Optimal control of a vectored plant disease model for a crop with continuous replanting.

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.

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 Risk Assessment Framework for Seed Degeneration: Informing an Integrated Seed Health Strategy for Vegetatively Propagated Crops.

Pathogen buildup in vegetative planting material, termed seed degeneration, is a major problem in many low-income countries. When smallholder farmers use seed produced on-farm or acquired outside certified programs, it is often infected. We introduce a risk assessment framework for seed degeneration, evaluating the relative performance of individual and combined components of an integrated seed health strategy. The frequency distribution of management performance outcomes was evaluated for models incorporating biological and environmental heterogeneity, with the following results. (1) On-farm seed selection can perform as well as certified seed, if the rate of success in selecting healthy plants for seed production is high; (2) when choosing among within-season management strategies, external inoculum can determine the relative usefulness of 'incidence-altering management' (affecting the proportion of diseased plants/seeds) and 'rate-altering management' (affecting the rate of disease transmission in the field); (3) under severe disease scenarios, where it is difficult to implement management components at high levels of effectiveness, combining management components can be synergistic and keep seed degeneration below a threshold; (4) combining management components can also close the yield gap between average and worst-case scenarios. We also illustrate the potential for expert elicitation to provide parameter estimates when empirical data are unavailable. [Formula: see text] Copyright © 2017 The Author(s). This is an open access article distributed under the CC BY-NC-ND 4.0 International license .

Theoretical modeling suggests that synergy may result from combined use of two biocontrol agents for controlling foliar pathogens under spatial heterogeneous conditions.

There has been a trend for combined use of several biocontrol agents (BCAs) with an expectation of synergistic interactions among BCAs. However, previous modeling studies suggested that, under homogeneous and temporal-fluctuating conditions, combined use of two BCAs, in most cases, only results in efficacies similar to the more efficacious one used alone; a result consistent with published experimental data. The present modeling study investigated whether combined use of two mycoparasitic BCAs, two competitive BCAs, or a mycoparasitic and a competitive BCA leads to synergistic interactions under spatially heterogeneous conditions. In the model, there were two patches with varying relative sizes and two BCAs differentially adapted to the two patches. Within the range of model parameter values considered, combined use of two BCAs is more effective than the more efficacious BCA used alone in 72% of the simulated cases. There was also a considerable proportion (≈21%) of model simulations in which combined use of two BCAs led to synergy (i.e., efficacy was greater than expected under the assumption of Bliss independence, especially when each of the two BCAs can only survive in one [different] patch). Combined use of a mycoparasitic BCA with a competitive one is more likely to result in synergy than the other two BCA combinations. When biocontrol activities of individual BCAs are low or moderate, biocontrol efficacy arising from combined use of two BCAs does not depend greatly on biocontrol mechanisms. However, for high BCA activities, combined use with at least one competitive BCA resulted in better control than combined use of two mycoparasitic BCAs. The present modeling study emphasized the need for understanding the degree of spatial patchiness and quantitative relationships between biocontrol activities and external conditions in order to apply commercial BCAs effectively.

Combined use of two biocontrol agents with different biocontrol mechanisms most likely results in less than expected efficacy in controlling foliar pathogens under fluctuating conditions: a modeling study.

Effective use of biocontrol agents (BCAs) is a potentially important component of sustainable agriculture; recently, there has been a trend for combined use of several BCAs, with an expectation of synergistic interactions among them. A previous numerical study suggested that, under homogenous conditions in which two BCAs occupied the same host tissue as the pathogen, combined use of two BCAs with different biocontrol mechanisms resulted, in most cases, in efficacies similar to using the more efficacious one alone; this result is consistent with published experimental results. The present study investigates whether combined use of a mycoparasitic and a competitive BCA leads to greater efficacy than that expected when the model is modified to allow for fluctuating temperature regimes and the effects of temperature on the pathogen and BCAs. Within the range of parameter values considered, combined use of two BCAs is shown to be less effective than that expected under the assumption of Bliss independence, and to result in a level of efficacy similar to that achieved by the more efficacious component used alone, indicating antagonistic interactions between the two BCAs. Nevertheless, combined use of two BCAs resulted in a slightly longer delay in epidemic development than did individual use of BCAs. Stochastic variability in simulated hourly temperatures did not result in a high level of variability in efficacy among replicates; nevertheless, the among-replicate variability appeared to be greater for the combined use of BCAs than for individual BCAs used alone. In contrast, there were greater effects of varying BCA-temperature relationships and application time (reflected in the temperature profile) on efficacy, suggesting the importance of characterizing the relationship between BCA activity and environmental conditions in future research.

Impact of the spatial scale of grass-legume mixtures on sheep grazing behaviour, preference and intake, and subsequent effects on pasture.

The benefits of using white clover in pastures have been widely recognised for many years. However, clover is perceived as being unreliable because of its typically low content, which is spatially and temporally variable, in mixed pastures. One proposed solution to increase the proportion of clover in the diet of grazing animals and composition in the pasture is to spatially separate clover from grass within the same field. In a field experiment ryegrass and white clover were grown in fine mixtures, and in pure alternating strips of ryegrass and clover of 1.5 m, 3 m or 18 m width within a field. Pastures were grazed for two grazing periods of 9 and 12 weeks, and measurements of sward surface height (SSH), herbage mass and composition and clover morphology were taken. Grazing behaviour was also observed. Results showed that spatial separation in the long term, when compared with a fine mixture, increased clover availability (18% to 30% v. 9%, based on standing dry matter) and was not grazed to extinction. Ewes maintained their preference for clover throughout the experiment (selection coefficient 2 to 5), which resulted in a reduction in the SSH of clover in monocultures to <3 cm and significant changes to the morphology of clover (smaller leaves, shorter petioles and thicker stolon), at the expense of maximising their intake. Spatial separation in the short term may therefore allow grazing animals to select their preferred diet; however, in the long term in continuously grazed pasture, their preference for clover depletes its availability.

Modelling virus- and host-limitation in vectored plant disease epidemics.

Models of plant virus epidemics have received less attention than those caused by fungal pathogens. Intuitively, the fact that virus diseases are systemic means that the individual diseased plant can be considered as the population unit which simplifies modelling. However, the fact that a vector is required in the vast majority of cases for virus transmission, means that explicit consideration must be taken of the vector, or, the involvement of the vector in the transmission process must be considered implicitly. In the latter case it is also important that within-plant processes, such as virus multiplication and systemic movement, are taken into account. In this paper we propose an approach based on the linking of transmission at the population level with virus multiplication within plants. The resulting models are parameter-sparse and hence simplistic. However, the range of model outcomes is representative of field observations relating to the apparent limitation of epidemic development in populations of healthy susceptible plants. We propose that epidemic development can be constrained by virus limitation in the early stages of an epidemic when the availability of healthy susceptible hosts is not limiting. There is an inverse relationship between levels of transmission in the population and the mean virus titre/infected plant. In the case of competition between viruses, both virus and host limitation are likely to be important in determining whether one virus can displace another or whether both viruses can co-exist in a plant population. Lotka-Volterra type equations are derived to describe density-dependent competition between two viruses multiplying within plants, embedded within a population level epidemiological model. Explicit expressions determining displacement or co-existence of the viruses are obtained. Unlike the classical Lotka-Volterra competition equations, the co-existence requirement for the competition coefficients to be both less than 1 can be relaxed.

Interactions in a host plant-virus-vector-parasitoid system: modelling the consequences for virus transmission and disease dynamics.

A full understanding of plant virus epidemiology requires studies at different scales of integration: from within-plant cell processes to vector population dynamics, behaviour and broader ecological interactions. Vectors respond to cues derived from plants (both healthy and virus-infected), from natural enemies and from other environmental influences, and these directly affect the temporal and spatial patterns of disease development. The key element in linking these scales is the transmission process and the determining factors involved. We use a mathematical model to show how the presence of natural enemies, by increasing virus transmission, can increase the rate of virus disease development while at the same time reducing vector population size, supporting recent empirical evidence obtained in microcosm studies. The implication of this work is that biological control of arthropod pests, which are also virus vectors, using parasitoid wasps, may have unanticipated and negative effects in terms of increased incidence of virus disease.

Combined use of biocontrol agents to manage plant diseases in theory and practice.

Effective use of biological control agents (BCAs) is a potentially important component of sustainable agriculture. Recently, there has been an increasing interest among researchers in using combinations of BCAs to exploit potential synergistic effects among them. The methodology for investigating such synergistic effects was reviewed first and published results were then assessed for available evidence for synergy. Correct formulation of hypotheses based on the theoretical definition of independence (Bliss independence or Loewe additivity) and the subsequent and statistical testing for the independence-synergistic-antagonistic interactions have rarely been carried out thus far in studies on biocontrol of plant diseases. Thus, caution must be taken when interpreting reported "synergistic" effects without assessing the original publications. Recent theoretical modeling work suggested that disease suppression from combined use of two BCAs was, in general, very similar to that achieved by the more efficacious one, indicating no synergistic but more likely antagonistic interactions. Only in 2% of the total 465 published treatments was there evidence for synergistic effects among BCAs. In the majority of the cases, antagonistic interactions among BCAs were indicated. Thus, both theoretical and experimental studies suggest that, in combined use of BCAs, antagonistic interactions among BCAs are more likely to occur than synergistic interactions. Several research strategies, including formulation of synergy hypotheses in relation to biocontrol mechanisms, are outlined to exploit microbial mixtures for uses in biocontrol of plant diseases.

A numerical study of combined use of two biocontrol agents with different biocontrol mechanisms in controlling foliar pathogens.

Effective use of biocontrol agents is an important component of sustainable agriculture. A previous numerical study of a generic model showed that biocontrol efficacy was greatest for a single biocontrol agent (BCA) combining competition with mycoparasitism or antibiosis. This study uses the same mathematical model to investigate whether the biocontrol efficacy of combined use of two BCAs with different biocontrol mechanisms is greater than that of a single BCA with either or both of the two mechanisms, assuming that two BCAs occupy the same host tissue as the pathogen. Within the parameter values considered, a BCA with two biocontrol mechanisms always outperformed the combined use of two BCAs with a single but different biocontrol mechanism. Similarly, combined use of two BCAs with a single but different biocontrol mechanism is shown to be far less effective than that of a single BCA with both mechanisms. Disease suppression from combined use of two BCAs was very similar to that achieved by the more efficacious one. As expected, a higher BCA introduction rate led to increased disease suppression. Incorporation of interactions between two BCAs did not greatly affect the disease dynamics except when a mycoparasitic and, to a lesser extent, an antibiotic-producing BCA was involved. Increasing the competitiveness of a mycoparasitic BCA over a BCA whose biocontrol mechanism is either competition or antibiosis may lead to improved biocontrol initially and reduced fluctuations in disease dynamics. The present study suggests that, under the model assumptions, combined use of two BCAs with different biocontrol mechanisms in most cases only results in control efficacies similar to using the more efficacious one alone. These predictions are consistent with published experimental results, suggesting that combined use of BCAs should not be recommended without clear understanding of their main biocontrol mechanisms and relative competitiveness, and experimental evaluation.

Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent.

A previously published generic mathematic model has been used in a numerical study to understand the dynamics of foliar pathogens in relation to mechanisms, and timing and coverage of biocontrol agent (BCA) applications. With the model parameter values used, it was demonstrated that a BCA possessing either competition or induced resistance as the main mechanism of biological control was more effective in reducing disease development than a BCA with either mycoparasitism or antibiosis as its mechanism. Application coverage, ranging from 50 to 90%, had little effect on biocontrol efficacy, particularly for a BCA with competition and induced resistance as the main mechanism of biocontrol. Conversely, delayed application of BCA had more profound effects on biocontrol efficacy for those with competition or induced resistance as their main mechanism than those with mycoparasitism and antibiosis. Biocontrol efficacy was greatest for a single BCA combining competition with mycoparasitism or antibiosis. The efficacy for a single BCA combining induced resistance with competition critically depended on application time; the efficacy was greatly reduced for delayed applications. The present study suggests that development of an effective strategy for BCA application is critically dependent upon our quantitative understanding of several key biocontrol processes and their interactions. Without reliable quantitative estimation of these processes, it is impossible to make quantitative predictions about biological control and hence to optimize BCA application strategies.

A generic theoretical model for biological control of foliar plant diseases.

We have developed a generic modelling framework to understand the dynamics of foliar pathogen and biocontrol agent (BCA) populations in order to predict the likelihood of successful biocontrol in relation to the mechanisms involved. The model considers biocontrol systems for foliar pathogens only and, although it is most applicable to fungal BCA systems, does not address a specific biocontrol system. Four biocontrol mechanisms (competition, antibiosis, mycoparasitism and induced resistance) were included within the model rubric. Because of the wide range of mechanisms involved we use Trichoderma/Botrytis as an exemplar system. Qualitative analysis of the model showed that the rates of a BCA colonising diseased and/or healthy plant tissues and the time that the BCA remains active are two of the more important factors in determining the final outcome of a biocontrol system. Further evaluation of the model indicated that the dynamic path to the steady-state population levels also depends critically on other parameters such as the host-pathogen infection rate. In principle, the model can be extended to include other potential mechanisms, including spatio-temporal heterogeneity, fungicide effects, non-fungal BCA and strategies for BCA application, although with a cost in model tractability and ease of interpretation.

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.

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.

Epidemiology of cercospora leaf spot on sugar beet: modeling disease dynamics within and between individual plants.

ABSTRACT Disease dynamics of Cercospora leaf spot (CLS) of sugar beet was analyzed at two hierarchical scales: as vertical profiles within individual plants and in relation to disease on neighboring plants. The relative contribution of different leaf layers to increase in CLS was analyzed using a simple continuous-time model. The model was fitted to data from two field trials in the Netherlands: one in an area with a long history of CLS, the other in an area where CLS has only recently established; in each case these were unsprayed and twice-sprayed treatments. There were differences in the relative contribution of different leaf layers to disease increase on the target leaf layer according to the CLS history and whether the plants were sprayed or unsprayed. In both field trials, parameter estimates giving the relative contribution of the target leaf layer to disease increase at that leaf layer were higher than those for the lower leaf layer. On only a few occasions the contribution of an upper leaf layer to disease increase at the target leaf layer was significant. Thus, CLS increase at the target leaf layer was determined mainly by disease severity at that leaf layer and to a lesser extent by disease at the lower leaf layer. Our continuous-time model was also used to analyze CLS increase on an individual sugar beet plant in relation to its own and its neighbor's level of disease in field trials at five locations in the two CLS areas over two years. In all field trials, the contribution of the target plant itself to disease increase (auto-infection) was larger than that of its neighboring plants (allo-infection). The overall analysis in the two CLS areas also indicated a larger contribution of the target plant to its disease increase than of neighboring plants, and this pattern was also apparent in a pooled analysis across all sites. Thus, CLS increase on a sugar beet plant was mainly determined by the disease severity on that plant and to a lesser extent by its within-row neighboring plants.

Analysis of summer epidemic progress of apple scab at different apple production systems in the Netherlands and hungary.

ABSTRACT Two, 4-year studies on summer epidemic progress of apple scab were conducted at Randwijk, the Netherlands, from 1998 until 2001 and at Eperjeske, Hungary, from 2000 until 2003. Disease assessments were made on scab-susceptible cv. Jonagold. A range of nonlinear growth functions were fitted to a total of 96 disease progress curves (3 treatment classes x 2 plant parts x 2 disease measures x 4 years x 2 locations) of apple scab incidence and severity. The three-parameter logistic model gave the most consistent fit across three treatment classes in the experiment (integrated, organic-sprayed, and organic-unsprayed). Parameters estimated or calculated from the three-parameter logistic function were used to analyze disease progress. These were disease incidence and severity on the day of the first assessment (Y(s)); final disease incidence or upper asymptote for incidence (Y(if)) or severity (Y(sf)); fruit incidence and severity on day 40, after which no new lesions on fruits appeared (Y(40)); leaf incidence and severity on day 75, at which shoot growth stopped (Y (75)); relative (beta) and "absolute" (theta) rates of disease progress; inflection point (M); and area under the disease progress curve (AUDPC(S)) standardized by the duration of the total epidemic. Comparisons among disease progress curves were made by correlation and factor analysis followed by Varimax rotation. There were large differences but high positive correlations among the parameters Y(s), Y(f), theta, and AUDPC(S) across the three treatment classes. In the factor analysis, two factors accounted for more than 85% of the total variance for both incidence and severity. Factor 1 gave an overall description of epidemic progress of both scab incidence and severity and included the parameters Y(f), Y(40), Y(75), theta, and AUDPC(S). Factor 2 identified a relationship between the relative rate parameter (beta) and the inflection point (M) for severity and a relationship between disease incidence and severity. For an integrated or an organic orchard, theta, AUDPC(S), and one of Y(f) or Y(75) (because of the link with host phenology) can characterize apple scab epidemics during summer. Based on these findings, improved scab management approaches were provided for integrated and organic apple production systems.

Analysis of disease progress as a basis for evaluating disease management practices.

The relationship between epidemiology and disease management is long-standing but sometimes tenuous. It may seem self-evident that improved understanding of epidemic processes will lead to more effective control practices but this remains a testable proposition rather than demonstrated reality. A wide range of models differing in mathematical sophistication and computational complexity has been proposed as a means of achieving a greater understanding of epidemiology and carrying this through to improved management. The potential exists to align these modeling approaches to evaluation of control practices and prediction of the consequent epidemic outcomes, but these have yet to make a major impact on practical disease management. For the immediate future simpler pragmatic approaches for analysis of disease progress, using nonlinear growth functions and/or integrated measures such as area under disease progress curves, will play a key role in informing tactical and strategic decisions on control treatments. These approaches have proved useful in describing control effectiveness and, in some cases, optimizing or changing control practices.

Overwintering of Conidia of Venturia inaequalis and the Contribution to Early Epidemics of Apple Scab.

Overwintering of conidia of Venturia inaequalis associated with shoots and buds was determined, and the contribution to early spring epidemics of apple scab was evaluated during three consecutive seasons (1999 to 2001) in the Netherlands. Examinations of shoot samples collected before bud break showed that the percentage of shoots with superficial black fungal mycelia or conidia was above 65%, and the mean number of conidia on a 1-cm piece of shoot length ranged from 581 to 1,033. However, germination tests showed that the viability of conidia on shoots was less than 1.5%. No macroscopic scab lesions were detected on the scales of dormant buds. However, microscopic examinations of individual bud tissues demonstrated that the number of conidia was >3,000 per 100 buds in each year. The mean viability of conidia associated with buds ranged from 0.7 to 1.9% and from 3.7 to 10.5% for the outer and inner bud tissues, respectively. Results of field assessments at tight-cluster phenological stage showed that the percentage of infection caused by the viable overwintered conidia ranged from 0.3 to 3.8% in the various treatments. Our results indicated that conidia were unlikely to overwinter on the surface of shoots or outer bud tissues, where they were exposed to fluctuating environmental conditions, and, consequently, were unlikely to play a role in initiating an early epidemic of apple scab in the spring. However, our results indicated a risk from overwintered conidia in the inner bud tissues arising from a high level of scab the previous autumn. Therefore, orchards with high levels of apple scab, where ascosporic inoculum is much reduced, e.g., by sanitation, should be protected in early spring by means of fungicide treatment at green tip.