VARIATIONAL ITERATION METHOD WITH GAUSS-SEIDEL TECHNIQUE FOR SOLVING AVIAN HUMAN INFLUENZA EPIDEMIC MODEL

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 08, No. 01 (2016), pp. 29–41 http://jurnal.bull-math.org

  

VARIATIONAL ITERATION METHOD WITH

GAUSS-SEIDEL TECHNIQUE FOR SOLVING

AVIAN HUMAN INFLUENZA EPIDEMIC

MODEL

  

Yulita Molliq Rangkuti, Essa Novalia, Sri Marhaini, &

_Abstract.

Silva Humaira

A new method is built to solve a Susceptible infected -Susceptibel Infected

Removed (SI-SIR) model of the Avian Human influenza epidemic. The new method

is made from a hybrid of both variational iteration method and Gauss-Seidel method,

so-called Gauss-seidel variational iteration method (GSVIM). GSVIM based on the

Lagrange Multiplier and run by successive displacement technique. This new method

has success to solve SI-SIR model of the Avian Human influenza epidemic. From

the result, GSVIM more accurate than VIM.By only using second iterate of GSVIM

and fourth iterate of VIM, GSVIM solution is closer than VIM solution when both

solution compared to the fouth order Runge Kutta method (RK4).

  1. INTRODUCTION In 2007, [1] modelled Avian-Human Influenza epidemic by SI-SIR model. To describe the spreading dynamic of the susceptible bird and the infected bird by SI model, they modelled as: _′

  X = c − X − ωXY (1) Received 13-01-2016, Accepted 24-05-2016.

  2010 Mathematics Subject Classification: 37J45, 90C33

Key words and Phrases: Gauss-Seidel Method; Variational Iteration Method; Lagrange Multiplier,

Avian Human Influenza Epidemic, Susceptible Infected -Susceptibel Infected Removed model.

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel _′

  Y = ωXY − (b + m)Y (2) where c is the rate at which new birds are born, ω is the rate at which avian influenza is contracted from an average bird individual, provided it is infective, X is susceptible birds and Y infected birds with avian influenza.

  Next, [1]. considered the human system. They assumed that the human infected with avian influenza cannot infect susceptible humans and this disease also has the high virulence for humans, the infected human with mutant avian influenza can infect a susceptible human with mutant avian influenza (and cannot infect a human who infected with avian influenza be- cause of immunity against the same kind of virus) and this mutant avian influenza has the lower virulence for human than avian influenza. In ad- dition, the human infected with mutant avian influenza can recover with eternal immunity against avian influenza and mutant avian influenza. That is to say, avian influenza is spread by infected birds and mutant avian in- fluenza is spread by infected humans in the human world. Thus, The SIR model is considered as: _′

  S = λ − µS − β SY − β SH (3) _′

  1

  2 B SY

  = β − (µ + d + ε)B (4) _′

  1 H SH

  = β − εB − (µ + α + γ)H (5)

  2 _′

  R = γH − µR (6)

  Finally, by combining the above 2 systems, it arrived at the following SI-SIR avian-human influenza model as: _′

  X = c − X − ωXY (7) _′ Y _′ = ωXY − (b + m)Y (8)

  S SY SH = λ − µS − β − β (9) _′

  1

  2 B SY

  = β − (µ + d + ε)B (10) _′

  1 H SH

  = β − εB − (µ + α + γ)H (11)

  2 _′

  R = γH − µR (12) subject to the initial conditions

  X (0) = 20, Y (0) = 2, S(0) = 100, B(0) = 0, H(0) = 0andR(0 = 0) (13)

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  Here c, ω, b, m, β, β

  1 , β 2 , d, ε, µ and γ are constants, positive

  and sufficiently small. b is sufficiently larger than µ (b >> µ), α is less than d and d less than m (m > d > α) because of the differences of the virulence. In [1] paper, the population is divided into six classes: susceptible birds and infected birds with avian influenza, susceptible human, infected human with avian influenza, infected human with mutant avian influenza, recovered human from mutant avian influenza, with size X, Y, S, B, H, R , respectively.

  The spreading number of Avian-Human influenza epidemic which mod- elled by SI-SIR model is important to investigate. It can determine by find- ing the solution of the model either numerically or analytically. Several numerical analytical methods for solving epidemic problems were applied, such as the Adomian decomposition method (ADM), the homotopy pertur- bation method (HPM) and the homotopy analysis method (HAM), differen- tial transform method (DTM). For example, [2] obtained the approximately spreading number of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic using ADM. [3] used HPM to find the approximate solution of epidemic model. [4] applied HAM to solve the SIR epidemic model.[5] solved epidemic problem using DTM to obtain the approximate solution.

  Another powerful method which can also give explicit form for the so- lution is the variational iteration method (VIM). It was proposed by [6], and other researchers have applied VIM to solve various problems [15]. For exam- ple, [7], [10] solved fractional Zhakanov-Kuznetsov and fractional heat-and wave-like and Fractional Rosenau-Hyman Equations using VIM to obtain the approximate solution have shown the accuracy and efficiently of VIM. Based on VIM, [11]-[13], has modidied the VIM for finding the approximate solution of chaotic systems, Fractional Biochemical Reaction and nonlinear problems. Nevertheless, VIM is only valid for short-time interval for solving the kind of systems of differential equation.

  Our motivation of this work is to overcome the weakness of VIM. The Gauss-Seidel technique [15] is adapted to VIM. The new values are used to calculate other variables. On the other hand, once you have determined from the first equation, its value is then used in the second equation to obtain the new similarly, the new and are used in the third equation to obtain the new and so on. We modify the standard VIM by adopting the Gauss-Seidel tech- nique in this paper and call it the Gauss-Seidel variational iteration method (GSVIM). In this paper, GSVIM was used to solve dynamical system which is modelled by SI-SIR model [1] and the fourth-order Runge-Kutta method Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel (RK4) and VIM are used for comparison.

  2. Variational Iteration Method Consider the following general system of first-order ordinary differen- tial equations (ODEs):

  Lu i (t) + Ru i (t) + Nu i (t) − g i (t) = 0, (14) where L is n the order differential of u respecting to t, R is a linear operator, N is a nonlinear operator and g is an inhomogeneous term. According to

  VIM, one can construct a correction functional as follows: Z t u (t) = u (t) + λ (s)[Lu (s) + Ru (s) + Nu (s) − g (s)]ds, (15)

  i,n +1 i,n i i i i i

  where λ(s) is a general Lagrangian multiplier which can be optimally identi- u fied via the variational theory [14]. ˜ i,n is considered as a restricted variation

  ˜ [14], that is, δu i,n = 0 and the subscript n indicates the n the approximation. We have

  Z t δu λ u

  i,n (t) = δu i,n (t) + δ i (s)[Lu i (s) + Ru i (s) + N˜ i,n (s) − g i (s)]ds (16)

• 1

  where ˜ u is considered as restricted variations, that is, δ ˜ u = 0, we

  i,n i,n

  have Z t

  δu λ

  n +1 (t) = δu n (t) + δ i (s)[Lu i,n (s) + Ru i,n (s)]ds, (17)

  Z t _′ δu (t) = [1 + λ (t)]δRu (t) − δ (λ (s) − λ (s))Ru (s)ds (18)

  n +1 i n i i,n i

  Thus, we obtain the following stationary conditions: δu

  n (t) : [1 − λ i (t)]| s = 0 (19) _′ +t

  δu

  n (s) : λ (s) − λ i (s) = 0 (20) i

  Solving this system of equations yields λ

  i (s) = −Exp(C i (s − t)), (21)

  where C i is any function for i = 1, 2, .... Furthermore, substituting (21) to (15), the iteration formula of VIM can be written as follows:

  Z t u u

  i,n = u i,n − Exp(C i (s − t))[Lu i + Ru i + N˜ i,n − g i (s)]ds. (22)

• 1

  u The solution is given by u i = lim n→∞ i,n (t)

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  3. The Gauss-Seidel Method Gauss-Seidel method is proposed by Carl Friedrich Gauss (1777-1855) and Philipp L. Seidel (1821-1896). Gauss-Seidel method is modification of Jacobi algorithm for solving the linear system. In this technique, the new values are used to calculate other variables. On the other hand, once you have determined from the first equation, its value is then used in the second equation to obtain the new similarly, the new and are used in the third equation to obtain the new and so on. A possible improvement to the Jacobi Algorithm can be seen by reconsidering

  n

  X

  1 u u

  i,k = [ (−a ij j,k− 1 ) + g i ], f ori = 1, 2, ..., n. (23)

  a

  i,i _{j =1} j6 =1

  The components of u j,k− are used to compute all the components u j,k

  1

  , ...., u of u k . But, for i > 1, the components u i− of u k have already

  

1,k 1,k

  been computed and are expected to be better approximations to the actual , ...., u , ...., u solutions u

  1 i− 1 than are u i− . Instead of using 1,k−1 1,k−1 n

  X

  1 u u

  i,k = [ (−a ij j,k− ) + g i ], f ori = 1, 2, ..., n. (24)

  1

  a

  i,i _{j =1} j6 =1

  It seems reasonable, then, to compute u using these most recently

  i,k

  calculates values. The Gauss-Seidel Iterative Technique

  i− 1 n

  X X

  1 u u u

  i,k = [− (a ij j,k− ) − (a ij j,k− ) + g i ], f oreachi = 1, 2, ..., n.

  1

  1

  a

  i,i j =1 j =i+1

  (25)

  4. Gauss Seidel Variational Iteration Method (GSVIM) In this section, we shall now look at how this new modification of

  VIM works to find the accurate approximate solution. Here, a new solution of first variable is used to calculate the other variables (Gauss and Seidel in Gaub and Loan 1996). The new formulas will be calculated iteratively. Thus, the formula can be written as:

  i− n

  X X u Exp

  

1

Z t

  

i,n (t) = u i,n (t)− (C i (s−t))[Lu i,n (s)+ (u j,k )(s)+ (u j,k− )(s)+g i (s)]dsf ori = 1, 2, ...,

• 1

  1 j =1 j =i+1

  (26)

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  The solution is given by u i = lim u i,n (t) (27)

  

n→∞

  5.GSVIM’S IMPLEMENTATION FOR SI-SIR MODEL OF AVIAN-HUMAN INFLUENZA EPIDEMIC

  To demonstrate the accuracy of GSVIM for solving SI-SIR model of Avian-Human Influenza epidemic, firstly, we consider the SI-SIR model in (7)-(12) as follow _{′ ′ ′}

  X = c − X − ωXY Y = ωXY − (b + m)Y S = λ − µS − β SY − _{′ ′ ′}

  1

  β SHB SY SH = β −(µ+d+ε)BH = β −εB−(µ+α+γ)HR = γH −µR

  2

  1

  

2

  where X, Y, S, B, H, R are indicated as the susceptible birds, in- fected birds with avian influenza, susceptible human, infected human with avian influenza, infected human with mutant avian influenza, recovered hu- man from mutant avian influenza, respectively and c, ω, b+m, λ,µ,ε, β , β ,

  1

  2

  α , β, γ are positive constants.

  To solve (7) to (12), GSVIM will be applied through 4 steps as follows which are made lucid for the SI-SIR model. Step 1. First, the correction functional is constructed as used by VIM to find the Lagrange multiplier in the following forms:

  Z t dX

  n

• X n = X n λ (s)[ − c + X n + ωX n Y n ]ds,
• 1

  1

  ds Z t dY

  

n

  n = Y n (s)[ − ωX n n − (b + m)Y n ]ds, (29)

• Y λ Y
• 1

  2

  ds Z t dS

  n

  n = S n (s)[ − λ + µS n + β n n + β n n ]ds, (30)

• S λ S Y S H
• 1

  3

  1

  2

  ds Z t dB

  n

  B λ S Y

• n = B n (s)[ − β n n + (µ + d + ε)B n ]ds,
• 1

  4

  1

  ds Z t dH

  n

  H λ S H

• n = H n (s)[ − β n n + εB n + (µ + α + γ)H n ]ds,
• 1

  5

  2

  ds Z t dR

  n

  n = R n (s)[ − γH n + µR n ]ds, (33)

• R λ
• 1

  6

  ds where λ , i = 1, 2, 3, 4, 5 are the general Lagrange multipliers and con-

  i

  sidered as restricted variations, i.e. by taking variation with respect to the

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  ˜ X , ˜ Y , ˜ S , ˜ Y S H independent variables ˜ n n n n and ˜ n n the following forms can be obtained below :

  Z t dX

  n

  ˜

• δX n = δX n λ (s)[δ − δc + δX n + ωδ ˜ X n Y n ]ds,
• 1

  1

  ds Z t dY

  

n

  ˜

• δY n = δY n λ (s)[δ − δω ˜ X n Y n − δ(b + m)Y n ]ds,
• 1

  2

  ds Z t dS

  n

  ˜ ˜ ˜ ˜

• δS = δS λ (s)[δ − δλ + µδS + δβ S Y + δβ S H ]ds, (36)

  n +1 n 3 n 1 n n 2 n n

  ds Z t dB

  n

  ˜ ˜ δB λ S Y

• n +1 = δB n

  4 (s)[δ − δβ 1 n n + δ(µ + d + ε)B n ]ds, (37)

  ds Z t dH

  n

  ˜ ˜

  λ S

  n = H n (s)[δ −δβ n n +δεB n +δ(µ+α+γ)H n ]ds, (38)

• δH H
• 1

  5

  2

  ds Z t dR

  n

• δR n = δR n λ (s)[δ − δγH n + δµR n ]ds,
• 1

  6

  ds Making each of the correction functional in (34) to (38) stationary and also observe that therefore the three sets of stationary conditions can be

  Z t dX n δX = δX λ (s)[δ + δX ]ds, + (40)

  n +1 n 1 n

  ds Z t dY

  n

  δY λ

• n = δY n (s)[δ − δ(b + m)Y n ]ds, (41)
• 1

  2

  ds Z t dS

  n

  δS λ

  n = δS n (s)[δ + µδS + n ]ds, (42)

• 1

  

3

  ds Z t dB

  n

• δB n = δB n λ (s)[δ + δ(µ + d + ε)B n ]ds,
• 1

  4

  ds Z t dH

  n

• δH = H λ (s)[δ + δ(µ + α + γ)H ]ds, (44)

  n +1 n 5 n

  ds Z t dR

  n

• δR λ

  n +1 = δR n 6 (s)[δ + δµR n ]ds, (45)

  ds The obtained λ for each, i = 1, 2, 3, 4, 5and6. The general Lagrange

  i

  multipliers therefore can be easily identified as

  b (s−t)

  λ , = −e (46) Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel (b+m)(s−t)

  λ ,

  2 = −e (47) µ (s−t)

  λ ,

  3 = −e (48) (µ+d+ε)(s−t)

  λ ,

  4 = −e (49) (µ+α+γ)(s−t)

  λ ,

  5 = −e (50) γ (s−t)

  λ , = −e (51)

  (46) to (51) then, the following iteration formulas will be obtained: Z t dX

  n b (s−t)

  X = X − e [ − c + X + ωX Y ]ds, (52)

  

n +1 n n n n

  ds Z t dY

  n (b+m)(s−t)

  Y n = Y n − e [ − ωX n Y n − (b + m)Y n ]ds, (53)

• 1

  ds Z t dS

  n µ (s−t)

  S n = S n − e [ − λ + µS n + β S n Y n + β S n H n ]ds, (54)

• 1

  1

  2

  ds Z t dB

  

n

(µ+d+ε)(s−t)

  B n = B n − e [ − β S n Y n + (µ + d + ε)B n ]ds, (55)

• 1

  1

  ds Z t dH

  n (µ+α+γ)(s−t)

  H e S H

  n = H n − [ −β n n +εB n +(µ+α+γ)H n ]ds, (56)

• 1

  2

  ds Z t dR

  n γ (s−t)

  R e

  n = R n − [ − γH n + µR n ]ds, (57)

• 1

  ds Step 3: furthermore, apply the Gauss-Seidel on VIM by the new values are used to calculate other variables. On the other hand, once you have determined from the first equation, its value is then used in the second equation to obtain the new similarly, the new and are used in the third equation to obtain the new and so on. The iteration formula of GSVIM can be

  Z t dX

  n b (s−t)

  X e Y

  n = X n − [ − c + X n + ωX n n ]ds, (58)

• 1

  ds Z t dY

  n (b+m)(s−t)

  Y e Y

  n +1 = Y n − [ − ωX n +1 n − (b + m)Y n ]ds, (59)

  ds Z t dS

  n µ (s−t)

  S e S Y S H

  n +1 = S n − [ − λ + µS n + β 1 n n +1 + β 2 n n ]ds, (60)

  ds Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  Z t dB

  n (µ+d+ε)(s−t)

  B n = B n − e [ − β S n Y n + (µ + d + ε)B n ]ds, (61)

• 1 1 +1 +1

  ds Z t dH

  n (µ+α+γ)(s−t)

  H n = H n − e [ − β S n H n + εB n + (µ + α + γ)H n ]ds,

• 1 2 +1

  ds (62)

  Z t dR n

  γ (s−t)

  R e

  n +1 = R n − [ − γH n +1 + µR n ]ds, (63)

  ds Step 4. The other components are obtained as follow:

  − 5t

  X = −2.7 + 12.7e , (64)

  1 − 10t 5t 10t

  Y = 2 + 0.04(−177 − 77e + 254e )e , (65)

  1 − − − 0.015t 10t 5t

  S = 1640 − 1566.581015e − 14.18127191e + 40.7622868e , (66)

  1 − − 1.016t 10t

  B = −348.6614173 + 41.29840250e + 248.9256321e

  1 − − − 5t 0.015t 10.015t

  −834.2558599e + 338.0434558e − 246.5028022e

  − 5.015t − 20t − 15t

  ,

• 796.0221612e − 1.057768701e + 6.188196704e (67)

  − − 0.085t 1.016t

  H = −4.101899027 − 0.6905620680e − 0.04435918636e

  1 − 10t 5t − 0.015t

  −0.0251059639e + 0.16973669658e + 4.829192226e

  − − 10.015t 5.015t

• 0.02482404856e − 0.1614649414e

  − 20t − 15t

  ,

• 0.00005311417027e − 0.000414897533e (68)

  − − 0.015t 0.015t

  R = −2.734599351 + 2.635521474e + 0.098651724e

  1 − 1.016t − 10t

• 0.0004431487149e + 0.00002514367942e

  − − 5t 0.015t

  −0.00034049487149e + 0.04829192226te

  − 10.015t − 5.015t

  −0.00002482404856e + 0.0003229298828e

  − − 20t 15t

  −0.2657701790e + 0.2768752305e , (69) and so on. In this work, the second iterate of GSVIM is chosen to show its accuracy whereas, the fourth iterate of VIM is chosen to show the limitation of this method.

  6. Result and Discussion Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel

  The accuracy of GSVIM for the solution of SI-SIR model was presented in this paper. The GSVIM algorithm was coded in the computer algebra package Maple. The simulations were done in this paper for the time span t ∈ [0, 1] and comparison was done by the fourth-order Runge Kutta (RK4) method. First, the parameters are fixed at c = 26.5, b = 5, ω = 2, m =

  −

  3

  5, λ = 3, µ = 0.015, β = 0.2, β = 0.003, ε = 10 , d = 1, α = 0.01, andγ =

  1

  2

  0.01. The solution with an initial value in predicts the spread of avian influenza and mutant avian influenza in the human world. Thus the initial value is fixed at X(0) = 10,Y(0) = 2,S(0) = 100,B(0) = 0,H(0) = 0 and (0) = 0 . Figure 1 shows the approximate solution of SI-SIR model using second iterate of GSVIM, fourth iterate of VIM and RK4 with ? t=0.001. From figure, GSVIM solution is closer to RK4 than VIM solution, it shows that GSVIM is more accurate than VIM only by choosing second iterate of GSVIM while VIM reach accurate in fourth iteration. Figure 1. the approximate solution of SI-SIR model of Avian-Human influenza epidemic for such as: (a) susceptible birds, (b) infected birds with avian influenza, (c), susceptible human (d) infected human with avian influenza, (e ) infected human with mutant avian influenza, (f) recovered human from mutant avian influenza using GSVIM, VIM and RK4.

  7. Conclusion In this paper, the algorithm for solving chaotic systems via Gauss-Seidel variational iteration method (GSVIM) was developed. For computations and plots, the Maple package was used. The conclusions of GSVIM are the GSVIM was a suitable technique to solve the SI-SIR model. This modi- fied method yields a Gauss-Seidel analytical solution in easily computable terms by choosing second iterate of GSVIM while VIM reach accurate in fourth iteration. Comparison between GSVIM, VIM and RK4 was made; the GSVIM was found to be more accurate than the VIM. It has potential for solving more complex systems which may arise in various fields of pure and applied sciences.

  

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  G, Golub, H. Loan C. F. V. 1996. Matrix Computations (3rd ed.), Balti- more, Johns Hopkins

  Yulita Molliq Rangkuti et al., – Variational iteration method with Gauss-Seidel Yulita Molliq Rangkuti : Department of Mathematics, Faculty of Mathematics

and Natural Science, Universitas Negeri Medan (UNIMED), Medan 20221, Sumat-

era Utara, Indonesia

  E-mail: yulitamolliq@yahoo.com

  Essa Novalia : Department of Mathematics, Faculty of Mathematics and Natural

Science, Universitas Negeri Medan (UNIMED), Medan 20221, Sumatera Utara, In-

donesia

  E-mail: novaliaessa@gmail.com

  Sri marhaini : 1Department of Mathematics, Faculty of Mathematics and Natural

Science, Universitas Negeri Medan (UNIMED), Medan 20221, Sumatera Utara, In-

donesia

  E-mail: sri marhaini@ymail.com

  Silva Humaira : Department of Mathematics, Faculty of Mathematics and Natu-

ral Science, Universitas Negeri Medan (UNIMED), Medan 20221, Sumatera Utara,

  Indonesia

  E-mail: silvahumaira@gmail.com

  FIGURES Figure 1: a Figure 2: b