# Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19

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## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

#### 2.1. Graph

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

**Definition**

**5.**

**Definition**

**6.**

#### 2.2. Graph Classes and Basic Network Models

#### 2.3. SIR Model

## 3. Model Description

#### 3.1. Graph from a Database

**Definition**

**7.**

**Definition**

**8.**

#### 3.2. Weighting Variables

**Definition**

**9.**

**Lemma**

**1.**

**Proof.**

**Definition**

**10.**

#### 3.3. Weighted Link

**Definition**

**11.**

**Example**

**1.**

- 1
- $REL=\{{\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3},{\mathcal{X}}_{4}\}$ and
- 2
- $CHAR=\{{\mathcal{X}}_{5},{\mathcal{X}}_{6},{\mathcal{X}}_{7},{\mathcal{X}}_{8},{\mathcal{X}}_{9}\}$.

#### 3.4. Graph-Based SIR Model

- 1
- The probability (${P}_{I}\left({v}_{i}\right)$) that a susceptible vertex ${v}_{i}$ is infected by one of its neighbors is given by$${P}_{I}\left({v}_{i}\right)=\sum _{{v}_{j}\in {N}_{I}\left({v}_{i}\right)}\rho \mathsf{\Delta}t\xb7{\tilde{w}}_{ij},$$
- 2
- The probability (${P}_{R}\left({v}_{i}\right)$) that a infected vertex ${v}_{i}$ at time t will recover is given by$${P}_{R}\left({v}_{i}\right)=\delta \mathsf{\Delta}t,$$

## 4. Simulation of Disease Spread

#### 4.1. Order of the Graph

#### 4.2. Representative Factor of the Disease ($\rho $)

#### 4.3. The Mean Degree of Vertices

#### 4.4. Amount of Relation Variables

#### 4.5. The Amount of Classes

## 5. Deterministic Approximation

## 6. Modeling

## 7. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Fraction of infected for different values of $\rho $ (

**a**); end of the disease for different values of $\rho $ (

**b**).

**Figure 24.**Graph obtained from database of Olmué city, Chili, with 3866 vertices and 6,841,470 edges.

**Figure 25.**Simulations on graph obtained in Figure 24.

Person | City | Workplace | E. C. Activity | Address | Sm. | Dri. | Gen. | M. S | Age |
---|---|---|---|---|---|---|---|---|---|

1 | A | Workplace 1 | Theater | y | Y | Y | F | IC | 35 |

2 | A | Workplace 3 | Cinema | y | Y | Y | M | IC | 35 |

3 | B | School B | Football | z | N | N | F | S | 10 |

4 | B | Workplace 1 | Photography | x | N | N | F | M | 48 |

5 | A | Workplace 5 | Does not have | u | Y | N | F | W | 65 |

6 | A | Workplace 4 | Does not have | v | Y | Y | M | IC | 27 |

7 | B | Workplace 2 | Does not have | x | Y | N | M | M | 46 |

8 | A | University 1 | Photography | v | N | N | M | IC | 29 |

9 | A | University 2 | Does not have | w | Y | Y | M | IC | 19 |

10 | B | School B | Karate | x | N | N | M | S | 10 |

11 | A | Workplace 4 | Ping-pong | r | Y | Y | F | M | 54 |

12 | A | School A | Football | s | N | N | M | S | 8 |

13 | A | Workplace 5 | Dance | r | Y | Y | F | M | 57 |

14 | B | School A | Handball | q | N | N | M | S | 11 |

15 | A | University 1 | Does not have | t | N | N | F | S | 25 |

16 | A | Workplace 7 | Singing | p | Y | Y | F | S | 60 |

17 | A | Workplace 8 | Music | k | N | Y | F | S | 28 |

18 | A | Workplace 3 | Does not have | d | N | N | M | S | 47 |

19 | B | School A | Music | g | N | N | F | S | 8 |

20 | A | Workplace 6 | Does not have | h | Y | Y | M | S | 30 |

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**MDPI and ACS Style**

Manríquez, R.; Guerrero-Nancuante, C.; Martínez, F.; Taramasco, C. Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19. *Int. J. Environ. Res. Public Health* **2021**, *18*, 4432.
https://doi.org/10.3390/ijerph18094432

**AMA Style**

Manríquez R, Guerrero-Nancuante C, Martínez F, Taramasco C. Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19. *International Journal of Environmental Research and Public Health*. 2021; 18(9):4432.
https://doi.org/10.3390/ijerph18094432

**Chicago/Turabian Style**

Manríquez, Ronald, Camilo Guerrero-Nancuante, Felipe Martínez, and Carla Taramasco. 2021. "Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19" *International Journal of Environmental Research and Public Health* 18, no. 9: 4432.
https://doi.org/10.3390/ijerph18094432