  Multivariate count time series models are an important tool for the analysis
and prediction of infectious disease spread. We consider the endemic-epidemic
framework, an autoregressive model class for infectious disease surveillance
counts, and replace the default autoregression on counts from the previous time
period with more flexible weighting schemes inspired by discrete-time serial
interval distributions. We employ three different parametric formulations, each
with an additional unknown weighting parameter estimated via a profile
likelihood approach, and compare them to an unrestricted nonparametric
approach. The new methods are illustrated in a univariate analysis of dengue
fever incidence in San Juan, Puerto Rico, and a spatio-temporal study of viral
gastroenteritis in the twelve districts of Berlin. We assess the predictive
performance of the suggested models and several reference models at various
forecast horizons. In both applications, the performance of the
endemic-epidemic models is considerably improved by the proposed weighting
schemes.
