# Modeling COVID-19 Disease with Deterministic and Data-Driven Models Using Daily Empirical Data in the United Kingdom

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Mathematical Model Formulation

#### 2.3. Statistical Fitting of Model (1)

#### 2.4. Statistical Predictors and Principal Component Analysis (PCA)

**Principal component analysis (PCA)**is a technique widely used for dimensionality reduction, feature extraction, and data visualization [19] commonly used in the field of machine learning and statistics. It is used to transform high-dimensional data into a lower-dimensional representation while retaining as much of the original data’s variability as possible. PCA achieves this by finding a set of orthogonal axes, called principal components, along which the data varies the most. The principal component analysis can be applied in the statistical modeling of infectious diseases to help analyze and understand complex datasets related to disease dynamics, transmission patterns, and other epidemiological factors [20]. The first principal component explains the most variance in the data, the second principal component explains the second most, and so on. The $kth$ principal component of a data (for instance UK COVID-19 cases) vector ${x}_{\left(i\right)}$ can therefore be given as a score ${t}_{k\left(i\right)}$ = ${x}_{\left(i\right)}\xb7{w}_{\left(k\right)}$ in the transformed coordinates or as the corresponding vector in the space of the original variables, ${x}_{\left(i\right)}\xb7{w}_{\left(k\right)}w\left(k\right)$, where ${w}_{\left(k\right)}$ is the $kth$ eigenvector of ${X}^{T}X$ [21].

**skewness**. The distributional characteristics of this data require appropriate preprocessing steps to ensure that PCA results accurately capture the underlying structure of the data. If data contains significant outliers or skewness that cannot be easily addressed through data preprocessing, one might consider using robust PCA techniques that are less sensitive to extreme values and skewed distributions.

**Kurtosis**, on the other hand, in the context of principal component analysis (PCA), refers to the distribution of data points in terms of their peakedness or the presence of heavy tails in the data’s probability distribution. It measures the degree to which the data deviates from a normal distribution (Gaussian distribution). There are several different measures of kurtosis, but they all essentially assess the tails of the distribution relative to a normal distribution. Kurtosis can have an impact on PCA in the following ways: interpretability of principal components, robustness to outliers, data transformation, and so on. If your data exhibits high kurtosis due to extreme outliers, you might consider using robust PCA techniques that are less sensitive to outliers and heavy-tailed distributions. In as much as kurtosis can impact the results of PCA by affecting the distributional characteristics of the data, appropriate data preprocessing techniques and transformations can help address these issues and lead to more reliable and interpretable principal components.

**coefficient of variation**(CV) is a statistic used to measure the relative variability or spread of data points in a dataset. It is expressed as a percentage and is calculated as the ratio of the standard deviation ($\sigma $) to the mean ($\mu $) of the data, multiplied by $100\left(CV=\left(\frac{\sigma}{\mu}\right)\times 100\right)$. The coefficient of variation is often used to compare the variation in datasets with different units or scales. A higher CV indicates greater relative variability, while a lower CV indicates less relative variability. The coefficient of variation can be a helpful tool in the context of PCA for data preprocessing, feature selection, and interpreting the significance of individual variables in the principal components. It helps ensure that PCA is applied appropriately, especially when dealing with datasets with varying scales and levels of variability.

**entropy**. It is used to gain and interpret information and is also useful in making decisions about splitting data at each node in the (decision trees/forests).

**Kolmogorov–Smirnov (KS) test**is a statistical test used to compare the distribution of a sample data set with a known distribution or to compare two sample data sets. It assesses whether a sample is drawn from a particular distribution, such as a normal distribution [23]. While the KS test itself is not typically used directly within principal component analysis (PCA). It can be used to identify potential outliers.

**dispersion index (ID)**), particularly variance and explained variance, play a crucial role in the technique of PCA. PCA aims to maximize the variance of the data along its principal components, and the analysis often involves assessing how much variance is retained or explained by each component to make informed decisions about dimensionality reduction.

## 3. Basic Reproduction Number

#### 3.1. Derivation of Basic Reproduction Number of Model (1)

#### 3.2. Time Varying Reproduction Number

#### 3.3. Sensitivity Index

## 4. Numerical Simulation

## 5. Statistical Modeling and Analysis

#### 5.1. Statistical Analysis of the Entire Dataset

- Choose the same length of moving window for the predictor indicator calculation (14 days).
- Use the same time step as for moving the window (1 day).
- Move the window from the start to the end of the COVID-19 outbreak observed between January 2020 and July 2022 for both daily cases and daily deaths.

#### 5.2. Statistical Analysis before Vaccination Started

#### 5.3. Statistical Analysis after Vaccination Has Started

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Mathematical Analysis of Model (1)

#### Appendix A.1.1. Positivity of Model (1)

**Theorem A1.**

**Proof.**

#### Appendix A.1.2. Equilibrium Points

#### Appendix A.1.3. Stability Analysis

#### Appendix A.1.4. Analytical Solution of Model (1) Using Homotopy Perturbation Method (HPM)

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**Figure 1.**(

**a**) Daily new cases and (

**b**) daily deaths of COVID-19 outbreak in the UK (from [17]) as at year 2023 with 3-day moving average (in blue and yellow, respectively).

**Figure 2.**(

**a**) Cumulative cases and (

**b**) cumulative deaths with their moving average (in blue) as at July 2022.

**Figure 3.**Proportion of those aged 12 years and over who have received one, two, or three or more doses of a COVID-19 vaccine in the UK from 10 January 2021 to 31 August 2022 (from [18]).

**Figure 5.**(

**a**) Graph of fitted cases vs. real COVID-19 cases in the UK from the beginning of the pandemic till 31 July 2022. (

**b**) Graph of fitted deaths vs. real COVID-19 deaths in the UK from the beginning of the pandemic till 31 July 2022.

**Figure 6.**Graph of (

**a**) time-varying reproduction number and (

**b**) average effective reproduction number.

**Figure 8.**(

**a**) Susceptible human population against time (t is in months) for various ${\theta}_{3}$ and (

**b**) infected human population against time (t is in months) for various $\alpha $.

**Figure 9.**(

**a**) Death population against time (t is in months) for various $\varphi $ and (

**b**) recovered unreported human population against time (t is in months) for various $\omega $.

**Figure 10.**(

**a**) Recovered reported human population against time (t is in months) for various ${\theta}_{2}$ and (

**b**) recovered reported human population against time (t is in months) for various $\omega $.

**Figure 11.**(

**a**) Exposed unreported human population against time (t is in months) for various ${\theta}_{4}$ and (

**b**) exposed reported human population against time (t is in months) for various ${\theta}_{5}$.

**Figure 12.**Various statistical predictor indicators for the COVID-19 pandemic in the UK for (

**a**) daily new cases and (

**b**) daily deaths.

**Figure 13.**The Index of dispersion (in blue) as a predictor of the epidemic waves for the UK COVID-19 outbreak, with (

**a**) daily new cases superimposed (in green) and (

**b**) daily deaths superimposed (in green).

**Figure 14.**Various statistical predictor indicators for the COVID-19 pandemic in the UK before vaccination started for (

**a**) daily new cases and (

**b**) daily deaths.

**Figure 15.**The index of dispersion (in blue) as a predictor of the epidemic waves for the UK COVID-19 outbreak before vaccination was introduced in the population, with (

**a**) daily new cases superimposed (in green) and (

**b**) daily deaths superimposed (in green).

**Figure 17.**Various statistical predictor indicators for the COVID-19 pandemic in the UK after vaccination has started for (

**a**) daily new cases and (

**b**) daily deaths.

**Figure 18.**The index of dispersion (in blue) as a predictor of the epidemic waves for the UK COVID-19 outbreak after vaccination campaign has started, with (

**a**) daily new cases superimposed (in green) and (

**b**) daily deaths superimposed (in green).

Parameter | Description |
---|---|

$\mathsf{\Lambda}$ | Recruitment into susceptible |

$\beta $ | Effective contact rate |

$\varphi $ | Death rate of infectious individuals |

$\mu $ | Natural death rate |

$\gamma $ | Progression rate of infectious individuals to recovered unreported class |

${\theta}_{1}$ | Progression rate of vaccinated individuals to recovered unreported class |

${\theta}_{2}$ | Progression rate of vaccinated individuals to recovered reported class |

${\theta}_{3}$ | Rate of vaccination |

${\theta}_{4}$ | Progression rate of vaccinated individuals to exposed unreported class |

${\theta}_{5}$ | Progression rate of vaccinated individuals to exposed reported class |

$\alpha $ | Progression rate of exposed reported individuals to infectious class |

$\sigma $ | Progression rate of exposed unreported individual to exposed reported class |

${\delta}_{1}$ | Loss of immunity by unreported recovered individual |

${\delta}_{2}$ | Loss of immunity by reported recovered individual |

$\omega $ | Progression rate of recovered unreported individual to recovered reported class |

Parameter | References | Values | Best Fit | Unit |
---|---|---|---|---|

$\mathsf{\Lambda}$ | Estimated | 20,000 | 20,000 | Fixed |

$\beta $ | Estimated | $0.1$ | $0.1$ | 1/day |

$\varphi $ | Assumed | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ | $0.245$ | 1/day |

$\mu $ | Assumed | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $0.489$ | 1/day |

$\gamma $ | [15] | $0.1$ | $0.1$ | 1/day |

${\theta}_{1}$ | Varied | $(0,1)$ | $0.67$ | 1/day |

${\theta}_{2}$ | Varied | $(0,1)$ | $0.72$ | 1/day |

${\theta}_{3}$ | Varied | $(0,1)$ | $0.84$ | 1/day |

${\theta}_{4}$ | Varied | $(0,1)$ | $0.51$ | 1/day |

${\theta}_{5}$ | Varied | $(0,1)$ | $0.611$ | 1/day |

$\alpha $ | Fitted | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $0.5$ | 1/day |

$\sigma $ | Fitted | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $0.5$ | 1/day |

${\delta}_{1}$ | Assumed | $0.9$ | $0.91$ | 1/day |

${\delta}_{2}$ | Assumed | $0.1$ | $0.111$ | 1/day |

$\omega $ | Fitted | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$11$}\right.$ | $0.1$ | 1/day |

Parameters | $\mathbf{\Lambda}$ | $\mathit{\beta}$ | $\mathit{\varphi}$ | $\mathit{\mu}$ | $\mathit{\gamma}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | ${\mathit{\theta}}_{3}$ |
---|---|---|---|---|---|---|---|---|

Sensitivity Index | $1.00$ | $1.00$ | $-0.6944$ | $-0.1932$ | $-0.2778$ | $-0.0208$ | $0.0196$ | $-0.8333$ |

Kurtosis | Entropy | Skewness | CV | ID | KStest | NormalizedID | |
---|---|---|---|---|---|---|---|

0 | $-1.560422$ | $2.277053$ | $-0.429678$ | $0.744472$ | $8.788635$ | $3.225218\times {10}^{-9}$ | $0.000163$ |

1 | $-1.147522$ | $2.368861$ | $-0.656162$ | $0.639084$ | $7.410055$ | $1.320401\times {10}^{-11}$ | $0.000137$ |

2 | $-0.494423$ | $2.438590$ | $-0.913383$ | $0.543306$ | $6.030124$ | $1.320401\times {10}^{-11}$ | $0.000110$ |

3 | $0.607783$ | $2.505568$ | $-1.222075$ | $0.441749$ | $4.432511$ | $8.637110\times {10}^{-15}$ | $0.000080$ |

4 | $2.486025$ | $2.569515$ | $-1.497753$ | $0.327226$ | $2.676923$ | $1.989779\times {10}^{-23}$ | $0.000046$ |

PC0 | PC1 | PC2 | |
---|---|---|---|

0 | $-3657.958091$ | $-0.418547$ | $0.040527$ |

1 | $-3659.336674$ | $-0.259834$ | $0.478428$ |

2 | $-3660.716608$ | $0.073121$ | $1.083968$ |

3 | $-3662.314227$ | $0.725743$ | $2.000770$ |

4 | $-3664.069825$ | $1.998592$ | $3.358590$ |

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## Share and Cite

**MDPI and ACS Style**

Agbaje, J.O.; Babasola, O.; Adeyemo, K.M.; Zhiri, A.B.; Adigun, A.J.; Lawal, S.A.; Nuga, O.A.; Abah, R.T.; Adam, U.M.; Oshinubi, K.
Modeling COVID-19 Disease with Deterministic and Data-Driven Models Using Daily Empirical Data in the United Kingdom. *COVID* **2024**, *4*, 289-316.
https://doi.org/10.3390/covid4020020

**AMA Style**

Agbaje JO, Babasola O, Adeyemo KM, Zhiri AB, Adigun AJ, Lawal SA, Nuga OA, Abah RT, Adam UM, Oshinubi K.
Modeling COVID-19 Disease with Deterministic and Data-Driven Models Using Daily Empirical Data in the United Kingdom. *COVID*. 2024; 4(2):289-316.
https://doi.org/10.3390/covid4020020

**Chicago/Turabian Style**

Agbaje, Janet O., Oluwatosin Babasola, Kabiru Michael Adeyemo, Abraham Baba Zhiri, Aanuoluwapo Joshua Adigun, Samuel Adefisoye Lawal, Oluwole Adegoke Nuga, Roseline Toyin Abah, Umar Muhammad Adam, and Kayode Oshinubi.
2024. "Modeling COVID-19 Disease with Deterministic and Data-Driven Models Using Daily Empirical Data in the United Kingdom" *COVID* 4, no. 2: 289-316.
https://doi.org/10.3390/covid4020020