Fractional model of COVID-19 applied to Galicia, Spain and Portugal
Faical Ndairou, Ivan Area, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres
aa r X i v : . [ q - b i o . P E ] J a n Fractional model of COVID-19 applied to Galicia, Spain and Portugal
Fa¨ı¸cal Nda¨ırou a,b, ∗ , Iv´an Area b , Juan J. Nieto c , Cristiana J. Silva a , Delfim F. M. Torres a, ∗ a Center for Research and Development in Mathematics and Applications (CIDMA),Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal b Departamento de Matem´atica Aplicada II, E. E. Aeron´autica e do Espazo, Campus de Ourense,Universidade de Vigo, 32004 Ourense, Spain c Instituto de Matematicas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Abstract
A fractional compartmental mathematical model for the spread of the COVID-19 disease is pro-posed. Special focus has been done on the transmissibility of super-spreaders individuals. Numer-ical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order ofthe Caputo derivative takes a different value, that is not close to one, showing the relevance ofconsidering fractional models.
Key words: mathematical modeling of COVID-19 pandemic, Galicia, Spain and Portugal casestudies, fractional differential equations, numerical simulations.
1. Introduction
Coronavirus disease 2019 (COVID-19), the outbreak due to severe acute respiratory syndromecoronavirus 2 (SARS-CoV-2), has taken on pandemic proportions in 2020, affecting several millionsof individuals in almost all countries [12]. An integrated science and multidisciplinary approach isnecessary to fight the COVID-19 pandemic [17, 18]. In particular, mathematical and epidemiolog-ical simulation plays a crucial role in predicting, anticipating, and controlling present and futureepidemics.As for the mathematical modelling of coronavirus disease COVID-19, it has been shown to beextremely useful for governments in order to define appropriate policies [19]. In this direction, anumber of papers has been recently published related with modelling of this pandemic (see, e.g.,[6, 9], just to cite some of them).In [19], a model including the super-spreader class has been presented, and applied to givean estimation of the infected and death individuals in Wuhan. The collaboration with Galiciangovernment [3] has allowed to understand some important considerations in order to performanalysis. In particular, due to the pandemic, some cases have not been reported as expected, butwith some days of delay. As a consequence, in this paper we propose to consider not the dailyreported cases, but the means in the previous 5 days of daily reported cases. As a result, it seemsappropriate to consider fractional derivatives, which have been intensively used to obtain modelsof infectious diseases since they take into account the memory effect, which is now bigger due tothe aforementioned mean of the five previous days of daily reported cases. Having estimates apriori of infected individuals of COVID-19, obtained by using mathematical models, has helpedto predict the number of required beds both for hospitalized individuals and mainly at intensivecare units [3]. ∗ Corresponding author: Delfim F. M. Torres (delfi[email protected])
Email addresses: [email protected] (Fa¨ı¸cal Nda¨ırou), [email protected] (Iv´an Area), [email protected] (Juan J. Nieto), [email protected] (Cristiana J. Silva), [email protected] (Delfim F. M. Torres)
Final form published by ’Chaos Solitons Fractals’. Submitted 4/Jun/2020; Revised 22/Jul and 2/Aug 2020; Accepted 4/Jan/2021. ractional calculus and fractional differential equations have recently been applied in numerousareas of mathematics, physics, engineering, bio-engineering, and other applied sciences. We referthe reader to the monographs [7, 11, 13, 22, 24, 25, 27] and the articles [1, 2, 20, 26]. In this workwe shall consider the Caputo fractional derivative [4] (see also [8]). A fractional model using theCaputo–Fabrizio fractional derivative of COVID-19 in Wuhan (China) has been developed in [21].The structure of this work is as follows. In Section 2, we introduce a fractional model by usingCaputo fractional derivatives on the classical compartmental model presented in [19], and wherethe fractional order of differentiation α can be used to describe different strains and genomes of thecoronavirus and vary with mutations. In Section 3, some numerical results are presented for threedifferent territories: Galicia, Spain, and Portugal. Galicia is an autonomous community of Spainand located in the northwest Iberian Peninsula and having a population of about 2,700,000 anda total area of 29,574 km2. Spain (officially, the Kingdom of Spain) is a country mostly locatedon the Iberian Peninsula, in southwestern Europe, with a population of about 47,000,000 peopleand a total area of 505,992 km2. Portugal (officially, the Portuguese Republic) is also a countrylocated mostly on the Iberian Peninsula with a population of about 10,276,000 individuals and atotal area of 92,212 km2. We end with Section 4 of conclusions and discussion.
2. The Proposed COVID-19 Fractional Model
In what follows we shall assume that we have a constant population divided in 8 epidemiologicalclasses, namely:1. susceptible individuals ( S ),2. exposed individuals ( E ),3. symptomatic and infectious individuals ( I ),4. super-spreaders individuals ( P ),5. infectious but asymptomatic individuals ( A ),6. hospitalized individuals ( H ),7. recovery individuals ( R ), and8. dead individuals ( F ) or fatality class.Our model is based on the one presented in [19] and substituting the first order derivative bya derivative of fractional order α . We use the fractional derivative in the sense of Caputo: for anabsolutely continuous function f : [0 , ∞ ) → R the Caputo fractional derivative of order α > D α f ( t ) = 1Γ(1 − α ) Z t ( t − s ) − α f ′ ( s ) ds. Fractional calculus and fractional differential equations are an active area of research and, insome cases, adequate to incorporate the history of the processes [1, 10, 14, 15, 16, 23, 27]. Thefractional proposed model takes the form D α S ( t ) = − β IN S − lβ HN S − β ′ PN S,D α E ( t ) = β IN S + lβ HN S + β ′ PN S − κE,D α I ( t ) = κρ E − ( γ a + γ i ) I − δ i I,D α P ( t ) = κρ E − ( γ a + γ i ) P − δ p P ,D α A ( t ) = κ (1 − ρ − ρ ) E,D α H ( t ) = γ a ( I + P ) − γ r H − δ h H,D α R ( t ) = γ i ( I + P ) + γ r H,D α F ( t ) = δ i I ( t ) + δ p P ( t ) + δ h H ( t ) , (1)2n which we have the following parameters:1. β quantifies the human-to-human transmission coefficient per unit time (days) per person,2. β ′ quantifies a high transmission coefficient due to super-spreaders,3. l quantifies the relative transmissibility of hospitalized patients,4. κ is the rate at which an individual leaves the exposed class by becoming infectious (symp-tomatic, super-spreaders or asymptomatic),5. ρ is the proportion of progression from exposed class E to symptomatic infectious class I ,6. ρ is a relative very low rate at which exposed individuals become super-spreaders,7. 1 − ρ − ρ is the progression from exposed to asymptomatic class,8. γ a is the average rate at which symptomatic and super-spreaders individuals become hospi-talized,9. γ i is the recovery rate without being hospitalized,10. γ r is the recovery rate of hospitalized patients,11. δ i denotes the disease induced death rates due to infected individuals,12. δ p denotes the disease induced death rates due to super-spreaders individuals,13. δ h denotes the disease induced death rates due to hospitalized individuals.A flowchart of model (1) is presented in Figure 2. For additional details and particular values ofthe parameters we refer the reader to [19]. Figure 1: Flowchart of model (1).
3. Numerical Simulations
Next, we shall show the numerical simulations in three territories: Galicia, Portugal, andSpain. For all these cases we have considered the official data published by the correspondingauthorities and we have computed the means of the five previous reports. As it has been observedduring this pandemic, the output of the laboratories has had some delays due to the big pressureand collapse of the public health systems. In this way, some cases have been reported withsome delay and some updates have been published days later of the corresponding dates. Inorder to reduce these problems, we consider the mean of the five previous reported cases, alwaysfollowing the official data. Moreover, in each of the territories there are specificities such asterritorial dispersion/concentration, use of public transportation, and mainly the date of startingthe confinement, as compared with the initial spread of the COVID-19. These factors imply tinyadjustments in the factor to divide the total population as well as in the value of the fractionalparameter α . For solving the system of fractional differential equations (1) we have used [5], byusing Matlab in a MacBook Pro computer with a 2.3 GHz Intel Core i9 processor and 16GB of2400 MHz DDR4 memory. 3 .1. The Case Study of Galicia In the autonomous region of Galicia, we have the values given in Table 3.1 as for the cumulativecases, the new daily infected individuals, as well as the mean of the 5 previous days.Date Confirmed Newconfirmed 5 daysmean Date Confirmed Newconfirmed 5 daysmean03-08 6 1 1 04-03 5625 406 380,403-09 22 16 4,2 04-04 5944 319 38103-10 35 13 6,4 04-05 6151 207 343,803-11 35 0 6,4 04-06 6331 180 297,803-12 85 50 16 04-07 6538 207 263,803-13 115 30 21,8 04-08 6758 220 226,603-14 195 80 34,6 04-09 6946 188 200,403-15 245 50 42 04-10 7176 230 20503-16 292 47 51,4 04-11 7336 160 20103-17 341 49 51,2 04-12 7494 158 191,203-18 453 112 67,6 04-13 7597 103 167,803-19 578 125 76,6 04-14 7708 111 152,403-20 739 161 98,8 04-15 7873 165 139,403-21 915 176 124,6 04-16 8013 140 135,403-22 1208 293 173,4 04-17 8084 71 11803-23 1415 207 192,4 04-18 8185 101 117,603-24 1653 238 215 04-19 8299 114 118,203-25 1915 262 235,2 04-20 8468 169 11903-26 2322 407 281,4 04-21 8634 166 124,203-27 2772 450 312,8 04-22 8805 171 144,203-28 3139 367 344,8 04-23 8932 127 149,403-29 3723 584 414 04-24 9116 184 163,403-30 4039 316 424,8 04-25 9176 60 141,603-31 4432 393 422 04-26 9238 62 120,804-01 4842 410 414 04-27 9328 90 104,604-02 5219 377 416
Table 1: Data of the autonomous region of Galicia. The list of 51 days includes the cumulative, new infected andmean of the previous 5 days.
The data includes 51 values starting 7th March since after that date (27th April) the way ofofficially computing individuals has changed.By considering the fractional order α = 0 .
85 and the same values of the parameters as in [19],the results of the numerical simulation are shown in Figure 3.1. The green line denotes the real datawhile the black line is the numerical solution of the fractional system (1), with total population N = 2 , , / N = S + E + I + P + A + H + R + F , since the population of Galiciais widely dispersed in the territory with very few big cities and low use of public transportation. As for the Kingdom of Spain, the data of 82 days is collected in Table 3.2, as for the cumulativecases, the new daily infected individuals, as well as the mean of the 5 previous days, starting 25thFebruary.By considering again the fractional order α = 0 .
85 and the same values of the parameters asin [19], the results of the numerical simulation are shown in Figure 3.2. The green line denotesthe real data while the black line is the numerical solution of the fractional system (1), with N = 47 , , /
425 since in some parts of Spain there is more concentrated population andintensive use of public transportation. 4 ime(in days, starting 7th march) -10 0 10 20 30 40 50 60 D a il y ne w c on f i r m ed c a s e s Figure 2: Number of confirmed cases per day in Galicia. The green line corresponds to the real data given inTable 3.1 while the black line ( I + P + H ) has been obtained by solving numerically the system of fractionaldifferential equations (1), by using [5]. Time(in days, starting 7th march) -10 0 10 20 30 40 50 60 70 80 90 D a il y ne w c on f i r m ed c a s e s Figure 3: Number of confirmed cases per day in Spain. The green line corresponds to the real data given Table3.2 while the black line ( I + P + H ) has been obtained by solving numerically the system of fractional differentialequations (1), by using [5]. As for the Republic of Portugal, the data of 56 days starting 3rd March for the cumulativecases, the new daily infected individuals, as well as the mean of the 5 previous days is collected inTable 3.3.By considering now the fractional order α = 0 .
75 and the same values of the parameters as in[19], the results of the numerical simulation are shown in Figure 3.3. As in the previous figures,the green line denotes the real data while the black line is the numerical solution of the fractionalsystem (1), with N = 10 , , /
4. Conclusions and Discussion
In this paper, we have shown the importance of considering a fractional Caputo differentialsystem, where the order of the derivative α plays a crucial role to fit the number of confirmedcases in the regions of Galicia, Spain and Portugal. In fact, the considered values of α = 0 .
85 forGalicia and Spain and α = 0 .
75 for Portugal, are not close to 1 (the classical derivative), as it5ate Confirmed Newconfirmed 5 daysmean Date Confirmed Newconfirmed 5 daysmean02-25 10 6 1,4 04-06 147717 5213 5676,202-26 18 8 3 04-07 153303 5586 5337,402-27 36 18 6,6 04-08 159051 5748 5151,402-28 55 19 10,4 04-09 163591 4540 4951,802-29 83 28 15,8 04-10 168151 4560 5129,403-01 138 55 25,6 04-11 172054 3903 4867,403-02 195 57 35,4 04-12 175087 3033 4356,803-03 270 75 46,8 04-13 178224 3137 3834,603-04 352 82 59,4 04-14 182662 4438 3814,203-05 535 183 90,4 04-15 186484 3822 3666,603-06 769 234 126,2 04-16 190308 3824 3650,803-07 1101 332 181,2 04-17 194150 3842 3812,603-08 1536 435 253,2 04-18 193437 713 3042,603-09 2309 773 391,4 04-19 195655 2218 2598,603-10 3285 976 550 04-20 198614 2959 242603-11 4442 1157 734,6 04-21 200968 2354 213203-12 5976 1534 975 04-22 203888 2920 1947,603-13 7659 1683 1224,6 04-23 206002 2114 251303-14 9806 2147 1499,4 04-24 208507 2505 2570,403-15 11515 1709 1646 04-25 210148 1641 2306,803-16 14018 2503 1915,2 04-26 211807 1659 2167,803-17 17713 3695 2347,4 04-27 213338 1531 189003-18 21764 4051 2821 04-28 214215 877 1642,603-19 26333 4569 3305,4 04-29 215470 1255 1392,603-20 31779 5446 4052,8 04-30 216757 1287 1321,803-21 36645 4866 4525,4 05-01 217992 1235 123703-22 41291 4646 4715,6 05-02 218894 902 1111,203-23 48984 7693 5444 05-03 219338 444 1024,603-24 57546 8562 6242,6 05-04 220362 1024 978,403-25 66503 8957 6944,8 05-05 221236 874 895,803-26 75691 9188 7809,2 05-06 222145 909 830,603-27 83944 8253 8530,6 05-07 223305 1160 882,203-28 90371 6427 8277,4 05-08 224048 743 94203-29 96184 5813 7727,6 05-09 224755 707 878,603-30 104332 8148 7565,8 05-10 227659 2904 1284,603-31 111745 7413 7210,8 05-11 228373 714 1245,604-01 119336 7591 7078,4 05-12 228978 605 1134,604-02 126616 7280 7249 05-13 229471 493 1084,604-03 133294 6678 7422 05-14 230228 757 1094,604-04 138832 5538 6900 05-15 230929 701 65404-05 142504 3672 6151,8 05-16 231350 421 595,4
Table 2: Data of the Kingdom of Spain. The list of 82 days includes the cumulative, new infected and mean of theprevious 5 days. happens in many of the proposed fractional compartmental models in the literature. Note thatthe same values of the parameters in the differential system (1), taken from [19], were used forthe three regions. Therefore, we may conclude that model (1) can be used to approximate theconfirmed cases of COVID-19 in regions with different economic, geographical, social and epidemiccharacteristics, as it happens for the three considered regions in this paper.6ate Confirmed Newconfirmed 5 daysmean Date Confirmed Newconfirmed 5 daysmean03-03 4 2 4 03-31 7443 1035 725,903-04 6 2 2 04-01 8251 808 750,903-05 9 3 3 04-02 9034 783 784,303-06 13 4 4 04-03 9886 852 802,603-07 21 8 8 04-04 10524 638 764,903-08 30 9 9 04-05 11278 754 759,403-09 39 9 5,5 04-06 11730 452 760,303-10 41 2 5,3 04-07 12442 712 714,103-11 59 18 7,6 04-08 13141 699 698,603-12 78 19 9,9 04-09 13956 815 703,103-13 112 34 14,1 04-10 15472 1516 79803-14 169 57 21,1 04-11 15987 515 780,403-15 245 76 30,7 04-12 16585 598 758,103-16 331 86 41,7 04-13 16934 349 743,403-17 448 117 58,1 04-14 17448 514 715,103-18 642 194 83,3 04-15 18091 643 707,103-19 785 143 101 04-16 18841 750 697,903-20 1020 235 129,7 04-17 19022 181 507,103-21 1280 260 158,7 04-18 20206 1184 602,703-22 1600 320 193,6 04-19 20863 657 611,103-23 2060 460 247 04-20 21379 516 63503-24 2362 302 273,4 04-21 21982 603 647,703-25 2995 633 336,1 04-22 22353 371 608,903-26 3544 549 394,1 04-23 22797 444 565,103-27 4268 724 464 04-24 23392 595 624,303-28 5170 902 555,7 04-25 23864 472 522,603-29 5962 792 623,1 04-26 24027 163 45203-30 6408 446 621,1 04-27 24322 295 420,4
Table 3: Data of the Republic of Portugal. The list of 56 days includes the cumulative, new infected and mean ofthe previous 5 days.
Time(in days) C on f i r m ed c a s e s pe r da y Figure 4: Number of confirmed cases per day in Portugal. The green line corresponds to the real data given inTable 3.3 while the black line ( I + P + H ) has been obtained by solving numerically the system of fractionaldifferential equations (1), by using [5]. FracPECE [5] toapproximate numerically the solution of the proposed fractional system of differential equations.Our numerical simulations show a good agreement between the output of the fractional modelgiven by the sum of the symptomatic and infectious individuals, super-spreaders, and hospitalizedindividuals and the data collected from the health authorities in Spain, Portugal and Galicia. Weplan to consider other countries and regions in our future studies and also, of course, an updateof the data. In the future, we also plan to study the stability of the possible equilibrium point,the bifurcation of solutions depending on the parameters, and the role of the basic reproductionnumber.Our fractional model is novel and in the future we will study the optimal fractional order ofdifferentiation for the study of the COVID-19 epidemic in different contexts. The system has aunique solution for given initial conditions and a detailed mathematical analysis study will beperformed. A crucial point is, of course, to determine the optimal fractional order α adequate foreach process and, in this case, each region.The results obtained here allow us to conjecture that the strains and genomes of the newcoronavirus present in Spain and Portugal are different than those that initially hit China: theproposed mathematical model is good to describe the outbreak that was first identified in Wuhanin December 2019 with α = 1; to describe the spread in Spain and its autonomous community ofGalicia, where the virus was first confirmed on January 31 and March 4 2020, respectively, with α = 0 .
85; and the COVID-19 situation in Portugal with α = 0 .
75, where the first cases of COVID-19 were recorded in March 2, 2020. We will continue our research using this and other futuremodels, as well as considering different approaches as the COVID-19 evolves and new insights andconjectures emerge.
Funding
This research was partially supported by the Portuguese Foundation for Science and Technology(FCT) within “Project n. 147 – Controlo ´Otimo e Modela¸c˜ao Matem´atica da Pandemia COVID-19:contributos para uma estrat´egia sist´emica de interven¸c˜ao em sa´ude na comunidade”, in the scopeof the “RESEARCH 4 COVID-19” call financed by FCT; and by the Instituto de Salud Carlos III,within the Project COV20/00617 “Predicci´on din´amica de escenarios de afectaci´on por COVID-19a corto y medio plazo (PREDICO)”, in the scope of the “Fondo COVID” financed by the Ministeriode Ciencia e Innovaci´on of Spain. The work of Nda¨ırou, Silva and Torres was also partiallysupported within project UIDB/04106/2020 (CIDMA); the work of Area and Nieto has beenpartially supported by the Agencia Estatal de Investigaci´on (AEI) of Spain, cofinanced by theEuropean Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyearfinancial framework, project MTM2016-75140-P. Moreover, Nda¨ırou is also grateful to the supportof FCT through the Ph.D. fellowship PD/BD/150273/2019; Nieto also thanks partial financialsupport by Xunta de Galicia under grant ED431C 2019/02. Silva is also supported by nationalfunds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017,of July 19.
Acknowledgment
The authors are grateful to the anonymous reviewers for their suggestions and invaluablecomments. 8 eferences [1] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. M. Torres, and Y. Zhou. A survey on fuzzyfractional differential and optimal control of nonlocal evolution equations. J. Comput. Appl.Math. 339 (2018), 3–29. doi:10.1016/j.cam.2017.09.039. arXiv:1709.07766 [2] A. Alshabanat, M. Jleli, S. Kumar, and B. Samet. Generalization of Caputo-Fabrizio frac-tional derivative and applications to electrical circuits. Front. Phys. 8 (2020), Art. 64, 10 pp.doi:10.3389/fphy.2020.00064.[3] I. Area, X. Hervada Vidal, J. J. Nieto, M.J. Purri˜nos Hermida. Determination in Galicia ofthe required beds at Intensive Care Units. Alexandria Engineering Journal 60 (2021), no. 1,559–564. doi:10.1016/j.aej.2020.09.034.[4] M. Caputo. Linear model of dissipation whose Q is almost frequency in-dependent. II. Geophysical Journal International 13 (1967), no. 5, 529–539.doi:doi:10.1111/j.1365-246x.1967.tb02303.x.[5] K. Diethelm, A. D. Freed. The FracPECE subroutine for thenumerical solution of differential equations of fractional order. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.2444&rep=rep1&type=pdf (2002).[6] M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi and A. Rinaldo, Spreadand dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures,PNAS 117 (2020), no. 19, 10484–10491.[7] F. Ge, Y. Q. Chen, and C. Kou.
Regional analysis of time-fractional diffusion processes .Springer, Cham, 2018.[8] A. N. Gerasimov. A generalization of linear laws of deformation and its application to problemsof internal friction. Akad. Nauk SSSR, Prikladnaya Matematika i Mekhanika 12 (1948), 251–259.[9] G. Giordano, F. Blanchini, R. Bruno, et al., Modelling the COVID-19 epidemic and imple-mentation of population-wide interventions in Italy, Nature Medicine 26 (2020), 855–860.[10] A. Goswami, J. Singh, D. Kumar, and Sushila. An efficient analytical approach for fractionalequal width equations describing hydro-magnetic waves in cold plasma. Physica A 524 (2019),563–575. doi:10.1016/j.physa.2019.04.058.[11] R. Hilfer.
Applications of Fractional Calculus in Physics . World Scientific Publishing Co.,Inc., River Edge, NJ, 2000. doi:10.1142/9789812817747.[12] Johns Hopkins Coronavirus Resource Center. Coronavirus COVID-19 Global Cases by theCenter for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU). 2020,May 27, 2020.[13] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo.
Theory and Applications of FractionalDifferential Equations . Amsterdam: Elsevier Science, 2006.[14] S. Kumar, Ali Ahmadian, Ranbir Kumar, D. Kumar, J. Singh, D. Baleanu, and M. Salimi.An efficient numerical method for fractional SIR epidemic model of infectious disease by usingBernstein wavelets. Mathematics 8 (2020), no. 4, Art. 558, 10 pp. doi:10.3390/math8040558.[15] D. Kumar, J. Singh, M. Al Qurashi, and D. Baleanu. A new fractional SIRS-SI malariadisease model with application of vaccines, anti-malarial drugs, and spraying. Advances inDifference Equations 2019 (2019), Art. 278, 19 pp. doi:10.1186/s13662-019-2199-9.916] D. Kumar, J. Singh, K. Tanwar, and D. Baleanu. A new fractional exothermicreactions model having constant heat source in porous media with power, exponen-tial and Mittag-Leffler Laws. Int. J. Heat and Mass Transfer 138 (2019), 1222–1227.doi:10.1016/j.ijheatmasstransfer.2019.04.094.[17] K. Mohamed, E. Rodr´ıguez-Rom´an, F. Rahmani, H. Zhang, M. Ivanovska, S.A. Makka SA, etal. Borderless collaboration is needed for COVID-19; a disease that knows no borders. Infect.Control Hosp. Epidemiol. 16 (2020), no. 5, 465–470. doi:10.1080/1744666X.2020.1750954.[18] N. Moradian et al. The urgent need for integrated science to fight COVID?19 pandemic andbeyond. J. Transl. Med. 18 (2020), Art. 205, 7 pp. doi:10.1186/s12967-020-02364-2.[19] F. Nda¨ırou, I. Area, J. J. Nieto, and D. F. M. Torres. Mathematical Modeling of COVID-19Transmission Dynamics with a Case Study of Wuhan. Chaos, Solitons & Fractals 135 (2020),Art. 109846, 6 pp. doi:10.1016/j.chaos.2020.109846. arXiv:2004.10885 [20] K. S. Nisar. Generalized Mittag-Leffler Type Function: Fractional Integrations andApplication to Fractional Kinetic Equations. Front. Phys. 8 (2020), Art. 33, 7 pp.doi:10.3389/fphy.2020.00033.[21] R. Prasad and R. Yadav. A numerical simulation of Fractional order mathematical modeling ofCOVID-19 disease in case of Wuhan China. Chaos, Solitons & Fractals 140 (2020), Art. 110124,17 pp. doi:10.1016/j.chaos.2020.110124.[22] S. G. Samko, A. A. Kilbas, and O. I. Marichev.
Fractional Integrals and Derivatives. Theoryand Applications . Amsterdam: Gordon and Breach, 1993.[23] J. Singh, D. Kumar, and D. Baleanu. A new analysis of fractional fish farm model associ-ated with Mittag-Leffler type kernel. Int. J. Biomath. 13 (2020), no. 2, Art. 2050010, 17 pp.doi:0.1142/S1793524520500102.[24] V. E. Tarasov.
Fractional Dynamics: Application of Fractional Calculus to Dynamics of Par-ticles, Fields and Media . Springer, Heidelberg, Higher Education Press, Beijing, 2010.[25] D. Val´erio and J. S´a da Costa.
An introduction to fractional control . London: Institution ofEngineering and Technology (IET), 2013.[26] T. A. Yıldız. Optimal control problem of a non-integer order waterborne pathogenmodel in case of environmental stressors. Frontiers in Physics 7 (2019), Art. 95, 10 pp.doi:10.3389/fphy.2019.00095.[27] Y. Zhou.