# Nowcasting COVID-19 Statistics Reported with Delay: A Case-Study of Sweden and the UK

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## Abstract

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## 1. Introduction

#### Reporting of COVID-19 Statistics in Sweden

## 2. Materials and Methods

- It does not relay any previous trend in the data,
- It allows the generation of prediction intervals for the uncertainty of daily frequencies,
- These uncertainty estimates can be carried over to epidemiological models to increase realism.

## 3. Applying the Model to COVID-19 in Sweden and the UK

#### Model Performance

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Model

#### Appendix A.1. Notation

Variable Name | Dimension | Description |

$\mathit{d}$ | $T\times 1$ | ${d}_{i}$ is the number of deaths that occurred on the day i. |

$\mathit{r}$ | $T\times T$ | ${r}_{ij}$ is number of death recorded for day i at day j. Note that ${r}_{ij}$ for $i<j$ is not defined. |

$\mathit{p}$ | $T\times T$ | ${p}_{ij}$ is the probability of that a death for day i not yet recorded is recorded at day j. Note that ${p}_{ij}$ for $i<j$ is not defined. |

$\mathit{\alpha}$ | $K\times 1$ | latent prior parameter for $\mathit{p}$ |

$\mathit{\beta}$ | $K\times 1$ | latent prior parameter for $\mathit{p}$ |

${\mathit{\alpha}}^{H}$ | $2\times 1$ | parameter for the probability, $\mathit{p}$ for holiday adjustment |

${\mathit{\beta}}^{H}$ | $2\times 1$ | parameter for the probability, $\mathit{p}$ for holiday adjustment |

$\mu $ | $T\times 1$ | ${\mu}_{i}$ is the intensity of the expected number of deaths at day i |

${\sigma}^{2}$ | $1\times 1$ | variation of the random walk prior of the log intensity |

$\varphi $ | $1\times 1$ | overdispersion parameter for the negative binomial distribution |

${p}_{0}$ | $1\times 1$ | probability of reporting for a low reporting event |

$\pi $ | $1\times 1$ | probability of a low reporting event |

#### Appendix A.2. Likelihood

Reported Date | |||||
---|---|---|---|---|---|

Death Date | ${r}_{11}$ | ${r}_{12}$ | ⋯ | ⋯ | ${r}_{1T}$ |

${r}_{22}$ | ⋯ | ⋯ | ${r}_{2T}$ | ||

${r}_{33}$ | ⋯ | ${r}_{3T}$ | |||

⋱ | ⋮ | ||||

${r}_{TT}$ |

#### Appendix A.3. Priors

#### Appendix A.4. Full Model

## Appendix B. Inference

- First, we sample $\mathit{\alpha},\mathit{\beta},{\mathit{\alpha}}^{H},{\mathit{\beta}}^{H}|\mathit{d},\mathit{r}$. Using the fact that one can integrate out p in the model, $\mathit{d}|\mathit{\alpha},\mathit{\beta},{\mathit{\alpha}}^{H},{\mathit{\beta}}^{H},\mathit{r},\mathit{\lambda}$ follows a Beta-Binomial distribution. Here, we also use an adaptive MALA [23] to sample from these parameters.
- To sample $\mathit{d}|\mathit{\alpha},\mathit{\beta},{\mathit{\alpha}}^{H},{\mathit{\beta}}^{H},\mathit{r},\mathit{\lambda}$, we assume that each death, ${d}_{i}$ is conditionally independent, and use a Metropolis Hastings random walk.
- To sample $\mathit{\lambda}|\mathit{d},{\sigma}^{2}$, we again use an adaptive MALA.
- Finally, we sample ${\sigma}^{2}|\mathit{d}$,and ${p}_{0},\pi $ directly, since this distribution is explicit, and $\varphi $ using an MH-RW.

#### Simplified Model

- We fit $\mathit{\alpha},\mathit{\beta}$ for the retain sampling, treating ${d}_{i}$ as known after 30 days, using a MAP estimate.
- We then run an MCMC chain on the likelihood part of ${d}_{i}$ for the fixed parameters in the previous step;$${d}_{i}|{r}_{i,i:j}\propto BB({r}_{i,i:j},{d}_{i}\mathit{\alpha},\mathit{\beta}).$$This MCMC has no mixing problem as each ${d}_{i}$ is independent of each other and one can adapt each chain separately. Note that this is the model from the previous section but with no prior on ${d}_{i}$, i.e., no log-Gaussian Cox processes.
- Since the number of deaths approximately follows a Poisson distribution, we make a Box–Cox transformation and assume that the square root of the number of deaths is approximately normal:$$\sqrt{\mathit{d}}\sim \mathcal{N}\left(\mathit{\mu},\mathrm{\Sigma}\right).$$To link the catch retain model to the Box–Cox model, we make a normal approximation of the posterior distribution. We approximate the posterior distribution with$${\sqrt{d}}_{i}|{r}_{i,i:j}\approx \mathcal{N}\left({\widehat{\mu}}_{ij},{\widehat{\sigma}}_{ij}^{2}\right).$$
- Using the above model, we get an explicit posterior distribution of the square root of the number of deaths. In order to generate predictions of the number of deaths, we simulate the square number of deaths, setting negative values to zero.

## Appendix C. Model Benchmark

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**Figure 1.**Screenshot from the daily press conference on 8 May by the Swedish Public Health Agency. The headline translates to “Number of deceased per day”. The bars show the number of deceased individuals who have been reported so far, assigned to their true date of death. The sections colored green show those who have been reported dead within the last 24 hours, while purple-colored bars represent earlier reported deaths. A 7-day rolling average of daily deaths is plotted as a black line. It stops on April 28, 10 days before the relevant date, because of the delay in reporting.

**Figure 3.**Model accuracy over time for three randomly chosen dates, compared to the constant benchmark. The gray dots indicate the actual number of reported dead until that point in time. The solid line indicates the total number that will have been reported after 30 days.

**Figure 4.**Average (January 2020 to May 2021) model and benchmark accuracy as more information becomes available.

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## Share and Cite

**MDPI and ACS Style**

Altmejd, A.; Rocklöv, J.; Wallin, J.
Nowcasting COVID-19 Statistics Reported with Delay: A Case-Study of Sweden and the UK. *Int. J. Environ. Res. Public Health* **2023**, *20*, 3040.
https://doi.org/10.3390/ijerph20043040

**AMA Style**

Altmejd A, Rocklöv J, Wallin J.
Nowcasting COVID-19 Statistics Reported with Delay: A Case-Study of Sweden and the UK. *International Journal of Environmental Research and Public Health*. 2023; 20(4):3040.
https://doi.org/10.3390/ijerph20043040

**Chicago/Turabian Style**

Altmejd, Adam, Joacim Rocklöv, and Jonas Wallin.
2023. "Nowcasting COVID-19 Statistics Reported with Delay: A Case-Study of Sweden and the UK" *International Journal of Environmental Research and Public Health* 20, no. 4: 3040.
https://doi.org/10.3390/ijerph20043040