Fractional Order SEIRS Model

Transcription

1 Advaced Studies i Biology Vol o HKAR Ltd ttp://dx.doi.org/0.2988/asb Fractioal Order SRS Model Muammad Ozair* Umer Saeed ad Takasar Hussai Scool of Natural Scieces Natioal Uiversity of Scieces ad Tecology H-2 Campus slamabad Pakista Correspodig autor Copyrigt 204 Muammad Ozair Umer Saeed ad Takasar Hussai. Tis is a ope access article distributed uder te Creative Commos Attributio Licese wic permits urestricted use distributio ad reproductio i ay medium provided te origial work is properly cited. Abstract tis paper we itroduce a fractioal order SRS model. Te local asymptotic stability of equilibrium poits is studied. We utilized te Adams-Basfort predictor corrector metod for solvig te proposed model. Numerical simulatios are preseted to sow te advatage of itroducig a fractioal model.. troductio Matematical modelig of te spread of ifectious diseases cotiues to be a area of active researc. t as become a importat tool i uderstadig te dyamics of diseases ad i decisio makig processes regardig itervetio programs for cotrollig tese diseases i may coutries. We replace te real world peomea of te disease by a abstract model i order to employ te tools of matematical aalysis. Tis model typically takes a matematical formulatio tat ivolves basic variables ad relatiosips correspodig to te etities ad te laws of ature or beavior beig observed. t usually takes te form of o-liear ordiary differetial equatios. Te subsequet solutio is i te matematical form ad ca be reiterpreted back i te origial real world settig. Matematical modelig of complex biological processes is a maor callege for cotemporary scietists. Te complex systems are caracterized by te variability of structures i tem multiscale beavior ad oliearity i te matematical descriptio of te mutual relatiosip

2 48 Muammad Ozair Umer Saeed ad Takasar Hussai betwee parameters []. Terefore te teory of iteger-order differetial equatios is ot sufficiet to describe teir dyamics. recet years fractioal differetial equatios become more popular because of its powerful potetial applicatios. A large umber of ew differetial equatios models tat ivolve fractioal calculus are developed. Tere ave foud a umber of works especially i mecaics biology cemistry electrical egieerig [ ]. te literature a umber of metods ave bee developed for te umerical or aalytical solutios for fractioal differetial equatios. We listed some of tese metods as follows: Adomia decompositio metod [7] te collocatio metod [8] te fractioal differetial trasform metod [9 0] omotopy aalysis metod [ 2] omotopy perturbatio metod [3]. biology it as bee deduced tat te membraes of cells of biological orgaism ave fractioal-order electrical coductace ad te are classified i groups of o-iteger order models. Fractioal derivatives embody essetial features of te beavior of te patter formatio i bacterial coloies. Also it as bee sow tat modelig te beavior of braistem vestibule-oculumotor euros by fractioal ordiary differetial equatios as more advatages ta classical iteger-order modelig. Fractioal derivatives are aturally related to systems wit memory wic exists i most biological systems [4]. For more details about usig fractioal calculus i modelig complex biomaterials we refer te reader to [] ad te refereces terei. Adam Basfort predictor corrector metod as bee proposed i [5 6] for solvig fractioal differetial equatios. Te metod ca be used for liear as well as oliear fractioal differetial equatios. te preset work we use it for te umerical simulatio of biological system wic is fractioal order. We itroduce te Caputo derivative of order i SRS model wic was discussed i [7] ad we discuss te oscillatory beavior of te edemic level of ifected populatio. To te best of our kowledge tis work represets te first umerical solutio for disease model of fractioal order by applyig te Adams-Basfort-Moulto metod. Te rest of te paper is orgaized as follows. sectio 2 some ecessary defiitio ad otatios related to fractioal calculus are preseted. Te formulatio of te model ad stability of equilibrium poits is discussed i sectio 3. Te geeral procedure for implemetatio of te predictor corrector metod for te fractioal model is discussed i sectio 4 ad te umerical results are preseted grapically i sectio 5. Fially cocludig remarks are give i sectio Prelimiaries First we review some basic defiitios of fractioal differetiatio ad fractioal itegratio [3]. Riema-Liouville fractioal itegral operator of order Te operator x defied o L [0 b] by x x y( x) = ( x t) y( t) dt Γ( ) 0 + for 0 x b were R is called Riema-Liouville fractioal itegral of order.

3 Fractioal order SRS model 49 Riema-Liouville ad Caputo fractioal derivative operator of order Te operator D defied by x d x ( ) = ( ) ( ) 0 Dx y x x t y t dt Γ( ) dx + for0 x b were R ad = order.te Caputo fractioal derivative of a fuctio y L [0 b] c x d ( ) = ( ) ( ) 0 Dx y x x t y t dt Γ( ) dt + x b were R ad =. for0 is called Riema-Liouville fractioal itegral of is defied as 3. Matematical Model Te iteger order model wic is reported i [7] is give by ds dt S = π + θ R β µ S + d S = β σ µ dt + d = σ σ µ δ dt dr = σ θ R µ R. dt Tis model as bee discussed i te biologically feasible regio 4 D = {( S R) R S R π µ } wit iitial coditios S (0) > 0 (0) > 0 (0) > 0 R(0) > 0. tis paper we discuss te model cosistig of fractioal system of equatios wic is obtaied by ust replacig a iteger order derivative by a fractioal derivative of order 0 <. Tus te fractioal order system is give by (

4 50 Muammad Ozair Umer Saeed ad Takasar Hussai S Dt S( t) = π + θ R β µ S + S Dt ( t) = β σ µ (2) + D ( t) = σ σ µ δ t D R( t) = σ θ R µ R. t 3.. Asymptotic stability of quilibrium Poits To calculate te equilibrium poits we put te rigt ad side of system (2) equal to zero ad obtai After simple calculatio we ave S π + θ R β µ S = 0 + S β σ µ = 0 + σ σ µ δ = 0 N R ad is te solutio of te equatio 2 A B C were + + = 0 S σ θ R µ R = 0. π δ = µ σ θ µ = + ( σ + µ + δ ) = σ (3) ( σ + µ )( σ + µ + δ )( µ + ω( π δ )) = µβσ

5 Fractioal order SRS model 5 ( ) ( )( )( )( ) ( ( ( ) ( ))) 2 ( )( ) 2 ( )( ) ( )( )( )( )( ) A = θβµδωσ σ + δω θ + µ µ + σ βµ µδω µ + δ + σ B = βµσ δω πθ + πµ θσ µ + πω ( )( ) ( )( ) ( ) θ + µ µ + σ µ + πω βµ µδω δω µ + πωµ µ + δ + σ C = βµσ πθ + πµ µ + πω θ + µ µ + σ µ + πωµ µ + πω µ + δ + σ For positive equilibrium it is clear from system (3) tat π δ 0 > wic meas π 0 < <. (4) δ By usig te coditio (4) ad te parameter values give i [7] te correspodig approximate ad R are give i te followig table. equilibrium values of S π θ β ω µ Table : quilibrium Values for differet θ ad δ. σ σ δ S R Te Adams-Basfort-Moulto Metod We utilize Adams-Basfort-Moulto predictor corrector metod [5] for solvig system of fractioal order differetial equatios. Te implemetatio of te adams-basfort-moulto metod accordig to te cocered fractioal system is as follows: D x( t) = g ( t x y z u) D y( t) = g ( t x y z u) (5) 2 D z( t) = g ( t x y z u) 3 D u( t) = g ( t x y z u) 4

6 52 Muammad Ozair Umer Saeed ad Takasar Hussai wit x(0) = x0 y(0) = y0 z(0) = z0 u(0) = u0 were 0 <. Cosider te T uiform grid t = = 0... N wit some iteger N ad = N weret is te upper boud of te iterval[0 T ]. mplemetatio of te Adams-Basfort Moulto metod [5] o te proposed model are as follows: Predictor values for (5) are x = x g( t x y z u Γ ( + 2) = 0a + g( t x y z u ) Γ ( + 2) y = y g2( t x y z u Γ ( + 2) = 0a + g2( t x y z u ) Γ ( + 2) z = z g3( t x y z u Γ ( + 2) = 0a + g3( t x y z u ) Γ ( + 2) u = u g4( t x y z u Γ ( + 2) = 0a + g4( t x y z u ) Γ ( + 2) were b = (( + ( ) + ). Corrector values are obtaied by usig Predictor values as

7 Fractioal order SRS model 53 x+ = x0 + g( t+ x+ y+ z+ u+ Γ ( + 2) Γ ( + 2) a g ( t x y z u ) = 0 + y+ = y0 + g2( t+ x+ y+ z+ u+ Γ ( + 2) Γ ( + 2) a g ( t x y z u ) = z+ = z0 + g3( t+ x+ y+ z+ u+ Γ ( + 2) Γ ( + 2) a g ( t x y z u ) = u+ = u0 + g4( t+ x+ y+ z+ u+ Γ ( + 2) were a Γ ( + 2) a g ( t x y z u ) = ( )( + = 0 = ( + 2) + ( ) 2( Accordig to te matematical aalysis of tis metod i [6] we ave order of accuracy`` " were p = mi(2 + ). p 5. Numerical results tis sectio we sall discuss te oscillatory beavior of te edemic level of ifected populatio of model (2) o te basis of te umerical results wic are obtaied by usig te parameter

8 54 Muammad Ozair Umer Saeed ad Takasar Hussai values give i Table. t is clear from figure tat we te recovered idividuals lose teir ifectio-acquired immuity ad i te absece of disease related deat tere are damped oscillatios of te ifected populatios for =. Te edemic equilibrium level is evetually attaied ad tis coverges to a steady state tat is asymptotically stable. Te oscillatios decrease for fractioal values of ad we obtai te edemic level at te early stage as compared to te iteger order. Similar beavior as bee observed we we suppose permaet immuity ad disease related deat rate of ifected idividuals. Tis beavior of ifected populatio is sow i figure 2. Figure. Oscillatory beavior of ifected populatio we θ = 0.02 adδ = 0. Figure2. Oscillatory beavior of ifected populatio we θ = 0 ad δ =

9 Fractioal order SRS model Discussio We recosidered SR model wic was discussed i [7] by itroducig te Caputo derivative of order i place of ordiary derivative. We derive te relatios of equilibrium values i terms of ifected populatios ad te equilibrium poits are obtaied by usig parameter values. Numerical simulatios are carried out by usig Adams-Basfort-Moulto metod ad it is sow tat fractioal model provides better stability results ad te edemic level is obtaied at te early stage ta te ordiary differetial equatios model. Tus te preset success of te proposed fractioal order model verifies tat it is a useful tool for tese kids of models. Refereces J. S. Leszczyski A troductio to Fractioal Mecaics Czestocowa Uiversity of Tecology 20. 2) K. B. Oldam & J. Spaier Te fractioal calculus Academic Press New York (974). 3). Podluby Fractioal differetial equatios Academic Press Sa Diego (999). 4) M. F. Slesiger G. M. Zaslavsky & J. Klafter Strage kietics Nature Vol. 363 No. 6pp (993). 5) S. G. Samko A. A. Kilbas & O.. Maricev Fractioal tegrals ad Derivatives: Teory ad Applicatios Gordo ad Breac Lodo (993). 6) Y. Dig ad H. Ye A fractioal-order differetial equatio model of HV ifectio of CD4 T-Cells Matematical ad Computer Modellig Volume 50 ssues 3-4 August 2009 pages ) S. Momai ad R. Qaralle A efficiet metod for solvig systems of fractioal itegro-differetial equatios Computers & Matematics wit Applicatios vol. 52 o. 3-4 pp ). A. Rawasde Numerical solutio of fractioal itegro-differetial equatios by collocatio metod Applied Matematics ad Computatio vol. 76 o. pp ) A. Secer M. A. Akilar ad A. Cevikel fficiet solutios of systems of fractioal PDs by te differetial trasform metod Advaces i Differece quatios vol. 202 article ) A. Alawe Applicatio of te Multistep Geeralized Differetial Trasform Metod to Solve a Time-Fractioal zyme Kietics Discrete Dyamics i Nature ad Society Volume 203 (203) Article D pages. M. Kurulay A. Secer ad M. A. Akilar A ew approximate aalytical solutio of Kuramoto-Sivasisky equatio usig omotopy aalysis metod Applied Matematics & formatio Scieces vol. 7 o. pp

10 56 Muammad Ozair Umer Saeed ad Takasar Hussai 2) S. Z. Rida A. A. M. Arafa ad M. Kalil Solutios of Fractioal model of uma T-cell lympotropic virus (HTLV-) fectio of CD4 T-Cells usig HAM teratioal Joural of Basic ad Applied Scieces ( (202) -4. 3) Q.Wag Homotopy perturbatio metod for fractioal KdVBurgers equatio Caos Solitos & Fractals vol. 35 o. 5 pp ) A. A. M. Arafa S. Z. Rida M. Kalil Fractioal Order Model of Huma T-cell Lympotropic Virus (HTLV-) fectio of CD4 T-Cells Advaced Studies i Biology Vol o ) K. Dietelm N. J. Ford A. D. Fareed A predictor-corrector approac for te umerical solutio of fractioal differetial equatios Noliear Dyamics 29: ) K. Dietelm N. J.Ford A. D. Fareed Detailed error aalysis for a fractioal Adams metod Numerical Algoritms 36: ) Suli Su Cuiua Guo Cegmi Li Global Aalysis of a SRS Model wit Saturatig Cotact Rate Applied Matematical Scieces Vol o Received: Marc 7 204