# Unreported Cases for Age Dependent COVID-19 Outbreak in Japan

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

#### 3.1. SIUR Model

#### 3.2. Comparison of the Model (1) with the Data

**Remark**

**1.**

#### 3.3. Model SIUR with Age Structure

## 4. Results

#### 4.1. Model without Age Structure

#### 4.2. Model with Age Structure

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Method to Fit of the Age Structured Model to the Data

**Remark**

**A1.**

**Figure A2.**We plot a comparison between the model (12)–(15) (without public intervention) and the age structured data from Japan. We set $1/\nu =1/\eta =7$ days, ${f}_{i}$ which actually depends on the age class, with ${f}_{1}=0.1$, ${f}_{2}=0.2$, ${f}_{3}=0.4$, ${f}_{4}=0.4$, ${f}_{5}=0.6$, ${f}_{6}=0.6$, ${f}_{7}=0.8$, ${f}_{8}=0.8$, ${f}_{9}=0.8$, and ${f}_{10}=0.9$. and we obtain ${\tau}_{1}=0.1264$, ${\tau}_{2}=0.1655$, ${\tau}_{3}=0.3538$, ${\tau}_{4}=0.2966$, ${\tau}_{5}=0.1513$, ${\tau}_{6}=0.1684$, ${\tau}_{7}=0.1251$, ${\tau}_{8}=0.1168$, ${\tau}_{9}=0.1015$, ${\tau}_{10}=0.1258$. The matrix $\varphi $ is the one defined in (18).

## Appendix B. Construction of the Contact Matrix

## References

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**Figure 1.**In this figure we plot in blue bars the age distribution of the Japanese population for 10,000 people and we plot in orange bars the age distribution of the number of reported cases of SARS-CoV-2 for 10,000 patient on 29 April (based on the total of 13,660 reported cases). We observe that $77\%$ of the confirmed patients belong to the 20–60 years age class.

**Figure 2.**In this figure we plot the number of infected patients for each age class per 10,000 individuals of the same age class (i.e., the number of infected individuals divided by the population of the age class times 10,000). The figure shows that the individuals are more or less likely to becomes infected depending on their age class. The bars describe the susceptibility of people to the SARS-CoV-2 depending on their age class.

**Figure 3.**Cumulative number of SARS-CoV-2-induced deaths per age class (red bars). We observe that $83\%$ of death occur in between 70 and 100 years old.

**Figure 4.**Time evolution of the cumulative number of reported cases of SARS-CoV-2 per age class. The vertical axis represents the total number of cumulative reported cases in each age class.

**Figure 5.**Time evolution of the cumulative number of reported cases of SARS-CoV-2 per age class. The vertical axis represents the total number of cumulative reported cases in each age class.

**Figure 7.**Graphical representation of the contact matrix $\varphi $. The intensity of blue in the cell $(i,j)$ indicates the conditional probability that, given a contact between an individual of age group i and another individual, the latter belongs to the age class j. The matrix was reconstructed from the data of Prem et al. [21], with the method described in Appendix B.

**Figure 8.**Cumulative number of cases. We plot the cumulative data (reds dots) and the best fits of the model $CR\left(t\right)$ (black curve) and $CU\left(t\right)$ (green curve). We fix $f=0.8$, $1/\eta =7$ days and $1/\nu =7$ and we apply the method described in Liu et al. [29]. The best fit is ${d}_{1}=$ 2 April, ${d}_{2}=$ 5 April, $D=$ 27 April, $\mu =0.6$, ${\chi}_{1}=179$, ${\chi}_{2}=0.085$, ${\chi}_{3}=1$ and ${t}_{0}=$ 13 January.

**Figure 9.**Daily number of cases. We plot the daily data (black dots) with $DR\left(t\right)$ (blue curve). We fix $f=0.8$, $1/\eta =7$ days and $1/\nu =7$ and we apply the method described in Liu et al. [29]. The best fit is ${d}_{1}=$ 2 April, ${d}_{2}=$ 5 April, $N=$ 27 April, $\mu =0.6$, ${\chi}_{1}=179$, ${\chi}_{2}=0.085$, ${\chi}_{3}=1$ and ${t}_{0}=$ 13 January.

**Figure 10.**In this figure we plot the data for the cumulative number of death (black dots), and our best fits for $D\left(t\right)$ (red curves).

**Figure 11.**We plot a comparison between the model (12)–(15) and the age structured data from Japan by age class. We took $1/\nu =1/\eta =7$ days for each age class. Our best fit is obtained for ${f}_{i}$ which depends linearly on the age class until it reaches 90%, with ${f}_{1}=0.1$, ${f}_{2}=0.2$, ${f}_{3}=0.3$, ${f}_{4}=0.4$, ${f}_{5}=0.5$, ${f}_{6}=0.6$, ${f}_{7}=0.7$, ${f}_{8}=0.8$, ${f}_{9}=0.9$, and ${f}_{10}=0.9$. The values we used for the first day of public intervention are ${D}_{i}=13\phantom{\rule{4.pt}{0ex}}\mathrm{April}$ for the 0–20 years age class $i=1,2$, ${D}_{i}=11\phantom{\rule{4.pt}{0ex}}\mathrm{April}$ for the age class going from $[20,30[$ to $[60,70[$$i=3,4,5,6,7$, and ${D}_{i}=16\phantom{\rule{4.pt}{0ex}}\mathrm{April}$ for the remaining age classes. We fit the data from 30 March to 20 April to derive the value of ${\chi}_{1}^{i}$ and ${\chi}_{2}^{i}$ for each age class. For the intensity of confinement we use the values ${\mu}_{1}={\mu}_{2}=0.4829$, ${\mu}_{3}={\mu}_{4}=0.2046$, ${\mu}_{5}={\mu}_{6}=0.1474$, ${\mu}_{7}=0.0744$, ${\mu}_{8}=0.1736$, ${\mu}_{9}={\mu}_{10}=0.1358$. By applying the method described in Appendix A, we obtain ${\tau}_{1}=0.1630$, ${\tau}_{2}=0.1224$, ${\tau}_{3}=0.3028$, ${\tau}_{4}=0.2250$, ${\tau}_{5}=0.1520$, ${\tau}_{6}=0.1754$, ${\tau}_{7}=0.1289$, ${\tau}_{8}=0.1091$, ${\tau}_{9}=0.1211$ and ${\tau}_{10}=0.1642$. The matrix $\varphi $ is the one defined in (18).

**Figure 13.**Rate of contact between age classes according to the fitted data. For each age class in the y-axis we plot the rate of contacts between one individual of this age class and another individual of the age class indicated on the x-axis. (

**a**) is the rate of contacts before the start of public measures (11 April). (

**b**) is the rate of contacts at the date of effect of the public measures for the last age class (16 April). (

**c**) is the rate of contacts one week later (23 April). (

**d**) is the rate of contacts one month later (16 May). In this figure we use ${\tau}_{1}=0.1630$, ${\tau}_{2}=0.1224$, ${\tau}_{3}=0.3028$, ${\tau}_{4}=0.2250$, ${\tau}_{5}=0.1520$, ${\tau}_{6}=0.1754$, ${\tau}_{7}=0.1289$, ${\tau}_{8}=0.1091$, ${\tau}_{9}=0.1211$ and ${\tau}_{10}=0.1642$, ${\mu}_{1}={\mu}_{2}=0.4829$, ${\mu}_{3}={\mu}_{4}=0.2046$, ${\mu}_{5}={\mu}_{6}=0.1474$, ${\mu}_{7}=0.0744$, ${\mu}_{8}=0.1736$, ${\mu}_{9}={\mu}_{10}=0.1358$, and ${D}_{1}={D}_{2}=13\phantom{\rule{4.pt}{0ex}}\mathrm{April}$, ${D}_{3}={D}_{4}={D}_{5}={D}_{6}={D}_{7}=11\phantom{\rule{4.pt}{0ex}}\mathrm{April}$, ${D}_{8}={D}_{9}={D}_{10}=16\phantom{\rule{4.pt}{0ex}}\mathrm{April}$.

**Table 1.**The age distribution of Japan is taken from the Statistics Bureau of Japan [24]. The number of cases and the number of death the data come from Prefectural Governments and Japan Ministry of Health, Labour and Welfare.

Age group | $[0,10[$ | $[10,20[$ | $[20,30[$ | $[30,40[$ | $[40,50[$ | $[50,60[$ | $[60,70[$ | $[70,80[$ | $[80,90[$ | $[90,100[$ |

Age class for 2019 | 9,859,515 | 11,171,044 | 12,627,964 | 14,303,042 | 18,519,755 | 16,277,853 | 16,231,582 | 15,926,926 | 8,939,954 | 2,309,313 |

Age class per 10,000 people | 781 | 885 | 1000 | 1133 | 1467 | 1290 | 1286 | 1262 | 709 | 183 |

Confirmed Cases | 211 | 327 | 2216 | 2034 | 2220 | 2355 | 1566 | 1289 | 857 | 304 |

Death | 0 | 0 | 0 | 2 | 6 | 4 | 7 | 37 | 49 | 9 |

**Table 2.**Statistical summary of the data from Table 1.

Dataset | Japanese Population | Infected | Deceased |
---|---|---|---|

First Quartile | 28 | 28 | 68 |

Median | 48 | 44 | 75 |

Third Quartile | 67 | 59 | 81 |

Symbol | Interpretation | Method | |
---|---|---|---|

${t}_{0}$ | Time at which the epidemic started | fitted | |

${S}_{0}$ | Number of susceptible at time ${t}_{0}$ | fixed | |

${I}_{0}$ | Number of asymptomatic infectious at time ${t}_{0}$ | fitted | |

${U}_{0}$ | Number of unreported symptomatic infectious at time ${t}_{0}$ | fitted | |

$\tau \left(t\right)$ | Transmission rate at time t | fitted | |

D | First day of public intervention | fitted | |

$\mu $ | Intensity of the public intervention | fitted | |

$1/\nu $ | Average time during which asymptomatic infectious are asymptomatic | fixed | |

f | Fraction of asymptomatic infectious that become reported symptomatic infectious | fixed | |

${\nu}_{1}=f\phantom{\rule{0.166667em}{0ex}}\nu $ | Rate at which asymptomatic infectious become reported symptomatic | fixed | |

${\nu}_{2}=(1-f)\phantom{\rule{0.166667em}{0ex}}\nu $ | Rate at which asymptomatic infectious become unreported symptomatic | fixed | |

$1/\eta $ | Average time symptomatic infectious have symptoms | fixed |

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**MDPI and ACS Style**

Griette, Q.; Magal, P.; Seydi, O.
Unreported Cases for Age Dependent COVID-19 Outbreak in Japan. *Biology* **2020**, *9*, 132.
https://doi.org/10.3390/biology9060132

**AMA Style**

Griette Q, Magal P, Seydi O.
Unreported Cases for Age Dependent COVID-19 Outbreak in Japan. *Biology*. 2020; 9(6):132.
https://doi.org/10.3390/biology9060132

**Chicago/Turabian Style**

Griette, Quentin, Pierre Magal, and Ousmane Seydi.
2020. "Unreported Cases for Age Dependent COVID-19 Outbreak in Japan" *Biology* 9, no. 6: 132.
https://doi.org/10.3390/biology9060132