# A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine

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## Abstract

**:**

## 1. Introduction

## 2. The Second Pandemic Wave and a Geometrical Approximation of the Form of Ukraine

## 3. Formulation of the Mathematical Model

## 4. Identification of Initial Profiles

## 5. Numerical Simulations: Preliminaries

^{2}/day. Thus, similar to the parameter $\gamma $, we used the fixed value of ${d}_{1}$ in what follows. It should be noted that a similar approach was used in works [12,13]. In [13], the diffusivity is the fixed value $0.5$ by assumption, while that is assumed to be a linear function of u with the fixed coefficient $\nu $ = 10

^{−4}km

^{2}/day/person in [12]. To the best of our knowledge, there are no papers suggesting a method for estimation of diffusion coefficients in the models devoted to the COVID-19 pandemic spread.

## 6. Results of Numerical Simulations with Four Variable Parameters in the Model

## 7. Results of Numerical Simulations with a Single Variable Parameter in the Model

^{2}/day, and $\phantom{\rule{4pt}{0ex}}{d}_{2}=0$ arising in the model, numerical simulations were conducted for a wide range of values of the parameter b. As a result, the most appropriate values of b for each region were identified as follows:

## 8. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The geometrical approximation of the shape of Ukraine. The numbers 1, 2, 3, and 4 are used for the regions West, Kyiv, East, and Rest, respectively.

**Figure 2.**3D plots for the density $u(t,x,y)$ at the time moments $t=0,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}100$, and 157 days.

**Figure 3.**3D plots for the density $v(t,x,y)$ at the time moments $t=0,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}100$, and 157 days.

**Figure 4.**Comparison of the modeling results (solid line) and official data (dashed line) for the number of infected persons U during the second wave of COVID-19 in the West, Kyiv, East, and Rest regions and in all of Ukraine.

**Figure 5.**Comparison of the modeling results (solid line) and official data (dashed line) for the number of deaths V during the second wave of COVID-19 in the West, Kyiv, East, and Rest regions, and in all of Ukraine.

**Figure 6.**3D plots for the density $u(t,x,y)$ at the time moments $t=0,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}100$, and 157 days.

**Figure 7.**3D plots for the density $v(t,x,y)$ at the time moments $t=0,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}100$, and 157 days.

**Figure 8.**Comparison of the modeling results (solid line) and official data (dashed line) for the number of infected persons U during the second wave of COVID-19 in the West, Kyiv, East, and Rest regions and in all of Ukraine.

**Figure 9.**Comparison of the modeling results (solid line) and official data (dashed line) for the number of deaths V during the second wave of COVID-19 in the West, Kyiv, East, and Rest regions and in all of Ukraine.

$u(t,x,y)$ | the total density of the COVID-19 cases at time t |

$v(t,x,y)$ | the total density of deaths at time t |

${d}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$ | the diffusivities (random movement) of the infected population |

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 COVID-19 cases is bounded in time |

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

$\mathit{t}=0$, 7 February 2021 | Ukraine (Without TOTs) | West | Kyiv | East | Rest |
---|---|---|---|---|---|

Initial value for ${U}_{0}$, persons | 148,206 | 15,296 | 61,100 | 12,560 | 59,250 |

Area of the region ${S}_{i}$, km^{2} | 550,000 | 49,557 | 49,876 | 92,142 | 358,430 |

Average value for ${u}_{0}^{i}$, persons/km^{2} | 0.2695 | 0.309 | 1.225 | 0.136 | 0.165 |

Initial value for ${V}_{0}$, persons | 81 | 12 | 16 | 12 | 41 |

Average value for ${v}_{0}^{i}$, persons/km^{2} | 0.000147 | 0.00024 | 0.00032 | 0.00013 | 0.000114 |

Data | |||||
---|---|---|---|---|---|

Region | ${U}_{T}\left(0\right)$ | ${V}_{T}\left(0\right)$ | ${U}_{0}$ | ${U}_{T}\left(0\right)-{U}_{0}$ | ${V}_{0}={V}_{T}\left(0\right)-{V}_{T}(-1)$ |

West | 162,758 | 3674 | 15,296 | 147,462 | 12 |

Kyiv | 247,219 | 4175 | 61,100 | 186,119 | 16 |

East | 192,241 | 4213 | 12,560 | 179,681 | 12 |

Rest | 642,631 | 11,535 | 59,250 | 583,381 | 41 |

Ukraine | 1,244,849 | 23,597 | 148,206 | 109,6643 | 81 |

West | Kyiv | East | Rest | |
---|---|---|---|---|

${a}_{i}$, $\frac{1}{\mathrm{day}}$ | 0.05556 | 0.04046 | 0.0471 | 0.049 |

${b}_{i}$, $\frac{1}{\mathrm{person}\times \mathrm{day}}$ | 18,520 | 8426 | 23,550 | 30,879 |

${{k}_{0}}_{i}$, $\frac{1}{\mathrm{day}}$ | 0.013333 | 0.0006985 | 0.0015241 | 0.0013287 |

${\alpha}_{i}$, $\frac{1}{\mathrm{day}}$ | 0.025 | 0.0175 | 0.0185 | 0.022 |

**Table 5.**The total number of infected persons U on the specified date: official data and computed data.

Day | Data Source | Ukraine | West | Kyiv | East | Rest |
---|---|---|---|---|---|---|

7 February 2021 | Official Data | 148,206 | 15,296 | 61,100 | 12,560 | 59,250 |

$t=0$ | Initial Data | 148,210 | 15,296 | 61,100 | 12,560 | 59,250 |

9 April 2021 | Official Data | 727,000 | 108,600 | 156,100 | 96,300 | 366,000 |

$t=61$ | Computed Data | 761,200 | 106,400 | 162,000 | 102,300 | 390,500 |

Accuracy | $5\%$ | $2\%$ | $4\%$ | $6\%$ | $7\%$ | |

19 May 2021 | Official Data | 1,063,500 | 139,800 | 218,000 | 169,000 | 536,600 |

$t=100$ | Computed Data | 1,050,900 | 140,400 | 210,000 | $163,500$ | 536,900 |

Accuracy | $1\%$ | $1\%$ | $4\%$ | $4\%$ | $1\%$ | |

14 July 2021 | Official Data | 1,145,600 | 147,100 | 240,000 | 183,200 | 575,600 |

$t=157$ | Computed Data | 1,132,700 | 146,600 | 231,000 | 182,600 | 572,500 |

Accuracy | $1\%$ | $1\%$ | $3\%$ | $1\%$ | $1\%$ |

Day | Data Source | Ukraine | West | Kyiv | East | Rest |
---|---|---|---|---|---|---|

7 February 2021 | Official Data | 81 | 12 | 16 | 12 | 41 |

$t=0$ | Initial Data | 81 | 12 | 16 | 12 | 41 |

9 April 2021 | Official Data | 12,865 | 1856 | 2307 | 1817 | 6885 |

$t=61$ | Computed Data | 14,167 | 1842 | 2660 | 2244 | 7422 |

Accuracy | $10\%$ | $1\%$ | $17\%$ | $22\%$ | $7\%$ | |

19 May 2021 | Official Data | 24,953 | 3003 | 4558 | 4388 | 13,004 |

$t=100$ | Computed Data | 22,439 | 2740 | 3923 | 4084 | 11,693 |

Accuracy | $10\%$ | $10\%$ | $15\%$ | $3\%$ | $10\%$ | |

14 July 2021 | Official Data | 29,149 | 3312 | 5126 | 5700 | 15011 |

$t=157$ | Computed Data | 28,122 | 3226 | 4891 | 5570 | 14,435 |

Accuracy | $3\%$ | $3\%$ | $4\%$ | $2\%$ | $4\%$ |

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**MDPI and ACS Style**

Cherniha, R.; Dutka, V.; Davydovych, V.
A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine. *Symmetry* **2024**, *16*, 1411.
https://doi.org/10.3390/sym16111411

**AMA Style**

Cherniha R, Dutka V, Davydovych V.
A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine. *Symmetry*. 2024; 16(11):1411.
https://doi.org/10.3390/sym16111411

**Chicago/Turabian Style**

Cherniha, Roman, Vasyl’ Dutka, and Vasyl’ Davydovych.
2024. "A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine" *Symmetry* 16, no. 11: 1411.
https://doi.org/10.3390/sym16111411