Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model

The aim of this work is to present a new fractional order model of novel coronavirus (nCoV-2019) under Caputo–Fabrizio derivative. We make use of fixed point theory and Picard–Lindelöf technique to explore the existence and uniqueness of solution for the proposed model. Moreover, we explore the generalized Hyers–Ulam stability of the model using Gronwall’s inequality.

In the early literature, fractional derivatives in the sense of Riemann-Liouville and of Caputo were used widely. Recent studies showed that at the boundary points of the interval on which the order of derivative is based, the kernels of these derivatives have a singularity.
After the outbreak of novel coronavirus (nCoV-2019) on December 31, 2020, researchers started working to find the cure of the virus. Due the importance of mathematical modeling, Chen et al. [24] and Khan and Atangana [25] proposed the coronavirus models independently. In this paper, we generalize the novel coronavirus (nVoC-2019) model proposed by Khan and Atangana [25] by utilizing the Caputo-Fabrizio fractional derivative and explore the existence and uniqueness of its solution using fixed point theory. Also, we present the generalized Hyers-Ulam stability of it.
We now give some basic definitions which are used in the sequel.
The definition of Caputo fractional derivative can be found in many books (see, e.g., [2]). Definition 1 For a differentiable function h, the Caputo derivative of order γ ∈ (0, 1) is defined by ( 1 ) , and γ ∈ (0, 1); then the γ th-order Caputo-Fabrizio derivative of h in the Caputo sense is given as where M(γ ) is a normalizing function depending on γ such that M(0) = M(1) = 1.

Definition 3 ([26])
The corresponding fractional integral in the Caputo-Fabrizio sense is given by (3)

Fractional model in the Caputo-Fabrizio sense
Very recently, Khan and Atangana [25] proposed a mathematical model of a novel corona virus (COVID-19) as follows: with the initial conditions They generalized the model to a fractional order model using Atangana-Baleanu derivative and solved the model numerically.
In this paper, we replace Atangana-Baleanu derivative with Caputo-Fabrizio fractional derivative and generalize model (4) in the following way: where γ denotes the fractional order parameter and the model variables in (4) are nonnegative, the initial conditions are given by Using the initial conditions and fractional integral operator, we convert model (5) into the following integral equations: For the sake of convenience, we assume the kernels and the functions Using (3), (7), and (8) in (6) and writing state variables in terms of kernels, we obtain The Picard iterations are given by In order to show the existence and uniqueness of solution of model (5), we make use of fixed point theory and Picard-Lindelöf technique. First, we re-write model (5) in the following way: The vector ψ(t) = (S p , E p , I p , A p , R p , M) and K in (10) represent the state variables and a continuous vector function respectively defined as follows: with the initial conditions ψ 0 (t) = (S p (0), E p (0), I p (0), A p (0), R p (0), M(0)). Corresponding to (11), the integral equation is given by Moreover, K satisfies the Lipschitz condition given by Theorem 1 Assuming (14), there exists a unique solution of (11) if Proof Consider A = [0, T], X = C(A, R 6 ) and the Picard operator T : X → X defined by which turns equation (13) to Together with the supremum norm · A on ψ given by X defines a Banach space.
It is to be noted that the solution of the fractional order novel coronavirus (nCoV-2019) model is bounded, i.e., Now using Picard operator equation (16), we have

This implies
Thus the defined operator T is a contraction, and hence model (11) has a unique solution.
Remark 1 We remark here that the stability by considering disease free equilibrium and the endemic equilibrium for model (11) can be proved on the same lines as given in [25].

Generalized Hyers-Ulam stability
In this section, we explore the stability analysis of model (11).