A New Modification of the Differential Transform Method for a SIRC Influenza Model

Transcription

1 A ew odificatio of the Differetial Traform ethod for a SIRC Iflueza odel S.F..IBRAHI -Kig Abdulaziz Uiverity Faculty of ciece For orth Jeddah Dept. of athematic Jeddah Saudi Arabia. -Ai Sham Uiverity Faculty of Educatio Departmet of athematic Roxy Cairo Egypt. S.. ISAIL ir Uiverity For Sciece &Techology Faculty of Egieerig Dept. of Baic ciece 6Th of October City Egypt. ABSTRACT I thi paper approximate aalytical olutio of SIRC model aociated with the evolutio of iflueza A dieae i huma populatio i acquired by the modified differetial traform method (DT). The differetial traform method (DT) i metioed i ummary. DT ca be obtaied from DT applied to Laplace ivere Laplace traform ad padé approximat. The DT i ued to icreae the accuracy ad accelerate the covergece rate of trucated erie olutio gettig by the DT. The aalytical-umerical techique ca be ued i order to produce imulatio with differet iitial coditio parameter value for differet value of the baic reproductio umber. Geeral Term Cro-immuity SIR model Laplace traform Iflueza Differetial traformatio method Keyword SIRC model Epidemic model odified Differetial traformatio method padé approximat.. ITRODUCTIO Iflueza i caued by a viru that attac maily the upper repiratory tract oe throat ad brochi ad rarely alo the lug. ay people recover withi - wee without requirig ay medical treatmet. I the very youg the elderly ad people ufferig from medical coditio uch a lug dieae diabete cacer idey or heart problem iflueza poe a eriou ri. I thee people the ifectio may lead to evere complicatio of uderlyig dieae peumoia ad death. I aual iflueza epidemic 5-5% of the populatio are affected with upper repiratory tract ifectio. Hopitalizatio ad death maily occur i high ri group (elderly chroically ill). Aual epidemic are poibly betwee three ad five millio cae of evere ille ad betwee ad death every year aroud the world. Iflueza i tramitted by a viru that ca be of three differet type amely A B ad C []. Amog thee taype the viru A i epidemiologically the mot importat oe for huma beig becaue it ca recombie it gee with thoe of trai circulatig i aimal populatio uch a bird wie hore etc. Ufortuately withi type a viru there are everal ubtype H H3 H5 etc. each oe of thee ha bee poited a the caual of recet pademic. uch evidece how that the atigeic ditace betwee two differet trai ifluece the degree of partial immuity ofte called cro-immuity coferred to a hot already ifected by oe of the trai with repect to the other[]. athematical model have prove to be ueful tool to tudy the dyamic of viral ifectio withi thee model compartmetal model have bee traformed of ordiary or partial differetial equatio. Over the lat two decade a umber of epidemic model for predictig the pread of iflueza through huma populatio have bee propoed baed o either the claical uceptible-ifected-removed (SIR) model developed by Kermac ad ckedric[3]. Caagradi et al.[] have itroduced SIRC model by addig a ew compartmet C which ca be called cro-immue compartmet to the SIR model. Thi cro-immue compartmet (C) decribe a itermediate tate betwee the fully uceptible (S) ad the fully protected (R)oe. They have tudied the dyamical behavior of thi model umerically [4]. Jodar et al. [5] developed two otadard fiite differece cheme to obtai umerical olutio of a iflueza A dieae model preeted by Caagradi et al.[]. Very recetly Samata[4] coidered a oautoomou SIRC epidemic model for Iflueza A with varyig total populatio ize ad ditributed time delay. Thi model aume o immue iterferece betwee the differet A viru ubtype that i why they oly coidered oe viru ubtype. I thi paper the modified differetial traform method (DT) ha bee applyed will be employed i a traightforward maer without ay eed of liearizatio or malle aumptio. DT wa firt applied i the egieerig domai by [6]. DT provide a efficiet explicit ad umerical olutio with high accuracy miimal calculatio parig of phyically urealitic aumptio. However DT ha ome drawbac. By uig DT a erie olutio which i obtaied i practice a trucated erie olutio. Thi erie olutio doe ot exhibit the periodic behavior which i characteritic of ocillator equatio ad give a good approximatio to the true olutio i a very mall regio. I order to develop the accuracy of DT a alterative techique i ued which modifie the erie olutio for oliear ocillatory ytem a follow: firt apply the Laplace traformatio to the trucated erie obtaied by DT the covert the traformed erie ito a meromorphic fuctio by formig it Padé approximat([7][8][9][0][]) ad fially accept a ivere Laplace traform to obtai a aalytic olutio which may be periodic or a better approximatio olutio tha the DT trucated erie olutio.. The SIRC ODEL Caagradi et al. [] coidered the model ds S C SI dt () 8

2 di dt SI SI I dr CI I R dt dc dt R CI C With iitial coditio () (3) (4) S 0 I 0 R 0 C 0. Where μ i the mortality rate 3 4 θ i the rate of progreio from ifective to recovered per year δ i the rate of progreio from recovered to cro-immue per year γ i the rate of progreio from recovered to uceptible per year σ i the recruitmet rate of cro-immue ito the ifective β i the cotact rate per year. The dieae free equilibrium i locally aymptotically table if ad oly if ad utable if. There exit a uique ad poitive edemic equilibrium poit if ad oly if (β/(μ+θ))> which i locally aymptotically table uder ome coditio o the coefficiet[] 3. PADÉ APPROXIATIOS Some techique exit to icreae the covergece of a give erie. Amog them the o- called padé techique i widely applied i thi ectio the otio of ratioal approximatio i itroduced for fuctio. The fuctio f(x) will be approximated over a mall portio of it domai. For example if f(x)=co(x) it i ufficiet to have a formula to geerate approximatio o the iterval [0π/]. The trigoometric idetitie ca be ued to compute co(x) for ay value x that lie outide [0π/]. A ratioal approximatio to P x f(x) o [ab] i the quotiet of two polyomial ad Q x of degree ad repectively. The otatio [/] (x) ca be ued to deote thi quotiet: / P x x for a x b Q x. (5) Our goal i to mae the maximum error a mall a poible. For a give amout of computatioal effort oe ca uually cotruct a ratioal approximatio that ha a maller overall error o [ab] tha a polyomial approximatio. The developmet i a itroductio ad will be limited to Padé approximatio. The method of Padé require that f(x) ad it derivative be cotiuou at x=0. There are two reao for the arbitrary choice of x=0. Firt it mae the maipulatio impler. Secod a chage of variable ca be ued to hift the calculatio over to a iterval that cotai zero. The polyomial ued i Eq. (5) are P x p p x p x p x Ad 0... Q x q x q x... q x The polyomial i (6) ad (7) are cotructed o that f(x) ad [/] (x) agree at x=0 ad their derivative up to + agree at x=0. I the cae Q₀ (x) = the approximatio i ut the aclauri expaio for f(x). For a fixed value of + P x Q the error i mallet whe ad x have the P x ame degree or whe ha degree oe higher tha Q x Q. otice that the cotat coefficiet of i q₀ =. Thi i permiible becaue it caot be 0 ad [/] (x) i ot chaged whe both (6) (7) P x Q ad x are divided by the ame cotat. Hece the ratioal fuctio [/](x) ha ++ uow coefficiet. Aume that f(x) i aalytic ad ha the aclauri expaio f x a0 ax a x... a x... Ad form the differece : f x Q x P x z x (8) a x q x p x c x (9) The lower idex =++ i the ummatio o the right ide of (9) i choe becaue the firt + derivative of f(x) ad [/](x) are to agree at x=0. Whe the left ide of (9) i multiplied out ad the coefficiet of the power of x are et equal to zero for = the reult i a ytem of ++ liear equatio: a p q a a p 0 0 q a q a a p 0 0 q a q a q a a p q a q a... a p 0 Ad 0 9

3 q a q a... q a a 0 q a q a... q a a 0 3 q a q a... q a a 0 otice that i each equatio the um of the ubcript o the factor of each product i the ame ad thi um icreae coecutively from 0 to +. The equatio i () q q... q ad mut be olved ivolve oly the uow firt. The the equatio i (0) are ued ucceively to fid P0 p... p 4. BASIC DEFIITIOS OF DIFFERETIAL TRASFORATIO ETHOD Puhov [3] propoed the cocept of differetial traformatio where the image of a traformed fuctio i computed by differetial operatio which i differet from the traditioal itegral traform a are Laplace ad Fourier. Thu thi method become a umerical-aalytic techique that formalize the Taylor erie i a totally differet maer. Differetial traformatio i a computatioal method that ca be ued to olve liear (or o-liear) ordiary (or partial) differetial equatio with their correpodig boudary coditio. A pioeer uig thi method to olve iitial value problem i Zhou [6] who itroduced it i a tudy of electrical circuit. Additioally differetial traformatio ha bee applied to olve a variety of problem that are modeled with differetial equatio ([4][5][6][7]) The method coit of give ytem of differetial equatio ad related iitial coditio; thee are traformed ito a ytem of recurrece equatio that fially lead to a ytem of algebraic equatio whoe olutio are the coefficiet of a power erie olutio. For the ae of clarity i the preetatio of the DT ad i order to help to the reader we ummarize the mai iue of the method that may be foud i [6]. Defiitio 4. A differetial traformatio Y() of fuctio y (x) i defied a follow [8] Y d y x! dx x 0 () I () y(x) i the Origial fuctio ad Y() i the traformed fuctio. Differetial ivere traform of Y() i defied a follow y x x Y (3) 0 I fact. From () ad (3) we obtai y x x d y x 0! dx (4) x 0 Equatio (4) implie that the cocept of differetial traformatio i derived from the Taylor erie expaio. From Equatio () ad (3) it i eay to obtai the followig mathematical operatio:. If y x g x h x the Y G H.. If y x cg x the Y cg cotat. 3. If y x d g x the dx Y G.! 4. If y x g x h x the. Y G l H l y x l 0 x 5. If the Y 0 Kroecer delta. 6. If y x u x v x w x δ i the the c i a. Y U V m W m 0 m 0 4.THE OPERATIO PROPERTIES OF DIFFERETIAL TRASFORATIO If x (t) ad y (t) are two ucorrelated fuctio with time t where X () ad Y(K) are the traformed fuctio correpodig to x(t) ad y(t) the the fudametal mathematic operatio ca be proved by differetial Traformatio ad are lited a follow [6]:() Liearity. If Y D y t X D x t ad c are idepedet of t ad the ad c D c x t c y t c X c Y (5) Thu if c i a cotat The D[ c] c where i the roecer delta fuctio. () Covolutio. if z t x t y t x t D X y t D Y ad deote the covolutio ad Symbol D deotig the differetial traformatio proce. The 0

4 l 0 D x t y t D z t X Y x l Y l (6) If y x y x y x y x y x... the 3... Y Y 0... Y Y Y (7) The proof of above propertie i deduced from the defiitio of the differetial traform 5. APPLICATIO OF THE SIRC IFlUEZA ODEL I thi ectio the differetial traformatio techique i applied to olve oliear differetial equatio ytem uch a SIRC iflueza model. The followig recurrece relatio to the SIRC iflueza model i obtaied By uig the fudametal operatio of differetial traformatio method. Applyig thi method the ytem i equatio ()-(4) ca be writte a follow: S S S l I l C l 0 (8) I S l I l C l I l I l0 l0 (9) R C l I l I R l 0 (0) C R C l I l C l 0 () with S 0 I 0 R 0 3 C 0 4. () Are differetial traform of repectively. S t I t R t C t Thu from a proce of ivere differetial traformatio it ca be obtaied the olutio i the power erie S t S t I t I t R t C t C t C t (3) Therefore 4 S t t t 4 4 I t t 4 4 t R t t t C t t t UERICAL ETHODS AD SIULATIOS I thi ectio the umerical reult are obtaied baed o the applicatio of the (DT) to SIRC iflueza model. Sice mot of the o-liear differetial equatio do ot have exact aalytic olutio o approximatio ad umerical techique mut be ued. 6. Dieae free equilibrium (R₀=(β/(μ+θ))<) For umerical tudy (for R₀ <) the followig parameter ca be ued a folow:

5 populatio uceptible / 50 y 73 y y 0.5 y Thi wa doe with the tadard parameter value give above ad iitial value ₁ =0.8 ₂ =0. ₃ =0.04 ₄ =0.06. Thee value correpod to table i []. By taig differetial traform method to iitial coditio i traformed a follow: S I R C Ad from equatio (8)-() The olutio erie ca be eaily writte a follow : S t S t t t t t 346. t 5877 t t (4) t.9550 t I t I t t t t t 877. t.660 t t t t R t R t t t t t 73.8t.3640 t t t t... C t C t t t t 7.98t 79.85t 4.9t t.7380 t t... I thi ectio Laplace traformatio i applied to (4) which yield L S t L I t (5) L R t L C t For implicity replacig = (/t) L S t t t t t t 4.30 t t t t t... L I t t t t t 8533t t t t t t... L R t t t t t t t (6) t t t t... L C t t 0.9t 9.06t t 683.5t t t t t t... t padé approximat [4/4] of (6) ad ubtitutig ca be obtaied [4/4] i term of S. Fially by uig the ivere Laplace traformatio the modified approximate olutio ca be expreed a: S t t t 0.057e e e 0.358e i i t it i e 7.98t 99.66t t I t e e e t i i t t.0303t R t 0.38e e e 6.080it i e t t 3.834t t C t e e e e Figure: S(t)for μ=0.0β=50δ=γ=0.5σ=0.05θ=73

6 Cro immuepopulatio populatio Recovered Ifected populatio Figure:I(t)for μ=0.0β=50δ=γ=0.5σ=0.05θ= S t S t t t t t 9906t t t t t I t I t t t t t t t t t t R t R t t t t t t t t t t... 0 C t C t t t t 7.35t 848.7t t t t t... (7) Figure3:R(t)forμ=0.0β=50δ=γ=0.5σ=0.05θ= Figure4:C(t)forμ=0.0β=50δ=γ=0.5σ=0.05θ=73 6. Edemic equilibrium R I thi ectio to illutrate the capability of the DT the variable ad parameter are coidered a follow: / 50 y 73 y y 0.5 y Thi wa doe with the tadard parameter value give above ad iitial value ₁=0.8 ₂=0. ₃=0.04 ₄=0.06. Thee value correpod to table i []. For the fourcompoet model. A approximatio for S(t) I(t)R(t)C(t) the olutio ca be eaily writte a: I thi ectio Laplace traformatio i applied to the erie olutio i (7)which yield L S t L I t (8) L R t L C t For implicity replacig = (/t) 3

7 Cro immuepopulatio populatio Rcovered Ifected populatio populatio uceptible L S t t t t t 3300t 38830t t t t t... L I t t t t t 4458t t t t t t... L R t t t t t 94.5t t (9) t t t.7980 t... L C t t t t t 733.t.4840 t t t t t... padé approximat [4/4] i applied of (9) ad ubtitutig t padé approximat [4/4] i obtaied i term of S. Fially by uig the ivere Laplace traformatio the modified approximate olutio ca be expreed a: S t t e e 8.700t e i t i t i i e I t e 9.903it i i e e i t R t e i t it t e i i t e i i i e e it t i e i t i e t C t e e e t i t i i e it Figure5:S(t)for μ=0.0β=00δ=γ=0.5σ=0.05θ=73] Figure6:I(t)for μ=0.0β=00δ=γ=0.5σ=0.05θ=73] Figure7:R(t)for μ=0.0β=00δ=γ=0.5σ=0.05θ=73] Figure8:C(t)forμ=0.0β=00δ=γ=0.5σ=0.05θ=73] 7. COCLUSIOS I thi paper the modified differetial traform method ha bee ued to obtai approximate aalytical olutio of oliear ordiary differetial equatio ytem uch a SIRC dyamical model. The accuracy ad efficiecy of thi method wa demotrated by olvig SIRC iflueza model. Laplace traformatio ad padé approximat are ued to obtai aalytic olutio ad to improve the accuracy of differetial traform method. The modified DT i a efficiet method for calculatig periodic olutio of oliear differetial equatio ytem. The advatage of the method i that the approximate olutio ca be calculated eaily i horter time with the computer program uch a atlap ad mathematica moreover thi method olve the problem without ay eed for dicretizatio perturbatio or liearizatio of the 4

8 variable. The computatio ad graph aociated with the example i thi paper were performed uig athematica ver ACKOWLEDGETS The author tha referee for fruitful commet ad uggetio for reviig the maucript. 9. REFERECES [] Palee P. ad Youg J. 98. Variatio of iflueza A B ad C virue. (Sciece 5) [] Caagradi R. Bolzoi L. Levi S.A. ad Adreae V The SIRC model ad iflueza A ath. Bioci. (00) [3] Kermac W. O. ad ckedric A. G. 97. Cotributio to the mathematical theory of epidemic Part I Proc. Roy. Sot. Ser. A [4] Samata G. P. 00. Global Dyamic of a oautoomou SIRC odel for Iflueza A with Ditributed Time Delay Differ Equ Dy Syt [5] Jodar L. Villaueva R. J. Area A. J. ad Goz alez G. C otadard umerical method for a mathematical model for iflueza dieae athematic ad Computer i Simulatio (79 ) [6] Zhou J. K Differetial Traform ad it Applicatio for Electrical CircuitWuha Huarug Uiverity Pre. [7] Baer G. A Eetial of padé approximat. Lodo: Academic pre;. [8] Chag S. H. ad Chag I. L. 008 A ew algorithm for calculatig oe-dimeioal differetial traform of oliear fuctio. Appl ath Comput ;95: [9] Jorda D. W. ad Smith P oliear Ordiary Differetial Equatio. Oxford Uiverity Pre. [0] Dehgha. Shaourifar. ad Hamidi A The olutio of liear ad oliear ytem of Volterra fuctioal equatio uig Adomia.Padé techique. Chao Solito Fract ;39:509-. [] Dehgha. Hamidi A. ad Shaourifar The olutio of coupled Burger equatio uig Adomia.Padé techique. Appl ath Comput ; 89: [] Chiviriyait W umerical modelig of the tramiio dyamic of iflueza The Firt Iter. Symp. o Optim. ad Sy. Biol. pp [3] Puhov G.E Differetial traformatio of fuctio ad equatio. auova Duma Kiev (i Ruia). [4] Che S. S. ad Che C. K. 009 Applicatio of the differetial traformatio method to the free vibratio of trogly o-liear ocillator oliear Aalyi: Real World Applicatio [5] Ye-Liag Yeh Cheg Chi Wag ig-jyi Jag 007. Uig fiite differece ad differetial traformatio method to aalyze of large deflectio of orthotropic rectagular plate problem. Appl. ath. Comput.; 90() [6] Abdel-Hialim Haa I. H. 008 Applicatio to differetial traformatio method for olvig ytem of differetial equatio Appl. ath odellig ; 3() [7] Jag. J. ad Che C. L Aalyi of the repoe of atrogly oliear damped ytem uig a differetial traformatio techique. Appl. ath Comput.; 88(-3): [8] Che Chao-Kuag Ho Shig-Huei Applicatio of differetial traformatio to eigevalue problem. Appl ath Comput ; 79: