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

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

Statistical analysis of shape of objects based on landmark data.

Two objects with homologous landmarks are said to be of the same shape if the configurations of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. The observations on an object are coordinates of its landmarks with reference to a set of orthogonal coordinate axes in an appropriate dimensional space. The origin, choice of units, and orientation of the coordinate axes with respect to an object may be different from object to object. In such a case, how do we quantify the shape of an object, find the mean and variation of shape in a population of objects, compare the mean shapes in two or more different populations, and discriminate between objects belonging to two or more different shape distributions. We develop some methods that are invariant to translation, rotation, and scaling of the observations on each object and thereby provide generalizations of multivariate methods for shape analysis.

An O(N log N) algorithm for shape modeling.

We present a shape-recovery technique in two dimensions and three dimensions with specific applications in modeling anatomical shapes from medical images. This algorithm models extremely corrugated structures like the brain, is topologically adaptable, and runs in O(N log N) time, where N is the total number of points in the domain. Our technique is based on a level set shape-recovery scheme recently introduced by the authors and the fast marching method for computing solutions to static Hamilton-Jacobi equations.