#
Forecasting COVID-19 Epidemic Trends by Combining a Neural Network with R_{t} Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Background

- lag order (p): the number of lag observations included in the model;
- degree of differencing (d): the number of times that the raw observations are differenced;
- order of moving average (q): the size of the moving average window.

## 3. Materials and Methods

#### 3.1. Dataset

#### 3.2. Preprocessing

#### 3.3. Model

#### 3.4. Postprocessing

#### 3.5. Implementation

- pandas [53]. This package is a software library for data analysis and manipulation. It includes a data structure to handle data frames efficiently. Furthermore, time-series are supported; for instance, it allows date range generation and frequency conversion, statistics, date shifting, and lagging.
- numpy [54]. This package provides support for large, multidimensional arrays and matrices, as well as a collection of high-level mathematical functions to manipulate these data structures.
- tensorflow [55]. This package is one of the most widely used end-to-end open source platforms for ML/DL.
- scikit-learn [56]. This package is a collection of tools for predictive data analysis, classification, regressions, and clustering. It supports the interoperability with other packages (e.g., numpy).
- epyestim [45]. This package is able to estimate time-varying reproduction numbers (i.e., ${R}_{t}$) from epidemic curves. It is provided in both software tool and package form. The latter also supports the Python languages.

#### 3.6. Key Performance Indicators

## 4. Results

- Type: n1-highmem-2 instance;
- CPU: 2vCPU @ 2.2 GHz;
- RAM: 13 GB;
- Backend: GCE, Python 3.

## 5. Discussion

- vs. proposed without ${R}_{t}$: 5.38%, 53.13%, 10.00%, and 14.63%;
- vs. LSTM: −10.00%, 68.75%, 18.18%, and 7.89%;
- vs. Simple RNN: 2.94%, 70.00%, 25.00%, and 20.45%;
- vs. GRU: 13.16%, 60.53%, 18.18%, and 16.67%,

## 6. Conclusions

## Key Points

- We combined an NN with the estimation of time-varying reproduction numbers during epidemics (i.e., using ${R}_{t}$ as a corrective index).
- We developed a solution that was able to handle the frequent restrictions adopted by a country (e.g., lockdowns and limitations) that could destabilize the evolution of a time series.
- We provided an effective methodology to forecast COVID-19 epidemic trends on a dataset consisting of a limited amount of information (e.g., Italy).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 2.**Figure reports the pipeline for our solution. We parsed data by retrieving the positive case registration as a time series.

**Figure 3.**The trends related to the ${R}_{t}$ estimation for Italy, the USA, France, the UK, and Sweden.

**Figure 5.**MAPE for the proposed model with and without the correction computed by using the ${R}_{t}$ estimation (i.e., proposed NN and proposed NN without ${R}_{t}$, respectively), as well as for the other models based on DL (i.e., LSTM, Simple RNN, GRU) and ARIMA. MAPE was oriented negatively (lower was better). Dataset: Italy.

**Figure 6.**RMSE, MAE, and MAPE were used to compare the proposed NN and an NN with the same configuration but without the adjustment based on ${R}_{t}$. Values related to this plot are reported in Table 5.

**Figure 7.**Figure shows a comparison between the proposed model with and without the correction computed by using the ${R}_{t}$ estimation (i.e., proposed NN and proposed NN w/o ${R}_{t}$, respectively), as well as from the test on the other models based on DL (i.e., LSTM, Simple RNN, GRU) and ARIMA. The comparison is based on MAPE, and it concerns all countries of interest: Italy, USA, France, UK, and Sweden. MAPE is oriented negatively (lower is better).

**Figure 8.**This plot shows a forecast related to the positive cases in Italy. The green line is related to the proposed solution that includes our adjustment based on ${R}_{t}$ estimation; the red line an NN based on a model built on the same configuration without ${R}_{t}$; the black line concerns the real trend for the period of interest. The X-axis reports the time points, while the Y-axis reports estimated positive cases.

**Table 1.**We obtained the information related to positive cases in Italy, the USA, France, the UK, and Sweden, through the “Our World in Data COVID-19 Cases” dataset. For the mentioned countries, it aggregated data from the sources accessed on 11 January 2022, and reported in this table. The related time series were built based on the number of daily new positive cases and whose timestamp was within the same date range (24 February 2020–11 January 2022).

Country | Organization | Data Source |
---|---|---|

Italy | Italian Civil Protection Department | github.com/pcm-dpc/COVID-19 |

USA | Center for Systems Science and Engineering, Johns Hopkins University | systems.jhu.edu |

France | French Ministry of Solidarity and Health and Public Health | data.gouv.fr/fr/datasets |

UK | Government of the UK | coronavirus.data.gov.uk |

Sweden | The Swedish Public Health Agency | experience.arcgis.com |

**Table 2.**The descriptive statistics both for new cases and total cases, in Italy, the USA, France, the UK, and Sweden. Date range: 24 February 2020–11 January 2022 (N is the number of time points for the time series, which is the number of evaluated days).

N | Mean | Std. Deviation | Std. Error | 95% Confidence Interval for Mean | Minimum | Maximum | |||
---|---|---|---|---|---|---|---|---|---|

Lower Bound | Upper Bound | ||||||||

Total Cases | Italy | 688 | 2,496,779.68 | 2,007,837.429 | 76,548.084 | 2,346,483.41 | 2,647,075.95 | 229 | 7,774,863 |

USA | 688 | 22,722,404.82 | 17,313,142.593 | 660,057.373 | 21,426,432.95 | 24,018,376.69 | 16 | 62,588,935 | |

France | 688 | 3,581,853.14 | 3,034,266.023 | 115,680.308 | 3,354,723.75 | 3,808,982.52 | 12 | 12,620,080 | |

UK | 688 | 3,729,463.11 | 3,563,954.756 | 135,874.501 | 3,462,683.98 | 3,996,242.24 | 30 | 14,766,757 | |

Sweden | 688 | 588,621.05 | 487,097.434 | 18,570.416 | 552,159.46 | 625,082.63 | 1 | 1,487,291 | |

New Cases | Italy | 688 | 11,317.11 | 22,531.009 | 859.612 | 9629.33 | 13,004.90 | 74 | 220,519 |

USA | 688 | 90,972.27 | 115,946.169 | 4420.406 | 82,293.14 | 99,651.39 | 0 | 1,383,898 | |

France | 688 | 19,238.68 | 38,431.340 | 1475.947 | 16,340.70 | 22,136.67 | 0 | 368,379 | |

UK | 688 | 21,536.37 | 29,864.863 | 1140.246 | 19,297.58 | 23,775.17 | 2 | 219,290 | |

Sweden | 688 | 2161.76 | 5199.062 | 198.212 | 1772.58 | 2550.93 | 0 | 70,641 |

**Table 3.**The descriptive statistics related to the study of the ${R}_{t}$ index in the countries of interest.

N | Mean | Std. Deviation | Minimum | 25% | 50% | 75% | Maximum | |
---|---|---|---|---|---|---|---|---|

Italy | 688 | 1.05 | 0.23 | 0.70 | 0.89 | 1.00 | 1.14 | 2.98 |

USA | 688 | 1.08 | 0.33 | 0.74 | 0.91 | 1.03 | 1.13 | 3.74 |

France | 688 | 1.32 | 2.99 | 0 | 0.91 | 1.06 | 1.24 | 56.87 |

UK | 688 | 1.07 | 0.32 | 0.59 | 0.90 | 1.04 | 1.18 | 5.95 |

Sweden | 688 | 1.11 | 0.622 | 0.02 | 0.93 | 1.06 | 1.20 | 10.04 |

**Table 4.**The MAPE indicator was used to compare the proposed NN and the following models: LSTM, GRU, Simple RNN, ARIMA. It was oriented negatively (lower was better). Dataset: Italy.

95% Confidence Interval for Mean | ||||||||
---|---|---|---|---|---|---|---|---|

N | Mean | Std. Deviation | Std. Error | Lower Bound | Upper Bound | Minimum | Maximum | |

LSTM | 10 | 0.2110 | 0.03281 | 0.01038 | 0.1875 | 0.2345 | 0.18 | 0.28 |

GRU | 10 | 0.2490 | 0.04533 | 0.01433 | 0.2166 | 0.2814 | 0.20 | 0.31 |

Simple RNN | 10 | 0.1960 | 0.05125 | 0.01621 | 0.1593 | 0.2327 | 0.16 | 0.33 |

Proposed NN w/o${R}_{t}$ | 10 | 0.1840 | 0.01713 | 0.00542 | 0.1717 | 0.1963 | 0.16 | 0.21 |

Proposed NN | 10 | 0.1510 | 0.01101 | 0.00348 | 0.1431 | 0.1589 | 0.14 | 0.17 |

**Table 5.**RMSE, MAE, and MAPE were used to compare the proposed NN and an NN with the same configuration but without the adjustment based on ${R}_{t}$. It was oriented negatively (lower was better). The figures show the proposed solution in green, and the other models are in blue. Furthermore, we reported our solution by excluding the use of ${R}_{t}$ in yellow to demonstrate the benefits of using the latter. Dataset: Italy.

95% Confidence Interval for Mean | |||||||||
---|---|---|---|---|---|---|---|---|---|

N | Mean | Std. Deviation | Std. Error | Lower Bound | Upper Bound | Minimum | Maximum | ||

RMSE | |||||||||

Proposed NN w/o${R}_{t}$ | 10 | 8040.9180 | 1181.94026 | 373.76233 | 7195.4089 | 8886.4271 | 6224.14 | 9763.09 | |

Proposed NN | 10 | 4522.5080 | 2645.23581 | 836.49701 | 2630.2203 | 6414.7957 | 1407.01 | 7123.64 | |

MAE | |||||||||

Proposed NN w/o${R}_{t}$ | 10 | 3218.7000 | 922.85482 | 291.83232 | 2558.5294 | 3878.8706 | 2122.75 | 4813.67 | |

Proposed NN | 10 | 2112.4220 | 1169.72987 | 369.90106 | 1275.6477 | 2949.1963 | 918.19 | 4010.12 | |

MAPE | |||||||||

Proposed NN w/o${R}_{t}$ | 10 | 0.1840 | 0.01713 | 0.00542 | 0.1717 | 0.1963 | 0.16 | 0.21 | |

Proposed NN | 10 | 0.1510 | 0.01101 | 0.00348 | 0.1431 | 0.1589 | 0.14 | 0.17 |

**Table 6.**MAPE was reported for the proposed model with and without the correction computed by using the ${R}_{t}$ estimation, as well as for the other models based on DL (i.e., LSTM, Simple RNN, GRU). The forecasting was performed 10 times for each model and the resulting average value was reported. MAPE was oriented negatively (lower was better).

Proposed NN | Proposed NN w/o ${\mathit{R}}_{\mathit{t}}$ | LSTM | Simple RNN | GRU | |
---|---|---|---|---|---|

USA | 0.33 | 0.39 | 0.30 | 0.34 | 0.38 |

France | 0.30 | 0.64 | 0.96 | 1.00 | 0.76 |

UK | 0.09 | 0.10 | 0.11 | 0.12 | 0.11 |

Sweden | 0.35 | 0.41 | 0.38 | 0.44 | 0.42 |

**Table 7.**The paired-sample t-test before and after applying the ${R}_{t}$ adjustment to the proposed solution. Dataset: Italy.

Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | p-Value | |
---|---|---|---|---|---|

Lower | Upper | ||||

−646.23 | 2690.34 | 247.67 | −1136.72 | −155.74 | 0.01 |

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Cinaglia, P.; Cannataro, M.
Forecasting COVID-19 Epidemic Trends by Combining a Neural Network with *R _{t}* Estimation.

*Entropy*

**2022**,

*24*, 929. https://doi.org/10.3390/e24070929

**AMA Style**

Cinaglia P, Cannataro M.
Forecasting COVID-19 Epidemic Trends by Combining a Neural Network with *R _{t}* Estimation.

*Entropy*. 2022; 24(7):929. https://doi.org/10.3390/e24070929

**Chicago/Turabian Style**

Cinaglia, Pietro, and Mario Cannataro.
2022. "Forecasting COVID-19 Epidemic Trends by Combining a Neural Network with *R _{t}* Estimation"

*Entropy*24, no. 7: 929. https://doi.org/10.3390/e24070929