1 Institute of Applied Sciences, University of North Texas, Denton, Texas 76203 USA
2 Center for Spatial Analysis and Mapping, University of North Texas, Denton, Texas 76203 USA

Abstract The progression of an airborne fungal disease within an agricultural landscape is an important ecological process that is influenced by landscape pattern. Plant epidemiology research has led to the creation of several regression-based probabilistic models of disease-focus expansion. These models have no explicit spatial component and do not account for heterogeneity at the site of investigation. To better understand the relationship between the pattern of an agricultural landscape and the spread of airborne fungal disease, a raster-based GIS simulation model was developed and implemented in Arc/Info(R) GRID. The model treats infection dynamics as a diffusion process and vegetation as spore-filtering media. The GIS-based model allowed for investigation of the affect of variations in vegetation patterns on disease spread. Using the GIS-based model, a two-crop experiment was conducted. Disease spread was simulated for one-hectare fields with varying arrangements of wheat and maize or fallow. Landscape metrics sensitive to boundary shapes and fragmentation were calculated for each crop raster. Increasing connectivity of wheat resulted in increased infection in the maize simulations. Introducing narrow fallow patches to homogeneous wheat fields increased spread.

INTRODUCTION

The patterns of landscape development in time and space result from complex interactions of physical, biological, and social forces (Risser 1984). The alteration of natural land cover patterns results in a mixture of natural and human-managed patches. The size-shape relationships of the altered land cover can influence a number of important ecological phenomena (Beasley 1981, Burgess 1981), including migration, speciation, and the spread of disturbance (Turner 1987). Gradients of disturbance frequency and severity are often controlled by physical or vegetational features.

Goals and Objectives

The goal of this research has been to explore relationships between landscape pattern and the spread of disease by modeling the spatial and temporal development of an airborne fungal disease in varying patterns of susceptible and non-susceptible plant communities. Specifically, our objective was the formulation and GIS implementation of a spatial allocation model for airborne crop disease. Simulations were performed to study the response of fungal crop disease to small-area landscape pattern. Statistical analyses of this relationship were performed. An additional objective was to assess the usefulness and applicability of an off-line response surface design which would allow parameter searches and optimization techniques to be applied to simulation results. Response surface analysis is proposed as a method to minimize the cost of finding target landscape configurations that will minimize transmission of infection.

Rationale

Heterogeneity of the landscape may enhance or retard the spread of disturbance (Turner 1987). The incidence of diseases is related to the spatial arrangement of crops and cultivars (Perrin 1980). According to Alexander (1989), spatial heterogeneity, as it applies to phytopathology, is the variation in plant density, plant genetic composition, and plant location with respect to the physical environment. A plant pathogen may be readily transported over long distances and placed in an uninvaded area, but unless it can establish and maintain itself there, its geographical range will not be increased (Roberts 1975). Murdoch (1975) postulated that it may be possible to design agroecosystems that minimize pest problems and reduce the need for active control measures. Landscape connectivity may be important for the persistence of organisms (Turner 1989, Wegner 1979). Therefore, by fragmenting or structuring the landscape to minimize connectivity, the spread of disturbance may be minimized. Alexander (1989) added that "by mimicking aspects of these natural populations and communities, it was postulated that disease control could be achieved in more natural, long-lasting ways." It is axiomatic that plant pathogens cannot establish themselves in new areas unless susceptible plants occur there. Barriers in the form of fallow or non-susceptible species may therefore be important in retarding the spread of an airborne disease. The dispersal of pests may be impeded where host and non-host grow together (Perrin 1980).

Understanding the role of landscape pattern in the spread of disturbance is fundamental to the development of functional landscape management scenarios. GIS-based modeling incorporates the spatial component of disease spread, allowing spatial scales and patterns to be easily altered, and allowing flexibility in modeling different cover types, disturbances, spatial patterns, and weather conditions.

Background

Phytopathological modeling predicts infection probability or severity within a stand of vegetation given that the severity of a source is known at a point in time. Severity is measured as lesion density or spore coverage per unit area. For fungal rusts and blights, probability of infection and infection severity depend on spore dispersal mechanisms. Methods for modeling the spread of fungal plant disease approach the problem from different perspectives: (1) a mechanistic approach, (2) gradient modeling, (3) stochastic simulation, and (4) spatio-temporal autoregression modeling.

Aylor (1986) proposed a mechanistic model for regional spread of uredeosproes over distances on the order of 1000 km. Probability of infection at a site is based on a rigorous accounting of spores from production through deposition onto the crop. This is a compartmental model that provides a useful framework for understanding the local processes of infection spread (e.g., spore release and transport by wind). Some processes, such as spore loss to UV-radiation, that are significant at regional scales do not influence local dispersion. A disadvantage of the mechanistic approach for simulation modeling is the expense of obtaining precise spore counts at extreme concentrations. Furthermore, a high degree of computational precision is required to implement a complex mechanistic model which incorporates meteorological factors.

Gradient models such as the Gregory model (Gregory 1968, Mundt 1989) use an epidemiological gradient to express the relationship of infection severity to distance. The relationship is usually linearized by applying a logarithmic transformation to the measurement of distance from an infection source and a gompit, probit, or corrected logarithmic transformation to the severity metric. The power of the gradient breaks down with increasing complexity of terrain topology and interacting infection sources. Furthermore, predicting severity a set distance from a source requires intensive spore inventory, and extrapolation from off-center points is not robust (Headrick 1988).

Minogue (1989) advises stochastic simulation of the spread of infection based on sampling from a realistic distribution of dispersal distances. This is a spore-conservative approach. Distributions that result in a wavelike diffusion pattern are to be avoided. Infinite-mean distributions such as the Pareto distribution tend to produce new foci and result in more realistic patterns of dispersal than do finite-mean distributions such as the exponential distribution. Statistical spore-distance distributions resemble the shapes of gradient curves. In fact, the Pareto distribution and the Gregory model have the same form, . In order to be practical for simulation, the spore map should be aggregated into raster cells. Otherwise, tables of disease lesions necessary for computing release at time t+1 become enormous.

The most easily calibrated models are the spatio-temporal autoregression models (Reynolds 1988). A linear combination of distance- and time-lag terms is used to compute the severity or infection probability at a point within a stand of vegetation. Different linear time-series functions can be fit to along-row and across-row spread. These methods have the advantage of applying statistically legitimate techniques to observations with time-autocorrelated error terms. The rigid treatment of spatial configuration causes these models to be less suitable than other techniques for theoretical simulations of disease spread in heterogeneous landscapes.

New lesions formed on a single plant, on adjacent plants, and on plants some distance away are the result of common dispersion processes. The simulation model described herein partitions the increase of disease severity into two processes: severity increase within a cell and the spread to other cells. The logistic growth equation was used to model the increase in disease severity, while a stochastic model was used to model the transmission of infection between cells. An all-or-none distinction is made between infected and uninfected raster cells. A cell-based model is used to reduce computational requirements.

METHODS

Model

Crop disease was chosen as an ecological disturbance whose spatio-temporal progress could be modeled in a straightforward manner. The model was chosen and formulated so that the effect of patches of non-susceptible crops and buffers of unplanted space on disease spread could be investigated. Landscape topology had to be communicated in a modular manner so that patterns could be varied without changing the simulation engine.

A raster-based data structure for stochastic simulation of local spore dispersal mechanisms was devised. The rationale for this approach were that: (1) the analytical models describing the advance of an infection frontier using regressions between logarithmic transformations of distance and disease severity (Headrick 1988, Mundt 1989) are spatially too general for exploration of various landscape patterns; and (2) at the landscape scale, ideal mixture models are functionally homogeneous (Bosch 1990). The equations used to predict rates of focus expansion in homogeneous landscapes are not applicable to heterogeneous sites.

Disease propagation was partitioned into three sources, exposure, infection, and severity. Exposure is the event that the number of spores accumulating in a cell exceeds an arbitrary threshold. Infection occurs when a cell becomes a host to the disease and is able to expose other cells. Severity is a measure of the amount of infection within a cell.

Our model is a simple stochastic simulation model based on the deterministic probabilities of airborne spore dispersal. It partitions disease severity into two proscesses: (1) severity increase within a cell (modeled by a logistic growth equation), and (2) the spread to other cells (modeled stochastically).

The simulation model must generate probabilities of exposure and infection at each timestep over the entire crop raster. The spatial model was adapted from Minogoue's (1989) model of disease spread as a diffusion of spores through a filter of vegetation. The probability of exposure drops off exponentially with distance. The gradient of disease spread depends on the media (i.e., vegetation height and density) through which it is filtered. The input crop rasters were coded so that each cover type had associated with it a "distance to 50% probability of infection." This spore-filtering capacity is analogous to the directional gradient of the Gregory (1968) model, but applies to both susceptible and non-susceptible vegetation.

Exposure A raster cell becomes infected during timestep t if both exposure and infection occur within the cell. The event that both exposure and infection occur is affected by the susceptibility of the crop within a cell, distance to neighboring infected cells, and the severity of infection in neighboring cells. Susceptibility is defined as the probability of infection given exposure. If we let (i,j) be the coordinates of an uninfected susceptible raster cell and (m,n) be the coordinates of the infected cell nearest (i,j), the probability that cell (i,j) becomes infected is then given by

The simulation model must generate probabilities for exposure and infection events at each timestep over the entire crop raster. The spatial model was adapted from a treatment of disease spread modeled as diffusion of spores through a filter of vegetation (Minogue 1989). The probability E of sufficient exposure at (i,j) by (m,n) is given by

Figure 1. Probability of exposure among cells.

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Figure 2. Input preprocessing: steps necessary for calculation of exposure.

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Severity Based on the assumption that a host's propensity to expose nearby potential hosts is related to its severity, the intercept of equation (3) is adjusted to include a response to severity:

The progress of disease severity through time at a fixed location was modeled by the discrete-time logistic growth equation

Model Implementation

The GRID module of Arc/Info(R) allows for the creation of input crop-cover maps and formula-based implementation of the model. Spatial interactions within the model were accomplished using GRID's cost-allocation functions. It was required that the distance and angle between (i,j) and (m,n) be calculated and that the parameter be allocated to (i,j) at runtime. A diagram of the processing structure of the simulation model is given in Figure 3.

Figure 3. Arc/Info GRID implementation of the nearest-infected-neighbor model for local spread. click on image for higher resolution

Let C be a matrix representing the arrangement of crops. Let c represent a particular crop type. Let be the mean of the exponential distribution in equation (3) which characterizes the spore-filtering capability of crop c. Let M be a matrix of parameters corresponding to C.

In order to preprocess the input raster C, raster M must be generated by merging raster C with a database of parameters . From M, four rasters, , must be generated. These are computed by smoothing M with wedge-shaped convolution window . has central angle and is populated by

Parameter is the radius of the wedge (5m used) and determines the distance decay of interactivity. The parameter is derived from at runtime from the expression

which holds since is the mean of an exponential distribution.

Since GRID's spatial interaction functions EUCALLOCATION, EUCDISTANCE, and EUCDIRECTION take integer rasters as arguments, an internal magnification of was used. To reduce calculations at a timestep, a cutoff value of 10m was applied to the spatial interaction modeling as suggested by Minogue (1989). Given , the 10m cutoff value results in discarding only of the probability.

Simulations

Three replicates of the homogeneous wheat simulation and five simulations each of the input patterns, varying barrier width and type, were conducted. The homogeneous simulations, HOMO, were used as zero barrier width patterns. Fallow patches model the situation where there is little or no obstruction to the spore cloud. Maize patches represent a barrier that is taller and more dense than wheat. In order to determine barrier width, a test strip of alternating bands of susceptible crop and increasing barrier widths was designed. The model was run until no further spread was observed. Spread ceased at barrier width of 10m in fallow and 7m in maize. Therefore, barrier widths of 3m, 6m, and 9m were chosen for the simulations. Rasters containing five patterns (ROWS, WAVES, CIRCLES, CHECK, and FRAGCHECK) were input to the model with barrier widths of 3, 6, and 9 meters (Figure 4). Barrier widths were the length in the x-direction of the barrier patches. FRAGCHECK has less diagonal connectivity than CHECK. ROWS are linear swaths of alternating susceptible and nonsusceptible cover. WAVES is a sinusoidal variation of ROWS. CIRCLES consists of closed annuli arranged concentrically about the center of the crop raster. Simulation rasters were 100 cells on a side, or one hectare. Generation of the patterns was facilitated by a graphical user interface (GUI) to algorithms developed in the Arc/Info AML script language. The homogeneous simulations were considered the zero barrier width point for each pattern type.

Figure 4. Crop patterns input to the simulation model.

Simulations were conducted for 10 timesteps of equal length. Infection foci were a single cell near the center of each raster. For the CIRCLES pattern, infections were started in outer annuli because it was undesirable to confine infection to the center circle, a patch atypical of the overall pattern.

Pattern Quantification

Landscape pattern was quantified so that is influence on local airborne crop disease mechanisms could be evaluated. Input patterns were analyzed using Landstat descriptive landscape statistics software (Riitters 1994). Perimeter/area ratio was used as a simple measure of shape complexity. As the complexity of the landscape increases, the ratio also increases.

Response Surface Design

Optimal response surface designs for resolving crossed effects in mixtures have design points at vertices of a regular k-dimensional simplex, a geometric figure with facets created by intersecting hyperplanes. k represents the dimension of the model or number of independent variables. Mixture designs, having all positive component concentrations bounded by a ceiling of 100% or less are confined to the surface of a figure no larger than a k-simplex. Block designs can be combined with mixture designs to allow process variables into the model.

A response surface was fit to the data to analyze the dependency of areal spread after 10 timesteps on the landscape pattern metrics. The model form is derived by Khuri and Cornell (1987), and combines process variables and mixture component variables. The model fit to the data was composed of linear and crossproduct terms only, with no intercept term. To reduce the number of terms in the model, a stepwise algorithm was applied. Standardized parameter coefficients were computed by dividing the estimate for each parameter by the ratio of the independent parameter standard error to the standard deviation of the dependent variable. Standardized parameter coefficients were used as unitless estimators of the effect of the landscape metrics on the response variable. Lack-of-fit tests were applied to response surfaces in an iterative manner, with order increasing until the hypothesis of zero lack-of-fit could not be rejected at 95% significance (following Khuri and Cornell 1987).

RESULTS

One simulation run for each variation (i.e., changing barrier width and type) of the CHECK pattern is illustrated in figure 5. The simulation shown in figure 5 was repeated for each pattern. To compare patterns, a normalized infected area (NIA) was determined for each pattern. NIA was calculated as total infected area divided by total susceptible area. The NIA was calculated for each pattern-barrier-width combination. Because the simulations were stochastic mean NIA was determined for each pattern. The HOMO and CHECK matrices are used to demonstrate the range of variance associated with the stochastic simulations (Figure 6). Bartlett's test applied to infected area of the landscape patterns suggested that variances were not equal among the maize and fallow observations (P= 0.95). Mean NIA, for both fallow and maize barriers, and P/A for the five landscape patterns are given in figures 7 and 8.

Figure 5. Infection severity after 10 timesteps.

Figure 6. Variance associated with multiple simulations after 10 timesteps.

Fallow Barrier

The mean NIA after 10 time steps for the fallow simulations was 30.1% of total susceptible area, with a standard deviation of 16.3%. The fallow infection data were normally distributed (Shapiro-Wilk W= 0.8, p > 0.05). NIA for single fallow runs ranged from 4.5% to 80% of susceptible cover. The maximum mean of three replicates for this measurement was observed in the CHECK pattern with 3m-fallow barrier; the minimum mean infected area occurred in the ROW-pattern with 9m barrier. For CHECK, a trifold increase in NIA over homogeneous is seen in the 3m-fallow barrier runs but, NIA the 6m and 9m fallow barriers was less than for HOMO. For all patterns except ROWS, the NIA decreased with increased barrier width. The 3m-fallow barrier for the ROWS pattern had lower NIA than did the 6m-fallow barrier, with NIA for the 9m-fallow barrier less than the 3m-fallow barrier (Figure 8).

Maize Barrier

The mean NIA was lower for the maize barriers than for the fallow barriers, with mean and standard deviation of 12.5% and 5.3%, respectively. The maize data were normally distributed (Shapiro-Wilk W= 0.937, p < 0.05). Single observations of NIA for the maize barrier simulations ranged from 4.5% to 80.5%. Maximum and minimum means were observed in the homogeneous simulations and the 3m-maize CHECK patterns, respectively. The maize CIRCLES, FRAGCHECK, ROWS, and WAVES patterns had a decrease in areal spread with increasing barrier width. The exception was CHECK, which had an increase in NIA with barrier width. The range of NIA for CHECK was small and may represent no change with barrier width (Figure 8).

Pattern Quantification

Pattern metrics (perimeter to area, P/A, ratios) for resource patches were used to quantify fragmentation and to compare simulation results (Figures 7 and 8). Pattern complexity (i.e., resource patch fragmentation) increases as NIA decreases. For all three barrier widths, CHECK and WAVE had the highest P/A ratios. ROWS and CIRCLE had the lowest P/A ratios, with FRAGCHECK having intermediate values. Comparison of absolute P/A values is complicated by the influence of raster boundaries on the area of resource patches. For example, P/A for CIRCLE grid with 9m barrier is greater than P/A in the 6m barrier. This results from the number of resource patches at the edge.

Figure 7. Perimeter/Area ratio, Normalized Infected Area, by pattern type. click on image for higher resolution

Figure 8. Perimeter/Area ratio, Normalized Infected Area, by barrier width. click on image for higher resolution

DISCUSSION

Pattern-Disease Spread Relationships

As shown by the plots of normalized infection spread (Figures 7 and 8), the fallow barrier seems to exhibit a dual function. Greater areal spread was observed in all three simulations of the 3m fallow CHECK pattern than in any of the homogeneous simulations (Figure 5). This may be attributed to the reduced filtering capacity of fallow as opposed to wheat. A spore is transmitted more readily through the fallow than wheat barrier. At the 3m barrier width, there is greater potential for dramatic spread due to the small patches of open space.

Contiguous susceptible crop favors the spread of infection until local saturation of the vegetation with disease spores occurs. At saturation, spores released within the patch are largely "wasted" by landing on infected vegetation. When barriers become wide enough, however, infection becomes lower than in homogeneous simulations. The low mean areal spread observed for the 3m ROWS pattern is difficult to explain. The maximum spread observed for WAVES at 6m results from the normalization of susceptible area. A monotonically decreasing trend is seen in the raw WAVES infection data. For the maize WAVES runs, maximum spread also occurs for 6m barriers in the normalized, but not in the raw data. The decrease in normalized areal spread for the other patterns at 6m and 9m barrier widths suggests that interruption of the susceptible crop with large areas of bare patches or swaths of nonsusceptible crops reduces spread by local mechanisms.

Barrier width and amount of susceptible crop are positively correlated. The CHECK pattern is exceptional in that barrier cells always occupy half the raster, with the other half allocated to susceptible crop. For the fallow scenario, a trifold increase in normalized spread is seen in the 3m run relative to the homogeneous run. Adding connected patches of a barrier, such as maize, lessens the areal spread of infection. An increase in normalized spread with barrier width is evident in the 3m, 6m, and 9m runs of CHECK. Only for the CHECK pattern is there a positive relationship between barrier width and connectivity among susceptible patches. For the other patterns, increasing the barrier size increases the fragmentation of the susceptible areas.

Response Surface

Against Khuri and Cornell's (1987) criteria, our simulation design was suboptimal for a response surface analysis. Two caveats apply when the response model is suspected of being unstable due to imperfect design: (1) response estimates are not robust outside the boundaries of the design points and (2) the parameter estimates may not resolve the effects of interaction among terms. The model may fit the design points well, but the relationships indicated by the parameter estimates may not apply in general. Searches for optimal parameter combinations were forgone in this phase of the study because of the unbalanced design.

The work presented in this study could be used to design a more robust response surface analysis. A variable transformation which equalizes variance in the areal infection parameter would increase the power of response model significance testing. It is important to take advantage of what is known about the behavior of the variance when formulating an experimental design so that the power of statistical tests for model significance can be optimized.

Ongoing and Future Research

A simple disease dispersion model was implemented on a GIS platform. Rigorous validation of this model was not conducted based on the assumption that broad generalizations could be made using a model which captured the essential nature of biological processes. Calibration is necessary in order to prescribe landscape configurations that contain or impede the spread of airborne rust.

Further study is required to refine and execute the research designs described herein. The model was designed to be flexible enough that important environmental factors (e.g., wind, moisture, topography) could be incorporated. A Pareto distribution to model the distance-probabilities of exposure may help reduce any wave-like properties of simulated disease spread. The model should be refined and calibrated with measured field data, including integrating more realistic landscape patterns and environmental factors. We have shown that GIS-based simulation models can be used to explore the relationship between landscape pattern and ecological disturbance.

This paper and research is dedicated to the memory of Fred Bogs. He was a bright and talented young student who is deeply missed by all who knew and worked with him.

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Jeffrey Fitzgerald
Department of Geography
University of North Texas
Denton, TX 76203
fitz@unt.edu