The results presented here extend an initial (non-spatial) analysis as published by Batchelor et al 
. A spatial analysis was conducted, in which a generalised linear geostatistical model was applied with Bayesian implementation, as described by Diggle et al 
. This method allows the assessment of covariate effects while modelling the residual spatial autocorrelation explicitly and incorporating uncertainty in the input data and model parameters. This approach allowed a more robust assessment of covariate effects, with more accurate parameter and significance estimates than those obtained using non-spatial methods. The results provide significant support for the hypothesis that Rhodesian HAT was introduced into Kaberamaido and Dokolo districts via the movement of infected livestock. In addition, the significant relationships between HAT prevalence and environmental, climatic and social factors detected using the non-spatial regression have been clarified.
Following on from the non-spatial logistic regression methods discussed in Batchelor et al 
, many of the covariates that retained significance in the multivariate non-spatial regression model of HAT prevalence lost significance in the Bayesian implementation of a spatial logistic regression model. As a starting point for the spatial model, it would have been preferable to include all covariates from the final fitted non-spatial logistic regression model. Any covariates which did not retain statistical significance when accounting for residual spatial autocorrelation would then be removed prior to the final fitting of the model. However, when including all covariates, problems with the convergence and mixing of the MCMC algorithms were encountered. It is likely that the problematic MCMC performance with the full multivariate model was due to correlation and redundancy of some of the covariates and potentially also difficulties in estimating a large number of parameters at the same time.
Five covariates did not retain statistical significance during the univariate spatial regression and one (NDVI phase of annual cycle) did not retain significance during the multivariate spatial regression, indicating that the non-spatial model may have inflated the significance of covariates and produced inaccurate parameter estimates. The final spatial model included three covariate effects: distance to the closest livestock market, distance to the closest health centre and minimum LST. These results, using a more robust assessment of covariate effects, provide considerable strength to the hypothesis that the movement of infected, untreated livestock from endemic areas resulted in the introduction of T. b. rhodesiense
to Kaberamaido and Dokolo districts. Previous research has established that the introduction of Rhodesian HAT transmission within Soroti district (which neighbours the study area) was due to movements of untreated cattle from endemic areas through a local livestock market 
. The results discussed here, supported by the findings discussed in Batchelor et al 
, strongly indicate a similar occurrence in Kaberamaido and Dokolo districts; T. b. rhodesiense
is likely to have been introduced to Dokolo and Kaberamaido via the continued movement of untreated livestock, despite the introduction of a law requiring the treatment of livestock from endemic areas, prior to sale 
Within the study area, it is problematic to separate the effects of differential utilisation of the HAT treatment centre, where those living closer are more likely to travel there for diagnosis and treatment than those living further away, from the purposeful siting of the treatment centre within the area most affected by HAT. Following the detection of a number of cases in Kaberamaido district in 2004, appropriate training and equipment were provided to one hospital within the area. The facility was selected based on a number of criteria, including the location within the affected area. Due to this difficulty, the distance to the closest health centre of any kind was used rather than distance to the HAT treatment centre. The significance of this variable in the spatial regression model highlights the importance of accessibility to health services as has been shown previously 
. The observed protective effect of living further from a health centre may indicate a confounding effect, with individuals living in more remote areas and further from health care services being less likely to access treatment and, thus, be diagnosed with and treated for HAT.
Minimum LST was observed to be a risk factor for HAT, with higher prevalence in areas with higher minimum LST. Minimum LST is calculated using measurements of radiance modified by the atmosphere in several spectral wavebands and varies depending on climate and also landcover properties (e.g. amount of vegetation, urbanisation or soil moisture) 
. The size of the study area (approximately 60 km by 60 km) suggests that the observed correlations are more likely to be due to the heterogeneous landcover profile and soil and vegetation moisture content than to climatic variability across the two districts, although the precise interpretation of this mechanism is not clear. Further work is planned to disentangle the effects of climate and landcover; utilising finer spatial resolution climatic data and encompassing a larger study area, the research will investigate the dynamic nature of the distribution of HAT and relate this to climatic, environmental and social covariates (including temperature, rainfall and landcover classes).
When the performance of the spatial regression model was compared with that of the non-spatial model (one-step model of prevalence as discussed in Batchelor et al 
), the predictions from the spatial model are seen to be more accurate. The correlation between observed and fitted prevalence for the non-spatial model was 0.58, compared with a correlation of 0.95 for the spatial model. The absolute mean error for the non-spatial model was 0.13%; double that of the spatial model (0.064%, calculated based on prevalence per 100 population). Despite the increase in accuracy gained by modelling the residual spatial autocorrelation after accounting for covariate effects, there was still a tendency to over-predict in zero prevalence villages and also to under-predict in high prevalence villages. The over-prediction in zero prevalence villages indicates the presence of extra-binomial variation (greater variability in the observations than can explained by the model) whereby additional unmeasured factors may be influencing the spatial heterogeneity of HAT prevalence within small areas. From the observed prevalence it can be seen that within the main ‘focus’ of infection there are several zero prevalence villages interspersed amongst high prevalence villages, which are not explained adequately by the spatial regression model. The estimates of model uncertainty (95% CrIs) also highlight this, with larger predictive uncertainty in the areas with higher predicted prevalence as can be seen in . This non-constant variance in the error is known as heteroscedasticity. Future research as described above aims to deal with these issues by utilising a wider range of covariate datasets, with finer spatial resolutions.
Although these methods have taken into account the effect of health care accessibility on the spatial distribution of reported HAT, underreporting is well documented 
, with evidence suggesting that for every Rhodesian HAT case that dies within the health care system, another 11 cases will go undetected and therefore untreated, resulting in death 
. Underreporting of HAT causes serious problems for the estimation of disease burden, determination of the spatial extent of disease transmission and the prioritisation of resources, and also impacts on research conducted using data acquired from passive case detection. However, the lack of a rapid, cheap and easy to use diagnostic test for T. b. rhodesiense
, combined with the very low prevalence of disease in affected areas, makes active screening a difficult and expensive task for the detection of very few cases of disease. Further work which is currently being planned includes active population screening in a sample of villages; this data will be compared with hospital records to ascertain the proportion of Rhodesian HAT patients that are not accessing treatment and to allow estimation of the true burden of disease in affected areas.
The research described utilised a variety of data sources providing information relevant to the distribution of the tsetse fly vector and, thus, also the distribution of Rhodesian HAT. However, accurate tsetse distribution or density data were not available for the study area, although the explicit inclusion of information on the spatial distribution of tsetse may have resulted in improved predictive power and provided further information on the determinants influencing the spatial heterogeneity in HAT prevalence within the main focus of disease. Additional factors that may play an important role in the observed spatial heterogeneity of HAT within Uganda include demographic factors, migration and human movement and behaviour patterns, due to their influence on the frequency of interaction between humans, tsetse and livestock. Although human migration has the potential to introduce T. b. rhodesiense
to previously unaffected areas, in this situation it seems unlikely to have occurred due to the strong evidence supporting the theory of introduction via livestock movements. Additionally, the transmission of T. b. rhodesiense
normally occurs between reservoir hosts (i.e. cattle) with only sporadic transmission to humans 
The current research has demonstrated the application of Bayesian geostatistical modelling to the spatial distribution of HAT within a small area of Uganda. The more robust results provide strengthened evidence of the role of livestock trade in the continued spread of Rhodesian HAT within Uganda and the utility of this methodology for the prediction of HAT prevalence based on external covariates has also been demonstrated. The dataset used in this situation covered a relatively small area (two districts) with as complete a dataset as possible (all but two villages were geo-referenced, and all HAT cases that could be matched to a village of residence were used). The predictive power of this model over larger areas (i.e. out with the initial study area) is constrained due to the limited area from which the observed data came. To allow the full exploitation of these methods, future work will focus on a larger study area using a sample of villages. This will allow an investigation of HAT prevalence in relation to wider covariate ranges and will allow extrapolation over larger areas. The Bayesian implementation of model-based geostatistics as described here is computationally expensive and can be time consuming, but the application of such methods to epidemiological research is being assisted by a growing base of knowledge and expertise, along with the creation of more efficient algorithms 
. The utility of such methods for the accurate estimation of disease burden and the spatial targeting of control measures has been demonstrated in the literature by a variety of applications at local, national, regional and continental scales including malaria 
, schistosomiasis 
and trachoma 
The research presented here illustrates the importance of spatial autocorrelation in epidemiological variables; the use of non-spatial logistic regression analysis resulted in a model with a large number of covariates, complicating the interpretation of their effects. The use of a generalised linear geostatistical modelling framework, which models the residual autocorrelation after accounting for covariate effects, gave more precise and less biased parameter and significance estimates, with only three covariates retaining significance in the final model. The Bayesian implementation of the method allowed the incorporation of uncertainty in each of the model parameters from the posterior distributions and from the definition of a random variable. By carrying out the spatial-regression analysis, the quantified relationships between HAT prevalence and significant covariates can be more confidently described and interpreted. The predictive accuracy was also increased by using the spatial regression when compared to the non-spatial logistic regression analysis. These results strengthen the evidence in support of the hypothesis generated by the analysis discussed in Batchelor et al 
; that the movement of untreated, infected livestock from endemic areas resulted in the introduction of Rhodesian HAT to the study area.