# A New Compartment Model of COVID-19 Transmission: The Broken-Link Model

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. The $\delta $ Epidemic Surge in Japan

#### 3.2. The $o$ Epidemic Surge in Japan

#### 3.3. The Status of the $\delta $ and $o$ Surges in Other Countries

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A suggested formula of the broken-link model. (

**a**) Cartoon of the suppression of secondary infection in a transmission tree ($k=1/2$ case). When the transmission link from the primary infected individual A to the secondary candidate C is unconnected at time ${t}_{1}$, subsequent transmissions starting from C are not generated as denoted by the shaded area. (

**b**) Compartments of the broken-link model. The temporary removed compartment $S\prime $ is introduced due to the suppression of the secondary and higher-order transmissions. Contrary to the SIR model, the coupling between susceptible $S$ and infected $I$ becomes time dependent in the broken-link model.

**Figure 2.**The epicurve of the $\delta $ surge of COVID-19 spread in Japan from June to September 2021. (

**a**) The logarithmic plot of the number of daily confirmed cases (one week average) and the fit result (solid curve). The fit was performed with three partial waves and a baseline denoted by dashed lines. (

**b**) The observed data and fit result of the K-value.

**Figure 3.**The $o$ surge in Japan from January to February 2022. (

**a**) The logarithmic plot of daily confirmed cases (one week average) and fit result. The fit was performed with a single partial wave. (

**b**) The observed data and fit result for the K-value.

**Figure 4.**The logarithmic plot of daily confirmed cases (one week average) and the fit results for the $\delta $ and $o$ surges in South Africa, the United States of America, France and Denmark from November 2021 to February 2022. (

**a**) South Africa; (

**b**) United States; (

**c**) France; (

**d**) Denmark.

**Figure 5.**The predicted $k$ dependence of the relative values of the number of daily confirmed cases at peak positions in the logarithmic scale. The relative value is normalized at $k=0.92$, which was obtained in the 1st epidemic surge in April 2020 in Japan [19]. The value is proportional to $-\left(\mathrm{ln}k\right)\mathrm{exp}\left(-a/\mathrm{ln}k\right)$ in the model. The case with $a=0.5$ is shown.

**Table 1.**The parameters and functions for the K-value and phenomenological constant attenuation model.

Parameters/Functions | Descriptions |
---|---|

$t$ | day |

$R\left(t\right)$ | cumulative number of confirmed cases |

$K\left(t\right)=1-R\left(t-7\right)/R\left(t\right)$ | K-value (indicator) |

$a\left(t\right)=\mathrm{exp}\left[-\left(1-k\right)\right]a\left(t-1\right)$ | geometric progression |

$k$ | constant attenuation factor |

**Table 2.**The parameters and functions for the broken-link model. For the basic reproduction number ${R}_{0}$, the constant parameter $a$ is determined from the fit to the actual data. The analytic expressions of ${R}_{0}$, ${N}_{\infty}$ and $\gamma $ are obtained by solving Equations (3) and (4).

Parameters/Functions | Descriptions |
---|---|

$t$ | time |

$S\left(t\right)$ | number of susceptible people |

$I\left(t\right)$ | number of infected people |

$R\left(t\right)$ | cumulative number of confirmed cases |

$k$ | probability of connected transmission links |

${R}_{0}=-a/\mathrm{ln}k$ | basic reproduction number |

${N}_{\infty}=R\left(0\right)\mathrm{exp}\left({R}_{0}\right)$ | cumulative number of infected people in each infection wave generated by ${R}_{0}$ |

$\gamma =-\mathrm{ln}k$ | removal rate from transmission trees |

**Table 3.**The parameters of the Gompertz curves in the $\delta $ surge in Japan. The ${N}_{\infty}$, ${R}_{0}$ and $k$ are the cumulative number of infected people, basic reproduction number and connected probability of transmission links, respectively. The “shift” stands for the onset of a partial wave from the reference date (25 June 2021). The statistical errors evaluated by the jackknife method are represented in the parentheses.

Partial Wave | ${\mathit{N}}_{\mathit{\infty}}$ | ${\mathit{R}}_{0}$ | $\mathit{k}$ | Shift (Days) |
---|---|---|---|---|

1st | 75 (12) k | 6.49 (20) | 0.918 (4) | 7.2 (3) |

2nd | 340 (23) k | 6.98 (16) | 0.907 (3) | 24.5 (3) |

3rd | 375 (2) k | 4.40 (15) | 0.892 (1) | 47.7 (3) |

**Table 4.**The parameters of the Gompertz curve for the $o$ surge in Japan. The definition of the parameters is the same as in Table 3, but the reference date is 1 January 2022.

Partial Wave | ${\mathit{N}}_{\mathit{\infty}}$ | ${\mathit{R}}_{0}$ | $\mathit{k}$ | Shift (Days) |
---|---|---|---|---|

1st | 4332 (6) k | 10.4 (1) | 0.944 (1) | −4.3 (1) |

**Table 5.**The parameters of the Gompertz curves for other countries from November 2021 to February 2022. The definition of the parameters is the same as in Table 3, but the reference dates are 24 November 2021, 30 November 2021, 1 November 2021 and 1 November 2021 for South Africa, United States, France and Denmark, respectively.

Region | Partial Wave | ${\mathit{N}}_{\mathit{\infty}}$ | ${\mathit{R}}_{0}$ | $\mathit{k}$ | Shift (Days) |
---|---|---|---|---|---|

South Africa | 1st | 592 (1) k | 8.98 (11) | 0.905 (1) | −3.7 (2) |

United States | 1st | 22,411 (269) k | 9.68 (6) | 0.922 (1) | 13.6 (4) |

France | 1st | 2233 (235) k | 7.18 (22) | 0.949 (2) | 4.3 (6) |

2nd | 13,330 (16) k | 9.92 (1) | 0.935 (1) | 42.4 (1) | |

Denmark | 1st | 84 (1) k | 6.16 (64) | 0.924 (1) | −3.0 (6) |

2nd | 1094 (124) k | 9.93 (28) | 0.956 (2) | 19.4 (1.0) | |

3rd | 1401 (6) k | 8.00 (25) | 0.937 (1) | 63.3 (5) |

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**MDPI and ACS Style**

Ikeda, Y.; Sasaki, K.; Nakano, T. A New Compartment Model of COVID-19 Transmission: The Broken-Link Model. *Int. J. Environ. Res. Public Health* **2022**, *19*, 6864.
https://doi.org/10.3390/ijerph19116864

**AMA Style**

Ikeda Y, Sasaki K, Nakano T. A New Compartment Model of COVID-19 Transmission: The Broken-Link Model. *International Journal of Environmental Research and Public Health*. 2022; 19(11):6864.
https://doi.org/10.3390/ijerph19116864

**Chicago/Turabian Style**

Ikeda, Yoichi, Kenji Sasaki, and Takashi Nakano. 2022. "A New Compartment Model of COVID-19 Transmission: The Broken-Link Model" *International Journal of Environmental Research and Public Health* 19, no. 11: 6864.
https://doi.org/10.3390/ijerph19116864