# An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Model and Assimilation Scheme

#### 2.1. Extended SEIR Model

#### 2.2. Ensemble Kalman Filter (EnKF)

## 3. Mathematical Analysis of the COVID-19 Model

#### 3.1. Non-Negativity of the Model

**Theorem 1.**

**Proof of**

**Theorem 1.**

#### 3.2. Boundedness of the Model

**Theorem 2.**

**Proof of**

**Theorem 2.**

#### 3.3. Epidemic Equilibrium and Basic Reproduction Number of the Model

**Theorem 3.**

**Proof of**

**Theorem 3.**

#### 3.4. Existence and Uniqueness of the Endemic Equilibrium

## 4. Results and Discussion

#### 4.1. Assimilation Settings

#### 4.2. Sensitivity to Ensemble Size

#### 4.3. Predictability Experiment

#### 4.4. Impact of Vaccination

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Time series of the estimated model parameters using the Joint-EnKF with 200 ensemble members. Initial and converged values for each parameter are also displayed.

**Figure 6.**Predicted numbers of: (

**a**) Recovered cases. (

**b**) Death cases. (

**c**) Confirmed cases for the first two weeks of December.

**Figure 7.**Impact of vaccination rate on the total number of: (

**a**) Active cases. (

**b**) Recovered cases. (

**c**) Death cases. (

**d**) Confirmed cases. $\alpha =0$ corresponds to the model output without vaccination.

Parameter | Initial Value | Description | Reference |
---|---|---|---|

$\Lambda $ | 2300 persons/day | New births and new residents | [51] |

${\beta}_{1}$ | $8.58\times {10}^{-9}$ day${}^{-1}$ | Transmission rate before intervention | Assumed |

${\beta}_{2}$ | $3.43\times {10}^{-9}$ day${}^{-1}$ | Transmission rate during and after intervention | Assumed |

$\alpha $ | $3.5\times {10}^{-4}$ day${}^{-1}$ | Vaccination rate | [52] |

$\mu $ | $3\times {10}^{-5}$ persons/day | Natural death rate | [51] |

${\gamma}^{-1}$ | 5.5 days | Incubation period | [42] |

$\sigma $ | 0.05 | Vaccine inefficacy | [53] |

${\delta}^{-1}$ | 3.8 days | Infection time | [42] |

$\kappa $ | 0.014 | Case fatality rate | [50] |

${\lambda}^{-1}$ | 10 days | Recovery time | [42] |

${\rho}^{-1}$ | 15 days | Time until death | [42] |

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**MDPI and ACS Style**

Ghostine, R.; Gharamti, M.; Hassrouny, S.; Hoteit, I.
An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter. *Mathematics* **2021**, *9*, 636.
https://doi.org/10.3390/math9060636

**AMA Style**

Ghostine R, Gharamti M, Hassrouny S, Hoteit I.
An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter. *Mathematics*. 2021; 9(6):636.
https://doi.org/10.3390/math9060636

**Chicago/Turabian Style**

Ghostine, Rabih, Mohamad Gharamti, Sally Hassrouny, and Ibrahim Hoteit.
2021. "An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter" *Mathematics* 9, no. 6: 636.
https://doi.org/10.3390/math9060636