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.

An induction or flare of arthritis and/or sacroiliitis by vedolizumab in inflammatory bowel disease: a case series.

BACKGROUND: In inflammatory bowel disease (IBD), a new biological therapy has recently been approved. Vedolizumab is a humanised IgG1 monoclonal antibody to α4ß7 integrin that modulates gut lymphocyte trafficking. Although an exclusively local effect of vedolizumab could be expected based on the restricted presence of the α4ß7-mucosal vascular addressin cell adhesion molecule 1 complex in the gut, past combined success with anti-tumour necrosis factor, and previous demonstration of α4ß7 integrin in the joint, led to the expectation of a therapeutic efficacy in spondyloarthritis. Nonetheless, the effect of vedolizumab on extraintestinal manifestations-and especially the joint-has not been reported so far. CASE REPORT: A series of five patients with IBD who were treated with vedolizumab and promptly developed new onset or exacerbation of sacroiliitis or arthritis are reported. CONCLUSIONS: Vedolizumab therapy does not seem to show any efficacy in and might even induce arthritis and/or sacroiliitis. However, larger cohort studies are needed to provide information on the prevalence, the evolution and underlying mechanism.

Identifying circumstances under which high insecticide dose increases or decreases resistance selection.

Insect management strategies for agricultural crop pests must reduce selection for insecticide resistant mutants while providing effective control of the insect pest. One management strategy that has long been advocated is the application of insecticides at the maximum permitted dose. This has been found, under some circumstances, to be able to prevent the resistance allele frequency from increasing. However this approach may, under different circumstances, lead to rapid selection for resistance to the insecticide. To test when a high dose would be an effective resistance management strategy, we present a flexible deterministic model of a population of an insect pest of agricultural crops. The model includes several possible life-history traits including sexual or asexual reproduction, diploid or haplodiploid genetics, univoltine or multivoltine life cycle, so that the high dose strategy can be tested for many different insect pests. Using this model we aim to identify the key characteristics of pests that make either a high dose or a low dose of insecticide optimal for resistance management. Two outputs are explored: firstly whether the frequency of the resistance allele increases over time or remains low indefinitely; and secondly whether lowering the dose of insecticide applied reduces or increases the rate of selection for the resistance allele. It is demonstrated that with high immigration resistance can be suppressed. This suppression however, is rarely lost if the insecticide dose is reduced, and is absent altogether when individuals move from the treated population back into an untreated population. Reducing the dose of insecticide often resulted in slower development of resistance, except where the population combined a high influx of less resistant individuals into the treated population, a recessive resistance gene and a high efficacy, in which case reducing the dose of insecticide could result in faster selection for resistance.

MR signal in the sacroiliac joint space in spondyloarthritis: a new sign.

OBJECTIVES: To determine the diagnostic value of MR signal within the sacroiliac (SI) joint space in spondyloarthritis (SpA). METHODS: A retrospective analysis of MRIs of SI joints was performed in 363 patients, aged 16-45 years, clinically suspected of sacroiliitis. Intra-articular SI joint MR signals were categorized as normal, high T1 signal, fluid signal, ankylosis or vacuum phenomenon (VP). These MRI findings were correlated with the final diagnosis, according to the ASAS criteria. Sensitivity, specificity, and positive and negative likelihood ratios (LR) and predictive values were calculated. RESULTS: Presence of intra-articular high T1 signal, fluid signal and ankylosis had a specificity of 95.8 %, 95.3 % and 99.5 % for SpA. High T1 signal, fluid signal and ankylosis were present in 38.4 %, 19.2 % and 17.9 % of SpA patients and in 4.2 %, 4.7 % and 0.5 % of patients without SpA, resulting in LR+ of 9.0, 4.1 and 37.9, respectively. VP was present in 13.2 % of SpA patients and in 20.8 % of patients without SpA, resulting in an LR+ of 0.6. CONCLUSIONS: Presence of high T1 signal, fluid signal and ankylosis within the SI joint on MRI have high specificity for SpA. High T1 signal is the most sensitive MRI feature within the SI joint for SpA. KEY POINTS: • MRI of the SI joints is typically obtained for diagnosis of spondyloarthritis. • The MR signal within the SI joint itself reflects features of spondyloarthritis. • Intra-articular high T1 signal, fluid signal and ankylosis are seen in spondyloarthritis. • The vacuum phenomenon makes spondyloarthritis less likely.

Physiological Traits Determining Yield Tolerance of Wheat to Foliar Diseases.

Tolerance is defined as the ability of one cultivar to yield more than another cultivar under similar disease severity. If both cultivars suffer an equal loss in healthy (green) leaf area duration (HAD) over the grain filling period due to disease presence, then the yield loss per unit HAD loss is smaller for a more tolerant cultivar. Little is understood of what physiological and developmental traits of cultivars determine disease tolerance. In this study, we use a mathematical model of wheat to investigate the effect of a wide range of wheat phenotypes on tolerance. During the phase from stem extension to anthesis, the model calculates the assimilate source and sink potential, allowing for dynamic changes to the source-sink balance by partitioning assimilates between ear development and storage of water-soluble carbon (WSC) reserves, according to assimilate availability. To quantify tolerance, rates of epidemic progress were varied on each phenotype, leading to different levels of HAD loss during the postanthesis, grain-filling period. Model outputs show that the main determinant of tolerance is the total amount of assimilate produced per grain during the rapid grain-fill period, leading to a strong positive correlation between HAD per grain and tolerance. Reductions in traits that affect carbon assimilation rate and increases in traits that determine the amount of structural biomass in the plant increase disease tolerance through their associated reduction in number of grains per ear. Some of the most influential traits are the canopy green area index, carbon use efficiency, and leaf specific weight. Increased WSC accumulation can either increase or decrease tolerance. Furthermore, a cultivar is shown to be maximally tolerant when a crop is able to just fill its total sink size in the presence of disease. The model has identified influential functional traits and established that their associations with tolerance have a mechanistic basis.

Epidemiology: Past, Present, and Future Impacts on Understanding Disease Dynamics and Improving Plant Disease Management-A Summary of Focus Issue Articles.

Epidemiology has made significant contributions to plant pathology by elucidating the general principles underlying the development of disease epidemics. This has resulted in a greatly improved theoretical and empirical understanding of the dynamics of disease epidemics in time and space, predictions of disease outbreaks or the need for disease control in real-time basis, and tactical and strategic solutions to disease problems. Availability of high-resolution experimental data at multiple temporal and spatial scales has now provided a platform to test and validate theories on the spread of diseases at a wide range of spatial scales ranging from the local to the landscape level. Relatively new approaches in plant disease epidemiology, ranging from network to information theory, coupled with the availability of large-scale datasets and the rapid development of computer technology, are leading to revolutionary thinking about epidemics that can result in considerable improvement of strategic and tactical decision making in the control and management of plant diseases. Methods that were previously restricted to topics such as population biology or evolution are now being employed in epidemiology to enable a better understanding of the forces that drive the development of plant disease epidemics in space and time. This Focus Issue of Phytopathology features research articles that address broad themes in epidemiology including social and political consequences of disease epidemics, decision theory and support, pathogen dispersal and disease spread, disease assessment and pathogen biology and disease resistance. It is important to emphasize that these articles are just a sample of the types of research projects that are relevant to epidemiology. Below, we provide a succinct summary of the articles that are published in this Focus Issue .

Elevated calprotectin levels reveal bowel inflammation in spondyloarthritis.

INTRODUCTION: Microscopic bowel inflammation is present in up to 50% of patients with spondyloarthritis (SpA) and is associated with more severe disease. Currently no reliable biomarkers exist to identify patients at risk. Calprotectin is a sensitive marker of neutrophilic inflammation, measurable in serum and stool. OBJECTIVES: To assess whether serum and faecal calprotectin in addition to C-reactive protein (CRP) can be used to identify patients with SpA at risk of microscopic bowel inflammation. METHODS: Serum calprotectin and CRP were measured in 125 patients with SpA. In 44 of these patients, faecal samples were available for calprotectin measurement. All 125 patients underwent an ileocolonoscopy to assess the presence of microscopic bowel inflammation. RESULTS: Microscopic bowel inflammation was present in 53 (42.4%) patients with SpA. Elevated serum calprotectin and CRP were independently associated with microscopic bowel inflammation. Faecal calprotectin was also significantly higher in patients with microscopic bowel inflammation. Patients with CRP and serum calprotectin elevated had a frequency of bowel inflammation of 64% vs 25% in patients with low levels of both. When either CRP or serum calprotectin was elevated, the risk was intermediate (40%) and measuring faecal calprotectin provided further differentiation. Hence we suggest a screening approach where initially serum calprotectin and CRP are assessed and, if necessary, faecal calprotectin. The model using this scenario provided an area under the ROC curve of 74.4% for detection of bowel inflammation. CONCLUSIONS: Calprotectin measurements in stool and serum, in addition to CRP, may provide a promising strategy to identify patients with SpA at risk of bowel inflammation and could play a role in overall patient stratification.

European League Against Rheumatism (EULAR) recommendations for the management of psoriatic arthritis with pharmacological therapies: 2015 update.

BACKGROUND: Since the publication of the European League Against Rheumatism recommendations for the pharmacological treatment of psoriatic arthritis (PsA) in 2012, new evidence and new therapeutic agents have emerged. The objective was to update these recommendations. METHODS: A systematic literature review was performed regarding pharmacological treatment in PsA. Subsequently, recommendations were formulated based on the evidence and the expert opinion of the 34 Task Force members. Levels of evidence and strengths of recommendations were allocated. RESULTS: The updated recommendations comprise 5 overarching principles and 10 recommendations, covering pharmacological therapies for PsA from non-steroidal anti-inflammatory drugs (NSAIDs), to conventional synthetic (csDMARD) and biological (bDMARD) disease-modifying antirheumatic drugs, whatever their mode of action, taking articular and extra-articular manifestations of PsA into account, but focusing on musculoskeletal involvement. The overarching principles address the need for shared decision-making and treatment objectives. The recommendations address csDMARDs as an initial therapy after failure of NSAIDs and local therapy for active disease, followed, if necessary, by a bDMARD or a targeted synthetic DMARD (tsDMARD). The first bDMARD would usually be a tumour necrosis factor (TNF) inhibitor. bDMARDs targeting interleukin (IL)12/23 (ustekinumab) or IL-17 pathways (secukinumab) may be used in patients for whom TNF inhibitors are inappropriate and a tsDMARD such as a phosphodiesterase 4-inhibitor (apremilast) if bDMARDs are inappropriate. If the first bDMARD strategy fails, any other bDMARD or tsDMARD may be used. CONCLUSIONS: These recommendations provide stakeholders with an updated consensus on the pharmacological treatment of PsA and strategies to reach optimal outcomes in PsA, based on a combination of evidence and expert opinion.

Early detection surveillance for an emerging plant pathogen: a rule of thumb to predict prevalence at first discovery.

Emerging plant pathogens are a significant problem for conservation and food security. Surveillance is often instigated in an attempt to detect an invading epidemic before it gets out of control. Yet in practice many epidemics are not discovered until already at a high prevalence, partly due to a lack of quantitative understanding of how surveillance effort and the dynamics of an invading epidemic relate. We test a simple rule of thumb to determine, for a surveillance programme taking a fixed number of samples at regular intervals, the distribution of the prevalence an epidemic will have reached on first discovery (discovery-prevalence) and its expectation E(q*). We show that E(q*) = r/(N/Δ), i.e. simply the rate of epidemic growth divided by the rate of sampling; where r is the epidemic growth rate, N is the sample size and Δ is the time between sampling rounds. We demonstrate the robustness of this rule of thumb using spatio-temporal epidemic models as well as data from real epidemics. Our work supports the view that, for the purposes of early detection surveillance, simple models can provide useful insights in apparently complex systems. The insight can inform decisions on surveillance resource allocation in plant health and has potential applicability to invasive species generally.

Diagnostic value of MRI features of sacroiliitis in juvenile spondyloarthritis.

AIM: To determine the diagnostic utility of magnetic resonance imaging (MRI) features of sacroiliitis in juvenile spondyloarthritis (JSpA). MATERIALS AND METHODS: This was a prospective study of 80 paediatric patients who underwent MRI of the sacroiliac joints that were clinically suspected to have sacroiliitis. The prevalence of MRI features of active and structural lesions of sacroiliitis was recorded. Patients were classified according to the International League of Association for Rheumatology criteria. The MRI findings were compared to the final clinical diagnosis. RESULTS: Sacroiliitis was seen in 25/80 (31%) patients. MRI showed active inflammation in 23 patients (29%): synovial enhancement (28%), high short tau inversion recovery (STIR)-signal in the joint space (29%), bone marrow oedema (BMO; 20%), and capsulitis (8%). Structural changes were present in 14 patients (18%): erosion (14%), fat infiltration (13%), sclerosis (8%), and ankylosis (1%). Of all MRI features, ankylosis (100%), capsulitis (98%), BMO (96%), and erosion (96%) had the highest specificity for JSpA; global diagnostic impression (55%) and synovial enhancement (52%) were the MRI features with the highest sensitivity. The likelihood ratios (LR+) for diagnosis of JSpA were high for BMO (10.5), capsulitis (7.5), global diagnostic impression (6.9), and erosions (6.75), but greater for BMO concomitant with synovial enhancement (LR+ 19.5) and for erosion concomitant with BMO (LR+ 12) or synovial enhancement (LR+ 13.5). CONCLUSION: There are multiple features of active inflammation and structural damage visible at MRI of the sacroiliac joints that can provide a specific diagnosis of JSpA when present in children with suspected sacroiliitis. Synovial enhancement is the MRI feature with the highest sensitivity for JSpA. If BMO is seen concomitant with synovial enhancement or erosion, the diagnosis of JSpA is very likely. Ankylosis, capsulitis, bone marrow oedema, and erosion all have a high specificity for JSpA. Absence of MRI findings of sacroiliitis does not exclude the diagnosis of JSpA.

Which spinal lesions are associated with new bone formation in patients with ankylosing spondylitis treated with anti-TNF agents? A long-term observational study using MRI and conventional radiography.

OBJECTIVE: To study the relationship of spinal inflammation and fatty degeneration (FD) as detected by MRI and new bone formation seen on conventional radiographs (CRs) in ankylosing spondylitis (AS). METHODS: CRs at baseline, 2 years and 5 years and spinal MRIs at baseline and 2 years of 73 AS patients treated with infliximab in European AS Infliximab Cohort were available. Relative risks (RR) were calculated with a general linear model after adjustment for within-patient variation. RESULTS: In a total of 1466 vertebral edges (VEs) without baseline syndesmophytes, 61 syndesmophytes developed at 5 years, the majority of which (57.4%) had no corresponding detectable MRI lesions at baseline. VEs with both inflammation and FD at baseline had the highest risk (RR 3.3, p=0.009) for syndesmophyte formation at 5 years, followed by VEs that developed new FD or did not resolve FD at 2 years (RR=2.3, p=0.034), while inflammation at baseline with no FD at 2 years had the lowest risk for syndesmophyte formation at 5 years (RR=0.8). Of the VEs with inflammation at baseline, >70% resolved completely, 28.8% turned into FD after 2 years, but only 1 syndesmophyte developed within 5 years. CONCLUSIONS: Parallel occurrence of inflammation and FD at baseline and development of FD without prior inflammation after 2 years were significantly associated with syndesmophyte formation after 5 years of anti-tumour necrosis factor (TNF) therapy. However, the sequence 'inflammation-FD-new bone formation' was rarely observed, an argument against the TNF-brake hypothesis. Whether an early suppression of inflammation leads to a decrease of the risk for new bone formation remains to be demonstrated.

Diagnostic value of pelvic enthesitis on MRI of the sacroiliac joints in spondyloarthritis.

OBJECTIVE: To determine the prevalence and diagnostic value of pelvic enthesitis on MRI of the sacroiliac (SI) joints in spondyloarthritis (SpA). MATERIALS AND METHODS: A retrospective study in 444 patients aged 17-45 years old with MRI of the SI joints and with clinically suspected sacroiliitis was performed. Patients were classified as having SpA if they fulfilled the Assessment of Spondyloarthritis International Society (ASAS) criteria. Pelvic enthesitis on MRI was correlated with the final diagnosis. Sensitivity, specificity, positive and negative likelihood ratio (LR) and predictive values (PV) of pelvic enthesitis for the diagnosis of SpA were calculated. RESULTS: MRI showed pelvic enthesitis in 24.4 % of patients with SpA and in 7.1 % of patients without SpA. Presence of any enthesitis had sensitivity, specificity, LR+, LR-, PPV and NPV of 24.4 %, 92.9 %, 3.45, 0.81, 69.4 % and 65.2 % for the diagnosis of SpA, respectively. The most commonly affected entheses were the longitudinal ligament insertion (4.5 %), the retroarticular ligaments (4.1 %) and the pubic symphysis (4.1 %). The sites of enthesitis with the highest PPV for SpA were the iliac crest/wing (85.7 %) and the retroarticular ligaments (81.3 %). CONCLUSION: Nearly one fourth of SpA patients with suspected sacroiliitis showed pelvic enthesitis on MRI. Such pelvic enthesitis has a high specificity for the diagnosis of spondyloarthritis. KEY POINTS: • Enthesitis is the primary clinical feature of spondyloarthritis. • Magnetic resonance imaging of the sacroiliac joints can demonstrate pelvic enthesitis. • Pelvic enthesitis has a high specificity for the diagnosis of spondyloarthritis.

A generic risk-based surveying method for invading plant pathogens.

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

Optimal fungicide application timings for disease control are also an effective anti-resistance strategy: a case study for Zymoseptoria tritici (Mycosphaerella graminicola) on wheat.

Strategies to slow fungicide resistance evolution often advocate early "prophylactic" fungicide application and avoidance of "curative" treatments where possible. There is little evidence to support such guidance. Fungicide applications are usually timed to maximize the efficiency of disease control during the yield-forming period. This article reports mathematical modeling to explore whether earlier timings might be more beneficial for fungicide resistance management compared with the timings that are optimal for efficacy. There are two key timings for fungicide treatment of winter wheat in the United Kingdom: full emergence of leaf three (counting down the canopy) and full emergence of the flag leaf (leaf 1). These timings (referred to as T1 and T2, respectively) maximize disease control on the upper leaves of the crop canopy that are crucial to yield. A differential equation model was developed to track the dynamics of leaf emergence and senescence, epidemic growth, fungicide efficacy, and selection for a resistant strain. The model represented Zymoseptoria tritici on wheat treated twice at varying spray timings. At all fungicide doses tested, moving one or both of the two sprays earlier than the normal T1 and T2 timings reduced selection but also reduced efficacy. Despite these opposing effects, at a fungicide dose just sufficient to obtain effective control, the T1 and T2 timings optimized fungicide effective life (the number of years that effective control can be maintained). At a higher dose, earlier spray timings maximized effective life but caused some reduction in efficacy, whereas the T1 and T2 timings maximized efficacy but resulted in an effective life 1 year shorter than the maximum achievable.

The usefulness of fungicide mixtures and alternation for delaying the selection for resistance in populations of Mycosphaerella graminicola on winter wheat: a modeling analysis.

A fungicide resistance model (reported and tested previously) was amended to describe the development of resistance in Mycosphaerella graminicola populations in winter wheat (Triticum aestivum) crops in two sets of fields, connected by spore dispersal. The model was used to evaluate the usefulness of concurrent, alternating, or mixture use of two high-resistance-risk fungicides as resistance management strategies. We determined the effect on the usefulness of each strategy of (i) fitness costs of resistance, (ii) partial resistance to fungicides, (iii) differences in the dose-response curves and decay rates between fungicides, and (iv) different frequencies of the double-resistant strain at the start of a treatment strategy. Parameter values for the quinine outside inhibitor pyraclostrobin were used to represent two fungicides with differing modes of action. The effectiveness of each strategy was quantified as the maximum number of growing seasons that disease was effectively controlled in both sets of fields. For all scenarios, the maximum effective lives achieved by the use of the strategies were in the order mixtures ≥ alternation ≥ concurrent use. Mixtures were of particular benefit where the pathogen strain resistant to both modes of action incurred a fitness penalty or was present at a low initial frequency.

Accounting for the economic risk caused by variation in disease severity in fungicide dose decisions, exemplified for Mycosphaerella graminicola on winter wheat.

A method is presented to calculate economic optimum fungicide doses accounting for the risk aversion of growers responding to variability in disease severity between crops. Simple dose-response and disease-yield loss functions are used to estimate net disease-related costs (fungicide cost plus disease-induced yield loss) as a function of dose and untreated severity. With fairly general assumptions about the shapes of the probability distribution of disease severity and the other functions involved, we show that a choice of fungicide dose which minimizes net costs, on average, across seasons results in occasional large net costs caused by inadequate control in high disease seasons. This may be unacceptable to a grower with limited capital. A risk-averse grower can choose to reduce the size and frequency of such losses by applying a higher dose as insurance. For example, a grower may decide to accept "high-loss" years 1 year in 10 or 1 year in 20 (i.e., specifying a proportion of years in which disease severity and net costs will be above a specified level). Our analysis shows that taking into account disease severity variation and risk aversion will usually increase the dose applied by an economically rational grower. The analysis is illustrated with data on Septoria tritici leaf blotch of wheat caused by Mycosphaerella graminicola. Observations from untreated field plots at sites across England over 3 years were used to estimate the probability distribution of disease severities at mid-grain filling. In the absence of a fully reliable disease forecasting scheme, reducing the frequency of high-loss years requires substantially higher doses to be applied to all crops. Disease-resistant cultivars reduce both the optimal dose at all levels of risk and the disease-related costs at all doses.

Estimating the incidence of an epidemic when it is first discovered and the design of early detection monitoring.

The early detection of an invading epidemic is crucial for successful disease control. Although models have been used extensively to test control strategies following the first detection of an epidemic, few studies have addressed the issue of how to achieve early detection in the first place. Moreover, sampling theory has made great progress in understanding how to estimate the incidence or spatial distribution of an epidemic but how to sample for early detection has been largely ignored. Using a simple epidemic model we demonstrate a method to calculate the incidence of an epidemic when it is discovered for the first time (given a monitoring programme taking samples at regular intervals). We use the method to explore how the intensity and frequency of sampling influences early detection. In particular, we find that for epidemics characterised by high population growth rates it is most effective to spread sampling resources evenly in time. In addition we derive a useful approximation to our method which results in a simple equation capturing the relation between monitoring and epidemic dynamics. Not only does this provide valuable new insight but it provides a simple rule of thumb for the design of monitoring programmes in practice.

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.