Table 1.

Description of the 12 main models that were considered for the comparison in terms of the specification of the logarithm of the relative risk, $log(r_{\mathit{it}})$, corresponding to region i ($i=1,\dots ,42$) on day t ($t=1,\dots ,182$)

Model	$log(r_{\mathit{it}})$
Model 1	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^3{\beta }_j{x}_{\mathit{jit}}$
Model 2	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}$
Model 3	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_{w(t)}+{\varphi }_{w(t)}$
Model 4	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_t+{\varphi }_t$
Model 5	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_{w(t)}+{\varphi }_{w(t)}+{\delta }_{iw(t)}$ (I)
Model 6	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_{w(t)}+{\varphi }_{w(t)}+{\delta }_{iw(t)}$ (II)
Model 7	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_{w(t)}+{\varphi }_{w(t)}+{\delta }_{iw(t)}$ (III)
Model 8	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_{w(t)}+{\varphi }_{w(t)}+{\delta }_{iw(t)}$ (IV)
Model 9	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_t+{\varphi }_t+{\delta }_{\mathit{it}}$ (I)
Model 10	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_t+{\varphi }_t+{\delta }_{\mathit{it}}$ (II)
Model 11	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_t+{\varphi }_t+{\delta }_{\mathit{it}}$ (III)
Model 12	$\mu +log(E_{\mathit{it}})+{\sum }_{j=1}^4{\beta }_j{x}_{\mathit{jit}}+u_i+v_i+{\gamma }_t+{\varphi }_t+{\delta }_{\mathit{it}}$ (IV)

For all the models, $\mu$ denotes the intercept of the model, E the number of expected cases, and $x_j$ ($j=1,\dots ,4$) the covariates. In addition, $u_i$ and $v_i$ represent the structured and unstructured random spatial effect of the model, ${\gamma }_t$ and ${\varphi }_t$ the structured and unstructured random temporal effect, and ${\delta }_{\mathit{it}}$ the random spatio-temporal effect. The symbols I, II, III, IV denote the type of spatio-temporal interaction (for either ${\delta }_{iw(t)}$ or ${\delta }_{\mathit{it}}$) considered in Models 5 to 12, according to Table 2