Skip to main content

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

Abstract

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.

Introduction and preliminaries

Fractional calculus plays an important role for the mathematical modeling in many scientific and engineering disciplines. For detailed study, we refer the readers to [114] and the references cited therein.

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. To overcome such problems, fractional derivatives have been generalized in many other ways. For details, we refer to [1523].

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 \(\gamma\in(0, 1)\) is defined by

$$ {}^{\mathrm{C}}\mathscr{D}^{\gamma}h(t)= \frac{1}{\varGamma(n-\gamma)} \int _{0}^{t}h{'}(s) \frac{1}{(t-s)^{\gamma}}\,ds. $$
(1)

Definition 2

([17])

Let \(h \in H^{1}(a, b)\), \(a< b\), \(a\in(-\infty, t)\), and \(\gamma\in (0, 1)\); then the γth-order Caputo–Fabrizio derivative of h in the Caputo sense is given as

$$ {}^{\mathrm{CF}}\mathscr{D}^{\gamma}h(t)= \frac{M(\gamma)}{(1-\gamma)} \int _{a}^{t}h{'}(s)\exp\biggl[- \frac{\alpha}{1-\alpha}(t-s)\biggr]\,ds, $$
(2)

where \(M(\gamma)\) 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

$$ {}^{\mathrm{CF}}\mathscr{I}^{\gamma}h(t)= \frac{(1-\gamma)}{M(\gamma)}h(t)+\frac {\gamma}{M(\gamma)} \int_{a}^{t}h(s)\,ds,\quad 0< \gamma< 1. $$
(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:

$$\begin{aligned}& \begin{gathered}\frac{d\mathscr{S}_{p}}{dt}=\bigwedge_{p}- \omega_{p}\mathscr {S}_{p}-\frac{\zeta_{p}\mathscr{S}_{p}(\mathscr{I}_{p}+\varPsi\mathscr {A}_{p})}{\mathscr{N}_{p}}- \omega_{w}\mathscr{S}_{p}\mathscr {M}, \\ \frac{d\mathscr{E}_{p}}{dt}=\frac{\zeta_{p}\mathscr{S}_{p}(\mathscr {I}_{p}+\varPsi\mathscr{A}_{p})}{\mathscr{N}_{p}}+\omega_{w}\mathscr {S}_{p}\mathscr{M}-(1-\varTheta_{p}) \eta_{p}\mathscr{E}_{p}-\varTheta _{p} \varrho_{p}\mathscr{E}_{p}-\varTheta_{p} \varrho_{p}\mathscr {E}_{p}-\omega_{p} \mathscr{E}_{p}, \\ \frac{d\mathscr{I}_{p}}{dt}=(1-\varTheta_{p})\eta_{p} \mathscr {E}_{p}-(\tau_{p}-\omega_{p}) \mathscr{I}_{p}, \\ \frac{d\mathscr{A}_{p}}{dt}=\varTheta_{p}\varrho_{P} \mathscr {E}_{p}-(\tau_{ap}-\omega_{p}) \mathscr{A}_{p}, \\ \frac{d\mathscr{R}_{p}}{dt}=\tau_{p}\mathscr{I}_{p}+ \tau_{ap}\mathscr {A}_{p}-\omega_{p} \mathscr{R}_{p}, \\ \frac{d\mathscr{M}}{dt}=\phi_{p}\mathscr{I}_{p}+ \varpi_{p}\mathscr {A}_{p}-\varphi\mathscr{M}, \end{gathered} \end{aligned}$$
(4)

with the initial conditions

$$\begin{aligned}& \mathscr{S}_{p}(0)=\mathscr{S}_{p}(0)\geq0,\qquad \mathscr{E}_{p}(0)=\mathscr {E}_{p}(0)\geq0,\qquad \mathscr{I}_{p}(0)=\mathscr{I}_{p}(0)\geq0, \\& \mathscr {A}_{p}(0)=\mathscr{A}_{p}(0)\geq0,\qquad \mathscr{R}_{p}(0)=\mathscr {R}_{p}(0)\geq0,\qquad \mathscr{M}(0)=\mathscr{M}(0)\geq0. \end{aligned}$$

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:

$$\begin{aligned}& {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{S}_{p}= \bigwedge_{p}-\omega _{p} \mathscr{S}_{p}-\frac{\zeta_{p}\mathscr{S}_{p}(\mathscr {I}_{p}+\varPsi\mathscr{A}_{p})}{\mathscr{N}_{p}}-\omega_{w}\mathscr {S}_{p}\mathscr{M}, \\& \begin{gathered}{}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{E}_{p}= \frac{\zeta_{p}\mathscr {S}_{p}(\mathscr{I}_{p}+\varPsi\mathscr{A}_{p})}{\mathscr{N}_{p}}+\omega _{w}\mathscr{S}_{p} \mathscr{M}-(1-\varTheta_{p})\eta_{p}\mathscr {E}_{p}-\varTheta_{p}\varrho_{p} \mathscr{E}_{p}-\varTheta_{p}\varrho _{p} \mathscr{E}_{p}-\omega_{p}\mathscr{E}_{p},\end{gathered} \\& \begin{gathered}{}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{I}_{p}=(1- \varTheta_{p})\eta _{p}\mathscr{E}_{p}-( \tau_{p}-\omega_{p})\mathscr{I}_{p}, \\ {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{A}_{p}= \varTheta_{p}\varrho _{P}\mathscr{E}_{p}-( \tau_{ap}-\omega_{p})\mathscr{A}_{p},\end{gathered} \\& \begin{gathered}{}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{R}_{p}= \tau_{p}\mathscr{I}_{p}+\tau _{ap} \mathscr{A}_{p}-\omega_{p}\mathscr{R}_{p}, \\ {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\mathscr{M}=\phi_{p} \mathscr{I}_{p}+\varpi _{p}\mathscr{A}_{p}- \varphi\mathscr{M},\end{gathered} \end{aligned}$$
(5)

where γ denotes the fractional order parameter and the model variables in (4) are nonnegative, the initial conditions are given by

$$\begin{aligned}& \mathscr{S}_{p}(0)=\mathscr{S}_{p}(0)\geq0,\qquad \mathscr{E}_{p}(0)=\mathscr {E}_{p}(0)\geq0,\qquad \mathscr{I}_{p}(0)=\mathscr{I}_{p}(0)\geq0, \\& \mathscr {A}_{p}(0)=\mathscr{A}_{p}(0)\geq0,\qquad \mathscr{R}_{p}(0)=\mathscr {R}_{p}(0)\geq0,\qquad \mathscr{M}(0)=\mathscr{M}(0)\geq0. \end{aligned}$$

Using the initial conditions and fractional integral operator, we convert model (5) into the following integral equations:

$$\begin{aligned}& \begin{gathered}\mathscr{S}_{p}(t)-\mathscr{S}_{p}(0)={}^{\mathrm{CF}} \mathscr{I}^{\gamma} \biggl[\bigwedge_{p}- \omega_{p}\mathscr{S}_{p}-\frac{\zeta_{p}\mathscr {S}_{p}(\mathscr{I}_{p}+\varPsi\mathscr{A}_{p})}{\mathscr{N}_{p}}-\omega _{w}\mathscr{S}_{p}\mathscr{M} \biggr], \\ \mathscr{E}_{p}(t)-\mathscr{E}_{p}(0)={}^{\mathrm{CF}} \mathscr{I}^{\gamma} \biggl[\frac{\zeta_{p}\mathscr{S}_{p}(\mathscr{I}_{p}+\varPsi\mathscr {A}_{p})}{\mathscr{N}_{p}}+\omega_{w} \mathscr{S}_{p}\mathscr {M}-(1-\varTheta_{p}) \eta_{p}\mathscr{E}_{p} \\ \phantom{\mathscr{E}_{p}(t)-\mathscr{E}_{p}(0)={}}-\varTheta_{p}\varrho_{p}\mathscr{E}_{p}- \varTheta_{p}\varrho _{p}\mathscr{E}_{p}- \omega_{p}\mathscr{E}_{p} \biggr], \\ \mathscr{I}_{p}(t)-\mathscr{I}_{p}(0)={}^{\mathrm{CF}} \mathscr{I}^{\gamma }\bigl[(1-\varTheta_{p}) \eta_{p}\mathscr{E}_{p}-(\tau_{p}- \omega_{p})\mathscr {I}_{p}\bigr], \\ \mathscr{A}_{p}(t)-\mathscr{A}_{p}(0)={}^{\mathrm{CF}} \mathscr{I}^{\gamma }\bigl[\varTheta_{p}\varrho_{P} \mathscr{E}_{p}-(\tau_{ap}-\omega_{p}) \mathscr {A}_{p}\bigr], \\ \mathscr{R}_{p}(t)-\mathscr{R}_{p}(0)={}^{\mathrm{CF}} \mathscr{I}^{\gamma}[\tau _{p}\mathscr{I}_{p}+ \tau_{ap}\mathscr{A}_{p}-\omega_{p}\mathscr {R}_{p}], \\ \mathscr{M}(t)-\mathscr{M}(0)={}^{\mathrm{CF}}\mathscr{I}^{\gamma}[\phi _{p}\mathscr{I}_{p}+\varpi_{p} \mathscr{A}_{p}-\varphi\mathscr {M}].\end{gathered} \end{aligned}$$
(6)

For the sake of convenience, we assume the kernels

$$\begin{aligned}& \begin{gathered}\mathscr{K}_{1}(t,\mathscr{S}_{p})= \bigwedge_{p}-\omega_{p}\mathscr {S}_{p}(t)-\frac{\zeta_{p}\mathscr{S}_{p}(t)(\mathscr{I}_{p}(t)+\varPsi \mathscr{A}_{p}(t))}{\mathscr{N}_{p}(t)}-\omega_{w}\mathscr {S}_{p}(t)\mathscr{M}(t), \\ \mathscr{K}_{2}(t,\mathscr{E}_{p})= \frac{\zeta_{p}\mathscr {S}_{p}(t)(\mathscr{I}_{p}(t)+\varPsi\mathscr{A}_{p}(t))}{\mathscr {N}_{p}(t)}+\omega_{w}\mathscr{S}_{p}(t) \mathscr{M}(t)-(1-\varTheta _{p})\eta_{p} \mathscr{E}_{p}(t)-\varTheta_{p}\varrho_{p} \mathscr {E}_{p}(t)\hspace{-12pt} \\ \phantom{\mathscr{K}_{2}(t,\mathscr{E}_{p})=} {}-\varTheta_{p}\varrho_{p}\mathscr {E}_{p}(t)-\omega_{p}\mathscr{E}_{p}(t), \\ \mathscr{K}_{3}(t,\mathscr{I}_{p})=(1- \varTheta_{p})\eta_{p}\mathscr {E}_{p}(t)-( \tau_{p}-\omega_{p})\mathscr{I}_{p}(t), \\ \mathscr{K}_{4}(t,\mathscr{A}_{p})= \varTheta_{p}\varrho_{P}\mathscr {E}_{p}(t)-( \tau_{ap}-\omega_{p})\mathscr{A}_{p}(t), \\ \mathscr{K}_{5}(t,\mathscr{R}_{p})= \tau_{p}\mathscr{I}_{p}(t)+\tau _{ap} \mathscr{A}_{p}(t)-\omega_{p}\mathscr{R}_{p}(t), \\ \mathscr{K}_{6}(t,\mathscr{M})=\phi_{p} \mathscr{I}_{p}(t)+\varpi _{p}\mathscr{A}_{p}(t)- \varphi\mathscr{M}(t)\end{gathered} \end{aligned}$$
(7)

and the functions

$$ \varUpsilon(\gamma)=\frac{1-\gamma}{M(\gamma)},\qquad \varPhi(\gamma)= \frac {\gamma}{M(\gamma)}. $$
(8)

Using (3), (7), and (8) in (6) and writing state variables in terms of kernels, we obtain

$$\begin{aligned}& \begin{gathered} \mathscr{S}_{p}(t)=\mathscr{S}_{p}(0)+ \varUpsilon(\gamma)\mathscr {K}_{1}(t,\mathscr{S}_{p})+ \varPhi(\gamma) \int_{a}^{t}\mathscr {K}_{1}(x, \mathscr{S}_{p})\,dx, \\ \mathscr{E}_{p}(t)=\mathscr{E}_{p}(0)+\varUpsilon( \gamma)\mathscr {K}_{2}(t,\mathscr{E}_{p})+\varPhi( \gamma) \int_{a}^{t}\mathscr {K}_{2}(x, \mathscr{E}_{p})\,dx, \\ \mathscr{I}_{p}(t)=\mathscr{I}_{p}(0)+\varUpsilon( \gamma)\mathscr {K}_{3}(t,\mathscr{I}_{p})+\varPhi( \gamma) \int_{a}^{t}\mathscr {K}_{3}(x, \mathscr{I}_{p})\,dx, \\ \mathscr{A}_{p}(t)=\mathscr{A}_{p}(0)+\varUpsilon( \gamma)\mathscr {K}_{4}(t,\mathscr{A}_{p})+\varPhi( \gamma) \int_{a}^{t}\mathscr {K}_{4}(x, \mathscr{A}_{p})\,dx, \\ \mathscr{R}_{p}(t)=\mathscr{R}_{p}(0)+\varUpsilon( \gamma)\mathscr {K}_{5}(t,\mathscr{R}_{p})+\varPhi( \gamma) \int_{a}^{t}\mathscr {K}_{5}(x, \mathscr{R}_{p})\,dx, \\ \mathscr{M}(t)=\mathscr{M}(0)+\varUpsilon(\gamma)\mathscr {K}_{6}(t, \mathscr{M})+\varPhi(\gamma) \int_{a}^{t}\mathscr {K}_{6}(x, \mathscr{M})\,dx.\end{gathered} \end{aligned}$$
(9)

The Picard iterations are given by

$$\begin{aligned}& \begin{gathered}\mathscr{S}_{p}^{j+1}(t)=\varUpsilon( \gamma)\mathscr{K}_{1}\bigl(t,\mathscr {S}_{p}^{j} \bigr)+\varPhi(\gamma_{1}) \int_{a}^{t}\mathscr{K}_{1}\bigl(x, \mathscr {S}^{j}_{p}\bigr)\,dx, \\ \mathscr{E}_{p}^{j+1}(t)=\varUpsilon( \gamma_{2})\mathscr {K}_{2}\bigl(t,\mathscr{E}_{p}^{j} \bigr)+\varPhi(\gamma_{2}) \int_{a}^{t}\mathscr {K}_{2} \bigl(x,\mathscr{E}_{p}^{j}\bigr)\,dx, \\ \mathscr{I}_{p}^{j+1}(t)=\varUpsilon( \gamma_{3})\mathscr {K}_{3}\bigl(t,\mathscr{I}_{p}^{j} \bigr)+\varPhi(\gamma_{3}) \int_{a}^{t}\mathscr {K}_{3} \bigl(x,\mathscr{I}_{p}^{j}\bigr)\,dx, \\ \mathscr{A}_{p}^{j+1}(t)=\varUpsilon( \gamma_{4})\mathscr {K}_{4}\bigl(t,\mathscr{A}_{p}^{j} \bigr)+\varPhi(\gamma_{4}) \int_{a}^{t}\mathscr {K}_{4} \bigl(x,\mathscr{A}_{p}^{j}\bigr)\,dx, \\ \mathscr{R}_{p}^{j+1}(t)=\varUpsilon( \gamma_{5})\mathscr {K}_{5}\bigl(t,\mathscr{R}_{p}^{j} \bigr)+\varPhi(\gamma_{5}) \int_{a}^{t}\mathscr {K}_{5} \bigl(x,\mathscr{R}_{p}^{j}\bigr)\,dx, \\ \mathscr{M}^{j+1}(t)=\varUpsilon(\gamma_{6}) \mathscr{K}_{6}\bigl(t,\mathscr {M}^{j}\bigr)+\varPhi( \gamma_{6}) \int_{a}^{t}\mathscr{K}_{6}\bigl(x, \mathscr {M}^{j}\bigr)\,dx.\end{gathered} \end{aligned}$$
(10)

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:

$$ \textstyle\begin{cases} {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\psi(t)=\mathscr{K}(t, \psi(t)), \\ \psi(0)=\psi_{0},\quad 0< t< T< \infty. \end{cases} $$
(11)

The vector \(\psi(t)=(\mathscr{S}_{p}, \mathscr{E}_{p}, \mathscr{I}_{p}, \mathscr{A}_{p}, \mathscr{R}_{p}, \mathscr{M})\) and \(\mathscr{K}\) in (10) represent the state variables and a continuous vector function respectively defined as follows:

$$ \begin{aligned}[b]\mathscr{K}&=\left ( \textstyle\begin{array}{c} \mathscr{K}_{1} \\ \mathscr{K}_{2} \\ \mathscr{K}_{3} \\ \mathscr{K}_{4} \\ \mathscr{K}_{6} \\ \mathscr{K}_{6} \end{array}\displaystyle \right )\\&=\left ( \textstyle\begin{array}{c} \bigwedge_{p}-\omega_{p}\mathscr{S}_{p}(t)-\frac{\zeta_{p}\mathscr {S}_{p}(t)(\mathscr{I}_{p}(t)+\varPsi\mathscr{A}_{p}(t))}{\mathscr {N}_{p}(t)}-\omega_{w}\mathscr{S}_{p}(t)\mathscr{M}(t) \\ \frac{\zeta_{p}\mathscr{S}_{p}(t)(\mathscr{I}_{p}(t)+\varPsi\mathscr {A}_{p}(t))}{\mathscr{N}_{p}(t)}+\omega_{w}\mathscr{S}_{p}(t)\mathscr {M}(t)-(1-\varTheta_{p})\eta_{p}\mathscr{E}_{p}(t)-\varTheta_{p}\varrho _{p}\mathscr{E}_{p}(t) \\ -\varTheta_{p}\varrho_{p}\mathscr {E}_{p}(t)-\omega_{p}\mathscr{E}_{p}(t) \\ (1-\varTheta_{p})\eta_{p}\mathscr{E}_{p}(t)-(\tau_{p}-\omega _{p})\mathscr{I}_{p}(t) \\ \varTheta_{p}\varrho_{P}\mathscr{E}_{p}(t)-(\tau_{ap}-\omega _{p})\mathscr{A}_{p}(t) \\ \tau_{p}\mathscr{I}_{p}(t)+\tau_{ap}\mathscr{A}_{p}(t)-\omega _{p}\mathscr{R}_{p}(t) \\ \phi_{p}\mathscr{I}_{p}(t)+\varpi_{p}\mathscr{A}_{p}(t)-\varphi\mathscr{M}(t) \end{array}\displaystyle \right )\end{aligned} $$
(12)

with the initial conditions \(\psi_{0}(t)=(\mathscr{S}_{p}(0), \mathscr {E}_{p}(0), \mathscr{I}_{p}(0), \mathscr{A}_{p}(0), \mathscr{R}_{p}(0), \mathscr{M}(0))\). Corresponding to (11), the integral equation is given by

$$ \psi(t)=\psi_{0}+\varUpsilon(\gamma)\mathscr{K} \bigl(t,\psi(t)\bigr)+\varPhi (\gamma) \int_{a}^{t}\mathscr{K}\bigl(x,\psi(x)\bigr)\,dx. $$
(13)

Moreover, \(\mathscr{K}\) satisfies the Lipschitz condition given by

$$ \bigl\Vert \mathscr{K}\bigl(t, \psi_{1}(t)\bigr)- \mathscr{K}\bigl(t, \psi_{2}(t)\bigr) \bigr\Vert \leq \varOmega \bigl\Vert \psi_{1}(t)-\psi_{2}(t) \bigr\Vert . $$
(14)

Theorem 1

Assuming (14), there exists a unique solution of (11) if

$$ \bigl(\varUpsilon(\gamma)+T\varPhi(\gamma)\bigr)\varOmega< 1. $$
(15)

Proof

Consider \(A=[0, T]\), \(\mathcal{X}=\mathcal{C}(A, \mathbb{R}^{6})\) and the Picard operator \(\mathcal{T}:\mathcal{X}\to\mathcal{X}\) defined by

$$ \mathcal{T}\bigl[\psi(t)\bigr]=\psi_{0}+ \varUpsilon(\gamma)\mathscr{K}\bigl(t,\psi (t)\bigr)+\varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x,\psi(x)\bigr)\,dx, $$
(16)

which turns equation (13) to

$$ \psi(t)=\mathcal{T}\bigl[\psi(t)\bigr]. $$
(17)

Together with the supremum norm \(\|\cdot\|_{A}\) on ψ given by

$$ \bigl\Vert \psi(t) \bigr\Vert _{A}=\sup _{t\in A} \bigl\Vert \psi(t) \bigr\Vert ,\quad \psi(t)\in \mathcal{X}, $$
(18)

\(\mathcal{X}\) 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.,

$$\begin{aligned} \bigl\Vert \mathcal{T}\bigl[\psi(t)\bigr]-\psi_{0} \bigr\Vert _{A} =& \biggl\Vert \varUpsilon(\gamma ) \bigl(\mathscr{K}\bigl(t, \psi(t)\bigr)\bigr) + \varPhi(\gamma) \int_{0}^{t}\mathscr {K}\bigl(x,\psi(x)\bigr)\,dx \biggr\Vert _{A} \\ \leq& \varUpsilon(\gamma) \bigl\Vert \mathscr{K}\bigl(t,\psi(t)\bigr) \bigr\Vert _{A} + \varPhi (\gamma) \int_{0}^{t} \bigl\Vert \mathscr{K}\bigl(x, \psi(x)\bigr) \bigr\Vert _{A}\,dx \\ \leq& \bigl(\varUpsilon(\gamma)+T\varPhi(\gamma)\bigr)\varOmega< 1. \end{aligned}$$

Now using Picard operator equation (16), we have

$$\begin{aligned}& \bigl\Vert \mathcal{T}\bigl[\psi_{1}(t)\bigr]-\mathcal{T}\bigl[ \psi_{2}(t)\bigr] \bigr\Vert _{A} \\& \quad= \biggl\Vert \varUpsilon(\gamma) \bigl(\mathscr{K}\bigl(t, \psi_{1}(t)\bigr)-\mathscr {K}\bigl(t,\psi_{2}(t)\bigr) \bigr) + \varPhi(\gamma) \int_{0}^{t}\bigl(\mathscr{K}\bigl(x,\psi _{1}(x)\bigr)-\mathscr{K}\bigl(x,\psi_{1}(x)\bigr) \bigr)\,dx \biggr\Vert _{A} \\& \quad\leq \varUpsilon(\gamma) \bigl\Vert \mathscr{K}\bigl(t,\psi_{1}(t) \bigr)-\mathscr {K}\bigl(t,\psi_{2}(t)\bigr) \bigr\Vert _{A} + \varPhi(\gamma) \int_{0}^{t} \bigl\Vert \mathscr {K}\bigl(x, \psi_{1}(x)\bigr)-\mathscr{K}\bigl(x,\psi_{1}(x)\bigr) \bigr\Vert _{A}\,dx \\& \quad\leq \varUpsilon(\gamma)\varOmega \bigl\Vert \psi_{1}(t)- \psi_{2}(t) \bigr\Vert _{A} + \varPhi(\gamma)\varOmega \int_{0}^{t} \bigl\Vert \psi_{1}(x)- \psi_{2}(x) \bigr\Vert _{A}\,dx \\& \quad\leq \bigl(\varUpsilon(\gamma)+T\varPhi(\gamma)\bigr)\varOmega \bigl\Vert \psi _{1}(t)-\psi_{2}(t) \bigr\Vert _{A} \\& \quad= \mathcal{A} \bigl\Vert \psi_{1}(t)-\psi_{2}(t) \bigr\Vert _{A}, \end{aligned}$$

where

$$ \mathcal{A}=\bigl(\varUpsilon(\gamma)+T\varPhi(\gamma)\bigr)\varOmega. $$

This implies

$$ \bigl\Vert \mathcal{T}\bigl[\psi_{1}(t)\bigr]-\mathcal{T}\bigl[ \psi_{2}(t)\bigr] \bigr\Vert _{A}\leq\mathcal {A} \bigl\Vert \psi_{1}(t)-\psi_{2}(t) \bigr\Vert _{A}. $$
(19)

Thus the defined operator \(\mathcal{T}\) 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).

Definition 4

Let \(0<\gamma<1\) and \(\mathscr{K}: [0, T]\times\mathbb{R}^{6}\to \mathbb{R}^{6}\) be a continuous function. Then (11) is Hyers–Ulam stable if there exist \(L>0\) and \(\epsilon>0\) such that, for each solution \(\psi\in\mathcal{C}([0, T], \mathbb{R}^{6})\) of

$$ \bigl\vert {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\psi(t)-\mathscr{K}\bigl(t, \psi(t)\bigr) \bigr\vert \leq \epsilon\quad \forall~ t \in[0, T], $$
(20)

there exists a solution \(\psi' \in\mathcal{C}([0, T], \mathbb{R}^{6})\) of (11) with

$$ \bigl\vert \psi(t)-\psi'(t) \bigr\vert \leq L\epsilon \quad\forall ~t \in[0, T]. $$
(21)

Definition 5

Let \(0<\gamma<1\) and \(\mathscr{K}: [0, T]\times\mathbb{R}^{6}\to \mathbb{R}^{6}\) and \(\varPi:[0, T]\to\mathbb{R_{+}}\) be a continuous function. Then (11) is generalized Hyers–Ulam–Rassias stable with respect to Π if there exists a constant \(C_{\mathscr{K}, \varPi}>0\) such that, for each solution \(\psi\in\mathcal{C}([0, T], \mathbb{R}^{6})\) of

$$ \bigl\vert {}^{\mathrm{CF}}\mathscr{D}^{\gamma}\psi(t)- \mathscr{K}\bigl(t, \psi(t)\bigr) \bigr\vert \leq\varPi (t) \quad\forall ~t \in[0, T], $$
(22)

there exists a solution \(\psi' \in\mathcal{C}([0, T], \mathbb{R}^{6})\) of (11) with

$$ \bigl\vert \psi(t)-\psi'(t) \bigr\vert \leq C_{\mathscr{K}, \varPi}\varPi(t) \quad\forall ~t \in [0, T]. $$
(23)

Assume the following:

[\(\mathcal{A}_{1}\)]:

\(\mathscr{K}: [0, T]\times\mathbb{R}^{6}\to \mathbb{R}^{6}\) is continuous;

[\(\mathcal{A}_{2}\)]:

there exists \(C_{\mathscr{K}}>0\) such that

$$\bigl\vert \mathscr{K}(t, \psi)-\mathscr{K}\bigl(x, \psi'\bigr) \bigr\vert \leq C_{\mathscr{K}} \bigl\vert \psi -\psi' \bigr\vert $$

for all \(\psi, \psi' \in\mathbb{R}^{6}\), \(t\in[0, T]\);

[\(\mathcal{A}_{3}\)]:

let \(\varPi\in\mathcal{C}([0, T], \mathbb {R}_{+})\) be an increasing function, and let there exist \(\lambda _{\varPi}>0\) such that

$$ \int_{0}^{t}\varPi(x)\,dx \leq \lambda_{\varPi}\varPi(t) $$
(24)

for all \(x\in[0, T]\).

Theorem 2

Assuming [\(\mathcal{A}_{1}\)][\(\mathcal{A}_{3}\)] hold, (11) is generalized Ulam–Hyers–Rassias stable with respect toΠon\([0, T]\)provided that\(\varUpsilon(\gamma)\mathcal{C}_{\mathscr{K}}<1\).

Proof

Let \(\psi'\in\mathcal{C}([0, T], \mathbb{R}^{6})\) be a solution of (11). Then, from Theorem 1, model (11) has the unique solution

$$ \psi(t)=\psi_{0}+\varUpsilon(\gamma)\mathscr{K} \bigl(t,\psi(t)\bigr)+\varPhi (\gamma) \int_{0}^{t}\mathscr{K}\bigl(x, \psi(x)\bigr)\,dx,\quad t\in[0, T]. $$
(25)

From (22), we have

$$\begin{aligned}& \biggl\vert \psi'(t)-\psi_{0}+\varUpsilon(\gamma)\mathscr{K} \bigl(t,\psi '(t)\bigr)+\varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x, \psi'(x)\bigr)\,dx \biggr\vert \\& \quad\leq \varUpsilon(\gamma)\varPi(t) + \varPhi(\gamma) \int_{0}^{t}\varPi (x)\,dx \\& \quad\leq \bigl(\varUpsilon(\gamma)+\varPhi(\gamma)\lambda_{\varPi}\bigr) \varPi(t). \end{aligned}$$

Thus

$$\begin{aligned}& \begin{gathered} \bigl\vert \psi'(t)-\psi(t) \bigr\vert \\ \quad\leq \biggl\vert \psi'(t)-\psi_{0}-\varUpsilon(\gamma) \mathscr{K}\bigl(t,\psi(t)\bigr)-\varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x, \psi(x)\bigr)\,dx \biggr\vert \\ \quad\leq \biggl\vert \psi'(t)-\psi_{0}-\varUpsilon(\gamma) \mathscr{K}\bigl(t,\psi '(t)\bigr)- \varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x, \psi'(x)\bigr)\,dx - \varUpsilon(\gamma)\mathscr{K}\bigl(t,\psi(t) \bigr) \\ \qquad{}-\varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x,\psi(x)\bigr)\,dx + \varUpsilon (\gamma)\mathscr{K}\bigl(t,\psi'(t)\bigr)+\varPhi(\gamma) \int_{0}^{t}\mathscr {K}\bigl(x,\psi'(x)\bigr)\,dx \biggr\vert \\ \quad\leq \biggl\vert \psi'(t)-\psi_{0}-\varUpsilon(\gamma) \mathscr{K}\bigl(t,\psi'(t)\bigr)-\varPhi(\gamma) \int_{0}^{t}\mathscr{K}\bigl(x,\psi'(x)\bigr)\,dx \biggr\vert \\\qquad{}+ \varUpsilon(\gamma) \bigl\vert \mathscr{K}\bigl(t,\psi(t)\bigr)- \mathscr{K}\bigl(t,\psi'(t)\bigr) \bigr\vert \\ \qquad{}+\varPhi(\gamma) \int_{0}^{t} \bigl\vert \mathscr{K}\bigl(x, \psi(x)\bigr)- \mathscr {K}\bigl(x,\psi'(x)\bigr) \bigr\vert \, dx \\ \quad\leq \bigl(\varUpsilon(\gamma)+\varPhi(\gamma)\lambda_{\varPi}\bigr) \varPi(t) + \varUpsilon(\gamma)C_{\mathscr{K}} \bigl\vert \psi(t)- \psi'(t) \bigr\vert +\varPhi(\gamma )C_{\mathscr{K}} \int_{0}^{t} \bigl\vert \psi(x)- \psi'(x) \bigr\vert \, dx.\end{gathered} \end{aligned}$$

Now, \(\varUpsilon(\gamma)C_{\mathscr{K}}<1\), so

$$ \bigl\vert \psi(t)-\psi'(t) \bigr\vert \leq \frac{(\varUpsilon(\gamma)+\varPhi(\gamma)\lambda _{\varPi})\varPi(t)}{1-\varUpsilon(\gamma)}+\frac{\varPhi(\gamma )C_{\mathscr{K}}}{1-\varUpsilon(\gamma)} \int_{0}^{t} \bigl\vert \psi(x)- \psi'(x) \bigr\vert \,dx. $$
(26)

From Gronwall’s inequality, we have

$$\bigl\vert \psi(t)-\psi'(t) \bigr\vert \leq \biggl[ \frac{\varUpsilon(\gamma)+\varPhi(\gamma )\lambda_{\varPi}}{1-\varUpsilon(\gamma)}\exp(t) \biggr]\varPi(t). $$

Setting \(C_{\mathscr{K}, \varPi}= [\frac{\varUpsilon(\gamma )+\varPhi(\gamma)\lambda_{\varPi}}{1-\varUpsilon(\gamma )}\exp(t) ]\), we arrived at

$$\bigl\vert \psi(t)-\psi'(t) \bigr\vert \leq C_{\mathscr{K},\varPi}\varPi(t). $$

This completes the proof. □

Conclusion

In this paper, we discussed the novel corona virus model given in [25] within the Caputo–Fabrizio fractional model, and we showed the existence and uniqueness of its solution by applying the Banach contraction principle and Picard–Lindelöf technique. Utilizing Gronwall’s inequality, we presented the generalized Hyers–Ulam stability of the fractional model.

References

  1. 1.

    Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)

    Google Scholar 

  2. 2.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  3. 3.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    Google Scholar 

  4. 4.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  5. 5.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon (1993)

    Google Scholar 

  6. 6.

    Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)

    Google Scholar 

  7. 7.

    Kumar, D., Singh, J., Baleanu, D.: Numerical computation of a fractional model of differential-difference equation. J. Comput. Nonlinear Dyn. 11(6), Article ID 061004 (2016)

    Article  Google Scholar 

  8. 8.

    Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A.: On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015(1), Article ID 278 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ma, M., Baleanu, D., Gasimov, Y.S., Yang, X.J.: New results for multidimensional diffusion equations in fractal dimensional space. Rom. J. Phys. 61, 784–794 (2016)

    Google Scholar 

  10. 10.

    Ullah, S., Khan, M.A., Farooq, M., Hammouch, Z., Baleanu, D.: A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative. Discrete Contin. Dyn. Syst., Ser. S 13(3), 975–993 (2020)

    MATH  Google Scholar 

  11. 11.

    Khan, M.A., Atangana, A.: Dynamics of Ebola disease in the framework of different fractional derivatives. Entropy 21(3), Article ID 303 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Khan, M.A., Hammouch, Z., Baleanu, D.: Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative. Math. Model. Nat. Phenom. 14(3), Article ID 311 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Khan, M.A., Ullah, S., Okosun, K.O., Shah, K.: A fractional order pine wilt disease model with Caputo–Fabrizio derivative. Adv. Differ. Equ. 2108, Article ID 410 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Atangana, A., Khan, M.A.: Validity of fractal derivative to capturing chaotic attractors. Chaos Solitons Fractals 126, 50–59 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Atangana, A., Alkahtani, B.S.T.: Analysis of the Keller–Segel model with a fractional derivative without singular kernel. Entropy 17(6), 4439–4453 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Alsaedi, A., Nieto, J.J., Venktesh, V.: Fractional electrical circuits. Adv. Mech. Eng. (2015). https://doi.org/10.1177/1687814015618127

    Article  MATH  Google Scholar 

  17. 17.

    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  18. 18.

    Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015)

    Google Scholar 

  19. 19.

    Tateishi, A.A., Ribeiro, H.V., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. 5, Article ID 52 (2017)

    Article  Google Scholar 

  20. 20.

    Kumar, D., Singh, J., Al Qurashi, M., Baleanu, D.: Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel. Adv. Mech. Eng. (2017). https://doi.org/10.1177/1687814017690069

    Article  Google Scholar 

  21. 21.

    Owolabi, K.M., Atangana, A.: Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos Solitons Fractals 105, 111–119 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kumar, D., Tchier, F., Singh, J., Baleanu, D.: An efficient computational technique for fractal vehicular traffic flow. Entropy 20, Article ID 259 (2018)

    Article  Google Scholar 

  23. 23.

    Singh, J., Kumar, D., Baleanu, D.: New aspects of fractional Biswas–Milovic model with Mittag-Leffler law. Math. Model. Nat. Phenom. 14(3), Article ID 303 (2019)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Chen, T., Rui, J., Wang, Q., Zhao, Z., Cui, J., Yin, L.: A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infect. Dis. Poverty 9, Article ID 24 (2020)

    Article  Google Scholar 

  25. 25.

    Khan, M.A., Atangana, A.: Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex. Eng. J. (2020, in press)

  26. 26.

    Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete version. Rep. Math. Phys. 80, 11–27 (2017)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the referees for their helpful suggestions.

Availability of data and materials

Not applicable.

Funding

There is no funding.

Author information

Affiliations

Authors

Contributions

All authors contributed equally. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Azhar Hussain.

Ethics declarations

Competing interests

The authors declare that they have no conflict of interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hussain, A., Baleanu, D. & Adeel, M. Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model. Adv Differ Equ 2020, 384 (2020). https://doi.org/10.1186/s13662-020-02845-0

Download citation

Keywords

  • Fractional Caputo–Fabrizio derivative
  • Novel coronavirus (nCoV-2019)
  • Picard–Lindelöf technique