# A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease

^{1}

^{2}

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*Fractal Fract’s*Editorial Board Members)

## Abstract

**:**

## 1. Introduction

## 2. Analysis of the Model

**Theorem**

**1.**

- 1.
- There exists a unique solution to system (1), and the solution is nonnegative.
- 2.
- The set Ω is invariant with respect to system (1).
- 3.
- ${lim}_{t\to \infty}N\left(t\right)={A}^{\alpha}/{\mu}^{\alpha}$.
- 4.
- For all $t>0$, $I\left(t\right)\le {I}_{0}+{\sigma}^{\alpha}{\parallel E\parallel}_{\infty}/({\gamma}^{\alpha}+{\mu}^{\alpha})$.

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Numerical Simulations

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Profile of infected population from both classical and fractional measles model for the real monthly data with monotonically increasing (

**a**) and decreasing (

**b**) fashion.

$\mathit{\beta}$ | disease transmission rate | Estimated |

A | birth rate | Fixed |

$\mu $ | natural mortality rate | Fixed |

$\rho $ | percentage of vaccinated individuals | Fixed |

$\sigma $ | rate at which an exposed person becomes infective | Fixed |

$\gamma $ | rate an infected recovers | Fixed |

$\alpha $ | fractional order parameter | Estimated |

Parameter | Sensitivity Indices |
---|---|

$\alpha $ | $-1.411652563$ |

$\beta $ | $+0.5$ |

A | $+0.5$ |

$\mu $ | $-0.5169256402$ |

$\rho $ | $-4$ |

$\sigma $ | $+0.007971768371$ |

$\gamma $ | $-0.4910461282$ |

Parameter | Sensitivity Indices |
---|---|

$\alpha $ | $-12.92457802$ |

$\beta $ | $+0.5$ |

A | $+0.5$ |

$\mu $ | $-0.5169256402$ |

$\rho $ | $-4$ |

$\sigma $ | $+0.007971768371$ |

$\gamma $ | $-0.4910461282$ |

Monotonicity | Classical | Fractional |
---|---|---|

Increasing | $\beta =1.220959\times {10}^{-8}$ | $\beta =1.428694\times {10}^{-22}$ |

$\alpha =1.0000$ | $\alpha =3.557048\times {10}^{-1}$ | |

$E=1.619008\times {10}^{2}$ | $E=8.407327\times {10}^{1}$ | |

Decreasing | $\beta =4.0415\times {10}^{-9}$ | $\beta =4.5623\times {10}^{-11}$ |

$\alpha =1.0000$ | $\alpha =8.3680\times {10}^{-1}$ | |

$E=9.1481\times {10}^{2}$ | $E=4.3283\times {10}^{2}$ |

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**MDPI and ACS Style**

Almeida, R.; Qureshi, S.
A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease. *Fractal Fract.* **2019**, *3*, 53.
https://doi.org/10.3390/fractalfract3040053

**AMA Style**

Almeida R, Qureshi S.
A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease. *Fractal and Fractional*. 2019; 3(4):53.
https://doi.org/10.3390/fractalfract3040053

**Chicago/Turabian Style**

Almeida, Ricardo, and Sania Qureshi.
2019. "A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease" *Fractal and Fractional* 3, no. 4: 53.
https://doi.org/10.3390/fractalfract3040053