# Dynamics of a Fractional-Order Delayed Model of COVID-19 with Vaccination Efficacy

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Mathematical Formulation of the Model

**Remark**

**1.**

#### 2.1. Positivity, Boundedness, and Uniqueness of the Solution

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 2.2. Equilibrium Points (Disease-Free and Endemic)

#### 2.3. Basic Reproduction Number ${R}_{0}$

## 3. Stability and Hopf Bifurcation Analyses

**Theorem**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

- (i)
- The stability of the bifurcating periodic solution is determined by ${\beta}_{2}$: when ${\beta}_{2}<0\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}({\beta}_{2}>0)$, the bifurcating periodic solutions are stable or (unstable).
- (ii)
- The direction of the Hopf bifurcation is determined by ${k}_{2}$: when ${k}_{2}>0$ or $({k}_{2}<0)$, the Hopf bifurcation is supercritical (subcritical), and for $\tau >{\tau}_{0}(\tau <{\tau}_{0})$, bifurcating periodic solutions exist.
- (iii)
- The period of the bifurcating periodic solution is determined by ${T}_{2}$: when ${T}_{2}>0({T}_{2}<0)$, the period increases (decreases).

## 4. Simulation Results and Discussion

**Remark**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphical representation of the interactions between the various elements in the proposed model.

**Figure 2.**(

**a**) Bifurcating parameter ${\tau}_{0}$. (

**b**) Critical frequency ${\omega}_{0}$. (

**c**) Basic reproduction number ${R}_{0}$.

**Figure 3.**Vaccination model (2) without time delay. (

**a**) Susceptible $S\left(t\right)$. (

**b**) Infected $I\left(t\right)$. (

**c**) Recovered $R\left(t\right)$. (

**d**) Vaccination rate of 0.5%. (

**e**) Vaccination rate of 0.8%. (

**f**) Vaccination rate of 1.2%.

**Figure 4.**Fitting infected cases of model (2) for COVID-19 versus real observations of infected cases in the UAE. (

**a**) Model fitting without the vaccinated population. (

**b**) Model fitting with the vaccinated (after 50%) population. (

**c**) Model fitting with the vaccinated (after 100%) population. (

**d**) Time-delayed model fitting without the vaccinated population. (

**e**) Time-delayed model fitting with the vaccinated (after 50%) population. (

**f**) Time-delayed model fitting with the vaccinated (after 100%) population.

**Figure 5.**The trend of infected individuals with respect to different vaccination rates ${V}_{ac}.$ (

**a**) Fractional-order $\alpha =0.62$. (

**b**) Fractional-order $\alpha =0.83$. (

**c**) Fractional-order $\alpha =0.94$.

**Figure 6.**Vaccination model (2) with time delay $\tau =1.5\in [0,{\tau}_{0}).$ (

**a**) Susceptible $S\left(t\right)$. (

**b**) Infected $I\left(t\right)$. (

**c**) Recovered $R\left(t\right)$.

**Figure 7.**The trend of infected individuals regarding different vaccination efficacy rates $\nu .$ (

**a**) Fractional-order $\alpha =0.62$. (

**b**) Fractional-order $\alpha =0.83$. (

**c**) Fractional-order $\alpha =0.94$.

**Figure 8.**Fitting the recovered cases of model (2) for COVID-19 versus real observations of recovered cases in the UAE. (

**a**) Model fitting without the vaccinated population. (

**b**) Model fitting with the vaccinated (after 50%) population. (

**c**) Model fitting with the vaccinated (after 100%) population. (

**d**) Time-delayed model fitting without the vaccinated population. (

**e**) Time-delayed model fitting with the vaccinated (after 50%) population. (

**f**) Time-delayed model fitting with the vaccinated (after 100%) population.

**Figure 9.**Vaccination model (2) with time delay $\tau >{\tau}_{0}.$ (

**a**) Susceptible $S\left(t\right)$. (

**b**) Infected $I\left(t\right)$. (

**c**) Recovered $R\left(t\right)$.

**Figure 10.**The trend of infected individuals regarding different transmission rates $\delta $ with a vaccination rate of $0.5\%$. (

**a**) Fractional-order $\alpha =0.62$. (

**b**) Fractional-order $\alpha =0.83$. (

**c**) Fractional-order $\alpha =0.94$.

**Figure 11.**The trend of infected individuals regarding different transmission rates $\delta $ without a vaccination rate. (

**a**) Fractional-order $\alpha =0.62$. (

**b**) Fractional-order $\alpha =0.83$ (

**c**) Fractional-order $\alpha =0.94$.

**Table 1.**Descriptions of model (1)’s variables and parameters.

Parameter | Description | Value Range | References |
---|---|---|---|

$\delta $ | Transmission rate of symptomatic individuals | $[0,1)$ | [48,49] |

${V}_{ac}$ | Vaccination rate | 0.01/day | Assume |

$\nu $ | Vaccine efficacy rate | $[0.8,0.9]$ | Assume |

$\beta $ | Symptomatic infectious disease recovery rate | $[0,1)$ | [50] |

Transmission Rate $\mathit{\delta}$ | Recovery Rate $\mathit{\beta}$ | Reproduction Number ${\mathit{R}}_{0}$ | ${\mathit{\omega}}_{0}$ | ${\mathit{\tau}}_{0}$ |
---|---|---|---|---|

0.30 | 0.15 | 2.00 | 0.6269 | 3.8192 |

0.32 | 0.15 | 1.77 | 0.7430 | 3.7078 |

0.35 | 0.20 | 1.75 | 0.8290 | 3.6818 |

0.39 | 0.23 | 1.6956 | 0.9422 | 3.7368 |

0.42 | 0.25 | 1.6800 | 1.0237 | 3.8759 |

0.45 | 0.28 | 1.6071 | 1.1477 | 4.5782 |

Fractional-Order $\mathit{\alpha}$ | Critical Frequency ${\mathit{\omega}}_{0}$ | Bifurcating Point ${\mathit{\tau}}_{0}$ |
---|---|---|

0.810 | 1.6075 | 5.1136 |

0.815 | 1.5756 | 4.4790 |

0.820 | 1.3602 | 3.7445 |

0.825 | 1.2538 | 3.681 |

0.830 | 1.1472 | 3.6714 |

0.835 | 1.10864 | 3.6808 |

Vaccination Rate ${\mathit{V}}_{\mathbf{ac}}$ | Critical Frequency ${\mathit{\omega}}_{0}$ | Bifurcating Point ${\mathit{\tau}}_{0}$ |
---|---|---|

0.75% | 1.000 | 6.108 |

0.78% | 1.002 | 5.305 |

0.81% | 1.002 | 4.862 |

0.84% | 1.007 | 4.554 |

0.87% | 1.009 | 4.317 |

0.90% | 1.012 | 4.125 |

Vaccine Efficacy Rate $\mathit{\nu}$ | Critical Frequency ${\mathit{\omega}}_{0}$ | Bifurcating Point ${\mathit{\tau}}_{0}$ |
---|---|---|

90% | 1.005 | 4.768 |

86% | 1.002 | 5.272 |

82% | 0.999 | 6.313 |

88% | 1.004 | 4.990 |

84% | 1.001 | 5.665 |

80% | 0.998 | 8.642 |

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**MDPI and ACS Style**

Rihan, F.A.; Kandasamy, U.; Alsakaji, H.J.; Sottocornola, N. Dynamics of a Fractional-Order Delayed Model of COVID-19 with Vaccination Efficacy. *Vaccines* **2023**, *11*, 758.
https://doi.org/10.3390/vaccines11040758

**AMA Style**

Rihan FA, Kandasamy U, Alsakaji HJ, Sottocornola N. Dynamics of a Fractional-Order Delayed Model of COVID-19 with Vaccination Efficacy. *Vaccines*. 2023; 11(4):758.
https://doi.org/10.3390/vaccines11040758

**Chicago/Turabian Style**

Rihan, Fathalla A., Udhayakumar Kandasamy, Hebatallah J. Alsakaji, and Nicola Sottocornola. 2023. "Dynamics of a Fractional-Order Delayed Model of COVID-19 with Vaccination Efficacy" *Vaccines* 11, no. 4: 758.
https://doi.org/10.3390/vaccines11040758