On fractal-fractional Covid-19 mathematical model

Abstract

In this article, we are studying a Covid-19 mathematical model in the fractal-fractional sense of operators for the existence of solution, Hyers-Ulam (HU) stability and computational results. For the qualitative analysis, we convert the model to an equivalent integral form and investigate its qualitative analysis with the help of iterative convergent sequence and fixed point approach. For the computational aspect, we take help from the Lagrange’s interpolation and produce a numerical scheme for the fractal-fractional waterborne model. The scheme is then tested for a case study and we obtain interesting results.

1. Introduction

As we know the Covid-19 is violent acute aspiration syndrome, and it is also a pandemic [1]. After the end of 2019, he Covid-19 has caused significant economic loss and destruction, and a few million people also died from this virus.

Due to the enormous public health problems and the need to direct health measures, many researchers have focused their efforts on the Covid-19 modeling and its spread in the population [2], [3], [4], [6], [8], [12], [14], [15], [16], [17], [18], [19]. Both mathematical models [20], [21], [22] and statistical approaches [32] were used.

During the recent years, numerous mathematical models of fractal and fractional order to the Covid-19 have also been constructed by researchers [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. Now, we would like to summarize some works on the Covid-19 briefly. Also, for some very interesting and recent works on Covid-19, we referee the readers to the papers of [33], [34], [35].

In Kolabje et al. [4], the authors investigated the time-series evolution of the cumulative number of confirmed cases of Covid-19, the novel coronavirus disease for some African countries.

In Ullah et al. [6], the transmission dynamics of a Covid-19 pandemic model with vertical transmission has been improved for nonsingular kernel type of fractional differentiation. Here, numerical simulations have also been given depending upon based on real data of Covid-19 in Indonesia to show the plots of the impacts of the fractional order derivative with the expectation. The constructed model gives better than classical models.

Das and Samanta [7] discussed transmission dynamics of the Covid -19 in Italy 2020. Here, taking into account the uncertainty due to the limited information about the Covid -19, the authors have taken the modified susceptible-asymptomatic-infectious-recovered compartmental model under fractional order framework. The validity of the Covid -19 model is justified by comparing real data with the results obtained from simulations.

In Baba and Nasidi [9] presented a fractional order SIR model incorporating individual with mild cases as a compartment to become SMIR model. Here, it was shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. Hence, it is notified that to curtail the spread of the disease, there is need to take care of the mild cases as well.

Omame et al. [25] considered and analyzed a fractional order model for Covid-19 and tuberculosis co-infection, using the Atangana-Baleanu derivative. The model was simulated using data relevant to both diseases in New Delhi, India. Simulations of the fractional order model revealed that reducing the risk of Covid-19 infection by latently-infected TB individuals will not only bring down the burden of Covid-19, but will also reduce the co-infection of both diseases in the population. Rezapour et al. [30] provided a SEIR epidemic model for the spread of Covid-19 using the Caputo fractional derivative. Using the fractional Euler method, they have got an approximate solution to the model. To predict the transmission of ovid-19 in Iran and in the world, they provided a numerical simulation based on real data.

Tuan et al. [31] gave a mathematical model for the transmission of Covid-19 by the Caputo fractional-order derivative. Using the generalized Adams-Bashforth-Moulton method, they solved the system and obtain the approximate solutions. They also presented a numerical simulation for the transmission of Covid -19 in the world. Here, the reproduction number was also obtained as which shows that the epidemic continues.

Using the fractal-fractional sense of differential and integral operators we get the following the Covid-19 model:

$$\{\begin{array}{c}{}_0^{FF}{D}_t^{\alpha ,\beta }S\left(t\right)={\Lambda }_1-(\mu +{\theta }_1)S\left(t\right)-{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}-{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N},\hfill \\ {}_0^{FF}{D}_t^{\alpha ,\beta }E\left(t\right)={\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}-(\mu +{\alpha }_1+{\theta }_2)E\left(t\right),\hfill \\ {}_0^{FF}{D}_t^{\alpha ,\beta }I\left(t\right)={\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I\left(t\right),\hfill \\ {}_0^{FF}{D}_t^{\alpha ,\beta }R\left(t\right)={\alpha }_{2}I\left(t\right)-\mu R\left(t\right),\hfill \\ {}_0^{FF}{D}_t^{\alpha ,\beta }{Q}_T\left(t\right)={\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T\left(t\right),\hfill \end{array}$$ (1)

where $S\left(0\right)\ge 0$, $E\left(0\right)\ge 0$, $I\left(0\right)\ge 0$, $R\left(0\right)\ge 0$ and $Q_T\left(0\right)\ge 0$ are the initial state.

• •S represented the susceptible people that may be infected with COVID-19 disease.
• •E represented the infected without symptoms which COVID19 disease in the incubation period.
• •I represented the infected with symptoms.
• •R represented the recovered people after infected with the COVID-19 disease.
• •$Q_T$ represented the people who are not infected with the virus in the quarantine period. complications.
• •${\Lambda }_1$: The recruitment rate of susceptible.
• •$\mu$: Natural mortality rate.
• •${\beta }_1$:The rate of people who were infected by contact with the infected without symptoms.
• •${\beta }_2$:The rate of people who were infected by contact with the infected with symptoms.
• •${\alpha }_1$: The rate of people become normaly infected with symptoms.
• •${\alpha }_2$: The rate of recovered from the virus.
• •${\theta }_1$:The rate of susceptible who have been in quarantine total.
• •${\theta }_2$:The rate of infected without symptoms who have been in quarantine total.
• •${\theta }_3$:The rate of infected with symptoms who have been in quarantine total.
• •${\delta }_1$: Mortality rate due to complications.

We should mention that here we used the fractal-fractional differential and integral operators, then we obtain the Covid-19 model given by the system (1). This system is different from that given in [40] and those in the literature.

Definition 1.1

[2], [3] Consider $\varphi \in C\left(\right(a,b),\mathbb{R})$ which is fractal differentiable on $(a,b)$ of order $0<{\varrho }^{*}\le 1$. The fractal-fractional derivation operator for $\varphi$ in the Atangana-Baleanu settings of order $0<{\kappa }_1\le 1$, with the generalized kernel of the Mittag-Leffler type is introduced as

$${}_0^{FFM}{D}_t^{{\kappa }_1,{\varrho }^{*}}\varphi \left(t\right)=\frac{AB\left({\kappa }_1\right)}{1-{\kappa }_1}\frac{d}{d{t}^{{\varrho }^{*}}}{\int }_0^t\varphi \left(s\right)E_{{\kappa }_1}[-\frac{{\kappa }_1}{1-{\kappa }_1}{(t-s)}^{{\kappa }_1}]ds,$$

where, $AB\left({\kappa }_1\right)=1-{\kappa }_1+{\displaystyle \frac{{\kappa }_1}{\Gamma {\kappa }_1}}$, and

$$\frac{d\varphi \left(s\right)}{d{s}^{{\varrho }^{*}}}=\underset{t\to s}{\lim }\frac{\varphi \left(t\right)-\varphi \left(s\right)}{t^{{\varrho }^{*}}-s^{{\varrho }^{*}}}.$$

Definition 1.2

[2], [3] Let $\varphi$ be the same function considered above. Then, the fractal-fractional integration operator in the Atangana-Baleanu settings for $\varphi$ of order $0<{\kappa }_1\le 1$ with the kernel of Mittag-Leffler type is given by

$$\begin{array}{ccc}\hfill {}_0^{FFM}{I}_t^{{\kappa }_1,{\varrho }^{*}}\varphi \left(t\right)& =& \frac{{\kappa }_1{\varrho }^{*}}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{s}^{{\varrho }^{*}-1}\varphi \left(s\right){(t-s)}^{{\kappa }_1-1}ds\hfill \\ & & +\frac{{\varrho }^{*}(1-{\kappa }_1)t^{{\varrho }^{*}-1}}{AB\left({\kappa }_1\right)}\varphi \left(t\right),\hfill \end{array}$$

where, $AB\left({\kappa }_1\right)=1-{\kappa }_1+{\displaystyle \frac{{\kappa }_1}{\Gamma {\kappa }_1}}.$

In Section 2 the existence criteria are given by Theorem 2.1 and Theorem 2.2. In the next section, Section 3 uniqueness of solutions of system (1) is given by Theorem 3.1. In Section 4, Hyers-Ulam stability of system (1) is discussed by Theorem 4.1. In Section 5, numerical results are given. Finally, in Section 6, the conclusion of the paper is presented.

2. Existence criteria

With the help of fixed point procedure we check the existence of fractal fractional model (1), We have,

$$\begin{array}{cc}\hfill S\left(t\right)-S\left(0\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{S}^{{\kappa }_2-1}{(t-s)}^{{\kappa }_1-1}[{\Lambda }_1-(\mu +{\theta }_1)S\left(t\right)\hfill \\ & -{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}-{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)t^{{\kappa }_2-1}}{AB\left({\kappa }_1\right)}[{\Lambda }_1-(\mu +{\theta }_1)S\left(t\right)-{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}\hfill \\ & -{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}],\hfill \\ \hfill E\left(t\right)-E\left(0\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{S}^{{\kappa }_2-1}{(t-s)}^{{\kappa }_1-1}[{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}\hfill \\ & -(\mu +{\alpha }_1+{\theta }_2)E\left(t\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)t^{{\kappa }_2-1}}{AB\left({\kappa }_1\right)}[{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}\hfill \\ & -(\mu +{\alpha }_1+{\theta }_2)E\left(t\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill I\left(t\right)-I\left(0\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{S}^{{\kappa }_2-1}{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times [{\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I\left(t\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)t^{{\kappa }_2-1}}{AB\left({\kappa }_1\right)}[{\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I\left(t\right)],\hfill \\ \hfill R\left(t\right)-R\left(0\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{S}^{{\kappa }_2-1}{(t-s)}^{{\kappa }_1-1}[{\alpha }_{2}I\left(t\right)-\mu R\left(t\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)t^{{\kappa }_2-1}}{AB\left({\kappa }_1\right)}[{\alpha }_{2}I\left(t\right)-\mu R\left(t\right)],\hfill \\ \hfill Q_T\left(t\right)-Q_T\left(0\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{S}^{{\kappa }_2-1}{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times [{\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T\left(t\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)t^{{\kappa }_2-1}}{AB\left({\kappa }_1\right)}[{\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T\left(t\right)].\hfill \end{array}$$ (2)

Now, we define some functions $Q_i$ and constant ${\eta }_i$, $i\in N_1^5$ is below

$$\{\begin{array}{c}{Q}_1(t,S)={\Lambda }_1-(\mu +{\theta }_1)S\left(t\right)-{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}-{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N},\hfill \\ Q_2(t,E)={\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}-(\mu +{\alpha }_1+{\theta }_2)E\left(t\right),\hfill \\ Q_3(t,I)={\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I\left(t\right),\hfill \\ Q_4(t,R)={\alpha }_{2}I\left(t\right)-\mu R\left(t\right),\hfill \\ Q_5(t,Q_T)={\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T\left(t\right).\hfill \end{array}$$

For proving our results, we assume the following assumptions:

$\left(G^{*}\right)$ The continuous functions $S\left(t\right),S^{★}\left(t\right),E\left(t\right),E^{★}\left(t\right),I\left(t\right),I^{★}\left(t\right),R\left(t\right),R^{★}\left(t\right),$ $Q_T\left(t\right),Q_T^{★}\left(t\right)\in L[0,1]$, such that $\parallel S\parallel \le {\psi }_1,\parallel I\parallel \le {\psi }_2$ for ${\psi }_1,{\psi }_2>0$.

Theorem 2.1

The kernels $Q_1,Q_2,Q_3,Q_4,Q_5$ are satisfying the Lipschitz condition if the assumption $G^{*}$ holds and satisfies ${\varphi }_i<1$ for $i\in N_1^5$ and are contractions provided that ${\psi }_i<1$ for every $i\in$ $N_1^5$ .

Proof

First, we prove that $Q_1(t,S)$ satisfies Lipschitz condition.Using $S\left(t\right)$ and $S^{★}\left(t\right)$, we have

$$\begin{array}{cc}\hfill \parallel Q_1(t,S)-Q_1(t,S^{*})\parallel =& \parallel {\Lambda }_1-(\mu +{\theta }_1)S\left(t\right)-{\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}-{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}\hfill \\ & -({\Lambda }_1-(\mu +{\theta }_1)S^{*}\left(t\right)-{\beta }_1\frac{S^{*}\left(t\right)E\left(t\right)}{N}-{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N})\parallel \hfill \\ \hfill \le & [\mu +{\theta }_1+{\beta }_1\frac{1}{N}\parallel E\left(t\right)\parallel +{\beta }_2\frac{1}{N}\parallel I\left(t\right)\parallel ]\parallel S-S^{*}\parallel \hfill \\ \hfill \le & [\mu +{\theta }_1+{\beta }_1{c}_1+{\beta }_2{c}_2]\parallel S_1-S_1^{*}\parallel \hfill \\ \hfill \le & {\psi }_1\parallel S-S^{*}\parallel ,\hfill \end{array}$$

where    ${\varphi }_1=\mu +{\theta }_1+{\beta }_1{c}_1+{\beta }_2{c}_2<1$.

Hence, $Q_1$ satisfies Lipschitz condition and ${\varphi }_1<1$. Next, we prove that $Q_2(t,E)$ satisfies Lipschitz condition for this $E,E^{*}$ we have

$$\begin{array}{cc}\hfill \parallel Q_2(t,E)-Q_2(t,E^{*})\parallel =& \parallel {\beta }_1\frac{S\left(t\right)E\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}-(\mu +{\alpha }_1+{\theta }_2)E\left(t\right)\hfill \\ & -({\beta }_1\frac{S\left(t\right)E^{*}\left(t\right)}{N}+{\beta }_2\frac{S\left(t\right)I\left(t\right)}{N}-(\mu +{\alpha }_1+{\theta }_2)E^{*}\left(t\right))\parallel \hfill \\ \hfill \le & [{\beta }_1\frac{1}{N}\parallel S\left(t\right)\parallel +{\lambda }_1+{\theta }_1+\mu ]\parallel E-E^{*}\parallel \hfill \\ \hfill \le & [{\beta }_1{c}_3+{\lambda }_1+{\theta }_1+\mu ]\parallel E-E^{*}\parallel \hfill \\ \hfill \le & {\psi }_2\parallel E-E^{*}\parallel ,\hfill \end{array}$$

where, ${\psi }_2={\beta }_1{c}_3+{\lambda }_1+{\theta }_1+\mu <1$. Hence, $Q_2$ satisfies Lipschitz condition and ${\psi }_2<1$.

Next, we prove that $Q_3(t,S_3)$ satisfies Lipschitz condition for this using $I,I^{*}$ we have

$$\begin{array}{cc}\hfill \parallel Q_3(t,I)-Q_3(t,I^{*})\parallel =& \parallel {\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I\left(t\right)\hfill \\ & -({\alpha }_{1}E\left(t\right)-({\alpha }_2+{\theta }_3+\mu +{\delta }_1)I^{*}\left(t\right))\parallel \hfill \\ \hfill \le & [{\alpha }_2+{\theta }_3+\mu +{\delta }_1]\parallel I-I^{*}\parallel \hfill \\ \hfill \le & {\psi }_3\parallel I-I^{*}\parallel ,\hfill \end{array}$$

where, ${\psi }_3={\alpha }_2+{\theta }_3+\mu +{\delta }_1<1$. Hence, $Q_3$ satisfies Lipschitz condition and ${\psi }_3<1$ Next, we prove that $Q_4(t,R)$ satisfies Lipschitz condition. For this we have

$$\begin{array}{cc}\hfill \parallel Q_4(t,R)-Q_4(t,R^{*})\parallel =& \parallel {\alpha }_{2}I\left(t\right)-\mu R\left(t\right)\hfill \\ & -({\alpha }_{2}I\left(t\right)-\mu R^{*}\left(t\right))\hfill \\ \hfill \le & \mu |R-R^{*}\parallel \hfill \\ \hfill \le & {\psi }_4\parallel R-R^{*}\parallel ,\hfill \end{array}$$

where, ${\psi }_4=\mu <1$.

Hence, $Q_4$ satisfies Lipschitz condition and ${\psi }_4<1$.

Next, we prove that $Q_5(t,Q_T)$ satisfies Lipschitz condition. For this we have

$$\parallel Q_5(t,Q_T)-Q_5(t,Q_T^{*})\parallel =\parallel {\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T\left(t\right)$$

$$\begin{array}{cc}& -({\theta }_{1}S\left(t\right)+{\theta }_{2}E\left(t\right)+{\theta }_{3}I\left(t\right)-\mu Q_T^{*}\left(t\right))\parallel \hfill \\ \hfill \le & \mu \parallel Q_T-Q_T^{*}\parallel \hfill \\ \hfill \le & {\psi }_4\parallel Q_T-Q_T^{*}\parallel .\hfill \end{array}$$

Hence, $Q_5$ satisfies Lipschitz condition and ${\psi }_4<1$.

Ultimately all the functions satisfies Lipschitz conditions with ${\psi }_i<1$ for $i\in N_1^4$ complete the proof. □

We rewrite the system in the following form by using the kernels $Q_i,i\in N_1^5$ and initial condition $S\left(0\right)=E\left(0\right)=I\left(0\right)=R\left(0\right)=0=Q_T\left(0\right)$, we have

$$\{\begin{array}{c}S\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_1(s,S\left(s\right))ds\hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_1(t,S\left(t\right))\hfill \\ E\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_2(s,E\left(s\right))ds\hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_2(t,E\left(t\right))\hfill \end{array}$$

$$\{\begin{array}{c}{I}_3\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_3(s,I\left(s\right))ds\hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_3(t,I\left(t\right))\hfill \\ R\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_4(s,R\left(s\right))ds\hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_4(t,R\left(t\right))\hfill \\ Q_T\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_5(s,Q_T\left(s\right))ds\hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_5(t,Q_T\left(t\right)).\hfill \end{array}$$

Now, we define the following recursive formulas:

$$\begin{array}{ccc}\hfill S_n\left(t\right)& =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_1(s,S_{n-1}\left(s\right))ds\hfill \\ & & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_1(t,S_{n-1}\left(t\right)),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill E_n\left(t\right)& =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_2(s,E_{n-1}\left(s\right))ds\hfill \\ & & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_2(t,E_{n-1}\left(t\right)),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_n\left(t\right)& =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_3(s,I_{n-1}\left(s\right))ds\hfill \\ & & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_3(t,I_{n-1}\left(t\right))\hfill \end{array}$$

$$\begin{array}{ccc}\hfill R_n\left(t\right)& =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_4(s,R_{n-1}\left(s\right))ds\hfill \\ & & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_4(t,R_{n-1}\left(t\right)),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {Q_T}_n\left(t\right)& =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_5(s,{Q_T}_{n-1}\left(s\right))ds\hfill \\ & & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_5(t,{Q_T}_{n-1}\left(t\right))\hfill \end{array}$$

Next, we consider the differences as follow:

$$\begin{array}{cc}\hfill \Delta S_{n+1}\left(t\right)=& S_{n+1}\left(t\right)-S_n\left(t\right)\hfill \\ \hfill =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\left[Q_1\right(s,S_n\left(s\right)-Q_1(s,S_{n-1}\left(s\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\left[Q_1\right(t,S_n\left(t\right)-Q_1(t,S_{n-1}\left(t\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta E_{n+1}\left(t\right)& =E_{n+1}\left(t\right)-E_n\left(t\right)\hfill \\ & =\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\left[Q_2\right(s,E_n\left(s\right)-Q_2(s,E_{n-1}\left(s\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\left[Q_2\right(t,E_n\left(t\right)-Q_2(t,E_{n-1}\left(t\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta I_{n+1}\left(t\right)=& I_{n+1}\left(t\right)-I_n\left(t\right)\hfill \\ & +\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\left[Q_3\right(s,I_n\left(s\right)-Q_3(s,I_{n-1}\left(s\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\left[Q_3\right(t,I_n\left(t\right)-Q_3(t,I_{n-1}\left(t\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta R_{n+1}\left(t\right)=& R_{n+1}\left(t\right)-R_n\left(t\right)\hfill \\ \hfill =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\left[Q_4\right(s,R_n\left(s\right)-Q_4(s,R_{n-1}\left(s\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\left[Q_4\right(t,R_n\left(t\right)-Q_4(t,R_{n-1}\left(t\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta {Q_T}_{n+1}\left(t\right)=& {Q_T}_{n+1}\left(t\right)-{Q_T}_n\left(t\right)\hfill \\ \hfill =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\left[Q_5\right(s,{Q_T}_n\left(s\right)-Q_5(s,{Q_T}_{n-1}\left(s\right)]ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\left[Q_5\right(t,{Q_T}_n\left(t\right)-Q_5(t,{Q_T}_{n-1}\left(t\right)].\hfill \end{array}$$

Now, taking norm of the above system on both sides,

$$\begin{array}{cc}\hfill \parallel \Delta S_{n+1}\left(t\right)\parallel =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\parallel [Q_1(s,S_n\left(s\right)-Q_1(s,S_{n-1}\left(s\right)]\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel [Q_1(t,S_n\left(t\right)-Q_1(t,S_{n-1}\left(t\right)]\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel \Delta E_{n+1}\left(t\right)\parallel =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\parallel Q_2(s,E_n\left(s\right)-Q_2(s,E_{n-1}\left(s\right)\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_2(t,E_n\left(t\right)-Q_2(t,E_{n-1}\left(t\right)\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel \Delta I_{n+1}\left(t\right)\parallel =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\parallel Q_3(s,I_n\left(s\right)-Q_3(s,I_{n-1}\left(s\right)\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_3(t,I_n\left(t\right)-Q_3(t,I_{n-1}\left(t\right)\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel \Delta R_{n+1}\left(t\right)\parallel =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\parallel Q_4(s,R_n\left(s\right)-Q_4(s,R_{n-1}\left(s\right)\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_4(t,R_n\left(t\right)-Q_4(t,R_{n-1}\left(t\right)\parallel ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel \Delta {Q_T}_{n+1}\left(t\right)\parallel =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}\hfill \\ & \times S^{{\kappa }_2-1}\parallel Q_5(s,{Q_T}_n\left(s\right)-Q_5(s,{Q_T}_{n-1}\left(s\right)\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_5(t,{Q_T}_n\left(t\right)-Q_5(t,{Q_T}_{n-1}\left(t\right)\parallel .\hfill \end{array}$$

Theorem 2.2

The fractal fractional order COVID-19 model(1)has a solution if the following holds true,

$$\Delta =\max [{\Psi }_1,{\psi }_2,{\psi }_3,{\psi }_4,{\psi }_5]<1.$$

Proof

We define the function

$$\{\begin{array}{c}\mathcal{H}{1}_n\left(t\right)=S_{n+1}\left(t\right)-S\left(t\right)\hfill \\ \mathcal{H}{2}_n\left(t\right)=E_{n+1}\left(t\right)-E\left(t\right),\hfill \\ \mathcal{H}{3}_n\left(t\right)=I_{n+1}\left(t\right)-I\left(t\right),\hfill \\ \mathcal{H}{4}_n\left(t\right)=R_{n+1}\left(t\right)-R\left(t\right),\hfill \\ \mathcal{H}{5}_n\left(t\right)={Q_T}_{n+1}\left(t\right)-Q_T\left(t\right).\hfill \end{array}$$

Taking norm of the above system we have,

$$\begin{array}{cc}\hfill \parallel \mathcal{H}{1}_n\left(t\right)\parallel =& \parallel S_{n+1}\left(t\right)-S\left(t\right)\parallel \hfill \\ \hfill =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}\parallel [Q_1(s,S_n\left(s\right)-Q_1(s,S\left(s\right)]\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel [Q_1(t,S_n\left(t\right)-Q_1(t,S\left(t\right)]\parallel \hfill \\ \hfill \le & \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{\psi }_1\parallel S_n-S\parallel ds\hfill \\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{\psi }_1\parallel S_n-S\parallel \hfill \\ \hfill \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\psi }_1\parallel S_n-S\parallel \hfill \\ \hfill \le & {[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]}^n{\delta }^n\parallel S_n-S\parallel ,\hfill \end{array}$$

where $\delta <1$ and $n⟶\infty$.

$S_n\to S$ similarly we have

$$\parallel \mathcal{H}{2}_n\left(t\right)\parallel {[\frac{{\kappa }_1{\kappa }_2\Gamma {\kappa }_2}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]}^n{\delta }^n\parallel E_n-E\parallel$$

$$\parallel \mathcal{H}{3}_n\left(t\right)\parallel {[\frac{{\kappa }_1{\kappa }_2\Gamma {\kappa }_2}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]}^n{\delta }^n\parallel I_n-I\parallel$$

$$\parallel \mathcal{H}{4}_n\left(t\right)\parallel {[\frac{{\kappa }_1{\kappa }_2\Gamma {\kappa }_2}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]}^n{\delta }^n\parallel R_n-R\parallel$$

$$\parallel \mathcal{H}{5}_n\left(t\right)\parallel {[\frac{{\kappa }_1{\kappa }_2\Gamma {\kappa }_2}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]}^n{\delta }^n\parallel {Q_T}_n-Q_T\parallel .$$

Thus, we find that ${\mathcal{H}}_{i}n\left(t\right)\to 0$ as $n\to \infty$ for $i\in N_1^5$ and $\delta <1$ which complete the proof. □

3. Uniqueness solutions

For our suggested (1), we study the analysis of the uniqueness of solution.

Theorem 3.1

The fractal fractional(1)has unique solution if the following holds true:

$$[\frac{{\kappa }_1{\kappa }_2\Gamma {\kappa }_2}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_i\le 1,i\in N_1^5.$$

Proof

Let us consider contradiction that there exists another solution of fractal fractional of model (1) such that $\dot{S}\left(t\right),\dot{E}\left(t\right),\dot{I}\left(t\right),\dot{R}\left(t\right),\dot{Q_T}\left(t\right)$ such that

$$\begin{array}{cc}\hfill \dot{S}\left(t\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_1(s,\dot{S}\left(s\right))ds\\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_1(t,\dot{S}\left(t\right))\end{array}$$

$$\dot{E}\left(t\right)=\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_2(s,\dot{E}\left(s\right))ds+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_2(t,\dot{E}\left(t\right))$$

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_3(s,\dot{I}\left(s\right))ds\\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_3(t,\dot{I}\left(t\right))\end{array}$$

$$\begin{array}{cc}\hfill \dot{R}\left(t\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_4(s,\dot{R}\left(s\right))ds\\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_4(t,\dot{R}\left(t\right))\end{array}$$

$$\begin{array}{cc}\hfill \dot{Q_T}\left(t\right)=& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_5(s,\dot{Q_T}\left(s\right))ds\\ & +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_5(t,\dot{Q_T}\left(t\right)).\end{array}$$

Now taking differences of $S\left(t\right),\dot{S\left(t\right)}$ and then take norm, we have

$$\begin{array}{c}\parallel S\left(t\right)-\dot{S}\left(t\right)\parallel =\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}\parallel Q_1(S,S\left(t\right))-Q_1(S,\dot{S}\left(t\right))\parallel \hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_1(S,S\left(t\right))-Q_1(S,\dot{S}\left(t\right))\parallel \hfill \\ \le \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{\varphi }_1\parallel S-\dot{S}\parallel \hfill \\ +\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{\varphi }_1\parallel S-\dot{S}\parallel \hfill \\ \le [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_1\parallel S-\dot{S}\parallel \hfill \\ \parallel S-\dot{S}\parallel -[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_1\parallel S-\dot{S}\parallel \le 0\hfill \\ [1-[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}\left]{\varphi }_1\right]\parallel S-\dot{S}\parallel \le 0.\hfill \end{array}$$

The above inequality is true if

$$\begin{array}{c}\hfill \parallel S-\dot{S}\parallel =0\\ \hfill \Rightarrow S=\dot{S}.\end{array}$$

$$\begin{array}{ccc}\hfill \parallel E-\dot{E}\parallel & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_2\parallel E-\dot{E}\parallel \hfill \\ & & [1-[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}\left]{\varphi }_2\right]\parallel E-\dot{E}\parallel \le 0.\hfill \end{array}$$

The above inequality is true if

$$\begin{array}{c}\hfill \parallel E-\dot{E}\parallel =0\\ \hfill \Rightarrow E=\dot{E},\end{array}$$

$$\begin{array}{ccc}\hfill \parallel I-\dot{I}\parallel & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_3\parallel I-\dot{I}\parallel \hfill \\ & & [1-[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}\left]{\varphi }_3\right]\parallel I-\dot{I}\parallel \le 0.\hfill \end{array}$$

The above inequality is true if

$$\parallel I-\dot{I}\parallel =0,\text{this}\text{implies}I=\dot{I}.$$

$$\begin{array}{ccc}\hfill \parallel R-\dot{R}\parallel & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_4\parallel R-\dot{R}\parallel \hfill \\ & & [1-[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}\left]{\varphi }_4\right]\parallel R-\dot{R}\parallel \le 0.\hfill \end{array}$$

The above inequality true if

$$\begin{array}{c}\hfill \parallel R-\dot{R}\parallel =0\\ \hfill \Rightarrow R=\dot{R}.\end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \parallel Q_T-\dot{Q_T}\parallel & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_5\parallel Q_T-\dot{Q_T}\parallel \hfill \\ & & [1-[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}\left]{\varphi }_5\right]\parallel Q_T-\dot{Q_T}\parallel \le 0.\hfill \end{array}$$

The above inequality true if

$$\begin{array}{c}\hfill \parallel Q_T-\dot{Q_T}\parallel =0\\ \hfill \Rightarrow Q_T=\dot{Q_T}.\end{array}$$

Thus the (1) has unique solution. □

4. Hyers-Ulams stability

Definition 4.1

The fractal fractional integral system (1) is to be Hyers-Ulam stability if there exist a constant ${\phi }_i>0,i\in N_1^5$ satisfying for every ${\beta }_i>0,i\in N_1^5$

Definition 4.2

$$\begin{array}{cc}\hfill \parallel S\left(t\right)& -\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_1(s,S\left(s\right))ds\\ & -\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_1(t,S\left(t\right))\le {\beta }_1\end{array}$$

$$\begin{array}{cc}\hfill \parallel E\left(t\right)& -\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_2(s,E\left(s\right))ds\\ & -\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_2(t,E\left(t\right))\le {\beta }_2\end{array}$$

$$\begin{array}{cc}\hfill \parallel I\left(t\right)& -\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_3(s,I\left(s\right))ds\\ & -\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_3(t,I\left(t\right))\le {\beta }_3\end{array}$$

$$\begin{array}{cc}\hfill \parallel R\left(t\right)& -\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_4(s,E\left(s\right))ds\\ & -\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_4(t,R\left(t\right))\le {\beta }_4\end{array}$$

$$\begin{array}{cc}\hfill \parallel Q_T\left(t\right)& -\frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}{Q}_5(s,Q_T\left(s\right))ds\\ & -\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}{Q}_5(t,Q_T\left(t\right))\le {\beta }_5.\end{array}$$

There exist approximate solution of the model (1)$S^{*}\left(t\right),E^{*}\left(t\right),I^{*}\left(t\right),R^{*}\left(t\right),Q_T^{*}\left(t\right)$ that satisfies the given model,such that

$$\begin{array}{ccc}\hfill \parallel S\left(t\right)-S^{*}\left(t\right)\parallel & =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}\hfill \\ & \times & \parallel Q_1(t,S\left(t\right))-Q_1(t,S^{*}\left(t\right))\parallel \hfill \\ & +& \frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_1(t,S\left(t\right))-Q_1(t,S^{*}\left(t\right))\parallel \hfill \\ & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_1\parallel S-S^{*}\parallel .\hfill \end{array}$$

Let

$${\varsigma }_1=[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]\parallel S-S^{*}\parallel ,$$

${\eta }_1={\varphi }_1$ the above an inequalities become

$$\parallel S\left(t\right)-S^{*}\left(t\right)\parallel \le {\eta }_1{\varsigma }_1.$$

Similarly we have

$$\parallel E\left(t\right)-E^{*}\left(t\right)\parallel \le {\eta }_2{\varsigma }_2$$

$$\parallel I\left(t\right)-I^{*}\left(t\right)\parallel \le {\eta }_3{\varsigma }_3$$

$$\parallel R\left(t\right)-R^{*}\left(t\right)\parallel \le {\eta }_4{\varsigma }_4$$

$$\parallel Q_T\left(t\right)-Q_T^{*}\left(t\right)\parallel \le {\eta }_5{\varsigma }_5.$$

Theorem 4.1

If the above assumptions hold, then the fractal fractional COVID-19 model (1) is HU stable

Proof

We know that the fractal fractional COVID-19 model (1) has unique solution let $S^{*}\left(t\right),E^{*}\left(t\right),I^{*}\left(t\right),R^{*}\left(t\right),Q_T^{*}\left(t\right)$ be approximate solution of model (1) which satisfy the model then we have

.

$$\begin{array}{ccc}\hfill \parallel S\left(t\right)-S^{*}\left(t\right)\parallel & =& \frac{{\kappa }_1{\kappa }_2}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^t{(t-s)}^{{\kappa }_1-1}{S}^{{\kappa }_2-1}\parallel Q_1(t,S\left(t\right))-Q_1(t,S^{*}\left(t\right))\parallel \hfill \\ & +& \frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}{t}^{{\kappa }_2-1}\parallel Q_1(t,S\left(t\right))-Q_1(t,S^{*}\left(t\right))\parallel \hfill \\ & \le & [\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_1\parallel S-S^{*}\parallel .\hfill \end{array}$$

Let

$${\alpha }_1=[\frac{{\kappa }_1{\kappa }_2\Gamma \left({\kappa }_2\right)}{AB\left({\kappa }_1\right)\Gamma ({\kappa }_1+{\kappa }_2)}+\frac{{\kappa }_2(1-{\kappa }_1)}{AB\left({\kappa }_1\right)}]{\varphi }_1\parallel S-S^{*}\parallel$$

and

$${\beta }_1={\varphi }_1,$$

so the above inequality become

$$\parallel S-S^{*}\parallel \le {\alpha }_1{\beta }_1$$

similarly

$$\begin{array}{c}\hfill \parallel E-E^{*}\parallel \le {\alpha }_2{\beta }_2\\ \hfill \parallel I-I^{*}\parallel \le {\alpha }_3{\beta }_3\\ \hfill \parallel R-R^{*}\parallel \le {\alpha }_4{\beta }_4\\ \hfill \parallel Q_T-Q_T^{*}\parallel \le {\alpha }_5{\beta }_5\end{array}$$

consequently by definition the COVID-19 model (1) is hyers-ulams stable which is complete the proof. □

5. Numerical scheme

Let us consider

$${}_0^{FFM}{D}_t^{{\kappa }_1,{\kappa }_2}\varrho \left(t\right)=V(t,\varrho \left(t\right)),\text{where}\varrho \left(0\right)={\varrho }_0,$$

The above equation can be written in Antangana-Baleanu fractional derivative as following

$${}_0^{FFR}{D}_t^{{\kappa }_1}\varrho \left(t\right)={\kappa }_2{t}^{{\kappa }_2-1}L(t,\varrho \left(t\right))=V(t,\varrho \left(t\right)).$$

Taking Antangana-Baleanu integral, we get

$$\begin{array}{ccc}\hfill \varrho \left(t\right)& =& \varrho \left(0\right)+\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}V(t,\varrho \left(t\right))+\frac{{\kappa }_1}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}\hfill \\ & & {\int }_0^t{(t-\zeta )}^{{\kappa }_1-1}{\zeta }^{{\kappa }_2-1}V(\zeta ,\varrho \left(\zeta \right))d\zeta .\hfill \end{array}$$

Replacing (t) by $t_n+1$ we have

$$\begin{array}{cc}\hfill {\varrho }^{n+1}=& \varrho \left(0\right)+\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}V(t_n+1,\varrho \left(t\right))\hfill \\ & +\frac{{\kappa }_1}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}{\int }_0^{t_n+1}{(t_n+1-\zeta )}^{{\kappa }_1-1}{\zeta }^{{\kappa }_2-1}V(\zeta ,\varrho \left(\zeta \right))d\zeta .\hfill \end{array}$$

By applying two step Lagrange Polynomial we obtain

$$\begin{array}{c}\hfill V(x,\varrho \left(t\right))=\frac{(x-t_{k-1})V(t_k,\varrho \left(t_k\right))}{t_k-t_{k-1}}-\frac{(x-t_k)V(t_{k-1},\varrho \left(t_{k-1}\right)}{t_k-t_{k-1}}\\ \hfill =\frac{V(t_k,\varrho \left(t_k\right))(x-t_{k-1})}{t_k-t_{k-1}}-\frac{V(t_{k-1},\varrho \left(t_{k-1}\right)(x-t_k)}{t_k-t_{k-1}}\\ \hfill =\frac{V(t_k,\varrho \left(t_k\right))(x-t_{k-1})}{h}-\frac{V(t_{k-1},\varrho \left(t_{k-1}\right)(x-t_k)}{h}.\end{array}$$

Applying Lagrange Polynomial to considering equation, we get

$$\begin{array}{cc}\hfill {\varrho }^{n+1}=& \varrho \left(0\right)+\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}V(t_n,\varrho \left(t_n\right))\hfill \\ & +\frac{{\kappa }_1}{AB\left({\kappa }_1\right)\Gamma {\kappa }_1}\sum _{i=1}^n[\frac{V(t_i,\varrho \left(t_i\right))}{h}\hfill \\ & {\int }_{t_k}^{t_k+1}(\zeta -t_i-1){(t_n+1-\zeta )}^{{\kappa }_1-1}d\zeta \hfill \\ & -\frac{V(t_{i-1},\varrho \left(t_{i-1}\right))}{h}{\int }_{t_k}^{t_n+1}(\zeta -t_i){(t_n+1-\zeta )}^{{\kappa }_1-1}d\zeta ].\hfill \end{array}$$

Now solving the integral we get

$${\varrho }^{n+1}=\varrho \left(0\right)+\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}V(t_n,\varrho \left(t_n\right))+\frac{{\kappa }_1{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[V(t_i,\varrho \left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}^{{\kappa }_1}(n-i+2+2{\kappa }_1))-V(t_{i-1},{\varrho }_{i-1})({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

Now Replacing the value of $V(x,\varrho (t\left)\right)$ we get

$${\varrho }^{n+1}=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}I(t_i,\varrho \left(t_i\right))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[V(t_i,\varrho \left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}_1^{\kappa }(n-i+2+2{\kappa }_1))-V(t_{i-1},{\varrho }_{i-1})({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

Thus, the numerical scheme for the above system is

$$S_{n+1}=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}{Q}_1(t_n,S_{(}{t}_n))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[Q_1(t_i,S\left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}^{{\kappa }_1}(n-i+2+2{\kappa }_1))-Q_1(t_{i-1},S_{i-1})({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

$$E_{n+1}=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}{Q}_2(t_n,E_{(}{t}_n))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[Q_2(t_i,E_{n+1}\left(t_i\right))\left({(n+1-i)}^{{\kappa }_1}\right(n-i+2+{\kappa }_1{(n-i)}^{{\kappa }_1}\times (n-i+2+2{\kappa }_1))-Q_2(t_{i-1},E_{i-1})({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

$$In+1=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}{Q}_3(t_n,I_{(}{t}_n))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}$$

$$\times \sum _{i=1}^n[Q_3(t_i,I_{n+1}\left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}^{{\kappa }_1}(n-i+2+2{\kappa }_1))-Q_3(t_{i-1},Ii-1))({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

$$R_{n+1}=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}{Q}_4(t_n,R_{(}{t}_n))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[Q_4(t_i,R_{n+1}\left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}^{{\kappa }_1}(n-i+2+2{\kappa }_1))-Q_4(t_{i-1},R_{i-1}))({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

$${Q_T}_{n+1}=\varrho \left(0\right)+{\kappa }_2{t}^{{\kappa }_2-1}\frac{1-{\kappa }_1}{AB\left({\kappa }_1\right)}{Q}_5(t_n,{Q_T}_{(}{t}_n))+{\kappa }_2{t}^{{\kappa }_2-1}\frac{{\kappa }_2{h}^{{\kappa }_1}}{\Gamma ({\kappa }_1+2)}\times \sum _{i=1}^n[Q_5(t_i,{Q_T}_{n+1}\left(t_i\right))({(n+1-i)}^{{\kappa }_1}(n-i+2+{\kappa }_1)-{(n-i)}^{{\kappa }_1}(n-i+2+2{\kappa }_1))-Q_5(t_{i-1},Q_{T_i-1})({(n+1-i)}^{{\kappa }_1+1}-(n-i+1+{\kappa }_1){(n-i)}^{{\kappa }_1})].$$

5.1. Numerical results

In this portion, we present the numerical description of the Covid-19 model 1. The numerical values are taken from the article [4], where $\mu =0.02,{\beta }_2=0.3,{\alpha }_2=0.1,{\theta }_1=0.9,{\alpha }_1=0.4,{\beta }_1=0.4,{\theta }_2=0.2,{\theta }_3=0.2,{\delta }_1=0.1,{\delta }_2=0.4,\sigma =0.1,{\Lambda }_1=8\times {10}^8,{\Lambda }_2=5\times {10}^5,$ and the initial values are: $S\left(0\right)=3*{10}^9;E\left(0\right)=2.5\times {10}^6,I\left(0\right)=6*{10}^5,R\left(0\right)=5000,Q_T\left(0\right)={10}^{1}0,$

From the numerical results which are explained via graphs we have observed that the population of the susceptible people and exposed people are suddenly reduced and are transferred in to the infected class. As they are quarantined, the infection is control. After few days, the recovery begins and the infection is then reduced. This shows that in the controlling of the infection, the quarantine has an important role.

The numerical description of the model is presented with the help of eight figures. The first Fig. 1 is the numerical data for the susceptible class of the model, the Fig. 2 is the exposed people to the Covid-19 patients, the Fig. 3 is the Covid-19 infected class, Fig. 4 are the simulations for the recovered class from the Covid-19, Fig. 5 represents the graphical data about quarantined people.

Fig. 1.

Susceptible class for the different fractional orders of the model (1).

Fig. 2.

Infected class for the different fractional orders of the model (1).

Fig. 3.

$W\left(t\right)$ class for the different fractional orders of the model (1).

Fig. 4.

$D\left(t\right)$ class for the different fractional orders of the model (1).

Fig. 5.

$R\left(t\right)$ class for the different fractional orders of the model (1).

6. Conclusion

In this work, a fractal-fractional order Covid-19 model based on five classes of a population was taken into consideration. The classes of the considered populations are the susceptible, exposed, infected, quarantined and recovered. Here, the existence of solutions, stability of the considered system and numerical simulation based on an iterative numerical scheme were investigated for the considered model. The scheme is supported by the Lagrange’s interpolation polynomial. We also applied the numerical scheme to the available data in literature and got very much interesting results for different fractional orders. We suggest the readers for reconsideration of the fractal-fractional Covid-19 model (1) for other fractional derivatives and stability results. They can also generate numerical schemes with the help of other polynomials. The researchers can also consider the model for variable order and develop more general results as a continuation of this study.

Availability of data and material

Not applicable.

Funding source

There is no source of funding this article.

Authors contributions

The first author (H.K) worked in the conceptualization, formal analysis and methodology. The second author (F.A) investigated the suggested model and helped in the writing. The third author (O.T) worked in the methodology, software and validation. The fourth author (M.I) worked in the project administration and supervision.

Declaration of Competing Interest

Authors declare that they have no conflict of interest.