Seasonal patterns, as they occur in time series of infectious disease surveillance counts, are frequently modelled using a superposition of sine and cosine functions. However, in some cases this might be too simple. We propose the use of circular second order random walks instead and extend this approach to multivariate time series of counts. A correlated Gaussian Markov random field
approach combines a uniform correlation matrix with a circular random walk to allow the seasonal pattern to be similar across regions, say, but not identical.
Thus, spatially-varying disease onsets may be accounted for. The methodology is applied to weekly number of deaths from in uenza and pneumonia in nine major regions of the USA.