Incorporating an Asymptotic Parameter into the Weibull Model to Describe Plant Disease Progress

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Abstract
The Weibull model is a flexible growth model that describes both general population growth and plant disease progress. However, lack of an asymptotic parameter has limited its wider application. In the present study, an asymptotic parameter K was introduced into the original Weibull model, written as; y=K{1 − exp [− (t−a)c]}, in which a, b, c and K are location, scale, shape, and asymptotic parameters, respectively, y is the proportion of disease and t is time. A wide range of simulated disease progress data sets were generated using logistic, Gompertz and monomolecular models by specifying different parameter values, and fitted to both original and modified Weibull models. The modified model provided statistically better fits for all data than the original model. The modified model can thus improve the curve‐fitting ability of the original model which often failed to converge, especially when the asymptote is less than 1.0. Actual disease progress data on wheat leaf rust and tomato root rot with different asymptotic values were also used to compare the original and modified Weibull models. The modified model provided a statistically better fit than the original model, and model estimates of asymptotic parameter K were nearly identical to the actual disease maxima reflecting the characteristics of the host‐pathosystem. Comparison of logistic, Gompertz, and Weibull models including parameter K by fitting to the observed data on wheat leaf rust and tomato root rot revealed the applicability of the modified Weibull model, which in a majority of cases provided a statistically superior fit.