Control of COVID-19 dynamics through a fractional-order model
aa r X i v : . [ m a t h . O C ] F e b Control of COVID-19 dynamics through a fractional-order model
Samia Bushnaq a, ∗ , Tareq Saeed b , Delfim F. M. Torres c, ∗ , Anwar Zeb d a Department of Basic Sciences, Princess Sumaya University for Technology, 11941 Amman, Jordan b Department of Mathematics, King Abdulaziz University, 41206 Jeddah, Kingdom Saudi Arabia c Center for Research and Development in Mathematics and Applications (CIDMA),Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal d Department of Mathematics, COMSATS University Islamabad, 22060 Abbottabad, Pakistan
Abstract
We investigate, through a fractional mathematical model, the effects of physical distance on theSARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of thespread of COVID-19: media education, through campaigns explaining the importance of socialdistancing, use of face masks, etc., towards all population, while the second one is quarantinesocial isolation of the exposed individuals. A general fractional order optimal control problem,and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimizethe number of susceptible and infected while maximizing the number of recovered. The extremalsare then numerically obtained.
Key words:
COVID-19 mathematical model, isolation, fractional order derivatives, optimalcontrol theory, numerical simulations
1. Introduction
The availability of easy-to-use precise estimation models are essential to get an insight intothe effects of transferable infectious diseases. In outbreak diseases, policy makers and institutionsmake decisions based on forecasting models to decide on future policies and to check the efficiencyof existing policies [1].Coronaviruses are a group of viruses that can be transmitted between humans, livestock andwild animals. Person to person spread of COVID-19 happens through close contact, up to six feet.This group of viruses mainly affects the hepatic, neurological and respiratory systems [2, 3, 4].In the end of 2019, the World Health Organization (WHO) reported a novel coronavirus inChina, which causes severe damage to the respiratory system. The virus was first found in Wuhancity, and was named as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [5]. Afterthe outbreak of the virus, the Chinese government put several cities on lock down [6, 7]. However,the number of affected people increased daily within China and in other countries. In March11, 2020, COVID-19 was declared as a global pandemic by WHO. At the time we write theselines, January 6, 2021, approximately 87 million people were infected with 1.9 million of deaths,worldwide [8].Recently, many mathematical models were proposed to understand disease transmission andproject handful controls: see, e.g., [9, 10, 11] and references therein. Simultaneously, all healthorganizations are trying to drive the most lethal infectious diseases towards eradication, using ∗ Corresponding author
Email addresses:
[email protected] (Samia Bushnaq), [email protected] (Tareq Saeed), [email protected] (Delfim F. M. Torres), [email protected] (Anwar Zeb)
URL: https://orcid.org/0000-0002-2427-7704 (Samia Bushnaq), https://orcid.org/0000-0002-0170-5286 (Tareq Saeed), https://orcid.org/0000-0001-8641-2505 (Delfim F. M. Torres), https://orcid.org/0000-0002-5460-3718 (Anwar Zeb)
Final form published by ’Alexandria Engineering Journal’. Submitted 6/Jan/2021; Revised 28/Jan/2021; Accepted 12/Feb/2021. ducational and enlightenment campaigns, vaccination, treatment, etc. However, many of theseinfectious diseases will become eventually endemic because of interventions to mitigate the spreadin time and lack of adequate policies. For control of infectious diseases, proactive steps are required,specially for diseases having vaccine and cure. Indeed, some times it is more difficult to controlthe spread of an infectious disease than to cure it. Regarding COVID-19, several vaccines beginto be available [12].Optimal control theory is a branch of mathematical optimization that deals with finding acontrol for a dynamical system, over a period of time, such that an objective function is minimizedor maximized. Along the years, optimal control theory has found applications in several fields,containing process control, aerospace, robotics, economics, bio-engineering, management sciences,finance, and medicine [13, 14, 15]. In particular, the study of epidemic models is strongly relatedto the study of control strategies, as screening and educational campaigns [16], vaccination [17],and resource allocation [18].In the current pandemic situation of COVID-19, due to best presentation of memory effects andits usefulness in many different and widespread phenomena [19, 20, 21, 22, 23, 24, 25], fractional(non-integer order) models are receiving the attention of many researchers: see, e.g., [26, 27, 28,29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Here, with the purpose to control the current pandemic,we follow two control variables, in the form of media (education) campaigns, social distance,and use of mask for protection of susceptible individuals; and quarantine (social isolation) for theexposed. For a general non-integer order optimal control problem, necessary optimality conditionsare presented, with the help of Caputo derivatives. One of the great advantages of the Caputofractional derivative is that it allows traditional initial or boundary conditions to be included inthe formulation of the problem. More concretely, we minimize the number of susceptible andinfected, while maximizing the number of recovered population from COVID-19. The optimallevels of the proposed two controls are characterized using the fractional version of Pontryagin’smaximum principle. The resulting optimality system is then solved numerically with
Matlab .The rest of the paper is arranged as follows. In Section 2, we present our fractional mathemat-ical model. Section 3 recalls the fundamental definitions and the main result of fractional optimalcontrol. We then derive an optimal control problem in Section 4, while parameter estimation withnumerical results are discussed in Section 5. We finish with some remarks and conclusions, inSection 6.
2. The fractional model
Our model consists of four classes: S ( t ), which represents the vulnerable individuals (healthypeople but who may get the disease in a near future); E ( t ), representing the exposed populationor individuals who are infected but not yet infectious; the group I ( t ), devoted to the populationwho are confirmed infected (individuals who have contracted the disease and are now sick withit and are infectious); the group R ( t ), defined as the recovered population (individuals who haverecovered from COVID-19). For the dynamics of this base model, see [28]. Thus, the fractionalorder model we consider here is given by C D αt S ( t ) = Λ − β S ( t ) E ( t ) − β S ( t ) I ( t ) − µS ( t ) + τ R ( t ) , C D αt E ( t ) = β S ( t ) E ( t ) + β S ( t ) I ( t ) − ( µ + ρ ) E ( t ) , C D αt I ( t ) = ρE ( t ) − ( γ + d + µ ) I ( t ) , C D αt R ( t ) = γI ( t ) − ( µ + τ ) R ( t ) , (1)where Λ is the recruitment rate, β and β are the incidence rates, τ is the relapse rate, µ is thenatural death rate, ρ the rate at which the exposed population of COVID-19 join the infectiousclass, γ the recovery rate of infected population, and d is the death rate of infected class due tothe SARS-CoV-2 virus. The total population N ( t ) is given, at each instant of time, by N ( t ) = S ( t ) + E ( t ) + I ( t ) + R ( t ) . (2)By adding all the equations of system (1), we have C D αt N ( t ) = Λ − µN ( t ) − dI ( t ) ≤ Λ − µN ( t ) . (3)2 . Basics of fractional control theory In this section we recall the basic definitions of Caputo fractional calculus and the centralresult of fractional optimal control theory [23, 39], which are required for the coming sections.
Definition 1.
For f ∈ C m , m ∈ N , the left-sided Caputo fractional derivative is given by Ca D αt f ( t ) = 1Γ( m − α ) Z ta ( t − τ ) m − α − (cid:18) ddτ (cid:19) m f ( τ ) dτ, (4) while the right-sided Caputo fractional derivative is given by Ct D αb f ( t ) = 1Γ( m − α ) Z bt ( τ − t ) m − α − (cid:18) − ddτ (cid:19) m f ( τ ) dτ, (5) where α stands for order of the derivative, m − < α ≤ m . Definition 2.
For f an integrable function, the left-sided Riemann–Liouville fractional derivativeis defined by a D tα f ( t ) = 1Γ( m − α ) (cid:18) ddt (cid:19) m Z ta ( t − τ ) m − α − f ( τ ) dτ, (6) while the right-sided Riemann–Liouville derivative of f is given by t D αb f ( t ) = 1Γ( m − α ) (cid:18) − ddt (cid:19) m Z bt ( τ − t ) m − α − f ( τ ) dτ, (7) where α is the order of the derivative with m − < α ≤ m , m ∈ N . Our control system is described by a fractional differential system (FDS) with a given/fixedinitial condition as follows: ( C D αt X ( t ) = f ( X ( t ) , u ( t ) , t ) ,X (0) = X , (8)where α ∈ (0 , n -dimensional X ( t ) is the state vector, f is a given vector-valued function, t ∈ [0 , t f ] with t f > m -dimensional u ( t ) is the controlvector. A fractional optimal control problem consists to minimize or maximize a performance index J [ u ( · )] = θ ( X ( t f ) , t f ) + t f Z φ ( X ( t ) , u ( t ) , t ) dt (9)subject to the control system (8) (see, e.g., [39, 40]). Functions θ and φ will be specified inSection 4. Note that here t f is fixed but X ( t f ) is free. For finding the optimal control law u ( t ) solution to the optimal control problem (8)–(9), we use the fractional version of Pontryaginmaximum principle, which coincides with the classical Pontryagin maximum principle when α = 1: Theorem 3 (See, e.g., [39, 40]) . For the optimality of (8) – (9) , a necessary condition is given by ∂φ∂u ( X ( t ) , u ( t ) , t ) + λ T ∂f∂u ( X ( t ) , u ( t ) , t ) = 0 , C D αt X ( t ) = f ( X ( t ) , u ( t ) , ( t )) , X (0) = X , t D αt f λ ( t ) = ∂φ∂X ( X ( t ) , u ( t ) , t ) + λ T ∂f∂X ( X ( t ) , u ( t ) , t ) , λ ( t f ) = ∂θ∂X ( X ( t f ) , t f ) . In the next section, we compute the optimal control strategy for the fractional order COVID-19model, which is a hot topic in current times. 3 . Fractional-order model with controls
We implement an optimal control technique to the fractional order model (1). With the purposeto control the spread of the COVID-19 pandemic in the world, we use two control variables in theform of media (educational) campaigns, social distance, and use of masks — the control u ( t ) —applied to the susceptible class; and quarantine (social isolation) — the control u ( t ) — appliedto the exposed class. Then, the new system with controls is given by C D αt S ( t ) = Λ − β S ( t ) E ( t ) − β S ( t ) I ( t ) − µS ( t ) + τ R ( t ) − u ( t ) S ( t ) , C D αt E ( t ) = β S ( t ) E ( t ) + β S ( t ) I ( t ) − ( µ + ρ ) E ( t ) − u ( t ) E ( t ) , C D αt I ( t ) = ρE ( t ) − ( γ + d + µ ) I ( t ) + (1 − p ) u ( t ) E ( t ) , C D αt R ( t ) = γI ( t ) − ( µ + τ ) R ( t ) + u ( t ) S ( t ) + pu ( t ) E ( t ) , (10)where the fractional order α is a real number in the interval (0 ,
1] and p can be interpreted as theprobability of infected individuals to recover by quarantine. In vector form, the system (10) canbe written as C D αt X ( t ) = f ( X ( t ) , u ( t )) , (11)where X ( t ) = ( S ( t ) , E ( t ) , I ( t ) , R ( t )) represents the state-vector and u ( t ) = ( u ( t ) , u ( t )) standsfor the control-vector.Our optimal control problem consists to minimize the spread of COVID-19 and maximize thenumber of recovered population. The following objective functional is defined with this purpose: J [ u ( · )] = A S ( t f ) + A E ( t f ) + Z t f A I ( t ) − A R ( t ) + 12 (cid:0) r u ( t ) + r u ( t ) (cid:1) dt −→ min , (12)where the positive weights A i , i = 1 , , ,
4, and r i , i = 1 ,
2, are used to balance the control factors.The objective functional (12) is a particular case of the general form (9) discussed in Section 3,and can be written as J [ u ( · )] = θ ( X ( t f )) + Z t f φ ( X ( t ) , u ( t )) dt (13)with θ ( X ( t f )) = A S ( t f ) + A E ( t f ) and φ ( X ( t ) , u ( t )) = A I ( t ) − A R ( t ) + 12 (cid:0) r u ( t ) + r u ( t ) (cid:1) . Similar functionals (13) to be optimized, e.g. for optimal control problems in the combat of Zikaand Ebola, have been previously considered in the literature, see [41, 42] and references therein.By using Theorem 3, the following necessary optimality conditions can be written: the controlsystem and its initial condition, ( C D αt X = f ( X, u ) ,X (0) = X , (14)the adjoint system and its transversality condition, ( t D αt f λ ( t ) = ∂φ∂X + λ T ∂f∂X ,λ ( t f ) = ∂θ∂X (cid:12)(cid:12) t f , (15)and the stationary condition ∂φ∂u + λ T ∂f∂u = 0 , (16)where λ ( t ) = ( λ ( t ) , λ ( t ) , λ ( t ) , λ ( t )) and f = ( f , f , f , f ) with f = α − β S ( t ) E ( t ) − β S ( t ) I ( t ) − µS ( t ) + τ R ( t ) − u ( t ) S ( t ) ,f = β S ( t ) E ( t ) + β S ( t ) I ( t ) − ( µ + ρ ) E ( t ) − u ( t ) E ( t ) ,f = ρE ( t ) − ( γ + d + µ ) I ( t ) + (1 − p ) u ( t ) E ( t ) ,f = γI ( t ) − ( µ + τ ) R ( t ) + u ( t ) S ( t ) + pu ( t ) E ( t ) . t D αt f λ ( t ) = − λ β E ( t ) − λ β I ( t ) − λ µ + λ β E ( t ) + λ β I ( t ) , t D αt f λ ( t ) = − λ β S ( t ) + λ β S ( t ) − λ µ − λ ρ + λ ρ, t D αt f λ ( t ) = A − β λ S ( t ) + β λ S ( t ) − ( γ + d + µ ) λ + γλ , t D αt f λ ( t ) = τ λ − λ ( τ + µ ) − A , (17)subject to the transversality conditions λ ( t f ) = A ,λ ( t f ) = A ,λ ( t f ) = 0 ,λ ( t f ) = 0 . (18)The optimal control variables are given by the stationary conditions: ( u ( t ) = ( λ ( t ) − λ ( t )) S ( t ) r ,u ( t ) = ( λ − (1 − ρ ) λ − ρλ ) E ( t ) r . (19)These analytic necessary optimality conditions are solved numerically in Section 5.
5. Numerical simulations
To illustrate the theoretical results presented in previous sections, here we use numerical simu-lations. For this purpose, a program was developed in
Matlab to integrate the necessary optimalityconditions and, with the help of a number of simulations, a detailed output is comprehensivelyverified. As explained in Section 4, we obtain the optimality system for the proposed optimalcontrol problem from the state and adjoint equations subject to suitable boundary conditions: theinitial conditions X (0) = X on the state variables, see (14); and the terminal conditions on theadjoint variables provided by the transversality conditions, see (18). Furthermore, we obtain theoptimal control strategies from the stationary system, see (19). We use a forward time/backwardspace finite-difference numerical method. Beginning with an initial guess for the adjoint variables,a forward time and backward space finite-difference method is used to solve the state equations.The key is to rewrite the control system (11) into the equivalent integral form X ( t ) = X (0) + 1Γ( α ) t Z ( t − τ ) α − f ( X ( τ ) , u ( τ )) dτ and then use the generalized Adams-type predictor-corrector method [21, 22] for solution. Further,these state values are used for the solution of the adjoint equations by a backward time and forwardspace finite-difference method, because of the transversality conditions. System (15) is written, inan equivalent way, as the integral equation λ ( t ) = ∂θ∂X (cid:12)(cid:12)(cid:12)(cid:12) t f + 1Γ( α ) t f Z t ( τ − t ) α − (cid:20) ∂φ∂X + λ T ∂f∂X (cid:21) dτ. Using a steepest-method to generate a successive approximation of the optimal control form, wecontinue iterating until convergence is achieved. For illustrative purposes, take the initial valuesas S (0) = 220, E (0) = 100, I (0) = 3, R (0) = 0 and parameter values as Λ = 0 . β = 0 . β = 0 . µ = 0 . ρ = 0 . γ = 0 . τ = 0 . p = 0 .
3, and d = 0 . S in the uncontrolled system (1), without controls, while the solid lines5 S u sc ep t i b l e c l a ss Figure 1: The susceptible population S ( t ), with and without controls, respectively solid and doted lines, for α = 0 . , . , . , E x po s ed c l a ss Figure 2: The exposed class E ( t ) of individuals, with and without controls, respectively solid and doted lines, for α = 0 . , . , . , S ( t ) in the controlled system (10), under optimal controls for α = 0 . .
85, 0 .
95 and 1.Figure 2 represents the exposed population of both systems (1) and (10). The doted lines showthat there are more exposed individuals when no control measures are implemented.Figure 3 illustrates the infectious population I ( t ) of system (1), without any control, andthat of system (10) with controls. The doted lines make it clear that there are more infectiousindividuals when no control is implemented. I n f e c t ed c l a ss Figure 3: Infected population for systems with and without controls, respectively solid and doted lines, for α =0 . , . , . , Finally, Figure 4 illustrates the recovered population R ( t ). We see that there are more recoveredindividuals in the case one uses optimal control theory (because there is less susceptible, exposedand infected).
6. Conclusion
The current pandemic situation due to COVID-19 affects the whole world on an unprece-dented scale. In this work, we implemented optimal control techniques to the COVID-19 pandemicthrough a fractional order model. For the eradication of virus spread throughout the world, weapplied two controls in the form of media (education) campaigns, social distance, use of masksand protection for the susceptible class; and quarantine (social isolation) for the exposed individ-uals. We discussed necessary optimality conditions for a general fractional optimal control prob-lem, whose fractional system is described in the Caputo sense while the adjoint system involvesRiemann–Liouville derivatives. In the COVID-19 setting, we minimize the number of susceptibleand infected population, while maximizing the number of recovered population from SARS-CoV-2virus. Using the fractional version of Pontryagin’s maximum principle, we characterize the opti-mal levels of the proposed controls. The resulting optimality system is solved numerically in the
Matlab numerical computing environment. Our numerical experiments were based on data of [27].In a future work, we plan to use real data of Africa, USA and UK.
Acknowledgement
D.F.M.T. was supported by The Center for Research and Development in Mathematics andApplications (CIDMA) through FCT, project UIDB/04106/2020. The authors are very grateful7 R e c o v e r ed c l a ss Figure 4: Recovered individuals for systems with and without controls, respectively solid and doted lines, for α = 0 . , . , . , to two Reviewers for several useful comments, suggestions and questions, which helped them toimprove the original manuscript. References [1] Chatterjee, K., Chatterjee, K., Kumar, A., & Shankar, S. (2020). Healthcare impact ofCOVID-19 epidemic in India: A stochastic mathematical model. Medical Journal ArmedForces India, 76(2), 147–155. https://doi.org/10.1016/j.mjafi.2020.03.022 [2] Duan, L., & Zhu, G. (2020). Psychological interventions for people af-fected by the COVID-19 epidemic. The Lancet Psychiatry, 7(4), 300–302. https://doi.org/10.1016/S2215-0366(20)30073-0 [3] Schett, G., Sticherling, M., & Neurath, M. F. (2020). COVID-19: risk for cytokine tar-geting in chronic inflammatory diseases? Nature Reviews Immunology, 20(5), 271–272. https://doi.org/10.1038/s41577-020-0312-7 [4] Seah, I., & Agrawal, R. (2020). Can the coronavirus disease 2019 (COVID-19) affect the eyes?A review of coronaviruses and ocular implications in humans and animals. Ocular Immunologyand Inflammation, 28(3), 391–395. https://doi.org/10.1080/09273948.2020.1738501 [5] Lau, H., Khosrawipour, V., Kocbach, P., Mikolajczyk, A., Schubert, J., Bania, J.,& Khosrawipour, T. (2020). The positive impact of lockdown in Wuhan on contain-ing the COVID-19 outbreak in China. Journal of Travel Medicine, 27(3), taaa037. https://doi.org/10.1093/jtm/taaa037 [6] Liang, R., Lu, Y., Qu, X., Su, Q., Li, C., Xia, S., Liu, Y., Zhang, Q., Cao, X., Chen, Q., &Niu, B. (2020) Prediction for global African swine fever outbreaks based on a combinationof random forest algorithms and meteorological data. Transbound Emerg. Dis., 67, 935–946. https://doi.org/10.1111/tbed.13424
87] Liu, W., Zhang, Q., Chen, J., Xiang, R., Song, H., Shu, S., & You, L. (2020). Detectionof Covid-19 in children in early January 2020 in Wuhan, China. New England Journal ofMedicine, 382(14), 1370–1371. https://doi.org/10.1056/NEJMc2003717 [8] Worldometer, COVID-19 Coronavirus Pandemic, Last accessed: 06-Jan-2021. [9] Lemos-Pai˜ao, A.P., Silva, C.J., & Torres, D.F.M. (2020) A new compartmental epidemio-logical model for COVID-19 with a case study of Portugal, Ecological Complexity, 44, Art.100885, 8 pp. https://doi.org/10.1016/j.ecocom.2020.100885 arXiv:2011.08741 [10] Nda¨ırou, F., Area, I., Nieto, J.J., & Torres, D.F.M. (2020) Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos Solitons Fractals, 135, 109846,6 pp. https://doi.org/10.1016/j.chaos.2020.109846 arXiv:2004.10885 [11] Zine, H., Boukhouima, A., Lotfi, E.M., Mahrouf, M., Torres, D.F.M., & Yousfi,N. (2020) A stochastic time-delayed model for the effectiveness of Moroccan COVID-19 deconfinement strategy, Math. Model. Nat. Phenom., 15, Paper No. 50, 14 pp. https://doi.org/10.1051/mmnp/2020040 arXiv:2010.16265 [12] The New York Times, Coronavirus Vaccine Tracker, Last accessed: 30-Nov-2020. [13] Kostylenko, O., Rodrigues, H.S., & Torres, D.F.M. (2019) The risk of contagion spread-ing and its optimal control in the economy, Stat. Optim. Inf. Comput., 7(3), 578–587. https://doi.org/10.19139/soic.v7i3.833 arXiv:1812.06975 [14] Lemos-Pai˜ao, A.P., Silva, C.J., Torres, D.F.M., & Venturino, E. (2020) Optimal control ofaquatic diseases: a case study of Yemen’s cholera outbreak, J. Optim. Theory Appl., 185(3),1008–1030. https://doi.org/10.1007/s10957-020-01668-z arXiv:2004.07402 [15] Sidi Ammi, M.R., & Torres, D.F.M. (2019) Optimal control of a nonlocal thermistorproblem with ABC fractional time derivatives, Comput. Math. Appl., 78(5), 1507–1516. https://doi.org/10.1016/j.camwa.2019.03.043 arXiv:1903.07961 [16] Castilho, C. (2006) Optimal control of an epidemic through educa-tional campaigns, Electron. J. Differential Equations, 2006(125), 11 pp. https://ejde.math.txstate.edu/Volumes/2006/125/castilho.pdf [17] Brandeau, M.L., Zeric, G.S., & Richter, A. (2003) Resource allocation for control of infectiousdiseases in multiple independent populations: Beyond cost-effectiveness analysis, Journal ofHealth Economics 22(4), 575–598. https://doi.org/10.1016/S0167-6296(03)00043-2 [18] Ball, F., & Becker, N. G. (2006) Control of transmission with two types of infection, Math.Biosci., 200(2), 170–187. https://doi.org/10.1016/j.mbs.2005.12.024 [19] Anastasio, T.J. (1994) The fractional order dynamics of bainstem vestibulooculomotor neu-rons, Biolog. Cybern., 72, 69–79. https://doi.org/10.1007/BF00206239 [20] Agrawal, O.P. (2004) A general formulation and solution scheme for fractional optimal controlproblems, Nonlinear Dyn., 38, 323–337. https://doi.org/10.1007/s11071-004-3764-6 [21] Diethelm, K., Ford, N.J., & Freed, A.D. (2002) A predictor-corrector approach forthe numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3–22. https://doi.org/10.1023/A:1016592219341 [22] Diethelm, K. , Ford, N.J., & Freed, A.D. (2004), Detailed error anal-ysis for a fractional Adams method, Numer. Algo., 36(1), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be https://doi.org/10.1115/1.2814055 [24] Tricaud, C., & Chen, Y. (2010) An approximation method for numerically solving frac-tional order control problems of general form, Comput. Math. Appl., 59(5), 1644–1655. https://doi.org/10.1016/j.camwa.2009.08.006 [25] Biswas, R.K., & Sen, S. (2011) Fractional optimal control problems with specified final time,J. Comput. Non. Dyn., 6(2), 021009, 6 pp. https://doi.org/10.1115/1.4002508 [26] Ali, S.A., Baloch, M., Ahmed, N., Ali, A.A., & Iqbal, A. (2020) The outbreak of coronavirusdisease 2019 (COVID-19)—An emerging global health threat, Journal of Infection and PublicHealth., 13(4), 644–646. https://doi.org/10.1016/j.jiph.2020.02.033 [27] Nda¨ırou, F., Area, I., Nieto, J.J., Silva, C.J., & Torres, D.F.M. (2021) Fractional modelof COVID-19 applied to Galicia, Spain and Portugal, Chaos Solitons Fractals 144 (2021),Art. 110652, 7 pp. https://doi.org/10.1016/j.chaos.2021.110652 arXiv:2101.01287 [28] Zhang, Z. (2020) A novel COVID-19 mathematical model with fractional deriva-tives: Singular and nonsingular kernels, Chaos Solitons Fractals, 139, 110060, https://doi.org/10.1016/j.chaos.2020.110060 [29] Zeb, A., Alzahrani, E., Erturk, V.S., & Zaman, G. (2020) Mathematical model for coronavirusdisease 2019 (COVID-19) containing isolation class, BioMed Research International, ArticleID 3452402, 7 pp. https://doi.org/10.1155/2020/3452402 [30] Yousaf, M., Muhammad, S.Z., Muhammad, R.S., & Shah, H.K. (2020) Statistical analysisof forecasting COVID-19 for upcoming month in Pakistan, Chaos, Solitons & Fractals, 138,109926. https://doi.org/10.1016/j.chaos.2020.109926 [31] Shah, K., Abdeljawad, T., Mahariq, I., & Jarad, F. (2020) Qualitative analy-sis of a mathematical model in the time of COVID-19, 2020, ID 5098598, 11 pp. https://doi.org/10.1155/2020/5098598 [32] Din, R.U., Shah, K., Ahmad, I., & Abdeljawad, T. (2020) Study of transmission dynamics ofnovel COVID-19 by using mathematical model, Advances in Difference Equations, 2020(323), https://doi.org/10.1186/s13662-020-02783-x [33] Zhang, Z., Zeb, A., Egbelowo, O.F., & Erturk, V.S. (2020) Dynamics of a fractional or-der mathematical model for COVID-19 epidemic, Advances in Difference Equations, 2020,Art. 420, 16 pp. https://doi.org/10.1186/s13662-020-02873-w [34] Zhang, Z., Zeb, A., Hussain, S., & Alzahrani, E. (2020) Dynamics of COVID-19 mathematicalmodel with stochastic perturbation, Advances in Difference Equations, 2020, Art. 451, 12 pp. https://doi.org/10.1186/s13662-020-02909-1 [35] Atangana, A. (2018) Blind in a commutative world: Simple illustrations withfunctions and chaotic attractors, Chaos, Solitons & Fractals, 114, 347–363. https://doi.org/10.1016/j.chaos.2018.07.022 [36] Atangana, A. (2020) Fractional discretization: The African’s tortoise walk, Chaos, Solitons& Fractals, 130, 109399. https://doi.org/10.1016/j.chaos.2019.109399 [37] Atangana, A. (2017) Fractal-fractional differentiation and integration: Connecting fractalcalculus and fractional calculus to predict complex system, Chaos, Solitons & Fractals, 102,396–406. https://doi.org/10.1016/j.chaos.2017.04.027 [38] Ghanbari, B., & Atangana, A. (2020) Some new edge detecting techniques based on fractionalderivatives with non-local and non-singular kernels, Advances in Difference Equations, 2020,Article 435, 19 pp. https://doi.org/10.1186/s13662-020-02890-9 https://doi.org/10.1016/j.cam.2017.09.039 arXiv:1709.07766 [40] Almeida, R., Pooseh, S., & Torres, D.F.M. (2015) Computational methods in the fractionalcalculus of variations, Imperial College Press, London. https://doi.org/10.1142/p991 [41] Nda¨ırou, F., Area, I., Nieto, J. J., Silva, C. J., & Torres, D.F.M. (2018) Mathematical model-ing of Zika disease in pregnant women and newborns with microcephaly in Brazil, Math. Meth-ods Appl. Sci., 41(18), 8929–8941. https://doi.org/10.1002/mma.4702 arXiv:1711.05630 [42] Area, I., Nda¨ırou, F., Nieto, J. J., Silva, C. J., & Torres, D.F.M. (2018) Ebola Model andOptimal Control with Vaccination Constraints, J. Ind. Manag. Optim. 14 (2018), no. 2, 427–446. https://doi.org/10.3934/jimo.2017054 arXiv:1703.01368https://doi.org/10.3934/jimo.2017054 arXiv:1703.01368