# A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### A Fractional-Order Model

**Definition**

**1.**

**Lemma**

**1.**

## 3. Equilibria and Its Stability

#### Existence of Endemic Equilibria

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Estimation of Parameters

## 5. Numerical Results and Discussion

#### Numerical Scheme for Model (4)

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Simulation of model variables with different values of $\varpi =1,0.97,0.95,0.93$, where sub-figures describe, (

**a**) susceptible individuals, (

**b**) exposed individuals, (

**c**) asymptomatic individuals, (

**d**) symptomatic individuals, (

**e**) hospitalized individuals, (

**f**) recovered individuals.

**Figure 3.**The impact of contact parameters on the infected population, where sub-figures describe, (

**a**) ${\beta}_{1}=0.7085,0.6985,0.6885,0.6785$, (

**b**) ${\beta}_{2}=0.03,0.028,0.024,0.02$, (

**c**) ${\beta}_{3}=0.2784,0.1784,0.0784$, (

**d**) ${\beta}_{4}=0.03,0.02,0.01$.

**Figure 4.**The impact of parameters $\eta $, $\nu $ and $\kappa $ on the infected individuals, where sub-graphs describe (

**a**) $\eta =0.8036,0.7836,0.7436$, (

**b**) $\nu =0.8887,0.8487,0.8087$, (

**c**) $\kappa =0.2,0.3,0.4$.

Symbol | Definition | Value/per Day | Source |
---|---|---|---|

$\mathrm{\Lambda}$ | Birth rate | $\mu \times N\left(0\right)$ | Estimated |

$\mu $ | Natural death rate | $\frac{1}{67.7\times 365}$ | [26] |

$\eta $ | Incubation period | 0.8036 | Fitted |

$\nu $ | Individuals progress to A | 0.8887 | Fitted |

${q}_{1}$ | Recovery from asymptomatic class | 1/5.1 | [27] |

${q}_{2}$ | Recovery from symptomatic class | 0.1 | [27] |

${q}_{3}$ | Recovery of hospitalized individuals | 1/8 | [27] |

$\omega $ | Disease contact rate of hospitalized individuals | 0.01 | Fitted |

$\psi $ | Disease death rate of symptomatic individuals | 0.015 | [27] |

$\kappa $ | Hospitalization of symptomatic people | 0.4000 | Fitted |

${\beta}_{1}$ | Contact rate due to exposed | 0.7085 | Fitted |

${\beta}_{2}$ | Contact rate due to asymptomatic | 0.0300 | Fitted |

${\beta}_{3}$ | Contact rate due to symptomatic | 0.2784 | Fitted |

${\beta}_{4}$ | Contact rate due to hospitalized | 0.0100 | Fitted |

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**MDPI and ACS Style**

Gu, Y.; Khan, M.A.; Hamed, Y.S.; Felemban, B.F.
A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative. *Fractal Fract.* **2021**, *5*, 271.
https://doi.org/10.3390/fractalfract5040271

**AMA Style**

Gu Y, Khan MA, Hamed YS, Felemban BF.
A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative. *Fractal and Fractional*. 2021; 5(4):271.
https://doi.org/10.3390/fractalfract5040271

**Chicago/Turabian Style**

Gu, Yu, Muhammad Altaf Khan, Y. S. Hamed, and Bassem F. Felemban.
2021. "A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative" *Fractal and Fractional* 5, no. 4: 271.
https://doi.org/10.3390/fractalfract5040271