# Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

- ${A}_{1}$.
- All the variables and parameters of the system are non-negative.
- ${A}_{2}$.
- The susceptible people transfer to the infectious compartment with a constant susceptible inflow into population.
- ${A}_{3}$.
- Originally infectious or susceptible persons transfer to the quarantined class while reported cases return to the infected class from quarantined classes.

## 3. Preliminary Results

**Definition**

**1**

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**1**

**Theorem**

**1**

- (1)
- $\mathbf{F}u+\mathbf{G}u\in \mathbf{B}$$\forall u\in \mathbf{B}$,
- (2)
- $\mathbf{F}$ is a contraction, and
- (3)
- $\mathbf{G}$ is compact and continuous.

## 4. Qualitative Analysis of the Proposed Model

- (H1)
- there is ${C}_{\mathsf{\Phi}}$ and ${D}_{\mathsf{\Phi}}$ such that$$\left|\mathsf{\Phi}(t,\mathcal{A}\left(t\right))\right|\le {C}_{\mathsf{\Phi}}\parallel \mathcal{A}\parallel +{D}_{\mathsf{\Phi}};$$
- (H2)
- there is ${L}_{\mathsf{\Phi}}>0$ such that ∀ $\mathcal{A},\phantom{\rule{4pt}{0ex}}\overline{\mathcal{A}}\in \mathbf{Z}$ one has$$|\mathsf{\Phi}(t,\mathcal{A})-\mathsf{\Phi}(t,\overline{\mathcal{A}})|\le {L}_{\mathsf{\Phi}}[\parallel \mathcal{A}\parallel -\parallel \overline{\mathcal{A}}\parallel ].$$

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Construction of an Algorithm for Deriving the Solution of the Model

## 6. Numerical Interpretation and Discussion

#### 6.1. Case Study with Real Data: Khyber Pakhtunkhawa (Pakistan)

## 7. Sensitivity Analysis

**Definition**

**2.**

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamical nature of susceptible, infected and quarantined individuals of the fractional ABC Model (1) for different values of the fractional-order $\theta $. (

**a**) $P\left(t\right)$—susceptible individuals along time t; (

**b**) $I\left(t\right)$—infected individuals along time t; (

**c**) $Q\left(t\right)$—quarantined individuals along time t.

**Figure 2.**Real data of infected individuals by COVID-19 from Khyber Pakhtunkhwa, Pakistan, from 9 April to 2 June 2020.

**Figure 3.**Comparison of infected individuals by COVID-19: Model (1) output (in blue) versus real data of Khyber Pakhtunkhawa, Pakistan, from 9 April to 2 June 2020 (in red).

**Figure 4.**Real data of infected individuals by COVID-19 in Khyber Pakhtunkhwa, Pakistan (first 1.8 months, in red) and prediction from Model (1) during a period of 8 months (in blue).

**Figure 5.**Sensitivity of the basic reproduction number ${R}_{0}$ (3) for relevant parameters of Model (1). (

**a**) ${R}_{0}$ versus $\gamma $ and d; (

**b**) ${R}_{0}$ versus $\gamma $ and $\mu $; (

**c**) ${R}_{0}$ versus $\gamma $ and $\eta $; (

**d**) ${R}_{0}$ versus h and d; (

**e**) ${R}_{0}$ versus h and $\mu $; (

**f**) ${R}_{0}$ versus h and $\eta $; (

**g**) ${R}_{0}$ versus d and $\sigma $; (

**h**) ${R}_{0}$ versus d and $\eta $.

**Table 1.**Parameters description defined in the given Model (1).

Notation | Description |
---|---|

$\lambda $ | Rate of recruitment |

$\gamma $ | Transmission rate of disease |

${d}_{0}$ | Natural death rate |

$\eta $ | Transmission rate of infected to quarantine |

$\mu $ | Deaths in quarantined zone |

$\sigma $ | Transmission flow of quarantined to become infectious |

h | Rate of deaths in infected zone |

**Table 2.**Numerical values for the parameters of Model (1).

Notation | Parameters Description | Numerical Value |
---|---|---|

$\lambda $ | Rate of recruitment | $0.003$ |

$\gamma $ | Transmission rate of disease | $0.009$ |

${d}_{0}$ | Natural death rate | $0.009$ |

$\eta $ | Transmission rate of infected to quarantine | $0.004$ |

$\mu $ | Death rate in quarantine | $0.004$ |

$\sigma $ | Transmission flow of quarantined to infectious | $0.003$ |

h | Rate of death for infected | $0.007$ |

${P}_{0}$ | Initial population of susceptible | 10 millions |

${I}_{0}$ | Initially infected population | $0.01$ millions |

${Q}_{0}$ | Quarantined population at $t=0$ | $0.0011$ millions |

Notation | Value | Reference |
---|---|---|

$\lambda $ | 0.028 | [55] |

$\gamma $ | 0.2 | Estimated |

${d}_{0}$ | 0.011 | [55] |

$\mu $ | 0.2 | Estimated |

h | 0.06 | [55] |

$\sigma $ | 0.04 | Estimated |

$\eta $ | 0.3 | Estimated |

Parameters | Sensitivity | Value | Parameters | Sensitivity | Value |
---|---|---|---|---|---|

$\gamma $ | ${S}_{\gamma}$ | 1.00000000 | h | ${S}_{h}$ | 0.63636363 |

${d}_{0}$ | ${S}_{{d}_{0}}$ | −1.48944805 | $\mu $ | ${S}_{\mu}$ | −0.00974026 |

$\sigma $ | ${S}_{\sigma}$ | 0.03165584 | $\eta $ | ${S}_{\eta}$ | −0.16883117 |

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Din, A.; Khan, A.; Zeb, A.; Sidi Ammi, M.R.; Tilioua, M.; Torres, D.F.M. Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application. *Axioms* **2021**, *10*, 290.
https://doi.org/10.3390/axioms10040290

**AMA Style**

Din A, Khan A, Zeb A, Sidi Ammi MR, Tilioua M, Torres DFM. Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application. *Axioms*. 2021; 10(4):290.
https://doi.org/10.3390/axioms10040290

**Chicago/Turabian Style**

Din, Anwarud, Amir Khan, Anwar Zeb, Moulay Rchid Sidi Ammi, Mouhcine Tilioua, and Delfim F. M. Torres. 2021. "Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application" *Axioms* 10, no. 4: 290.
https://doi.org/10.3390/axioms10040290