# Risk Interactions of Coronavirus Infection across Age Groups after the Peak of COVID-19 Epidemic

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{j,t}) followed a generalized Poisson distribution to account for over-dispersion of case counts (i.e., observed variance is larger than expected variance) [25]. The model also included case counts from previous days (lags) across age groups as predictors to form a dynamic model (see Appendix A for details). Therefore, the current risk of infection in each age group was predicted not only by previous case counts in its own group but also by previous counts from other age groups.

_{j,t}could be viewed as a random effect to account for the correlation of daily counts between age groups. The b

_{j,t}was assumed a multivariate normal distribution.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SARS-CoV-2 | severe acute respiratory syndrome coronavirus 2 |

COVID-19 | coronavirus infectious disease 2019 |

GAM | generalized additive model |

VAR | vector autoregressive regression |

NB | negative binomial model |

REML | restricted maximum likelihood |

PMF | probability mass function |

## Appendix A

#### Appendix A.1. Obtained Smoothed Predicted Daily Cases with Generalized Additive Model (GAM)

_{ij}represents the observed case counts of day i and group j, and E(Y

_{ij}) is the expected (predicted) value. The variable time

_{i}represents day (1,…,I), b

_{k}( ) represents a basis function for the kth term to the smooth temporal trend, and β

_{j,k}s are regression coefficients for the smooth term k and group j. The restricted maximum likelihood (REML) approach was used in the parameter estimation. The R mgcv package was used [23], and smooth terms were fitted using a thin plate regression spline with 16 knots. The number of knots is typically determined empirically, commonly with 8–20 knots. A smaller number of knots yields a smoother curve while a larger number of knows results in a rougher curve. We observed that 16 knots are reasonable to capture the nonlinearity of the epidemic curve without introducing too many spikes.

#### Appendix A.2. Model Setups and Comparisons

**ξ**[43] as quoted in Hilbe JM. 2014 [27]. The PMF is

_{j,t}that can be viewed as a random effect. The b

_{j,t}was assumed a multivariate normal distribution.

## Appendix B. Additional Table

**Table A1.**Risk interactions in coronavirus infection across age groups based on negative binomial models, COVID-19, South Korea.

Model Outcomes | Predictors | Lags of Predictors | ||||
---|---|---|---|---|---|---|

Lag 1 | Lag 2 | Lag 3 | Lag 4 | Lag 5 | ||

Aged 60 or above | ||||||

60 or above | 1.93 (1.14–3.19) * | 1.09 (0.66–1.80) | 1.02 (0.61–1.82) | 0.99 (0.57–1.67) | 1.11 (0.68–1.87) | |

40–59 | 1.03 (0.53–2.02) | 1.66 (0.83–3.01) | 0.69 (0.36–1.46) | 1.26 (0.72–2.25) | 1.00 (0.57–1.71) | |

20–39 | 0.94 (0.54–1.69) | 1.30 (0.66–2.42) | 1.42 (0.73–2.68) | 0.50 (0.25–1.01) | 1.21 (0.63–2.37) | |

0–19 | 1.10 (0.72–1.78) | 1.29 (0.79–1.98) | 0.85 (0.51–1.35) | 0.62 (0.40–0.98) # | 1.52 (0.88–2.55) | |

Aged 40–59 | ||||||

60 or above | 1.64 (1.12–2.41) * | 0.75 (0.50–1.09) | 0.97 (0.67–1.49) | 1.01 (0.69–1.53) | 1.02 (0.69–1.53) | |

40–59 | 1.15 (0.68–1.96) | 1.55 (0.92–2.46) | 1.14 (0.64–2.24) | 0.90 (0.54–1.39) | 0.95 (0.63–1.47) | |

20–39 | 1.10 (0.59–1.73) | 1.13 (0.70–1.81) | 1.54 (0.95–2.47) | 0.63 (0.37–1.27) | 0.89 (0.44–1.56) | |

0–19 | 1.09 (0.77–1.53) | 1.23 (0.87–1.78) | 1.16 (0.82–1.64) | 1.16 (0.82–1.60) | 1.15 (0.79–1.68) | |

Aged 20–39 | ||||||

60 or above | 1.00 (0.71–1.44) | 0.93 (0.65–1.39) | 1.00 (0.69–1.48) | 1.58 (1.10–2.26) * | 0.90 (0.64–1.29) | |

40–59 | 1.12 (0.69–1.86) | 1.09 (0.67–1.72) | 0.96 (0.58–1.53) | 0.97 (0.65–1.44) | 1.08 (0.72–1.61) | |

20–39 | 1.63 (1.09–2.48) * | 1.04 (0.66–1.64) | 1.35 (0.86–2.11) | 0.99 (0.58–1.56) | 0.96 (0.60–1.56) | |

0–19 | 1.04 (0.74–1.44) | 0.95 (0.69–1.29) | 0.89 (0.63–1.23) | 0.95 (0.67–1.40) | 0.73 (0.51–1.04) | |

Aged 0–19 | ||||||

60 or above | 1.77 (1.17–2.76) * | 1.35 (0.82–2.14) | 1.00 (0.62–1.61) | 0.81 (0.50–1.32) | 1.52 (0.97–2.44) | |

40–59 | 0.76 (0.39–1.42) | 0.71 (0.39–1.27) | 0.86 (0.43–1.64) | 0.94 (0.53–1.66) | 0.96 (0.58–1.60) | |

20–39 | 1.53 (0.87–2.72) | 1.46 (0.82–2.69) | 1.05 (0.57–1.92) | 0.78 (0.41–1.41) | 0.90 (0.49–1.74) | |

0–19 | 0.94 (0.63–1.44) | 0.83 (0.57–1.25) | 0.88 (0.58–1.33) | 0.92 (0.59–1.44) | 1.52 (0.95–2.42) |

## References

- Zhu, N.; Zhang, D.; Wang, W.; Li, X.; Yang, B.; Song, J.; Zhao, X.; Huang, B.; Shi, W.; Lu, R.; et al. A Novel Coronavirus from Patients with Pneumonia in China, 2019. N. Engl. J. Med.
**2020**, 382, 727–733. [Google Scholar] [CrossRef] [PubMed] - Li, Q.; Guan, X.; Wu, P.; Wang, X.; Zhou, L.; Tong, Y.; Ren, R.; Leung, K.S.M.; Lau, E.H.Y.; Wong, J.Y.; et al. Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia. N. Engl. J. Med.
**2020**, 382, 1199–1207. [Google Scholar] [CrossRef] [PubMed] - Guan, W.J.; Ni, Z.Y.; Hu, Y.; Liang, W.H.; Ou, C.Q.; He, J.X.; Liu, L.; Shan, H.; Lei, C.L.; Hui, D.S.C.; et al. Clinical Characteristics of Coronavirus Disease 2019 in China. N. Engl. J. Med.
**2020**. [Google Scholar] [CrossRef] [PubMed] - Richardson, S.; Hirsch, J.S.; Narasimhan, M.; Crawford, J.M.; McGinn, T.; Davidson, K.W.; Barnaby, D.P.; Becker, L.B.; Chelico, J.D.; Cohen, S.L.; et al. Presenting Characteristics, Comorbidities, and Outcomes Among 5700 Patients Hospitalized With COVID-19 in the New York City Area. JAMA
**2020**. [Google Scholar] [CrossRef] [PubMed] - Wu, Z.; McGoogan, J.M. Characteristics of and Important Lessons From the Coronavirus Disease 2019 (COVID-19) Outbreak in China: Summary of a Report of 72314 Cases From the Chinese Center for Disease Control and Prevention. JAMA
**2020**. [Google Scholar] [CrossRef] - Garg, S.; Kim, L.; Whitaker, M.; O’Halloran, A.; Cummings, C.; Holstein, R.; Prill, M.; Chai, S.J.; Kirley, P.D.; Alden, N.B.; et al. Hospitalization Rates and Characteristics of Patients Hospitalized with Laboratory-Confirmed Coronavirus Disease 2019—COVID-NET, 14 States, March 1–30, 2020. Mmwr Morb Mortal Wkly. Rep.
**2020**, 69, 458–464. [Google Scholar] [CrossRef] - Yu, X. Urban-rural inequalities during the COVID-19 pandemic among elderly people in Florida, US. medRxiv
**2020**. [Google Scholar] [CrossRef] - Hay, J.A.; Haw, D.J.; Hanage, W.P.; Metcalf, C.J.E.; Mina, M.J. Implications of the Age Profile of the Novel Coronavirus. Available online: https://dash.harvard.edu/handle/1/42639493 (accessed on 4 July 2020).
- Chavan, P.; Kedia, S.; Yu, X. Physical and Functional Limitations in US Older Cancer Survivors. J. Palliat. Care Med.
**2017**. [Google Scholar] [CrossRef] [Green Version] - Ward, B.W.; Schiller, J.S.; Goodman, R.A. Multiple chronic conditions among US adults: A 2012 update. Prev. Chronic Dis.
**2014**, 11, E62. [Google Scholar] [CrossRef] [Green Version] - CDC. Severe Outcomes Among Patients with Coronavirus Disease 2019 (COVID-19)—United States, February 12–March 16, 2020. MMWR Morb Mortal Wkly. Rep.
**2020**, 343–346. [Google Scholar] - Imai, N.; Cori, A.; Dorigatti, I.; Baguelin, M.; Donnelly, C.A.; Riley, S.; Ferguson, N.M. Report 3: Transmissibility of 2019-nCoV; Imperial College of London: London, UK, 2020. [Google Scholar]
- Lv, H.; Wu, N.C.; Tsang, O.T.-Y.; Yuan, M.; Perera, R.A.P.M.; Leung, W.S.; So, R.T.Y.; Chan, J.M.C.; Yip, G.K.; Chik, T.S.H.; et al. Cross-reactive Antibody Response between SARS-CoV-2 and SARS-CoV Infections. Cell Rep.
**2020**, 31, 107725. [Google Scholar] [CrossRef] [PubMed] - Lipsitch, M.; Swerdlow, D.L.; Finelli, L. Defining the Epidemiology of Covid-19—Studies Needed. N. Engl. J. Med.
**2020**, 382, 1194–1196. [Google Scholar] [CrossRef] - Shim, E.; Tariq, A.; Choi, W.; Lee, Y.; Chowell, G. Transmission potential and severity of COVID-19 in South Korea. Int. J. Infect. Dis
**2020**, 93, 339–344. [Google Scholar] [CrossRef] - Park, S.Y.; Kim, Y.M.; Yi, S.; Lee, S.; Na, B.J.; Kim, C.B.; Kim, J.I.; Kim, H.S.; Kim, Y.B.; Park, Y.; et al. Coronavirus Disease Outbreak in Call Center, South Korea. Emerg. Infect. Dis.
**2020**, 26. [Google Scholar] [CrossRef] [PubMed] - Kim, S.; Castro, M.C. Spatiotemporal pattern of COVID-19 and government response in South Korea (as of May 31, 2020). Int. Ernational J. Infect. Dis.
**2020**. [Google Scholar] [CrossRef] [PubMed] - Yu, X.; Duan, J.; Jiang, Y.; Zhang, H. Distinctive trajectories of COVID-19 epidemic by age and gender: A retrospective modeling of the epidemic in South Korea. Int.Ernational J. Infect. Dis.
**2020**. [Google Scholar] [CrossRef] - Chowell, G.; Mizumoto, K. The COVID-19 pandemic in the USA: What might we expect? Lancet
**2020**, 395, 1093–1094. [Google Scholar] [CrossRef] - Kissler, S.M.; Tedijanto, C.; Goldstein, E.; Grad, Y.H.; Lipsitch, M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science
**2020**. [Google Scholar] [CrossRef] - Anderson, R.M.; Heesterbeek, H.; Klinkenberg, D.; Hollingsworth, T.D. How will country-based mitigation measures influence the course of the COVID-19 epidemic? Lancet
**2020**, 395, 931–934. [Google Scholar] [CrossRef] - Kim, Y.J.; Seo, M.H.; Yeom, H.E. Estimating a breakpoint in the spread pattern of COVID-19 in South Korea. Int. J. Infect. Dis.
**2020**. [Google Scholar] [CrossRef] - Wood, S.N. Generalized Additive Models: An Int.roduction with R, 2nd ed.; Chapman and Hall/CRC: Boca Raton, FL, USA, 2017. [Google Scholar]
- Brandt, P.T.; Sandler, T. A bayesian Poisson vector autoregressive model. Political Anal.
**2012**, 20, 23. [Google Scholar] [CrossRef] [Green Version] - Yang, Z.; Hardin, J.W.; Addy, C.L.; Vuong, Q.H. Testing approaches for overdispersion in poisson regression versus the generalized poisson model. Biom. J.
**2007**, 49, 565–584. [Google Scholar] [CrossRef] [PubMed] - Lauer, S.A.; Grantz, K.H.; Bi, Q.; Jones, F.K.; Zheng, Q.; Meredith, H.R.; Azman, A.S.; Reich, N.G.; Lessler, J. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Ann. Int. Ern. Med.
**2020**. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hilbe, J.M. Modeling Count Data; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Carpenter, B.; Gelman, A.; Hoffman, M.D.; Lee, D.; Goodrich, B.; Betancourt, M.; Brubaker, M.; Guo, J.; Li, P.; Riddell, A. Stan: A probabilistic programming language. J. Stat. Softw.
**2017**, 76. [Google Scholar] [CrossRef] [Green Version] - Davies, N.G.; Klepac, P.; Liu, Y.; Prem, K.; Jit, M.; Eggo, R.M. Age-dependent effects in the transmission and control of COVID-19 epidemics. medRxiv
**2020**. [Google Scholar] [CrossRef] - Zhang, J.; Litvinova, M.; Liang, Y.; Wang, Y.; Wang, W.; Zhao, S.; Wu, Q.; Merler, S.; Viboud, C.; Vespignani, A.; et al. Changes in contact patterns shape the dynamics of the COVID-19 outbreak in China. Science
**2020**. [Google Scholar] [CrossRef] - Mossong, J.; Hens, N.; Jit, M.; Beutels, P.; Auranen, K.; Mikolajczyk, R.; Massari, M.; Salmaso, S.; Tomba, G.S.; Wallinga, J.; et al. Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med.
**2008**, 5, e74. [Google Scholar] [CrossRef] - Tenforde, M.W.; Billig Rose, E.; Lindsell, C.J.; Shapiro, N.I.; Files, D.C.; Gibbs, K.W.; Prekker, M.E.; Steingrub, J.S.; Smithline, H.A.; Gong, M.N.; et al. Characteristics of Adult Outpatients and Inpatients with COVID-19-11 Academic Medical Centers, United States, March-May 2020. MMWR Morb Mortal Wkly. Rep.
**2020**, 69, 841–846. [Google Scholar] [CrossRef] - Cowling, B.J.; Ho, L.M.; Leung, G.M. Effectiveness of control measures during the SARS epidemic in Beijing: A comparison of the Rt curve and the epidemic curve. Epidemiol Infect.
**2008**, 136, 562–566. [Google Scholar] [CrossRef] [Green Version] - Gostic, K.; Gomez, A.C.; Mummah, R.O.; Kucharski, A.J.; Lloyd-Smith, J.O. Estimated effectiveness of symptom and risk screening to prevent the spread of COVID-19. Elife
**2020**, 9. [Google Scholar] [CrossRef] - Pan, A.; Liu, L.; Wang, C.; Guo, H.; Hao, X.; Wang, Q.; Huang, J.; He, N.; Yu, H.; Lin, X.; et al. Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China. JAMA
**2020**. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yu, X. Modeling Return of the Epidemic: Impact of Population Structure, Asymptomatic Infection, Case Importation and Personal Contacts. medRxiv
**2020**. [Google Scholar] [CrossRef] - Bai, Y.; Yao, L.; Wei, T.; Tian, F.; Jin, D.Y.; Chen, L.; Wang, M. Presumed Asymptomatic Carrier Transmission of COVID-19. JAMA
**2020**. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Ji, F.; Wang, L.; Wang, L.; Hao, J.; Dai, M.; Liu, Y.; Pan, X.; Fu, J.; Li, L.; et al. Asymptomatic and Human-to-Human Transmission of SARS-CoV-2 in a 2-Family Cluster, Xuzhou, China. Emerg. Infect. Dis.
**2020**, 26. [Google Scholar] [CrossRef] - Li, G.; Li, W.; He, X.; Cao, Y. Asymptomatic and Presymptomatic Infectors: Hidden Sources of COVID-19 Disease. Clin. Infect. Dis.
**2020**. [Google Scholar] [CrossRef] - Kucharski, A.J.; Russell, T.W.; Diamond, C.; Liu, Y.; Edmunds, J.; Funk, S.; Eggo, R.M. Centre for Mathematical Modelling of Infectious Diseases, C.-w.g. Early dynamics of transmission and control of COVID-19: A mathematical modelling study. Lancet Infect. Dis.
**2020**. [Google Scholar] [CrossRef] [Green Version] - IHME. Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilatordays and deaths by US state in the next 4 months. IHME COVID-19 health service utilization forecasting team. MMWR Morb Mortal Wkly. Rep.
**2020**, 69, 343–346. [Google Scholar] - Woody, S.; Garcia Tec, M.; Dahan, M.; Gaither, K.; Lachmann, M.; Fox, S.; Meyers, L.A.; Scott, J.G. Projections for first-wave COVID-19 deaths across the US using social-distancing measures derived from mobile phones. medRxiv
**2020**. [Google Scholar] [CrossRef] - Consul, P. Generalized Poisson Distribution: Properties and Applications; Marcel Decker: New York, NY, USA, 1989. [Google Scholar]

**Figure 1.**Observed daily new cases of COVID-19 and predicted epidemic curve from the generalized additive model, South Korea, from 10 March to 30 April 2020, by age groups.

**Table 1.**Risk ratios in the coronavirus infection across age groups during the COVID-19 pandemic, South Korea, 10 March to 30 April 2020.

Model Outcomes | Predictors | Lags of Predictors | ||||
---|---|---|---|---|---|---|

Lag 1 | Lag 2 | Lag 3 | Lag 4 | Lag 5 | ||

Aged 60 or above | ||||||

60 or above | 2.09 (1.28–3.17) * | 1.21 (0.78–1.80) | 0.90 (0.60–1.39) | 1.03 (0.60–1.67) | 0.96 (0.63–1.53) | |

40–59 | 0.95 (0.49–1.84) | 1.81 (0.98–3.29) | 0.59 (0.30–1.18) | 1.37 (0.85–2.23) | 1.03 (0.61–1.70) | |

20–39 | 0.89 (0.53–1.46) | 2.02 (1.12–3.47) * | 1.28 (0.72–2.31) | 0.41 (0.22–0.78) # | 1.11 (0.58–2.22) | |

0–19 | 0.90 (0.61–1.43) | 1.36 (0.95–1.95) | 0.82 (0.57–1.19) | 0.61 (0.44–0.85) # | 1.64 (1.03–2.58) * | |

Aged 40–59 | ||||||

60 or above | 1.66 (1.19–2.29) * | 0.79 (0.55–1.12) | 0.89 (0.64–1.23) | 0.98 (0.70–1.40) | 1.02 (0.71–1.40) | |

40–59 | 1.01 (0.62–1.63) | 1.76 (1.16–2.66) * | 1.23 (0.76–1.96) | 0.89 (0.62–1.29) | 0.88 (0.62–1.28) | |

20–39 | 1.13 (0.75–1.73) | 1.12 (0.76–1.67) | 1.59 (1.03–2.50) * | 0.60 (0.37–0.95) | 0.88 (0.53–1.48) | |

0–19 | 1.04 (0.79–1.37) | 1.28 (0.97–1.68) | 1.17 (0.88–1.57) | 1.14 (0.88–1.49) | 1.17 (0.84–1.62) | |

Aged 20–39 | ||||||

60 or above | 0.95 (0.69–1.29) | 0.89 (0.65–1.23) | 1.01 (0.72–1.39) | 1.54 (1.11–2.12) * | 0.97 (0.69–1.31) | |

40–59 | 1.17 (0.73–1.88) | 1.18 (0.78–1.78) | 0.98 (0.60–1.52) | 0.90 (0.64–1.26) | 1.04 (0.75–1.47) | |

20–39 | 1.56 (1.05–2.36) * | 1.02 (0.68–1.55) | 1.45 (0.96–2.24) | 1.06 (0.68–1.64) | 0.85 (0.55–1.32) | |

0–19 | 1.04 (0.79–1.37) | 0.96 (0.75–1.27) | 0.88 (0.66–1.20) | 0.97 (0.75–1.26) | 0.74 (0.54–1.00) | |

Aged 0–19 | ||||||

60 or above | 1.78 (1.23–2.61) * | 1.34 (0.86–2.06) | 0.99 (0.67–1.51) | 0.82 (0.52–1.25) | 1.55 (1.02–2.30) * | |

40–59 | 0.76 (0.41–1.38) | 0.71 (0.42–1.18) | 0.85 (0.47–1.56) | 0.91 (0.55–1.57) | 0.97 (0.62–1.54) | |

20–39 | 1.51 (0.92–2.55) | 1.50 (0.85–2.57) | 1.04 (0.61–1.80) | 0.78 (0.44–1.38) | 0.91 (0.51–1.62) | |

0–19 | 0.93 (0.64–1.36) | 0.85 (0.60–1.21) | 0.85 (0.60–1.27) | 0.92 (0.62–1.38) | 1.51 (0.99–2.31) |

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**MDPI and ACS Style**

Yu, X.
Risk Interactions of Coronavirus Infection across Age Groups after the Peak of COVID-19 Epidemic. *Int. J. Environ. Res. Public Health* **2020**, *17*, 5246.
https://doi.org/10.3390/ijerph17145246

**AMA Style**

Yu X.
Risk Interactions of Coronavirus Infection across Age Groups after the Peak of COVID-19 Epidemic. *International Journal of Environmental Research and Public Health*. 2020; 17(14):5246.
https://doi.org/10.3390/ijerph17145246

**Chicago/Turabian Style**

Yu, Xinhua.
2020. "Risk Interactions of Coronavirus Infection across Age Groups after the Peak of COVID-19 Epidemic" *International Journal of Environmental Research and Public Health* 17, no. 14: 5246.
https://doi.org/10.3390/ijerph17145246