## Introduction

The total impact of the COVID-19 pandemic on mortality should be the least controversial outcome to measure. However, its analysis is complicated by the lack of real time cause specific data; a potential additional issue concerns the quality of coding on the death certificates, particularly in the earlier stages of the pandemic. Furthermore, there are important differences in the recording systems, both across and within countries (e.g. in the UK, England and Wales have consistently reported data on daily deaths based on different time of recording in comparison to Scotland and Northern Ireland). In this context, estimating excess deaths for all causes at national level, with respect to past year trends has been used in several countries as an effective way to evaluate the total burden of the COVID-19 pandemic, including direct COVID-19-related, as well as indirect effects (e.g. people not being able to access health-care). At the same time, all-cause mortality is not affected by mis-coding on the death certificates.

Despite these positive features, this approach can only present global pictures of the total burden of the first wave of the infection. However, to understand the dynamics of the pandemic, we need to analyse data at sub-national level; this would allow to account for geographical differences due to the infectious nature of the disease, as well as those in the population characteristics and health system provision. Additionally, time trends can vary substantially, rendering comparisons even more complex.

The modelling is generally based on the following structure.

\(y_{jtsk}=\) number of all-cause deaths in week \(j\) of year \(t\) for NUTS3 area \(s\) and age-sex group \(k\)

\(k=1,\ldots,K=10=\) age-sex group (male/female and \(<\)40, 40-59, 60-69, 70-79, \(\)80)

\(P_{jtsk}=\) population at risk in week \(j\) of year \(t\) for NUTS3 area \(s\) and age-sex group \(k\)

\(\rho_{jtsk}=\) the risk of death (mortality rate) in week \(j\) of year \(t\) for NUTS3 area \(s\) and age-sex group \(k\)

\(z_{j}=\) dummy variable for public holiday

\(x_{jts}=\) average weekly temperature in each area

The observed counts are modelled as

\[y_{jtsk}\sim \mbox{Poisson}\left(\rho_{jtsk}P_{jtsk}\right) \qquad \qquad \log \left(\rho_{jtsk} \right) = \beta_{0t} + \beta_1 z_{j} + f(x_{jts}) + b_s + w_j\]

and

**“Unstructured” effects \(\boldsymbol\beta\)**

- \(\beta_{0t}=\beta_0+\varepsilon_t\): year-specific intercept

\(\beta_0\sim \mbox{Normal}(0, 10^{3})\): global intercept

\(\varepsilon_t\sim \mbox{Normal}(0,\tau_\varepsilon^{-1})\): unstructured random effect

\(\beta_1\sim \mbox{Normal}(0, 10^{3})\): effect of public holidays

**Non-linear effect of average weekly temperature \(f(x_{jts})\)** - RW2 model: \(x_{jts} \mid x_{(j-1)ts}, x_{(j-2)ts}, \tau_x \sim \mbox{Normal}\left(2x_{(j-1)ts}+x_{(j-2)ts},\tau_x^{-1}\right)\)

**Spatial component** - \(b_s=\frac{1}{\sqrt{\tau_b}}\left(\sqrt{1-\phi}\tau_v^{0.5} v_s+\sqrt{\phi}\tau_u^{0.5} u_s\right)\): Besag-York-Mollié (BYM)-type model

- \(v_s\sim \mbox{Normal}(0,\tau_v^{-1})\): unstructured random effect
- \(u_s \mid \boldsymbol{u}_{-s}\sim \mbox{Normal}\left(\frac{\sum_{r=1}^Rn_{rs}u_r}{\sum_{r=1}^R n_{rs}},\frac{1}{\tau_u \sum_{r=1}^R n_{rs}}\right)\): spatially structured random effect
- \(\phi \in [0,1]\): mixing parameter (measures proportion of variance explained by the structured effect)

**Temporal component (non-linear weekly effect)** - RW1 model (accounts for seasonality): \(w_j \mid w_{j-1}, \tau_w \sim \mbox{Normal}(w_{j-1},\tau_w^{-1})\)

Last updated: 20 November 2023