Infectious disease counts from surveillance systems are typically observed in several administrative geographical areas. In this paper, a non-linear model for the analysis of such multiple time series of counts is discussed. To account for heterogeneous incidence levels or varying transmission of a pathogen across regions, region-specific and possibly spatially correlated random effects are introduced. Inference is based on penalized likelihood methodology for mixed models. Since the use of classical model choice criteria such as AIC or BIC can be problematic in the presence of random effects, models are compared by means of one-step-ahead predictions and proper scoring rules. In a case study, the model is applied to monthly counts of meningococcal disease cases in 94 departments of France (excluding Corsica) and weekly counts of influenza cases in 140 administrative districts of Southern Germany. The predictive performance improves if existing heterogeneity is accounted for by random effects.