Numerical Solution of SIR Model using Differential Transformation Method and Variational Iteration Method

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Ge. Mah. Noe, Vol., No., Jue, pp. 8-9 ISSN 9-8; Copygh ICSRS Publcao,.-c.og Avalable fee ole a hp://.gea. Nuecal Soluo of SIR Model ug Dffeeal Tafoao Mehod ad Vaaoal Ieao Mehod F.S. Aboo, S.

1 Ge. Mah. Noe, Vol., No., Jue, pp. 8-9 ISSN 9-8; Copygh ICSRS Publcao,.-c.og Avalable fee ole a hp://.gea. Nuecal Soluo of SIR Model ug Dffeeal Tafoao Mehod ad Vaaoal Ieao Mehod F.S. Aboo, S. Alao ad F.O. Apelu,, Depae of Pue & Appled Maheac LAUTECH Ogbooo, Ngea E-al: folexboo@gal.co E-al: aheedalao@yahoo.co E-al: foapelu.co.u Receved: -- / Acceped: 6-- Abac Th udy vegae he applcao of dffeeal afoao ehod ad vaaoal eao ehod fdg he appoxae oluo of Epdeology SIR odel. SIR odel ae olea ye of oday dffeeal equao ha ha o aalyc oluo. VIM ue he geeal Lagage ulple o couc he coeco fucoal fo he poble hle DTM ue he afoed fuco of he ogal olea ye. The eul evealed ha boh ehod ae coplee ageee, accuae ad effce fo olvg ye of ODE. Keyod: Dffeeal Tafoao ehod, Epdeology odel, Lagage ulple, Vaaoal Ieao ehod, Tafoed fuco. Ioduco Epdeology ude he pead of deae populao ad paly he hua populao. Ofe he o of a aheacal epdeolog co of

2 Nuecal Soluo of SIR Model ug 8 odel buldg, eao of paaee ad vegao of he evy of odel o chage he paaee ad uecal ulao. All hee acve ae expeced o ell u oehg abou he pead of he deae he populao, he pobly o cool h pead ad aybe ho o ae he deae dappea fo he populao []. Thee deae odeled o ofe ae he o called fecou deae,.e. deae ha ae coagou ad ca be afeed fo oe dvdual o aohe hough coac. Exaple of uch deae ae he chldhood deae: eale, ubella, chce pox ad up he exually aed deae: HIV/AIDS, goohea, yphl ad ohe. The S-I-R odel a oduced by W.O. Keac ad ha played a ajo ole aheacal epdeology. I he odel, a populao dvded o hee goup: he ucepble, he fecve, ad he ecoveed, h ube,, ad epecvely. The oal populao The ucepble ae hoe ho ae o feced ad o ue, he fecve ae hoe ho ae feced ad ca a he deae, ad he ecoveed ae hoe ho have bee feced, have ecoveed ad ae peaely ue. Code a epdec ha occu o a ecale ha uch hoe ha ha of he populao, ohe od egad he populao a havg a coa ze ad goe bh ad deah he e have he follog ye of olea oday dffeeal equao d cβ d d cβ d d d Th odel appopae o val deae uch a eale, up ad ubella. The SIR odel ha o aalycal oluo, bu e ca coduc uecal ulao fo he appoxae oluo. Icludg bh ad deah he adad S-I-R odel fo epdec he eulg odel ll allo u o loo a eve of loge duao. I h cae, e code a odel h a coa bh ae ad a coa pe-capa deah ae. We aue ha all bh ae o he ucepble ad he deah ae equal fo ebe of all hee clae. The bh ad deah ae ae equal o ha populao aoay he e have he ye

3 8 F.S. Aboo e al. d α d d α d d d Whee,, epee he ube of ucepble, fecve ad ecoveed dvdual a e epecvely, c ube of coac pe u e, β Pobably of deae ao pe coac, Pe-capa ecovey ae, Pe-capa eoval ae adα he avy. I h pape, equao ad ae olved ug vaaoal eao ehod ad he dffeeal afoao ehod fo uecal copao. Dffeeal afoao ehod oe of he ell-o echque o olve lea ad olea equao. I a f oduced by Zhou [5] fo olvg lea ad olea al value poble eleccal ccu aaly. Th ehod ha bee ued o olve dffeeal algebac equao [], Schödge equao [], facoal dffeeal equao [], Lae-Ede ype equao, fee vbao aaly of oag bea, ueady ollg oo of phee equao cled ube. The a advaage of h ehod ha ca be appled decly o lea ad olea ODE hou equg leazao, dcezao o peubao. Aohe poa advaage ha h ehod capable of gealy educg he ze of copuaoal o hle ll accuaely povdg he ee oluo h fa covegece ae. Alo, he bac dea of he vaaoal eao ehod [9, ] o couc a eao ehod baed o coeco fucoal ha clude a geealzed Lagage ulple [8]. The VIM a popoed by J.H He 999 hee he value of he ulple a choe ug vaaoal heoy o ha he each pove he accuacy of he oluo. The al appoxao.e. al fuco uually clude uo coeffce hch ca be deeed o afy ay bouday ad al codo. VIM doe o eque pecfc afoao fo olea e a equed by ohe echque ad o dely ued by ay eeache o udy auooou oday dffeeal equao[], Iegodffeeal ye[],lea Helholz paal dffeeal equao [] ad ohe feld[,9,5,,]. I h ehod he oluo gve a fe ee uually covege o a accuae oluo.

4 Nuecal Soluo of SIR Model ug 85. The Dffeeal Tafoao Mehod A abay fuco f x ca be expaded Taylo ee abou a po x a x d f f x! dx x The dffeeal afoao of f x defed a F x d f! dx x The he vee dffeeal afo f x x F The fudaeal aheacal opeao pefoed by dffeeal afo ehod ae led Table. Table : The fudaeal opeao of dffeeal afoao ehod DTM Ogal Fuco Tafoed Fuco y x g x ± h x Y G ± H y x α g x Y α G dg x Y G y x dx d g x Y G y x dx d g x Y... G y x dx y x Y δ y x x Y δ y x x {, Y δ y x g x h x y x e x Y, H G Y!

5 86 F.S. Aboo e al. y x x Y...!. Vaaoal Ieao Mehod Accodg o he vaaoal eao ehod of [9,, ] e code he follog geeal dffeeal equao L u N u gx hee, L a lea opeao, N a olea opeao ad gx a hoogeeou e. We ca couc a coecoal fuco a follo x { Lu Nu g } d u u x Whee a Lagaga ulple [] hch ca be defed opally va vaaoal heoy. The ubcp deoe he h appoxao ad u codeed a a eced vaao.e δ u.code he aoay codo of he above coeco fucoal, he he Lagage ulple ca be expeed a x 5! Whee he hghe ode of he dffeeal equao. Applcao of Dffeeal Tafoao Mehod Ug he afoed fuco of he ogal fuco able, e obaed he ecuece elao of equao a cβ S cβ I R [ I ] S I S I I 6 Wh al codo,, ad paaee c 6, β.5,., 5, he S, I, R Applyg he codo 6 e oba

6 Nuecal Soluo of SIR Model ug 8 S, S., S.68, S.9, I, I.6, I.8968, I.8, R, R., R.58, R.9, The, he cloed fo of he oluo hee ca be e a I S R Alo, ug he afoao able, equao gve S I α R S I R α S I I I [ I R ] S I S 8 h al codo 99,, ad paaee α.,,.5,. Fo he al codo, S 99, I, R he equao 8 gve S 9., S., S 6.5, S 6.8, I 9., I.99, I.688, I.56, R, R 9.5, R 5.866, R.5599, Theefoe, he cloed fo of he oluo hee ca be e a

7 88 F.S. Aboo e al R I S. Applcao of he Vaaoal Ieao Mehod Applyg he vaaoal eao ehod, e deve he coecoal fucoal a follo d d c d c β β Whee,, ae geeal Lagage ulple,,, ae codeed eced vaao hch ea ha δ δ δ. Fo equao 5,, he equao becoe d d N c d N c β β Wh al appoxao, hch afy he al codo, o gve

8 Nuecal Soluo of SIR Model ug I ca be obeved ha he eul of he epdec ye of equao coplee ageee h he eul obaed by he dffeeal afoao ehod. Alo, he ae appoach applyg VIM o equao he coeco fucoal a follo d d d 5 α α The Lagage ulple,,,, ubug he ulple o he coeco fucoal above, e have he follog eao foula d d d 6 α α

9 9 F.S. Aboo e al. Gve he al codo ad paaee 99,,, α.,,.5,, h al appoxao 99,, hch afy he al codo, e oba The eul above dcae ha boh ehod ae coplee ageee hch ple ha boh ae poeful ool fo olvg ye of dffeeal equao. Cocluo I h pape, dffeeal afoao ehod DTM ad vaaoal eao ehod VIM ha bee uccefully eployed o oba he appoxae oluo of SIR odel h al codo. The peeed ehod have bee appled a dec ay hou leazao, dcezao o peubao. Reul obaed by h ehod ho ha boh ae excelle ageee hch dcae he effecvee ad elably. Thee o ehod ca be codeed a a aleave ehod fo olvg a de cla of lea ad o-lea poble hch ae vaou feld of udy. Refeece [] A. Aoglu ad I. Ozol, Soluo of facoal dffeeal equao by ug dffeeal afo ehod, Chao Solo ad Facal,, -8. [] B. Baha ad R. Baha, A e appoach o he oluo of he pey ad pedao poble, Ieaoal Joual of Bac ad Appled Scece, I, 5-5. [] J. Baza, Soluo of he epdec odel by Adoa decopoo ehod, Appled Maheac ad Copuao, 6, - 6.

10 Nuecal Soluo of SIR Model ug 9 [] C.K. Che ad S.H. Ho, Applcao of dffeeal afoao o Ege-value poble, Appled Maheac ad Copuao, 9 996, -88. [5] E.M. Abdulafa, M.A. Abdou ad A.A. Mahoud, The oluo of olea coagulao poble h a lo, Chao Solo & Facal, 66, -. [6] E.S. Mouafa, Applcao of dffeeal afoao ehod o olea ocllaoy ye, Co. Nolea Sc. Nue. Sul., 8, -. [] F. Ayaz, Applcao of dffeeal afo ehod o dffeealalgebac equao, Appled Maheac ad Copuao, 5, [8] J. Baza ad M. Ela, Dffeeal afo ehod fo quadac Rcca dffeeal equao, Ieaoal Joual of Nolea Scece, 9, -. [9] J.H. He, A e appoach o olea paal dffeeal equao, Co. Nolea Sc. Nue. Sulao, 999, -5. [] J.H. He, A vaaoal appoach o olea poble ad applcao, Mech. Applc., 998, -. [] J.H. He, Soe aypoc ehod fo ogly olea equao, I. J. Mode Phy., B 6, -99. [] J.H. He, Vaaoal eao ehod a d of olea aalycal echque: Soe exaple, I. J. Nolea Mech., 999, [] J.H. He, Vaaoal eao ehod fo auooou oday dffeeal ye, Appl. Mah. Copu., 8-, 5-. [] J.H. He, The vaaoal eao ehod fo egh-ode al bouday value poble, Phy. Sc., 6, [5] J.K. Zhou, Dffeeal Tafoao ad Applcao fo Eleccal Ccu, Huazhog Uvey Pe, Wuha, Cha, 986 Chee. [6] M. Abdul ad B. Baha, Dffeeal afoao ehod fo a elable eae of he olea bochecal eaco odel, Adv. Sude Bology, 8, [] M.A. Abdou ad A.A. Sola, Ne applcao of vaaoal eao ehod, Phyca, D- 5, -8. [8] M. Iou e al., Geeal ue of he Lagage ulple olea aheacal phyc, I: S. Nea-Naed ed., Vaaoal Mehod he Mechac of Sold, Pegeo Pe, 98, [9] N. Bld ad A. Koualp, The ue of vaaoal eao ehod, dffeeal afo ad Adoa decopoo ehod fo olvg dffee ype of olea paal dffeeal equao, I. J. Nolea Sc. Nue. Sul., 6, 65-. [] S. Abbabady ad E. Shvaa, Applcao of he vaaoal eao ehod fo ye of olea Volea ego-dffeeal equao, Maheacal ad Cop. Applc., 9, -58. [] S. Moa ad S. Abuaad, Applcao of He vaaoal eao ehod o Helholz equao, Chao, Solo & Facal, 6, 9-.

11 9 F.S. Aboo e al. [] S.V. Kah ad K. Aua, To-deoal dffeeal afo ehod fo olvg lea ad o-lea Schodge equao, Chao Solo ad Facal, 5 9, -8.

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# % # & ( # ) +,!../ )4 5! + 3 )2 + 4 # # 2 ! # % # & #!../0 4 5! 4 # # 6 The Paame haaess o euey Resose uos o Nolea Sysems. J. Jg Z. Q. ag S. A. Bllgs ad G. R. Tomlso Deame o Auoma ool ad Sysems Egeeg The Uvesy o Sheeld Ma See Sheeld S JD U Reseah