# A Mathematical Model for the COVID-19 Outbreak and Its Applications

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Application for the COVID-19 Outbreak in China

**Remark**

**1.**

## 4. Application for the COVID-19 Outbreak in Austria and Poland

**Example**

**1.**

**Remark**

**2.**

**Example**

**2.**

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The comparison of the exact solution $u\left(t\right)$ (11) with $a=0.28,\phantom{\rule{4pt}{0ex}}b=\frac{7}{\mathrm{2,000,000}},\phantom{\rule{4pt}{0ex}}{u}_{0}=571$ (blue curve) and the measured data of the COVID-19 cases in

**China**(red dots).

**Figure 2.**The comparison of the exact solution $v\left(t\right)$ (11) with $a=0.28,\phantom{\rule{4pt}{0ex}}b=\frac{7}{\mathrm{2,000,000}},\phantom{\rule{4pt}{0ex}}{u}_{0}=571,\phantom{\rule{4pt}{0ex}}{k}_{0}=0.0094,\phantom{\rule{4pt}{0ex}}\alpha =0.07,\phantom{\rule{4pt}{0ex}}{v}_{0}=17$ (brown curve) and the measured data of the total number of deaths in

**China**(green dots).

**Figure 3.**The solution $u\left(t\right)$ of Equation (2) for $\gamma =0.3$ (red curve), $\gamma =0.5$ (green curve), $\gamma =1$ (blue curve), $\gamma =1.5$ (black curve) and the measured data of the COVID-19 cases in

**China**(red dots).

**Figure 4.**The comparison of the exact solutions (red curves) $u\left(t\right)$ from (11) with $a=0.275,\phantom{\rule{4pt}{0ex}}b=\frac{a}{\mathrm{15,000}},\phantom{\rule{4pt}{0ex}}{u}_{0}=104$ (

**left**) and $u\left(t\right)$ from (12) with $\gamma =2/5,\phantom{\rule{4pt}{0ex}}a=0.383,\phantom{\rule{4pt}{0ex}}b=\frac{a}{{\mathrm{15,000}}^{2/5}},\phantom{\rule{4pt}{0ex}}{u}_{0}=104$ (

**right**) and the measured data of the COVID-19 cases in

**Austria**(black dots).

**Figure 5.**The comparison of the exact solutions (brown curve) $v\left(t\right)$ from (12) with $\gamma =2/5,\phantom{\rule{4pt}{0ex}}a=0.383,\phantom{\rule{4pt}{0ex}}b=\frac{a}{{\mathrm{15,000}}^{2/5}},\phantom{\rule{4pt}{0ex}}{u}_{0}=104,\phantom{\rule{4pt}{0ex}}{v}_{0}=6,\phantom{\rule{4pt}{0ex}}{k}_{0}=0.0224.\phantom{\rule{4pt}{0ex}}\alpha =0.1$ and the measured data of the total number of deaths in

**Austria**(green dots).

**Figure 6.**The comparison of the exact solution $u\left(t\right)$ (red curve) and $v\left(t\right)$ (brown curve) from (12) with $\gamma =2/5,\phantom{\rule{4pt}{0ex}}a=0.2835,\phantom{\rule{4pt}{0ex}}{u}_{0}=100,\phantom{\rule{4pt}{0ex}}{t}^{*}=34,\phantom{\rule{4pt}{0ex}}{k}_{0}=0.02,\phantom{\rule{4pt}{0ex}}\alpha =0.036,\phantom{\rule{4pt}{0ex}}{v}_{0}=33$ (parameter b is calculated by Formula (14)) and the measured data of the COVID-19 cases (black dots) and the total number of deaths (green dots) in

**France**.

$u\left(t\right)$ | the total number of the COVID-19 cases at the time t |

$v\left(t\right)$ | the total number of deaths at the time t |

$w\left(t\right)$ | the total number of recovered patients and those under treatment at the time t |

a | the coefficient for the virus transmission mechanism |

b | the coefficient for the effectiveness of the government restrictions (quarantine rules) |

$\gamma $ | the exponent, which guarantees that the total number of the COVID-19 cases is bounded in time |

$k\left(t\right)$ | the coefficient for effectiveness of the health care system during the epidemic process |

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**MDPI and ACS Style**

Cherniha, R.; Davydovych, V.
A Mathematical Model for the COVID-19 Outbreak and Its Applications. *Symmetry* **2020**, *12*, 990.
https://doi.org/10.3390/sym12060990

**AMA Style**

Cherniha R, Davydovych V.
A Mathematical Model for the COVID-19 Outbreak and Its Applications. *Symmetry*. 2020; 12(6):990.
https://doi.org/10.3390/sym12060990

**Chicago/Turabian Style**

Cherniha, Roman, and Vasyl’ Davydovych.
2020. "A Mathematical Model for the COVID-19 Outbreak and Its Applications" *Symmetry* 12, no. 6: 990.
https://doi.org/10.3390/sym12060990