Bayesian Analysis for Inference of an Emerging Epidemic: Citrus Canker in Urban Landscapes

Neri, Franco M.; Cook, Alex R.; Gibson, Gavin J.; Gottwald, Tim R.; Gilligan, Christopher A.

Models with short-range dispersal (exponential kernel) and longrange dispersal (Cauchy kernel) together with external primary infection were compared using DIC tests (DIC 6 , cf. Table S1 and Text S1). Table S1 shows no significant differences between the exponential and Cauchy models, except for site D1, for which the exponential model is favoured. For the other census sites, the two models are essentially equivalent. This result can be explained in two steps, first by analysing dispersal at short distances (Figure 7) , then by considering the contribution of external infection at longer distances ( Figure S1 ). Figure 7 shows a direct comparison of estimated exponential and Cauchy kernels, plotted as a function of distance for each census site. The pattern is qualitatively similar for all census sites: the two kernels are substantially identical up to distances of a few hundred metres (''plus'' signs in Figure 7) : 250-300 m for all the sites bar D1, and ,150 m for site D1 ( Figure 7C : this may be a reason why the DIC tests favours the exponential kernel for this site). Beyond those distances, which correspond to a fraction of the size of the census site (1 km-4 km), the relative difference between the two kernels increases rapidly. Hence, in principle it should still be possible to detect the effect of such difference in estimates from spatio-temporal maps of disease. However, the long-distance divergence between the two kernels is balanced by the primary infection rate e. This is shown with an illustrative example in Figure S1 (see also Text S1 for details), where exponential and Cauchy kernels are used to generate spatial maps of the infectious pressure from a given experimental snapshot of site D2 ( Figure S1(A) ). When only secondary infection is considered, clear differences between the two kernels emerge at long distances ( Figures S1(B-C) ), but the differences disappear, yielding virtually identical maps, when adding the effect of the external infection rate e (Figures S1(D-E) ). We draw the following conclusion: that the scale of our observations is too small to choose unambiguously between the two dispersal kernels, as the potential effect of long-range dispersal within a census site is confounded by the presence of external infection. Gottwald et al. [53] found that a power law dispersal model was superior to an exponential model for the spread of ACC in 203 citrus plots in Brazil, following the introduction of the leaf miner. In the absence of the leaf miner, however, dispersal of ACC was adequately described by an exponential model, which is in agreement with our findings; moreover, none of the models considered in [53] included external infection.