Models of Plant Disease Epidemics. II: Gradients of Bean Rust

The spread of bean rust from three types of inoculum sources (area, line, and point) was studied under field conditions in two seasons. Disease intensity (y) over time was quantified as incidence of diseased plants, incidence of diseased leaves, and disease severity at distances (d) of 0.3, 0.7, 1.5, 2.7, 4.3, 6.5, and 8.7 m from the inoculum sources. Seven curvilinear models were compared and the models that best explained the gradients of bean rust were y = ad^?b exp(cd), y = a exp(–bd^c) and y = a exp(–bd); where a was proportion of disease near the source, d was distance in metres, b was the slope, and c was a shape parameter. For general analysis to compare the slopes of the gradients, the gradient curves were linearized by ln(y) versus d and the parameters ln(a) (intercept) and b (slope) were estimated. The three measures of disease intensity and the three types of inoculum sources had estimated gradient parameters that were statistically different. The incidence of diseased plants rapidly approached y = 1.0 and the gradient slopes were near zero. Also with the ln(y) linearization, the gradients of the incidence of diseased leaves had slopes flatter than the gradients of disease severity. The gradient slopes from area sources of inoculum were flatter than from line sources, which in turn were flatter than from point sources. Late in the epidemics, the b-values became similar for all combinations of inoculum sources × assessment types. The interpretation of flattening of the gradients was confounded because aberrations in transformations occurred with most models. That is, when the intensity of disease (incidence or severity) approached the maximum, the increases in the intercepts were restricted and this restriction was responsible for most of the flattening. No flattening of gradients occurred over time when the gompit gradient model [–ln(–ln(y/y_max)) = –ln(–ln(a))?b ln(d)] was used for these same disease values. The fitting of models to the gradients served little purpose except to classify the curves in a general way. A new statistic, the area under the disease gradient curve (AUDGC), is proposed to compare epidemics.