key: cord-0878494-xoau3z0e authors: Atangana, Abdon; Araz, Seda İğret title: A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton polynomial date: 2021-02-16 journal: nan DOI: 10.1016/j.aej.2021.02.016 sha: e35b832f5da92ade76ee1713e3fa0e11d92d349d doc_id: 878494 cord_uid: xoau3z0e To capture more complexities associated to the spread of Covid-19 within a given population, we considered a system to nine differential equations that include a class of susceptible, 5 sub-classes of infected population, recovered, death and vaccine. The mathematical model was suggested with a lockdown function such that after the lockdown, the function follows a fading memory rate, a concept that is justified by the effect of social distancing that suggests, susceptible class should stay away from infected objects and humans. We presented a detailed analysis that includes reproductive number and stability analysis. Also, we introduced the concept of fractional Lyapunov function for Caputo, Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives. We established the sign of the fractional Lyapunov function in all cases, additionally we proved that, if the fractional order is one, we recover the results for the model with classical differential operators. With the nonlinearity of the differential equations depicting the complexities of the Covid-19 spread especially the cases with nonlocal operators, and due to the failure of existing analytical methods to provide exact solution to the system, we employed the newly introduced numerical method based on the Newton polynomial to derive numerical solutions for all cases and numerical simulations are provided for different values of fractional orders and fractal dimensions. Collected data from Turkey case for a period of 90 days were compared with the suggested mathematical model with Atangana-Baleanu fractional derivative and a agreement was reached for alpha 1.009 . information as possible. As for example, very recently, a complex mathematical model comprising nine components 39 was suggested by Atangana and Seda [5] . In their model, they considered a total population that can be divided in 9 40 classes including: S (t) is the class of individuals that are susceptible to contact Covid-19 at time t: I (t) is the class of 41 individuals that are susceptible to contacted Covid-19, but have no symptoms and have not been tested. I A (t) is the 42 class of individuals that have some symptoms but not tested yet. I D (t) is the class of individuals that have contacted 43 Covid-19, have been tested positive, but no symptoms. I R (t) is the class of individuals that have contacted Covid19, 44 have been tested positive and have symptoms. I T (t) is the class of individuals that have contacted Covid-19 and 45 one is critical condition. R (t) is the class of recovered individuals at time t. D (t) is the number of death at time t: 46 V (t) is the class of individuals that have been vaccination, while their model has included parameters, they assumed 47 no transmission from class of individuals that have contacted Covid-19 and one is critical condition to susceptible 48 class. In this work, we assume that there is transmission between these two classes. Additionally, we will introduce 49 the concept of fractional Lyapunov function. 50 Many mathematical models about Covid-19 spread were investigated to understand, analyze and interpret the analysis including stability analysis, optimal control for this model. 59 The organization of this paper is as follows. In section 2, some definitions of differential and integral operators are 60 presented. In section 3, the comprehensive mathematical model presented in [5] has been modified with the addition of 61 the lockdown function. In section 4, the equilibrium points for both disease-free and endemic are obtained. In section 62 5, we calculate the reproductive number by using next generation technique. In section 6, the global asymptotic 63 stability for disease-free equilibrium is presented considering the Lyapunov function for classical differentiation and To accommodate researchers that are not acquainted to the concept of non-local differential operators with singular 73 and non-singular kernels, also, to differential and integral operators with fractal dimension and fractional orders, 74 we present in this section, some definitions of differential and integral operators starting with Caputo fractional Caputo-Fabrizio fractional derivative Atangana-Baleanu fractional derivative 78 ABC 0 The fractal-fractional derivative with power-law kernel, exponential decay and Mittag-Leffler kernel are given by; The fractal-fractional integral with power-law, exponential decay and Mittag-Leffler kernel are as below; This assumption could be argued as there is interaction between these classes and medical staffs, thus, a possibility 87 of transmission from these class to susceptible belonging to medical staffs. In this section, we modified their model 88 by introducing transmission between the class of individuals that have contacted and one is critical condition with 89 susceptible. 90 · S = Λ − {δ (t) (αI + w (βI D + γI A + δ 1 I R + δ 2 I T ) + γ 1 + µ 1 )} S · I = δ (t) (αI + w (βI D + γI A + δ 1 I R + δ 2 I T )) S − (ε + ξ + λ + µ 1 ) I · I A = ξI − (θ + µ + χ + µ 1 ) I A Here, δ 2 is the contact rate of I T (t) class and S (t) class and we take as S = S1 N . For example, medical staff that are 91 in charge of the patients of class I T (t) .We consider δ (t) to be function of time as the containment rule become law 92 at time t 0 thus, according to [21] , we can take 93 δ (t) = δ 0 for t ≤ t 0 δ 0 (1 − α r ) e − t−t 0 ∆t for t ≥ t 0 where α r is a fractional number accounting for asymptotic decrease of infection rate afforded by the lockdown measures α r ∈ [0, 1] . The descriptions of the parameters for the considered model are given in Table 1 . 4 Equilibrium points: Disease free and endemic 95 In this section, we derive the equilibrium points for both disease-free and endemic, but the death class is not 96 considered in this analysis. The disease-free equilibrium is given as The endemic equilibrium is obtained by solving the following system For simplicity, we put Therefore, Also, However Thus, replacing I * , we get 5 Reproductive number 108 We derive the reproductive number Here, using the next generation matrix [19] . To achieve this, we consider the 109 following equation From the above, the following matrices are derived Also, we have there is no endemic equilibrium if R 0 = 1. However the disease is endemic 118 δ (t) (αI + w (βI D + γI A + δ 1 I R + δ 2 I T )) S − l 2 I > 0, Therefore, a unique endemic exists when R 0 > 1. The Jacobian matrix associated to disease-free equilibrium is given Thus, T r (J (E * )) = − (δ 0 l 1 + l 2 + l 3 + l 4 + l 5 + l 6 + l 7 + l 8 ) < 0 (30) and 122 Det (J (E * )) = δ 0 l 1 l 2 l 3 l 4 l 5 l 6 l 7 l 8 . system are studied within the framework of classical mechanic especially when using classical differential operators. In the last decades several authors studies fractional dynamical system for example epidemiological model. To study 127 the stability of a fractional system , almost all the researchers use the classical Lyapunov function. This has been 128 going on in many papers but the classical differentiation generate an energy with no memory effect. While the 129 system with fractional derivatives generate an energy with accumulative information. It is perhaps unfair to evaluate 130 a classical Lyapunov for a fractional system. In this section, we will introduce an analysis with fractional Lyapunov. Now, we present the Lyapunov for different cases when the model is with classical differentiation and fractional 132 derivatives 133 · L = 1 l 2 With fractional derivative, we consider the case Caputo derivative We start with the classical case Dividing by (I + I D + I A + I R + I T ) , we have 136 = S l2 (δ (t) (αI + w (βI D + γI A + δ 1 I R + δ 2 I T ))) + 1 l3 ξI + 1 l4 εI Thus, dL dt ≤ S l2 (δ (t) (αI + w (βI D + γI A + δ 1 I R + δ 2 I T ))) + 1 l3 ξI + 1 l4 εI Hence, the function L is the Lyapunov function on a largest compact ∆ invariant set in S, I, I D , I A , I R , I T , R, D, V ∈ ∆ : dL dt ≤ 0 139 is the point E * . Thus, using Lasalle's invariance principle all solution of the system with initial condition in ∆ tends We consider a general fractional derivative, we have the following Replacing each fractional class by its value, we obtain again Therefore, following the routine presented earlier, we obtain We compute first the Jacobian matrix of the Covid-19 model for endemic equilibrium case We now construct a characteristic equation associated to this model From the above, we obtain the following polynomial 150 K (λ) = λ 8 + k 1 λ 7 + k 2 λ 6 + k 3 λ 5 + k 4 λ 4 + k 5 λ 3 + k 6 λ 2 + k 7 λ + k 8 . The Hurwitz matrix for the characteristic polynomial K (λ) is written as Then, we have 1 k 2 k 6 − k 2 1 k 2 4 − k 1 k 2 2 k 5 + k 1 k 2 k 3 k 4 − k 1 k 2 k 7 − k 1 k 3 k 6 + 2k 1 k 4 k 5 +k 2 k 3 k 5 − k 2 3 k 4 + k 3 k 7 − k 2 5 > 0 H 5 = k 3 1 k 4 k 8 − k 3 1 k 2 6 − k 2 1 k 2 k 3 k 8 − k 2 1 k 2 k 4 k 7 + 2k 2 1 k 2 k 5 k 6 + k 2 1 k 3 k 4 k 6 +k 1 k 2 2 k 3 k 7 − k 1 k 2 2 k 2 5 − k 1 k 2 k 2 3 k 6 + k 1 k 2 k 3 k 4 k 5 − k 2 1 k 5 k 8 + 2k 2 1 k 6 k 7 −k 1 k 2 k 5 k 7 − 3k 1 k 3 k 5 k 6 + 2k 1 k 4 k 2 5 − k 2 k 2 3 k 7 + k 2 k 3 k 2 5 +k 1 k 2 3 k 8 − k 2 1 k 2 4 k 5 + k 3 3 k 6 − k 2 3 k 4 k 5 − k 1 k 2 7 + 2k 3 k 5 k 7 − k 3 5 > 0 −2k 2 1 k 4 k 7 k 8 − 2k 2 1 k 5 k 6 k 8 + 3k 2 1 k 2 6 k 7 − 3k 1 k 2 k 3 k 7 k 8 + k 2 3 k 2 4 k 7 − 3k 1 k 6 k 2 7 −k 3 5 k 6 + 3k 1 k 2 k 4 k 2 7 − k 1 k 2 k 5 k 7 k 6 + k 1 k 2 3 k 6 k 8 + 2k 1 k 4 k 3 k 5 k 8 − k 3 k 8 k 2 5 +k 2 k 3 k 6 k 2 5 + k 1 k 4 k 3 k 6 k 7 − 3k 1 k 3 k 5 k 2 6 − 2k 1 k 2 4 k 5 k 7 + 2k 1 k 4 k 2 5 k 6 +k 2 k 2 3 k 5 k 8 − 2k 2 k 2 3 k 6 k 7 − k 2 k 3 k 4 k 5 k 7 − k 3 3 k 4 k 8 + k 3 3 k 2 6 + k 3 7 −k 2 3 k 4 k 5 k 6 + 2k 1 k 7 k 5 k 8 − k 2 k 2 7 k 5 + k 2 3 k 7 k 8 − 2k 4 k 3 k 2 7 + 3k 5 k 3 k 6 k 7 > 0 H 7 = k 4 1 k 3 8 − 3k 3 1 k 2 k 7 k 2 8 − k 3 1 k 3 k 6 k 2 8 − 2k 3 1 k 2 8 k 4 k 5 + 3k 3 1 k 4 k 6 k 8 k 7 − k 3 3 k 5 k 6 k 8 +k 3 1 k 5 k 2 6 k 8 − k 3 1 k 3 6 k 7 + 3k 2 1 k 2 2 k 2 7 k 8 + 3k 2 1 k 2 k 3 k 5 k 2 8 − k 2 1 k 2 k 3 k 6 k 7 k 8 + k 4 3 k 2 8 +k 2 1 k 2 k 5 k 4 k 7 k 8 − 3k 2 1 k 2 k 4 k 6 k 2 7 − 2k 2 1 k 2 k 8 k 6 k 2 5 − k 2 k 3 k 4 k 5 k 2 7 + k 3 3 k 2 6 k 7 +2k 2 1 k 2 k 5 k 2 6 k 7 + k 2 1 k 4 k 2 3 k 2 8 − 2k 2 1 k 2 4 k 3 k 7 k 8 − k 2 1 k 3 k 4 k 5 k 6 k 8 − 2k 3 3 k 4 k 7 k 8 +k 2 1 k 2 6 k 3 k 4 k 7 + k 2 1 k 3 4 k 2 7 + k 2 1 k 2 4 k 2 5 k 8 − k 2 1 k 2 4 k 5 k 6 k 7 − k 1 k 3 2 k 3 7 + 4k 1 k 5 k 2 7 k 8 −3k 1 k 2 2 k 3 k 5 k 7 k 8 + 2k 1 k 2 2 k 3 k 6 k 2 7 + k 1 k 2 2 k 4 k 5 k 2 7 + k 1 k 2 2 k 3 5 k 8 + 3k 3 k 5 k 6 k 2 7 −k 1 k 2 2 k 2 5 k 6 k 7 − k 1 k 2 k 3 3 k 2 8 + 2k 1 k 2 k 2 3 k 4 k 7 k 8 + k 1 k 2 k 2 3 k 5 k 6 k 8 − k 1 k 2 k 2 3 k 2 6 k 7 −k 1 k 2 k 3 k 2 4 k 2 7 − k 1 k 2 k 3 k 4 k 2 5 k 8 + k 1 k 2 k 3 k 4 k 5 k 6 k 7 − k 2 k 3 k 3 5 k 8 + k 2 k 3 k 2 5 k 6 k 7 +4k 2 1 k 3 k 7 k 2 8 − 3k 2 1 k 4 k 2 7 k 8 + 2k 2 1 k 2 5 k 2 8 + k 2 2 k 3 k 3 7 + 3k 2 k 2 3 k 5 k 7 k 8 + k 2 3 k 2 4 k 2 7 −5k 2 1 k 5 k 6 k 7 k 8 + 3k 2 1 k 2 6 k 2 7 − 5k 1 k 2 k 3 k 2 7 k 8 + 3k 1 k 2 k 4 k 3 7 + k 1 k 2 k 2 5 k 7 k 8 −k 1 k 2 k 5 k 6 k 2 7 − 4k 1 k 2 3 k 5 k 2 8 + k 1 k 2 3 k 6 k 7 k 8 + 4k 1 k 3 k 4 k 5 k 7 k 8 + k 1 k 3 k 4 k 6 k 2 7 +3k 1 k 3 k 2 5 k 6 k 8 − 3k 1 k 3 k 5 k 2 6 k 7 − 2k 1 k 2 4 k 5 k 2 7 − 2k 1 k 4 k 3 5 k 8 + 2k 1 k 4 k 2 5 k 6 k 7 −2k 2 k 2 3 k 6 k 2 7 + k 2 3 k 2 Theorem 1. If R 0 ≥ 1, then the endemic point E * is globally asymptotically stable. Proof. We prove thus using the idea of a fractional Lyapunov function. We start by defining the Lyapunov 154 function associated the system We apply a fractional differential derivative on both sides, in particular without loss of generality, we use the Caputo where 0 < α L ≤ 1. Of course the power law kernel can be replaced by AB(α) Thus, ) 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 and 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 Thus, in the case of ABC derivative, we obtain and Caputo-Fabrizio, we have Therefore, for all three cases the fractional Lyapunov It is worth noting that when the fractional order α L = 1, we recover the case of classical model. In this section, we present error analysis for a general Cauchy problem for some fractional differential operators which 177 are Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu fractional operators. 178 We start with a general Riemann-Liouville case We convert this equation as follows; 180 For derivation numerical solution based on the Newton polynomial, we write Therefore, we approximate Cauchy problem as follows 184 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 with the following error Then, we next evaluate Replacing the functions f by its value, we have Thus, we write We calculate We consider (t l ) m l=0 and assume that (t l ) are equally-spaced. At the point So, in general, we obtain such that 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 s ∈ (0, m) . Taking the absolute value on both sides, we get For any given s ∈ (0, m) , let a be an integer such that a < s < a + 1. It then follows that and Therefore, For s > a, we get the following Thus, we conclude that It follows that Putting everything together, we obtain With Atangana-Baleanu derivative, we consider the following problem Integrating the above equation, we can have the following Following the methodology presented to derive numerical solution using the Newton polynomial, we have 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 Therefore, the considered problem can be written with the following error Thus, we have we conclude that Finally, we present error analysis for Caputo-Fabrizio case Applying the Caputo-Fabrizio integral, we get 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 Using the Newton polynomial, we have Therefore, the general Cauchy problem can be approximated with the following error Therefore, we evaluate the error Therefore, the following is written We know that Then, putting the above we have the following 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 For simplicity, we write above equation as follows; After applying fractional integral with exponential kernel and putting Newton polynomial into these equations, we 227 can solve our model as follows 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 We can have the following numerical scheme for Mittag-Leffler case 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 Finally, we can have the following numerical approximation with Caputo derivative 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 64 252 After applying fractal-fractional integral with exponential kernel, we have the following scheme for this model For Mittag-Leffler kernel, we can get the following numerical scheme For power-law kernel, we can get the following numerical scheme 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 this section, using the numerical solutions obtained the previous section, we present numerical method for all cases. 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 initial conditions are 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 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 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 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 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 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 63 64 65 The comparison is performed between the suggested model with Atangana-Baleanu derivative and collected data 295 in Figure 16 for 30 days, in Figure 17 for 60 days and finally in Figure 18 for 90 days. In the first 30 days, the 296 37 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 daily numbers of new infections followed an exponential spread, after 60 to 90 days the daily numbers followed a 297 lognormal distribution. The comparison between collected data and mathematical model although not exactly in 298 perfect agreement but most of the points are depicted by the mathematical models. This comparison was obtained 299 for a fractional order 1.009. We stress on the fact that our model is the perfect one. However the model can be used 300 to provide a trend of the spread in different countries. 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 The authors have declared no conflict of interest. 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 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 Modelling the spread of Covid-19 with new fractal-fractional operators: Can the lockdown save 317 mankind before vaccination? 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