Background: Severe acute respiratory syndrome Coronavirus 2019 (COVID-19) has been firstly detected in China at the end of 2019 and it spread in few months all over the world. Italy is the second country in the world for number of cases. In this country, the diffusion of COVID-19 has followed a peculiar spatial pattern. However, the interest of scientific community has been devoted almost exclusively to the prediction of the disease evolution over time so far. Methods: Official freely available data about the number of infected at the finest possible level of spatial areal aggregation (Italian provinces) are used to model the spatio-temporal distribution of COVID-19 infections at local level. Data time horizon ranges from 20 February 2020, which is the date when the first case not directly connected with China has been discovered in northern Italy, to 18 March 2020. An endemic-epidemic multivariate time-series mixed-effects generalized linear model for areal disease counts has been implemented to understand and predict spatio-temporal diffusion of the phenomenon. Previous literature has shown that these class of models provide reliable predictions of infectious diseases in time and space. Findings: Three subcomponents characterize the estimated model. The first is related to the evolution of the disease over time; the second is characterized by transmission of the illness among inhabitants of the same province; the third remarks the effects of spatial neighbourhood and try to capture the contagion effects of nearby areas. Focusing on the aggregated time-series of the daily counts in Italy, the contribution of any of the three subcomponents do not dominate on the others and our predictions are excellent for the whole country, with an error of 3 ‰ compared to the late available data. At local level, instead, interesting distinct patterns emerge. In particular, the provinces first concerned by containment measures are those that are not affected by the effects of spatial neighbours. On the other hand, for the provinces the are currently strongly affected by contagions, the component accounting for the spatial interaction with surrounding areas is prevalent. Moreover, the proposed model provides good forecasts of the number of infections at local level while controlling for delayed reporting. Interpretation: Social and demographic aspects can explain the dramatic involvement of some provinces. A strong evidence is found that strict control measures implemented in some provinces efficiently break contagions and limit the spread to nearby areas. While policies to contain the 1 spreading of the disease may potentially be more effective if planned considering the peculiarities of local territories, the effective and homogeneous enforcement of control measures at national level is imperative to prevent the disease control being delayed or missed as a whole. This may also apply at international level where, as it is for the EU or the USA, the internal border checks among states have largely been abolished. Funding Statement: None. Declaration of Interests: None of the authors of this paper has a financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of the paper. No competing interests are at stake and there is no conflict of interest with other people or organizations that could inappropriately influence or bias the content of the paper. Ethics Approval Statement: Data is publicly available.

This paper aims at investigating empirically whether and to what extent the containment measures adopted in Italy had an impact in reducing the diffusion of the COVID-19 disease across provinces. For this purpose, we extend the multivariate time-series model for infection counts proposed in Paul and Held (Stat Med 30(10):118–1136, 2011) by augmenting the model specification with B-spline regressors in order to account for complex nonlinear spatio-temporal dynamics in the propagation of the disease. The results of the model estimated on the time series of the number of infections for the Italian provinces show that the containment measures, despite being globally effective in reducing both the spread of contagion and its self-sustaining dynamics, have had nonlinear impacts across provinces. The impact has been relatively stronger in the northern local areas, where the disease occurred earlier and with a greater incidence. This evidence may be explained by the shared popular belief that the contagion was not a close-to-home problem but rather restricted to a few distant northern areas, which, in turn, might have led individuals to adhere less strictly to containment measures and lockdown rules.

When a regression analysis is performed on a variable that exhibits some form of spatial dependence, Gauss–Markov assumptions about the exogeneity of regressors and independence amongst error terms may be violated. If this is the case and no information is available about the location of units, point estimators and standard error estimators are typically heavily biased. The paper proposes an empirical solution that helps to mitigate spatial dependence in regression residuals when no information is available about unit positions. It illustrates the method, discusses some practical issues, provides some evidence through Monte Carlo simulations and includes an analysis of actual data from the real estate market.

The application of spatial Cliff–Ord models requires information about spatial coordinates of statistical units to be reliable, which is usually the case in the context of areal data. With micro-geographic point-level data, however, such information is inevitably affected by locational errors, that can be generated intentionally by the data producer for privacy protection or can be due to inaccuracy of the geocoding procedures. This unfortunate circumstance can potentially limit the use of the spatial autoregressive modelling framework for the analysis of micro data, as the presence of locational errors may have a non-negligible impact on the estimates of model parameters. This contribution aims at developing a strategy to reduce the bias and produce more reliable inference for spatial models with location errors. The proposed estimation strategy models both the spatial stochastic process and the coarsening mechanism by means of a marked point process. The model is fitted through the maximisation of a doubly-marginalised likelihood function of the marked point process, which cleans out the effects of coarsening. The validity of the proposed approach is assessed by means of a Monte Carlo simulation study under different real-case scenarios, whereas it is applied to real data on house prices.