HOMOTOPY PERTURBATION METHOD FOR SOLVING A MODEL FOR HIV INFECTION OF CD4 + T CELLS

Transcription

1 İtanbul icaret Üniveritei Fen Bilimleri Dergii Yıl: 6 Sayı: Güz 7/. 9-5 HOMOOPY PERURBAION MEHOD FOR SOLVING A MODEL FOR HIV INFECION OF CD4 + CELLS Mehmet MERDAN ABSRAC In thi article, homotopy perturbation method i implemented to give approximate and analytical olution of nonlinear ordinary differential equation ytem uch a a model for HIV infection of CD4 + cell. A modification of the homotopy perturbation method (HPM), baed on the ue of Pade approximant, i propoed. Some plot are preented to how the reliability and implicity of the method. Keyword: Pade Approximant, Homotopy Perturbation Method, A Model for HIV Infection of CD4 + Cell CD4 + HÜCRELERİNİN BİR HIV ENFEKSİYONLU MODELİNİN HOMOOPY PERURBAION YÖNEMİ İLE ÇÖZÜMÜ ÖZE Bu makalede, CD4 + hücrelerinin bir HIV enfekiyonlu modeli gibi lineer olmayan adi difereniyel denklem iteminin yaklaşık analitik çözümünü bulmak için homotopy perturbation yöntemi (HPY) uygulanmıştır. Homotopy perturbation yöntemine pade yaklaşımı uygulanmıştır. Yöntemlerin baitliğini ve doğruluğunu götermek için birkaç grafik göterilmiştir. Anahtar Kelimeler: Pade Yaklaşımı, Homotopy Perturbation Yöntemi, CD4 + Hücrelerinin Bir HIV Enfekiyonlu Modeli Karadeniz echnical Univerity, Engineering Faculty of Gümüşhane, Civil Engineering, Gümüşhane

2 Mehmet MERDAN. INRODUCION Dynamic of a model for HIV infection of CD4 + cell i examined (Liancheng and Michael, 6) at the tudy. he component of the baic four-component model are the concentration of CD4 + cell, the concentration of infected CD4 + cell by the HIV virue and free HIV viru particle are denoted repectively by (t), I(t), and V(t). hee quantitie atify d I r kv, dt max di kv I, dt dv N I V, dt with the initial condition: () () r, I() r and V () r. hroughout thi paper, we et;.,.,., r,.4, k.7, 5, N. he motivation of thi paper i to extend the application of the analytic homotopyperturbation method (HPM) and variational iteration method (He, 998/a, 998/b, 999/a, 6) to olve a model for HIV infection of CD4 + cell (). he homotopy perturbation method (HPM) wa firt propoed by Chinee mathematician He (998/a, 998/b, 999/a, 999/b,, 6). he firt connection between erie olution method uch a an Adomian decompoition method and Padé approximant wa etablihed in. he tranmiion and dynamic of HLV-I feature everal biological characteritic that are of interet to epidemiologit, mathematician, and biologit, ee for example, Aquith and Bangham (), Finlayon (97), Abdou and Soliman (5), etc. Like HIV, HLV-I target CD4 + -cell, the mot abundant white cell in the immune ytem, decreaing the body ability to fight infection. We will ue Laplace tranform and Pade approximant to deal with the truncated erie. max. PADÉ APPROXIMAON A rational approximation to (x) on a, b i the quotient of two polynomial P N (x) and Q M (x) of degree N and M, repectively. We ue the notation R N,M (x) to denote 4

3 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ thi quotient. he R N,M (x) Padé approximation to a function (x) are given by Baker (975). PN ( x) RN, M ( x) for a x b. Q ( x) () M he method of Padé require that (x) and it derivative be continuou at x =. he polynomial ued in () are N PN ( x) p p x px... pn x () M QM ( x) q x qx... qm x (4) he polynomial in () and () are contructed o that (x) and R N,M (x) agree at x = and their derivative up to N+M agree at x =. In the cae Q (x) =, the approximation i jut the Maclaurin expanion for (x). For a fixed value of N+M the error i mallet when P N (x) and Q M (x) have the ame degree or when P N (x) ha degree one higher then Q M (x). Notice that the contant coefficient of Q M i q =. hi i permiible, becaue it notice be and R N,M (x) i not changed when both P N (x) and Q M (x) are divided by the ame contant. Hence the rational function R N,M (x) ha N+M+ unknown coefficient. Aume that (x) i analytic and ha the Maclaurin expanion f x a a x a x a x (5) k ( )... k..., And from the difference f ( x) Q ( x) P ( x) Z ( x) : M N M N i i i i ai x qi x pi x ci x, i i i in M (6) he lower index j = N+M+ in the ummation on the right ide of (6) i choen becaue the firt N+M derivative of (x) and R N,M (x) are to agree at x =. When the left ide of (6) i multiplied out and the coefficient of the power of x i are et equal to zero for k,,,..., N M, the reult i a ytem of N+M+ linear equation: 4

4 Mehmet MERDAN a p q a a p q a q a a p q a q a q a a p q a q a a p M N M M N M N N and q a q a... q a + a M N M M N M N N q a q a... q a + a M N M M N M N N q a q a... q a + a M N M N N M N M (7) (8) Notice that in each equation the um of the ubcript on the factor of each product i the ame, and thi um increae conecutively from to N+M. he M equation in (8) involve only the unknown q, q, q,, q M and mut be olved firt. hen the equation in (7) are ued ucceively to find p, p, p,, p N (Baker, 975).. HOMOOPY PERURBAION MEHOD o illutrate the homotopy perturbation method (HPM) for olving non-linear differential equation, He (999/a, ) conidered the following non-linear differential equation: A( u) f ( r), r (9) ubject to the boundary condition u B u,, r n () where A i a general differential operator, B i a boundary operator, f(r) i a known analytic function, i the boundary of the domain and / n denote differentiation along the normal vector drawn outward from. he operator A can generally be divided into two part M and N. herefore, (9) can be rewritten a follow: 4

5 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ M ( u ) N ( u ) f ( r ), r () He (999/a, ) contructed a homotopy v( r, p) : x, atifie which H ( v, p) ( p) M ( v) M ( u ) p A( v) f ( r), () which i equivalent to H( v, p) M ( v) M ( u ) pm ( v ) p N( v) f ( r), () where, p i an embedding parameter, and u i an initial approximation of (). Obviouly, we have H ( v,) M ( v) M ( u ), H ( v,) A( v) f ( r). (4) he changing proce of p from zero to unity i jut that of H (v,p) from M ( v) M ( v ) to A( v) f ( r). In topology, thi i called deformation and M ( v) M ( v ) and A( v) f ( r) are called homotopic. According to the homotopy perturbation method, the parameter p i ued a a mall parameter, and the olution of Eq. () can be expreed a a erie in p in the form v v pv p v p v (5)... When p, Eq. () correpond to the original one, Eq. () and (5) become the approximate olution of Eq. (), i.e., u lim v v v v v... (6) p he convergence of the erie in Eq. (6) i dicued by He (999/a and ). 4. APPLICAIONS In thi ection, we will apply the homotopy perturbation method to nonlinear ordinary differential ytem (). 4

6 Mehmet MERDAN 4.. Homotopy Perturbation Method to A Model for HIV Infection of CD4 + Cell According to homotopy perturbation method, we derive a correct functional a follow: v v p v x p v v rv kv v, max, pv y pv kv v v pv z pv Nv v, (7) where dot denote differentiation with repect to t, and the initial approximation are a follow: v ( t) x ( t) () r,, v ( t) y ( t) I() r,, v ( t) z ( t) V () r., (8) and v v pv p v p v...,,,,, v v pv p v p v...,,,,, v v pv p v p v...,,,,, (9) where,,,,,,... i j v i j are function yet to be determined. Subtituting Eq. (8) and (9) into Eq. (7) and arranging the coefficient of p power, we have 44

7 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ r r v, r r r r r kr r p m ax m ax r r v, v, r v, m ax r r v r v p m ax k r v, r v,,, r r v v v, r v, m ax r r v v v r v m ax k r v, v,v, r v, v k r r r p,, p,,,,, v, k r v, r v, v 4, v, p,,,, v, N v, v, p...,..., v k r v v v r v v v p,,,,, 4,,..., v N r r p v N v v p () In order to obtain the unknown vi, j ( t), i, j,,, we mut contruct and olve the following ytem which include nine equation with nine unknown, conidering the initial condition vi, j (), i, j,,, 45

8 Mehmet MERDAN r r v r r r r r kr r,, max max v v r rv, r r v, rv, rv,,,, max k rv, r v,, r rv, v, rv max r r v v v r v,,,, max k rv v v r v v kr r r,,,,,,, max v, k r v, r v, v 4, v,, v, k r v, v,v, r v, v4, v, v N r r,, v N v v,,,, v N v v,,,,, () From Eq. (6), if the three term approximation are ufficient, we will obtain: ( t ) lim v ( t ) v ( t ), p p p, k k I ( t ) lim v ( t ) v ( t ),, k k V ( t ) lim v ( t ) v ( t ),, k k () therefore 46

9 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ rr rr r ( t ) r r r kr r t m ax r r r r r r kr r k r k N r r r kr r k r r r r rr r k rr r r r rr rr t m ax rr r r k r r rr r m ax r r r r r r r r m ax () I ( t) r kr r r t kr r r kr N r r rr t rr r kr r r kr r max V ( t ) r N r r t N kr r r N r r t... Here () =., I() =, and V() =. for the three-component model. A few firt approximation for (t), I(t), and V(t) are calculated and preented below: hree term approximation: t t t t ( ) , I t t t t ( ) , V t t t t ( ) (4) Four term approximation: 47

10 Mehmet MERDAN ( t ) t t t t, * t, t. I ( t). 7 t t t V ( t ).. 4 t.8845 t t (5) Five term approximation: ( t ) t t t t t, I ( t).7 t t t * t * t, V ( t)..4 t.8845 t t t t. (6) Six term approximation: ( t) t t t t t t, I ( t).7 t t t * t * * t, V ( t)..4 t.8845 t t t t t. t (7) In thi ection, we apply Laplace tranformation to (7), which yield L ( )

11 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ L( I( )) (8) L( V ( )) For implicity, let ; then t L( ( t)).t.9795t.5988t 5.595t t t.568t 7 4 L( I( t)).7t t t.4448t.549t t 4 7 (9) L( V ( t)).t.4t.78656t t t t t Padé approximant 4/4] of (9) and ubtituting t = /, we obtain 4/4] in term of. By uing the invere Laplace tranformation, we obtain 7 ( t) e e t t e * t t e I ( t).44646e e e t t t () V ( t) e +.97e -.4 t t hee reult obtained by Padé approximation for (t), I(t), and V(t) are calculated and preented follow. 49

12 Mehmet MERDAN the concentration of CD4+ cell x 59 x - the concentration of infected CD4+ cell x -7-4 y t time free viru particle t time. v t time Figure. Plot of Padé Approximation for A Model for HIV Infection of CD4 + Cell hee reult obtained by homotopy perturbation method, three, four, five and ix term approximation for (t), I(t), and V(t) are calculated and preented follow. x 5 hree term approximation I V Four term approximation I V -.5 t time Five term approximation I V -.5 t time.5 t time Six term approximation I V.5 t time Figure. Plot of hree, Four, Five and Six erm Approximation for A Model for HIV Infection of CD4 + Cell 5

13 İtanbul icaret Üniveritei Fen Bilimleri Dergii Güz 7/ 5. CONCLUSIONS In thi paper, homotopy perturbation method wa ued for finding the olution of nonlinear ordinary differential equation ytem uch a a model for HIV infection of CD4 + cell. We demontrated the accuracy and efficiency of thee method by olving ome ordinary differential equation ytem. We ue Laplace tranformation and Padé approximant to obtain an analytic olution and to improve the accuracy of homotopy perturbation method. We apply He homotopy perturbation method to calculate certain integral. It i eay and very beneficial tool for calculating certain difficult integral or in deriving new integration formula. he computation aociated with the example in thi paper were performed uing Maple 7 and Matlab REFERENCES Abdou, M. A., and Soliman, A. A., (5), Variational-Iteration Method for Solving Burger and Coupled Burger Equation, Journal of Computational and Applied Mathematic, 8,, Aquith, B., and Bangham, C. R. M., (), he Dynamic of -Cell Fratricide: Application of a Robut Approach to Mathematical Modeling in Immunology, Journal of heoretical Biology,, Baker, G. A., (975), Eential of Padé Approximant, Academic Pre, London. Finlayon, B. A., (97), he Method of Weighted Reidual and Variational Principle, Academic Pre, New York. He, J. H., (998/a), Approximate Analytical Solution for Seepage Flow with Fractional Derivative in Porou Media, Computer Method in Applied Mechanic and Engineering, 67,, He, J. H., (998/b), Approximate Solution of Nonlinear Differential Equation With Convolution Product Nonlinearitie, Computer Method in Applied Mechanic and Engineering, 67, -, He, J. H., (999/a), Variational Iteration Method-A Kind of Nonlinear Analytical echnique: Some Example, International Journal of Nonlinear Mechanic, 4, 4, He, J. H., (999/b), Homotopy Perturbation echnique, Computer Method in Applied Mechanic and Engineering, 78,

14 Mehmet MERDAN He, J. H., (), A Coupling Method of A Homotopy echnique and A Perturbation echnique For Non-Linear Problem, International Journal of Nonlinear Mechanic, 5,, 7-4. He, J. H., (6), Some Aymptotic Method For Strongly Non-Linear Equation, International Journal of Modern Phyic B.,,, Liancheng, W., and Michael, Y. L., (6), Mathematical Analyi of he Global Dynamic of a Model for HIV Infection of CD4 + Cell, Mathematical Biocience,,