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.

As the differences are mainly driven by changes in the transmission rate, b ( Figures 3D,H) , we tested whether the epidemics could be adequately predicted using model M V , which incorporates a long-term decreasing linear trend: b(t) = b 0 (12vt) (cf. Table 1 ). However, the linear trend is confounded by large monthly fluctuations (Figure 3H) , and a reliable estimate of the decay rate v was only possible when at least 12 snapshots were Table 1 ) are shown for four different intervals (each delimited by times t 0 and t 1 , with t 1 = t 0 +6 months). Parameter estimates obtained for each interval are used to run the model 1000 times between t 0 and t 1 , and summary statistics calculated from the output are compared with the data. A, C, F, I Distributions of simulated disease progress between t 0 and t 1 (shaded areas, with black corresponding to the median and different levels of gray to different quantiles) compared to observed disease progress (red circles; empty black circles mark data not used in the comparison). The total number of hosts in site D1 is N = 6056. B, D, G, J The autocorrelation function at time t 1 , C t1 (d), estimated from observed data (thick red line), together with the 95% bootstrapped confidence interval (thin red lines), is compared with the distribution of C t1 (d) estimated from simulated epidemics (shaded gray, same as for panels A, C, F, I). Dashed cyan lines represent the 95% significance interval found with random labelling techniques. E, H, K Time-lagged statistics calculated between times t 0 and t 1 , R t1 t0 (d). Thick red lines are R t1 t0 (d) estimated from observed data, thin red lines mark the 95% confidence interval, dashed cyan lines mark the 95% significance intervals, and distributions of R t1 t0 (d) estimated from simulated epidemics are shown in shaded gray. doi:10.1371/journal.pcbi.1003587.g005 from the ''full'' estimation, cf. solid gray line in Figures 3D, H) , and estimating only a, b 0 , and e. While very early predictions ( Figure 6B1 ) slightly under-estimate disease (with a very large credible interval), including more snapshots for estimation leads to consistent improvement of the forecast (Figures 6B2,B3) . Hence, information about a single parameter, v, leads to a stark improvement of disease prediction. We remark, however, that it was not possible to identify a single, clear environmental factor responsible for the overall decreasing trend of the time series (henceforth, we refer to the monthly series only, cf. Figure 3H) . Hence, knowing v implies advance knowledge of the behaviour of b t along the whole course of the epidemic. It is desirable to test epidemic predictions under alternative, more parsimonious assumptions about our prior information on b t . Scenario C. (Figures 6C1-C3 , 6D1-D3) We assumed to have prior information about the time of occurrence and values of the three peaks of b t (cf. Figure 3H) ; no prior information was given about the drought period. We fitted to the observation windows a Table 1 ). A1-A3 Predictions based on model M 0 , assuming no prior information. The probability distributions for predicted trajectories are shown by gray shading, with intensity of shading representing probability of occurrence. The observational data (disease snapshots) used for prediction are marked by orange circles, the last snapshot used (the prediction time) by a larger red circle, and the observational data to be predicted by white circles. The total number of hosts