# On the Modeling of COVID-19 Transmission Dynamics with Two Strains: Insight through Caputo Fractional Derivative

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Caputo Fractional-Order Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. The Model

## 4. Mathematical Analyses

#### 4.1. Basic Properties of the Fractional Model

**Proof.**

#### 4.2. Existence and Uniqueness of Solutions

**Proof.**

#### 4.3. Model Equilibrium and the Basic Reproduction Number

#### 4.3.1. Stability Analysis of the Equilibrium Point

#### 4.3.2. Local Stability of DFE

#### 4.3.3. Local Stability of the Endemic Equilibrium for Strain 1

- (i).
- $\left(\frac{{\mathcal{R}}_{01}}{{R}_{2}}-1\right)>0\u27fa{\mathcal{R}}_{01}>{R}_{2}$.
- (ii).
- $\left(\frac{{\mathcal{R}}_{01}}{{\mathcal{R}}_{02}}-1\right)>0\u27fa{\mathcal{R}}_{01}>{\mathcal{R}}_{02}$.

#### 4.3.4. Local Stability of the Endemic Equilibrium for Strain 2

## 5. Numerical Simulations

#### 5.1. Parameter Estimation

#### 5.2. Local Sensitivity Analysis

#### 5.3. Simulation Results Using Caputo Operator

## 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.**Contour plot of (

**a**) ${\theta}_{1}$ versus ${\beta}_{1}$ and (

**b**) ${\theta}_{2}$ versus ${\beta}_{2}$ as a function of ${\mathcal{R}}_{0}.$

**Figure 3.**Numerical simulation for the fractional model (9) when $\sigma =1,0.9,0.8,0.7$ and $0.6$, where (

**a**) Susceptible individuals, (

**b**) Exposed individuals to strain 1, (

**c**) Infected individuals to strain 1, (

**d**) Exposed individuals to strain 2, (

**e**) Infected individuals to strain 2, and (

**f**) Recovered individuals.

**Figure 4.**Effect of the model parameters (

**a**) ${\beta}_{1},\left(\mathbf{b}\right)\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2},\left(\mathbf{c}\right)\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}$, and (

**d**) ${\theta}_{2}$ on ${I}_{1}$ individuals with fractional order $\sigma =0.95.$.

**Figure 5.**Effect of the model parameters (

**a**) ${\beta}_{1},\left(\mathbf{b}\right)\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2},\left(\mathbf{c}\right)\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}$, and (

**d**) ${\theta}_{2}$ on ${I}_{2}$ individuals with fractional order $\sigma =0.95.$.

Symbol | Description | Units |
---|---|---|

$\Lambda $ | Recruitment rate | $\frac{Individual}{time}$ |

$\mu $ | Natural death rate | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\alpha}_{1}$ | Transmission rate by ${E}_{1}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\alpha}_{2}$ | Transmission rate by ${E}_{2}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\beta}_{1}$ | Transmission rate by ${I}_{1}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\beta}_{2}$ | Transmission rate by ${I}_{2}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

$\delta $ | Progression rate from ${E}_{1}$ to ${I}_{1}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

$\mathcal{E}$ | Progression rate from ${E}_{2}$ to ${I}_{2}$ | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\sigma}_{1}$ | Death rate due to strain 1 | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\sigma}_{2}$ | Death rate due to strain 2 | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\theta}_{1}$ | Recovery rate of infected with strain 1 | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

${\theta}_{2}$ | Recovery rate of infected with strain 2 | $\frac{1}{\phantom{\rule{4pt}{0ex}}time}$ |

Parameters | Value | Source |
---|---|---|

$\Lambda $ | 1930 | [15] |

$\mu $ | $\frac{1}{70\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}365}$ | [15] |

${\alpha}_{1}$ | $2\phantom{\rule{3.33333pt}{0ex}}\times $${10}^{-11}$ | [21] |

${\beta}_{1}$ | $1.2644\times {10}^{-8}$ | [15] |

$\delta $ | $0.0784$ | [15] |

${\sigma}_{1}$ | $0.015$ | [15] |

${\theta}_{1}$ | $0.13978$ | [15] |

${\alpha}_{2}$ | $2.65\times {10}^{-11}$ | Assumed |

${\beta}_{2}$ | $1.3\times {10}^{-8}$ | Assumed |

$\mathcal{E}$ | $0.0564$ | Assumed |

${\sigma}_{2}$ | $0.0143$ | Assumed |

${\theta}_{2}$ | $0.1$ | Assumed |

Parameter | Sensitivity Index (${\mathcal{R}}_{01}$) | Parameter | Sensitivity Index (${\mathcal{R}}_{02}$) |
---|---|---|---|

$\Lambda $ | 0.9999 | $\Lambda $ | 1 |

$\mu $ | −1 | $\mu $ | −1 |

${\alpha}_{1}$ | 0.0031 | ${\alpha}_{2}$ | 0.0041 |

${\beta}_{1}$ | 0.9969 | ${\beta}_{2}$ | 0.9959 |

$\delta $ | −0.0026 | $\epsilon $ | −0.0034 |

${\sigma}_{1}$ | −0.0966 | ${\sigma}_{2}$ | −0.1245 |

${\theta}_{1}$ | −0.9 | ${\theta}_{2}$ | −0.871 |

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**MDPI and ACS Style**

Fatmawati; Yuliani, E.; Alfiniyah, C.; Juga, M.L.; Chukwu, C.W.
On the Modeling of COVID-19 Transmission Dynamics with Two Strains: Insight through Caputo Fractional Derivative. *Fractal Fract.* **2022**, *6*, 346.
https://doi.org/10.3390/fractalfract6070346

**AMA Style**

Fatmawati, Yuliani E, Alfiniyah C, Juga ML, Chukwu CW.
On the Modeling of COVID-19 Transmission Dynamics with Two Strains: Insight through Caputo Fractional Derivative. *Fractal and Fractional*. 2022; 6(7):346.
https://doi.org/10.3390/fractalfract6070346

**Chicago/Turabian Style**

Fatmawati, Endang Yuliani, Cicik Alfiniyah, Maureen L. Juga, and Chidozie W. Chukwu.
2022. "On the Modeling of COVID-19 Transmission Dynamics with Two Strains: Insight through Caputo Fractional Derivative" *Fractal and Fractional* 6, no. 7: 346.
https://doi.org/10.3390/fractalfract6070346