# Applying Regressive Machine Learning Techniques in Determination of COVID-19 Vaccinated Patients’ Influence on the Number of Confirmed and Deceased Patients

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## Abstract

**:**

## 1. Introduction

- Is there a correlation between the number of vaccinated, fully vaccinated, and boosted patients and the number of new confirmed and deceased cases?
- Can the above be modeled using AI-based regression methods?
- Does the use of cross-correlation determined lags (the time-shifts of discrete data points between the input and output datasets) enable better performance when regressing with AI-based regression methods?

## 2. Materials and Methods

#### 2.1. Dataset

#### Cross-Correlation Analysis

#### 2.2. Regression Methods

#### 2.2.1. Linear Regression

#### 2.2.2. LASSO

#### 2.2.3. Logistic Regression

#### 2.2.4. Multilayer Perceptron

#### 2.2.5. Support Vector Regression

#### 2.3. Evaluation

#### Cross-Validation

## 3. Results and Discussion

- Vaccinated Patients and Confirmed Patients (VC),
- Vaccinated Patients and Deceased Patients (VD),
- Fully Vaccinated Patients and Confirmed Patients (FVC),
- Fully Vaccinated Patients and Deceased Patients (FVD),
- Boosted Patients and Confirmed Patients (BC), and
- Boosted Patients and Deceased Patients (BD).

#### 3.1. USA

#### 3.1.1. Correlation Analysis Results

#### 3.1.2. Regression Results

#### 3.2. United Kingdom

#### 3.2.1. Correlation Analysis Results

#### 3.2.2. Regression Results

#### 3.3. Germany

#### 3.3.1. Correlation Analysis Results

#### 3.3.2. Regression Results

#### 3.4. India

#### 3.4.1. Correlation Analysis Results

#### 3.4.2. Regression Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MLP | Multilayer Perceptron |

SVR | Support Vector Regressor |

FVC | Fully Vaccinated Patients—Confirmed Patients Data Pair |

FVD | Fully Vaccinated Patients—Deceased Patients Data Pair |

GER | Germany |

IND | India |

LASSO | Least Absolute Shrinkage and Selection Operator |

LogR | Logistic Regression |

LR | Linear Regression |

MAPE | Mean Average Percentage Error |

OWID | Our World in Data |

UK | United Kingdom |

USA | United States of America |

VC | Vaccinated Patients—Confirmed Patients Data Pair |

VD | Vaccinated Patients—Deceased Patients Data Pair |

BC | Boosted Patients—Confirmed Patients Data Pair |

BD | Boosted patients—Deceased patients Data Pair |

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**Figure 1.**The display of the data contained within the dataset for all used countries, including confirmed cases, deceased patients, vaccinated and fully vaccinated patients. (

**a**) Data for Germany; (

**b**) Data for India; (

**c**) Data for United Kingdom; (

**d**) Data for United States.

**Figure 3.**Cross-correlation data for USA. (

**a**) Cross-correlation results, data for confirmed cases and vaccinations in the United States; (

**b**) cross-correlation results, data for deceased patients and vaccinations in the United States; (

**c**) cross-correlation results, data for confirmed cases and full vaccinations in the United States; (

**d**) cross-correlation results, data for deceased patients and full vaccinations in the United States; (

**e**) cross-correlation results, data for confirmed cases and boosted patients in the United States; (

**f**) cross-correlation results, data for deceased patients and boosted patients in the United States.

**Figure 4.**The results of ML methods per goal for USA (VC—vaccinated-confirmed, VD—vaccinated-deceased, FVC—fully vaccinated-confirmed, FVD—fully vaccinated-deceased, BC—boosted-confirmed, BD—boosted-deceased; lower is better).

**Figure 5.**Cross-correlation data for the UK. (

**a**) Cross-correlation results, data for confirmed cases and vaccinations in the United Kingdom; (

**b**) cross-correlation results, data for deceased patients and vaccinations in the United Kingdom; (

**c**) cross-correlation results, data for confirmed cases and full vaccinations in the United Kingdom; (

**d**) cross-correlation results, data for deceased patients and full vaccinations in the United Kingdom; (

**e**) cross-correlation results, data for confirmed cases and boosted patients in the United Kingdom; (

**f**) cross-correlation results, data for deceased patients and boosted patients in the United Kingdom.

**Figure 6.**The results of ML methods per goal for the UK (VC—vaccinated-confirmed, VD—vaccinated-deceased, FVC—fully vaccinated-confirmed, FVD—fully vaccinated-deceased, BC—boosted-confirmed, BD—boosted-deceased; lower is better).

**Figure 7.**Cross-correlation data for Germany. (

**a**) Cross-correlation results, data for confirmed cases and vaccinations in Germany; (

**b**) cross-correlation results, data for deceased patients and vaccinations in Germany; (

**c**) cross-correlation results, data for confirmed cases and full vaccinations in Germany; (

**d**) cross-correlation results, data for deceased patients and full vaccinations in Germany; (

**e**) cross-correlation results, data for confirmed cases and boosted patients in Germany; (

**f**) cross-correlation results, data for deceased patients and boosted patients in Germany.

**Figure 8.**The results of ML methods per goal for Germany (VC—vaccinated-confirmed, VD—vaccinated-deceased, FVC—fully vaccinated-confirmed, FVD—fully vaccinated-deceased, BC—boosted-confirmed, BD—boosted-deceased; lower is better).

**Figure 9.**Cross-correlation data for India. (

**a**) Cross-correlation results, data for confirmed cases and vaccinations in India; (

**b**) cross-correlation results, data for deceased patients and vaccinations in India; (

**c**) cross-correlation results, data for confirmed cases and full vaccinations in India; (

**d**) cross-correlation results, data for deceased patients and full vaccinations in India; (

**e**) cross-correlation results, data for confirmed cases and boosted patients in India; (

**f**) cross-correlation results, data for deceased patients and boosted patients in India.

**Figure 10.**The results of ML methods per goal for India (VC—vaccinated-confirmed, VD—vaccinated-deceased, FVC—fully vaccinated-confirmed, FVD—fully vaccinated-deceased, BC—boosted-confirmed, BD—boosted-deceased; lower is better).

Paper | Goal | Results | Drawbacks |
---|---|---|---|

[13] | Epidemiology curve metrics, globally | ${R}^{2}>0.9$ globally | Early in pandemic, low amount of data. |

[14] | Epidemiology curve metrics, 10-day prediction | ${R}^{2},{R}_{Adjusted}^{2}>0.95$ | Early in pandemic, low amount of data. |

[15] | Incidence rates, USA | Getis-Ord Gi* (p < 0.05) | Only USA is explored. |

[16] | Spread and influence modeling | ∼95% variance explained | Only focuses on LR and SVM method variants. |

[17] | ROC | Prediction of increase | Only focussed on the sub-Saharan Africa region. |

**Table 2.**The starting date of vaccination data in the dataset and the number of used data points per country.

Country | Starting Date | Number of Data Points |
---|---|---|

Germany | 27 December 2020 | 564 |

India | 16 January 2021 | 544 |

United Kingdom | 10 January 2021 | 550 |

United States | 13 December 2020 | 578 |

Hyperparameter Name | Possible Values | Count |
---|---|---|

Fit Intercept | True, False | 2 |

Normalize | True, False | 2 |

Positive | True, False | 2 |

Hyperparameter Name | Possible Values | Count |
---|---|---|

Regularization Parameter | 0.1, 0.3, 0.5, 0.7, 1.0 | 5 |

Normalization | True, False | 2 |

Fit Intercept | True, False | 2 |

Positive | True, False | 2 |

Hyperparameter Name | Possible Values | Count |
---|---|---|

Fit Intercept | True, False | 2 |

Normalize | True, False | 2 |

Positive | True, False | 2 |

C | 0.1, 0.3, 0.5, 0.7, 1.0 | 5 |

Solver | newton-cg, LBFGS, Liblinear, SAG, SAGA | 5 |

Hyperparameter Name | Possible Values | Count |
---|---|---|

Hidden Layer Sizes | (50,
50, 50, 50), (50, 50, 50), (50, 50), (50), (25, 25, 25, 25), (25, 25, 25), (25, 25), (25), (10, 10, 10, 10), (10, 10, 10), (10, 10), (10), (5, 5, 5, 5), (5, 5, 5), (5, 5), (5), (50, 25, 10, 5), (25, 10, 5), (50, 25, 10), (25, 10) | 20 |

Activation function | ReLU, Identity, Logistic, tanh | 4 |

Solver | Adam, LBFGS | 2 |

Learning Rate Type | Constant, Adaptive, Inversely Scaling | 3 |

Initial Learning Rate | 0.1, 0.01, 0.5, 0.00001 | 4 |

L2 Regularization Parameter | 0.01, 0.1, 0.001, 0.0001 | 4 |

Hyperparameter Name | Possible Values | Count |
---|---|---|

Kernel | Linear, Poly, RBF, Sigmoid, Precomputed | 5 |

Gamma | Scale, Auto | 2 |

Degree | 1, 2, 3, 4, 5 | 5 |

C | 0.1, 0.3, 0.5, 0.7, 1.0 | 5 |

coef0 | 0.0, 0.1, 0.2, 0.3, 0.4, 0.5 | 6 |

Goal | Method | MAPE | ${\mathit{\sigma}}_{\overline{\mathbf{MAPE}}}$ | Hyperparameters |
---|---|---|---|---|

VC | LR | 0.007894757 | 0.000182385 | ’fit_intercept’: True, ’normalize’: True, ’positive’: False |

VD | LR | 0.272645679 | 0.030292848 | ’fit_intercept’: True, ’normalize’: False, ’positive’: False |

FVC | LR | 0.022135412 | 0.002929293 | ’fit_intercept’: True, ’normalize’: True, ’positive’: False |

FVD | LR | 0.238485828 | 0.020128384 | ’fit_intercept’: True, ’normalize’: True, ’positive’: False |

BC | MLP | 0.054622943 | 0.018534421 | ’activation’: ’identity’, ’L2 Regularization’: 0.001, ’hidden_layer_sizes’: (25, 25, 25), ’learning_rate’: ’adaptive’, ’learning_rate_init’: 0.01, ’solver’: ’lbfgs’ |

BD | LR | 0.239913949 | 0.027688232 | ’fit_intercept’: True, ’normalize’: False, ’positive’: False |

Goal | Method | MAPE | ${\mathit{\sigma}}_{\overline{\mathbf{MAPE}}}$ | Hyperparameters |
---|---|---|---|---|

VC | LR | 0.019928482 | 0.017283747 | ’fit_intercept’: True, ’normalize’: True, ’positive’: True |

VD | MLP | 0.448848236 | 0.517283746 | ’activation’: ’logistic’, ’L2 Regularization’: 0.001, ’hidden_layer_sizes’: (50, 50, 50, 50), ’learning_rate’: ’constant’, ’learning_rate_init’: 0.01, ’solver’: ’adam’ |

FVC | LR | 0.021928348 | 0.017274727 | ’fit_intercept’: True, ’normalize’: False, ’positive’: True |

FVD | MLP | 0.392838295 | 0.450293876 | ’activation’: ’logistic’, ’L2 Regularization’: 0.0001, ’hidden_layer_sizes’: 25, ’learning_rate’: ’adaptive’, ’learning_rate_init’: 0.5, ’solver’: ’adam’ |

BC | LR | 0.202194939 | 0.090513952 | ’fit_intercept’: True, ’normalize’: False, ’positive’: True |

BD | LR | 0.244092882 | 0.078351545 | ’fit_intercept’: True, ’normalize’: True, ’positive’: False |

Goal | Method | MAPE | ${\mathit{\sigma}}_{\overline{\mathbf{MAPE}}}$ | Hyperparameters |
---|---|---|---|---|

VC | LR | 0.099382736 | 0.019283747 | ’fit_intercept’: False, ’normalize’: False, ’positive’: False |

VD | MLP | 0.449293021 | 0.041937453 | ’activation’: ’logistic’, ’L2 Regularization’: 0.001, ’hidden_layer_sizes’: 25, ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.5, ’solver’: ’adam’ |

FVC | MLP | 0.138294921 | 0.009283746 | ’activation’: ’tanh’, ’L2 Regularization’: 0.01, ’hidden_layer_sizes’: (25, 10), ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.5, ’solver’: ’lbfgs’ |

FVD | MLP | 0.364042302 | 0.033928144 | ’activation’: ’tanh’, ’L2 Regularization’: 0.0001, ’hidden_layer_sizes’: 10, ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.5, ’solver’: ’adam’ |

BC | MLP | 0.168827331 | 0.065944293 | ’activation’: ’identity’, ’L2 Regularization’: 0.0001, ’hidden_layer_sizes’: (10, 10), ’learning_rate’: ’adaptive’, ’learning_rate_init’: 0.01, ’solver’: ’adam’ |

BD | MLP | 0.380012828 | 0.060623841 | ’activation’: ’logistic’, ’L2 Regularization’: 0.01, ’hidden_layer_sizes’: (25, 10, 5), ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.1, ’solver’: ’adam’ |

Goal | Method | MAPE | ${\mathit{\sigma}}_{\overline{\mathbf{MAPE}}}$ | Hyperparameters |
---|---|---|---|---|

VC | LR | 0.089727374 | 0.012938482 | ’fit_intercept’:True, ’normalize’: False, ’positive’: False |

VD | MLP | 0.391827932 | 0.039283742 | ’activation’: ’relu’, ’L2 Regularization’: 0.01, ’hidden_layer_sizes’: (10, 10, 10, 10), ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.01, ’solver’: ’adam’ |

FVC | LR | 0.059982834 | 0.005674237 | ’fit_intercept’: True, ’normalize’: True, ’positive’: False |

FVD | MLP | 0.446372182 | 0.059283875 | ’activation’: ’logistic’, ’L2 Regularization’: 0.0001, ’hidden_layer_sizes’: (25, 25, 25, 25), ’learning_rate’: ’adaptive’, ’learning_rate_init’: 0.5, ’solver’: ’adam’ |

BC | MLP | 0.213498520 | 0.031304591 | ’activation’: ’tanh’, ’L2 Regularization’: 0.1, ’hidden_layer_sizes’: (50, 50), ’learning_rate’: ’adaptive’, ’learning_rate_init’: 0.1, ’solver’: ’lbfgs’ |

BD | MLP | 0.314889279 | 0.028250913 | ’activation’: ’tanh’, ’L2 Regularization’: 0.01, ’hidden_layer_sizes’: (25, 25, 25), ’learning_rate’: ’invscaling’, ’learning_rate_init’: 0.5, ’solver’: ’adam’ |

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## Share and Cite

**MDPI and ACS Style**

Baressi Šegota, S.; Lorencin, I.; Anđelić, N.; Musulin, J.; Štifanić, D.; Glučina, M.; Vlahinić, S.; Car, Z. Applying Regressive Machine Learning Techniques in Determination of COVID-19 Vaccinated Patients’ Influence on the Number of Confirmed and Deceased Patients. *Mathematics* **2022**, *10*, 2925.
https://doi.org/10.3390/math10162925

**AMA Style**

Baressi Šegota S, Lorencin I, Anđelić N, Musulin J, Štifanić D, Glučina M, Vlahinić S, Car Z. Applying Regressive Machine Learning Techniques in Determination of COVID-19 Vaccinated Patients’ Influence on the Number of Confirmed and Deceased Patients. *Mathematics*. 2022; 10(16):2925.
https://doi.org/10.3390/math10162925

**Chicago/Turabian Style**

Baressi Šegota, Sandi, Ivan Lorencin, Nikola Anđelić, Jelena Musulin, Daniel Štifanić, Matko Glučina, Saša Vlahinić, and Zlatan Car. 2022. "Applying Regressive Machine Learning Techniques in Determination of COVID-19 Vaccinated Patients’ Influence on the Number of Confirmed and Deceased Patients" *Mathematics* 10, no. 16: 2925.
https://doi.org/10.3390/math10162925