses semi-spatial models and pair approximation to take explicit account of spatial clustering of disease without requiring spatial data. The approach is demonstrated for data from experiments with plant populations invaded by a common soilborne fungus, Rhizoctonia solani. Model inferences concerning the number of sources of disease and primary and secondary infections are tested against independent measures from spatio-temporal data. The applicability of the method to a wide range of host-pathogen systems is discussed.

Epidemiological dynamics and the efficiency of biological control of soil-borne disease during consecutive epidemics in a controlled environment

•  A combination of experimentation and modelling is used to examine the role of epidemiological dynamics on the production and infectivity of inoculum and the efficiency of biocontrol by Trichoderma viride during consecutive epidemics of damping-off disease caused by the pathogen Rhizoctonia solani in crops of radish. •  Changes in the net infectivity of inoculum at the beginning of first and second crops caused a switch in epidemiological dynamics. Epidemics of first crops were dominated by secondary infection leading to amplification of inoculum so that epidemics of second crops were overwhelmingly determined by primary infection. •  The biocontrol agent reduced primary infection and hence parasitic amplification of inoculum in both first and second crops but the efficiency of control dropped from 91.7% in first crops to 64.8% in second crops, with sudden outbreaks of disease in second crops which had previously been disease-free. •  We conclude that parasitic amplification can cause a rapid build-up of disease and inoculum over consecutive crops, leading to loss in the efficiency of biocontrol. This form of inoculum production is supplemented by saprotrophic infestation which can result in sudden outbreaks of disease in protected crops where control of disease had previously been fully successful.

Response of a deterministic epidemiological system to a stochastically varying environment

Fluctuations in the natural environment introduce variability into the biological systems that exist within them. In this paper, we develop a model for the influence of random fluctuations in the environment on a simple epidemiological system. The model describes the infection of a dynamic host population by an environmentally sensitive pathogen and is based on the infection of sugar beet plants by the endoparasitic slime-mold vector Polymyxa betae. The infection process is switched on only when the temperature is above a critical value. We discuss some of the problems inherent in modeling such a system and analyze the resulting model by using asymptotic techniques to generate closed-form solutions for the mean and variance of the net amount of new inoculum produced within a season. In this way, the variance of temperature profile can be linked with that of the inoculum produced in a season and hence the risk of disease. We also examine the connection between the model developed in this paper and discrete Markov-chain models for weather.

Measures of durability of resistance

ABSTRACT Conventional models for the durability of resistant cultivars focus on the dynamics of the frequency of resistance genes. This leads to a definition of the durability of resistance as the time from introduction of the cultivar to the time when the frequency of the virulence gene reaches a preset threshold. It is questionable whether this is the most appropriate way to measure durability. Here we use a simple epidemiological model to link population dynamics and population genetics to compare three measures of durability: (i) the expected time until invasion of the virulent genotype, by mutation or immigration, and subsequent establishment of a population (T(invasion)); (ii) the virulence frequency related measure of the time for the virulent genotype to take-over the pathogen population ( T(take-over)); and (iii) the additional yield, measured by the additional number of uninfected host growth days (T(additional)). Specifically, we show how the measures of durability are affected by deployment and epidemiological parameters. We use a combination of numerical solution and analytical approximation of a model for the population dynamics of avirulent and virulent genotypes of a pathogen growing in dynamically changing populations of resistant and susceptible cultivars. The three measures of durability are compared. Some consequences of the results for durable resistance in multilines and mixtures and the regional deployment of resistant cultivars are discussed.

Solution of epidemic models with quenched transients

We consider a model for single-season disease epidemics, with a delay (latent period) in the onset of infectivity and a decay ("quenching") in host susceptibility described by time-varying rates of primary and secondary infections. The classical susceptible-exposed-infected (SEI) model of epidemiology is a special case with constant rates. The decaying rates force the epidemics to slow down, and eventually stop in a "quenched transient" state that depends on the full history of the epidemic including its initial state. This equilibrium state is neutrally stable (i.e., has zero-value eigenvalues), and cannot be studied using standard equilibrium analysis. We introduce a method that gives an approximate analytical solution for the quenched state. The method uses an interpolation between two exactly solvable limits and applies to the whole, five-dimensional parameter space of the model. Some applications of the solutions for analysis of epidemics are given.

The effect of cultivation on the size, shape, and persistence of disease patches in fields

Epidemics of soil-borne plant disease are characterized by patchiness because of restricted dispersal of inoculum. The density of inoculum within disease patches depends on a sequence comprising local amplification during the parasitic phase followed by dispersal of inoculum by cultivation during the intercrop period. The mechanisms that control size, shape, and persistence have received very little rigorous attention in epidemiological theory. Here we derive a model for dispersal of inoculum in soil by cultivation that takes account into the discrete stochastic nature of the system in time and space. Two parameters, probability of movement and mean dispersal distance, characterize lateral dispersal of inoculum by cultivation. The dispersal parameters are used in combination with the characteristic area and dimensions of host plants to identify criteria that control the shape and size of disease patches. We derive a critical value for the probability of movement for the formation of cross-shaped patches and show that this is independent of the amount of inoculum. We examine the interaction between local amplification of inoculum by parasitic activity and subsequent dilution by dispersal and identify criteria whereby asymptomatic patches may persist as inoculum falls below a threshold necessary for symptoms to appear in the subsequent crop. The model is motivated by the spread of rhizomania, an economically important soil-borne disease of sugar beet. However, the results have broad applicability to a very wide range of diseases that survive as discrete units of inoculum. The application of the model to patch dynamics of weed seeds and local introductions of genetically modified seeds is also discussed.

Bubonic plague: a metapopulation model of a zoonosis

Bubonic plague (Yersinia pestis) is generally thought of as a historical disease; however, it is still responsible for around 1000-3000 deaths each year worldwide. This paper expands the analysis of a model for bubonic plague that encompasses the disease dynamics in rat, flea and human populations. Some key variables of the deterministic model, including the force of infection to humans, are shown to be robust to changes in the basic parameters, although variation in the flea searching efficiency, and the movement rates of rats and fleas will be considered throughout the paper. The stochastic behaviour of the corresponding metapopulation model is discussed, with attention focused on the dynamics of rats and the force of infection at the local spatial scale. Short-lived local epidemics in rats govern the invasion of the disease and produce an irregular pattern of human cases similar to those observed. However, the endemic behaviour in a few rat subpopulations allows the disease to persist for many years. This spatial stochastic model is also used to identify the criteria for the spread to human populations in terms of the rat density. Finally, the full stochastic model is reduced to the form of a probabilistic cellular automaton, which allows the analysis of a large number of replicated epidemics in large populations. This simplified model enables us to analyse the spatial properties of rat epidemics and the effects of movement rates, and also to test whether the emergent metapopulation behaviour is a property of the local dynamics rather than the precise details of the model.

Metapopulation dynamics of bubonic plague

Bubonic plague is widely regarded as a disease of mainly historical importance; however, with increasing reports of incidence and the discovery of antibiotic-resistant strains of the plague bacterium Yersinia pestis, it is re-emerging as a significant health concerns. Here we bypass the conventional human-disease models, and propose that bubonic plague is driven by the dynamics of the disease in the rat population. Using a stochastic, spatial metapopulation model, we show that bubonic plague can persist in relatively small rodent populations from which occasional human epidemics arise, without the need for external imports. This explains why historically the plague persisted despite long disease-free periods, and how the disease re-occurred in cities with tight quarantine control. In a contemporary setting, we show that human vaccination cannot eradicate the plague, and that culling of rats may prevent or exacerbate human epidemics, depending on the timing of the cull. The existence of plague reservoirs in wild rodent populations has important public-health implications for the transmission to urban rats and the subsequent risk of human outbreaks.

Population dynamics of plant-parasite interactions: thresholds for invasion

Thresholds are derived for the invasion of plant populations by parasites. The theory is developed for a generic model that takes into account two features characteristic of plant-parasite interactions: a dual source of inoculum (infection from primary or externally introduced inoculum and secondary infection from contact between susceptible and infected host tissue) and a host response to infection load. Each of the threshold criteria is shown to be the sum of the individual components for primary and secondary infection. This indicates that if parasite invasion is not possible through primary or secondary infection alone, when the two modes of transmission are combined, the parasite may be able to invade. The invasion criteria demonstrate that there is a threshold population of susceptible hosts below which the parasite is unable to invade. If there are nonlinearities in the population dynamics (arising through either the transmission process or the host response), there are also threshold densities for the infected hosts and parasite populations below which invasion does not occur. The implications of the results for the control of plant disease are discussed.

Quantitative analysis and model simplification of an epidemic model with primary and secondary infection

Models of particular epidemiological systems can rapidly become complicated by biological detail which can obscure their essential features and behaviour. In general, we wish to retain only those components and processes that contribute to the dynamics of the system. In this paper, we apply asymptotic techniques to an SEI-type model with primary and secondary infection in order to reduce it to a much simpler form. This allows the identification of parameter groupings discriminating between regions of contrasting dynamics and leads to simple approximations for the model's transient behaviour. These can be used to follow the evolution of the developing infection process. The techniques examined in this paper will be applicable to a large number of similar models.

Invasion thresholds for fungicide resistance: deterministic and stochastic analyses

Fungicide resistance is an important practical problem, but one that is poorly understood at the population level. Here we introduce a simple nonlinear model for fungicide resistance in botanical epidemics which includes the dynamics of the chemical control agent and the host population, while also allowing for demographic stochasticity in the host-parasite dynamics. This provides a mathematical framework for analysing the risk of fungicide resistance developing by including the parameters for the amount applied, longevity and application frequency of the fungicide. The model demonstrates the existence of thresholds for the invasion of the resistant strain in the parasite population which depend on two quantities: the relative fitness of the resistant strain and the effectiveness of control. This threshold marks a change from definite elimination of the resistant strain below the threshold to a finite probability of invasion which increases above the threshold. The fungicide decay rate, the amount of fungicide applied and the period between applications affect the effectiveness of control and, consequently, they influence whether or not resistance develops and the time taken to achieve a critical frequency of resistance. All three parameters are amenable to control by the grower or by coordinating the activity of a population of growers. Providing crude estimates of the effectiveness of control and relative fitness are available, the results can be used to predict the consequences of changing these parameters for the risk of invasion and the proportion of sites at which this might be expected to occur. Although motivated for fungicide resistance, the model has broader application to herbicide, antibiotic and antiviral resistance. The modelling approach and results are discussed in the context of resistance to chemical control in general.

Design of an intravaginal ring for the controlled delivery of 17 beta-estradiol as its 3-acetate ester

Suitable ester prodrugs of 17beta-estradiol are identified, thus permitting effective sustained and controlled estrogen replacement therapy (ERT) from an elastomeric, silicone intravaginal ring (IVR). IVR devices of reservoir design were prepared by blending silicone elastomer base with n-propylorthosilicate (cross-linker) and 10% w/w of 17beta-estradiol or an ester prodrug, the mix being activated with 0.5% w/w stannous octoate and cured at 80 degrees C for 2 min. A rate-controlling membrane was similarly prepared, without the active agent. IVR devices were of cross-sectional diameter 9 mm, outer diameter 54 mm, with core cross-sectional diameter of 2 mm and core length varied as required. Sink conditions were evident for the 17beta-estradiol esters in 1.0% aqueous benzalkonium chloride solution. The low release rates into 0.9% w/v saline of the lipophilic valerate and benzoate esters were due to their intrinsically low aqueous solubilities. In vivo, these esters failed to raise plasma estradiol above baseline levels in postmenopausal human volunteers, despite good in vitro release characteristics under sink conditions. The best release rates under sink conditions, in combination with substantial aqueous solubilities as indicated by the release rates into saline, were observed for the acetate and propionate esters. A combination of drug release characteristics, short plasma half-life and a toxicologically acceptable hydrolysis product indicated that 17beta-estradiol-3-acetate was the prodrug of choice for IVR delivery of ERT. In vivo, an IVR device releasing 100 microg/day of estradiol as its 3-acetate ester maintained over 84 days a circulating plasma concentration in the region of 300 pmoll(-1), within the clinically desirable range for ERT.

Predicting variability in biological control of a plant-pathogen system using stochastic models

A stochastic model for the dynamics of a plant-pathogen interaction is developed and fitted to observations of the fungal pathogen Rhizoctonia solani (Kühn) in radish (Raphanus sativus L.), in both the presence and absence of the antagonistic fungus Trichoderma viride (Pers ex Gray). The model incorporates parameters for primary and secondary infection mechanisms and for characterizing the time-varying susceptibility of the host population. A parameter likelihood is developed and used to fit the model to data from microcosm experiments. It is shown that the stochastic model accounts well for observed variability both within and between treatments. Moreover, it enables us to describe the time evolution of the probability distribution for the variability among replicate epidemics in terms of the underlying epidemiological parameters for primary and secondary infection and decay in susceptibility. Consideration of profile likelihoods for each parameter provides strong evidence that T. viride mainly affects primary infection. By using the stochastic model to study the dependence of the probability distribution of disease levels on the primary infection rate we are therefore able to predict the effectiveness of a widely used biological control agent.

The role of sub-milankovitch climatic forcing in the initiation of the northern hemisphere glaciation

Mechanisms responsible for the initiation of major glaciation in the Northern Hemisphere at about 2.75 million years ago are poorly understood. A laminated terrestrial sequence from Pula maar, Hungary, containing about 320,000 years in annual layers between 3.05 and 2. 60 million years ago, provides a detailed record of rates of climatic change across this dramatic transition. An analysis of the record implies that climatic variations at sub-Milankovitch frequencies (less than or equal to 15,000 years) were an important driving force during this transitional interval and that, as the threshold was approached, these increased in frequency and amplitude, possibly providing the final trigger for the amplification of Northern Hemisphere ice sheets.