On the Homotopy Analysis Method for an Seir Tuberculosis Model

Transcription

1 Aerican Journal of Applied Maheaics and Saisics, 3, Vol., No. 4, 7-75 Available online a hp://pubs.sciepub.co/ajas/4/4 Science and Educaion Publishing DOI:.69/ajas--4-4 On he Hooopy Analysis Mehod for an Seir Tuberculosis Model M.O. Ibrahi, S.A. Egbeade,* Deparen of Maheaics, Universiy of Ilorin, Ilorin, Nigeria Deparen of Maheaics & Saisics, The Polyechnic, Ibadan, Nigeria *Corresponding auhor: egbeades@yahoo.co Received Augus 3, 3; Revised Augus, 3; Acceped Sepeber, 3 Absrac In his paper, we provide a very accurae, non-perurbaive, sei-analyical soluion o a syse of nonlinear firs-order differenial equaions odeling he ransission of uberculosis (TB) in a hoogeneous populaion. Our analysis is based on Hooopy Analysis Mehod (HAM). Maple 5 sofware is used o carry ou he copuaions. Our resuls show he validiy and poenial of HAM for copuing he soluion of nonlinear equaions. Keywords: uberculosis, hooopy analysis ehod, series soluion, nonlinear equaions, aheaical odel Cie This Aricle: M.O. IBRAHIM, and S.A. EGBETADE, On he Hooopy Analysis Mehod for an Seir Tuberculosis Model. Aerican Journal of Applied Maheaics and Saisics, no. 4 (3): doi:.69/ajas Inroducion Infecion wih uberculosis (TB) is caused by a bacerial known as Mycobaceriu Tuberculosis [,,3]. Globally, TB is one of he greaes diseases of public concern because he pandeic is a subsanial hrea o socioeconoic developen iposing a heavy burden on failies, couniies and econoies [4,5,6]. In 993, he World Healh Organisaion (WHO) declared TB a global eergency and abou billion people were esiaed o be globally infeced wih TB ha year [7]. However, wih drasic global reaen easures, he incidence of TB has reduced across he globe. According o a recen global TB repors, 9.4 illion people acquired he disease in 8 resuling in.8 illion deahs while he nuber of acive cases has reduced o 5.7 illion in wih.4 illion deahs showing ha TB oraliy has decreased by % globally since 8 [8,9]. Maheaical odels have been widely used in differen fors for sudying he ransission dynaics of TB epideics [-7]. However, he dilea wih any odels in epideiology is soeies how o obain analyic soluions of he nonlinear equaions describing he dynaics of hese diseases [8]. In 99, a non-perurbaive ehod known as hooopy analysis ehod (HAM) was proposed by Liao [9].This ehod was based on hooopy, an iporan par of opology []. HAM is a general analyic echnique developed for he purpose of obaining approxiae analyic series soluions o differen ypes of nonlinear equaions especially hose wih srong nonlineariy. This ehod has been successfully applied o solve any ypes of nonlinear probles arising in he field of science, engineering and finance [-38]. The HAM offers cerain advanages over previous non-perurbaive ehods. Firsly, is validiy does no depend upon sall paraeers of he considered nonlinear proble. Secondly, i provides a siple way o ensure he convergence of series soluions. Furherore, we have grea freedo o choose auxiliary linear operaor so ha one can approxiae a nonlinear equaion ore efficienly by eans of beer base funcions. Equal iporanly, a few new soluions of soe nonlinear probles which are negleced by all oher analyic and nuerical echniques are found using HAM. In addiion, as proved in [5], HAM logically conains he hree radiional non-perurbaion ehods such as Lyapunov arificial sall paraeer ehod [39], δ- expansion ehod [4] and Adoian decoposiion ehod [4]. The hooopy perurbaion ehod developed in [4] is also a special case of HAM as poined ou by Sajid and Haya [43], Liang and Jeffery [44] and oher researchers.. Maheaical Forulaion In his paper, we consider he following TB epideic odel proposed by Egbeade and Ibrahi [45] ( ) S = γ π + si β IS µ S (.) ( ρ) β ( µ υ) E = IS + E (.) I = d ρis + Dυ E µ + µ + s I (.3) T R = εi si β IR µ R (.4) S= nuber of suscepible who do no have he disease bu could ge i

2 7 Aerican Journal of Applied Maheaics and Saisics E= nuber of exposed who are infeced bu are ye o show any sign of sypos I= nuber of infecives who have he disease and can ransi i o ohers R= nuber of recovered or reoved who can no ge he disease or ransi i. γ= proporion of recruien due o iigraion π= rae of recruien of suscepible individuals S= reaen rae of TB β= ransission rae of TB µ= naural deah rae µ T = deah rae of TB ν= rae of slow progression ρ= rae of fas progression D= deecion rae of TB ε= rae a which suscepible individuals recover. In secion 3, we shall apply he hooopy analysis ehod described in he nex secion o solve equaions (.) (.4). 3. Hooopy Analysis Mehod For he sake of copleeness and readabiliy of he presen work, we give below a syseaic descripion of he procedures of HAM. Consider a nonlinear equaion of he for N u = (3.) N is a nonlinear operaor, denoes he ie and u() is an unknown funcion. Le u () denoe an iniial approxiaion of u() and L denoe an auxiliary linear operaor, Liao [] consrucs he zero-order deforaion equaion. ( L φ ( ; u = phh N ( ; (3.) p [,] is he ebedding paraeer, h is a nonzero auxiliary paraeer, H() is a non-zero auxiliary funcion. When p= and p=, he zero-order deforaion equaions becoes respecively and ( ;) u φ = (3.3) ( ;) u φ = (3.4) Thus, as p increases fro o, he soluion φ(; varies coninuously fro he iniial approxiaion u () o he exac soluion u(). Such a kind of coninuous variaion is called deforaion in opology. Expanding φ(; by Taylor s series in power series of p, we have φ p = u + u p (3.5) = ( ; ) u is he deforaion derivaive. ( ; φ =! p (3.6) If he auxiliary linear operaor N, he iniial approxiaion u (), he auxiliary paraeer h and he auxiliary funcion H() are properly chosen so ha φ ; p of he zero-order deforaion () he soluion equaion (3.) exiss for all p [,]. () he deforaion derivaive (3.6) exiss for all =,, (3) he series (3.5) converges a p=. Then, we have he series soluion φ ; = + (3.7) = ( ) u u Define he vecor u = { u, u,, u } (3.8) According o he definiion (3.6), he governing equaion can be derived fro he zero-order deforaion equaion (3.). Differeniaing (3.) ies wih respec o he ebedding paraeer p, hen seing p = and finally dividing by!, we obain he h order deforaion equaion L y χu = hh Q( u ) (3.9) and ( ; N φ Q( u ) = (! ) p, χ =, > (3.) (3.) Noe ha according o he definiion (3.), he righ hand side of (3.9) depends only on u. Thus, we easily gain he series u, u, by solving he linear high-order deforaion equaion (3.9) using sybolic copuaion sofware such as Malab, Maple or Maheaica. 4. Soluion of SEIR Model by HAM To solve he odel equaion (.) (.4) by HAM, we consider equaion (.) and choose he linear operaor wih he propery ha ( ; ) ds p N S( ; = (4.) d N[ c ] = (4.) c is a consan of inegraion. The inverse operaor N is given by N = d (4.3) Le he nonlinear operaor be defined as

3 Aerican Journal of Applied Maheaics and Saisics 73 ( ; ) ds p N S( ; = ; d + β I p S p + ( γ) π si( ( ; ) ( ; ) us( ; By consrucing he zero-order deforaion equaion ( N S( ; s ( ; = phh N S( ; (4.4) (4.5) we have ha for S ; = s p =, hen p =, hen S( ;) = s Then, we have he h order deforaion equaion N [ S χs( A) ] = hh Q( S ), (4.6) ds Q( S ) = si d + β I S + µ S ( γ) π (4.7) The soluion of he h order deforaion equaion (4.6) for and using h = and H( ) = is given by S S S d si + I S d, + µ S Following earlier seps, we ge E E ( ) ( γ) π β E d I S d, + + I I ( ) ( ρ) β ( µ υ) E I d I S d d E T d, + + ρβ υ ( µ µ T + ε) I si R R R I d + si d, + si R + R ε µ 5. Nuerical Resuls and Discussion (4.8) (4.9) (4.) (4.) For nuerical resuls, he following values for paraeers are considered. Table. Paraeer values for he series soluions Paraeer Assigned values S E I 5 R 5 β. γ.8 s. µ. µ T.3 U.4 ε.3 d.4 π.3 ρ.5 For high accuracy of resuls, we use Maple 5 copuaion sofware [46].For he graphs, do lines: Suscepibles; dash lines: Exposed; dashdo lines: Infecives; longdash lines: Recovered. The 5 h, 6 h, 7 h and 8 h ers approxiaions for S(), E(), I() and R() are calculaed and presened below. 5 h ers approxiaions = S = E = I = R h ers approxiaions = S = E = I = R h ers approxiaions

4 74 Aerican Journal of Applied Maheaics and Saisics = S , = E = I = R h ers approxiaions = S = E Figure. Plos of 6 h ers approxiaions for S(), I() and R() agains ie() Figure 3. Plos of 7 h ers approxiaions for S(), E(), I() and R() agains ie() = I = R Figure 4. Plos of 8 h ers approxiaions for S(), E(), I() and R() agains ie() Fro he various order of approxiaions, he HAM yields convergen series soluions ha are reasonable and easy o express. The plos show ha while he nuber of suscepible (S) decreases he populaion who are infecives (I) increases in he period of he epideic. Meanwhile, he nuber of exposed (E) increases while he nuber of recovered (R) decreases. However, fro figure 4, as he infecion dies ou (i.e. as I, he nuber of suscepible, exposed and recovered increases. In paricular, as I, S approaches soe posiive value S=6.53 which is he evenual populaion who were never infecive. 6. Conclusion Figure. Plos of 5 h ers approxiaions for S(), I() and R() agains ie() In his paper, he HAM has been successfully applied o approxiaely solve a syse of nonlinear equaions in

5 Aerican Journal of Applied Maheaics and Saisics 75 uberculosis dynaics. The resuls show he poenial and efficiency of HAM in solving nonlinear probles. We hus conclude ha, cobined wih high perforance copuer and sybolic copuaion sofware such as Maple and so on, he hooopy analysis ehod igh becoe a new powerful analyic ool o ge saisfacory approxiaions for nonlinear probles in science and engineering. References [] Blower, S.M., McLean, A.R., Porco, T.C., Sall, P.M., Hopewell, P.C., Sanchez, M.A. and Moss, A.R., The inrinsic ransission dynaics of uberculosis epideics, Na. Med., (8) [] Song, B., Casillo-Chavez, C. and Aparicio, J-P., Tuberculosis wih fas and slow dynaics: he role of close and casual conac, Mah. Biosci., [3] Egbeade, S.A. and Ibrahi, M.O., Global sabiliy resuls for a uberculosis epideic odel, Res. J. Mahs & Sa., 4() [4] Colijn, C., Cohen, T. and Murray, M. (6) Maheaical odels of uberculosis: accoplishens and fuure challenges. Proc. Nal. Acad. USA 3(8), -8. 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