# A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment

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## Abstract

**:**

## 1. Introduction

## 2. Basic Tools about Discrete Fractional Calculus

**Definition**

**1**

**.**Let $\alpha >0$. Then, the $\alpha -th$ fractional sum of $f:{\mathbb{N}}_{a}\to \mathbb{R}$ is defined by

**Riemann–Liouville**as follows:

**Definition**

**2**

**.**Let $0<\alpha <1$. Then, the ${\alpha}^{th}$-order

**Riemann–Liouville**fractional difference of a function f is defined by:

**Definition**

**3**

**.**Let $0<\alpha \le 1$. Then, the α-order

**Caputo**fractional difference of a function f defined on ${\mathbb{N}}_{a},$ is defined by

**Proposition**

**1**

**.**Let $0<\alpha \le 1$ and let f be defined on ${\mathbb{N}}_{a}.$ Then,

**Definition**

**4**

**.**For a function $f:{N}_{a,h}\to \mathbb{R}$, the nabla fractional sum of order $\alpha >0$ is defined by

**Definition**

**5**

**.**The nabla

**Riemann–Liouville**fractional difference of order $0<\alpha \le 1$ (starting from a) is defined by

**Definition**

**6**

**.**Assume that $0<\alpha \le 1$, $a\in \mathbb{R}$ and f is defined on ${N}_{a}$. Then, the left

**Caputo**fractional difference of order α starting at a is defined by

**Definition**

**7**

**.**For $t\in {\mathbb{N}}_{a},\alpha >0,$ and $\left|\lambda \right|<1,$ define the one parameter Mittag–Leffler function of fractional nabla calculus by

**Proposition**

**2**

**.**If $0<\lambda <1,$ then ${F}_{\alpha}(-\lambda ,{\left(t-a\right)}^{\overline{\alpha}})\to 0$, as $t\to \infty .$

**Proposition**

**3**

**.**${}^{C}{\nabla}_{a}^{\alpha}{F}_{\alpha}(\lambda ,{\left(t-a\right)}^{\overline{\alpha}})=\lambda {F}_{\alpha}(\lambda ,{\left(t-a\right)}^{\overline{\alpha}}),$$t\in {\mathbb{N}}_{a+1}.$

**Theorem**

**1**

**.**Assume ${}^{C}{\nabla}_{a}^{\alpha}x\left(t\right){\ge}^{C}{\nabla}_{a}^{\alpha}y\left(t\right)$, $t\in {\mathbb{N}}_{a+1}$, $x\left(a\right)\ge y\left(a\right),$ and $\alpha \in (0,1]$. Then, we have $x\left(t\right)\ge y\left(t\right)$ for $t\in {\mathbb{N}}_{a}.$

## 3. A New Discrete Fractional Model including the Vaccinated Class

**Susceptible class:**This class acquires a number of people, which is the number of people entering the studied area, and in the event that the studied area is isolated, then represents the birth rate in this area. This class loses people who are exposed to infection, people who have been vaccinated against the disease, and natural deaths.

**Recovered class:**This class is acquired at the rate of new recovered persons and loses people who are exposed to infection, as well as losing people who have been vaccinated against the disease and natural deaths.

**Vaccinated class:**This class is acquired at the rate of new vaccinated persons, and class loses people who are infected and natural deaths.

**Infection class:**This class consists of acquired new infection. This class loses people who are recovered and natural deaths (we assume that in this class there are no deaths due to the epidemic).

**Infection dangerous class:**This class consists of acquired new infected persons. This class loses people who recover, natural deaths, and deaths due to infection.

**the forward difference system**and

**the backward difference system**.

## 4. Fixed Points and Basic Reproduction Number

## 5. Stability Analysis of the Disease-Free Fixed Point

**Theorem**

**2**

**.**Let M be the lowest common multiple (LCM) of the denominators ${u}_{i}$ of ${\alpha}_{i}$’s, where $\alpha =\frac{{v}_{1}}{{u}_{1}},\beta =\frac{{v}_{2}}{{u}_{2}},({u}_{i},{v}_{i})=1,{u}_{i},{v}_{i}\in {\mathbb{Z}}_{+}$ for $i=1,2.$ Then, (14) has a unique solution for all initial vectors close enough to ${E}_{0}$; moreover, ${E}_{0}$ is asymptotically stable if any zero solution of the polynomial equation

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

## 6. A Condition for the Disappearance of the Pandemic

**Theorem**

**4.**

**Remark**

**2.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Remark**

**3.**

## 7. Numerical Simulations and Application

## 8. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The number of active infections in Germany in the period 26 April to 30 May 2022 [35].

**Figure 3.**Numerical simulation of infected class with integer order and comparison with the real data.

**Figure 4.**Numerical simulation of infected class with fractional-order system and comparison with real data.

Variable | Description |
---|---|

S | Susceptible class |

R | Recovered class |

V | Vaccinated class |

I | Infection class |

${I}_{d}$ | Infection dangerous class |

$\mathsf{\Omega}$ | The birth rate |

$\mu $ | Natural death rate |

${r}_{1},{r}_{2},{r}_{3}$ | Infection rates |

$\rho $ | Recovered rate |

$\upsilon $ | Vaccinated rate |

$\delta $ | Death rate due to infection |

_{i}is the probability of contagion (p

_{1}> p

_{2}> p

_{3}), and N is the total population and can be considered as the maximum value of the population. In certain cases, we take $N=\frac{\mathsf{\Omega}}{\mu}$ .

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**MDPI and ACS Style**

Dababneh, A.; Djenina, N.; Ouannas, A.; Grassi, G.; Batiha, I.M.; Jebril, I.H.
A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment. *Fractal Fract.* **2022**, *6*, 456.
https://doi.org/10.3390/fractalfract6080456

**AMA Style**

Dababneh A, Djenina N, Ouannas A, Grassi G, Batiha IM, Jebril IH.
A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment. *Fractal and Fractional*. 2022; 6(8):456.
https://doi.org/10.3390/fractalfract6080456

**Chicago/Turabian Style**

Dababneh, Amer, Noureddine Djenina, Adel Ouannas, Giuseppe Grassi, Iqbal M. Batiha, and Iqbal H. Jebril.
2022. "A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment" *Fractal and Fractional* 6, no. 8: 456.
https://doi.org/10.3390/fractalfract6080456