While analytical methods are adequate to provide the exact solution of a giving equation, or systems of equations, it is important to note that when dealing with nonlinear equations, analytical methods cannot be used. In particular, the model of Covid-19 suggested in this work either with classical or nonlocal operators contains nonlinear components and therefore analytical methods are ineffective. Very recently, Atangana and Seda [16] made use of Newton polynomial to introduce an alternative numerical scheme that can be used to solving nonlinear equations arising in many fields of science, technology, and engineering. The method has been recognized to be very efficient and accurate. In this section, we will make use of the Atangana–Seda scheme to solve the suggested mathematical model for Covid-19 for different differential operators.
We start with the classical case for numerical solution of Covid-19 model:
$$\begin{aligned} &\overset{\cdot }{S} = \Lambda - \bigl( \alpha ( x ) + \gamma _{1}+ \mu _{1} \bigr) S, \\ &\overset{\cdot }{I}= \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &\overset{\cdot }{I_{A}}= \xi I- ( \theta +\mu +\chi + \mu _{1} ) I_{A}, \\ &\overset{\cdot }{I_{D}}= \varepsilon I- ( \eta + \varphi + \mu _{1} ) I_{D}, \\ &\overset{\cdot }{I_{R}}= \eta I_{D}+\theta I_{A}- ( v+ \xi +\mu _{1} ) I_{R}, \\ &\overset{\cdot }{I_{T}}= \mu I_{A}+vI_{R}- ( \sigma + \tau +\mu _{1} ) I_{T}, \\ &\overset{\cdot }{R}= \lambda I+\varphi I_{D}+\chi I_{A}+ \xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &\overset{\cdot }{D}= \tau I_{T}, \\ &\overset{\cdot }{V}= \gamma _{1}S+\Phi R-\mu _{1}V. \end{aligned}$$
(5.1)
For simplicity, we write the above equations as follows:
$$\begin{aligned} &\overset{\cdot }{S} = \widetilde{S} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{I}= \widetilde{I} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{I_{A}} = \widetilde{I_{A}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{I_{D}}= \widetilde{I_{D}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{I_{R}}= \widetilde{I_{R}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{I_{T}}= \widetilde{I_{T}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{R}= \widetilde{R} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{D}= \widetilde{D} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &\overset{\cdot }{V}= \widetilde{V} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \end{aligned}$$
(5.2)
where
$$\begin{aligned} &\widetilde{S} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \Lambda - \bigl( \alpha ( x ) +\gamma _{1}+\mu _{1} \bigr) S, \\ &\widetilde{I} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &\widetilde{I_{A}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \xi I- ( \theta +\mu +\chi +\mu _{1} ) I_{A}, \\ &\widetilde{I_{D}}( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \varepsilon I- ( \eta +\varphi +\mu _{1} ) I_{D}, \\ &\widetilde{I_{R}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \eta I_{D}+\theta I_{A}- ( v+\xi +\mu _{1} ) I_{R}, \\ &\widetilde{I_{T}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \mu I_{A}+vI_{R}- ( \sigma +\tau +\mu _{1} ) I_{T} , \\ &\widetilde{R} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \lambda I+\varphi I_{D}+\chi I_{A}+\xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &\widetilde{D}( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \tau I_{T}, \\ &\widetilde{V}( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) = \gamma _{1}S+\Phi R-\mu _{1}V. \end{aligned}$$
(5.3)
After applying fractal-fractional integral with the exponential kernel, we have the following:
$$\begin{aligned} S ( t_{p+1} ) ={}& S ( t_{p} ) + \begin{bmatrix} \widetilde{S} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{S} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I ( t_{p+1} ) ={}& I ( t_{p} ) + \begin{bmatrix} \widetilde{I} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{A} ( t_{p+1} ) ={}& I_{A} ( t_{p} ) + \begin{bmatrix} \widetilde{I_{A}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{A}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{D} ( t_{p+1} ) = {}&I_{D} ( t_{p} ) + \begin{bmatrix} \widetilde{I_{D}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{D}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \end{aligned}$$
(5.4)
$$\begin{aligned} I_{R} ( t_{p+1} ) = {}&I_{R} ( t_{p} ) + \begin{bmatrix} \widetilde{I_{R}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{R}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{T} ( t_{p+1} ) = {}&I_{T} ( t_{p} ) + \begin{bmatrix} \widetilde{I_{T}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{T}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ R ( t_{p+1} ) = {}&R ( t_{p} ) + \begin{bmatrix} \widetilde{R} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{R} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ D ( t_{p+1} ) = {}&D ( t_{p} ) + \begin{bmatrix} \widetilde{D} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{D} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ V ( t_{p+1} ) ={}& V ( t_{p} ) + \begin{bmatrix} \widetilde{V} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{V} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+ \int _{t_{p}}^{t_{p+1}}\widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau. \end{aligned}$$
We can have the following scheme for this model:
(5.5)
Now, we handle the following model with classical derivative:
$$\begin{aligned} &\overset{\cdot }{S} = \Lambda - \bigl( \alpha ( x ) + \gamma _{1}+ \mu _{1} \bigr) S, \\ &\overset{\cdot }{I} = \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &\overset{\cdot }{I_{A}} = \xi I- ( \theta +\mu +\chi +\mu _{1} ) I_{A}, \\ &\overset{\cdot }{I_{D}} = \varepsilon I- ( \eta +\varphi +\mu _{1} ) I_{D}, \\ &\overset{\cdot }{I_{R}} = \eta I_{D}+\theta I_{A}- ( v+\xi + \mu _{1} ) I_{R}, \\ &\overset{\cdot }{I_{T}} = \mu I_{A}+vI_{R}- ( \sigma +\tau + \mu _{1} ) I_{T}, \\ &\overset{\cdot }{R} = \lambda I+\varphi I_{D}+\chi I_{A}+\xi I_{R}+ \sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &\overset{\cdot }{D} = \tau I_{T}, \\ &\overset{\cdot }{V} = \gamma _{1}S+\Phi R-\mu _{1}V, \end{aligned}$$
(5.6)
where initial conditions are
$$\begin{aligned} &S ( 0 ) = 57780000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.7)
Also the parameters are chosen as follows:
$$\begin{aligned} \begin{aligned} &\Lambda = 57000000,\qquad k=3,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4, \qquad\theta = 0.301,\qquad \gamma =0.09,\qquad \beta =0.013,\\ & \lambda =0.0345,\qquad \varphi =0.0345, \qquad \delta _{1}=0.01,\qquad \gamma _{1} = 0.4,\qquad\mu _{1}=0.3,\\ &\varepsilon =0.161,\qquad\xi =0.015,\qquad\sigma =0.015,\qquad \tau =0.0199,\qquad\Phi =0.2. \end{aligned} \end{aligned}$$
(5.8)
We present a numerical simulation for Covid-19 model in Figs. 33 and 34.
In Figs. 35 and 36, the initial conditions are chosen as
$$\begin{aligned} &S ( 0 ) = 81000000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.9)
Also the parameters are
$$\begin{aligned} \begin{aligned} &\Lambda = 80000000,\qquad k=2,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4, \qquad\theta =0.301, \qquad \gamma = 0.09,\qquad \beta =0.013,\\ & \gamma _{1}=0.4,\qquad \mu _{1}=0.3,\qquad \varepsilon =0.161,\qquad \xi =0.015,\qquad \sigma =0.015,\\ & \tau = 0.0199,\qquad \Phi =0.2,\qquad \lambda =0.0345,\qquad \varphi =0.0345,\qquad \delta _{1}=0.01. \end{aligned} \end{aligned}$$
(5.10)
We present a numerical simulation for Covid-19 model in Figs. 35 and 36.
Now, we replace the classical differential operator by the operator with power-law, exponential decay, and Mittag-Leffler kernels. We start with the exponential decay kernel:
$$\begin{aligned} &{}_{0}^{CF}D_{t}^{\alpha }S = \Lambda - \bigl( \alpha ( x ) +\gamma _{1}+\mu _{1} \bigr) S, \\ &{}_{0}^{CF}D_{t}^{\alpha }I= \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{A} = \xi I- ( \theta +\mu + \chi +\mu _{1} ) I_{A}, \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{D} = \varepsilon I- ( \eta + \varphi +\mu _{1} ) I_{D}, \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{R} = \eta I_{D}+\theta I_{A}- ( v+\xi +\mu _{1} ) I_{R}, \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{T} = \mu I_{A}+vI_{R}- ( \sigma +\tau +\mu _{1} ) I_{T}, \\ &{}_{0}^{CF}D_{t}^{\alpha }R= \lambda I+\varphi I_{D}+ \chi I_{A}+\xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &{}_{0}^{CF}D_{t}^{\alpha }D= \tau I_{T}, \\ &{}_{0}^{CF}D_{t}^{\alpha }V= \gamma _{1}S+\Phi R-\mu _{1}V. \end{aligned}$$
(5.11)
For simplicity, we write the above equations as follows:
$$\begin{aligned} &{}_{0}^{CF}D_{t}^{\alpha }S = \widetilde{S} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }I= \widetilde{I} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{A} = \widetilde{I_{A}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{D} = \widetilde{I_{D}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{R} = \widetilde{I_{R}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }I_{T} = \widetilde{I_{T}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }R= \widetilde{R} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }D= \widetilde{D} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{CF}D_{t}^{\alpha }V= \widetilde{V} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ). \end{aligned}$$
(5.12)
After applying fractal-fractional integral with the exponential kernel, we have the following:
$$\begin{aligned} S ( t_{p+1} ) ={}& S ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{S} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{S} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I ( t_{p+1} ) = {}&I ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{I} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{A} ( t_{p+1} ) = {}&I_{A} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{I_{A}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{A}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{D} ( t_{p+1} ) = {}&I_{D} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{I_{D}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{D}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{R} ( t_{p+1} ) ={}& I_{R} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{I_{R}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{R}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ I_{T} ( t_{p+1} ) ={}& I_{T} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{I_{T}} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{I_{T}} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ R ( t_{p+1} ) = {}&R ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{R} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{R} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ D ( t_{p+1} ) ={}& D ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{D} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{D} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau, \\ V ( t_{p+1} ) = {}&V ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} \widetilde{V} ( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -\widetilde{V} ( t_{p-1},S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \,d\tau. \end{aligned}$$
(5.13)
We can have the following scheme for this model:
(5.14)
For the Mittag-Leffler kernel, we have the following:
$$\begin{aligned} S^{p+1} ={}& S^{p}+\frac{1-\alpha }{AB ( \alpha ) } \widetilde{S} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ I^{p+1} ={}& I^{p}+\frac{1-\alpha }{AB ( \alpha ) } \widetilde{I} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ I_{A}^{p+1} ={}& I_{A}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }\widetilde{I_{A}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ I_{D}^{p+1} ={}& I_{D}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }\widetilde{I_{D}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ I_{R}^{p+1} ={}& I_{R}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }\widetilde{I_{R}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ I_{T}^{p+1} ={}& I_{T}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }\widetilde{I_{T}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ R^{p+1} ={}& R^{p}+\frac{1-\alpha }{AB ( \alpha ) } \widetilde{R} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ D^{p+1} ={}& D^{p}+\frac{1-\alpha }{AB ( \alpha ) } \widetilde{D} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ V^{p+1} ={}& V^{p}+\frac{1-\alpha }{AB ( \alpha ) } \widetilde{V} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau. \end{aligned}$$
(5.15)
We can get the following numerical scheme:
$$\begin{aligned} S^{p+1} = {}&\frac{1-\alpha }{AB ( \alpha ) }\widetilde{S} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{S} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{S} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }\widetilde{I} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{I} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \end{aligned}$$
(5.16)
$$\begin{aligned} &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \\ &{}\times \begin{bmatrix} \widetilde{I} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{A}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) } \widetilde{I_{A}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{I_{A}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{ r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \\ &{}\times \begin{bmatrix} \widetilde{I_{A}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{D}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) } \widetilde{I_{D}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{I_{D}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{ r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \end{aligned}$$
$$\begin{aligned} &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{D}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{R}^{p+1} = {}&\frac{1-\alpha }{AB ( \alpha ) } \widetilde{I_{R}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{I_{R}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{ r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{R}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ &{}+6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{T}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) } \widetilde{I_{T}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{I_{T}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{ r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{T}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ R^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }\widetilde{R} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{R} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{R} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ D^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }\widetilde{D} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{D} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \sum_{r=2}^{p} \begin{bmatrix} \widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{D} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \end{aligned}$$
$$\begin{aligned} &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ V^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }\widetilde{V} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{ p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p} \widetilde{V} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} \widetilde{V} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}. \end{aligned}$$
For the power-law kernel, we have the following:
$$\begin{aligned} &S^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &I^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &I_{A}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &I_{D}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &I_{R}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &I_{T}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &R^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &D^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau, \\ &V^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\,d\tau. \end{aligned}$$
(5.17)
We can get the following numerical scheme:
$$\begin{aligned} S^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p} \widetilde{S} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{S} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p} \widetilde{I} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{A}^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}\widetilde{I_{A}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{A}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{D}^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}\widetilde{I_{D}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{D}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{R}^{p+1} = {}&\frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}\widetilde{I_{R}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{R}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{T}^{p+1} ={}& \frac{\alpha ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) }\sum _{r=2}^{p}\widetilde{I_{T}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{I_{T}} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ R^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p} \widetilde{R} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{R} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ D^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p} \widetilde{D} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{D} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ V^{p+1} = {}&\frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p} \widetilde{V} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{ r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} \widetilde{V} ( t_{r},S^{r},I^{r},I_{A}^{r},I_{D}^{ r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}. \end{aligned}$$
(5.18)
Now, we handle the following model:
$$\begin{aligned} &{}_{0}^{ABC}D_{t}^{\alpha }S = \Lambda - \bigl( \alpha ( x ) +\gamma _{1}+\mu _{1} \bigr) S, \\ &{}_{0}^{ABC}D_{t}^{\alpha }I = \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &{}_{0}^{ABC}D_{t}^{\alpha }I_{A} = \xi I- ( \theta +\mu +\chi + \mu _{1} ) I_{A}, \\ &{}_{0}^{ABC}D_{t}^{\alpha }I_{D} = \varepsilon I- ( \eta + \varphi +\mu _{1} ) I_{D}, \\ &{}_{0}^{ABC}D_{t}^{\alpha }I_{R} = \eta I_{D}+\theta I_{A}- ( v+ \xi +\mu _{1} ) I_{R}, \\ &{}_{0}^{ABC}D_{t}^{\alpha }I_{T} = \mu I_{A}+vI_{R}- ( \sigma + \tau +\mu _{1} ) I_{T}, \\ &{}_{0}^{ABC}D_{t}^{\alpha }R = \lambda I+\varphi I_{D}+\chi I_{A}+ \xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &{}_{0}^{ABC}D_{t}^{\alpha }D = \tau I_{T}, \\ &{}_{0}^{ABC}D_{t}^{\alpha }V = \gamma _{1}S+\Phi R-\mu _{1}V, \end{aligned}$$
(5.19)
where the initial conditions are
$$\begin{aligned} &S ( 0 ) = 57780000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.20)
Also the parameters are chosen as follows:
$$\begin{aligned} \begin{aligned} &\Lambda = 57000000,\qquad k=3,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4, \qquad \theta =0.301, \qquad \gamma = 0.09,\qquad \beta =0.013,\\&\gamma _{1}=0.4,\qquad\mu _{1}=0.3,\qquad \varepsilon =0.161, \qquad\xi =0.015,\qquad\sigma =0.015,\\ & \tau = 0.0199,\qquad\Phi =0.2,\qquad \lambda =0.0345,\qquad\varphi =0.0345,\qquad \delta _{1}=0.01. \end{aligned} \end{aligned}$$
(5.21)
We present a numerical simulation for Covid-19 model in Figs. 37 and 38.
In Figs. 39 and 40, the initial conditions are chosen as
$$\begin{aligned} &S ( 0 ) = 81000000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.22)
Also the parameters are
$$\begin{aligned} \begin{aligned} &\Lambda = 80000000,\qquad k=2,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4, \qquad \theta =0.301, \qquad \gamma = 0.09,\qquad\beta =0.013,\\ & \gamma _{1}=0.4,\qquad \mu _{1}=0.3,\qquad \varepsilon =0.161,\qquad \xi =0.015,\qquad \sigma =0.015,\\ & \tau = 0.0199,\qquad \Phi =0.2,\qquad \lambda =0.0345,\qquad \varphi =0.0345,\qquad \delta _{1}=0.01. \end{aligned} \end{aligned}$$
(5.23)
We present a numerical simulation for Covid-19 model in Figs. 39 and 40.
Now, we replace the classical differential operator by the operator with power-law, exponential decay, and Mittag-Leffler kernels. We start with the exponential decay kernel:
$$\begin{aligned} &{}_{0}^{FFE}D_{t}^{\alpha,\beta }S = \Lambda - \bigl( \alpha ( x ) +\gamma _{1}+\mu _{1} \bigr) S, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I= \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{A} = \xi I- ( \theta + \mu +\chi +\mu _{1} ) I_{A}, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{D} = \varepsilon I- ( \eta +\varphi +\mu _{1} ) I_{D}, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{R} = \eta I_{D}+\theta I_{A}- ( v+\xi +\mu _{1} ) I_{R}, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{T} = \mu I_{A}+vI_{R}- ( \sigma +\tau +\mu _{1} ) I_{T}, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }R= \lambda I+\varphi I_{D}+ \chi I_{A}+\xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }D= \tau I_{T}, \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }V= \gamma _{1}S+\Phi R-\mu _{1}V. \end{aligned}$$
(5.24)
For simplicity, we write the above equations as follows:
$$\begin{aligned} &{}_{0}^{FFE}D_{t}^{\alpha,\beta }S = \widetilde{S} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I= \widetilde{I} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{A} = \widetilde{I_{A}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{D} = \widetilde{I_{D}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{R} = \widetilde{I_{R}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }I_{T} = \widetilde{I_{T}} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }R= \widetilde{R} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }D= \widetilde{D} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ), \\ &{}_{0}^{FFE}D_{t}^{\alpha,\beta }V= \widetilde{V} ( t,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ). \end{aligned}$$
(5.25)
After applying fractal-fractional integral with the exponential kernel, we have the following:
$$\begin{aligned} S ( t_{p+1} ) ={}& S ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{S} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{S} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ I ( t_{p+1} ) ={}& I ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{I} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{I} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ I_{A} ( t_{p+1} ) ={}& I_{A} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{I_{A}} ( \tau,S^{p},I^{p},I_{A}^{p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{I_{A}} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1-\beta }\,d\tau, \\ I_{D} ( t_{p+1} ) ={}& I_{D} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{I_{D}} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{I_{D}} ( \tau,S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ I_{R} ( t_{p+1} ) ={}& I_{R} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{I_{R}} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{I_{R}} ( \tau,S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ I_{T} ( t_{p+1} ) ={}& I_{T} ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{I_{T}} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{I_{T}} ( \tau,S^{p-1},I^{p-1},I_{A}^{p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ R ( t_{p+1} ) ={}& R ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{R} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{R} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ D ( t_{p+1} ) ={}& D ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{D} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{D} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau, \\ V ( t_{p+1} ) ={}& V ( t_{p} ) + \frac{1-\alpha }{M ( \alpha ) } \begin{bmatrix} t_{p}^{1-\beta }\widetilde{V} ( \tau,S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} ) \\ -t_{p-1}^{1-\beta }\widetilde{V} ( \tau,S^{p-1},I^{p-1},I_{A}^{ p-1},I_{D}^{p-1},I_{R}^{p-1},I_{T}^{p-1},R^{p-1},D^{p-1},V^{p-1} )\end{bmatrix} \\ &{}+\frac{\alpha }{M ( \alpha ) } \int _{t_{p}}^{t_{p+1}} \widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) \tau ^{1- \beta }\,d\tau. \end{aligned}$$
(5.26)
We can have the following scheme for this model:
(5.27)
For the Mittag-Leffler kernel, we have the following:
$$\begin{aligned} S^{p+1} ={}& S^{p}+\frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{S} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr)\\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ I^{p+1} ={}& I^{p}+\frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ I_{A}^{p+1} ={}& I_{A}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1-\beta } \widetilde{I_{A}} \bigl( t_{p},S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d\tau, \\ I_{D}^{p+1} ={}& I_{D}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1-\beta } \widetilde{I_{D}} \bigl( t_{p},S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ I_{R}^{p+1} ={}& I_{R}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1-\beta } \widetilde{I_{R}} \bigl( t_{p},S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d\tau, \\ I_{T}^{p+1} ={}& I_{T}^{p}+ \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1-\beta } \widetilde{I_{T}} \bigl( t_{p},S^{p},I^{p},I_{A}^{ p},I_{D}^{p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ R^{p+1} = {}&R^{p}+\frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{R} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d\tau, \\ D^{p+1} ={}& D^{p}+\frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{D} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ V^{p+1} ={}& V^{p}+\frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{V} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}\times\frac{\alpha }{AB ( \alpha ) \Gamma ( \alpha ) }\sum_{r=2}^{p} \int _{t_{r}}^{t_{r+1}}\widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau. \end{aligned}$$
(5.28)
We can get the following numerical scheme:
$$\begin{aligned} S^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{S} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{S} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{S} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \end{aligned}$$
(5.29)
$$\begin{aligned} &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{A}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I_{A}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{A}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{ r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{A}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{D}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I_{D}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{D}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{ r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \end{aligned}$$
$$\begin{aligned} &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{D}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{R}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I_{R}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{R}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{ r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{R}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{T}^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{I_{T}} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{T}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{ r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \end{aligned}$$
$$\begin{aligned} &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{T}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ R^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{R} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{R} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{R} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \end{aligned}$$
$$\begin{aligned} &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ D^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{D} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{D} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \end{aligned}$$
$$\begin{aligned} &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{D} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ V^{p+1} ={}& \frac{1-\alpha }{AB ( \alpha ) }t_{p}^{1- \beta }\widetilde{V} \bigl( t_{p},S^{p},I^{p},I_{A}^{p},I_{D}^{ p},I_{R}^{p},I_{T}^{p},R^{p},D^{p},V^{p} \bigr) \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +1 ) }\sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{V} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{AB ( \alpha ) \Gamma ( \alpha +2 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{\alpha ( \Delta t ) ^{\alpha }}{2AB ( \alpha ) \Gamma ( \alpha +3 ) } \\ &{}\times \sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{V} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}. \end{aligned}$$
For the power-law kernel, we have the following:
$$\begin{aligned} &S^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{S} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &I^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &I_{A}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{A}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &I_{D}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{D}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &I_{R}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{R}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &I_{T}^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum _{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{I_{T}} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &R^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{R} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &D^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{D} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau, \\ &V^{p+1}=\frac{1}{\Gamma ( \alpha ) }\sum_{r=2}^{p}\int _{t_{r}}^{t_{r+1}}\widetilde{V} ( \tau,S,I,I_{A},I_{D},I_{R},I_{T},R,D,V ) ( t_{p+1}-\tau ) ^{\alpha -1}\tau ^{1-\beta }\,d \tau. \end{aligned}$$
(5.30)
We can get the following numerical scheme:
$$\begin{aligned} S^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{S} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{S} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{S} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{S} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{A}^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{A}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{A}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{A}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{A}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{D}^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{D}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{D}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{D}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{D}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{R}^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum _{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{R}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{R}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{R}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{R}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ I_{T}^{p+1} ={}& \frac{\alpha ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) }\sum _{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{I_{T}} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{ r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{I_{T}} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{I_{T}} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{I_{T}} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ R^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{R} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{R} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{R} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{R} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ D^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{D} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{D} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{D} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{D} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}, \\ V^{p+1} ={}& \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +1 ) } \sum_{r=2}^{p}t_{r-2}^{1-\beta } \widetilde{V} \bigl( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} \bigr) \\ &{}\times \bigl[ ( p-r+1 ) ^{\alpha }- ( p-r ) ^{ \alpha } \bigr] \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{\Gamma ( \alpha +2 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r-1}^{1-\beta }\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ -t_{r-2}^{1-\beta }\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } ( p-r+3+2\alpha ) \\ - ( p-r ) ^{\alpha } ( p-r+3+3\alpha )\end{bmatrix} \\ &{}+ \frac{ ( \Delta t ) ^{\alpha }}{2\Gamma ( \alpha +3 ) }\sum_{r=2}^{p} \begin{bmatrix} t_{r}^{1-\beta }\widetilde{V} ( t_{r},S^{r},I^{r},I_{A}^{ r},I_{D}^{r},I_{R}^{r},I_{T}^{r},R^{r},D^{r},V^{r} ) \\ -2t_{r-1}^{1-\beta }\widetilde{V} ( t_{r-1},S^{r-1},I^{r-1},I_{A}^{ r-1},I_{D}^{r-1},I_{R}^{r-1},I_{T}^{r-1},R^{r-1},D^{r-1},V^{r-1} ) \\ +t_{r-2}^{1-\beta }\widetilde{V} ( t_{r-2},S^{r-2},I^{r-2},I_{A}^{ r-2},I_{D}^{r-2},I_{R}^{r-2},I_{T}^{r-2},R^{r-2},D^{r-2},V^{r-2} )\end{bmatrix} \\ &{}\times \begin{bmatrix} ( p-r+1 ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 3\alpha +10 ) ( p-r ) \\ +2\alpha ^{2}+9\alpha +12\end{bmatrix} \\ - ( p-r ) ^{\alpha } \begin{bmatrix} 2 ( p-r ) ^{2}+ ( 5\alpha +10 ) ( p-r ) \\ +6\alpha ^{2}+18\alpha +12\end{bmatrix}\end{bmatrix}. \end{aligned}$$
(5.31)
Now, we handle the following model:
$$\begin{aligned} &{}_{0}^{FFM}D_{t}^{\alpha,\beta }S = \Lambda - \bigl( \alpha ( x ) +\gamma _{1}+\mu _{1} \bigr) S, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }I = \alpha ( x ) S- ( \varepsilon +\xi +\lambda +\mu _{1} ) I, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }I_{A} = \xi I- ( \theta +\mu + \chi +\mu _{1} ) I_{A}, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }I_{D} = \varepsilon I- ( \eta + \varphi +\mu _{1} ) I_{D}, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }I_{R} = \eta I_{D}+\theta I_{A}- ( v+\xi +\mu _{1} ) I_{R}, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }I_{T} = \mu I_{A}+vI_{R}- ( \sigma +\tau +\mu _{1} ) I_{T}, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }R = \lambda I+\varphi I_{D}+ \chi I_{A}+\xi I_{R}+\sigma I_{T}- ( \Phi +\mu _{1} ) R, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }D = \tau I_{T}, \\ &{}_{0}^{FFM}D_{t}^{\alpha,\beta }V = \gamma _{1}S+\Phi R-\mu _{1}V, \end{aligned}$$
(5.32)
where the initial conditions are
$$\begin{aligned} &S ( 0 ) = 57780000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.33)
Also the parameters are chosen as follows:
$$\begin{aligned} \begin{aligned} &\Lambda = 57000000,\qquad k=3,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4, \qquad \theta =0.301, \qquad \gamma = 0.09,\qquad \beta =0.013,\\ & \gamma _{1}=0.4,\qquad \mu _{1}=0.3,\qquad \varepsilon =0.161,\qquad \xi =0.015,\qquad \sigma =0.015,\\ & \tau = 0.0199,\qquad \Phi =0.2,\qquad \lambda =0.0345,\qquad\varphi =0.0345,\qquad \delta _{1}=0.01. \end{aligned} \end{aligned}$$
(5.34)
We present a numerical simulation for Covid-19 model in Figs. 41 and 42.
In Figs. 43 and 44, the initial conditions are chosen as
$$\begin{aligned} &S ( 0 ) = 81000000,\qquad I ( 0 ) =1,\qquad I_{A} ( 0 ) =1,\qquad I_{D} ( 0 ) =1,\qquad I_{R} ( 0 ) =1, \\ &I_{T} ( 0 ) = 1,\qquad R ( 0 ) =0,\qquad D ( 0 ) =0,\qquad V ( 0 ) =0. \end{aligned}$$
(5.35)
Also the parameters are
$$\begin{aligned} \begin{aligned} &\Lambda =80000000,\qquad k=2,\qquad p=0.5,\qquad \eta =0.12,\qquad \chi =0.015,\\ & v=0.027,\qquad x=0.4,\qquad \theta =0.301, \qquad \gamma = 0.09,\qquad \beta =0.013,\\ & \gamma _{1}=0.4,\qquad \mu _{1}=0.3,\qquad \varepsilon =0.161,\qquad \xi =0.015,\qquad\sigma =0.015,\\ & \tau = 0.0199,\qquad\Phi =0.2,\qquad \lambda =0.0345,\qquad\varphi =0.0345, \qquad\delta _{1}=0.01. \end{aligned} \end{aligned}$$
(5.36)
We present a numerical simulation for Covid-19 model in Figs. 43 and 44.