In general, inference problems for disease outbreak data are complicated by the facts that (i) the data are inherently dependent and (ii) the data are usually incomplete in the sense that the actual process of infection is not observed. We adopt a Bayesian approach and apply Markov Chain Monte Carlo (MCMC) methods in order to make inference for the parameters of interest (such as infection and removal rates). We show that once the size of the data set ncreases, the standard methods perform poorly. Therefore, apart from centered reparameterisation we extend the Non-Centered and partially Non-Centered algorithms presented in Neal and Roberts (2005). Finally, we adopt a fully Bayesian approach to analyze the Foot-and-Mouth disease occurred in 2001 in the UK and also discus modelling approaches for a potential Avian Influenza outbreak in the poultry industry of the UK.