On variable-order hybrid FracInt Covid-19 mathematical model: optimal control approach

Abstract

An optimal control problem for the variable-order fractional-integer mathematical model of vaccination Covid 19 is presented in this research, where the order of the derivatives varies during the course of the time interval, becoming fractional or classical when it is more favourable. The variable-order derivatives are defined here using both the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. The existence, uniqueness, boundedness and positivity of the solutions are given. In order to test the rate for the detection of symptomatic infected people, two control variables are introduced. The optimality conditions are derived. The fractional-integer operator is approximated using Grünwald–Letnikov. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.

Keywords:

• Coronavirus diseases
• Grünwald-Letnikov finite difference method
• Optimality conditions
• vaccination
• variable-order FracInt

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 37N25
• 49J15
• 26A33
• 65M06
• 65L05

1. Introduction

Coronavirus has been designated by the world health organization (WHO) as a variant of serious concern. According to scientists, COVID-19 is more contagious compared to severe acute respiratory syndrome (SARS), because there are more routes of transmission and a large number of susceptible people. COVID-19 broke out worldwide at a rapid pace since the end of 2019, which had a tremendous impact on the lives of people around the world where it has claimed the lives of millions. Fighting the COVID-19 epidemic is difficult and expensive, and to control and prevent its spread of it, many governments have implemented very strict Epidemic prevention and control strategies, such as social distancing, self-isolation, medical quarantine, contact tracing, travel restrictions and city shutdown (WHO Int, 2020).

Fractional order mathematical models give more information about diseases (Ain, Anjum, & He, 2021; Ain et al., 2022; Ali, Alshammari, Islam, Khan, & Ullah, 2021; Anjum, He, & He, 2021; Arenas, Gonzàlez-Parra, & Chen-Charpentierc, 2016; Singh, 2020; Singh, Ganbari, Kumar, & Baleanu, 2021), the constant proportional Caputo fractional derivate (CPC) is proposed by Baleanu, Fernandez, and Akgül (2020) and Sweilam, Al-Mekhlafi, and Baleanu (2021). In epidemic applications constructed in terms of the variable order derivatives see the work by Samko, Kilbas, and Marichev (1993), Samko and Ross (1993), Sun, Chen, Wei, and Chen (2011), Sweilam, Al‐Mekhlafi, and Shatta (2020), Sweilam, Al-Mekhlafi, Mohammed, and Baleanu (2020) and Sweilam and Al-Mekhlafi (2016). Recently, Rosa and Torres (2021) proposed a system of differential equations, the derivatives order varies along the interval of time, also, it is classical or fractional. This idea has many advantages. One such system, a variable order fractional system, is referred to herein as the FractInt system, which has been found to be useful in controlling the disease. Where the derivate order of the fractional-integer (FractInt) system varies accordingly (1) $$\alpha (t)=\{\begin{array}{c}0<{\alpha }_0\le 1,t^{\prime }\ge t\ge 0,\hfill \\ 1,T_{\mathrm{f}}\ge t>t^{\prime }.\hfill \end{array}$$(1)

We paid special attention in this article to study the variable-order FracInt mathematical model of vaccination Covid-19 and their optimal control (Olivares and Staffetti, 2021). The variable-order derivatives are defined here using both the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. Moreover, the solution properties such as the existence, uniqueness, boundedness and positivity will be studied. We add two-control variables for the testing rate for detecting symptomatic infected individuals. The optimality conditions are derived. The FracInt operator is approximated using Caputo-proportional-constant Grünwald–Letnikov finite difference method (CPC-GLFDM).

To our knowledge, no numerical investigations of variable-order FracInt for COVID-19 vaccination mathematical model utilizing CPC-GLFDM have been conducted.

This article is organized as follows: Some notations and definitions of variable order fractional derivatives are introduced in Section 2. In Section 3, the model with a hybrid variable-order FracInt with control variables is presented, moreover, the positivity, boundedness, existence, uniqueness of the solutions and the stability are discussed. The numerical method CPC-GLFDM are constructed, moreover, the stability analysis for the proposed method is studied in Section 4. In Section 5, numerical simulations are presented. The conclusions are ultimately outlined in Section 6.

2. Basic definitions

Definition 2.1.

Riemann–Liouville’s left and right (L-R) sides integral (Podlubny,1999): (2) $${}_a{I}_t^{{}^{\alpha }}f(t)=[{\int }_a^{t}f(s){(t-s)}^{\alpha -1}ds]\frac{1}{\Gamma (\alpha )},t>a,$$(2) (3) $${}_t{I}_b^{{}^{\alpha }}f(t)=[{\int }_t^b{(t-s)}^{\alpha -1}f(s)ds]{(\Gamma (\alpha ))}^{-1},t<b,$$(3) where $-\infty <a<b<+\infty ,$ $\alpha \in \mathbb{C},\mathfrak{R}(\alpha )>0,$ $f(t)$ is a continuous function and $0<\alpha <1.$

Definition 2.2.

The right-side and the left-side Caputo’s derivatives of order $\alpha ,$ for a function $f(t),$ $f\in A{C}^n[a,b]$ are defined, respectively by (Podlubny,1999) (4) $$\begin{array}{c}{(}^C{D}_{b-}^{\alpha }f)(x)={(}_t^C{D}_b^{\alpha }f)(t)=\frac{{(-1)}^n}{\Gamma (n-\alpha )}{\int }_t^b\frac{f^n(s)}{{(s-t)}^{1-n+\alpha }}ds,t<b.\hfill \end{array}$$(4) (5) $$\begin{array}{c}{(}^C{D}_{a+}^{\alpha }f)(t)={(}_a^C{D}_t^{\alpha }f)(t)=\frac{1}{\Gamma (n-\alpha )}{\int }_a^t\frac{f^n(s)}{{(t-s)}^{1-n+\alpha }}ds,t>a,\hfill \end{array}$$(5) where $n=[\mathfrak{R}(\alpha )]+1,$ $\mathfrak{R}(\alpha )\notin {\mathbb{N}}_0,$ $-\infty <a<b<+\infty ,$ $\alpha \in \mathbb{C}.$

Moreover, if $0<\alpha <1,$ (Almeida & Torres, 2011): $${\int }_a^{b}g{(t)}_a^C{D}_t^{\alpha }f(t)dt={\int }_a^{b}f{(t)}_t{D}_b^{\alpha }g(t)dt+f(t)I_t^{1-\alpha }g(t){|}_a^b,$$ $${\int }_a^{b}g{(t)}_t^C{D}_b^{\alpha }f(t)dt={\int }_a^{b}f{(t)}_a{D}_t^{\alpha }g(t)dt-f(t)I_t^{1-\alpha }g(t){|}_a^b.$$

Proposition 2.1

(Podlubny, 1999). Let L-R sides Caputo’s and Riemann–Liouville’s fractional derivatives of $f(t),$ given respectively ${}_a^C{D}_t^{{}^{\alpha }}f,{}_t^C{D}_b^{{}^{\alpha }}f,$ ${}_a{D}_t^{{}^{\alpha }}f,{}_t{D}_b^{{}^{\alpha }}f$ then (6) $${}_a{D}_t^{{}^{\alpha }}f(t)=\sum _{k=0}^{n-1}\frac{{(t-a)}^{(k-\alpha )}}{\Gamma (k-\alpha +1)}{f}^k(a){+}_a^C{D}_t^{{}^{\alpha }}f(t),$$(6) (7) $${}_t{D}_b^{{}^{\alpha }}f(t)={}_t^C{D}_b^{{}^{\alpha }}f(t)+\sum _{k=0}^{n-1}\frac{{(b-t)}^{(k-\alpha )}}{\Gamma (k-\alpha +1)}{f}^k(b),$$(7) for $n=[\mathfrak{R}(\alpha )]+1,$ $\alpha \notin \mathbb{N}.$ We can prove that $${}_a{D}_t^{{}^{\alpha }}f(t)={}_a^C{D}_t^{{}^{\alpha }}f(t),$$ if $$f(s)=f^{\prime }(s)=f^{″}(s)=\dots =f^{n-1}(s)=0,$$ where s = a or b, as in Definition 2.2.

Definition 2.3.

The variable order hybrid operator CPC (Baleanu et al., 2020): (8) $$\begin{array}{c}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)=({\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1(\alpha (t))+y^{\prime }(s)G_0(\alpha (t)))ds){(\Gamma (1-\alpha (t)))}^{-1}\hfill \\ =G_1{(\alpha (t))}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \end{array}$$(8) where, $G_0=Q^{(-\alpha (t)+1)}\alpha (t),$ $G_1=(-\alpha (t)+1)Q^{\alpha (t)}$, Q is a constant.

On the other side, we can define (Baleanu et al., 2020): (9) $$\begin{array}{c}{}_0^{\text{CPC}}{I}_t^{\alpha (t)}y(t)=({\int }_0^{t}exp{[G_1(\alpha (t)){(G_0(\alpha (t)))}^{-1}(t-s)]}_0^{RL}{D}_t^{1-\alpha (t)}y(s)ds)\frac{1}{G_0(\alpha (t))},\hfill \end{array}$$(9) where, ${}_0^{\text{CPC}}{I}_t^{\alpha (t)}y(t)$ is the CPC inverse operator.

3. Mathematical model of FractInt variable order

FractInt variable order COVID-19 model (Olivares and Staffetti, 2021) is developed here. Also, two control variables $v_1$ and $v_2$ are added; $v_1$ denotes the vaccination rate, ranging from 0 to $v_{1\mathrm{u}},$ where $v_{1\mathrm{u}}$ is an upper bound to be provided by the designers of the optimal control model. Therefore, the vaccination policy is represented in the model by the term $\varphi v_{1\mathrm{u}}{S}_t,$ where the parameter $\mathit{\varphi }$ represents the vaccine efficacy level. $v_2$ denotes the test rate for the detecting of asymptomatic infected people, which ranges from $v_{\text{2L}}$ to $v_{\text{2u}},$ where $v_{\text{2L}}$ and $v_{\text{2u}}$ represent a lower and an upper bound, respectively, be provided by the designers of the optimal control model. A new parameter $\sigma$ is presented in order to be consistent with the physical model problem. Moreover, we avoid dimensional mismatches by modifying the fractional model with an auxiliary parameter $\sigma .$ As a result, the left side possesses the dimension ${\text{day}}^{-1}$ (Ullah & Baleanu, 2021). (10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) subject to the following positive values: (11) $$\begin{array}{cc}S(0)=s_0,I(0)=i_0,D(0)=d_0,A(0)=a_0,\hfill & \hfill \\ R(0)=r_0,T(0)=t_{u0},H(0)=h_0,E(0)=e_0.\hfill & \hfill \end{array}$$(11) for more details on the variables meaning of the proposed model see Table 1.

Table 1. All variables definitions of Eq. (10(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) ).

Variable	Interpretation
R	The rate of detected symptomatic infected people.
I	The rate of pauci-symptomatic or asymptomatic infected people who go undetected
E	The rate of deaths
S	The rate of people susceptible to infection
A	The rate of symptomatic infections that go undetected
H	The rate of people who are infected but recover, or susceptible people who become immune through vaccination
T	The rate of infected population with life-threatening symptoms that are diagnosed
D	The rate of detected asymptomatic infected people.

By adding all equations of (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) , boundedness of the proposed model solution can be verified as follows: (12) $${}_0^{\text{CPC}}{D}_t^{\alpha (t)}{N}_H(t)=0,N_H(0)=A\ge 0,$$(12) where, $N_H$ is the total population summation and A is a constant. The solution of Eq. (12)(12) $${}_0^{\text{CPC}}{D}_t^{\alpha (t)}{N}_H(t)=0,N_H(0)=A\ge 0,$$(12) is given as follows: (13) $$N_H(t)\ge e^{\left(\frac{-G_{1(\alpha (t))}}{G_{0(\alpha (t))}}\right)t}A\ge 0.$$(13)

At $t\to \infty ,$ solutions of Eq. (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) are bounded.

Lemma 3.1.

All solutions of Eq. (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) are nonnegative under conditions (11)(11) $$\begin{array}{cc}S(0)=s_0,I(0)=i_0,D(0)=d_0,A(0)=a_0,\hfill & \hfill \\ R(0)=r_0,T(0)=t_{u0},H(0)=h_0,E(0)=e_0.\hfill & \hfill \end{array}$$(11) for $t\ge 0$ (Lin 2007).

Proof.

Using Eq. (11)(11) $$\begin{array}{cc}S(0)=s_0,I(0)=i_0,D(0)=d_0,A(0)=a_0,\hfill & \hfill \\ R(0)=r_0,T(0)=t_{u0},H(0)=h_0,E(0)=e_0.\hfill & \hfill \end{array}$$(11) , we have: (14) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}S|{}_{S=0}=\iota H\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}I|{}_{I=0}=S(\beta D+\gamma A+\delta R)\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D|{}_{D=0}=v_{2}I\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A|{}_{A=0}=\zeta I\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R|{}_{R=0}=\eta D+\theta A\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T|{}_{T=0}=A\mu +R\nu \ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H{|{}_{H=0}}_{}=I\lambda +D\rho +A\kappa +R\xi +T\sigma -\iota H+\varphi v_{1}S\ge 0,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E|{}_{E=0}=\tau T\ge 0.\hfill & \hfill \end{array}$$(14)

□

3.1. Existence and uniqueness

System (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) can be written as follows (15) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\varrho (y(t),t),y_0=y(0)\ge 0,\hfill \end{array}$$(15) where, the state variables are ${(A,I,D,S,T,H,R,E)}^{Tr}=y(t),$ and $\varrho$ is defined as follow

$$\left(\begin{array}{c}c{\varrho }_1\\ {\varrho }_2\\ {\varrho }_3\\ {\varrho }_4\\ {\varrho }_5\\ {\varrho }_6\\ {\varrho }_7\\ {\varrho }_8\end{array}\right)=\left(\begin{array}{c}c(-S(\alpha I+\beta D+\gamma A+\delta R)+\iota H-\varphi v_{1}S){\sigma }^{1-\alpha (t)}\\ (S(\delta R+\beta D+\alpha I+\gamma A)-(\lambda +v_2+\zeta )I){\sigma }^{1-\alpha (t)}\\ (-(\rho +\eta )D+v_{2}I){\sigma }^{1-\alpha (t)}\\ (-(\kappa +\theta +\mu )A+\zeta I){\sigma }^{1-\alpha (t)}\\ (\theta A+\eta D-(\xi +R\nu )){\sigma }^{1-\alpha (t)}\\ (-(\tau +\sigma )T+\nu R+\mu A){\sigma }^{1-\alpha (t)}\\ (\rho D+\xi R+\lambda I+\kappa A+\sigma T-\iota H+\varphi v_{1}S){\sigma }^{1-\alpha (t)}\\ (\tau T){\sigma }^{1-\alpha (t)}\end{array}\right),$$ with an initial condition ${ϵ}_0.$ Furthermore, Lipschitz requirement holds (Bonyah, Sagoe, Kumar, & Deniz, 2021): (16) $$\begin{array}{cc}\left|\right|\varrho ({ϵ}_1(t),t)-\varrho ({ϵ}_2(t),t)\left|\right|\hfill & \le G^0\left|\right|{ϵ}_1(t)-{ϵ}_2(t)\left|\right|.\hfill \end{array}$$(16)

Theorem 3.2.

If the following condition holds, (17) $$(G^0{A}_{\mathit{max}}^{\alpha (t)}{X}_{\mathit{max}}^{\alpha (t)}){(\Gamma (\alpha (t)-1)Q_0(\alpha (t)))}^{-1}<1,$$(17) the hybrid model (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) has a unique solution

Proof.

Using Eqs. (8)(8) $$\begin{array}{c}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)=({\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1(\alpha (t))+y^{\prime }(s)G_0(\alpha (t)))ds){(\Gamma (1-\alpha (t)))}^{-1}\hfill \\ =G_1{(\alpha (t))}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \end{array}$$(8) and (15)(15) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\varrho (y(t),t),y_0=y(0)\ge 0,\hfill \end{array}$$(15) , we have (18) $$\begin{array}{c}ϵ(t)=ϵ(t_0)+\frac{1}{Q_0(\alpha (t))}{\int }_0^{t}exp{(-\frac{G_1(\alpha (t))}{G_0(\alpha (t))}(t-s))}_0^{RL}{D}_t^{1-\alpha (t)}\varrho (ϵ(s),s)ds.\hfill \end{array}$$(18)

Let $B:C(Q,{\mathbb{R}}^8)\to C(Q,{\mathbb{R}}^8)$ and $Q=(0,T)$ achieved that: (19) $$\begin{array}{cc}B[ϵ(t)]\hfill & =ϵ(t_0)+\frac{1}{G_0(\alpha (t))}{\int }_0^{t}exp{(-\frac{G_1(\alpha (t))}{G_0(\alpha (t))}(t-s))}_0^{RL}{D}_t^{1-\alpha (t)}\varrho (ϵ(s),s)ds.\hfill \end{array}$$(19)

We have: $$B[ϵ(t)]=ϵ(t).$$

The supremum norm on Q is represented by $\left|\right|.|{|}_Q.$ Thus $$\left|\right|ϵ(t)|{|}_Q=\underset{t\in Q}{\sup }||ϵ(t)||,ϵ(t)\in C(Q,{\mathbb{R}}^8).$$

Then, $\left|\right|.|{|}_Q$ with $C(Q,{\mathbb{R}}^8):$ $$\Lambda \left|\right|\phi (s,t)\left|{|}_Q\right||ϵ(s)|{|}_Q\ge \left|\right|{\int }_0^t\varrho (s,t)\epsilon (s)ds\left|\right|,$$ with $\varrho (s,t)\in C(Q^2,{\mathbb{R}}^9)$ $ϵ(t)\in C(Q,{\mathbb{R}}^9),$ achieved that: $$\underset{t,s\in Q}{\sup }|\phi (s,t)|=\left|\right|\phi (s,t)|{|}_Q.$$

Also, relation (19)(19) $$\begin{array}{cc}B[ϵ(t)]\hfill & =ϵ(t_0)+\frac{1}{G_0(\alpha (t))}{\int }_0^{t}exp{(-\frac{G_1(\alpha (t))}{G_0(\alpha (t))}(t-s))}_0^{RL}{D}_t^{1-\alpha (t)}\varrho (ϵ(s),s)ds.\hfill \end{array}$$(19) can be written as (20) $$\begin{array}{cc}\left|\right|B[{ϵ}_1(t)]-B[{ϵ}_2(t)]|{|}_Q\le ||\frac{1}{G_0(\alpha (t))}{\int }_0^{t}exp(-\frac{G_1(\alpha (t))}{G_0(\alpha (t))}(t-s)){(}_0^{RL}{D}_t^{1-\alpha (t)}\varrho ({ϵ}_1(s),s)\hfill & \hfill \\ {-}_0^{RL}{D}_t^{1-\alpha (t)}\varrho ({ϵ}_2(s),s))ds|{|}_Q.\hfill & \hfill \\ \le \frac{A_{\max }^{\alpha (t)}(\Gamma {(\alpha (t)-1)}^{-1})}{G_0(\alpha (t))}||{\int }_0^t{(t-s)}^{\alpha (t)-2}(\varrho ({ϵ}_1(s),s)-\varrho ({ϵ}_2(s),s))ds|{|}_Q,\hfill & \hfill \\ \le \frac{A_{\max }^{\alpha (t)}{X}_{\max }^{\alpha (t)}(\Gamma {(\alpha (t)-1)}^{-1})}{G_0(\alpha (t))}\left|\right|\varrho ({ϵ}_1(t),t)-\varrho ({ϵ}_2(t),t)|{|}_Q,\hfill & \hfill \\ \le \frac{G^0{A}_{\max }^{\alpha (t)}{X}_{\max }^{\alpha (t)}(\Gamma {(\alpha (t)-1)}^{-1})}{G_0(\alpha (t))}\left|\right|{ϵ}_1(t)-{ϵ}_2(t)|{|}_Q.\hfill & \hfill \end{array}$$(20)

Then (21) $$\left|\right|B[{ϵ}_1(t)]-B[{ϵ}_2(t)]|{|}_Q\le L||{ϵ}_1(t)-{ϵ}_2(t)|{|}_Q,$$(21) where $$L=\frac{G^0{A}_{\max }^{\alpha (t)}{X}_{\max }^{\alpha (t)}}{G_0(\alpha (t))\Gamma (\alpha (t)-1)}.$$

So Eq. (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) has a unique solution. □

4. The control problem

Let, the admissible control set $$\Omega =\{(v_1(.),v_2(.))|v_1,v_2(.)\text{\hspace{0.17em}\hspace{0.17em}are\hspace{0.17em}\hspace{0.17em}Lebsegue\hspace{0.17em}\hspace{0.17em}measurable\hspace{0.17em}\hspace{0.17em}on\hspace{0.17em}\hspace{0.17em}}[0,1],$$ $$0\le v_1(.),v_2(.)\le 1,\forall t\in [0,T_f]\},$$

The objective functional: (22) $$\begin{array}{c}J(v_1,v_2)={\int }_0^{T_f}(C_1(T(t)+E(t))+C_2{v}_1^2(t)+C_3{v}_2^2(t))dt,\hfill \end{array}$$(22) $C_1,$ $C_2,$ and $C_3$ are the weight constants. The control problem is find $v_1(t),$ $v_2(t)$ such that (23) $$\begin{array}{c}J(v_1,v_2)={\int }_0^{T_f}\eta (t,A,I,S,R,D,T,E,H,v_1,v_2)dt,\hfill \end{array}$$(23) is minimum. It is subject to the constraints (24) $$\begin{array}{c}{}_a^{\text{CPC}}{D}_t^{\alpha (t)}{\Psi }_j={\xi }_i.\hfill \end{array}$$(24)

Where $$\begin{array}{cc}{\xi }_i={\xi }_i(t,H,T,I,D,S,R,E,A,v_1,v_2),i,j=1,\dots ,8,\hfill & \hfill \\ {\Psi }_j=\{H,T,I,D,S,R,E,A,v_1,v_2\},\hfill & \hfill \end{array}$$ ${\Psi }_1(0)=S_0,$ ${\Psi }_2(0)=I_0,$ ${\Psi }_3(0)=D_0,$ ${\Psi }_4(0)=A_0,$ ${\Psi }_5(0)=R_0,$

${\Psi }_6(0)=T_0,$ ${\Psi }_1(0)=H_0,$ ${\Psi }_1(0)=E_0,$

In the following using (Agrawal, 2008; Sweilam, Al-Mekhlafi, Alshomrani, & Baleanu, 2020; Sweilam, Al-Mekhlafi, Mohammed, et al., 2020; Sweilam, Al-Mekhlafi, Albalawi, & Tenreiro Machado, 2021) we extend numerically optimality conditions

We define the Hamiltonian: (25) $$\begin{array}{c}H(t,D,A,I,S,R,T,E,H,v_1,v_2{\lambda }_i)=\eta (t,D,S,I,A,T,R,E,H,v_1,v_2,{\lambda }_i)\\ +\sum _{i=1}^7{\lambda }_i{\xi }_i(t,D,A,I,S,R,T,E,H,v_1,v_2).\end{array}$$(25)

The necessary conditions of optimal control as in (Agrawal, 2008; Sweilam, Al-Mekhlafi, Alshomrani et al., 2020; Sweilam, Al-Mekhlafi, Mohammed et al., 2020; Sweilam et al., 2021) are (26) $${}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_{\iota }=\frac{\partial H}{\partial {\vartheta }_{\iota }},\iota =1,\dots ,8,$$(26) where, $${\vartheta }_{\iota }=\{t,S,I,D,A,R,T,H,E,v_1,v_2,\iota =1,\dots ,8\},$$ (27) $$0=\frac{\partial H}{\partial u_s},s=1,2,$$(27) (28) $${}_0^{\text{CPC}}{D}_t^{\alpha (t)}{\vartheta }_{\iota }=\frac{\partial H}{\partial {\lambda }_{\iota }},\iota =1,\dots ,8,$$(28) (29) $${\lambda }_{\iota }(T_f)=0,\iota =1,2,\dots ,8.$$(29)

Now, there exists $v_1^{\ast },$ $v_2^{\ast },$ that minimizes $J(v_1,v_2)$ over $\Omega$ with $I^{\ast },$ $E^{\ast },$ $D^{\ast },$ $A^{\ast },$ $S^{\ast },$ $R^{\ast },$ $T^{\ast },$ $H^{\ast }.$ Furthermore, there exists adjoint variables ${\lambda }_i,$ $i=1,2,3,\dots ,8,$ satisfy the following

1. Adjoint equations:

$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_1=(-{\lambda }_1\alpha I^{\ast }-{\lambda }_1\beta D-{\lambda }_1\gamma A^{\ast }-{\lambda }_1\delta R^{\ast }-{\lambda }_1\varphi v_1+{\lambda }_2\alpha I\hfill \\ +{\lambda }_1\beta D+{\lambda }_2\gamma A^{\ast }+{\lambda }_2\delta R^{\ast }+{\lambda }_7\varphi v_1,),\hfill & \hfill \end{array}$$$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_2=(-{\lambda }_1\alpha S^{\ast }+{\lambda }_2\alpha S^{\ast }-{\lambda }_2(v_2+\zeta +\lambda )+{\lambda }_3{v}_2+{\lambda }_4\zeta +{\lambda }_7\lambda ),\hfill \end{array}$$$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_3=(-{\lambda }_1\beta S^{\ast }+{\lambda }_2\beta S^{\ast }-{\lambda }_3(\eta +\rho )+{\lambda }_5\eta +{\lambda }_7\rho ),\hfill \end{array}$$$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_4=(-{\lambda }_1\gamma S+{\lambda }_2\gamma S-(\theta +\mu +\kappa ){\lambda }_4+\theta {\lambda }_5+\mu {\lambda }_6+\kappa {\lambda }_7),\hfill \end{array}$$(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_6=(c_1-{\lambda }_6(\sigma +\tau )+{\lambda }_7(\sigma )+{\lambda }_8(\tau )),\hfill \end{array}$$$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_7=(\iota {\lambda }_1-\iota {\lambda }_7),\hfill \end{array}$$$$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_8=(c_1),\hfill \end{array}$$(31) $$\begin{array}{c}{\lambda }_{\iota }(T_f)=0,\iota =1,2,\dots ,8,\hfill \end{array}$$(31)

i.e., this is transversality conditions.

• Optimality conditions: (32) $$\begin{array}{c}H(t,D,R,I,H,S,E,A,T,v_1,v_2,\lambda )=\\ \underset{0\le u_1,u_2\le 1}{\min }H(t,D,R,I,H,S,E,A,T,v_1,v_2,\lambda ).\end{array}$$(32)

Moreover: (33) $$v_1^{\ast }=\min \{1,\max \{0,\frac{{\varphi }^{\ast }{S}^{\ast }({\lambda }_1-{\lambda }_7)}{2C_2}\left\}\right\},$$(33) (34) $$v_2^{\ast }=\min \{1,\max \{0,\frac{I^{\ast }({\lambda }_2-{\lambda }_3)}{2C_3}\left\}\right\},$$(34)

Equations (30)(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) can be obtained from Eq. (26)(26) $${}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_{\iota }=\frac{\partial H}{\partial {\vartheta }_{\iota }},\iota =1,\dots ,8,$$(26) , where (35) $$\begin{array}{cc}H={\lambda }_1^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{S}^{\ast }+{\lambda }_2^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{I}^{\ast }+{\lambda }_3^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{D}^{\ast }\hfill & \hfill \\ +{\lambda }_4^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{A}^{\ast }+{\lambda }_5^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{R}^{\ast }+{\lambda }_6^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{T}^{\ast }+{\lambda }_7^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{H}^{\ast }\hfill \\ +{\lambda }_8^{\text{CPC}}{}_0{D}_t^{\alpha (t)}{E}^{\ast }+(C_1(T{(t)}^{\ast }+E{(t)}^{\ast })+C_2{v}_1^{2\ast }(t)+C_3{v}_2^{2\ast }(t)),\hfill & \hfill \end{array}$$(35)

is the Hamiltonian. ${\lambda }_s(T_f)=0,$ $s=1,\dots ,8.$ We can obtain Eqs. (33)(33) $$v_1^{\ast }=\min \{1,\max \{0,\frac{{\varphi }^{\ast }{S}^{\ast }({\lambda }_1-{\lambda }_7)}{2C_2}\left\}\right\},$$(33) and (34)(34) $$v_2^{\ast }=\min \{1,\max \{0,\frac{I^{\ast }({\lambda }_2-{\lambda }_3)}{2C_3}\left\}\right\},$$(34) from Eq. (32)(32) $$\begin{array}{c}H(t,D,R,I,H,S,E,A,T,v_1,v_2,\lambda )=\\ \underset{0\le u_1,u_2\le 1}{\min }H(t,D,R,I,H,S,E,A,T,v_1,v_2,\lambda ).\end{array}$$(32) .

We rewrite Eq. (10)(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) using Eqs. (33)(33) $$v_1^{\ast }=\min \{1,\max \{0,\frac{{\varphi }^{\ast }{S}^{\ast }({\lambda }_1-{\lambda }_7)}{2C_2}\left\}\right\},$$(33) and (34)(34) $$v_2^{\ast }=\min \{1,\max \{0,\frac{I^{\ast }({\lambda }_2-{\lambda }_3)}{2C_3}\left\}\right\},$$(34) (36) $$\begin{array}{cc}{\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{S}^{\ast }=-(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )S+\iota H^{\ast }-\varphi v_1^{\ast }{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{I}^{\ast }=S^{\ast }(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )-(v_2^{\ast }+\zeta +\lambda )I^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{D}^{\ast }=I^{\ast }{v}_2^{\ast }-(\rho +\eta )D^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{A}^{\ast }=I^{\ast }\zeta -A^{\ast }(\theta +\kappa +\mu ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{R}^{\ast }=D^{\ast }\eta +A^{\ast }\theta -(R^{\ast }\nu +\xi ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{T}^{\ast }=A\mu +R^{\ast }\nu -(\tau +\sigma )T^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{H}^{\ast }=I^{\ast }\lambda +D^{\ast }\rho +A^{\ast }\kappa +R^{\ast }\xi +T^{\ast }\sigma -\iota H^{\ast }+\varphi v_1{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{E}^{\ast }=\tau T^{\ast }.\hfill & \hfill \end{array}$$(36)

5. Numerical method for solving FracInt optimality systems

5.1. CPC-GLFDM

Let $1\ge \alpha (t)>0,$ (37) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\xi (t,y(t)),y_0=y(0).\hfill \end{array}$$(37)

Using Eq. (8)(8) $$\begin{array}{c}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)=({\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1(\alpha (t))+y^{\prime }(s)G_0(\alpha (t)))ds){(\Gamma (1-\alpha (t)))}^{-1}\hfill \\ =G_1{(\alpha (t))}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \end{array}$$(8) we have: (38) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\frac{1}{\Gamma (1-\alpha (t))}{\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1+y^{\prime }(s)G_0)ds,\hfill \\ =G_1{(\alpha )}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \\ =G_1{(\alpha )}_0^{RL}{D}_t^{\alpha (t)-1}y(t)+G0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t).\hfill & \hfill \end{array}$$(38)

Now by using GL-approximation, we have: (39) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t){|}_{t=t^n}\hfill & =\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})\hfill \\ +\frac{G_0(\alpha (t_n))}{{\tau }^{{\alpha }_n}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0),\hfill & \hfill \end{array}$$(39) (40) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{1-i+n}-y_0{q}_{n+1})\\ =\xi (y(t_n),t_n),\end{array}$$(40) ${\omega }_0=1,$ ${\omega }_i=(1-\frac{\alpha (t_n)}{i}){\omega }_{i-1},$ $t^n=n\tau , \tau =\frac{T_f}{N_n},$ $N_n$ is a natural number. $q_i=\frac{i^{\alpha (t_n)}}{\Gamma (1-\alpha (t_n))}$ $i=1,2,\dots ,n+1,$ ${\mu }_i=({}_i^{\alpha (t_n)}){(-1)}^{i-1},$ ${\mu }_1=\alpha (t_n),$ Also, let us assume that (Scherer, Kalla, Tang, & Huang, 2011): $$0<{\mu }_{i+1}<{\mu }_i<\dots <{\mu }_1=\alpha (t_n)<1,$$ $$0<q_{i+1}<q_i<\dots <q_1=\frac{1}{\Gamma (-\alpha (t_n)+1)}.$$

Equation (40)(40) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{1-i+n}-y_0{q}_{n+1})\\ =\xi (y(t_n),t_n),\end{array}$$(40) is used to solve Eq. (30)(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) . Moreover, using the same technique with backward in time, Eq. (36)(36) $$\begin{array}{cc}{\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{S}^{\ast }=-(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )S+\iota H^{\ast }-\varphi v_1^{\ast }{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{I}^{\ast }=S^{\ast }(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )-(v_2^{\ast }+\zeta +\lambda )I^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{D}^{\ast }=I^{\ast }{v}_2^{\ast }-(\rho +\eta )D^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{A}^{\ast }=I^{\ast }\zeta -A^{\ast }(\theta +\kappa +\mu ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{R}^{\ast }=D^{\ast }\eta +A^{\ast }\theta -(R^{\ast }\nu +\xi ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{T}^{\ast }=A\mu +R^{\ast }\nu -(\tau +\sigma )T^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{H}^{\ast }=I^{\ast }\lambda +D^{\ast }\rho +A^{\ast }\kappa +R^{\ast }\xi +T^{\ast }\sigma -\iota H^{\ast }+\varphi v_1{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{E}^{\ast }=\tau T^{\ast }.\hfill & \hfill \end{array}$$(36) with transitivity condition can be discretized.

Remark 1.

If $G_0(\alpha (t))=1$ and $G_1(\alpha (t))=0$ in (40)(40) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{1-i+n}-y_0{q}_{n+1})\\ =\xi (y(t_n),t_n),\end{array}$$(40) , this is the disceritization of Caputo opretor using finite differance method (C-GLFDM).

5.2. CPC-GLFDM stability analysis

Consider the following test problem: (41) $${(}_0^{\text{CPC}}{D}_t^{\alpha (t)})y(t)=Ay(t),0<\alpha (t)\le 1,t>0,A<0.$$(41)

Thanks to Eq. (38)(38) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\frac{1}{\Gamma (1-\alpha (t))}{\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1+y^{\prime }(s)G_0)ds,\hfill \\ =G_1{(\alpha )}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \\ =G_1{(\alpha )}_0^{RL}{D}_t^{\alpha (t)-1}y(t)+G0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t).\hfill & \hfill \end{array}$$(38) then, Eq. (41)(41) $${(}_0^{\text{CPC}}{D}_t^{\alpha (t)})y(t)=Ay(t),0<\alpha (t)\le 1,t>0,A<0.$$(41) can be discretized as follows: (42) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0)=A{y}_n,\hfill \end{array}$$(42) put $C=\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}},$ $B=\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}.$ Then from boundness theorem (Arenas et al., 2016) we have: (43) $$y_{n+1}=\frac{1}{C+B}(A{y}_n-C\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i}+B(\sum _{i=1}^{n+1}{\mu }_i{y}_{n+1-i}+q_{n+1}{y}_0))\le y_n,$$(43) then we have: $y_1<y_0$ and $y_0\ge y_1\ge \dots \ge y_{n-1}\ge y_n\ge y_{n+1}.$

Now by discretize Eq. (38)(38) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t)\hfill & =\frac{1}{\Gamma (1-\alpha (t))}{\int }_0^t{(t-s)}^{-\alpha (t)}(y(s)G_1+y^{\prime }(s)G_0)ds,\hfill \\ =G_1{(\alpha )}_0^{RL}{I}_t^{1-\alpha (t)}y(t)+G_0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t),\hfill & \hfill \\ =G_1{(\alpha )}_0^{RL}{D}_t^{\alpha (t)-1}y(t)+G0{(\alpha (t))}_0^C{D}_t^{\alpha (t)}y(t).\hfill & \hfill \end{array}$$(38) : (44) $$\begin{array}{cc}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}y(t){|}_{t=t^n}\hfill & =\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{-i+n+1})\hfill \\ +{\tau }^{-{\alpha }_n}{G}_0(\alpha (t_n))(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0),\hfill & \hfill \end{array}$$(44) (45) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0)\\ -\xi (y(t_n),t_n)=T_{R_n},\end{array}$$(45) where, $$\Vert T_{R_n}{\Vert }_{\infty }<M,M=C\underset{0\le i\le n+1}{\max }\left|y_{i+1}\right|,$$ $$C=({\tau }^{\alpha (t_i)-1}+{\tau }^{\alpha (t_i)}).$$

The proposed method is convergent because it is consistent and stable (Yuste & Quintana-Murillo, 2012).

6. Numerical experiments

This section examines the simulations of optimality systems Eqs. (30)(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) and (36)(36) $$\begin{array}{cc}{\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{S}^{\ast }=-(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )S+\iota H^{\ast }-\varphi v_1^{\ast }{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{I}^{\ast }=S^{\ast }(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )-(v_2^{\ast }+\zeta +\lambda )I^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{D}^{\ast }=I^{\ast }{v}_2^{\ast }-(\rho +\eta )D^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{A}^{\ast }=I^{\ast }\zeta -A^{\ast }(\theta +\kappa +\mu ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{R}^{\ast }=D^{\ast }\eta +A^{\ast }\theta -(R^{\ast }\nu +\xi ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{T}^{\ast }=A\mu +R^{\ast }\nu -(\tau +\sigma )T^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{H}^{\ast }=I^{\ast }\lambda +D^{\ast }\rho +A^{\ast }\kappa +R^{\ast }\xi +T^{\ast }\sigma -\iota H^{\ast }+\varphi v_1{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{E}^{\ast }=\tau T^{\ast }.\hfill & \hfill \end{array}$$(36) . The parameter values in Table 2 are used with the following initial conditions: $I(0)=0.0008,$ $S(0)=0.8474,$ $D(0)=0.0008,$ $A(0)=,$ $R(0)=0.00017,$ $H(0)=0.1489,$ $E(0)=0.0017,$ $T(0)=0.00003,$ $A(0)=0.0002,$ and transversality conditions ${\lambda }_i(T_f)=0,i=1,2,\dots ,8.$ CPC-GLNFDM (40)(40) $$\begin{array}{c}\frac{G_1(\alpha (t_n))}{{\tau }^{\alpha (t_n)-1}}(y_{n+1}+\sum _{i=1}^{n+1}{\omega }_i{y}_{n+1-i})+\frac{G_0(\alpha (t_n))}{{\tau }^{\alpha (t_n)}}(y_{n+1}-\sum _{i=1}^{n+1}{\mu }_i{y}_{1-i+n}-y_0{q}_{n+1})\\ =\xi (y(t_n),t_n),\end{array}$$(40) are presented to study Eqs. (30)(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) with (31)(31) $$\begin{array}{c}{\lambda }_{\iota }(T_f)=0,\iota =1,2,\dots ,8,\hfill \end{array}$$(31) . Equation (36)(36) $$\begin{array}{cc}{\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{S}^{\ast }=-(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )S+\iota H^{\ast }-\varphi v_1^{\ast }{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{I}^{\ast }=S^{\ast }(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )-(v_2^{\ast }+\zeta +\lambda )I^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{D}^{\ast }=I^{\ast }{v}_2^{\ast }-(\rho +\eta )D^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{A}^{\ast }=I^{\ast }\zeta -A^{\ast }(\theta +\kappa +\mu ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{R}^{\ast }=D^{\ast }\eta +A^{\ast }\theta -(R^{\ast }\nu +\xi ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{T}^{\ast }=A\mu +R^{\ast }\nu -(\tau +\sigma )T^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{H}^{\ast }=I^{\ast }\lambda +D^{\ast }\rho +A^{\ast }\kappa +R^{\ast }\xi +T^{\ast }\sigma -\iota H^{\ast }+\varphi v_1{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{E}^{\ast }=\tau T^{\ast }.\hfill & \hfill \end{array}$$(36) will be discretized using the same technique and backward in time, then we solve algebraic system of equations. The numerical results of the optimality systems (30)(30) $$\begin{array}{c}{\sigma }^{\alpha (t)-1}{}_t^{\text{CPC}}{D}_{t_f}^{\alpha (t)}{\lambda }_5=(-{\lambda }_1\delta S+{\lambda }_2\delta S-{\lambda }_5(\nu +\xi )+{\lambda }_6\nu +{\lambda }_6\xi ),\hfill \end{array}$$(30) and (36)(36) $$\begin{array}{cc}{\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{S}^{\ast }=-(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )S+\iota H^{\ast }-\varphi v_1^{\ast }{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{I}^{\ast }=S^{\ast }(I^{\ast }\alpha +D^{\ast }\beta +A^{\ast }\gamma +R^{\ast }\delta )-(v_2^{\ast }+\zeta +\lambda )I^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}{D}^{\ast }=I^{\ast }{v}_2^{\ast }-(\rho +\eta )D^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{A}^{\ast }=I^{\ast }\zeta -A^{\ast }(\theta +\kappa +\mu ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{R}^{\ast }=D^{\ast }\eta +A^{\ast }\theta -(R^{\ast }\nu +\xi ),\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{T}^{\ast }=A\mu +R^{\ast }\nu -(\tau +\sigma )T^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{H}^{\ast }=I^{\ast }\lambda +D^{\ast }\rho +A^{\ast }\kappa +R^{\ast }\xi +T^{\ast }\sigma -\iota H^{\ast }+\varphi v_1{S}^{\ast },\hfill & \hfill \\ {\frac{1}{{\sigma }^{1-\alpha (t)}}}_0^{\text{CPC}}{D}_t^{\alpha (t)}{E}^{\ast }=\tau T^{\ast }.\hfill & \hfill \end{array}$$(36) are shown graphically at different values of $0<{\alpha }_0\le 1$ if $0\le t\le t^{\prime },$ and ${\alpha }_1=1$ if $t^{\prime }<t\le T_f$ in Figures 1–5. Figure 1 shows the behavior of the solution of the proposed model before and after the optimal control when ${\alpha }_0=0.97,$ if $0\le t\le 10,$ ${\alpha }_1=1$ if $10<t\le 350.$ We found that, in the controlled case, the detected infected people with life-threatening symptoms is lower. Also, the number of deceased people are less in the controlled case. This has enabled us to reduce the number of confirmed infections with life-threatening symptoms and the number of deaths. Our results in Figure 2 show that controlling Covid-19 infection through optimal control is effective and that the FracInt model is the best one. Figure 3 shows the behavior of the control variables at variable-order FracInt $\alpha (t)$ when ${\alpha }_0=0.98,$ if $0\le t\le 10,$ ${\alpha }_1=1$ if $10<t\le 350$ and classical order when $\alpha =1$ and $\alpha =\mathrm{0.98.}$ Also, we noted the variable-order FracInt model is the best one. Figures 4 and 5 show how the behavior of the solutions in controlled case varies with the values of $\alpha (t)$ where $0<{\alpha }_0\le 1$ if $0\le t\le 10,$ ${\alpha }_1=1$ if $10<t\le 150.$ Tables 3 and 4 show the objective functional values in FracInt case and the classical case. Clear that values of objective functional 22 in case of the variable-order FracInt are better than the classical one.

Figure 1. Comparison of the solutions behivor with and without control using CPC-GLFDM.

Figure 2. CPC-GLFDM simulations at different $\alpha (t)$ in controlled case.

Figure 3. Approximations of $v_1,v_2$ using the method CPC-GLFDM.

Figure 4. Controlled soluations behivors at different $\alpha (t)$ of the proposed model by CPC-GLFDM.

Figure 5. Approximations of $v_1,v_2$ using the method CPC-GLFDM.

Table 2. System (10(10) $$\begin{array}{cc}{\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}S=-S(I\alpha +D\beta +A\gamma +R\delta )+H\iota -v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\mathit{CPC}}{D}_t^{\alpha (t)}I=S(\gamma A+\beta D+\alpha I+\delta R)-(\lambda +v_2+\zeta )I,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}D=v_{2}I-(\eta +\rho )D,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}A=I\zeta -(\mu +\kappa +\theta +)A,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}R=\eta D+\theta A-(R\nu +\xi ),\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}T=A\mu +\nu R-(\tau +\sigma )T,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}H=I\lambda +D\rho +A\kappa +\xi R+\sigma T-H\iota +v_1\varphi S,\hfill & \hfill \\ {\sigma }^{\alpha (t)-1}{}_0^{\text{CPC}}{D}_t^{\alpha (t)}E=T\tau ,\hfill & \hfill \end{array}$$(10) ) parameters.

Parameter	Definition	Value	Unit
$\alpha$	Transition contagious exposed	$(1/1.5)$	people per day
$\beta$	Transfer by superspreader	1.9	people per day
$\gamma$	Infectious contact rate	2.5	people per day
$\delta$	Recovery rate, Asymptomatic	$(1/7)$	people per day
$\theta$	Recovery rate, hospitalized	$(1/14)$	day^–1
$\zeta$	Disease rate induced death of infected class	$15\times {10}^{-3}$	day^–1
$\eta$	Recovery rate, symptomatic	$(1/7)$	day^–1
$\mu$	A part of infections that will be symptomatic	0.5	dimensionless
$\nu$	The rate of recovery	$(1/7)$	day^–1
$\tau$	Hospitalization rate	0.0025	day^–1
$\lambda$	Hospitalization rate	0.0025	day^–1
$\kappa$	Hospitalization rate	0.0025	day^–1
$\xi$	Hospitalization rate	0.0025	day^–1
$\rho$	Hospitalization rate	0.0025	day^–1
$\sigma$	Hospitalization rate	0.0025	day^–1
$\iota$	Hospitalization rate	0.0025	day^–1
$\varphi$	Hospitalization rate	0.0025	dimensionless

Table 3. Comparison between the J values and $T_{\mathrm{f}}=350,$ $r_1=0.5.$

$\alpha$	J	$\alpha (t)$	J
1	1.1411	$0<t\le 10$	0.9426
		${\alpha }_0=0.98$
		$10<t\le 350$
		${\alpha }_1=1$
0.98	1.1197	$0<t\le 15$	0.9660
		${\alpha }_0=0.98$
		$15<t\le 350$
		${\alpha }_1=1$
0.97	1.1057	$0<t\le 20$	0.9817
		${\alpha }_0=0.98$
		$20<t\le 350$
		${\alpha }_1=1$
0.95	1.0689	$0<t\le 25$	0.9931
		${\alpha }_0=0.98$
		$25<t\le 350$
		${\alpha }_1=1$
0.90	0.9060	$0<t\le 30$	0.0019
		${\alpha }_0=0.98$
		$30<t\le 350$
		${\alpha }_1=1$

Table 4. Comparison between the values of J, $T_{\mathrm{f}}=350,$ $r_1=0.5$ using different methods.

$\alpha$	J	$\alpha (t)$	J
1	01.1411	$0<t\le 10$	1.0291
		${\alpha }_0=0.99$
		$10<t\le 350$
		${\alpha }_1=1$
0.98	1.1197	$0<t\le 10$	0.9426
		${\alpha }_0=0.98$
		$10<t\le 350$
		${\alpha }_1=1$
0.95	1.0689	$0<t\le 10$	0.7625
		${\alpha }_0=0.95$
		$10<t\le 350$
		${\alpha }_1=1$
0.90	0.9060	$0<t\le 10$	0.5711
		${\alpha }_0=0.90$
		$10<t\le 350$
		${\alpha }_1=1$
0.85	0.6258	$0<t\le 10$	0.4337
		${\alpha }_0=0.85$
		$10<t\le 350$
		${\alpha }_1=1$

7. Conclusions

In the present article, we paid special attention to study the hybrid variable-order FracInt mathematical model of vaccination Covid-19 and their optimal control. The variable-order derivative is defined using both the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. It is more general than Caputo fractional operator. The stability analysis of the proposed model was discussed, as well as the existence of equilibrium points. In this model, boundedness, positivity and stability are proved. $\sigma$ is introduced to make the model compatible with the physical problem. The optimality conditions are numerically derived. CPC-GLNFDM is developed to study the optimality system. Generally, the numerical results show that the proposed FracInt system is effective in controlling the disease. Generally, the numerical findings demonstrate that the FOC system performs better only during a portion of the time period. In order to combat infection, we therefore suggest a system in which the derivative order changes over the course of the time interval, becoming fractional or integer as appropriate. This FracInt-named variable-order fractional model appears to be the most successful at controlling the illness. In the future, the present study can be extended to FracInt delay optimal control.

Disclosure statement

The authors have declared no conflict of interest.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.