  Our method extends the application of random spanning trees to cases where
the response variable belongs to the exponential family, making it suitable for
a wide range of real-world scenarios, including non-Gaussian likelihoods. The
proposed model addresses the limitations of previous spatial clustering methods
by allowing all within-cluster model parameters to be cluster-specific, thus
offering greater flexibility. Additionally, we propose a Bayesian inference
algorithm that overcomes the computational challenges associated with the
reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm by employing
composition sampling and the integrated nested Laplace approximation (INLA) to
compute the marginal distribution necessary for the acceptance probability.
This enhancement improves the mixing and feasibility of Bayesian inference for
complex models. We demonstrate the effectiveness of our approach through
simulation studies and apply it to real-world disease mapping applications:
COVID-19 in the United States of America, and dengue fever in the states of
Minas Gerais and S\~ao Paulo, Brazil. Our results highlight the model's
capability to uncover meaningful spatial patterns and temporal dynamics in
disease outbreaks, providing valuable insights for public health
decision-making and resource allocation.
