We obtained good chain convergence for all models (
< 1.1 for all variables in all models). Sampling the prior distributions for β
on a log-transformed scale was essential for obtaining reasonable estimates of the effective number of parameters (pD).
Of the 28 models tested (), the highest ranked model included a probability of transmission that was a function of both inter-village distance and the number of dogs in the village receiving infection, and where the probability of acquiring an external infection declined as a function of distance to neighbouring districts. However, three other models performed similarly well (ΔDIC < 2 relative to the top model) and therefore also warrant consideration. We infer from these four models that there is strong support for the role of village distances and the size of the village receiving infection in driving transmission dynamics (components of all four top-ranked models), but weaker support for the role of the size of the village transmitting infection (a component only of the models ranked third and fourth).
Table 1. Summary of competing patch-occupancy models, the number of parameters in the model, the effective number of parameters (pD) and the difference in the deviance information criteria value (ΔDIC) relative to the highest ranked model (DIC = 1839). (more ...)
Overall, there was most support for the district source of external infection (), especially at the lowest rate of infection (2 per year). At the higher rates of infection, the district and random-source models of external infection performed similarly: although the district model had a lower DIC value in five out of six comparisons (models 1–3, for rates 6 and 10 per year), the difference was generally less than 2. We found only weak support for the wildlife protected area source models, which consistently ranked lower than the other source models for each model and rate combination.
There was also strongest support for the highest rate of external infection (). On average, 10 external infections per year would account for 24.7 per cent (60 of 243 over the 6 year study period) of all observed occurrences (we reiterate that occurrence is not a direct measure of incidence). However, the inferences regarding the source of external infection and the important components of transmission dynamics were consistent among the three external infection rates ().
Although the highest ranked models have different structures and, therefore, do not all share the same set of parameters, there was high consistency in parameter values among these models ( and the electronic supplementary material, table A.2). The implications of the parameter values on the probability of transmission are shown in . Although this figure is based on the parameter values of the most complex model, which ranked second, it is representative for all four top-ranked models. For all models, the probability of transmission is negatively associated with the distance between villages and positively associated with the population size of dogs in the village receiving transmission. The probability of acquiring external infection declines with distance from the district boundary in the district-source models. Finally, the population size of dogs in the village transmitting infection is important only for small populations, whereby very small populations (less than 150 dogs) have a lower probability of transmission. Validation based on stochastic simulations indicate that, of the 75 villages, all but 3, 3, 1 and 2 villages in the four highest ranked models, respectively, fell within the 95% CI of the expected values (electronic supplementary material, figure A.1). These exceptions exceeded the upper 95% CI by a count of one occupancy in all cases.
Table 2. Estimated mean parameter values and 95% CI for the four highest ranked rabies disease transmission models. (A dash represents a model in which that parameter was omitted. The full table is provided in the electronic supplementary material, table A.2.) (more ...)
Figure 4. The relative probability of transmission between villages is negatively associated with (a) distance between villages and (b) distance to neighbouring districts, and positively associated with the population size of dogs in the village (c) receiving and (more ...)
Within the set of villages that were targeted for vaccination (all but five of the 75 villages in SD) and in the 12 month period following a vaccination campaign, vaccination reduced the occurrence of rabies by 57.3 per cent (59.0, 51.9, 60.0 and 58.1% for the four highest ranked models, respectively) relative to the occurrence predicted if no vaccination had occurred (). Under the alternative assumption that regional-scale vaccination occurred (thereby eliminating the external infection source after six months), vaccination reduced the occurrence of rabies by 80.9 per cent (81.7, 83.9, 79.0 and 78.9%, respectively) relative to the unvaccinated population. Vaccination also reduced the variance in the size of outbreaks (for instance, the standard deviation in the count of occurrences was reduced from 51.4, 38.6, 54.8 and 51.7 to 17.4, 16.0, 19.1 and 19.4, respectively, for each of the four highest ranked models). Over the entire district, and including all months following the first vaccination campaign, vaccination reduced the occurrence of rabies by 50.0 per cent (51.0, 44.9, 52.6 and 51.5%, respectively), and assuming regional-scale vaccination, the occurrence of rabies was reduced by 81.7 per cent (82.2, 84.1, 80.3 and 80.3%, respectively).
Figure 5. Frequency of disease occurrence based on 1000 stochastic simulations of the four highest ranking models (different line styles) among the 75 villages in Serengeti District. To quantify vaccination efficacy, rabies occurrence was quantified in unvaccinated (more ...)
Explicitly assuming that the reporting probability is only 60 or 80 per cent relative to perfect reporting (100%) resulted in a marginal increase in the estimates of all parameters (electronic supplementary material, figure A.2 and table A.3). This corresponds to a reduction in the spatial transmission kernel (a reduced probability of transmission over longer inter-village distances), and an increase in the probability of a village receiving infection as population size increases, for all four top-ranked models. For the models with the neighbouring district source of infection, there was a decrease in the spatial transmission kernel from that source. Finally, there was also a reduced effect of population size on the probability of transmission from a source village in model 3 (the full model).