The Conceprts of Delay Differential Equations and IT S Application

Transcription

1 Briish Journal of Mahemaics & Compuer Science 4(1): , 214 SCIENCEDOMAIN inernaional wwwsciencedomainorg The Conceprs of Delay Differenial Equaions and IT S Applicaion Adamu Wakili 1* 1 Deparmen of Mahemaical Sciences, Adamawa Sae Universiy, Mubi, Adamawa Sae, Nigeria Original Research Aricle Received: 31 July 213 Acceped: 26 November 213 Published: 25 March 214 Absrac All processes ake ime o complee While physical processes such as acceleraion and deceleraion ake lile ime compared o he imes need o ravel mos disances, he imes involved in biological processes such as gesaion and mauraion can be subsanial when compared o he daa-collecion imes in mos populaion sudies Therefore, i is ofen imperaive o explicily incorporae hese process imes in mahemaical models of populaion dynamics These process imes are ofen called delay imes, and he model ha incorporae such delay imes are referred as delay differenial equaion (DDE) models The models will examine some heoreical conceps and heir applicaions o real life siuaion The applicaion examines measles and he ime i akes o manifes or o is removal or reamen from he sysem The soluions of he models will be displayed in graphical forms using MATLAB mehod The analysis of he models indicae he imes delay and is characerisics Keywords: Time delay, mauraion, gesaion, oscillaion, periodic, parameer and dynamics 1 Inroducion The use of ordinary and parial differenial equaions o model biological sysems has a long hisory These sysems canno capure he rich variey of dynamic behaviour observed in naural sysems This usually lead o sysems wih more differenial equaions and having many parameers which canno be deermine experimenally Wih he inroducion of ime delay erms in he differenial equaions, his area is gaining ground more rapidly han expeced The delay or lags can represen gesaion imes, incubaion periods, ranspor delays or can simply lead o complicaed biological processes ogeher, accouning only for he ime required for process o occur These models have he advanage of combining a simple, inuiive derivaion wih a wide variey of possible behaviour rigid for a single sysem On he oherwise, hese models habour much of he deailed working of complex biological sysems and someimes hese deails are of ineres *Corresponding auhor: adamou_wakili@yahoocom;

2 Recenly delay models are becoming more popular, appearing in many branches of biological modeling They have been used for describing several aspecs of infecious disease dynamics; primary infecion [1], drug herapy [2] and immune response [3] Delays have also been exended o he sudy of chemosa models [4], circadian rhyhms [5], epidemiology [6], he respiraory sysems [7], umor growh [8] and neural neworks [9] In many species of populaion dynamics delay models have been applied in Saisical analysis of ecology daa [1, 11] 11 Basic Properies of Delay Differenial Equaions Like ordinary differenial equaions, delay differenial equaions have several feaures which make heir analysis more complicaed Consider he following delay differenial equaion x( ) = f ( x( ), x( τ ) (11) To begin wih an iniial value problem requires more informaion han an analogous problem for a sysem wihou delay For an ordinary differenial sysem, a unique soluion is deermined by an iniial poin in Euclidean space a an iniial ime For a delay differenial sysem, one requires informaion on he enire inerval [ τ ], To know he rae of change a, one needs o know x( ) and x( ), and for x( + ) So, in order for he iniial value problem o make sense, one needs o give an iniial funcion he value of x( ) for he inerval[ r,] Each such iniial funcion deermines a unique soluion o he delay differenial equaion If we require ha he iniial funcions o be coninuous, hen he soluion space has he same dimensionaliy as C ([ τ, ], ) τ R [12,13] The indefinie dimensional naure of delay differenial equaion is apparen in he sudy of linear sysems Jus as for ordinary differenial equaions, one seeks exponenial soluions and compues he characerisic equaion 2 Theoreical Conceps Delay differenial equaions ( DDE) provide a mahemaical model for physical sysems in which he rule of change of he sysems depends no only on heir presen sae, bu also on heir pas hisory (sae) Consider he DDE below; x( ) q( ) x( r) dx( ) (21) where q( ) d, d > ε 1382

3 In pracice q( ) will represen he nonlineariies of he equaion To undersand he behaviour of he sysem, we compare is dynamics wih hose of he sysem y( ) = dy( τ ) dy( ) (22) Lemma If x and y are defined as above, and x( ) = y( ), for [ e, e + ] for some e, hen x( ) y( ), [14]Ladas e al in 1983 has esablished a necessary and sufficien condiion under which all soluions of he rearded differenial equaion m x( ) + qix( τ i ) =, oscillaes (23) i= 1 where q i are posiive numbers and τ i are non-negaive numbers, i=1,2,,m 3 Applicaion of DDE The sudy of populaion dynamics in differenial equaion for single species populaion will be well esablished The mos common is he exponenial and logisic growh models This class of differenial equaion models will involve a ime delay par Consider he model below; x( ) = b( x( τ )) x( τ ) d( x( )) x( ) (31) Where b is non-increasing and d non-decreasing, which represens he populaion dynamics of a single species wih a delayed birh erm The basic properies of his model are he ype of funcions b and d which migh lead o he exisence of periodic soluions in equaion (31)We a specify o use of he case b( ) = be and d( ) consan I will prove he exisence of he periodic soluion of he equaion The delay-dependen erm is added o he parameer b and he effecs of his aleraion are explored, and condiions are given for he exisence of linear insabiliy of he posiive seady sae The model can be ransformed ino x( ) = [ b( x( τ )) d( x( ))] x( ) (32) The forms of b( x) and d( x) migh give rise o periodic soluions 1383

4 Consider ha b( x) is coninuous, decreasing funcion ha is he per capial growh rae of he populaion decreases wih increased populaion levels The delay can represen a gesaion or mauraion period, so he number of individuals enering he populaion depends on he levels of he populaion a he previous insance of ime The funcion d( x ) is non-decreasing and posiive This represens he per capia deah rae, which may be increased by inra specific compeiion [12,15,16,14] The ype of hese models has been exensively used in he mahemaical biology lieraure, especially when here is an ineres in modeling and oscillaion phenomena In populaion biology [16, 14] explored he model generally, while [17] is a specific applicaion o housefly populaion Oscillaory phenomena have few analyic resuls abou he exisence of periodic soluions [15] Theorem: Le b and d be posiive funcions Suppose ha here exiss x such ha sign( b( x) d ( x)) = si gn( x x), and x < d ( x) (34) Then x is a posiive seady sae, and he rivial seady sae is unsable If b ( x) x > 2 d( x) d ( x) x, hen x is linearly sable for all x Oherwise, here exiss ατ c > such ha x is sable forτ τ c <, and unsable for τ > τ c Proof To begin wih, x is a unique posiive seady sae, since b( x) d( x) = if and only if x = x I is he poin a which b( x) = d ( x) Line razing abou he seady sae yields he equaion which has he characerisic equaion x( ) = ( d( x) + b ( x) x) x( τ ) ( d( x) + d ( x) x) x( ) λ = α x( τ ) β x( ), where α = d( x) + b ( x) x and d( x) + d ( x) x since b ( x) < d ( x), α < β Furhermore, we know ha for α β = β, all roos of he characerisic equaion have negaive real par Since α < β, hen his condiion is saisfied if and only if α > β, bu his is exacly he condiion of he above equaion (34) If his is no he case, hen α < β I is clear ha for i=, he characerisic roo is λ = α β Thus, by he 1384

5 3 1 Applicaion of DDE on Measles We employed he basic sochasic model for epidemic processes This is he coninuous- infecion ype in which a suscepible becomes infecious immediaely afer he receip of infecion and coninues in his sae unil removal from circulaion by deah or isolaion If he ime elapsing beween receiving infecious maerial and he developmen of infeciousness were shor, and if he infecious period up o removal were approximaely negaive exponenial in disribuion, hen such a model would be quie appropriae Wheher his closely mimics any acual disease is sill uncerain, hough scarle fever and diphheria For example, he disease like measles a any rae, one version of his represenaion has me wih some success, he small amoun of daa available passing he appropriae goodness-of-fi ess [6] The daa for he model were colleced a Federal Medical Cenre Gombe 4 Mehod The mehod used in solving he delay differenial equaion is he use of MATLAB The delay differenial equaion is solved o is lowes level before applying MATLAB o generae resuls which are posiive a all ime, bounded and seady and sable The graphs below explained he resuls of he delay differenial equaion (41) Consider he equaion y ( ) = λ y( 1)(1 + y( )) for [, 4], y( ) = λ = 15, 2, 25,3 ( y( )) + λ y( )(1 + y( )) 1 y( ) C = ( λ ) 1 + l y( ) = l ( λ ) + 1 (41) 5 Resuls and Analysis When he model is ran using MATLAB, i produced graphical view ha varied from he parameer values, λ = 15, 2, 25 and 3 I will be realised ha as he value of he parameer is smaller he curve is clearer and has longer imes o cover han when he parameer value is larger This also agreed wih he paern of he model and reamen is faser and more feasible a he early sages of he disease han when he disease mus have sayed in he sysem The resuls are displayed graphical forms below: λ =

6 4 /( exp(15 ) - + 1) y() /( exp(2 ) - + 1) 25 2 y()

7 /( exp(25 ) - + 1) 25 2 y() /( exp(3 ) - + 1) 25 2 y() Conclusion The numerical scheme described above invesigaes he kind of ime delay involved and because he ime delay governed he dynamics of he sysem The model wih he ime delay embedded he echnique used o visualize he ime series generaed by he numerical inegraion of he model We see as he values of he parameer, λ, are varied from o 3, he curve changes paern, he peak of he curve also changes gradually from 32 o 22 ( he values of y( ) ) 1387

8 Compeing Ineress Auhor has declared ha no compeing ineress exis References [1] Ciupe SM, BL de Bivor, Borz DM, Nelson PW Esimaes of kineic parameers from HIV paien daa during primary infecion hrough he eyes of hree differen models Mah Biosci In press; 24 [2] Nelso PW, Murrey JD, Perelson AS A model of HIV-1 pahogenesis ha includes an ineracion delay Mah Biosci 2;163: [3] Cooke KK Kuang, B Li Analysis of an aniviral immune response model wih ime delays Canad Appl Mah Quar 1998;6(4): [4] Zhao T Global periodic soluions for a differenial delay sysem modeling a microbial populaion in he chemo sa J Mah Anal Appl 1995;193: [5] Smolen PD Baxer, J Byrne A reduced model clarifies he role of feedback loops and ime delays in he Drosophila circadian ascillaor Biophys J 22;83: [6] Cooke KL, P van den Driessche, Zou X Ineracion of mauraion delay and nonlinear birh in populaion and epidemic models J Mah Biol 1999;39: [7] Vielle B, G Chauve Delay equaion analysis of human respiraory sabiliy Mah Biol, 1998;47(2): [8] Villsana M, Radunskaya A A delay differenial equaion model for umor growh J Mah Biol 23;47(3): [9] Comphell SA, Ewards B, P van de Driessche Delayed coupling beween wo neural nework loops SIAM J Appl Mah 24;65(1): [1] Turchin P Rariy of densiy dependence or populaion regulaion wih lags Naure 199;344: [11] Turchin P, Taylor AD Complex dynamics in ecology ime series Ecology 1992;73: [12] Edelsein-Keshe L Mahemaical Models in Biology McGraw-Hill, New York; 1988 [13] El sgol s LE, Norkin SB An inroducion o he heory and applicaion of differenial equaions wih deviaing argumens Academic Press, New York; 1973 [14] Wangersky PJ, Cunningham WJ On ime lags in equaions of growh Proc Na Acad Sci USA 1956;42:

9 [15] Kuang K Delay Differenial equaions wih applicaion o populaion biology Academic Press, New York; 1993 [16] Blyhe SP Insabiliy and complex dynamic behaviour in populaion models wih long ime delays Theor Pop Biol 1982;22: [17] Taylor CE, Sokal AD Oscillaion in housefly populaions due o ime lags Ecology 1976;57: Wakili; This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License (hp://creaivecommonsorg/licenses/by/3), which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied Peer-review hisory: The peer review hisory for his paper can be accessed here (Please copy pase he oal link in your browser address bar) wwwsciencedomainorg/review-hisoryphp?iid=466&id=6&aid=