  The generalised network autoregressive (GNAR) model conceptualises time
series on the vertices of a network; it has an autoregressive component for
temporal dependence and a spatial autoregressive component for dependence
between neighbouring vertices in the network. Consequently, the choice of
underlying network is essential. This paper assesses the performance of GNAR
models on different networks in predicting COVID-19 cases for the 26 counties
in the Republic of Ireland, over two distinct pandemic phases (restricted and
unrestricted), characterised by inter-county movement restrictions. Ten static
networks are constructed, in which vertices represent counties, and edges are
built upon neighbourhood relations, such as railway lines. We find that a GNAR
model based on the fairly sparse Economic hub network explains the data best
for the restricted pandemic phase while the fairly dense 21-nearest neighbour
network performs best for the unrestricted phase. Across phases, GNAR models
have higher predictive accuracy than standard ARIMA models which ignore the
network structure. For county-specific predictions, in pandemic phases with
more lenient or no COVID-19 regulation, the network effect is not quite as
pronounced. The results indicate some robustness to the precise network
architecture as long as the densities of the networks are similar. An analysis
of the residuals justifies the model assumptions for the restricted phase but
raises questions regarding their validity for the unrestricted phase. While
generally performing better than ARIMA models which ignore network effects,
there is scope for further development of the GNAR model to better model
complex infectious diseases, including COVID-19.
