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The first part of this thesis is dedicated to the study of edge effects in maps of disease.  The aim of the analyses is to find out how the estimation of the risk from a disease near boundaries can be affected by the boundary position.  The behaviour of a selection of models for disease mapping is evaluated when different edge conditions exist in the data. Disease mapping plays an important role in monitoring the health of a community.  Plotting new cases on a map is a frequently used technique for monitoring the spread of infectious diseases and from a statistical point of view it is relevant to consider how statistical methods can be developed or employed to aid the task of surveillance.  In the second part of this thesis, methodological and practical issues in developing a rapid response in a spatial surveillance system are discussed.  In particular, I review and propose methods for the detection of changes.  A simulation study is set up to assess if these methods are good at detecting changes in risk over space and time.  An application to a real data set is also given. Surveillance should be performed as quickly as possible but complex Bayesian models require the use of sampling methods to provide estimates of posterior expectations, and these estimates may be computationally expensive to obtain.  To aid this, special computational approaches can be considered.  One option is to resample the output form initial iterations to provide reweighted estimates as time protocols.  This is known a filtration or sequential Monte Carlo.  In the third part of this thesis I review the use of sequential Monte Carlo methods (in particular, the Resample-Move algorithm) for dynamic systems, focusing on their use in a surveillance context.  This is followed by an application to a real data set where a comparison between the use of McMC methods and the Resample-Move algorithm is carried out.

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods.