# Modeling of Vaccination and Contact Tracing as Tools to Control the COVID-19 Outbreak in Spain

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. PDP COVID-19 Model

#### 2.2. Statistical Study and Sensitivity Analysis

#### 2.3. Ethical Statement

## 3. Results

#### 3.1. The Effects of Vaccination

#### 3.1.1. Response Surface

#### 3.1.2. Sensitivity Analysis

#### 3.1.3. Vaccination Activity

#### 3.2. Tracing Activity

#### 3.2.1. Response Surface

#### 3.2.2. Sensitivity Analysis

#### 3.2.3. Effect of Tracing Measures

#### 3.3. Analyzing Simultaneous Contact Tracing and Vaccine Protection

## 4. Discussion

_{0}= 4) is very similar to the value used by Colomer et al., (2021) [31] in their unmitigated scenario (R

_{0}= 5). On the other hand, the predicted unmitigated infection-fatality ratio was 0.63% in the UK [50], using a different modeling approach, and was higher than the value predicted in our model (0.41%). In fact, large variations in this parameter have been described in many studies, probably due to uncertainties in the key parameters chosen to run the models [51,52]. In any case, the differences that we found in the predicted numbers of people infected and dying in a baseline situation versus those described above are reasonable taking into account the different populations and models used to make these predictions that make direct comparisons between studies misleading.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphs (

**A**,

**B**) show the progress of the death toll in Spain depending on the percentage of the population protected by vaccination (from 0% (V0) to 40% (V40)), with and without the application of social measures, respectively using a population dynamic P system (PDP) model over an 80-day period. Graphs (

**C**,

**D**) show the number of people recovering depending on the percentage of the population protected by vaccination (from 0% (V0) to 40% (V40)) with and without the application of social measures, respectively using a PDP model over an 80-day period.

**Figure 2.**Graphs (

**A**,

**B**) show the progress of the death toll in Spain depending on the percentage of contacts traced (from 0% (T0) to 40% (T40)) with and without the application of social measures, respectively using a PDP model over an 80-day period. Graphs (

**C**,

**D**) show the number of people recovering depending on tracing contacts (from 0% (T0) to 40% (T40)) with and without the application of social measures, using a PDP model over an 80-day period. Graphs (

**E**,

**F**) show the number of diagnostic tests carried out depending on tracing contacts (from 0% (T0) to 40% (T40)) with and without the application of social measures, using a PDP model over an 80-day period.

**Figure 3.**Graphs (

**A**,

**B**) show the progress of the death toll in Spain depending on the percentage of traced contacts (from 0% (T0) to 40% (T40)) without and with the application of social measures, with 19.6% of the population protected by vaccination, using a PDP model over an 80-day period. Graphs (

**C**,

**D**) show the number of people recovering depending on contact tracing (from 0% (T0) to 40% (T40)) without and with the application of social measures, with 19.6% of the population protected by vaccination using a PDP model over an 80-day period. Graphs (

**E**,

**F**) show the number of diagnostic tests carried out depending on contact tracing (from 0% (T0) to 40% (T40)) without and with the application of social measures, with a 19.6% of the population protected by vaccination, using a PDP model over an 80-day period.

**Table 1.**Box–Behnken estimated values of the response surface parameters. Disease control measures, and vaccine and protection measures.

Parameter | $\mathbf{Died}\text{}{\mathit{R}}^{2}=0.96$ | $\mathbf{Recovered}\text{}{\mathit{R}}^{2}=0.97$ | ||
---|---|---|---|---|

Value (*) | p-Value | Value (*) | p-Value | |

(Intercept) | 118,259.5 | <0.001 | 17,077,712 | <0.001 |

Probability of disease transmission (Pd) | 27,333.1 | <0.001 | 4,390,755 | <0.001 |

% of the population protected by vaccination (V) | −37,108.5 | <0.001 | −4,674,854 | <0.001 |

Number of people infected at time 0 (F) | 5560.6 | 0.06991 | −19,057 | 0.9136 |

Pd × V | −3146.7 | 0.41215 | −318,566 | 0.2296 |

Pd × F | −1788 | 0.63444 | 64,945 | 0.7943 |

V × F | 3927.8 | 0.31357 | −13,346 | 0.9571 |

Pd^{2} | −6782.2 | 0.1063 | −915,654 | 0.0085 |

V^{2} | 3031 | 0.42859 | −303,586 | 0.2497 |

F^{2} | −3087.3 | 0.42054 | 331,471 | 0.2135 |

**Table 2.**Estimated sensitivity using response surfaces. Disease control measures: vaccine and protection measures affecting the probability of disease transmission from infected to non-infected people.

Deadpeople | Probability of disease transmission from infected to non-infected people | 10,933 | People who die if the probability of transmission of the disease is increased by 1% |

Percentage of people protected by vaccination | −1855 | Decrease in the number of people who died if the number of vaccinated people is increased by 1% | |

Recoveredpeople | Probability of disease transmission from infected to non-infected people | 1,756,302 | Number of people who recovered due to a 1% increase in the probability of disease transmission. |

Percentage of people protected by vaccination | −233,743 | Number of people who recovered due to a 1% increase in the population protected by vaccination. |

**Table 3.**Average number of expected people who died or recovered following infection depending on the level of population protection provided by vaccination. Percentages are with respect to population size (46,014,554 people).

Social Measures | Population Protected by Vaccination (%) | |||||
---|---|---|---|---|---|---|

0 | 10 | 20 | 30 | 40 | ||

With(p = 0.05) | Died | 0.33% | 0.22% (0.11%) | 0.17% (0.16%) | 0.17% (0.16%) | 0.10% (0.23%) |

Recovered | 45.34% | 30.73% (14.61%) | 26.93% (18.41%) | 22.75% (22.59%) | 14.46% (30.88%) | |

Without(p = 0.1) | Died | 0.41% | 0.32% (0.09%) | 0.29% (0.12%) | 0.22% (0.19%) | 0.16% (0.25%) |

Recovered | 56.11% | 50.43% (5.68%) | 44.55% (11.56%) | 39.29% (16.82%) | 33.55% (22.56%) |

**Table 4.**Box–Behnken estimated values of the response surface parameters. Disease control measures: positives traced and social control measures.

Parameter | Dead People ${\mathit{R}}^{2}=0.99$ | Recovered People ${\mathit{R}}^{2}=0.99$ | Number of Diagnostics Tests Required for Adequate Tracing ${\mathit{R}}^{2}=0.99$ | |||
---|---|---|---|---|---|---|

Value (*) | p-Value | Value (*) | p-Value | Value (*) | p-Value | |

(Intercept) | 48,039 | <0.001 | 13,511,678 | <0.001 | 27,364,890 | <0.001 |

Probability of disease transmission (Pd) | 17,049.88 | <0.001 | 3,660,762 | <0.001 | 5,478,335 | <0.001 |

Percentage of people being traced (T) | −61,096 | <0.001 | −5,373,225 | <0.001 | 20,264,444 | <0.001 |

Number of people infected at time 0 (F) | 433.12 | 0.8205 | −32,519 | 0.8367 | −92,972 | 0.8493 |

Pd × T | −14,757 | 0.0012 | −1,025,858 | 0.0030 | 4,683,341 | <0.001 |

Pd × F | 195.25 | 0.9422 | −365,194 | 0.1383 | −811,020 | 0.2671 |

T × F | −78 | 0.9769 | −17,971 | 0.9357 | 24,372 | 0.9719 |

Pd^{2} | −1053.62 | 0.6976 | −590,357 | 0.0327 | −954,013 | 0.2002 |

T^{2} | 43,716.62 | <0.001 | 2,515,438 | <0.001 | −6,884,809 | <0.001 |

F^{2} | 2329.38 | 0.402 | 88,975 | 0.692 | 522,739 | 0.460 |

**Table 5.**Estimated sensitivity using response surfaces. Disease control measures: tracing and social control measures.

Dead people | Probability of disease transmission from infected to non-infected people | 6820 | Increase in the number of people who died by increasing the probability of disease transmission by 1% |

Percentage of people being traced | −3055 | Decrease in the number of people who died by increasing tracing contacts by 1% | |

Recovered people | Probability of disease transmission from infected to non-infected people | 1,464,305 | Increase in the number of people who recovered due to a 1% increase in the probability of disease transmission. |

Percentage of people being traced | −268,661 | Decrease in the number of people who recovered due to a 1% increase in tracing contacts. | |

Diagnostic tests | Probability of disease transmission from infected to non-infected people | 2,191,334 | Increase in the number of tests by increasing the probability of disease transmission by 1% |

Percentage of people being traced | 1,013,222 | Increase in the number of tests by increasing the tracing contacts by 1% |

**Table 6.**Average number of expected people who died or recovered depending on the percentage of contact tracing carried out in the population. Percentages are with respect to population size (46,014,554 people).

Social Measures | People with Positive Contacts That Have Been Traced (%) | |||||
---|---|---|---|---|---|---|

0 | 10 | 20 | 30 | 40 | ||

With (p = 0.05) | Died | 0.33 | 0.12 (0.21) | 0.08 (0.25) | 0.06 (0.27) | 0.06 (0.27) |

Recovered | 45.34 | 25.88% (19.46) | 20.15 (25.19) | 18.26 (27.08) | 16.45 (28.89) | |

Without (p = 0.1) | Died | 0.41 | 0.21 (0.20) | 0.14 (0.27) | 0.10 (0.31) | 0.07 (0.34) |

Recovered | 56.11 | 44.15 (11.96) | 37.14 (18.97) | 32.24 (23.87) | 27.48 (28.63) |

Probability of Transmission of the Disease | Contacts Traced (%) | Number of People Traced | People Traced against the Total Population (%) |
---|---|---|---|

With social measures p = 0.05 | 0 | 0 | 0.00 |

10 | 13,106,761 | 28.48 | |

20 | 19,831,667 | 43.10 | |

30 | 26,495,337 | 57.58 | |

40 | 31,767,937 | 69.04 | |

Without social measures p = 0.1 | 0 | 0 | 0.00 |

10 | 21,943,371 | 47.69 | |

20 | 34,913,187 | 75.87 | |

30 | 43,462,608 | 94.45 | |

40 | 47,673,047 | 103.60 |

**Table 8.**Percentage of the total number of people who died or recovered depending on the percentage of contact tracing carried out in the population if there 19.56% of the population are protected by vaccination.

Percentage of the Population | No Vaccine | 19.56% of the Population Protected by Vaccination | |||||
---|---|---|---|---|---|---|---|

Positive Contact Tracing (%) | |||||||

0 | 10 | 20 | 30 | 40 | |||

With social measures (p = 0.05) | Died | 0.17 | 0.09 | 0.06 | 0.06 | 0.05 | |

Recovered | 26.96 | 19.60 | 14.46 | 13.26 | 10.62 | ||

Without social measures (p = 0.1) | Died | 0.4 | 0.31 | 0.16 | 0.11 | 0.08 | 0.08 |

Recovered | 56.1 | 45.06 | 36.06 | 30.37 | 26.83 | 28.68 |

**Table 9.**Diagnostic tests performed as a percentage of the Spanish population (46,014,554) when vaccination protects 19.6% of the population and contact tracing follows up 10% to 40% of contacts with an infected person.

Probability of Disease Transmission | Number of Diagnostic Tests Performed with Respect to Population Size (%) | |||
---|---|---|---|---|

10% Positive Contact Tracing (%) | 20% Positive Contact Tracing (%) | 30% Positive Contact Tracing (%) | 40% Positive Contact Tracing (%) | |

With social measures (p = 0.05) | 19.3 (8,876,461) | 27.7 (12,750,293) | 36.8 (16,927,530) | 39.0 (17,930,132) |

Without social measures p = 0.10 | 33.4 (15,346,800) | 52.5 (24,166,442) | 68.0 (31,294,462) | 78.8 (36,247,483) |

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Colomer, M.À.; Margalida, A.; Alòs, F.; Oliva-Vidal, P.; Vilella, A.; Fraile, L. Modeling of Vaccination and Contact Tracing as Tools to Control the COVID-19 Outbreak in Spain. *Vaccines* **2021**, *9*, 386.
https://doi.org/10.3390/vaccines9040386

**AMA Style**

Colomer MÀ, Margalida A, Alòs F, Oliva-Vidal P, Vilella A, Fraile L. Modeling of Vaccination and Contact Tracing as Tools to Control the COVID-19 Outbreak in Spain. *Vaccines*. 2021; 9(4):386.
https://doi.org/10.3390/vaccines9040386

**Chicago/Turabian Style**

Colomer, Mª Àngels, Antoni Margalida, Francesc Alòs, Pilar Oliva-Vidal, Anna Vilella, and Lorenzo Fraile. 2021. "Modeling of Vaccination and Contact Tracing as Tools to Control the COVID-19 Outbreak in Spain" *Vaccines* 9, no. 4: 386.
https://doi.org/10.3390/vaccines9040386