Analysis of Analytical and Numerical Methods of Epidemic Models

Transcription

1 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN Aalyi of Aalytical ad Nuerical Metod of Epideic Model Pooa Kuari Aitat Profeor, Departet of Mateatic Magad Maila College, Pata Uiverity Eail : pooauari85@gail.co Abtract I ti paper, we tudy SIR epideic odel for a give cotat populatio. Tee ateatical odel are decribed by oliear firt order differetial equatio. Firt, we fid te aalytical olutio by uig te Differetial Traforatio Metod. We te copute te uerical olutio by uig fourt-order Ruge-Kutta Metod ad copare te aalytical olutio wit te uerical. Te profile of te olutio are provided, fro wic we ifer tat te aalytical ad uerical olutio agreed very well. Keyword SIR epideic odel, Suceptible cla, Ifective cla. Recovered cla, Noliear Ordiary Differetial Equatio, Differetial Traforatio Metod, Ruge-Kutta Metod Itroductio Epideic odel are tool to aalye te pread ad cotrol of ifectiou dieae. Te odel tat decribe wat appe o te average at te populatio cale are called deteriitic or copartetal odel. Tey fit well large populatio. I ti paper, we tudy te SIR epideic odel, wic i a tadard copartetal odel ued to decribe ay epideiological dieae []. Ti odel wa forulated by A. G. McKedric ad W. O. Kerac i 97 []. We olve te reultig differetial equatio of te odel by aalytical a well a uerical etod. To fid te aalytical olutio, we ue Differetial Traforatio Metod, wic expree te depedet variable explicitly a fuctio of idepedet variable i te for of a coverget erie wit eaily coputable copoet. Objective Te purpoe of ti article i to tralate te real world proble of te pread of ifectiou dieae ito ateatical vocabulary ad to fid te olutio of it wit te elp of Mateatic. Siply foratio of ateatical odel of ifectiou dieae i ot eoug for a dieae cotrol. Ule we ow a efficiet etod to olve te ateatical odel, we caot elp i ay detectio or terapy progra. Ti wor i a effort to elp i variou ifectiou dieae cotrol progra by providig a practical, efficiet ad accurate etod to olve a ateatical odel. Forulatio of SIR Epideic Model I ti odel, a fixed populatio wit oly te followig tree copartet i coidered :. (t) : It repreet te uber of uceptible at tie t, i.e., te uber of idividual wo do ot ave te dieae at tie t but could get it.. i (t) : It repreet te uber of ifective at tie t, i.e., te uber of idividual wo ave te dieae at tie t ad ca trait it to oter.. r (t) : It repreet te uber of idividual wo ave bee ifected but recovered fro te dieae at tie t. Te idividual i ti category are ot able to be ifected agai or trait te ifectio to oter, i.e., tey acquire peraet iuity or tey ave bee placed i iolatio or tey ave died. If be te ize of te populatio at ay tie t, te te differetial equatio for te SIR odel are: d t d t d i t d t t it t it i...() t...() 447

2 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN d r t d t - i t...( ) wit te iitial coditio i r i 0 r (4) d It ca be ee tat t i t rt 0 d t Aalytical Solutio of SIR Epideic Model, terefore it i true tat te populatio ize i cotat, i.e., (t) + i(t) + r(t) =. To fid te aalytical olutio, we ue Differetial Traforatio Metod, wic wa firt itroduced by Zou [] for olvig liear ad oliear iitial value proble i electrical circuit aalyi. But, ow a day, te etod a bee applied to olve a variety of proble tat are odelled wit differetial equatio. Te cocept of differetial traforatio i derived fro te Taylor erie expaio. I ti etod, give yte of differetial equatio ad related iitial coditio are trafored ito a yte of recurrece equatio tat fially lead to a yte of algebraic equatio woe olutio are te coefficiet of a power erie olutio. Taylor erie expaio of a fuctio f (x) about te poit x = 0 i a follow : f x 0 x! d dx f x 0...(5) Defiitio. Te differetial traforatio F() of a fuctio f (x) i defied a follow : F d...()! dx f x 0 Defiitio. It follow fro equatio (5) ad () tat te differetial ivere traforatio f (x) of F () i give by : f x x F...(7) 0 Uig equatio () ad (7), te followig ateatical operatio ca be obtaied :. If f (x) = g(x) ± (x), te F () = G() ± H(). If f (x) = c g(x), te F () = c G(), were c i a cotat. If x 4. If x x, dg f te F () = ( +)G( +) dx x, d g f te Y () = ( +)( + )...( + )G( + ) dx 5. If f x, te F. If f x x, te F 7. If f x x, te F if 0, if 8. If f (x) = g(x) (x), te F HG 0

3 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN x 9. If f x e, te F! 0. If f x x, te F... Now, we coider te SIR Model give by equatio (), () ad () wit te iitial coditio (0) = 00, i(0) = 0, (0) = 0, =, β = 0. ad γ = 0. If S(), I() ad R() deote te differetial traforatio of (t), i(t) ad r(t) repectively, te te followig recurrece relatio of equatio (), () ad () ca be obtaied :..(8) S I...(9) S 0 S I I...(0) I 0 I...() R wit te iitial coditio S(0) = 00, I(0) = 0, R(0) = 0, =, β = 0. ad γ = 0. S S 0I S 0I 0 0.I I R S I 0. I () S 0I S I S 0I S I 0 0.I R S 0. I S 0 I S I S I S 0 I S I S I 0 0. I I

4 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN R 0. I Terefore, uig te equatio i r t t S S0 t S t S t S 0 t t I I0 t I t I t I 0 t t R R0 t R t R t R te cloed for of te olutio we = ca be writte a follow : i r t t t 00 t 0.5 t 0 4t 0 t 0.47 t 0.4 t t t t Nuerical Solutio of SIR Epideic Model Uig equatio (), (), () ad te iitial coditio (8), te value of (t), i(t) ad r(t) are calculated by fourt-order Ruge-Kutta Metod at t = 0., 0.4, 0., 0.8 ad.0, taig te iterval of differecig = 0..Te uerical reult are copared wit te reult obtaied by DTM. To evaluate (t) at t = 0., we ave 0i i i i ad Siilarly, to evaluate i (t) at t = 0., we ave 0i i 450

5 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN i 0 i i 0 i i i 0 i ad i 0. i 0 i Siilarly, to evaluate r (t) at t = 0., we ave i i i i r ad r 0. r 0 r Uig te above forulae, te value of (t), i(t) ad r(t) are calculated for oter value of t. Copario of Aalytical Solutio wit te Nuerical Solutio Te uerical reult are copared wit te reult obtaied by DTM ad diplayed below. t (t) by DTM (4 iterate) (t) by fourt-order Ruge -Kutta Metod Differece 45

6 (t) Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN Table : Nuerical Copario of (t) DTM Ruge-Kutta t Figure : Plot of (t) veru tie t t i(t) by DTM (4 iterate) i(t) by fourt-order Ruge -Kutta Metod Differece Table : Nuerical Copario of i(t) 45

7 r(t) i(t) Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN DTM Ruge-Kutta t Figure : Plot of i(t) veru tie t t r(t) by DTM (4 iterate) r(t) by fourt-order Ruge -Kutta Metod Differece Table : Nuerical Copario of r(t) t DTM Ruge-Kutta Figure : Plot of r(t) veru tie t 45

8 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN It ca be ee tat te value of (t) + i(t) + r(t) calculated by DTM i exactly equal to for every t, werea te value of (t) + i(t) + r(t) calculated by fourt-order Ruge -Kutta Metod differ ligtly fro for variou value of t.ti cofir te ability of DTM a a powerful tool for olvig o liear equatio. Cocluio I ti paper, Differetial Traforatio Metod (DTM) a bee ued to olve SIR Epideic Model wit give iitial coditio. A ti etod provide a explicit olutio of te odel, it i very ueful i udertadig ad aalyig a epideic. Te uerical copario of ti etod wit te fourt-order Ruge -Kutta Metod prove te efficiecy ad accuracy of te etod. Moreover, ti etod provide a direct cee for olvig differetial equatio witout te eed for liearizatio, perturbatio or ay traforatio. It ay, terefore, be cocluded tat it i a powerful ateatical tool for olvig epideic odel. REFERENCES : [] Hetcote H.W. Te Mateatic of Ifectiou Dieae, SIAM Review 4(4), (000). [] W. O. Kerac ad A. G. McKedric, Cotributio to te Mateatical Teory of Epideic, Proc. Roy. Soc. Lod. A 5, (97). [] J.K.Zou, Differetial Traforatio ad it Applicatio for Electrical Circuit, Huazog Uiverity Pre, Cia (98). [4] G. Adoia, A Review of te Decopoitio Metod i Applied Mateatic, J. Mat. Aal. Appl., 5, (988)

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10 Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN