Modelling approaches for addressing complexity in plant health management

CA Ingeniería de Biosistemas. División de Investigación y Postgrado. Facultad de Ingeniería, Universidad Autónoma de Querétaro; Cerro de las Campanas s/n, 76010, Querétaro, Qro. México. CA Instrumentación y Control. División de Investigación y Postgrado. Facultad de Ingeniería, Universidad Autónoma de Querétaro; Cerro de las Campanas s/n, 76010, Querétaro, Qro. México. Programa de Entomología. Campo Experimental General Terán, Km 31, Carr. Montemorelos-China, Col. Exhacienda las Anacuas, 67413, General Terán N. L., México. Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias (INIFAP). Programa de recursos genéticos. Campo experimental Bajío, INIFAP, Km 6.5, Carr. Celaya-San Miguel Allende, 38010. Celaya Gto. México.


INTRODUCTION
Plant health is one of the main concerns of nations worldwide.Attention is given not only to the economic implications of reductions in food production but also to the environmental impact that phytosanitary measure generate.Furthermore, the impacts on food security caused by inappropriate plant health practices account for almost 40% of annual crop losses (Flood, 2010).The complexity of this situation has stimulated the formation of interdisciplinary teams of scientists and technicians searching for a better understanding of the processes of introduction, dispersal and establishment of new pests in agricultural zones.Moreover, the teams have investigated the impacts of new pests on food production and biodiversity as well as the alternatives for management or eradication of these organisms on a sustainable basis (Jacobsen, 1997;Schrader, 2004;Banito et al., 2007).Current reviews of these objectives (Rafoss, 2003;Corrado and Quaglia, 2004;Baker et al., 2009) suggest that the analysis of the phytosanitary situations must change in several senses: (a) from qualitative to quantitative methods of assessing risk (b) from considering pest outbreaks as isolated events to the visualisation of the processes that originated the outbreaks and (c) from considering these situations as black-box processes, where only outputs are considered, to an explanatory approach where the focus is set on the understanding of the interactions that determine the behaviour of the processes.To support these changes, the parallel development of at least three items is required (Bouma, 2007): (1) Education and training for everyone involved with plant protection, including but not limited to decision makers, researchers, technicians and producers.
(2) A framework with scientific descriptions of the responses of organisms to the environment, in particular weather and climate.
(3) An infrastructure of weather and climate monitoring.
Mathematical models play a very important role in enhancing the capability to perform quantitative analysis of phytosanitary problems.They provide a very powerful language to understand and represent the interactions among weather, crop and pest variables (Allman and Rhodes, 2004).Mathematical models can help to assess the probabilities of the introduction, reproduction and dispersal of pests and the magnitude of pests' effects on crop yields and quality.Mathematical models are of particular interest for agricultural and ecological research because the construction and implementation of decision support systems require precision in analysis and conclusions.According to Thornley and France (2007), mathematics allows theory to be connected with experimentation and supports the progress of science from qualitative to quantitative methods of analysis.In this review, several mathematical approaches used to model the processes of plant health maintenance are compared.The review will emphasise data requirements and practical benefits.

Models as tools to understand predict and control systems
A model can be defined as a simplified representation of a particular domain of reality (Bossel, 1994).In terms of systems theory, a model is a representation of a system, where a system is defined as a portion of reality constructed by elements in interactions that show specific attributes or properties (Forrester, 1994).A general description of systems and their properties has been given by Bossel (1994) as follows: systems fulfil certain functions associated with purposes recognisable by an observer; systems have characteristic arrangements of their elements and a structure that determines their function, purpose and identity; and systems are not divisible, that is, if one or several elements are removed, the system's purpose cannot be realised.Haefner (2005) distinguishes four kinds of models according to their level of precision and complexity, including the conceptual or verbal model, which is composed of descriptions in natural language; diagrammatic models, which are graphical representations of objects and relations; physical models, which are physical mock-ups of a real system or object; and the formal model, which represents the mathematical approach.Models using algebraic or differential equations are scientific models because they provide numerical descriptions of systems.Biological systems are hierarchically organised, and in nature all elements are connected.Therefore, a key aspect of model creation is the definition of the boundaries of the system (Haefner, 2005).
Boundaries support the critical distinction between elements that are part of the system and elements that are external influences.Suppose that Figure 1 represents the approaches taken by two analysts to the same phytosanitary problem.Analyst 'A' sets system boundaries centred on the life cycle and behaviour of the pest and uses factors within these boundaries to look for alternatives to reduce its population.Analyst 'B' draws system boundaries centred on crop growth variables and will find completely different alternatives to protect production because the system boundaries include a much wider set of factors.
A model is thus a description of a problem in terms of systems, but there can be many different models of the same problem according to the limits assigned to the system.Descriptions of a system can refer to its different properties (Forrester, 1994).There are many classifications of mathematical models, and all of them are useful according to the criteria used to separate or classify models.In the scheme of Figure 2, a distinction is made between Mathematical and Non-Mathematical Models according to a classification provided by Haefner (2005).Specifically, a model is said to be mathematical if it contains in its structure a numerically expressed hypothesis about the system's behaviour that can be tested through comparison with observed data.Mathematical models may be static or dynamic.Some of the properties of a system may change over time in response to the system's own structure or to the influence of an external variable.Some other properties can be considered constants in time.Models referring to properties that do not change with time are classified as static models.Models considering the variations over time of some characteristics are classified as dynamic models (Kepés, 2007).Dynamic models can also be classified according to the ways in which they describe the changes in a system over time.Models describing only the outputs of a system are considered to be descriptive models, whereas models that explicitly describe the interactions of elements that generate system changes are classified as explanatory or ecological models (Haefner, 2005;Lima et al., 2009).

Static models, the first step towards a quantitative approach
Static models are characterised by the absence of a time variable from their structure (France and Thornley, 2007).These models are mathematical equations, e.g., linear regression equations that establish a relationship of proportionality among one or more variables considered to be "dependent" and a series of other variables classified as "independent" (Haefner, 2005).Bayesian networks that represent the statistical dependencies among a set of random variables (e.g., diseases and symptoms) are also static models (Kepés, 2007).Static models are useful for characterising the state of a system at a given moment.An implicit assumption made when a static model is used to represent the state of a system is that this state will not change during a certain period (Kepés, 2007).The entire world is experiencing a transformative period marked by the most accelerated changes in history.The globalisation process has spread not only products and technologies but also pests associated with some commodities (Lima and Berryman, 2006).Climate change is perceived mainly through the increased frequency and intensity of extreme meteorological events: drought, floods, warming, and other events that affect the natural distribution of many organisms (Rosenzweig et al., 2001;Ladányi and Horváth, 2010).As new conditions continue to appear, the lack of requisite knowledge is the main factor that limits the rational and effective management of phytosanitary problems.Under these circumstances, qualitative considerations, based on expert knowledge, represent the best option for supporting decisions about the risk that a specific pest represents and the identification of management alternatives.
Pest Risk Analysis is a good example of this situation.When a new exotic pest appears as a threat, there is little time and often very little information about the organism.In the original method established as the standard by the International Plant Protection Convention, IPPC (Food and Agriculture Organization, 2007a; b), the risk represented by a pest is categorised by using an ordinal classification of "High", "Medium" and "Low" levels.The criteria used for this purpose reflect a series of qualitative considerations about the likelihood of introduction and the estimated magnitude of the impact.A significant improvement to this methodology is the application of static models to derive quantitative estimates of the probability of entry, establishment or spread of the pest.The advent of such static models has led several authors to apply a correlative approach to assess the risk.These modelling techniques, called "comfort indexes", relate the potential distribution of a pest to climatological or meteorological variables.Pest risk maps, representing the risk levels of a region, are created to obtain a preliminary strategy in the process of planning the preventive measures to be used against exotic or quarantine pests (Magarey et al., 2007).These models are used to calculate probabilities for a specific criteria defined by model outputs; this criteria can be based on the number of favourable days for a given organism (Magarey et al., 2007;Moschini et al., 2010).
The most common static models fall into two general classes and yield estimates of two different kinds of variables.Infection models for plant pathogens are based on temperature-moisture response functions, whereas degree-days models are used to calculate the number of phenological stages and the number of generations for arthropods or other organisms.The structure of a general simple regression model is: where: Y = dependent variable, generally associated with the infection intensity or probability.a and b are constant and X coefficient, respectively.X is an independent variable, usually representing the effect of weather.Kang et al. (2010) have estimated the percentage of appressorium formation of the fungus Colletotrichum acutatum, the causal agent of anthracnose of chili pepper, using a multiple linear regression model as follows: P = -13.3+ 0.612T + 0.928W where: P = percentage of appressorium formation; T = level of temperature (°C); W = period of wetness (h).
The authors used this model in comparison with nonlinear models to forecast the time at which weather conditions are favourable to plant infections by the fungi.According to the possible relationships that define a phytosanitary problem presented in Figure 1, this approach only considers one pest-related variable: the reproduction rate, represented as a percentage.This variable is indirectly estimated as a function of the environmental influences represented by the temperature and the wetness period.
Computer systems based on static models have been developed to estimate the potential distributions of species.Some of these systems are used to create pest risk maps.The following systems are among the most extensively used.NAPPFAST (North Carolina State University APHIS Plant Pest Forecasting System) (Magarey et al., 2007), is an Internet-based system for developing plant pest risk maps.The system has a climate database using daily time intervals and has a biological template to create simple static models.CLIMEX (Climatic Index) (Sutherst et al., 1999) is a computer system that estimates the climatic regions that a species could potentially occupy in the absence of other physical or biological limiting factors.The system relates field and experimental data on the occurrence of a pest to a series of climatic indexes calculated on a weekly basis (Beddow et al., 2010;Sutherst et al., 2004).BIOCLIM (Bioclimatic Analysis and Prediction System) developed by Busby (1991) and MAXENT (Maximum Entropy Method) created by Phillips et al. (2006) are systems that estimate the geographic distribution of species by using observed data on species' occurrence and associated climatic indexes.These systems have been used to assess the potential distribution of plant pests.
Static models are very important for supporting plant protection decisions on the scale of countries or continents if site-specific data to characterise pest growth or development are not available or if the time for the decision is limited.Caution must be exercised if predictions are made using static models because correlations are usually valid only under very specific conditions (Faherty, 2008;Jackson, 2008).

Descriptive dynamic models: Changing the view from events to behaviour
Dynamic models allow scientists to connect events over time to describe the behaviour of a system (Thornley and France, 2007).Models that do not have an explicit representation of the mechanistic process that determine the behaviour of a system are classified as Descriptive or Phenomenological models (Haefner, 2005).In these models, the system is treated as a black box with a single input or stimulus and a single output or response (Bossel, 1994;Vázquez-Cruz et al., 2010).The development of Descriptive models utilises the analytic mathematical approach.These models have a high theoretical value, and the general mathematical structure that they embody can be applied to represent the evolution of many different processes.Biological, economic, social, and other kinds of applications are possible using Descriptive models.
Descriptive dynamic models must be fitted to suitable data and require large data sets based on time series observations.Therefore, they are valid only under the particular conditions for which the model parameters were estimated (Bossel, 1994;Haefner, 2005;Parry et al., 2005;Thornley and France, 2007).Consider the hypothetical conditions of Figure 3, representing the growth of a disease in two locations designated (a) and (b).The fitted models are the Exponential (dashed line) and the Logistic (solid line).Although the model structure in each case is the same, the fitting parameters are very different from one location to the other because of the effects of physical and biological factors related to local conditions.The variables that can be included in these models are Pest population, its Rate of reproduction and Time.These equations are generally focussed on the pest subsystem of a phytosanitary problem and can be used to forecast the time course of pest infestations and the progress of diseases (Arneson, 2006;Forbes et al., 2008).Frequently, they are also used to perform comparisons between patterns of disease progress for different cultivars and management strategies (Arneson, 2006;Madden et al., 2000).In this study, progress curves of incidence and severity are created by plotting a given variable (Y axis) as a function of time (X axis).Some frequently used descriptive growth curves are the Exponential, the Monomolecular, the Logistic, the Gompertz and the Weibull model.Descriptive dynamic equations are derived by assuming some characteristic relationships among the initial disease intensity, the initial inoculum availability and the absolute rate of change of disease intensity.This rate simultaneously represents the susceptibility of the host and the suitability of the environment (Haynes and Weingartner, 2003).If the rate of disease increase (dY/dt) is assumed to be proportional to disease intensity (Y), the curve follows the Exponential model (Thornley and France, 2007).The differential equation for this curve is: where c is a parameter known as the specific or relative growth rate.Integrating this equation: If the rate of change in disease level depends on diseased (Y) and healthy tissue (1-Y), the resulting curve is the well-known Logistic Model (Arneson, 2006;Tornley and France, 2007;Sparks et al., 2008).Its differential equation is: where K represents the total plant tissue.Integrating this equation: Descriptive dynamic models have been extensively used as a method for evaluating genetic resistance to several important crop diseases.The area under the Disease Progress Curve calculated from these models is considered to represent a direct measure of the resistance of a genotype (Haynes and Weingartner, 2003;Contreras et al., 2009).
The most important application of these models is the forecasting of disease progress for use in the planning of certain plant health management strategies (Cooke et al., 2006).For this purpose, a sufficient amount of experimental data must be available to ensure that the model parameters are accurately estimated (Bossel, 1994;Haefner, 2005).Descriptive models can be used to develop phytosanitary alert systems or to calculate the potential adaptation of a pest (Andersen, 2005).It is important to consider that these models generally fail to represent processes' behaviour if the weather conditions differ significantly from those prevailing when the fitting parameters were estimated (Bossel, 1994).

Explanatory dynamic models: An ecological approach
Explanatory or Ecological models are intended to identify and understand those processes that are decisive for systems' behaviour.In these types of models, structure and function are described in terms of differential equations directly related to the real system (Bossel, 1994).The equation's coefficients correspond to real characteristics of the process that can be measured, that is, they are not obtained indirectly, as in descriptive models (Bossel, 1994;Haefner, 2005).
To model the process, many characteristics of the systems must be known: their elements, connections and mutual influences.The differential equations formulated in explanatory systems must be solved using numerical methods (Thornley and France, 2007).The numerical solution facilitates the inclusion of as many variables as necessary to represent the behaviour of a process (Peart and Curry, 1998;Thornley and France, 2007).More than a single equation, an Ecological model is a system of equations.Influence or flow diagrams provide an overview of the model components and interactions.
In plant health management, ecological models are a useful tool to examine the interactions among plants, pests and the environment (Tixier et al., 2006;Ellner et al., 2002;Ladányi and Horváth, 2010).The elements or variables of the system are defined as quantities that change with time.Depending on the role that the variables play in the model, they are categorised as state or rate variables.State variables define the state of the system at a given moment in time.Examples of state variables are insect population and disease incidence.Rate variables define the rate or speed at which the state variables change.These variables have dimensions "per unit time" (Thornley and France, 2007).
A phenomenon termed "feedback" occurs if a state variable affects one or more of its rate variables and produces nonlinear behaviour.If the rate changes in direct proportion to the state, a positive feedback loop occurs and promotes an exponential growth or decay of the process.If the rate is inversely proportional to the state, negative feedback is present.The process behaves asymptotically (Haefner, 2005).The diagram of Figure 4 represents a model of a root pest, the Mexican rootworm, Diabrotica virgifera zeae (Quijano et al., 2010).The model includes soil water variables (SW, WC1, WC2, and SH) that influence larval hatching (H) and the plant's capacity to satisfy the evaporative demand of the environment (PCD).The root growth (RGR) process is also included to estimate the food availability (RB) for the larvae and the effect of root consumption (CR) on the plant's capacity to absorb water.The life cycle of the insect reflects the effect of temperature through the influence of the Degree-Days variable (DD) on the developmental rate.The names and descriptions of these variables are given in Table 1.The equations of state are numerically integrated over a time period (dt) of one day.Descriptive and static models can be used to calculate the values of the rate of variables.
The Ecological Modelling approach is very helpful when a strong interaction exists among the host, the pest and the environment.It facilitates the inclusion of as many variables as are necessary to represent the entire system.Two types of ecological models are of interest in plant-protection applications: models of population dynamics to represent the life cycle of arthropods, insects, mites, nematodes, etc., and epidemiological models to represent diseases caused by fungi or bacteria.Wang and Shipp (2001) developed a population dynamics model of Frankliniella occidentalis (Pergande) to determine the economic thresholds for this pest on greenhouse cucumber.A model to simulate phenology, growth, and plant-parasitic nematode/banana interactions (SIMBA) has been created by Tixier et al. (2006) to predict Radopholus similis and Pratylenchus coffeae population dynamics.The model considered the interspecific competition between these organisms and was used to optimise the nematicide applications.Tixier et al. (2008) used this model to evaluate the response of new banana hybrids to plant-parasitic nematodes in Martinique.By taking into account the pest-hostenvironment interactions, the authors could find alternatives to reduce the use of pesticides.Reji and Chander (2008) have created a mechanistic model of the population dynamics of the rice bug (Leptocorisa acuta, Thunb.) in India.The model was used to determine the population peaks of the rice bug to support the programming of control measures.One of the models used most widely for diseases is the Potato Late Blight prediction model, used to target fungicide application to times of the greatest need to reduce the use of chemicals.This model has been modified recently by Henshall et al. (2006).The model calculates infection periods as a function of hourly temperature, relative humidity and surface wetness.
A more complex model involving coconut palms (the host plant), the pathogenic microorganism -Phytomonas staheli McGhee & McGhee, and the insect vector -Lincus lobuliger Bred.(Hemiptera: Pentatomidae) has been developed by Sgrillo et al. (2005) to simulate the dynamics of the disease caused by the pathogen.The population dynamics of healthy and infected insect vectors were included as a subsystem.The results from the model help to support the conclusions that control techniques for the vector only delay the spread of the disease and that it would be more convenient not to apply control techniques for Lincus sp. in areas where the disease is absent.Miranda et al. (2008) proposed a theoretical model of the interaction of the Asian citrus psyllid (Diaphorina citri Kuwayama) with its natural enemies.The model has not yet been validated, but it was used in a preliminary investigation to calculate an ecological balance between the number of females of the natural enemy Tamarixia radiata and the quantity of healthy buds present in the tree.This model will be of great value for estimating the actual psyllid densities.A characteristic common to all of these models is that they need scientific descriptions of organisms' response to the environment and accurate input data to correctly represent the dynamic nature of the biological processes involved in pest-hostenvironment interactions.

Final considerations
Mathematical models provide a powerful language to represent and understand the complexity of the biological processes involved in plant health management.
Understanding the system and describing the structural relationships that govern the behaviour of the system in mathematical terms is the more effective and rational way to acquire the capacity to predict and control system outputs.Quantitative modelling gives a more detailed description of processes but requires many observations to minimise uncertainty (Thakar et al., 2010).Complexity increases with the number of components included in the model and with its temporal resolution.According to Javier and DiStefano (2009), a major new direction in systems biology modelling is the creation of multiscale models that combine different levels of detail as a function of the available data.Comprehensive models are of proven importance as a reference to identify gaps and to organise the available knowledge.In many cases, experimental data and mathematical descriptions are not available or are very difficult to obtain, so a simpler approach must be adopted.In some cases, a basic model yields sufficient detail, and comprehensive approaches are not required.

Figure 1 .
Figure 1.System boundaries definition of a phytosanitary problem by two different observers.

Figure 3 .
Figure 3. Examples of descriptive dynamic models fitted to two different locations.