In this section, using the suggested numerical scheme, we present its application to solve the mathematical model of COVID-19 with possibility of waves. The numerical scheme will be applied for all cases where the differential operators are with classical differential operators, modern fractional differential operators, and variable orders, although only few examples will be used for numerical simulations. Firstly, we shall use the Caputo–Fabrizio fractional derivative
$$\begin{aligned}& {}_{0}^{CF}D_{t}^{\alpha }S = \Lambda - \bigl( \delta ( t ) \bigl( \alpha I^{\ast }+w\beta I_{D}^{\ast }+ \gamma wI_{A}^{\ast }+w \delta _{1}I_{R}^{\ast }+w \delta _{2}I_{T}^{\ast } \bigr) +\gamma _{1}+ \mu _{1} \bigr) S, \\& {}_{0}^{CF}D_{t}^{\alpha }I = \bigl( \delta ( t ) \bigl( \alpha I^{\ast }+w\beta I_{D}^{\ast }+\gamma wI_{A}^{\ast }+w \delta _{1}I_{R}^{\ast }+w \delta _{2}I_{T}^{\ast } \bigr) \bigr) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\& {}_{0}^{CF}D_{t}^{\alpha }I_{A} = \xi I- ( \theta +\mu + \chi +\mu _{1} ) I_{A}, \\& {}_{0}^{CF}D_{t}^{\alpha }I_{D} = \varepsilon I- ( \eta + \varphi +\mu _{1} ) I_{D}, \\& {}_{0}^{CF}D_{t}^{\alpha }I_{R} = \eta I_{D}+\theta I_{A}- ( v+\xi +\mu _{1} ) I_{R}, \\& {}_{0}^{CF}D_{t}^{\alpha }I_{T} = \mu I_{A}+vI_{R}- ( \sigma +\tau +\mu _{1} ) I_{T}, \\& {}_{0}^{CF}D_{t}^{\alpha }R = \lambda I+ \varphi I_{D}+ \chi I_{A}+\xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\& {}_{0}^{CF}D_{t}^{\alpha }D = \tau I_{T}, \\& {}_{0}^{CF}D_{t}^{\alpha }V = \gamma _{1}S+\Phi R-\mu _{1}V. \end{aligned}$$
(110)
For simplicity, we rearrange the above equation as follows:
$$\begin{aligned}& {}_{0}^{CF}D_{t}^{\alpha }S = S^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }I = I^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }I_{A} = I_{A}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }I_{D} = I_{D}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }I_{R} = I_{R}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }I_{T} = I_{T}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }R = R^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }D = D^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{CF}D_{t}^{\alpha }V = V^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) . \end{aligned}$$
(111)
Thus, we can have the following scheme for our model:
(112)
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(120)
With the Atangana–Baleanu fractional derivative, we can solve numerically our model as follows:
(121)
where
(122)
With the Caputo fractional derivative, we can obtain the following:
(123)
We now do the same routine for fractal-fractional derivatives. We start with the Caputo–Fabrizio fractal-fractional derivative
$$\begin{aligned}& {}_{0}^{FFE}D_{t}^{\alpha }S = S^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }I = I^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }I_{A} = I_{A}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }I_{D} = I_{D}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }I_{R} = I_{R}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }I_{T} = I_{T}^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }R = R^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }D = D^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\& {}_{0}^{FFE}D_{t}^{\alpha }V = V^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) . \end{aligned}$$
(124)
After applying the fractional integral with exponential kernel and putting the Newton polynomial into these equations, we can solve our model as follows:
(125)
$$\begin{aligned}& I^{n+1} =S^{n}+\frac{1-\alpha }{M ( \alpha ) } \end{aligned}$$
(126)
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(133)
For the Atangana–Baleanu fractal-fractional derivative, we can have the following numerical scheme:
(134)
For the power-law kernel, we can have the following:
(135)
Now we apply
$$\begin{aligned}& {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }S = S^{\ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }I = I^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }I_{A} = I_{A}^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }I_{D} = I_{D}^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }I_{R} = I_{R}^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }I_{T} = I_{T}^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }R = R^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }D = D^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ & {}_{0}^{FFE}D_{t}^{\alpha ,\beta ( t ) }V = V^{ \ast } ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) . \end{aligned}$$
(136)
After applying the fractional integral with exponential kernel and putting the Newton polynomial into these equations, we can solve our model as follows:
(137)
(138)
For the Atangana–Baleanu fractal-fractional derivative, we can have the following numerical scheme:
(139)
For the power-law kernel, we can have the following:
(140)