Flexible Bayesian modeling of non-equidispersed counts with penalized complexity priors in disease incidence studies

Counts in epidemiology often deviate from equidispersion and exhibit spatial, temporal, and nonlinear structure that the Poisson model cannot accommodate. We introduce a gamma-count structured additive regression model that strategically integrates penalized complexity priors in two critical aspects: (i) a principled penalized complexity prior on the dispersion parameter of the gamma-count distribution, which naturally shrinks toward the base Poisson model when the data support equidispersion, and (ii) scale-dependent penalized complexity hyperpriors on the smoothing variances for nonlinear, spatial, and temporal effects. By formulating the model within a latent Gaussian framework, we enable efficient approximate Bayesian inference through integrated nested Laplace approximations. Simulation studies across under-, equi-, and over-dispersed regimes show that the penalized complexity prior for dispersion parameter combined with scale-dependent hyperpriors yields accurate estimation of dispersion and smooth effects, favorable predictive scores, and robust inference. In empirical applications to larynx cancer mortality in Germany, COVID-19 incidence in Georgia (USA), and lung and bronchus cancer in Iowa (USA), the gamma-count structured additive regression model exhibits competitive or enhanced fit relative to Poisson and negative binomial counterparts, while elucidating interpretable nonlinear and spatial structures. This framework delivers robust, spatially resolved estimates of disease burden in the presence of non-equidispersion, thereby facilitating evidence-based resource allocation, epidemiological surveillance, and monitoring of health disparities, contributing to Sustainable Development Goals (SDGs) 3 (Good Health and Well-Being) and 10 (Reduced Inequalities). For geographically targeted analyses, it further supports informed decision-making in urban and community planning, aligning with SDG 11 (Sustainable Cities and Communities).