# Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. The Model and Data

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

**Remark**

**1.**

**Remark**

**2.**

#### 2.3. Numerical Simulations

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Method to Estimate the Parameters of (1) from the Number of Reported Cases

**Step 1:**Since f and $\nu $, we know that

**Step 2:**By using Equation (3), we obtain

**Step 3:**In order to evaluate the parameters of the model, we replace $S\left(t\right)$ by ${S}_{0}=11.081\times {10}^{6}$ on the right-hand side of (1) (which is equivalent to neglecting the variation of susceptibles due to the epidemic, which is consistent with the fact that $t\to CR\left(t\right)$ grows exponentially). Therefore, it remains to estimate $\tau $ and $\eta $ in the following system:

**Remark**

**3.**

#### 4.2. Computation of the Basic Reproductive Number ${\mathcal{R}}_{0}$

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chinese Center for Disease Control and Prevention. Available online: http://www.chinacdc.cn/jkzt/crb/zl/szkb_11803/jszl_l11809/ (accessed on 30 January 2020).
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**2020**, 9, 462. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3525558 (accessed on 30 January 2020). [CrossRef] [PubMed] [Green Version] - Magal, P.; Webb, G. The parameter identification problem for SIR epidemic models: Identifying Unreported Cases. J. Math. Biol.
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_{0}in models for infectious diseases in heterogeneous populations. J. Math. Biol.**1990**, 28, 365–382. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Van den Driessche, P.; Watmough, J. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
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**Figure 2.**In the left side figures, the dots correspond to $t\to \mathrm{ln}\left(\right)open="("\; close=")">CR\left(t\right)+{\chi}_{3}$, and in the right side figures, the dots correspond to $t\to CR\left(t\right)$, where $CR\left(t\right)$ is taken from the cumulated confirmed cases in Table 2 (

**top**), in Table 3 (

**middle**), and in Table 4 (

**bottom**). The straight line in the left side figures corresponds to $t\to \mathrm{ln}\left(\right)open="("\; close=")">{\chi}_{1}$. We first estimate the value of ${\chi}_{3}$ and then use a least square method to evaluate ${\chi}_{1}$ and ${\chi}_{2}$. We observe that the data for China in Table 2 and Hubei in Table 3 provides a good fit for $CR\left(t\right)$ in (4), while the data for Wuhan in Table 4 does not provide a good fit for $CR\left(t\right)$ in (4).

**Figure 3.**In these figures, we use $f=0.8,\phantom{\rule{0.166667em}{0ex}}\eta =1/7,\phantom{\rule{0.166667em}{0ex}}\nu =1/7$, and ${S}_{0}=11.081\times {10}^{6}$. The remaining parameters are derived by using (6)–(8). In (

**a**), we plot the number of $t\to CR\left(t\right)$ (black solid line) and $t\to U\left(t\right)$ (blue dotted) and the data (red dotted) corresponding to the confirmed cumulated cases for mainland China in Table 2. We use ${\chi}_{1}=0.16$, ${\chi}_{2}=0.38$, ${\chi}_{3}=1.1$, ${t}_{0}=5.12$ and ${S}_{0}=11.081\times {10}^{6}$ which give $\tau =4.44\times {10}^{-08}$, ${I}_{0}=3.62$, ${U}_{0}=0.2$ and ${\mathcal{R}}_{0}=4.13$. In (

**b**), we plot the number of $t\to CR\left(t\right)$ (black solid line) and $t\to U\left(t\right)$ (blue dotted) and the data (red dotted) corresponding to the confirmed cumulated case for Hubei province in Table 3. We use ${\chi}_{1}=0.23$, ${\chi}_{2}=0.34$, ${\chi}_{3}=0.1$ and ${t}_{0}=-2.45$ and ${S}_{0}=11.081\times {10}^{6}$ which give $\tau =4.11\times {10}^{-08}$${I}_{0}=0.3$, ${U}_{0}=0.02$ and ${\mathcal{R}}_{0}=3.82$. In (

**c**), we plot the number of $t\to CR\left(t\right)$ (black solid line) and $t\to U\left(t\right)$ (blue dotted) and the data (red dotted) corresponding to the confirmed cumulated cases for Wuhan in Table 4. We use ${\chi}_{1}=0.36$, ${\chi}_{2}=0.28$, ${\chi}_{3}=0.1$, ${t}_{0}=-4.52$, and ${S}_{0}=11.08\times {10}^{6}$, which give $\tau =3.6\times {10}^{-08}$, ${I}_{0}=0.25$, ${U}_{0}=0.02$, and ${\mathcal{R}}_{0}=3.35$.

**Figure 4.**In this figure, we plot the graphs of $t\to CR\left(t\right)$ (black solid line), $t\to U\left(t\right)$ (blue solid line) and $t\to R\left(t\right)$ (red solid line). We use again $f=0.8,\phantom{\rule{0.166667em}{0ex}}\eta =1/7,\phantom{\rule{0.166667em}{0ex}}\nu =1/7$, and ${S}_{0}=11.081\times {10}^{6}$. In (

**a**), we use ${\chi}_{1}=0.16$, ${\chi}_{2}=0.38$, ${\chi}_{3}=1.1$, ${t}_{0}=5.12$ for the parameter values for China which give $\tau =4.44\times {10}^{-08}$ for $t\in [{t}_{0},25]$ and $\tau =0$ for $t>25$, ${I}_{0}=3.62$, ${U}_{0}=0.2$. In (

**b**), we use ${\chi}_{1}=0.23$, ${\chi}_{2}=0.34$, ${\chi}_{3}=0.1$ and ${t}_{0}=-2.45$, for the parameter values obtained from the data for Hubei province, which give $\tau =4.11\times {10}^{-08}$ for $t\in [{t}_{0},25]$ and $\tau =0$ for $t>25$, ${I}_{0}=0.3$, ${U}_{0}=0.02$. In (

**c**), we use ${\chi}_{1}=0.36$, ${\chi}_{2}=0.28$, ${\chi}_{3}=0.1$, and ${t}_{0}=-4.52$ for the parameter values obtained from the data for Wuhan, which give $\tau =3.6\times {10}^{-08}$ for $t\in [{t}_{0},25]$ and $\tau =0$ for $t>25$, ${I}_{0}=0.25$, ${U}_{0}=0.02$. The cumulated number of reported cases goes up to 7000 in (

**b**), 4000 in (

**b**) and 1400 in (

**c**), and the turning point is around 30 January in (

**a**–

**c**).

**Figure 5.**In this figure, we plot the graphs of $t\to CR\left(t\right)$ (black solid line), $t\to U\left(t\right)$ (blue solid line) and $t\to R\left(t\right)$ (red solid line). We use $f=0.8,\phantom{\rule{0.166667em}{0ex}}\eta =1/7,\phantom{\rule{0.166667em}{0ex}}\nu =1/7$, and ${S}_{0}=11.081\times {10}^{6}$. The remaining parameters are derived by using (6)–(8). We obtain $\tau =4.44\times {10}^{-08}$, ${I}_{0}=3.62$ and ${U}_{0}=0.2$. The cumulated number of reported cases goes up to $8.5$ million people and the turning point is day 54. Thus, the turning point is 23 February (i.e., 54–31).

**Figure 6.**In this figure, we use $1/\eta =7$ days, and we plot the basic reproductive number ${\mathcal{R}}_{0}$ as a function of f and $1/\nu $ using (9) with ${\chi}_{2}=0.38$, which corresponds to the data for China in Table 2. If both f and $1/\nu $ are sufficiently small, ${\mathcal{R}}_{0}<1$. The red plane is the value of ${\mathcal{R}}_{0}=4.13$.

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 $ | Transmission rate | 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 | fitted | |

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

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

**Table 2.**Reported case data 20–29 January 2020, reported for mainland China by the Chinese CDC [1].

Date January | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
---|---|---|---|---|---|---|---|---|---|---|

Confirmed cases (cumulated) for mainland China | 291 | 440 | 571 | 830 | 1287 | 1975 | 2744 | 4515 | 5974 | 7711 |

Mortality cases (cumulated) for mainland China | 9 | 17 | 25 | 41 | 56 | 80 | 106 | 132 | 170 |

**Table 3.**Reported case data 23–31 January 2020, reported for Hubei Province by the Wuhan Municipal Health Commission [6].

Date January | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
---|---|---|---|---|---|---|---|---|---|

Confirmed cases (cumulated) for Hubei | 549 | 729 | 1052 | 1423 | 2714 | 3554 | 4586 | 5806 | 7153 |

Mortality cases (cumulated) for Hubei | 24 | 39 | 52 | 76 | 100 | 125 | 162 | 204 | 249 |

**Table 4.**Reported case data 23–31 January 2020, reported for Wuhan Municipality by the Wuhan Municipal Health Commission [6].

Date January | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
---|---|---|---|---|---|---|---|---|---|

Confirmed cases (cumulated) for Wuhan | 495 | 572 | 618 | 698 | 1590 | 1905 | 2261 | 2639 | 3215 |

Mortality cases (cumulated) for Wuhan | 23 | 38 | 45 | 63 | 85 | 104 | 129 | 159 | 192 |

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**MDPI and ACS Style**

Liu, Z.; Magal, P.; Seydi, O.; Webb, G.
Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions. *Biology* **2020**, *9*, 50.
https://doi.org/10.3390/biology9030050

**AMA Style**

Liu Z, Magal P, Seydi O, Webb G.
Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions. *Biology*. 2020; 9(3):50.
https://doi.org/10.3390/biology9030050

**Chicago/Turabian Style**

Liu, Zhihua, Pierre Magal, Ousmane Seydi, and Glenn Webb.
2020. "Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions" *Biology* 9, no. 3: 50.
https://doi.org/10.3390/biology9030050