# Inference on COVID-19 Epidemiological Parameters Using Bayesian Survival Analysis

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. The Study Population

^{2}, considering the 300,000 km

^{2}of the entire Italian territory). However, the susceptible population of this area is about half a million of people, a sufficiently large number for obtaining statistically significant results. Furthermore, this empirical analysis can prove how data can be used to describe the evolution of the epidemic in a restricted area and support decisions of local institutions which can be easily extended to national level.

#### 2.1.1. Influence Analysis by Age Factor

#### 2.1.2. Influence Analysis by Period of Epidemic

#### 2.2. Statistical Analysis

- TVC ${T}_{i}^{E}$ is defined as the date of the first positive swab test; TVC ${T}_{i}^{*}$ is the recovery date for recovered people or death date for deceased people;
- LoS ${T}_{i}^{E}$ is defined as the hospitalization day; LoS ${T}_{i}^{*}$ is the discharge date for recovered people or death date for deceased people;
- DH ${T}_{i}^{E}$ is the first positive swab test and DH ${T}_{i}^{*}$ is the hospitalization day.

## 3. Results and Discussion

#### 3.1. TVC Results

#### 3.2. LoS Results

#### 3.3. DH Results

## 4. Application to Social SIR Model

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cucinotta, D.; Vanelli, M. WHO declares COVID-19 a pandemic. Acta Bio Med. Atenei Parm.
**2020**, 91, 157. [Google Scholar] - World Health Organization. COVID-19 Weekly Epidemiological Update, 6 April 2021; World Health Organization: Geneva, Switzerland, 2021. [Google Scholar]
- Hsiang, S.; Allen, D.; Annan-Phan, S.; Bell, K.; Bolliger, I.; Chong, T.; Druckenmiller, H.; Huang, L.Y.; Hultgren, A.; Krasovich, E.; et al. The effect of large-scale anti-contagion policies on the COVID-19 pandemic. Nature
**2020**, 584, 262–267. [Google Scholar] [CrossRef] [PubMed] - van Oosterhout, C.; Hall, N.; Ly, H.; Tyler, K. COVID-19 evolution during the pandemic-Implications of new SARS-CoV-2 variants on disease control and public health policies. Virulence
**2021**, 12, 507–508. [Google Scholar] [CrossRef] [PubMed] - Pellis, L.; Scarabel, F.; Stage, H.B.; Overton, C.E.; Chappell, L.H.; Lythgoe, K.A.; Fearon, E.; Bennett, E.; Curran-Sebastian, J.; Das, R.; et al. Challenges in control of Covid-19: Short doubling time and long delay to effect of interventions. arXiv
**2020**, arXiv:2004.00117. [Google Scholar] - Bonacini, L.; Gallo, G.; Patriarca, F. Identifying policy challenges of COVID-19 in hardly reliable data and judging the success of lockdown measures. J. Popul. Econ.
**2021**, 34, 275–301. [Google Scholar] [CrossRef] [PubMed] - Faes, C.; Abrams, S.; Van Beckhoven, D.; Meyfroidt, G.; Vlieghe, E.; Hens, N. Time between symptom onset, hospitalisation and recovery or death: Statistical analysis of belgian covid-19 patients. Int. J. Environ. Res. Public Health
**2020**, 17, 7560. [Google Scholar] [CrossRef] [PubMed] - Rees, E.M.; Nightingale, E.S.; Jafari, Y.; Waterlow, N.R.; Clifford, S.; Pearson, C.A.; Jombart, T.; Procter, S.R.; Knight, G.M.; CMMID Working Group. COVID-19 length of hospital stay: A systematic review and data synthesis. BMC Med.
**2020**, 18, 270. [Google Scholar] [CrossRef] [PubMed] - Vekaria, B.; Overton, C.; Wisniowski, A.; Ahmad, S.; Aparicio-Castro, A.; Curran-Sebastian, J.; Eddleston, J.; Hanley, N.; House, T.; Kim, J.; et al. Hospital length of stay for COVID-19 patients: Data-driven methods for forward planning. BMC Infect. Dis.
**2020**, 21, 700. [Google Scholar] - Raue, A.; Kreutz, C.; Maiwald, T.; Bachmann, J.; Schilling, M.; Klingmüller, U.; Timmer, J. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics
**2009**, 25, 1923–1929. [Google Scholar] [CrossRef] [Green Version] - Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. A
**1927**, 115, 700–721. [Google Scholar] - Hou, C.; Chen, J.; Zhou, Y.; Hua, L.; Yuan, J.; He, S.; Guo, Y.; Zhang, S.; Jia, Q.; Zhao, C.; et al. The effectiveness of quarantine of Wuhan city against the Corona Virus Disease 2019 (COVID-19): A well-mixed SEIR model analysis. J. Med. Virol.
**2020**, 92, 841–848. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fanelli, D.; Piazza, F. Analysis and forecast of COVID-19 spreading in China, Italy and France. Chaos Solitons Fractals
**2020**, 134, 109761. [Google Scholar] [CrossRef] [PubMed] - Roda, W.C.; Varughese, M.B.; Han, D.; Li, M.Y. Why is it difficult to accurately predict the COVID-19 epidemic? Infect. Dis. Model.
**2020**, 5, 271–281. [Google Scholar] [CrossRef] [PubMed] - Zanella, M.; Bardelli, C.; Azzi, M.; Deandrea, S.; Perotti, P.; Silva, S.; Cadum, E.; Figini, S.; Toscani, G. Social contacts, epidemic spreading and health system. Mathematical modeling and applications to COVID-19 infection. Math. Biosci. Eng.
**2021**, 18, 3384–3403. [Google Scholar] [CrossRef] [PubMed] - Giordano, G.; Blanchini, F.; Bruno, R.; Colaneri, P.; Di Filippo, A.; Di Matteo, A.; Colaneri, M. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med.
**2020**, 26, 855–860. [Google Scholar] [CrossRef] [PubMed] - Gatto, M.; Bertuzzo, E.; Mari, L.; Miccoli, S.; Carraro, L.; Casagrandi, R.; Rinaldo, A. Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proc. Natl. Acad. Sci. USA
**2020**, 117, 10484–10491. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dong, E.; Du, H.; Gardner, L. An interactive web-based dashboard to track COVID-19 in real time. Lancet Infect. Dis.
**2020**, 20, 533–534. [Google Scholar] [CrossRef] - Presidenza del Consiglio dei Ministri, Dipartimento della Protezione Civile. GitHub: COVID19 Italia—Monitoraggio Situazione. Available online: https://github.com/pcm-dpc/COVID-19 (accessed on 27 September 2021).
- Wu, J.; Li, W.; Shi, X.; Chen, Z.; Jiang, B.; Liu, J.; Wang, D.; Liu, C.; Meng, Y.; Cui, L.; et al. Early antiviral treatment contributes to alleviate the severity and improve the prognosis of patients with novel coronavirus disease (COVID-19). J. Intern. Med.
**2020**, 288, 128–138. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mueller, A.L.; McNamara, M.S.; Sinclair, D.A. Why does COVID-19 disproportionately affect older people? Aging
**2020**, 12, 9959. [Google Scholar] [CrossRef] [PubMed] - Hoffman, M.D.; Gelman, A. The No-U-Turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res.
**2014**, 15, 1593–1623. [Google Scholar]

(a) TVC | (b) LoS | ||||||
---|---|---|---|---|---|---|---|

Age group | Median | Q1 | Q3 | Age group | Median | Q1 | Q3 |

0–24 | 12 | 16 | 22 | 0–24 | 0 | 1 | 4 |

25–49 | 12 | 17 | 23 | 25–49 | 0 | 2 | 10 |

50–64 | 13 | 19 | 26 | 50–64 | 1 | 9 | 16 |

65–74 | 13 | 20 | 29 | 65–74 | 5 | 11 | 20 |

75+ | 11 | 23 | 40 | 75+ | 4 | 9 | 18 |

(c)DH | |||||||

Age group | Median | Q1 | Q3 | ||||

0–24 | 0 | 0 | 0 | ||||

25–49 | 0 | 0 | 2 | ||||

50–64 | 0 | 0 | 4 | ||||

65–74 | 0 | 1 | 5 | ||||

75+ | 0 | 1 | 5 |

(a) TVC | (b) LoS | ||||||
---|---|---|---|---|---|---|---|

Period | Median | Q1 | Q3 | Period | Median | Q1 | Q3 |

Period 1 | 13 | 25 | 42 | Period 1 | 3 | 8 | 16 |

Period 2 | 16 | 23 | 35 | Period 2 | 1 | 8 | 16 |

Period 3 | 12 | 17 | 23 | Period 3 | 3 | 10 | 17 |

(c) DH | |||||||

Period | Median | Q1 | Q3 | ||||

Period 1 | 0 | 0 | 1 | ||||

Period 2 | 0 | 0 | 2 | ||||

Period 3 | 0 | 1 | 7 |

(a) Age factor | (b) Period factor | ||
---|---|---|---|

Age Group | 95% C.I. | Period | 95% C.I. |

0–24 | $\left[17.2,18.9\right]$ | Period 1 | $\left[30.8,33.9\right]$ |

25–49 | $\left[17.8,21.3\right]$ | Period 2 | $\left[29.7,32.6\right]$ |

50–64 | $\left[20.4,23.9\right]$ | Period 3 | $\left[23.1,24.0\right]$ |

65–74 | $\left[21.4,24.7\right]$ | ||

75+ | $\left[28.5,32.8\right]$ |

(a) Age Factor | (b) Period Factor | ||
---|---|---|---|

Age Group | 95% C.I. | Period | 95% C.I. |

0–24 | $\left[4.4,6.2\right]$ | Period 1 | $\left[9.8,12.4\right]$ |

25–49 | $\left[6.0,8.7\right]$ | Period 2 | $\left[10.8,11.9\right]$ |

50–64 | $\left[10.2,13.0\right]$ | Period 3 | $\left[10.5,12.7\right]$ |

65–74 | $\left[12.6,16.1\right]$ | ||

75+ | $\left[11.3,13.9\right]$ |

(a) Age Factor | (b) Period Factor | ||
---|---|---|---|

Age Group | 95% C.I. | Period | 95% C.I. |

0–24 | $\left[0.9,1.6\right]$ | Period 1 | $\left[1.9,2.5\right]$ |

25–49 | $\left[2.1,3.3\right]$ | Period 2 | $\left[1.7,2.4\right]$ |

50–64 | $\left[2.4,3.9\right]$ | Period 3 | $\left[3.3,4.7\right]$ |

65–74 | $\left[2.8,4.1\right]$ | ||

75+ | $\left[2.7,3.5\right]$ |

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Bardelli, C.
Inference on COVID-19 Epidemiological Parameters Using Bayesian Survival Analysis. *Entropy* **2021**, *23*, 1262.
https://doi.org/10.3390/e23101262

**AMA Style**

Bardelli C.
Inference on COVID-19 Epidemiological Parameters Using Bayesian Survival Analysis. *Entropy*. 2021; 23(10):1262.
https://doi.org/10.3390/e23101262

**Chicago/Turabian Style**

Bardelli, Chiara.
2021. "Inference on COVID-19 Epidemiological Parameters Using Bayesian Survival Analysis" *Entropy* 23, no. 10: 1262.
https://doi.org/10.3390/e23101262