methods enables model fit. A common assumption in geostatistical modeling of malaria data is the stationarity, that is the spatial correlation is a function of distance between locations and not of the locations themselves. This hypothesis does not always hold, especially when modeling malaria over large areas, hence geostatistical models that take into account non-stationarity need to be assessed. Fitting geostatistical models requires repeated inversions of the variance-covariance matrix modeling geographical dependence. For very large number of data locations matrix inversion is considered infeasible. Methods for optimizing this computation are needed. In addition, the relation between environmental factors and malaria risk is often not linear and parametric functions may not be able to determine the shape of the relationship. Nonparametric geostatistical regression models that allow the data to determine the form of the environment-malaria relation need to be further developed and applied in malaria mapping. The aim of this thesis was to develop appropriate models for non-stationary and large geostatistical data that can be applied in the field of malaria epidemiology to produce accurate maps of malaria distribution. The main contributions of this thesis are the development of methods for: (i) analyzing non-stationary malaria survey data; (ii) modeling the nonlinear relation between malaria risk and environment/climatic conditions; (iii) modeling geostatistical mortality data collected at very large number of locations and (iv) adjusting for seasonality and age in mapping heterogeneous malaria survey data. Chapter 2 assessed the spatial effect of bednets on all-cause child mortality by analyzing data from a large follow-up study in an area of high perennial malaria transmission in Kilombero Valley, southern Tanzania. The results indicated a lack of community effect of bednets density possibly because of the homogeneous characteristic of nets coverage and the small proportion of re-treated nets in the study area. The mortality data of this application were collected over 7, 403 locations. To overcome large matrix inversion a Bayesian geostatistical model was developed. This model estimates the spatial process by a subset of locations and approximates the location-specific random effects by a weighted sum of the subset of location-specific random effects with the weights inversely proportional to the separation distance. In Chapter 3 a Bayesian non-stationary model was developed by partitioning the study region into fixed subregions, assuming a separate stationary spatial process in each tile and taking into account between-tile correlation. This methodology was applied on malaria survey data extracted from the MARA database and produced parasitaemia risk maps in Mali. The predictive ability of the non-stationary model was compared with the stationary analogue and the results showed that the stationarity assumption influenced the significance of environmental predictors as well as the the estimation of the spatial parameters. This indicates that the assumptions about the spatial process play an important role in inference. Model validation showed that the non-stationary model had better predictive ability. In addition, experts opinion suggested that the parasitaemia risk map based on the nonstationary model reflects better the malaria situation in Mali. This work revealed that non-stationarity is an essential characteristic which should be considered when mapping malaria. Chapter 4 employed the above non-stationary model to produce maps of malaria risk in West Africa considering as fixed tiles the four agro-ecological zones that partition the region. Non-linearity in the relation between parasitaemia risk and environmental conditions was assessed and it was addressed via P-splines within a Bayesian geostatistical model formulation. The model allowed a separate malaria-environment relation in each zone. The discontinuities at the borders between the zones were avoided since the spatial correlation was modeled by a mixture of spatial processes over the entire study area, with the weights chosen to be exponential functions of the distance between the locations and the centers of the zones corresponding to each of the spatial processes. The above modeling approach is suitable for mapping malaria over areas with an obvious fixed partitioning (i.e. ecological zones). For areas where this is not possible, a nonstationary model was developed in Chapter 5 by allowing the data to decide on the number and shape of the tiles and thus to determine the different spatial processes. The partitioning of the study area was based on random Voronoi tessellations and model parameters were estimated via reversible jump Markov chain Monte Carlo (RJMCMC) due to the variable dimension of the parameter space. In Chapter 6 the feasibility of using the recently developed mathematical malaria transmission models to adjust for age and seasonality in mapping historical malaria survey data was investigated. In particular, the transmission model was employed to translate age heterogeneous survey data from Mali into a common measure of transmission intensity. A Bayesian geostatistical model was fitted on the transmission intensity estimates using as covariates a number of environmental/climatic variables. Bayesian kriging was employed to produce smooth maps of transmission intensity, which were further converted to age specific parasitaemia risk maps. Model validation on a number of test locations showed that this transmission model gives better predictions than modeling directly the prevalence data. This approach was further validated by analyzing the nationally representative malaria surveys data derived from the Malaria Indicator surveys (MIS) in Zambia. Although MIS data do not have the same limitations with the historical data, the purpose of the analyzes was to compare the maps obtained by modeling 1) directly the raw prevalence data and 2) transmission intensity data derived via the transmission model. Both maps predicted similar patterns of malaria risk, however the map based on the transmission model predicted a slightly higher lever of endemicity. The use of transmission models on malaria mapping enables adjusting for seasonality and age dependence of malaria prevalence and it includes all available historical data collected at different age groups
