# On Using Piecewise Fractional Differential Operator to Study a Dynamical System

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

## 3. Qualitative Results

**Corollary**

**1.**

- (A
_{1}): - There exist constants ${\mathrm{L}}_{\mathrm{S}}>0$ and ${\mathrm{L}}_{\mathrm{I}}>0$ such that for every $(\mathrm{S},\mathrm{I}),(\overline{\mathrm{S}},\overline{\mathrm{I}})\in {\mathrm{Z}}_{1}\times {\mathrm{Z}}_{2}$$$\begin{array}{c}\hfill |{F}_{1}(t,\mathrm{S},\mathrm{I})-{F}_{1}(t,\overline{\mathrm{S}},\overline{\mathrm{I}})|\le {\mathrm{L}}_{\mathrm{S}}[\mathrm{S}-\overline{\mathrm{S}}|+|\mathrm{I}-\overline{\mathrm{I}}|],\end{array}$$$$\begin{array}{c}\hfill |{F}_{2}(t,\mathrm{S},\mathrm{I})-{F}_{2}(t,\overline{\mathrm{S}},\overline{\mathrm{I}})|\le {\mathrm{L}}_{\mathrm{I}}[\mathrm{S}-\overline{\mathrm{S}}|+|\mathrm{I}-\overline{\mathrm{I}}|].\end{array}$$
- (A
_{2}): - For constants ${C}_{{F}_{1}}$, ${D}_{{F}_{1}}$, ${C}_{{F}_{2}}$, ${D}_{{F}_{2}}$ and ${N}_{{F}_{1}},{N}_{{F}_{2}}>0$, the following assertions (also called the growth condition) are satisfied$$\begin{array}{c}\hfill |{F}_{1}(t,\mathrm{S},\mathrm{I})|\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{{F}_{1}}|\mathrm{S}|+{D}_{{F}_{1}}|\mathrm{I}|+{N}_{{F}_{1}},\\ \hfill |{F}_{2}(t,\mathrm{S},\mathrm{I})|\le \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{{F}_{2}}|\mathrm{S}|+{D}_{{F}_{2}}|\mathrm{I}|+{N}_{{F}_{2}}\end{array}$$

**Theorem**

**1.**

**Proof.**

**Cases-I**: At $t\in [0,{t}_{1}]$, then from (9)

**Cases-II**: Let $(\mathrm{S},\mathrm{I}),(\overline{\mathrm{S}},\overline{\mathrm{I}})\in {\mathrm{Z}}_{1}\times {\mathrm{Z}}_{2},$ then one has

**Proof.**

**Case-I**:At $t\in [0,{t}_{1}]:$

**Step-1**: If $(\mathrm{S},\mathrm{I})\in E,$ at $t\in [0,{t}_{1}],$ one has

**Case-II**: For $t\in ({t}_{1},T],$ for $(\mathrm{S},\mathrm{I})\in E$ at $t\in ({t}_{1},T],$ one has

**Step-2**: Since ${F}_{1},{F}_{2}\in C[0,T].$ Hence ${H}_{1},{H}_{2}$ are continuous in the same domain. Now to show that $H=({H}_{1},{H}_{2})$ is equi-continuous, Let $t\in [0,{t}_{1}],$ then ${t}_{m}<{t}_{n}\in [0,{t}_{1}],$ one has

## 4. Ulam–Hyers Stability Analysis

**Definition**

**3.**

**Remark**

**1.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Numerical Scheme

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 30%.

**Figure 2.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of 30%.

**Figure 3.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 30%.

**Figure 4.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of 30%.

**Figure 5.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 50%.

**Figure 6.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of $50\%$.

**Figure 7.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 50%.

**Figure 8.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of 50%.

**Figure 9.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 70%.

**Figure 10.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of 70%.

**Figure 11.**Numerical solution for susceptible class for the given days and using different fractional orders with isolation rate of 70%.

**Figure 12.**Numerical solution for infected class for the given days and using different fractional orders with isolation rate of 70%.

**Table 1.**Nomenclature involve in model (2) and their description.

Variables | Description |
---|---|

a | Birth rate |

k | Rate constant |

$\alpha $ | Rate of isolation in percent |

$\beta $ | Rate of protection |

$\gamma $ | Recovery rate |

$\mu $ | Natural birth rate |

${\mathrm{S}}_{0}$ | Initial papulation of susceptible compartment |

${\mathrm{I}}_{0}$ | Initial papulation of infected compartment |

**Table 2.**Nomenclature and numerical values are taken from [7].

Nomenclature | Numerical Value |
---|---|

a | 0.000090 |

k | 0.00090 |

$\alpha $ | 30%; 50%; 70% assumed |

$\beta $ | 0.000780 |

$\gamma $ | 100 |

$\mu $ | 0.05 |

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**MDPI and ACS Style**

Khan, S.; Khan, Z.A.; Alrabaiah, H.; Zeb, S.
On Using Piecewise Fractional Differential Operator to Study a Dynamical System. *Axioms* **2023**, *12*, 292.
https://doi.org/10.3390/axioms12030292

**AMA Style**

Khan S, Khan ZA, Alrabaiah H, Zeb S.
On Using Piecewise Fractional Differential Operator to Study a Dynamical System. *Axioms*. 2023; 12(3):292.
https://doi.org/10.3390/axioms12030292

**Chicago/Turabian Style**

Khan, Shahid, Zareen A. Khan, Hussam Alrabaiah, and Salman Zeb.
2023. "On Using Piecewise Fractional Differential Operator to Study a Dynamical System" *Axioms* 12, no. 3: 292.
https://doi.org/10.3390/axioms12030292