A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model

Graphical abstract

Highlights

• •A novel mathematical model of Corona virus with new hybrid fractional operator derivative are presented.
• •Three control variables are presented to minimize the number of infected population.
• •Necessary control conditions are derived.
• •Two numerical methods are constructed to study the behavior of the obtained fractional optimality system.
• •The stability of the proposed methods are proved.
• •Numerical simulations and comparative studies are given.

Abstract

Introduction

Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo’s fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions.

Objectives

A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived.

Methods

The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method.

Results

The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin’s maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator.

Conclusions

The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists.

Introduction

Coronavirus disease 2019 (COVID-19) is an infectious disease. In December 2019, the disease was first identified in China and rapidlied spread around that country and subsequently many others countries. It is reported that the virus might be bat origin, and the transmission of the virus might related to a seafood market (Huanan Seafood Wholesale Market) exposure. The genetic features and some clinical findings of the infection have been reported recently [10].

The spread of infectious diseases has serious effects on human society and healthy. The modeling study of infectious diseases is very useful in making strategies to control diseases [9]. Recently, many interesting papers in modeling the Coronavirus, see for example ([11], [12], [13], [14], [15]).

In general, mathematical models involved by the known ordinary differentiation could be used to capture dynamical systems of infectious disease, when only initial conditions are used to predict future behaviors of the spread. However, when the situation is unpredictable, which is the case of COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations (FODEs) models seem more consistent with the real phenomena than the integer order models ([2], [3], [4], [5], [6], [7]). This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential and integral equations. Moreover, the Caputo fractional derivative has been one of the most useful operators for modeling non-local behaviors by fractional differential equations [1].

Recently, Baleanu et. al., in [8] constructed a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral. This new operator is more general than the operator of Caputo fractional derivative. In this work we will use this new derivative with an efficient nonstandard finite difference method (NSFDM) to study numerically the obtained fractional systems. The technique of the NSFDM was firstly proposed by Mickens [19]. Using this technique, some interesting real life applications are studied in ([16], [17], [20]).

Moreover, one of the new topics in mathematics is the fractional optimal control (FOC). FOC can be defined using varieties types of fractional derivatives definitions. Riemann–Liouville and Caputo fractional derivatives [21], [22], [23] can be considered the most important fractional derivatives definitions. Interesting numerical schemes for FOC are given in ([25], [26], [27], [28]).

The main goal of this paper is to extend the mathematical model of Coronavirus given in [11] by using new hybrid fractional operator derivative. This operator can be written as a linear combination of a Riemann–Liouville integral with a Caputo derivative (CPC). We will introduce three control variables in order to minimize the number of the population of infected. Two numerical methods will be constructed to approximate the obtained fractional optimality system. These methods are: weighted average nonstandard finite difference method (WANFDM) and the Grünwald-Letnikov nonstandard finite difference method (GL-NSFDM). Stability analysis of the proposed methods will be proved. Comparative studies with Caputo derivative will be given.

To the best of our knowledge, a hybrid fractional optimal control for Coronavirus (2019-nCov) mathematical model has never been explored.

The organization of this article is as follows: The main mathematical formals will be given in Section ‘Preliminaries and notations’. The proposed model with new fractional order derivatives and three controls are presented in Section ‘Fractional order model of Coronavirus with control’. In Section ‘The FOCPs’, the formulation of the optimal control problem and the necessary optimality conditions are derived. In Section ‘Numerical method for solving FOCPs’, the numerical methods and there stability analysis are introduced. In Section ‘Numerical experiments’ numerical experiments with discussion are given. Finally, the conclusions are presented in Section ‘Conclusions’.

Preliminaries and notations

In this section, we recall some important definitions of the fractional calculus used throughout the remaining sections of this paper.

• •
  Let $0<\alpha <1,\mathrm{\Gamma }$ be the Euler gamma function, then the Caputo fractional order derivative is defined as follows [1]:

  $${}_0{}^{C}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t{(t-s)}^{-\alpha }{y}^{\prime }\left(s\right)\mathit{ds},$$ (1)
• •
  Let $y\left(t\right)$ be an integrable function, $0<\alpha <1$, then the Riemann–Liouville integral is defined as follows [1]:

  $${}_0{}^{\mathit{RL}}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}y\left(s\right)\mathit{ds},$$ (2)
• •
  The new type of fractional operator is defined as a hybrid fractional operator from combining the proprotional and Caputo definition [8]:

  $${}_0{}^{\mathit{CP}}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{-\alpha }\left(K_1\right(\alpha ,s\left)y\right(s)+K_0(\alpha ,s\left)y^{\prime }\right(s\left)\right)\mathit{ds}\text{.}$$ (3)

  Let the kernels are given as follows: $K_0(\alpha ,t)=\alpha C^{2\alpha }{t}^{(1-\alpha )}$,   $K_1(\alpha ,t)=(1-\alpha )t^{\alpha }$,

where $0<\alpha <1,C$ is constant. In the special case when $K_0$ and $K_1$ are independent of t, the new operators are given as follows:

Definition 2.1

The proprotial-Caputo hybrid operator is defined either as general way [8]:

$$\begin{array}{cc}\hfill & {}_0{}^{\mathit{CP}}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{-\alpha }\left(K_1\right(\alpha ,s\left)y\right(s)+K_0(\alpha ,s\left)y^{\prime }\right(s\left)\right)\mathit{ds},\hfill \\ \hfill & =\left(K_1\right(\alpha ,t\left)y\right(t)+K_0(\alpha ,t\left)y^{\prime }\right(t\left)\right)\times \left(\frac{t^{\alpha }}{\mathrm{\Gamma }(1-\alpha )}\right)\text{.}\hfill \end{array}$$ (4)

Or as the following simple expression [8]:

$$\begin{array}{cc}\hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{-\alpha }\left(K_1\right(\alpha \left)y\right(s)+K_0(\alpha \left)y^{\prime }\right(s\left)\right)\mathit{ds}\hfill \\ \hfill & =K_1\left(\alpha \right){}_0{}^{\mathit{RL}}I_t^{1-\alpha }y\left(t\right)+K_0\left(\alpha \right){}_0{}^{C}D_t^{\alpha }y\left(t\right),\hfill \end{array}$$ (5)

where, $K_1\left(\alpha \right),K_0\left(\alpha \right)$ are constants with respect to t and depending only on $\alpha$. Also, in this paper we consider the kernels as follows: $K_0\left(\alpha \right)=\alpha C^{2\alpha }{Q}^{(1-\alpha )}$,   $K_1\left(\alpha \right)=(1-\alpha )Q^{\alpha }$, where Q is constant and $C=1$.

Fractional order model of Coronavirus with control

Herein, we consider the recent Coronavirus spreading model given in [11] using a new hybrid fractional order derivative. This model consists of eight nonlinear differential equations. We change the order of the equations to $\alpha$, the dimension of the left-hand side would be ${\left(\mathit{time}\right)}^{-\alpha }$. In order to have the dimensions match we should change the dimensions of the parameters. Also, when $\alpha \to 1$ the fractional order system reduces to classical one. Three controls, $u_I,u_P,u_h$ are added in order to health care such as isolating patients in private health rooms and providing respirators and give them treatments soothing regularly. Let us assume that ${\delta }_i^{\alpha }={\delta }_p^{\alpha }={\delta }_h^{\alpha }=0$. The description of all the variables given in Table 1. Also, Table 2 describes the parameters. The CPC-modified model is then represented as follows:

$$\begin{array}{cc}\hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }S=-{\beta }^{\alpha }\frac{\mathit{IS}}{N}-L{\beta }^{\alpha }\frac{\mathit{HS}}{N}-{\beta }_1^{\alpha }\frac{\mathit{PS}}{N},\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }E={\beta }^{\alpha }\frac{\mathit{IS}}{N}+L{\beta }^{\alpha }\frac{\mathit{HS}}{N}+{\beta }_1^{\alpha }\frac{\mathit{PS}}{N}-K^{\alpha }E,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }I=K^{\alpha }{\rho }_{1}E-({\gamma }_a^{\alpha }+{\gamma }_i^{\alpha })I-{\delta }_i^{\alpha }I-\nu u_{I}I,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }P=K^{\alpha }{\rho }_{2}E-({\gamma }_a^{\alpha }+{\gamma }_i^{\alpha })P-{\delta }_p^{\alpha }P-\nu u_{P}P,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }A=K^{\alpha }(1-{\rho }_1-{\rho }_2)E,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }H={\gamma }_a^{\alpha }(I+P)-{\gamma }_r^{\alpha }H-{\delta }_h^{\alpha }H-\nu u_{h}H+0.5\nu u_{I}I+0.5\nu u_{P}P,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }R={\gamma }_i^{\alpha }(I+P)+{\gamma }_r^{\alpha }H+0.5\nu u_{I}I+0.5\nu u_{P}P+\nu u_{h}H,\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }F={\delta }_i^{\alpha }I+{\delta }_p^{\alpha }P+{\delta }_h^{\alpha }H,\hfill \end{array}$$ (6)

where, $0<\nu ⩽1$. The existence and uniqueness of the solutions of (6) follow from the results given in [29]. The basic reproduction number of the proposed model (6) is given as follows [11]:

$$R_0=\frac{{\beta }^{\alpha }{\rho }_1({\gamma }_a^{\alpha }L+{\chi }_h)}{{\chi }_i{\chi }_h}+\frac{{\beta }^{\alpha }{\rho }_2{\gamma }_a^{\alpha }L+{\rho }_2{\beta }_1^{\alpha }{\chi }_h}{{\chi }_p{\chi }_h}\text{.}$$ (7)

where, ${\chi }_i={\gamma }_a^{\alpha }+{\gamma }_i^{\alpha }+{\delta }_i^{\alpha },{\chi }_p={\gamma }_a^{\alpha }+{\gamma }_i^{\alpha }+{\delta }_p^{\alpha }$ and ${\chi }_h={\gamma }_r^{\alpha }+{\delta }_h^{\alpha }$. The endemic threshold is given at $R_0=1$ and indicates the minimal transmission potential that sustains endemic disease, that is, when $R_0<1$, the disease will die out and for $R_0>1$ the disease may become endemic [30]. In this work we consider $R_0>1$.

Table 1.

The variables of system (6)[11].

The variable	Description
R	The class of recovery
H	The class of hospitalized
E	The class of exposed
I	The class of symptomatic and infectious
S	The class of susceptible
F	The class of fatality
P	The class of super-spreaders
A	The class of infectious but asymptomatic

Table 2.

The parameters values for the Coronavirus model [11].

Parameter	Description	Value (per ${\mathit{day}}^{-\alpha }$)
${\beta }^{\alpha }$	Transmission coefficient from infected individuals	${2.55}^{\alpha }$
L	Relative transmissibility of hospitalized patients	$1.56$ dimensionless
${\beta }_1^{\alpha }$	Transmission coefficient due to super-spreaders	${7.65}^{\alpha }$
$K^{\alpha }$	Rate at which exposed become infectious	${0.25}^{\alpha }$
${\rho }_1$	Rate at which exposed people become infected I	$0.580$ dimensionless
${\rho }_2$	Rate at which exposed people become super-spreaders	$0.001$ dimensionless
${\gamma }_a^{\alpha }$	Rate of being hospitalized	${0.94}^{\alpha }$
${\gamma }_i^{\alpha }$	Recovery rate without being hospitalized	${0.27}^{\alpha }$
${\gamma }_r^{\alpha }$	Recovery rate of hospitalized patients	${0.5}^{\alpha }$
${\delta }_i^{\alpha }$	Disease induced death rate due to infected class	${3.5}^{\alpha }$
${\delta }_p^{\alpha }$	Disease induced death rate due to super-spreaders	$1^{\alpha }$
${\delta }_h^{\alpha }$	Disease induced death rate due to hospitalized class	${0.3}^{\alpha }$

The FOCPs

Consider the system (6) in ${\mathbb{R}}^8$, let

$$\mathrm{\Omega }=\left\{\right(u_I(.),u_P(.),u_h(.)\left)\right|u_I,u_P(.),u_h\mathit{are}\mathit{Lebsegue}\mathit{measurable}\mathit{on}[0,1],$$

$$0⩽u_I(.),u_P(.),u_h(.)\le 1,\forall t\in [0,T_f]\},$$

be the admissible control set. We will define the objective functional as follows:

$$J(u_I,u_P,u_h)={\int }_0^{T_f}\left(I\right(t)+H(t)+B_1{u}_I^2(t)+B_2{u}_P^2(t)+B_3{u}_h^2(t\left)\right)\mathit{dt}\text{.}$$ (8)

The aim now is to find $u_I\left(t\right),u_P\left(t\right)$ and $u_h\left(t\right)$ such that the following cost functional is minimum:

$$J(u_I,u_P,u_H)={\int }_0^{T_f}\eta (t,S,E,I,P,A,H,R,F,u_I,u_P,u_h)\mathit{dt},$$ (9)

subject to the constraints

$${}_a{}^{\mathit{CPC}}D_t^{\alpha }{\mathrm{\Psi }}_j={\xi }_i\text{.}$$ (10)

Where

$$\begin{array}{cc}\hfill & {\xi }_i={\xi }_i(t,S,E,I,P,A,H,R,F,u_I,u_P,u_h),i,j=1,\dots ,8,\hfill \\ \hfill & {\mathrm{\Psi }}_j=\{S,E,I,P,A,H,R,F\},\hfill \end{array}$$

${\Psi }_1\left(0\right)=S_0,{\Psi }_2\left(0\right)=E_0,{\Psi }_3\left(0\right)=I_0,{\Psi }_4\left(0\right)=P_0,{\Psi }_5\left(0\right)=A_0,{\Psi }_6\left(0\right)=H_0,{\Psi }_6\left(0\right)=R_0,{\Psi }_6\left(0\right)=F_0.$

We will use a kind of Pontryagin’s maximum principle in fractional order case, this idea is given by Agrwal in [23]:

Consider a modified cost functional as follows [25]:

$$\tilde{J}={\int }_0^{T_f}\left[H\right(t,S,E,I,P,A,H,R,F,u_I,u_P,u_h)-{\displaystyle \sum _{i=1}^8}{\lambda }_i{\xi }_i(t,S,E,I,P,A,H,R,F,u_I,u_P,u_h\left)\right]\mathit{dt}\text{.}$$ (11)

The Hamiltonian is define as follows:

$$\begin{array}{c}\hfill H(t,S,A,P,I,E,H,R,F,u_I,u_P,u_h,{\lambda }_i)=\eta (t,S,A,P,I,E,H,R,F,u_I,u_P,u_h,{\lambda }_i)\\ \hfill +{\displaystyle \sum _{i=1}^8}{\lambda }_i{\xi }_i(t,S,E,I,P,A,H,R,F,u_I,u_P,u_h).& \hfill \end{array}$$ (12)

From (11), (12), we have:

$${}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_{\iota }=\frac{\partial H}{\partial {\vartheta }_{\iota }},\iota =1,\dots ,8,$$ (13)

where,

$${\vartheta }_{\iota }=\{t,S,E,I,P,A,H,R,F,u_I,u_P,u_h,\iota =1,\dots ,8\},$$

$$0=\frac{\partial H}{\partial u_k},k=I,P,h,$$ (14)

$${}_0{}^{\mathit{CPC}}D_t^{\alpha }{\vartheta }_{\iota }=\frac{\partial H}{\partial {\lambda }_{\kappa }},\iota =1,\dots ,8,$$ (15)

and it is also required that the Lagrange multipliers satisfies:

$${\lambda }_{\iota }\left(T_f\right)=0,\iota =1,2,\dots ,8\text{.}$$ (16)

Theorem 4.1

There exists optimal control variables$u_I^{\ast },u_P^{\ast },u_h^{\ast }$with the corresponding solutions$S^{\ast },E^{\ast },I^{\ast },P^{\ast },A^{\ast },H^{\ast },R_p^{\ast },F^{\ast }$, that minimizes$J(u_I,u_p,u_h)$over$\mathrm{\Omega }$. Furthermore, there exists adjoint variables${\lambda }_i$,   $i=1,2,3,\dots ,8$, satisfy the following:

(i) adjoint equations:

$${}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_1={\lambda }_1(-{\beta }^{\alpha }\frac{I^{\ast }}{N}-L{\beta }^{\alpha }\frac{H^{\ast }}{N}-{\beta }_1^{\alpha }\frac{P^{\ast }}{N})+{\lambda }_2({\beta }^{\alpha }\frac{I^{\ast }}{N}+L{\beta }^{\alpha }\frac{H^{\ast }}{N}+{\beta }_1^{\alpha }\frac{P^{\ast }}{N}),$$

$${}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_2=-K^{\alpha }{\lambda }_2+{\lambda }_3{K}^{\alpha }{\rho }_1+K^{\alpha }{\rho }_2{\lambda }_4+K^{\alpha }(1-{\rho }_1-{\rho }_2){\lambda }_5,$$

$$\begin{array}{c}\hfill {}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_3=-{\lambda }_1{\beta }^{\alpha }\frac{S^{\ast }}{N}+{\lambda }_2{\beta }^{\alpha }\frac{S^{\ast }}{N}-{\lambda }_3({\gamma }_a+{\gamma }_i^{\alpha }+{\delta }_i^{\alpha }+\nu u_I^{\ast })\\ \hfill +{\lambda }_6({\gamma }_a^{\alpha }+0.5\nu u_I^{\ast })+({\gamma }_i^{\alpha }+0.5\nu u_I^{\ast }){\lambda }_7+{\delta }_i^{\alpha }{\lambda }_8,\end{array}$$

$$\begin{array}{c}\hfill {}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_4=-{\lambda }_1{\beta }_1^{\alpha }\frac{S^{\ast }}{N}+{\lambda }_2{\beta }_1^{\alpha }\frac{S^{\ast }}{N}-{\lambda }_4({\gamma }_a^{\alpha }+{\gamma }_i^{\alpha }+{\delta }_p^{\alpha }+\nu u_P^{\ast })+{\lambda }_6({\gamma }_a^{\alpha }+0.5\nu u_P^{\ast })\\ \hfill +({\gamma }_i^{\alpha }+0.5\nu u_P^{\ast }){\lambda }_7+{\delta }_p^{\alpha }{\lambda }_8,\end{array}$$

$$\begin{array}{cc}\hfill & {}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_6=-{\lambda }_1{\beta }^{\alpha }L\frac{S^{\ast }}{N}+{\lambda }_2{\beta }^{\alpha }L\frac{S^{\ast }}{N}-{\lambda }_6({\gamma }_r^{\alpha }+{\delta }_h^{\alpha }+({\gamma }_a^{\alpha }+\nu u_h^{\ast }\left)\right)\hfill \\ \hfill & +{\lambda }_7({\gamma }_r^{\alpha }+\nu u_P^{\ast })+{\delta }_h^{\alpha }{\lambda }_8,\hfill \end{array}$$ (17)

where

$${}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_5={}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_7={}_t{}^{\mathit{CPC}}D_{t_f}^{\alpha }{\lambda }_8=0\text{.}$$

(ii) The transversality conditions

$${\lambda }_{\iota }\left(T_f\right)=0,\iota =1,2,\dots ,8\text{.}$$ (18)

(iii) Optimality conditions:

$$\begin{array}{c}\hfill H(S,E,I,P,A,H,R,F,u_I,u_P,u_h,\lambda ,t)=\\ \hfill {\displaystyle \underset{0⩽u_I,u_P,u_h⩽1}{\min }}H(S,E,I,P,A,H,R,F,u_I,u_P,u_h,\lambda ,t)\text{.}& \hfill \end{array}$$ (19)

Moreover:

$$u_I^{\ast }=\min \{1,\max \{0,\frac{\nu I^{\ast }({\lambda }_3-0.5{\lambda }_6-0.5{\lambda }_7)}{B_1}\left\}\right\},$$ (20)

$$u_P^{\ast }=\min \{1,\max \{0,\frac{\nu P^{\ast }({\lambda }_4-0.5{\lambda }_6-0.5{\lambda }_7)}{B_2}\left\}\right\},$$ (21)

$$u_h^{\ast }=\min \{1,\max \{0,\frac{\nu H^{\ast }({\lambda }_6-{\lambda }_7)}{B_3}\left\}\right\}\text{.}$$ (22)

Proof

Eq. (17) can be obtained from (13, where:

$$\begin{array}{c}\hfill H={\lambda }_1{}_0{}^{\mathit{CPC}}D_t^{\alpha }{S}^{\ast }+{\lambda }_2{}_0{}^{\mathit{CPC}}D_t^{\alpha }{E}^{\ast }+{\lambda }_3{}_0{}^{\mathit{CPC}}D_t^{\alpha }{I}^{\ast }\\ \hfill +{\lambda }_4{}_0{}^{\mathit{CPC}}D_t^{\alpha }{P}^{\ast }+{\lambda }_5{}_0{}^{\mathit{CPC}}D_t^{\alpha }{A}^{\ast }+{\lambda }_6{}_0{}^{\mathit{CPC}}D_t^{\alpha }{H}^{\ast }\\ \hfill +{\lambda }_7{}_0{}^{\mathit{CPC}}D_t^{\alpha }{R}^{\ast }+{\lambda }_8{}_0{}^{\mathit{CPC}}D_t^{\alpha }{F}^{\ast }\\ \hfill +P^{\ast }+H^{\ast }+I^{\ast }+B_1{u}_I^{\ast }+B_2{u}_P^{\ast }+B_3{u}_h^{\ast },\end{array}$$ (23)

is the Hamiltonian. ${\lambda }_{\kappa }\left(T_f\right)=0$,  $\kappa =1,\dots ,8$, are hold. Eqs. (20), (21), (22) can be obtained from (19). □

Now, by substituting $u_I^{\ast },u_P^{\ast },u_h^{\ast }$ in (6):

$$\begin{array}{cc}\hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{S}^{\ast }=-{\beta }^{\alpha }\frac{I^{\ast }{S}^{\ast }}{N}-L{\beta }^{\alpha }\frac{H^{\ast }{S}^{\ast }}{N}-{\beta }_1^{\alpha }\frac{P^{\ast }{S}^{\ast }}{N},\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{E}^{\ast }={\beta }^{\alpha }\frac{I^{\ast }{S}^{\ast }}{N}+L{\beta }^{\alpha }\frac{H^{\ast }{S}^{\ast }}{N}+{\beta }_1^{\alpha }\frac{P^{\ast }{S}^{\ast }}{N}-K^{\alpha }{E}^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{I}^{\ast }=K^{\alpha }{\rho }_1{E}^{\ast }-({\gamma }_a^{\alpha }+{\gamma }_i^{\alpha })I^{\ast }-{\delta }_i^{\alpha }{I}^{\ast }-\nu u_I^{\ast }{I}^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{P}^{\ast }=K^{\alpha }{\rho }_2{E}^{\ast }-({\gamma }_a^{\alpha }+{\gamma }_i^{\alpha })P^{\ast }-{\delta }_p^{\alpha }{P}^{\ast }-\nu u_P^{\ast }{P}^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{A}^{\ast }=K(1-{\rho }_1-{\rho }_2)E^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{H}^{\ast }={\gamma }_a^{\alpha }(I^{\ast }+P^{\ast })-{\gamma }_r^{\alpha }{H}^{\ast }-{\delta }_h^{\alpha }{H}^{\ast }-\nu u_h{H}^{\ast }+0.5\nu u_I^{\ast }{I}^{\ast }+0.5\nu u_P^{\ast }{P}^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{R}^{\ast }={\gamma }_i^{\alpha }(I^{\ast }+P^{\ast })+{\gamma }_r^{\alpha }{H}^{\ast }+\nu u_h^{\ast }{H}^{\ast }+0.5\nu u_I^{\ast }{I}^{\ast }+0.5\nu u_P^{\ast }{P}^{\ast },\hfill \\ \hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }{F}^{\ast }={\delta }_i^{\alpha }{I}^{\ast }+{\delta }_p^{\alpha }{P}^{\ast }+{\delta }_h^{\alpha }{H}^{\ast }\text{.}\hfill \end{array}$$ (24)

Numerical method for solving FOCPs

NWAFDM

Let us consider the following fractional order differential equation with the hybrid fractional operator:

$${}_0{}^{\mathit{CP}}D_t^{\alpha }y\left(t\right)=\xi (t,y(t\left)\right),0<\alpha ⩽1,y\left(0\right)=y_0\text{.}$$ (25)

We can discretize (25) by using definition (3) as follows:

$$\begin{array}{c}\hfill \frac{1}{\varphi {\left(\tau \right)}^{\alpha -1}\mathrm{\Gamma }(2-\alpha )}{\displaystyle \sum _{i=0}^n}\left((1-\alpha )t_i^{\alpha }{y}_{n-i+1}+\alpha C^{2\alpha }{t}_i^{(1-\alpha )}\frac{y_{n-i+1}-y_{n-i}}{\varphi \left(\tau \right)}\right)\left[{(i+1)}^{(1-\alpha )}-{\left(i\right)}^{(1-\alpha )}\right]\\ \hfill =\mathrm{\Theta }\xi (t_{n+1},y(t_{n+1}\left)\right)+(1-\mathrm{\Theta })\xi (t_n,y(t_n\left)\right),\end{array}$$ (26)

where,

$$\varphi \left(\tau \right)=\tau +O\left({\tau }^2\right),0<\varphi \left(\tau \right)<1,\tau ⟶0\text{.}$$

Also, we can discretize (25) by using definition (5) and using GL-approximation to approximate the Caputo fractional derivatives:

$$\begin{array}{cc}\hfill & \frac{Q^{\alpha }(1-\alpha )}{\varphi {\left(\tau \right)}^{\alpha -1}\mathrm{\Gamma }(2-\alpha )}{\displaystyle \sum _{i=0}^{n+1}}{y}_{n-i+1}\left[{(i+1)}^{(1-\alpha )}-{\left(i\right)}^{(1-\alpha )}\right]+\frac{\alpha C^{2\alpha }{Q}^{(1-\alpha )}}{\varphi {\left(\tau \right)}^{\alpha }}\left(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0\right)\hfill \\ \hfill & =\mathrm{\Theta }\xi (t_{n+1},y(t_{n+1}\left)\right)+(1-\mathrm{\Theta })\xi (t_n,y(t_n\left)\right),\hfill \end{array}$$ (27)

where, $K_0\left(\alpha \right)=\alpha C^{2\alpha }{Q}^{(1-\alpha )}$,   $K_1\left(\alpha \right)=(1-\alpha )Q^{\alpha },{\omega }_0=1,{\omega }_i=(1-\frac{\alpha }{i}){\omega }_{i-1}$, $t^n=n\tau ,\tau =\frac{T_f}{N_n}$,    $N_n\in \mathbb{N}$. ${\mu }_i={(-1)}^{i-1}\left(\begin{array}{c}\hfill \alpha \hfill \\ \hfill i\hfill \end{array}\right)$,    ${\mu }_1=\alpha$,     $q_i=\frac{i^{\alpha }}{\mathrm{\Gamma }(1-\alpha )}$  and   $i=1,2,\dots ,n+1$. Additionally, consider ([18], [24]):

$$0<{\mu }_{i+1}<{\mu }_i<\dots <{\mu }_1=\alpha <1,$$

$$0<q_{i+1}<q_i<\dots <q_1=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\text{.}$$

The main advantage of this method is it can be explicit i. e., $(\mathrm{\Theta }=0)$ or implicit i. e., $(0<\mathrm{\Theta }<1)$ or fully implicit i. e., $(\mathrm{\Theta }=1)$, the advantage of implicit case is it has large stability regions by using the idea of the weighed step introduced by the nonstandard finite difference method. In this article we will use the method given in (27).

Remark 1

In (27), if we put $K_0\left(\alpha \right)=1$ and $K_1\left(\alpha \right)=0$, we obtained the discretization of the Caputo fractional derivative as follows:

$$\begin{array}{cc}\hfill & \frac{1}{\varphi {\left(\tau \right)}^{\alpha }}\left(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0\right)\hfill \\ \hfill & =\mathrm{\Theta }\xi (t_{n+1},y(t_{n+1}\left)\right)+(1-\mathrm{\Theta })\xi (t_n,y(t_n\left)\right),\hfill \end{array}$$ (28)

GL-NSFDM

We can rewrite the relation (5) in another way as follows:

$$\begin{array}{cc}\hfill & {}_0{}^{\mathit{CPC}}D_t^{\alpha }y\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{-\alpha }\left(K_1\right(\alpha \left)y\right(s)+K_0(\alpha \left)y^{\prime }\right(s\left)\right)\mathit{ds},\hfill \\ \hfill & =K_1\left(\alpha \right){}_0{}^{\mathit{RL}}I_t^{1-\alpha }y\left(t\right)+K_0\left(\alpha \right){}_0{}^{C}D_t^{\alpha }y\left(t\right),\hfill \\ \hfill & =K_1\left(\alpha \right){}_0{}^{\mathit{RL}}D_t^{\alpha -1}y\left(t\right)+K_0\left(\alpha \right){}_0{}^{C}D_t^{\alpha }y\left(t\right),\hfill \end{array}$$ (29)

where, $K_1\left(\alpha \right),K_0\left(\alpha \right)$ are constant with respect to t and depending only on $\alpha$. Using GL-approximation and NSFDM, we can discretize (29) as follows:

$${}_0{}^{\mathit{CPC}}D_t^{\alpha }y\left(t\right){|}_{t=t^n}=\frac{K_1\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha -1}}\left(y_{n+1}+{\displaystyle \sum _{i=1}^{n+1}}{\omega }_i{y}_{n+1-i}\right)+\frac{K_0\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha }}\left(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0\right),$$ (30)

$$\begin{array}{c}\hfill \frac{K_1\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha -1}}\left(y_{n+1}+{\displaystyle \sum _{i=1}^{n+1}}{\omega }_i{y}_{n+1-i}\right)+\frac{K_0\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha }}\left(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0\right)\\ \hfill =\xi \left(y\right(t_n),t_n),\end{array}$$ (31)

where,

$$\varphi \left(\tau \right)=\tau +O\left({\tau }^2\right),0<\varphi \left(\tau \right)<1,▵\left(t\right)⟶0,$$

${\omega }_0=1,{\omega }_i=(1-\frac{\alpha }{i}){\omega }_{i-1}$, $t^n=n\tau ,\tau =\frac{T_f}{N_n}$,    $N_n$ is a natural number. ${\mu }_i={(-1)}^{i-1}\left(\begin{array}{c}\hfill \alpha \hfill \\ \hfill i\hfill \end{array}\right)$,    ${\mu }_1=\alpha$,     $q_i=\frac{i^{\alpha }}{\mathrm{\Gamma }(1-\alpha )}$  and   $i=1,2,\dots ,n+1$. Additionally, let us assume that [18]:

$$0<{\mu }_{i+1}<{\mu }_i<\dots <{\mu }_1=\alpha <1,$$

$$0<q_{i+1}<q_i<\dots <q_1=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\text{.}$$

Stability of NWAFDM

In the following we will show that the NWAFDM in case $0<\mathrm{\Theta }⩽1$ (implicit case) is unconditionally stable. In order to investigate the stability of the proposed method when$(\mathrm{\Theta }\ne 0)$, consider the following test problem of linear fractional differential equation:

$$\left({}_0{}^{\mathit{CPC}}D_t^{\alpha }\right)y\left(t\right)=\mathit{Ay}\left(t\right),t>0,0<\alpha ⩽1,A<0\text{.}$$ (32)

Let $y\left(t_n\right)=y_n$ is the approximate solution of this equation then using GL-NFDM with (29) we rewrite Eq. (32) in the following form:

$$\begin{array}{cc}\hfill & =\frac{Q^{\alpha }(1-\alpha )}{\varphi {\left(\tau \right)}^{\alpha -1}\mathrm{\Gamma }(2-\alpha )}{\displaystyle \sum _{i=0}^{n+1}}{y}_{n-i+1}[{(i+1)}^{(1-\alpha )}-{\left(i\right)}^{(1-\alpha )}]+\frac{C^{2\alpha }\alpha Q^{(1-\alpha )}}{\left(\varphi {\left(\tau \right)}^{\left(\alpha \right)}\right)}(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0)\hfill \\ \hfill & =\mathrm{\Theta }{\mathit{Ay}}_{n+1}+(1-\mathrm{\Theta }){\mathit{Ay}}_n,\hfill \end{array}$$ (33)

put,

$$g_1=\frac{Q^{\alpha }(1-\alpha )}{\varphi {\left(\tau \right)}^{\alpha -1}\mathrm{\Gamma }(2-\alpha )},W^{\alpha }=\left[{(i+1)}^{(1-\alpha )}-{\left(i\right)}^{(1-\alpha )}\right],g_2=\frac{C^{2\alpha }\alpha Q^{(1-\alpha )}}{\left(\varphi {\left(\tau \right)}^{\left(\alpha \right)}\right)}\text{.}$$

We can write (33) as follows:

$$\begin{array}{cc}\hfill & g_1{\displaystyle \sum _{i=0}^{n+1}}{y}_{n-i+1}{W}^{\alpha }+g_2(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0)\hfill \\ \hfill & =\mathrm{\Theta }{\mathit{Ay}}_{n+1}+(1-\mathrm{\Theta }){\mathit{Ay}}_n\text{.}\hfill \end{array}$$ (34)

Then,

$$\begin{array}{cc}\hfill & g_1{y}_{n+1}+g_1{\displaystyle \sum _{i=1}^n}{y}_{n-i+1}{W}^{\alpha }+g_2(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0)\hfill \\ \hfill & =\mathrm{\Theta }{\mathit{Ay}}_{n+1}+(1-\mathrm{\Theta }){\mathit{Ay}}_n,\hfill \end{array}$$ (35)

$$y_{n+1}=\frac{\left(\right(1-\mathrm{\Theta }){\mathit{Ay}}_i-g_1{\displaystyle \sum _{i=1}^n}{y}_{n-i+1}{W}^{\alpha }+g_2({\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}+q_{n+1}{y}_0\left)\right)}{(g_1+g_2-\mathrm{\Theta }A)},$$ (36)

we have $\frac{1}{(g_1+g_2-\mathrm{\Theta }A))}<1$, hence

$$y_1⩽y_0,$$

$$y_0⩾y_1⩾\dots ⩾y_{n-1}⩾y_n⩾y_{n+1}\text{.}$$

So the proposed implicit scheme is stable.

Stability of GL-NSFDM

In order to investigate the stability of the proposed method 37 consider the test problem of linear fractional differential Eq. (32). Using GL-approximation and NSFDM (29) we can discretize (32) as follows:

$$\frac{K_1\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha -1}}\left(y_{n+1}+{\displaystyle \sum _{i=1}^{n+1}}{\omega }_i{y}_{n+1-i}\right)+\frac{K_0\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha }}\left(y_{n+1}-{\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}-q_{n+1}{y}_0\right)={\mathit{Ay}}_n,$$ (37)

put $C=\frac{K_1\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha -1}}$,    $B=\frac{K_0\left(\alpha \right)}{\varphi {\left(\tau \right)}^{\alpha }}$. Then, we have:

$$y_{n+1}=\frac{1}{C+B}\left({\mathit{Ay}}_n-C{\displaystyle \sum _{i=1}^{n+1}}{\omega }_i{y}_{n+1-i}+B\left({\displaystyle \sum _{i=1}^{n+1}}{\mu }_i{y}_{n+1-i}+q_{n+1}{y}_0\right)\right),$$ (38)

since, $C+B>1$ then we have: $y_1<y_0$ and $y_0⩾y_1⩾\dots ⩾y_{n-1}⩾y_n⩾y_{n+1}$. So the proposed scheme is stable.

Numerical experiments

In the following, numerical simulations for the models (17), (24) without and with optimal control are presented. Two schemes (27), (31) are presented to solve the proposed model with the following initial conditions [11]: $S\left(0\right)=N-6,R\left(0\right)=0,A\left(0\right)=0,F\left(0\right)=0,E\left(0\right)=0,P\left(0\right)=5,I\left(0\right)=1,H\left(0\right)=0$. Then by using the nonstandard implicit finite difference method [27] we will solve the co-state Eq. (17) with the transversality conditions (18). The controls are updated by using a convex combination of the previous controls and the value from the characterizations of $u_I^{\ast },u_P^{\ast }$ and $u_h^{\ast }$ . This process is reiterated and the iteration is ended if the current state, the adjoint state, and the control values converge sufficiently. In this case we use different values of $0<\alpha ⩽1$ with $B_1=100,B_2=50$, and $B_3=100,\nu =1$. In the numerical simulations the time level is chosen in days.

Fig. 1 demonstrate the effective of three controls case for the proposed model (6) using the scheme WANFDM (27) at final time equal 15 and $\mathrm{\Theta }=0.5$ . We noted that in uncontrolled case, the number of the population of $I,P,H$ are increasing, while the number of these population are decreasing in controlled case in the same interval. Moreover, when the final time equal 100, as we can see in Fig. 2 the population number of $I,P,H$ are increasing in interval $(0,25)$, in uncontrolled case while the number of these population are decreasing in controlled case in the same interval.

Fig. 1.

Numerical simulations of the variables $I,P$ and H with and without controls at $\alpha =0.95,T_f=15$ and $\mathrm{\Theta }=0.5$ using scheme (27).

Fig. 2.

Numerical simulations of the variables I and H with and without controls at $\alpha =0.95$ and $T_f=100$ and $\mathrm{\Theta }=0.5$ using scheme (27).

Table 3 reports the cost functional values for the scheme (27) at fully implicit case with and without controls at different $\alpha$ and $\varphi \left(\tau \right)$. We have the best results in controlled case at $\varphi \left(\tau \right)=0.1(1-e^{-\tau })$.

Table 3.

Comparison between the values of objective functional with and without controls, for $T_f=60$, using scheme (27) and $\mathrm{\Theta }=1$.

$\alpha$	$J(u_I^{\ast },u_P^{\ast },u_h^{\ast })$	$J(u_I^{\ast },u_P^{\ast },u_h^{\ast })$ with 3 controls	$J(u_I^{\ast },u_P^{\ast },u_h^{\ast })$ with 3 controls
	without control	$\varphi \left(\tau \right)=(1-e^{-\tau })$	$\varphi \left(\tau \right)=0.1(1-e^{-\tau })$
1	$5.9739\times {10}^4$	$3.2372\times {10}^4$	$2.0261\times {10}^3$
$0.97$	$4.9343\times {10}^4$	$2.6898\times {10}^4$	$3.0983\times {10}^3$
$0.92$	$3.4850\times {10}^4$	$1.9303\times {10}^4$	$4.7571\times {10}^3$
$0.85$	$2.0857\times {10}^4$	$1.1886\times {10}^4$	$7.2660\times {10}^3$
$0.70$	$2.5082\times {10}^3$	$630.4559$	$373.6541$

A comparison between cost functional values derived by three schemes (27), (28), (31) with three controls at $T_f=90$, is given in Table 4, where the scheme (28) is a special case for the schemes (27), (31) when we put $K_0=1,K_1=0$. We concluded that when $\alpha =1$, all schemes give the same result of the objective functional, also in interval $0<\alpha <0.8$ the difference of the schemes are very small and almost the scheme (31) gives the best results, while at interval $0.8<\alpha <0.7$ the scheme (27) gives the best results. This mean that the new operator derivative CPC is more general and suitable to study the optimal control of the biological phenomena than the Caputo operator derivative.

Table 4.

Comparison between the values of objective functional with three controls, for $T_f=90$, using WANFDM (27), $\mathrm{\Theta }=0$.

The operator of fractional	$\alpha$	$J(u_I^{\ast },u_P^{\ast },u_h^{\ast })$ with 3 controls
CPC (27)	1	$3.2651\times {10}^4$
CPC (31)		$3.2651\times {10}^4$
Caputo(28)		$3.2651\times {10}^4$
CPC (27)	$0.99$	$3.0977\times {10}^4$
CPC (31)		$3.0911\times {10}^4$
Caputo(28)		$3.3816\times {10}^4$
CPC (27)	$0.80$	$6.7811\times {10}^3$
CPC (31)		$9.1050\times {10}^3$
Caputo(28)		$3.9292\times {10}^4$
CPC (27)	$0.75$	$4.1461\times {10}^3$
CPC (31)		$4.3852\times {10}^3$
Caputo(28)		$3.0509\times {10}^4$
CPC (27)	$0.7$	$816.10564$
CPC (31)		$2.0535\times {10}^3$
Caputo(28)		$1.7234\times {10}^4$

Fig. 3 shows how the behavior of I and R are change when we use the general scheme (27) with the new operator derivative CPC and the Caputo derivative. We noted that the results which obtained by (27) are the best, because the number of I which obtained by (27) is less than the number of I which obtained by (28), also, the number of R which obtained by (27) is bigger than the number of R which obtained by (28). This mean that, the new operator CPC is more suitable to describe the biological phenomena than the Caputo operator.

Fig. 3.

Numerical simulations of I and R with control case at $\alpha =0.98$ and $T_f=45$ and $\mathrm{\Theta }=1$ using schemes (27), (28).

Fig. 4 shows comparesion between the results obtained by the two schemes (27), (31) at $\alpha =0.98$ and $\alpha =0.8$. We noted that at $\alpha =0.8$, the number of R which obtained by scheme (27) is bigger than the number of R which obtained by (31), this mean that, the scheme (27) is the best to study the optimal control problems.

Fig. 4.

Numerical simulations of $R\left(t\right)$ with control case using schemes (31), (27).

Fig. 5 shows the behavior of the solutions for the proposed model (6) using (31) in controlled and uncontrolled cases. Fig. 6 shows how the behavior of the solutions in control case are changing by using different values of $\alpha$ and $T_f=90$ using (31). Fig. 7 shows how the behavior of the solutions $I,P$ and H in control case are changing by using different values of $\alpha$ and $T_f=300$ using (27).

Fig. 5.

Numerical simulations of the all variables of system (6) with and without controls at $\alpha =0.99$ using scheme (31).

Fig. 6.

Numerical simulations of the all variables of system (6) at different $\alpha$ and three controls case using scheme (31).

Fig. 7.

Numerical simulations of $I,P$ and H at different $\alpha$ and three controls case using scheme (27) and $\mathrm{\Theta }=1$.

Fig. 8 shows the evolution of the approximate solutions for the control variables with several values of $\alpha$. We noted that the range of the solutions remain between zero and one.

Fig. 8.

Numerical simulations of $u_I,u_P$ and $u_h$ for the system (6) at (a)$\alpha =0.8$ and $T_f=300$ and (b) $\alpha =0.7$ and $T_f=90$ using scheme (27) and $\mathrm{\Theta }=1$.

Table 5 reports the values of the objective functional obtained by the scheme (27) with and without controls at different values of $\alpha$,  $\mathrm{\Theta }=0.5$. Table 6 shows the CPU time for the optimality systems using NWAFDM (27) and GL-NSFDM(31) with CPC definition and NWAFDM (28) with Caputo definition at different values of $\alpha$. We noted that the second method GL-NSFDM is faster than the first and third methods.

Table 5.

Comparison between the values of $I,P$ and H with and without controls, $T_f=30$, using WANFDM (27), $\mathrm{\Theta }=0.5$.

The Controls	$\alpha$	I	P	H
With	1	$489.5754$	$5.1367\times {10}^3$	$633.4476$
Without		$581.1148$	$1.6621\times {10}^4$	$1.3685\times {10}^3$
With	$0.98$	$448.6273$	$3.7270\times {10}^3$	$474.3764$
Without		$636.5596$	$1.5491\times {10}^4$	$1.6026\times {10}^3$
With	$0.90$	$127.7800$	$1.0190\times {10}^3$	$127.7851$
Without		$1.1239\times {10}^3$	$1.0617\times {10}^3$	$1.9801\times {10}^3$
With	$0.8$	$28.0352$	$244.5881$	$27.5343$
Without		$85.8616$	$4.3370\times {10}^3$	$1.0642\times {10}^3$

Table 6.

CPU time in seconds for the solution of optimality systems at different values of $\alpha$ and $\mathrm{\Theta }=0$.

$\alpha$	CPU time of CPC (27)	CPU time of CPC (31)	CPU time of Caputo(28)
1	2.457034	2.445327	2.561605
0.98	4.661529	2.191416	4.807198
0.90	4.936312	2.176197	4.983086
0.8	4.984231	2.130180	4.965147

Conclusions

In the present work, the optimal control of Coronavirus model with new fractional operator is presented. This operator can be written as a linear combination of a Riemann–Liouville integral with a Caputo derivative. This dynamical model is more suitable to describe the biological phenomena with memory than the integer order model. Three control variables, $u_I\left(t\right),u_p\left(t\right)$ and $u_h\left(t\right)$ are introduced in order to health care such as isolating patients in private health rooms and providing respirators and give them treatments soothing regularly. These have been implemented to minimize the number of infected population. Necessary optimality conditions are derived. Also, the combination of fractional order derivative and optimal control in the model improves the dynamics and increases complexity of the model. Two schemes are constructed to study the behavior of the proposed problems. We can conclude from the obtained numerical results that the new operator derivative CPC is more general and suitable to study the optimal control of the biological phenomena than the Caputo operator derivative. Moreover, the WANFDM (27) is depending on the values of the factor $\mathrm{\Theta }$, it can be explicit or implicit with large stability regions. This scheme is the best for solve the obtained fractional optimality system. Numerical simulations are presented to support our theoretical findings. Moreover, the CPC fractional derivative provides best results and could be more useful for the researchers and scientists.

Declaration of Competing Interest

The authors have declared no conflict of interest.

Compliance with Ethics Requirements

This article does not contain any studies with human or animal subjects.

Biography

Nasser H. Sweilam, Professor of numerical analysis at the Department of Mathematics, Faculty of Science, Cairo University. He earned his M.Sc. in Mathematis, Faculty of Science, Cairo University. He was a channel system Ph.D. student between Cairo University, Egypt, and TU-Munich, Germany. He received his Ph.D. in “Optimal Control of Variational Inequalities, the Dam Problem”. He was the Head of the Department of Mathematics, Faculty of Science, Cairo University (May 2012–May 2018). Prof. Sweilam published more than 140 publications and supervised more than 33 thesis’ for Ph.D., M.Sc. He is referee and editor of several international journals, in the frame of pure and applied Mathematics. His main research interests are numerical analysis, optimal control of differential equations, fractional and variable order calculus, bio-informatics and cluster computing, ill-posed problems.