# Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (i)
- (ii)

## 2. Preliminaries on Fractional Calculus

**Lemma**

**1**

- $f\left(X\right)$ and $\frac{\partial f\left(X\right)}{\partial X}$ are continuous for all $X\in {\mathbb{R}}^{n}$;
- $\parallel f\left(X\right)\parallel \le \omega +\lambda \parallel X\parallel $ for all $X\in {\mathbb{R}}^{n}$, where ω and λ are two positive constants.

**Lemma**

**2**

**.**Suppose that the functions $x\left(t\right)$ and ${}^{C}{D}^{\alpha}x\left(t\right)$ are both continuous on $[0,b]$. Then,

## 3. The Considered Fractional-Order COVID-19 Model

- $\kappa $ is the rate at which an individual leaves the exposed class by becoming infectious (symptomatic, super-spreaders or asymptomatic);
- ${\rho}_{1}$ is the proportion of progression from exposed class E to symptomatic infectious class I;
- ${\rho}_{2}$ is a relative very low rate at which exposed individuals become super-spreaders;
- $1-{\rho}_{1}-{\rho}_{2}$ is the progression from exposed to asymptomatic class;
- ${\gamma}_{a}$ is the average rate at which symptomatic and super-spreaders individuals become hospitalized;
- ${\gamma}_{i}$ is the recovery rate without being hospitalized;
- ${\gamma}_{r}$ is the recovery rate of hospitalized patients;
- ${\delta}_{i}$ denotes the disease induced death rates due to infected individuals;
- ${\delta}_{p}$ denotes the disease induced death rates due to super-spreaders individuals;
- ${\delta}_{h}$ denotes the disease induced death rates due to hospitalized individuals.

## 4. Existence and Uniqueness of Positive Solution

**Theorem**

**1**

**Proof.**

## 5. Stability Analysis

**Theorem**

**2**

**.**Let $\alpha \in (0,1)$. The disease free equilibrium (DFE) of system (2) is globally asymptotically stable whenever ${R}_{0}<1$.

**Proof.**

## 6. Numerical Simulations

#### 6.1. Population Size, Initial Conditions, and Parameters

#### 6.2. Index of Memory’s Influence

#### 6.3. Infectivity Rate and Effect on the Basic Reproduction Number

## 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.**Evolution of infected populations by varying the order of differentiation $\alpha $: the data of Portugal is well described with $\alpha =0.75$; the data of Spain, and Galicia alone, with $\alpha =0.85$; the data of Wuhan with $\alpha =1$.

**Figure 2.**Evolution of infected populations ($I\left(t\right),P\left(t\right),H\left(t\right)$, and $I\left(t\right)+P\left(t\right)+H\left(t\right)$) by varying the infectivity rate $\beta $ by $1.55$, $2.55$, and $3.55$, corresponding, respectively, to the basic reproduction number $2.662$, $4.375$, and $6.088$, while fixing index memory $\alpha =1$ (Wuhan).

**Figure 3.**Evolution of infected populations ($I\left(t\right),P\left(t\right),H\left(t\right)$, and $I\left(t\right)+P\left(t\right)+H\left(t\right)$) by varying the infectivity rate $\beta $ by $1.55$, $2.55$, and $3.55$, corresponding, respectively, to the basic reproduction number $2.662$, $4.375$, and $6.088$, while fixing index memory $\alpha =0.85$ (Spain).

**Figure 4.**Evolution of infected populations ($I\left(t\right),P\left(t\right),H\left(t\right)$, and $I\left(t\right)+P\left(t\right)+H\left(t\right)$) by varying the infectivity rate $\beta $ by $1.55$, $2.55$, and $3.55$, corresponding, respectively, to the basic reproduction number $2.662$, $4.375$, and $6.088$, while fixing index memory $\alpha =0.75$ (Portugal).

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Ndaïrou, F.; Torres, D.F.M. Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal. *Axioms* **2021**, *10*, 135.
https://doi.org/10.3390/axioms10030135

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Ndaïrou F, Torres DFM. Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal. *Axioms*. 2021; 10(3):135.
https://doi.org/10.3390/axioms10030135

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Ndaïrou, Faïçal, and Delfim F. M. Torres. 2021. "Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal" *Axioms* 10, no. 3: 135.
https://doi.org/10.3390/axioms10030135