key: cord-0049336-bb93kckb
authors: Higazy, M.; Alyami, Maryam Ahmed
title: New Caputo-Fabrizio fractional order [Formula: see text] model for COVID-19 epidemic transmission with genetic algorithm based control strategy
date: 2020-08-31
journal: nan
DOI: 10.1016/j.aej.2020.08.034
sha: 29262c90d1e9ca864803bc4520e916183da497b8
doc_id: 49336
cord_uid: bb93kckb

Fractional derivative has a memory and non-localization features that make it very useful in modelling epidemics’ transition. The kernel of Caputo-Fabrizio fractional derivative has many features such as non-singularity, non-locality and an exponential form. Therefore, it is preferred for modeling disease spreading systems. In this work, we suggest to formulate COVID-19 epidemic transmission via [Formula: see text] paradigm using the Caputo-Fabrizio fractional derivation method. In the suggested fractional order COVID-19 [Formula: see text] paradigm, the impact of changing quarantining and contact rates are examined. The stability of the proposed fractional order COVID-19 [Formula: see text] paradigm is studied and a parametric rule for the fundamental reproduction number formula is given. The existence and uniqueness of stable solution of the proposed fractional order COVID-19 [Formula: see text] paradigm are proved. Since the genetic algorithm is a common powerful optimization method, we propose an optimum control strategy based on the genetic algorithm. By this strategy, the peak values of the infected population classes are to be minimized. The results show that the proposed fractional model is epidemiologically well-posed and is a proper elect.

COVID-19, the 2019 new corona-virus, 2019-nCoV or corona-virus of Wuhan are all names of the same transmissible virus causing respiratory contagion and highly transferred between humans. Until now, no one surely knows about the true source of COVID-19. The obvious truth is the existence of COVID-19 between us, and the pain it causes to us every hour. The pain caused to us every hour is either because a new injury or the death of someone we know due to Covid-19. The most likely source of COVID-19 is a lack of respect for and transgression of the laws of nature. These irresponsible actions of humans have caused many similar disasters like that created by the "Severe Acute Respiratory Syndrome" (SARS) outset in China in 2003 [1] , the "Middle East Respiratory Syndrome" (MERS) outset in Saudi Arabia in 2012 [2, 3] and the MERS outset in South Korea in 2015 [4, 5] . These epidemics have produced more than 8000 assured SARS patients and 2200 assured MERS patients, [6] .

COVID-19 pandemic caused many losses worldwide, in the other side it motivates researchers in all …elds to search for the nature of this virus, the way it spread, the symptoms it causes, how to deal with it, ultimately treat it permanently and prepare a vaccine for future prevention.

As realized by the WHO [7] , mathematical models, mainly those that are depending on a period of time, have a crucial function in assisting health decisionmakers and policy-makers. So far, only a few mathematical models have been announced that explain the nature of the transmission of COVID-19. For example the model that illustrates the relationship between bats, hosts, reservoir and people; and the model done for computing underestimated COVID-19 cases [8, 9] . In [10] , Khan and Atangana have studied COVID-19 via fractional order SEIARM -model. By applying Atangana-Balenau derivative, they have shown the e¤ect of the seafood market M on the other population classes. other applications of Atangana-Balenau derivative can be found for example in [11, 12, 13] . In [14] , Gao et al. produced a new ("bats, hosts, reservoir, people and COVID-19") nonlinear dynamical model that clear the dynamics of COVID-19 transition from the Bats to humans. Goa et al., in [15] , studied the epidemic prediction for COVID-19 epidemic by utilizing (q-HAT M ). The recorded cases information were used to determine the number of unrecorded cases in Wuhan, China. In [16] , the authors utilized the available data about COVID-19 to produce a comparative mathematical study in order to help the interested decision makers. They used HIV-COVID-19 general model to emulate the pandemic status in several countries. In [17] , using the Italian data, Atangana presented several statistical graphs to clear out the nature of COVID-19 transition. Utilizing these results he constructed a new model in which the e¤ect of lock-down was taken into account. Also, novel fractal-fractional operators were suggested and utilized to the proposed system. Via several numerical cases, the e¢ ciency of the lock-down was proved. In [18] , A new dynamical model of COVID-19 was proposed and studied with a nonlocal fractional derivative operator.

In [19] , Higazy generalized the integer order SIDARTHE COVID-19 epidemic model, that constructed …rstly in [20] , via fractional derivative scheme. In addition, certain optimum control strategies were constructed.

In [21] , Biao Tang et al. constructed the SEIAS q E q HR paradigm in order to predict the dynamics of the novel coronavirus transmission via the ordinary di¤erential equations.

Several problems in the real world can be formulated utilizing fractional-order mathematical paradigms, for example see [22] - [36] . Optimal control methods have been used by many researchers to minimize the total number of cases in the infected population, see for example [19, 37, 38, 39] . The control plans may be via giving treatments to the detected infected cases, giving vaccines to the uninfected people, decreasing the contact rate via social distances or isolation, and encouraging personal hygiene through educational programs, see for example [19, 37, 38, 40] and the related references cited therein.

As fractional derivative has a memory feature that makes it very useful in modelling epidemics'transition [10, 19, 41, 42, 43] , in the current research, we continue our study about modelling COVID-19 epidemic transition via fractional derivative schemes. We generalize the 8-dimensions COVID-19 SEIAS q E q HR model given in [21] utilizing Caputo-Fabrizio fractional operator. The reason of utilizing the Caputo-Fabrizio fractional operator to COVID-19 new model is that it has many features such as their kernel has useful features such as non-singularity, non-locality and an exponential form, and by utilizing this operator, the crossover action in the studied model can be good characterized via this fractional derivative operator. In the suggested fractional order COVID-19 SEIAS q E q HR paradigm, the impact of quarantining and contact rates are examined. The stability of the disease free equilibrium is studied and a new parametric rule for the fundamental reproduction number formula is given. The existence and uniqueness of stable solution for the proposed fractional order COVID-19 SEIAS q E q HR paradigm are proved. Since the genetic algorithm is a common powerful optimization method [44] , we propose an optimum control strategy based on the genetic algorithm, by which the optimum contact and quarantining rates can be estimated. By this strategy, the peak values of the infected population classes are to be minimized for the proposed fractional order COVID-19 SEIAS q E q HR paradigm. Research concerning Caputo-Fabrizio derivative and their utilization in several models in engineering and science can be found for example in [45, 46] and the related papers cited there.

The remaining parts of this paper are arranged as follows. Required basic notions about Caputo-Fabrizio fractional derivative will be recognized in Section 2. In Section 3, the suggested fractional order COVID-19 SEIAS q E q HR paradigm is introduced. The stability of the proposed fractional order COVID-19 SEIAS q E q HR paradigm will be discussed in Subsection 3.1. The existence and uniqueness of stable solution of the proposed Caputo-Fabrizio fractional order COVID-19 SEIAS q E q HR paradigm will be examined and proved in Subsection 3.2. Numerical simulations and results discussion will be recorded in Section 4. Genetic algorithm based three control examples of the proposed fractional order COVID-19 SEIAS q E q HR paradigm will be presented in Section 5. The concluding words will be recorded in Section 6.

Caputo's de…nition of the fractional derivative doesn't always describe the memory impact of the real time systems, this is because the kernel uniqueness in the Caputo concept of the fractional derivative [47] . In [48, 49] , another de…nition of the fractional di¤erential operator in their kernel with no singularity was proposed by Caputo and Fabrizio. In the new de…nition, the kernel takes an exponential form. The derivation of the new Caputo-Fabrizio fractional di¤erential operator was made in [50] by Losada and Nieto. The basic concepts and required properties for the studied model are displayed in the remaining part of this section.

where N ( ) is the the function normalization such that N (0) = N (1) = 1: If the function ' = 2 K(l; u), the de…nition takes the form

; then the de…nition takes the form Calculus in with fractional order and in speci…c Caputo de…nition of fractional order have several real applications, [48, 49, 47, 50] . The kernel (t y) is changed for Caputo and Fabrizio in [51] . Then the form becomes 

In [21] , Biao Tang et al. constructed the SEIAS q E q HR model (1) in order to predict the dynamics of the novel coronavirus transmission. This model is tested and its parameters are evaluated using the recorded real data from tenth to …fteenth of January 2020 of the recorded cases con…rmed by laboratory of 2019-nCoV cases that recorded in China by the report of situation from World I 0 (t) = lE(t) ( I + + I )I(t);

The (SEIAS q E q HR)-model is a generalization of the famous "Susceptible-Exposed-Infectious-Recovered" (SEIR) compartmentalized paradigm relied on the functional progress of the epidemic, epidemiological situation between people and intervention measures ( Figure 1 ). The parameters of the model were evaluated for the assured recorded cases of COVID-19 in China mainland and were used to calculate the fundamental reproduction number formula of the disease transmission [21] . In [21] , SEIR paradigm was generalized by estimating the impact of intervention actions, thorough closing o¤ and quarantine in addition to treatment.

Here, the model is generalized utilizing the Caputo-Fabrizio fractional order di¤erential operator and written in (2) as:

In the fractional order (SEIAS q E q HR)-model, (S) symbolizes the susceptible compartment, (E) symbolizes the exposed compartment, (A) symbolizes the symptomless infected compartment, (I) symbolizes the symptomatic infected compartment, (H) symbolizes the hospitalized cases compartment, (R) symbolizes the recuperated cases compartment, (S q ) symbolizes the quarantined susceptible compartment and (E q ) symbolizes the quarantined exposed. Via tracing the people's contact, q portion of the exposed humans to the virus are quarantined. The quarantined humans are splitted into two compartments E q or S q according to the power of infection [54] , however the second portion, 1 q, who contains humans that lost from tracing the people's contact, are moved to E population compartment or remain in S population compartment. Assume that be the probability of transmission and c be the rate of contact. Then, the isolated humans, if uninfected (or infected), moved into the population S q (or population E q ) with rate (1 )cq (or cq). Humans who are not isolated, if infected, will be put in the E compartment with rate c(1 q) . The sick humans could be determined and hence quarantined at a rate I and also due to recovery can be moved to the R population compartment. In our theoretical analysis, the fractional orders will be assumed to be (0 < i < 1; i = 1; 2; 3; :::; 8). The …rstorder derivatives in (1) is generalized by the fractional order Caputo-Fabrizio di¤erential operator in order to obtain the proposed fractional order derivative model. The other model parameters are explained as follows.

c is the rate of Contact, is the per contact transmission Probability, q is rate of quarantining the exposed humans, is rate of transition of exposed humans to the class of infected humans, is the rate of releasing the quarantined uninfected humans, l is the Probability of infected cases to have symptoms, I is the rate of transition from the group of infected with symptoms to the isolated infected group, q is The rate of transition from isolated exposed cases to the isolated infected group, I is the rate of recovery of the group of infected with symptoms cases, A is the rate of recovery of the symptomless infected cases, H is the rate of recovery of isolated infected cases, is the rate of isolating the symptomless infected cases and is the rate of death due to the disease. The computed values of the parameters in the model are recorded in the following table and taken from [21] and will be used later in the numerical simulations. All states of the model (2) are positive and bounded with the initial conditions:

By add up the eight equations of the model (2), the whole people's dynamics can be presented as follows.

where

The functional range for the studied model (2) is presented by

The current subsection examines the stability for the fractional order COVID-19 SEIAS q E q HR paradigm (2). The system has two equilibrium points, Disease Free Equilibrium (DFE) (E 0 = (S; 0; 0; 0; 0; 0; 0; 0)) and the endemic equilibrium point (EEP) (E e = (S e ; E e ; I e ; A e ; S qe ; E qe ; H e ; R e )).

To show the behavior of the model, we divided it to three sub-systems: the …rst subsystem consists of the variable S which corresponds to susceptible class, the second subsystem consists of E; I ; A; S q ; E q and H (the infected people classes), that have non-zero values only during the transient time, and the third subsystem consists of the variable R (for the recovered class). The second infected classes subsystem, denoted as EIAS q E q H subsystem. An remarkable notation is that the variables S and R are at equilibrium while (and only while) the infected individuals E + I + A + S q + E q + H are zero. Variable R (which is monotonous increase) approaches to its asymptotic values R, and S (which is monotonous decrease) approaches to S i¤ E; I; A; S q ; E q and H approach to null.

The whole model can be reconstructed in a feedback scheme, wherever the EIAS q E q H subsystem can be shown in a positive linear form with a feedback function f as following.

De…ne

Where

) and ! 7 = cq. The other state variables satisfy the following D.E.'s:

In view of the fact that the gain S(t) of feedback …nally approaches a constant value S, it is possible to go ahead with parameters based analysis according to the limit feedback gain S. The following result can be deduced.

Proof. Around the equilibrium DFE (E 0 = (S; 0; 0; 0; 0; 0; 0; 0)), the Jacobian matrix will be as follows.

The matrix has two zero eigenvalues, and three negative real eigenvalues f ! 2 ; ; q g, and three eigenvalues equal the roots of

Applying Routh-Hurwiz stability criteria [55] , the real parts of the eigenvalues of (13) will be negative if and only if all coe¢ cients b i of (13) are positive

From (14, 15, 16) , we can deduce that (13) is Hurwitz (all real parts of the roots are negative) if and only if expression (11) satis…ed. From (8), the transfer function between f and Y f in the system (4) to (10) is

Due to the positive property of the system, then the norm H 1 of G( ) equals to the static gain G(0) = N (0)=D(0) [20] . Due to the observation come from [20] , the basic reproduction

and the equilibrium stability occurs when SR 0 < 1 In addition, remark that R 0 = G(0) is the norm H 1 of the transfer function G( ) between f and Y f in the system (4) to (10) [20] .

The brink value S is of primary signi…cance. Since, S(t) approaches, asymptotically, monotonically to a limit S, such a limit S must guarantee convergence of the EIAS q E q H subsystem to zero (meaning stability; else, S could not reach S). Subsequently, we can deduce the following proposition. Proof. Since S(t) is non-negative and decreasing monotonically, which means that it has an end S 0. We can say that, after a proper large time t, it is clear that S(t) S. So, the system SEIAS q E q HR acts equivalently to the linearized scheme related to the linearization in S. let us assume the contradiction case, S causes instability for this paradigm, So I(t) diverges, because the Metzler matrix F +BSC has a +ve prevalent eigenvalue. Successively, that is mean that I(t) cannot reach to nil, therefore its ingredients stay positive, which means that ! 4 I(t) + ! 4 A(t) > 0 does not reach to nil. As a result, CF t S(t) = S(t)(! 4 I(t) + ! 4 A(t)) + S q (t) < 0 also does not approach to nil, then S(t) cannot approach to a non-negative limit S 0. a contradiction is obtained.

The stability gate value S of equation (11) has a profound meaning. The value S is stand for the part of population that has not been infected at no time in the past or future. This limit is a decreasing function of the parameters ; c; q; and l, that are the vital parameters. The act

has a destabilization e¤ect of the under study subsystem EIAS q E q H, that would be steady in the absence this feedback. In order to save the stabilization of the subsystem EIAS q E q H and con…rm that the equilibrium S is approached, the coe¢ cients of infection have to be little or the limit value S is little. Following [20] , and from (17) the basic reproduction number of the system can be de…ned as:

The stability of the DFE occurs while

At the start of the epidemic, we have S ' 1 (100 % of the population) then the stability is reached while R 0 < 1.

In the current subsection, the solutions'existence and uniqueness (see [56] ) for the Caputo-Fabrizio fractional order paradigm are studied taking into account the initial conditions of the system (2). By applying the operator of fractional Caputo-Fabrizio integral on the proposed studied model, we have

Then, kernels take the form

It is known that the system states are positive functions according to the structure of the model i.e. kS(t)k c 1 ; kE(t)k c 2 ; kI(t)k c 3 ; kA(t)k c 4 ; kS q (t)k c 5 ; kE q (t)k c 6 ; kH(t)k c 7 ; kR(t)k c 8 where c i (i = 1; 2; :::; 8) are certain non-negative constants.

Applying Caputo-Fabrizio fractional integral to equation (20), we get Proof. Consider the kernel K i for any state variable X i . Let X and X 1 be any two features.

Utilizing the triangle inequality on (24), we have

where 1 is de…ned in equation (22) . Following the same for the remaining kernels will satisfy the following:

Where i are de…ned in equation (22) . Then, the Lipschitz circumstances are satis…ed by all kernels.

Furthermore, since 0 U = maxf i : i = 1; 2; :::; 8g < 1 the kernels are contractions. From equation (25), we have

Utilizing Equation (23), The next recursive relations are being produced now.

Where S(0) = S 0 (t); E(0) = E 0 (t); I(0) = I 0 (t); A(0) = A 0 (t);

S q (0) = S q;0 (t); E q (0) = E q;0 (t); H(0) = H 0 (t); R(0) = R 0 (t):

The recursive relations for this system can be written as S n (t) = ( 1 )(K 1 (t; S n 1 ) K 1 (t; S n 2 )) (28)

S q;n (t) = ( 5 )(K 5 (t; S q;(n 1) ) K 5 (t; S q;(n 2) )) + !( 5 ) t Z 0 (K 5 (z; S q;(n 1) ) K 5 (z; S q;(n 2) ))dz;

E q (t) = ( 6 )(K 6 (t; E q;n 1 ) K 6 (t; E q;n 2 )) + !( 6 ) t Z 0 (K 6 (z; E q;n 1 ) K 6 (z; E q;n 2 )dz;

H n (t) = ( 7 )(K 7 (t; H n 1 ) K 7 (t; H n 2 ))

And kS n (t)k = kS n S n 1 k = (30)

By applying Equation (30) in triangle inequality, we get

So, because the kernel K 1 satis…es the Lipschitz condition with Lipschitz constant 1 , then we have

The following can be obtained by the similar reasoning Proof. Because all of the functions, S(t); E(t); I(t); A(t); S q (t); E q (t); H(t) and R(t), are bounded and all of the kernels meet a Lipschitz condition, the existence and smoothness of the solutions in (29) are proved. For the proof to be …nished, the convergence of S(t); E(t); I(t); A(t); S q (t); E q (t); H(t) and R(t); the solutions of (2), are going to be proved here. De…ne S n (t); E n (t); I n (t); A n (t); S q;n (t); E q;n (t); H n (t) and R n (t) such that

S q (t) S q (0) = S q;n (t) S q;n (t); E q (t) E q (0) = E q;n (t) E q;n (t);

For K 1 we get

Recursively, repeat the above steps, we obtain

So at t 0 , we have

Via taking the limit on Equation (44) while n ! 1, we get kS n (t)k ! 0. Similarly, we get kE n (t)k ! 0; kI n (t)k ! 0; kA n (t)k ! 0; S Qq;n (t) ! 0; E q;n (t) ! 0; kH n (t)k ! 0 and kR n (t)k ! 0. The existence of system solutions is hence proved. 

for i = 1; 2; :::; 8.

Proof. Suppose that S 1 (t); E 1 (t); I 1 (t); A 1 (t); S q;1 (t); E q;1 (t); H 1 (t) and R 1 (t) for the current solution in the previous theorem, so we have

Reorder (47)

Combine the condition in (45) for i = 1 with Equation (48), we get

and then S(t) = S 1 (t). Also, It is true for E 1 (t); I 1 (t); A 1 (t); S q;1 (t); E q;1 (t); H 1 (t) and R 1 (t). Consequently, the uniqueness of the solution scheme for the fractional derivative order system is proved.

The non-linear system of the fractional order di¤erential equations of COVID-19 (SEIAS q E q HR)-model has been modeled by utilizing the Caputo fraction derivative. The impacts of the system parameters on the dynamics of the fractional-order model (2) , can be done via (2) numerically using the parameters' values recorded in Table 1 with …nite time. The numerical integrations needed for the fractional order COVID-19 (SEIAS q E q HR)-model are executed in this paper via the predictor-corrector Adams-Bashforth-Moulton (ABM) technique for fractional order di¤erential equations, constructed by Diethelm et al. [57] . These simulations detect that changing the values of the fractional order, contact rate or quarantine rate a¤ects the dynamics of the model. Figure 2 shows the time history of COVID-19 fractional order (SEIAS q E q HR)model according to various fractional derivative orders = 0:7; 0:8; 0:9; 1; (a) Susceptible humans S(t) versus time in days, (b) Exposed humans E(t) versus time in days, (c) Infected with symptoms humans I(t), (d) Infected humans without symptoms versus time in days. The arrows directions indicate the effect of increasing the fractional order . Figure 3 shows the time history of COVID-19 (SEIAS q E q HR)-model according to various fractional derivative orders = 0:7; 0:8; 0:9; 1; (a) Quarantined Susceptible humans S q (t) versus time in days, (b) Quarantined exposed humans E q (t) versus time in days, (c) Hospitalised humans H(t) versus time in days, (d) Recuperated humans without symptoms versus time in days.The arrows directions indicate the e¤ect of increasing the fractional order . Figure 4 shows the impact of changing the contact rate c on the dynamics of all population classes at fractional order = 1. The arrow direction indicates increasing the contact rate where the estimated basic contact rate c = 14:781: From Figures 2, 3 and 4 , decreasing the fractional derivative order delays the peak values and makes the curves more ‡at. Figure 5 shows the impact of changing the contact rate c on the dynamics of all population classes at fractional order = 0:9. The direction of the arrow indicates increasing the contact rate where the estimated basic contact rate c = 14:781: Figure 6 shows the impact of changing the contact rate c on the dynamics of all population classes at fractional order = 0:8. The direction of the arrow indicates increasing the contact rate where the estimated basic contact rate c = 14:781: Figure 7 shows the impact of changing both contact rate and fractional order on all population classes at day 40. From which we can say that the contact rate has a great impact on the system dynamics of the system and has a great impact on the total number of infected classes. Figure 8 shows the impact of changing the quarantining rate q on the dynamics of all population classes at fractional order = 0:8. The direction of the arrow indicates increasing the quarantining rate where the estimated basic contact rate q = 1:8887 10^ 7: Figure 9 shows the impact of changing the quarantining rate q on the dynamics of all population classes at fractional order = 0:9. The direction of the arrow indicates increasing the quarantining rate where the estimated basic contact rate q = 1:8887 10^ 7: Figure 10 shows the impact of changing the quarantining rate q on the dynamics of all population classes at fractional order = 1. The arrow direction indicates increasing the quarantining rate where the estimated basic contact rate q = 1:8887 10^ 7: Figure 11 shows the impact of changing both quarantining rate and fractional order on all population classes at day 40. From which we can deduce that quarantining has a great impact on the total number of infected classes.

In the current section, we numerically study di¤erent control strategies for the proposed COVID-19 fractional order (SEIAS q E q HR)-model utilizing the contact rate, c; and quarantining rate, q; parameters. The logistic requirements may limit the minimum contact rate and the maximum quarantining rate. So, our optimum control problem is to search for the optimum values of the two control parameters that minimizes the peak values of the two infected classes within these limitations. Figure 12 shows the e¤ect of changing the two control parameters on the di¤erent classes of COVID-19 fractional order (SEIAS q E q HR)-model at day 40 with fractional derivative order = 0:95. Using the genetic algorithm toolbox of MATLAB [44] , the optimum values of the two control parameters that minimizes the sum of peak values of the two infected classes.

Let the system population classes vector be X = (S; E; I; A; S q ; E q ; H; R) tr : The under control reproduction number calculated in [21] , following [58] is R c = lc lcq

The control problem is to choose the optimum values of the two control parameters c and q in order to minimize the proposed …tness function ( P (I)+ P (A))as:

s.t.

Case 5.1 Using the genetic algorithm via MATLAB optimization toolbox, the optimum values of the contact rate is found as c = 3:4179 and quarantine rate q = 4:707 10 6 which equal the q max (25 times the quarantine rate that is estimated from the real data in [21] ), where R c max = 6:5, R c min = 1:5, c min = 1:5, c max = 15, q min = 1:8887 10 7 , q max = 25 q min , and the other system parameters and the initial values are as in the previous section. The minimum …tness value (sum of infected cases (49)) is about 1657 cases.

Case 5.2 Using the genetic algorithm via MATLAB optimization toolbox, the optimum values of the contact rate is found as c = 2:2787 and quarantine rate q = 4:72175 10 6 which equal the q max (25 times the quarantine rate that is estimated from the real data in [21] ), where R c max = 6:5, R c min = 1, c min = 2, c max = 15, q min = 1:8887 10 7 , q max = 25 q min , and the other system parameters and the initial values are as in the previous section. The minimum …tness value (sum of infected cases (49)) is about 4626 cases.

Using the genetic algorithm via MATLAB optimization toolbox, the optimum values of the contact rate is found as c = 2:2787 and quarantine rate q = 4:72175 10 6 which equal the q max (25 times the quarantine rate that is estimated from the real data in [21] ), where R c max = 6:5, R c min = 1, c min = 2, c max = 15, q min = 1:8887 10 7 , q max = 25 q min , and the other system parameters and the initial values are as in the previous section. The minimum …tness value is 23617 cases. Figure 13 displays the optimization process by genetic algorithm Case 5.1. Figure 14 Figure 16 shows the e¤ect of the two control parameters c and q on the controlled reproduction number R c . From Figure 16 , we see that the contact rate has a great impact on the reproduction number but the quarantine rate has a low impact on the reproduction number. The reproduction number at the optimum values of the two control parameters is R c (c ; q ) = 1:5; which equal the minimum possible reproduction number in our proposed optimization case. Figure 17 shows the dynamics of COVID-19 fractional order (SEIAS q E q HR)model with di¤erent fractional order = 0:8, 0:9 and 1. The direction of the arrows in Figure 17 indicate the direction of increasing the fractional derivative order. From Figure 17 , the number of individuals in the two infected classes are clearly minimized and the minimum of the sum of the infected classes is 1657.28 only. Figure 18 shows the dynamics of COVID-19 fractional order (SEIAS q E q HR)-model with di¤erent fractional order = 0:8, 0:9 and 1. The direction of the arrows in Figure 18 indicate the direction of increasing the fractional derivative order. From Figure 18 , the number of individuals in the two infected classes are clearly minimized and the minimum of the sum of the infected classes is 4626 cases.

We have studied theoretically and numerically a fractional order model for COVID-19 transmission under genetic algorithm based control strategies. The integer order SEIAS q E q HR model proposed by Biao Tang et al. (2020) in [21] has been generalized here to become a fractional order model via Caputo-Fabrizio fractional order di¤erential operator. The stability of the proposed generalized COVID-19 fractional order SEIAS q E q HR model has been discussed. The existence and uniqueness of solution of Caputo-Fabrizio fractional derivative order (SEIAS q E q HR) model have been investigated and proved. The impact of changing the fractional derivative order , contact rate parameter c and quarantining rate parameter q have been displayed numerically. Utilizing genetic algorithm MATLAB optimization toolbox, three control cases have been investigated. From inspection of the numerical solution graphs, we can conclude the following: as the fractional derivative order is decreased there is a related decrease in the number of infectious cases, and the peak values of the infected classes are decreased and delayed.

the number of cases in the infected classes is strongly dependent on the contact rate, as the contact rate c is decreased the number of infected cases is decreased.

the number of cases in the infected classes is weakly dependent on the quarantining rate, as the contact rate q is decreased the number of infected cases decreases.

from GA optimization results, we recommend to take decisions to minimize the contact rate to its possible minimum value and to maximize the quarantining rate to its possible maximum value.

These results are in agreement with those obtained by Biao Tang et al. (2020) for = 1 and suggest that the proposed fractional model is epidemiologically well-posed and is a proper elect.

For future work, we suggest to use the genetic algorithm to estimate the optimum values of the fractional derivative orders that minimize the errors between the real recorded data and the model solutions. In addition, we suggest to study the graph of the system via the graph theory concepts to discover more features of the model.

No competing interests.

This research did not receive any speci…c grant from funding agencies in the public, commercial, or not-for-pro…t sectors.

Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.

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Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy

Estimation of the Transmission Risk of the 2019-nCoV and Its Implication for Public Health Interventions

A generalization of truncated M-fractional derivative and applications to fractional di¤erential equations

A review on harmonic wavelets and their fractional extension

Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets

Complex solitons in the conformable (2+1)(2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation

A fractional epidemiological model for computer viruses pertaining to a new fractional derivative

New numerical results for the time-fractional Phi-four equation using a novel analytical approach

New wave solutions of time fractional Kadomtsev-Petviashvili equation arising in the evolution of nonlinear long waves of small amplitude. Erzincan Univ

Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative

Numerical solutions with linearization techniques of the fractional Harry Dym equation

Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Le-er type kernel

New analytical solutions of conformable time fractional bad and good modi…ed Boussinesq equations

A new technology for solving di¤usion and heat equations

A new analysis of fractional …sh farm model associated with Mittag-Le-er type kernel

An E¢ cient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by

A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying

Optimal control for a nonlinear Mathematical model of tumor under immune suppression a numerical approach

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study

Optimal Control of a delayed SIRS epidemic model with vaccination and treatment

SIR epidemic model with Mittag-Le-er fractional derivative

Ndolane Sene Second-grade ‡uid model with Caputo-Liouville generalized fractional derivative

Fractional di¤usion equation with new fractional operator

An Introduction to Genetic Algorithms

The dynamics of a new chaotic system through the Caputo-Fabrizio and Atanagan-Baleanu fractional operators

A Caputo Fabrizio fractional order model for control of glucose in insulin therapies for diabetes

Theory and applications of fractional di¤erential equations

A new de…nition of fractional derivative without singular kernel

On the analysis of fractional diabetes model with exponential law

Properties of the new fractional derivative without singular kernel

Assessment of model predictive and adaptive glucose control strategies for people with type 1 diabetes

Novel Coronavirus-China, Disease Outbreak News: Update

National Health Commission of the People's Republic of China

Mathematical models of isolation and quarantine

Control systems: principles and design

Singapore: World Scien-ti…c

Algorithms for the fractional calculus: a selection of numerical methods

Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission

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