# Protection Strategy against an Epidemic Disease on Edge-Weighted Graphs Applied to a COVID-19 Case

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## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Basic Definitions

#### 2.1. Graphs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7**

**.**The degree centrality of ${v}_{i}\in V$ of an edge-weighted graph $(G,w)$, denoted by ${C}_{D}^{w\alpha}\left({v}_{i}\right)$, is defined as

#### 2.2. DIL-W${}^{\alpha}$ Ranking

**Definition**

**8**

**.**The importance of an edge ${e}_{ij}\in E$, denoted by ${I}^{\alpha}\left({e}_{ij}\right)$, is defined as

**Definition**

**9**

**.**The contribution that ${v}_{i}\in V$ makes to the importance of the edge ${e}_{ij}$, denoted by ${W}^{\alpha}\left({e}_{ij}\right)$, is defined as

**Definition**

**10**

**.**The importance of a vertex ${v}_{i}\in V$, denoted by ${L}^{\alpha}\left({v}_{i}\right)$, is defined as

**Remark**

**1.**

#### 2.3. Graph from a Database

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**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. Method

#### Strategy Protection

**Definition**

**16.**

**Definition**

**17.**

## 4. Results

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

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Graph obtained from database of Olmué city, Chile, with 3866 vertices and 6,841,470 edges.

**Figure 4.**The left side shows a graph without protection or infection. On the right side, we see an infected node (red) and a protected node (blue) in the same graph.

**Figure 6.**In the left, the first 3 protected places generated by the DIL-W${}^{1}$ ranking. On the right, the first 3 protected places generated by Strength ranking.

**Figure 7.**Real infected data (black), fitted curve (red), and infected curve obtained in the spread on ${G}_{\mathcal{E}}$ (cyan).

**Figure 10.**Relationship between the real infected and those immunized according to the different DIL-W${}^{\alpha}$ ranking.

**Figure 11.**The different infected curves, considering the 10% protection according to the DIL-Wαranking and carried out in different weeks.

Notations | Definition and Description |
---|---|

G | Graph or network. |

$(G,w)$ | Edge-weighted graph. |

${v}_{i}$ | Vertex or node. |

$N\left({v}_{i}\right)$ | Neighborhood of a vertex v. |

${e}_{ij}$ | Edge between vertex ${v}_{i}$ and vertex ${v}_{j}$. |

${w}_{ij}$ | Weight of the edge ${e}_{ij}$. |

$deg\left({v}_{i}\right)$ | Degree of the vertex ${v}_{i}$. |

$S\left({v}_{i}\right)$ | Strength of the vertex ${v}_{i}$. |

$\alpha $ | Real number. Tuning parameter. |

${C}_{D}^{w\alpha}\left({v}_{i}\right)$ | Degree centrality of ${v}_{i}\in V$ of an edge-weighted graph $(G,w)$. |

DIL-W${}^{\alpha}$ | Ranking based on Degree and importance of line. |

${I}^{\alpha}\left({e}_{ij}\right)$ | Importance of edge ${e}_{ij}$. |

${W}^{\alpha}\left({e}_{ij}\right)$ | Contribution that ${v}_{i}$ makes to the importance of the edge ${e}_{ij}$. |

${L}^{\alpha}\left({v}_{i}\right)$ | The importance of a vertex ${v}_{i}$. |

$\mathcal{E}$ | Database. |

${\mathcal{X}}_{k}$ | Variable of a database. |

${p}_{k}$ | Weight of the variable ${\mathcal{X}}_{k}$. |

k | Protection budget (the number of nodes in graph G that can be protected). |

$\sigma $ | Ratio of surviving nodes. |

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.; Taramasco, C.
Protection Strategy against an Epidemic Disease on Edge-Weighted Graphs Applied to a COVID-19 Case. *Biology* **2021**, *10*, 667.
https://doi.org/10.3390/biology10070667

**AMA Style**

Manríquez R, Guerrero-Nancuante C, Taramasco C.
Protection Strategy against an Epidemic Disease on Edge-Weighted Graphs Applied to a COVID-19 Case. *Biology*. 2021; 10(7):667.
https://doi.org/10.3390/biology10070667

**Chicago/Turabian Style**

Manríquez, Ronald, Camilo Guerrero-Nancuante, and Carla Taramasco.
2021. "Protection Strategy against an Epidemic Disease on Edge-Weighted Graphs Applied to a COVID-19 Case" *Biology* 10, no. 7: 667.
https://doi.org/10.3390/biology10070667