Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 136263, 9 pages http://dx.doi.org/10.1155/2014/136263

Research Article

On Fractional SIRC Model with Salmonella Bacterial Infection

Fathalla A. Rihan,1,2 Dumitru Baleanu,3,4,5 S. Lakshmanan,1 and R. Rakkiyappan6

1 Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, AlAin, UAE

2 Faculty of Science, Helwan University, Cairo, Egypt

3 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 80204, Saudi Arabia

4 Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey

5 Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania

6 Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu 641046, India

Correspondence should be addressed to Fathalla A. Rihan; frihan@uaeu.ac.ae Received 3 March 2014; Accepted 25 March 2014; Published 22 April 2014 Academic Editor: Ali H. Bhrawy

Copyright © 2014 Fathalla A. Rihan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number R0 is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo's derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

1. Introduction

During the past three decades, the subject of fractional calculus has gained popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields of science and engineering. For example, fractional calculus has been successfully applied to system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance (see, e.g., [1-10] and the references therein). In many cases, the fractional order differential/integral equations models are more consistent with the real phenomena than the integer-order models because the fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence, there is a growing need to study and use the fractional order differential and integral equations. However, analytical and closed solutions of these types of fractional equations cannot generally be obtained. As a consequence, approximate and numerical techniques are playing important role in identifying the solution behavior of such fractional equations and exploring their applications (see, e.g., [9,11,12] and the references therein).

We recall that the Salmonella infection is a major zoonotic disease which is transmitted between humans and other animals. Most persons infected with Salmonella develop diarrhea, fever, and abdominal cramps 12 to 72 hours after infection. The illness usually lasts 4 to 7 days, and most persons recover without treatment. However, in some persons, the diarrhea may be so severe that the patient needs to be hospitalized. Salmonella live in the intestinal tracts of humans and other animals, including birds. Salmonella are usually transmitted to humans by eating foods contaminated with animal feces. Contaminated foods usually look and smell normal. Contaminated foods are often of animal origin, such as beef, poultry, milk, or eggs, but any food, including vegetables, may become contaminated [13]. Therefore, Salmonella is considered as a serious problem for the public health throughout the world. There are no doubts that mathematical modeling of Salmonella bacterial infection plays an important role in gaining understanding of the transmission of the disease in specific environment and in predicting the behavior of any outbreak. Furthermore, mathematical analysis leads to determining the nature of equilibrium states and to suggesting recommended actions

to be taken by decision makers to control the spreading of the disease. The objective of this work is to adopt the fractional order epidemic model to describe the dynamics of Salmonella infections in animal herds.

A large amount of work done on modelling biological systems has been restricted to integer-order ordinary (or delay) differential equations (see, e.g., [14-19]). In [20], the authors proposed the classical Susceptible-Infected-Recovered (SIR) model. The authors in [21] introduced a new compartment into SIR model, which is called cross-immune compartment to be called SIRC model. The new compartment cross-immune C(t) describes an intermediate state between the fully susceptible S(t) and the fully protected R(t) one. Recently, the fractional order SIRC model of influenza, a disease in human population, was discussed in [22]. In the present paper, we consider the fractional order SIRC model associated with evolution of Salmonella bacterial infection in animal herds. However, we will take into account the disease-induced mortality rate m in the model. Qualitative behavior of the fractional order SRIC model is then investigated. Numerical simulations of the fractional order SRIC model are provided to demonstrate the effectiveness of the proposed method by using implicit Euler's method.

We first give the definition of fractional order integration and fractional order differentiation [23-25]. Let L1 = L1 [a, b] be the class of Lebesgue integrable functions on [a,b], a < b < >x>.

Definition 1. The fractional integral of order v e R+ of the

function f(t), t>0 (f: R+

is defined by

IVJ(t) = 7T-, \ (t-s)v-1f(s)ds, t>0. (1) r (V) Ja

However, the fractional derivative of order a e (n - 1,n) of f(t) is defined by two ways.

(i) Riemann-Liouville fractional derivative: take fractional integral of order (n - a) and then take nth derivative:

DaJ(t) = Dna irf(t), D: = ^n, n=1,2,.... (2)

(ii) Caputo's fractional derivative: take nth derivative and then take a fractional integral of order (n- a):

Daaf(t) = i:-aDnJ(t), n= 1,2, —

We notice that the definition of time-fractional derivative of a function f(t) at t = tn involves an integration and calculating time-fractional derivative that requires all the past history, that is, all the values of f(t) from t = 0 to t = tn. Caputo's definition, which is a modification of the Riemann-Liouville definition, has the advantage of dealing properly with initial value problems. For more properties of the fractional derivatives and integrals, we refer to [8, 9, 24, 25] and references therein. The generalized mean value theorem is defined in the following Remark [26].

Remark 2. (i) Suppose that f(t) e C[a,b] and D"f(t) e C(a, b] for 0 < a < 1; then we have

f(t) = f(a) + -^D:f(Z)(t-aT,

r(a) (4)

with a < £ < t Vt e (a,b].

(ii) If (i) holds and DaJ(t) > 0Vt e [a, b], then f(t) is nondecreasing for each t e [a,b].If Da:f(t) < 0 Vt e [a,b], then f(t) is nonincreasing for each t e [a, b].

2. The SIRC Epidemic Model

Assume that the Salmonella infection spreads in animal herds which are grouped as four compartments, according to their infection status: S(t) is the proportion of susceptible individuals at time t (individuals that do not have the bacterial infection), I(t) is the proportion of infected individuals (that have the bacterial infection), R(t) is the proportion of recovered individuals (that recovered from the infection and have temporary immunity), and C(t) is the proportion of cross-immune individuals at time t. The total number of animals in the herd is given by N = S +1 + R + C. We consider that initially all the animals are susceptible to the infection. Once infected, a susceptible individual leaves the susceptible compartment and enters the infectious compartment where it then becomes infectious. The infected animals pass into the recovered compartment. The individuals who have recovered from the disease have temporary immunity and grouped into C(t) compartment. Therefore, we consider that the disease transmission model consists of nonnegative initial conditions together with system of equations:

S (t) = pN + qC (t) - (ßl (t) + (t), i (t) = ßS (t) I (t) + aßC (t) I(t)-(6 + m + p)I (t), R(t)= (1-a)ßC (t) I (t) + Ol (t) -(p + 8)R (t), C (t) = SR (t) - ßC (t) I (t) -(t] + ^)C (t).

Here, parameter ^ denotes the mortality rate in every compartment and is assumed to be equal to the rate of newborns in the population. p is the contact rate and also called transmission from susceptible to infected. rfl is the cross-immune period, 9- is the infectious period, S- is the total immune period, and a is the fraction of the exposed cross-immune individuals who are recruited in a unit time into the infective subpopulation [21, 27]. We also assume that the disease induces mortality rate m; see the diagram of Figure 1.

2.1. Fractional Order of SIRC Epidemic Model. Although a large number of work has been done in modeling the dynamics of epidemiological diseases, it has been restricted to integer-order (delay) differential equations. In recent years, it has turned out that many phenomena in different fields can be described very successfully by models using fractional order differential equations (FODEs) [1, 5, 28]. Now, we introduce

S p J I e i s C

(1 -a)p

Figure 1: Schematic diagram of SIRC model.

fractional order into model (5) and assume that s(t) = S(t)/N, i(t) = S(t)/N, r(t) = R(t)/N, c(t) = C(t)/N, where N is the total number of population. Then the model takes the form

D"1s (t) = y + qc (t) - (ßi (t) + y) s (t), D"2i (t) = ßs (t) i (t) + aßc (t) i (t)-(d + m + y) i (t), D% r (t) = (1 - a) ßc (t) i (t) + di (t) -(y + S)r (t), Da*c (t) = Sr (t) - ßc (t) i (t) -(t] + y)c (t) with initial conditions s(0) = s0, i(0) = i0, and r(0) = r0.

2.2. Stability Criteria for the Fractional Order SIRC Model. In model (6), assume that a1 = a2 = a3 = a4 = a. Then, to find the equilibria, we put Das = Dai = Dar = Dac = 0. _I

Consider

E0 = (1,0,0,0),

* d + m + y

Or? { * • * * * \

E+ = (s ,i ,r ,c ),

ß + ßi* + (y + S)(y + q)

di* (ßi* + q + y)

(y + Sa) ßi* + (y + S)(y + r()'

(y + Sa) ßi* + (y + S)(y + q)

The positive endemic equilibrium E+ = (s*,i*,r*,c*) satisfies (6) and i* is the positive root of A 1i* + A2i* + A3, where

Aj = - ß2 [m (y + Sa) + y(d + y + 5a)], A 2 = ß [ßy (y + Sa) + qdS - (d + m + y)

x [(y + S)(y + r() + y(y + 5a)] + yaSd], ' Q + m + ys

A 3 = ßy(y + S) (y + n)

The Jacobian matrix of model (6) is

/ -ßi (t) - y -ßs (t) 0 q

ßi (t) ßs (t) + aßc (t)-(d + m + y) 0 aßi (t)

0 (1-a)ßc(t) + 9 -(p + S) (1-a)ßi(t)

-ßc(t)

S -ßi(t)-(n + y),

2.3. The Reproduction Number R0. The basic reproduction number (the number of individuals infected by a single infected individual placed in a totally susceptible population) R0 that includes the indirect transmission may be obtained using next-generation matrix method. The spectral radius of the next-generation matrix (FV-1), which is the dominant eigenvalue of the same matrix, gives the value of R0 [29]. Then, the basic reproductive number R0 is obtained by the form

ro = P{FV-1),

where the matrices F

= [dFi(x)/dxj],

and v =

[dVi(x)/dxj]x=Xo. Fj(x), where x is the set of all disease free states in the compartment i, is the rate of appearance of new infections in compartment i, and Vt(x) is net transfer rate (other than infections) of compartment i. The net transfer rate is given by v; = v- - v+, where v- is the rate of transfer of individuals out of compartment i, and v+ is the rate of transfer of individuals into compartment i by all other means. Therefore, the disease transmission model consists

of nonnegative initial conditions, xt(0), together with the following system of equations:

x' = f, (x) = F (x) - V, j > 1.

From model (6), we have

di (t) dr(t)

di (t) dr(t)

dV1 dV1

di (t) dr(t)

dV2 dV2

di (t) dr (t)

ßs 0 0 0

d + m + y 0 -9 y + S

Since we have only two distinct stages, namely I(t) and R(t), it follows that both F and V are 2x2 square matrices. Further, it can be noticed that F is nonnegative and V is nonsingular. The basic reproductive number R0 is the

Unstable

> Re(A)

Unstable

Figure 2: Stability region of the fractional order system with a e (0,1].

The following theorem states that R0 is a threshold parameter for the stability of model (6).

Theorem 3. The disease-free equilibrium is locally asymptotically stable and the infection will die out if R0 < 1 and is unstable if R0 > 1.

Proof. The disease-free equilibrium is locally asymptotically stable if all the eigenvalues, Ai ,i = 1,2,3,4, of Jacobian matrix J(E0) satisfy condition [30]:

Iarg(Ai)\ >

an ~2'

J(Eo) =

0 ß- (9 + m + y) 0

0 0 -(y + S)

-{n + [i))

dominant eigenvalue of the matrix FV \ which is obtained by solving the characteristic equation (FV~ 1)I - AI = 0, where A is the eigenvalue and I(t) is the identity matrix. At the disease-free equilibrium E0 = (1,0,0, 0), we have

Ro -a-*-.

9 + m + p

Figure 2 depicts the stability region of the fractional order system, according to condition (15). The eigenvalues of Jacobian matrix J(E0) are At = -y,, A2 = fi-(9 + m + y), A3 = -(p + S), A4 = -(q + p). Hence, E0 is locally asymptotically stable if R0 < 1 and is unstable if R0 > 1.

Now, we extend the analysis to endemic equilibrium E+. Jacobian matrix J(E+) evaluated at the endemic equilibrium is

J(E+)-

f-ßi V

i -ßs* 0

ßi* ßs* + aßc* -(9 + m + y) 0

0 (l-°)ßc* + 9 -(p + 8)

0 -ßc* S

with characteristic equation

A4 + axA3 + a2k2 + a3A + aA — 0,

— (D1 +D3 +D5),

a2 — (D1D3 -D4S + D1D5 + D3D5 + ß2i*s* +aß2c*i* ), a3 — (D1D3D5 - D1D4S + D3ß2i*s* + D5ß2i*s* + ß2c*t]i*

-D2aßSi * + aß2 D1c*i* + aD3ß2c*i*), a4 — D3D5ß2i*s* - D2ß8qi* + D3ß2c*^i*

- D4ß2Si*s* - aßSD1 D2i* + aD1D3ß2c*i*, D1 — ßi + y, D2 — (l-a)ßc* +9, D3 — (p + 8), D4 — (l-a)ßi*, D5 — ßi* + (n + ^)-

If D(0) denotes the discriminant of the polynomial: O(A) = Aa + a1 A3 + a2A2 + a3A + a4, then denote

D(®) —

1 Ü1 a2 a3 a4 0 0

0 1 Ü1 Ü2 a3 a4 0

0 0 1 Ü1 Ü2 a3 a4

4 3a1 a3 0 0 0

III0 4 3a1 2&2 a3 0 0

0 0 4 3a1 2&2 a3 0

II0 0 0 4 3a1 2&2 a3

From [31], we have the proposition.

Proposition 4. Assume that E+ exists in R+.

(1) Let c1,c2,c3 be the Routh-Hurwitz determinants: c1 = a1 1 0

a1, C2 — la] a2 I c3 —

0 a4 a3

. Therefore, when a — 1,

the equilibrium point E+ is locally asymptotically stable if

c1 > 0, c2 > 0, c3 = 0, a4 > 0. (21)

However, conditions (21) are sufficient (not necessary) conditions for E+ to be locally asymptotically stable for all a e [0,1).

(2) If D(<&) > 0, a1 > 0, a2 < 0, and a > 2/3, then the equilibrium point E+ is unstable.

(3) If D(0) < 0, a1 > 0, a2 >0, a3 > 0, a4 > 0, and a < 1/3, then the equilibrium E+ is locally asymptotically stable. Also, if D(&) < 0, a1 < 0, a2 >0, a3 < 0, a4 > 0, then the equilibrium point E+ is unstable.

(4) If D(<&) < 0, a1 > 0, a2 > 0, a3 > 0, a4 > 0, and a2 = (a1a4/a3) + (a3/a1), then the equilibrium point E+ is locally asymptotically stable for all a e (0,1).

(5) a4 > 0 is the necessary condition for the equilibrium point E + to be locally asymptotically stable.

3. Implicit Euler's Scheme for FODEs

Since most of the FODEs do not have exact analytic solutions, so approximation and numerical techniques must be used. In addition, most of resulting biological systems are stiff (one definition of the stiffness is that the global accuracy of the numerical solution is determined by stability rather than local error and implicit methods are more appropriate for it). The stiffness often appears due to the differences in speed between the fastest and slowest components of the solutions and stability constraints. In addition, the state variables of these types of models are very sensitive to small perturbations (or changes) in the parameters which occur in the model. Therefore, efficient use of a reliable numerical method for dealing with stiff problems is necessary.

Consider the following fractional order differential equation:

Day(t) = f(t,y(t)), te[0,T], y(V (0) = y{k) (0), k = 0,1,2,... ,m- 1,

0 < a < 1.

Here, y(t) = [y1(t),y2(t),...,yn(t)]T and f(t,y(t)) satisfythe Lipschitz condition in variable y:

\\f (t, y (t)) - f(t, x (i))|| < K \\y (t) -x(t)\\, K>0,

where x(t) is the solution of the perturbed system.

Theorem 5. Problem (22) has a unique solution provided that Lipschitz condition (23) is satisfied and M = KTa/T(a + 1) < 1.

Proof. Using the definitions of Section 1, we can apply a fractional integral operator to the differential equation (22) and incorporate the initial conditions, thus converting the equation into the equivalent equation:

m-1 ,k i ¡-t

y(t)=lyo)U + FT-rl (f-s)a-1 f(s,y(s))ds, (24)

j=0 k\ i (a) ]0

which also is a Volterra equation of the second kind. Define operator l, such that

m-1 fk i ¡-t

= I *> ¡rwl «-mw)

Then, we have \\Ly (t) - Lx (

— I (t-S)a-1 \\/(5,7(s))-/(s,x(s))\\ds

(a) Jo

r(a) Jo

K " T(a)

I (t - s)a 1 sup | y (s) - x (s)| ds

Jo se[0,T]

K T(ä)

KTa r(a + 1)

' - x\

I sa 1ds

• - *H T

So, we obtain

\\Ly(t)- Lx(

<M\\y - x\\

By Banach contraction principle [32], we can deduce that l has a unique fixed point which implies that our problem has a unique solution. □

Several numerical methods have been proposed to solve the FODEs [11, 33]. Recently, the predictor-corrector algorithm is an efficient and powerful technique for solving the FODEs, which is a generalization of the Adams-Bashforth-Moulton method. The modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm [34, 35] to approximate the fractional order derivative. However, converted Volterra integral equation (24) is with a weakly singular kernel, such that a regularization is not necessary any more. It seems that there exist only a very small number of software packages for nonlinear Volterra equations. In our case, the kernel may not be continuous, and therefore the classical numerical algorithms for the integral part of (24) are unable to handle the solution of (22). Therefore, we implement the implicit Euler's scheme to approximate the fractional order derivative.

Given model (22) and mesh points T = |i0, t1,..., tN], such that t0 = 0 and tN = T, then a discrete approximation to the fractional derivative can be obtained by a simple quadrature formula, using Caputo's fractional derivative (3)

of order a, 0 < a < 1 and using implicit Euler's approximation as follows (see [12]):

n« 1 С r

Г(1 - a)

;tlJ (M)* 1

i i-1 Xj -

+ 0(h)

(nh - s) «ds

(1 -a)T(1 -a)

A -*i-1

+ 0(h)

x [(и- j+l)1 « -(n-j)1 "] 11

(1 -а)Г(1 - a) h"

k-i - 1 ] [(и - j + 1)1 « - (и - j)1 «] 1

xii*i -

(1 -а)Г(1 - a)

x > |xi - xi 1

x[(n-j+1)1-" - (и - j)1-"]o(h2-").

Setting

g (a, h) =

(1 -a)T(1 - a) h"

= J - (i - 1) , (where = 1),

then the first-order approximation method for the computation of Caputo's fractional derivative is then given by the expression

D>, (in) = g (a, h) X«; - + O (h) . (30)

From the analysis and numerical approximation, we also arrive at the following proposition.

Proposition 6. The presence of a fractional differential order in a differential equation can lead to a notable increase in the complexity of the observed behavior, and the solution continuously depends on all the previous states.

350 300 250 200 150 100 50

State trajectories of the proposed model with a = 0.8

■■ Vs( 1)

0 1 2 3 4 5 6 7

Figure 3: Numerical simulation of fractional order epidemic model (6), when a = 0.8, and R0 > 1 (each infected individual infects more than one other member of the population and a self-sustaining group of infectious individuals will propagate), with parameter values of Table 1.

3.1. Stability and Convergence. We here prove that fractional order implicit difference approximation (30) is unconditionally stable. It follows then that the numerical solution converges to the exact solution as h ^ 0. In order to study the stability of the numerical method, let us consider a test problem of linear scaler fractional differential equation

D>(f) = ftM(f)+A. U(0) = Uo (31)

such that 0 < a < 1, and <0, >0 are constants.

Theorem 7. The fully implicit numerical approximation (30), to test problem (31) for all t > 0, is consistent and unconditionally stable.

Proof. We assume that the approximate solution of (31) is of the form w(in) = Un = Cn; then (31) can be reduced to

1--^ R

= <-1 + W (k-i -^-i+0 + GH' и"2'

C-1 + ^=2 (c-i -c-i+1 ) + A/g«,* (1 - (A>/g«,*))

Since (1 - (flo/g«,*)) > 1, for all g«*, then C1 < Co,

n > 2.

С, < Си-1 + i«(«) (C„-i - C-i+1), и >2. (35)

Table 1: List of parameters.

Parameter Description Value Reference

^ Replacement and exit rate (day-1) 0.011 [36]

ß Contact (transmission) rate of suspectable to be infected (animal-1 day-1) 0.15 [36]

e Recovery rate of infected animals day -1 0.16 Assumed

m Disease-induced mortality rate (day-1) 0.041 Assumed

n Cross-immune period 0.5 [36]

a The average reinfection probability of C(i) 0.06 Assumed

S The average time of appearance of new dominant clusters 1 Assumed

N The total number ofpopulation 345 Assumed

0.4 0.5 0.6 Susceptible

T1 0.06

tc 0.05

/a = 0.75 / a 0.9 a = 1

0.4 0.5 0.6 0.7 0.8 Susceptible

Figure 4: Phase plane portrait for fractional order endemic model (6), in absence of C(t) and R(t) components, when a = 0.7 (a) and a = 0.9 (b) with R0 = 0.5 < 1. We note that solution paths approach the disease-free equilibrium E0 = (1,0,0).

Thus, for n = 2, the above inequality implies

c2 <c!(Co -co.

Using relation (34) and the positivity of the coefficients o>2, we get

C2 <ii-

Repeating the process, we have, from (35),

since each term in the summation is negative. Thus, < Cn-1 < C„-2 < • • • < (°. With the assumption that = < c0 = |U°|, which entails < P°||, we have stability. □

Of course this numerical technique can be used both for linear and for nonlinear problems, and it may be extended to multiterm FODEs.

3.2. Numerical Simulations. The approximate solutions of epidemic model (6) are displayed in Figures 3, 4, and 5, and sensitivity of R° to transmission coefficients is displayed in

Figure 6. The numerical simulations are performed by Euler's implicit scheme. We choose commensurate fractional order that with different fractional order

values and the parameter values given in Table 1.

4. Conclusions

In this paper, we provided a fractional order SIRC epidemic model with Salmonella bacteria infection. We derived the sufficient conditions to preserve the asymptotic stability of infection-free and endemic steady states. The threshold parameter (reproduction number) r0 has been evaluated in terms of contact rate, recovery rate, and other parameters in the model. We provided unconditionally stable method, using Euler's implicit method for the fractional order differential system. The solution of a fractional order model at any time i* continuously depends on all the previous states at t < i*. Fractional order dynamical models are more suitable to model biological systems with memory than their integer-orders. The presence of a fractional differential order into a corresponding differential equation leads to a notable increase in the complexity of the observed behavior and enlarges the stability region of the solutions. However, fractional order differential models have the same integerorder counterpart steady states, when a > 0.5.

0.4 0.5 0.6 Susceptible

0.4 0.35 0.3 0.25 0.2 0.15 0.1

a = 0.9

\ \ a = 0.75

0.1 0.2 0.3

0.4 0.5 0.6 Susceptible

0.7 0.8 0.9

Figure 5: Phase plane portrait for classic fractional order endemic model (6) when a = 1 (a) and a = 0.9 (b) with R0 that solution paths approach the endemic equilibrium E+ given by (7).

1.2 > 1. We note

1.3 1.2 1.1 1

0.9 « 0.8 0.7 0.6 0.5 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) (b)

Figure 6: Sensitivity of R0 with respect to the transmission coefficients fi and 0.

Conflict of Interests

The authors declare that they have no competing interests in

this paper.

Acknowledgments

This work was supported by UAE University (UAEU-NRF-7-

20886). The authors would like to thank Dr. Ali Bhrawy for

his valuable comments.

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