Comments regarding the binary power law for heterogeneity of disease incidence.

The binary power law (BPL) has been successfully used to characterize heterogeneity (overdispersion or small-scale aggregation) of disease incidence for many plant pathosystems. With the BPL, the log of the observed variance is a linear function of the log of the theoretical variance for a binomial distribution over the range of incidence values, and the estimated scale (?) and slope (b) parameters provide information on the characteristics of aggregation. When b = 1, the interpretation is that the degree of aggregation remains constant over the range of incidence values observed; otherwise, aggregation is variable. In two articles published in this journal in 2009, Gosme and Lucas used their stochastic simulation model, Cascade, to show a multiphasic (split-line) relationship of the variances, with straight-line (linear) relationships on a log-log scale within each phase. In particular, they showed a strong break point in the lines at very low incidence, with b considerably >1 in the first line segment (corresponding to a range of incidence values usually not observed in the field), and b being ?1 in the next segment (corresponding to the range of incidence values usually observed). We evaluated their findings by utilizing a general spatially explicit stochastic simulator developed by Xu and Ridout in 1998, with a wide range of median dispersal distances for the contact distribution and number of plants in the sampling units (quadrats), and through an assessment of published BPL results. The simulation results showed that the split-line phenomenon can occur, with a break point at incidence values of ?0.01; however, the split is most obvious for short median dispersal distances and large quadrat sizes. However, values of b in the second phase were almost always >1, and only approached 1 with extremely short median dispersal distances and small quadrat sizes. An appraisal of published results showed no evidence of multiple phases (although the minimum incidence may generally be too high to observe the break), and estimates of b were almost always >1. Thus, it appears that the results from the Cascade simulation model represent a special epidemiological case, corresponding primarily to a roughly nearest-neighbor population-dynamic process. Implications of a multiphasic BPL property may be important and are discussed.