  This paper aims to extend the Besag model, a widely used Bayesian spatial
model in disease mapping, to a non-stationary spatial model for irregular
lattice-type data. The goal is to improve the model's ability to capture
complex spatial dependence patterns and increase interpretability. The proposed
model uses multiple precision parameters, accounting for different intensities
of spatial dependence in different sub-regions. We derive a joint penalized
complexity prior for the flexible local precision parameters to prevent
overfitting and ensure contraction to the stationary model at a user-defined
rate. The proposed methodology can be used as a basis for the development of
various other non-stationary effects over other domains such as time. An
accompanying R package 'fbesag' equips the reader with the necessary tools for
immediate use and application. We illustrate the novelty of the proposal by
modeling the risk of dengue in Brazil, where the stationary spatial assumption
fails and interesting risk profiles are estimated when accounting for spatial
non-stationary.
