key: cord-1041925-mr0yzvp8 authors: Sweilam, N. H.; AL-Mekhlafi, S. M.; Almutairi, A.; Baleanu, D. title: A Hybrid Fractional COVID-19 Model with General Population Mask Use: Numerical Treatments date: 2021-02-05 journal: nan DOI: 10.1016/j.aej.2021.01.057 sha: d91e10ed5462963f1abb88df2f0240b71455038a doc_id: 1041925 cord_uid: mr0yzvp8 In this work, a novel mathematical model of Coronavirus (2019-nCov) with general population mask use with modified parameters. The proposed model consists of fourteen fractional-order nonlinear differential equations. Grünwald-Letnikov approximation is used to approximate the new hybrid fractional operator. Compact finite difference method of six order with a new hybrid fractional operator is developed to study the proposed model. Stability analysis of the used methods are given. Comparative studies with generalized fourth order Runge-Kutta method are given. It is found that, the proposed model can be described well the real data of daily confirmed cases in Egypt. In present time the whole globe is facing a threatful outbreak called COVID-19 which originated 22 from Wuhan; a large city in China. Since December 2019 to the 18th of July 2020, nearly 590,000 23 people died due to the mentioned disease and about 13.81 millions were all over the world. Also A novel coronavirus is a serious global issue and has a negative impact on the economy of 27 Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was 28 reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning 29 of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 The fractional mathematical models that are generalized model and considered useful for mod-52 eling purposes in epidemiology. Various benefits can be obtained from a fractional order system in 53 the sense of best data fitting, information about its memory and to identify the best possible value 54 of the fractional order that can best describe the model for the real cases . In addition, the heredi-55 tary properties make the models constructed in fractional derivatives stronger and more useful for where Γ is the Euler gamma function. • The Riemann-Liouville integral can be defined as follows [4]: where, 0 < α < 1 and y(t) is an integrable function. • The hybrid fractional operator is defined as follows [5]: Definition 2.1. [5] The Caputo proportional hybrid operator (CP) given in (3) is defined either as 97 general way : Or as the Caputo proportional constant hybrid operator (CPC) [5]: Definition 2.2. The inverse operators to the fractional CPC derivatives is given by[5]: The hybrid fractional COVID-19 model without any mask use is given: where, (E) denotes the exposed class, (I) denotes the symptomatic infectious class, (S) denotes the COVID-19 multi-group fractional order model given as follows: We can defined the feasible region for model (8) as follows: 112 Boundedness of the proposed model solution can be verified by adding all equations of system (8) as follows: where, A is a constant. The solution of 9 is given as follows [5] : It can be clearly seen from (10) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 In the following, we will use the fixed point theory. Let us rewrite (8)as follows: The vector , represents the state variables and b is a continuous vector function such that: g 5 g 6 g 7 g 8 g 9 g 10 g 11 g 12 g 13 with initial condition y 0 Moreover, the Lipschitz condition is satisfied by g, where g is a quadratic vector function i.e. there exists M 0 ∈ R, such that [27]: (12) 118 Theorem 3.1. The fractional proposed model (8) has unique solution if below condition holds: Proof. We apply the definition (5) on (11), we get: Let K = (0, T ) and the operator B : It gives: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Let . K denotes the supremum norm on K. Thus So, C(K, R 14 ) with . K is a Banach space. Moreover, the following relation holds: (15) can be written as: Finally, we obtain: where If L ≤ 1, then the operator B is called a contraction. Hence, the fractional system (8) . 127 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Then, where, R 0 is the basic reproduction number of the model, and ρ indicates the spectral radius of 128 F V −1 . 129 Let us consider the following fractional order differential equation: The first derivative approximation using the compact finite difference of six order method (CFD6M) is given as follows [11]: . (21). Then, we have: f (t j , y j ) + 1 60 y j−3 − 9 60 y j−2 + 45 60 y j−1 + 9 60 : Using the approximation of CPC-CFD6M (22) we can discretize (23) as follows: Then from boundness theorem [24] we have: Since, ( 45B 60 + C) > 1, then we have: |y 1 | < |y 0 | and |y 0 | ≥ |y 1 | ≥ ... ≥ |y n−1 | ≥ |y n | ≥ |y n+1 |. this means that, the proposed scheme is stable. Consider the following FODE: Then the approximate solution of (26) using GRK4M [8] is given as follows: 146 y n+1 = y n + 1 6 (K 1 + 2K 2 + 2K 3 + K 4 ), K 2 = κf (t n + 1 2 κ, y n + 1 2 K 1 ), 11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 where κ = τ α Γ(α + 1) . In order to study the stability of GRK4M. Consider for simplicity the following test problem: Using GRK4M [8], equation (28) can be written as follows: The stability analysis of GRK4M is similar to the GEM method [9], when the terms are regrouped, 153 the following equation is achieved: Then the stability condition [9] is given as follows: ) < 1. Table 2 . This section provides graphical results of the given study. Fig 1, shows the com- sion of COVID-19 for different subclasses of the total population. In Fig. 3-Fig. 10 , simulations the Table 2 and we put β = 0.5, η = 0.5 as given in 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 WHO Coronavirus Disease (COVID-19) To mask or not to mask: Modeling the potential for face mask use by the general 213 public to curtail the COVID-19 pandemic Fractional Differential Equations On a fractional operator combining proportional and 216 classical differintegrals Construction of nonstandard finite 218 difference schemes for the SI and SIR epidemic models of fractional order, Mathematics and 219 Computers in Simulation Reproduction numbers and sub-threshold endemic equilibria for 221 compartmental models of disease transmission Application of the Euler and Runge-Kutta gen-223 eralized methods for FDE and symbolic packages in the analysis of some fractional attrac-224 tors Optimal control for a nonlinear mathematical model of 226 tumor under immune suppression: a numerical approach Global existence theory and chaos control of fractional differential equations Numerical methods for fractional differentiation Optimal control for a fractional order 234 malaria transmission dynamics mathematical model A hybrid fractional optimal control for a 237 novel Coronavirus (2019-nCov) mathematical model Mathematical modeling of COVID-19 240 transmission dynamics with a case study of Wuhan Mathematical Modelling and Optimization of Engineer-243 ing Problems Fractional dynamics of an infection model with time-varying drug ex-245 An efficient numerical scheme for frac-248 tional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug, therapy A mathematical analy-251 sis of ongoing outbreak COVID-19 in India through nonsingular derivative The dynamics of COVID-19 with 254 quarantined and isolation -nCov) Mathematical Model; A Numerical Approach An efficient nonstandard finite difference scheme for 260 Rare and extreme events: the case of COVID-19 pandemic Construction of nonstandard finite 268 difference schemes for the SI and SIR epidemic models of fractional order, Mathematics and 269 Computers in Simulation On fractional SIRC model with 271 Salmonella bacterial infection Numerical Partial Differential Equations: Finite Difference Methods Coronavirus model with Mittag-Leffler law A mathematical model of COVID-19 using fractional 278 derivative: outbreak in India with dynamics of transmission and control Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using 282 A report on COVID-19 New results on existence in the framework of 287 Atangana-Baleanu derivative for fractional integro-differential equations The authors have declared no conflict of interest