  We introduce an extended generalised logistic growth model for discrete
outcomes, in which a network structure can be specified to deal with spatial
dependence and time dependence is dealt with using an Auto-Regressive approach.
A major challenge concerns the specification of the network structure, crucial
to consistently estimate the canonical parameters of the generalised logistic
curve, e.g. peak time and height. Parameters are estimated under the Bayesian
framework, using the {\texttt{ Stan}} probabilistic programming language. The
proposed approach is motivated by the analysis of the first and second wave of
COVID-19 in Italy, i.e. from February 2020 to July 2020 and from July 2020 to
December 2020, respectively. We analyse data at the regional level and,
interestingly enough, prove that substantial spatial and temporal dependence
occurred in both waves, although strong restrictive measures were implemented
during the first wave. Accurate predictions are obtained, improving those of
the model where independence across regions is assumed.
