Abstract

In this article, we consider a fractional model, denoted by the model, which aims to predict the outbreak of infectious diseases in general. In particular, we study the spread of COVID-19. The fractional order offers a flexible, appropriate, and reliable framework for pandemic growth characterization. Firstly, we analyze some elementary results of the model (boundedness and uniqueness of solutions). In addition, we establish certain conditions to ensure the local stability of the disease-free and endemic equilibrium points. Based on analytical and numerical results, we conclude that coronavirus infection (COVID-19) remains endemic, which requires long-term prevention and intervention strategies.

1. Introduction

COVID-19 represents the disease caused by a virus in the Coronaviridae family, and SARS-CoV-2 appeared at the end of 2019 in Wuhan, China [1]. It spread rapidly around the world, causing a worldwide epidemic [2, 3].

COVID-19 is a respiratory illness that can be fatal for old patients or other chronic diseases. It is transmitted through close contact with infected people. The disease could also be spread by asymptomatic patients [4], but scientific data are lacking to attest with certainty. In addition, the international health organization has imposed security measures to control the spread including isolation, quarantine, increased home confinement, promotion of wearing face masks, travel restrictions, the closure of public space, and the cancellation of events. The number of confirmed cases increased rapidly to reach more than 517 million cases, and approximately 6.25 million deaths were recorded worldwide as of May 2022.

There are several (susceptible-exposed-infectious-recovered) models that study infectious diseases in general [5–7]; since the appearance of the virus, the models proposed aim to study the behavior of this epidemic [8].

In the last decade, fractional models have been used to model diseases in general. For example, Singh [9] proposed a new fractional model of blood alcohol model using the Hilfer fractional operator. Kumar et al. [10] proposed a model to study the transmission dynamics of dengue, and they considered the generalized Caputo fractional operator. Since the apparition of COVID-19, researchers have used fractional derivatives used in the modeling of this infectious disease. As an example, the work in [11–15] has developed the classic model known by fractional models ( and ) for epidemic analysis of COVID-19 worldwide. This development is founded on the establishment of new quarantine conditions and the hospitalization of confirmed cases, which are considered as epidemic parameters for COVID-19. Recently, many types of fractional derivatives have been employed to model the propagation of COVID-19. For instance, Danane et al. proposed a fractional-order model of the disease (COVID-19) with government action and individual response by the Caputo operator. Bonyah et al. used the Atangana–Baleanu operator for investigating the fractional optimal control dynamics of a coronavirus model. Yadav et al. studied the dynamics of the fractional-order COVID-19 model with memory effect using the Liouville–Caputo operator, so they employed the Adams–Bashforth–Moulton approach to find an approximate solution. Consequently, this development shows us more precisely the behavior of this epidemic.

Recently, fractional calculus has shown wide applicability in many fields, which can be obtained by extracting a dynamic behavior of biological systems shown by a mathematical formulation of integer derivatives. Fractional models have been proposed to study several phenomena involving the memory effect, including epidemic behavior [16], and they offer more flexibility than classical integer-order models to fit the data accurately [17]. Several researchers presented the differential fractional theory [18–21] for this reason, and many papers studied fractional biological models [22, 23]. Recently, new types of fractional derivatives have been developed. For instance, Khalil et al. [24] proposed a new fractional derivative and its properties. Subsequently, Abdeljawad [25] developed the properties of the conformable fractional derivative. After that, Benmakhlouf et al. [26] studied the finite time stability (FTS) and finite time boundedness (FTB) of the conformable fractional derivative. Currently, Hattaf [27, 28] introduced a new definition of the fractional derivative with a non-singular kernel in the sense of Caputo to generalize the various types by making these properties, and he proposed an approach for studying the stability of the latter. After that, Hattaf et al. [29] proposed a new numerical method for approximating the generalized Hattaf fractional derivative involving a non-singular kernel based on Lagrange polynomial interpolation.

Inspired by the aforementioned work and previous literature, we provide the compartmental model of COVID-19 with a standard incidence rate explored in [30] considering fractional Caputo derivatives for a better insight into the disease. The aim of the epidemic model (4) is to describe the dynamic behavior of the disease and predict the tendency for the disease to spread. Therefore, we have proposed a model. First, we analyze the qualitative properties of this model, including the existence and uniqueness of the disease-free and endemic equilibrium points. Second, we establish the conditions to ensure local asymptotic stability of disease-free and endemic equilibrium points. Third, we study the contribution of parameters to reproductive numbers. At the end, a numerical simulation is proposed to show the behavior of the different compartments of our model (4), and we will give some clarification into the interpretation and role of fractional derivatives.

2. Preliminary Results

We begin by giving some definitions of fractional calculus.

Definition 1. (see [31]). The Caputo fractional derivate of function order is defined bywhere.

Definition 2. (see [32]). Let the fractional systemwhere,, and,. Iffulfill the local Lipschitz condition with respect to, there exists a unique solution of the above system.

Lemma 1 (see [33]). Let the fractional-order systemwith,. The equilibrium points of the above system are calculated by solving the following equation: .are locally asymptotically stable if all eigenvaluesof the Jacobian matrixatevaluated of the equilibrium points satisfy Matignon’s condition (see [34]) .

3. Description of Model

In this section, we present a mathematical formulation to model the behavior of compartments; first we will split our population into six categories Susceptible, Exposed, Infected, Quarantined, Hospitalized and Recovered. The description of the parameters is indicated in Table 1, so that the description of the interactions between the compartments is illustrated in Figure 1. The incidence rate plays an important role in describing the evolution of an infectious disease. Based on the spread of different diseases, there are many forms of incidence rates [30, 35–37]. In our model, we assume the bilinear incidence rates [30] ( and ). Thus, our proposed model (4) is in the following form:with initial conditionswhere is in the sense of Caputo fractional derivative and .

Figure 1 

Schematic diagram of (4).

Our model (4) is based on the following hypothesis. : the number of susceptible individuals added is constant each day. : the susceptible individuals move into the exposed class before the infected class. : when an individual is infected , he is either quarantined , hospitalized (severe case), or recovered. : a confined individual can be hospitalized if his situation is critical.

The detailed description of (4) is presented in the following schema.

4. Boundedness and Uniqueness of Solutions

This section is dedicated to demonstrate the boundedness of the solutions of (4).

Lemma 2. The set is a region of attraction for all solutions initiating in the interior of the positive octant, where .

Proof 1. We pose ; then,Using the theory of fractional inequality (see [31]), we obtainwhere is Mittag-Leffler function [31], is Euler’s gamma function, and if , and we have , proving this lemma.
Otherwise, model (4) is equivalent to the model:where

Theorem 1. Assume that the initial conditions . Then, there exists a unique solution of system (4) defined on.
The sufficient condition for the existence and uniqueness of the solution of system (4) in the region with initial conditions and is ‖F(X)−F(X’)‖‖1≤L‖X−X’‖1..

Proof 2. To prove the global existence and uniqueness of (4), consider the region , where .
For any , ,whereThus, satisfies Lipschitz’s condition (see [32] and Definition 2) with respect to .

5. Equilibria and Local Stability

In the first part of this section, we debate the existence of equilibria. It is evident that (4) has an infection-free equilibrium . Also, other endemic equilibrium point is defined after in (13). The basic reproduction number of (4) is

Theorem 2. (1)If , then system (4) has one infection-free equilibrium.(2)If , then (4) has endemic equilibrium point, whereare defined after in (9).

Proof 3. To get the endemic equilibrium point of the (4), in the interior of the equilibrium ,

Proof. i.e., for , we getThen, if , then (4) has endemic equilibrium point .
Now, we analyzed the local asymptotic stability of disease-free equilibrium point and endemic equilibrium point for (4).

Theorem 3. The disease-free equilibrium point is locally asymptotically stable, if and .

Proof 5. To prove the local stability of equilibria, the eigenvalues of the Jacobian matrix of (4) are evaluated bywhereThe characteristic equation of iswhere
The discriminant of equation is . Then, the eigenvalues of matrix (10) to the equilibrium point are reel roots, so , , , , and .
If and , then and . So, and . The proof is completed.

Theorem 4. The endemic equilibrium point is locally asymptotically stable if and condition (12) are realized.

Proof 6. In the same way as the previous proof, letwhereFrom the Jacobian matrix , the characteristic equation at iswhereUsing (13), we getFirstly, , , . Then, .
On the other hand, the discriminant of algebraic equation is this form (see [38]):Then, the equilibrium point is locally asymptotically stable in one of the following cases:(1)(2)(3)Then, is locally asymptotically stable if and (12) are realized.

6. Sensitivity Analysis

Sensitivity analysis shows us the impact of each parameter on the transmission of the disease. It is used to understand which parameters have a high impact on the threshold. More specifically, sensitivity indices allow us to measure the relative change in a variable when a parameter changes. If this variable is differentiable with respect to the parameter, the sensitivity index is defined as follows [39]:where represent the contribution to the basic reproductive number .