Feasible set in a discrete epidemic model

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1 Journal of Phic: Conference Serie Feaible et in a dicrete epidemic model To cite thi article: E-G Gu 008 J. Ph.: Conf. Ser View the article online for update and enhancement. Related content - Phic of Surface, Interface and Cluter Catali: Firt principle-baed kinetic Monte Carlo imulation in catali H Kaai and M C S Ecaño - Random walk, diffuion and mixing in imulation of calar tranport in fluid flow A Y Klimenko - Quantiation of the Kaplan-Yorke model N F de Godo, R Graham and W F Wrezinki Thi content wa downloaded from IP addre on 3//07 at :47

2 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 Feaible Set in a Dicrete Epidemic Model En-Guo Gu Department of Mathematic, South-Central Univerit for Nationalitie, Wuhan , P.R. China guenguo@63.com Abtract. Thi paper aim at the tud of the delimitation of the et of feaible trajectorie (dicrete trajectorie having a biological ene) generated b a famil of two-dimenional map T related to a dicrete epidemic model. The invere of T ha vanihing denominator giving rie to non-claical ingularitie: a non definition line, a focal point and a prefocal line. In particular, we give an anwer to one of the open problem propoed b Kocic and Lada.. Introduction Man claical epidemic model have been propoed and tudied [-4]. The determinitic epidemic model k Ax n xn + = ( xn j )( e ), n = 0,, j = (.) i the pecial cae of an epidemic model which i derived in [5, 6], where i the paraitoid population ize in generation n, A i the intantaneou earch rate i.e. average encounter per hot per unit of earch time. Two reearch project are propoed in [7]. One ha been partl olved in [8]. The other i to chooe the initial condition of Eq.(.) to keep the olution poitive for all n=0,,. The main purpoe of thi paper i to give an anwer to thi and other quetion concerning Eq.(.). For the ake of implicit we hall firtl focu on the implet cae, obtained for k= in (.). Axn xn+ = xn xn )( e ), = 0,, ( n However, man of the technique can be generated to recurrence (.) with higher value, i.e. recurrence order greater than. Some poible generation are given in the paper. We hall make ue of ome reult on iterated plane map with a denominator which can be vanih, given in [9-] where new kind of ingularitie, called focal point and prefocal curve, have been defined. A hown in the paper, ome intereting reult of the recurrence (.) can be evidenced in the light of thee concept. The plan of the work i a follow. In Sec. where ome propertie and the terminolog introduced in [9-] are briefl decribed and extenivel ued. In Sec.3 we preent the condition of exitence and tabilit for fixed point. Sec.4 concern ome definition of feaible et, feaible domain and bain, and the delimitation of them i preented in detail. Some extenion are given for higher order recurrence (.). We end thi paper with ome concluion and dicuion in Sec.5. x n (.) To whom an correpondence hould be addreed. c 008 IOP Publihing Ltd

3 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05. Propertie of the equivalent two-dimenional map In order to ue the concept of the ingularitie, let u rewrite the econd-order recurrence in (.) a a two-dimenional tem of the firt order, i.e. an iterated map of the plane. A uual, thi i obtained b letting ( x, ) (, ) o that (.) can be written a, where n xn = x M ( x, ) (, ) M ( x, ) ( x, n xn xn xn+ ) i the two-dimenional map given b The Jacobian matrix of map (.) i : x = M (.) = ( x )( e A ) J e 0 = A A Apparentl, it Jacobian determinant det( J ) = e map (.) i invertible except on the x axi, A( x ) e A (.) vanihe on the line given b =0, o that the the invere of M i x = x M : Ax e (.3) = x For the map (.3), the non definition etδ coinciding with the locu of point in which at leat one δ = ( x, ) R x = 0. In (here the firt component of Eq.(.3)) denominator vanihe i given b { } general, the two-dimenional recurrence equence obtained b the iteration of M i well defined provided that the initial condition belong to the good et G given b G = R \ Λ, where Λ (a et of zero Lebegue meaure in R ) i the union of the image of an rank of the line δ, that i Λ = k M ( δ ). In other word, equence generated b the recurrence (.3) can be obtained b the k =0 iteration of the two-dimenional map M : G G, which i not defined in the whole plane. Following the terminolog introduced in [9-], the line will be called et of non definition, and it point Q=(0,0), where the firt component of the map M aume the form 0/0 contituting a imple focal point of M, and the aociated prefocal etδ i the line of equation =0. Thi mean that a one-to-one Q correpondence exit between the lope m of arcη through the focal point Q and the point (x(m),0) where their preimage M ( η) cro the prefocal etδ. In thi cae uch correpondence i ver imple, given b m x( m) = m / A. Thi implie that the preimage b M of an arcη croing through Q, i an arc M ( η) which croe the prefocal line at the point (-m/a, 0), and converel, the image M (η ) of an arbitrar arcη which croe the line =0 at a point (x, 0) i an arc which croe through Q with a lope m=a(-x) in Q (ee the qualitative picture in Fig.(a)). Intead, if an arcγ croe the et of non definition δ at a non focal point, then it image b ( 0, ξ ), ξ 0 M i an unbounded arc, doubl amptotic to the line of equation =0. We hall ee that the propertie of focal point and prefocal et of M will help u to undertand the peculiar propertie of the recurrence (.). Q

4 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 Figure (a) how the noticeable qualitative change in the hape of the image and preimage, due to a contact between an arc and the ingular etδ and δ. (b) One dimenional bifurcation diagram of the Q tem (.). Exitence and tabilit of fixed point Propoition () Aume A>, the original point (0, 0) i addle fixed point. () Aume A>, map (.) ha a unique poitive fixed point ( x, x) and atifie 0 < x < / 3. (3) The poitive fixed point ( x, x) of map (.) i table for x A <. ( x)( 3x) Proof: () The multiplier (or eigenvalue) of the fixed point (0, 0) are A> and 0, therefore, the original point (0, 0) i addle fixed point. () The poitive fixed point ( x, x) of map (.) atifie the following equation: Ax ( x)( e ) = x (3.) From Eq.(3.) we have e Ax = ( 3x ) /(x ), thi implie that the necear condition of exitence for a poitive fixed point i 0 < (3x )/(x ) <. Therefore, the necear condition of exitence for a poitive fixed point i 0 < x < / 3. Let g(x)=(-x) (-e -Ax Ax Ax ), then g ( x) = + ( A Ax + ) e and g ( x) = A(4 + A Ax) e < 0 for 0<x</3, hence g(x) i a convex function. Noticing that g(0)=0 and g ( x) < g (0) = A >,b the propertie of g(x) we obtain that map (.) ha a unique poitive fixed point. (3) From Eq.(.) (Jacobain matrix of map(.)), the characteritic equation of the Jacobain matrix J at poitive fixed point ( x, x) can be given b Ax x λ A( x) e λ ( e A ) = 0 (3.) A imple anali how that the condition of tabilit for poitive fixed point i x [ A ( x) + ] e A < (3.3) Subtituting e Ax = ( 3x ) /(x ) into Eq.(3.3) we obtain that the poitive fixed point i table for 3

5 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 x A < ( x)( 3x). Simulation how that the poitive fixed point begin at A.65 then loe it tabilit giving birth to a 3-ccle through fold bifurcation for A > F( x) Thi can be een in the fig.(b). 3. The feaible et and feaible domain One of reearch project propoed in [7] i to chooe the initial condition of Eq.(.) o that the olution remain poitive for all n=0,. Thi i related to the following concept: Definition We call S the feaible et, the et of point whoe trajector tarting from S totall belong to the firt quadrant R + = {( x, ) R x 0, 0}. The full trajector i called a feaible trajector. Definition We call D the feaible domain, if a full trajector tarting from D totall belong to the firt quadrant R + = {( x, ) R x 0, 0} and converge to an one of the different bounded attractor belonging to R +. The full trajector i called a converging feaible trajector. Since onl converging feaible trajector can repreent reaonable time evolution of the epidemic model, the firt important problem to olve i the delimitation of the et of initial condition that generate a converging feaible trajector for given parameter, In thi ection, we hall give an anwer to thi quetion. Definition 3 We denote b B(A) the bain of attraction of an attractor A (or the bain), defined a the et of all the point whoe trajectorie converge to the bounded attractor A. From definition -3, we have D=B S, if there i onl one attractor in R +. So in uch cae, to determine the feaible domain, we onl need to know the feaible et and the attracting bain. The bain can be obtained b man claical method, o an exact determination of the feaible et S for the model (.) i the main goal of thi ection. From definition, we know that M k ( S) R+ k=0,,.thi implie S M -k ( R + ), k=0,,,. Therefore, we expect that the feaible et boundar can be determined b all the preimage of the two axe which are the boundarie of R + (ee[-6]). From the Eq.(.3) (the invere of the model (.)), the rank- preimage of Y-Q, doe not exit or i infinite and M - (Q)= {=0}, hence, we obtain n= 0 n S ( M ( X )) Y (4.) Thi mean that the feaible et boundar i mainl related to the axi X={=0}. Denoting bω the x-axi, we can analticall obtain the preimage ofω from the equation of (.). Here we onl give the rank- and rank- preimage ω x + : x+=; ω : e A = (4.) x ince the higher rank preimage are complex. Becaue theω croe the non definition lineδ at the focal point Q(0,0) with the lope m=0, we know, from the anali of Sec., it preimage ω interect the prefocal line δ Q : =0 at point (x(m),0)=(,0). However, theω alo croe the non definition line at non focal point (0, ), o it preimageω take the form of M ( γ ) in fig(a) and ha two branche amptotic to the line =0 (ee fig.(a)(c)). From Eq.(4.), We haveω : x=(-) e -A /(-e -A ). It i ea to find that the curvω ha other two amptotical line x=0, x+=. 4

007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/74-6596/96//05 Figure Feaible domain and bain in the model (.

6 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 Figure Feaible domain and bain in the model (.) To gain a good interpretation of the feaible et tructure, we firt give the following propoition: Propoition Conider a map T, If a et Ω entirel belong to the non feaible et C(S), then all it preimage belong to the et C(S) forever. -k Proof: Suppoe the aertion i not true, then there exit k uch that T (Ω ) S Φ, hence, there are point in S giving birth to the point of Ω C(S) after k iteration of T, which contradict the definition of the et S. Thi complete the proof. From propoition, we know that once a preimage ofω totall belong to C(S), all it preimage of higher order can not belong to the feaible et boundarie. From Eq.(.), we have M - ( R )={(x,) 0, x+ }, M({(x,) <0, x+<} {(x,) >0, + x+>})={(x,) <0}. Thi implie that onl the point in the triangle, bounded b ω, ω- and Y axi, denoted b D={(x,) x 0, 0,x+ }, can generate the feaible trajectorie. That i, the et C(D) i the non feaible et. Now we hall prove the curve ω - tall belong to the et C(D). The econd equation of Eq.(4.) implie that ω {(x,) (x+-)/x>0}, i.e. ω D D, where D={(x,) x>0,x+>}, D = {(x,) x<0, x+<}. Therefore,ω C(D) for A>. From propoition, the higher preimage ofω belong to C(D) forever thu can not become the boundarie of feaible et. Therefore, we gain the following propoition: Propoition 3 The feaible et of tem (.) are the triangle D bounded b x and axi and the line n M ( X )) Y = ω ω Y ω =M - (X): x+=, that i, D ( n= 0 5

7 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 The bain of attraction B(A) are illutrated b can region (ee Fig.(b)(d)). The boundarie of bain B(A) from B( ) can be determined b the inet (i.e. the table manifold) of the addle point O(0,0) which i on the bain boundarie. Becaue the original point i not onl the addle fixed point but alo the focal point, aociated prefocal line {=0}, i.e. M({=0})=O(0,0). Thi mean that x axi i jut the local table manifold of the addle fixed point. Therefore, the bain boundarie, a een in fig.(a)(d), can be determined b the preimage of x axi. The bain of poitive fixed point (0.974, 0.974) at A= i unbounded and non connected region (ee Fig.(b)). Becaue there i onl one attractor inr + for the epidemic model (.), the feaible domain i the feaible et, that i, D=B S=S. In fact, for man biological model, there i onl one attractor in R + + (denoted b A ), which implie that we uuall have S B (A + ). Therefore, the feaible domain i uuall the feaible et, namel D=S. But we ma have cae in which two different attractor exit in R +, and in uch cae D i different from B S (becaue S alo include table et of addle belonging tor + which are not in D). We have conidered the implet cae of the famil of recurrence given in (.), obtained for k=. However, from the proof of the propoition, we know that imilar reult continue to hold, with obviou change, even for an k>. The recurrence (.) of order k i equivalent to a tem of k equation of firt order. Identifing (x n-(k-),, xn-, x n )=(,, k-, k), we get a k dimenional map k k T( k) : R R,(,, k, k ) = T( k) (,, k, k ) We have the following propoition: Propoition 4 The feaible et of tem (.) are uper tetrahedron S = {(,,, ) 0,, 0, 0, = k k k k j = j = bounded b the uper plane =0,, =0, k=0 and j k-. Feaible domain are the feaible et, i.e. D=S. 4. Summar and dicuion In thi paper we invetigate the global propertie of a dicrete epidemic model b the interaction between the computer experiment and the mathematical anali. We give an anwer to one of the open problem, propoed b Kocic and Lada, from another perpective of the global dnamical behavior b a tud of the domain of feaible trajectorie and their bifurcation. For the biologit thee reult ma be intereting, ince the delimitation of the feaible domain and bain permit them to undertand which initial condition are uitable for the biological model, what kind of exogenou hock can be recovered b the endogenou dnamic of the biological tem, and which one will caue evere crahe and extinction of the biological tem, namel an irreverible departure from the attractor. Acknowledgment We are grateful for the upport from the doctoral tartup fund of South-Central Univerit for Nationalitie. Reference [] Li XY,Wang W. 005 Chao. Soliton. Fract [] Pang G, Chen L 007 Chao. Soliton. Fract [3] Li G, Wang W, Jin Z 006 Chao. Soliton. Fract.30 0 [4] Zhou Y, Xiao D,Li Y 007 Chao. Soliton. Fract [5] J. R. Beddington, C. A. Free & J. H. Lawton 975 Nature [6] Cooke K. L, Calef D. F. and Level E. V.977 Stabilit or chao in dicrete epidemic model, j = k = j j k }, 6

8 007 International Smpoium on Nonlinear Dnamic (007 ISND) IOP Publihing Journal of Phic: Conference Serie 96 (008) 05 doi:0.088/ /96//05 P.73 [7] Kocic V. I., Lada G. 993 Global behavior of nonlinear difference equation of higher order with application (Netherland: Kluwer Academic Publiher) [8] Zhang D.C, Shi B 003 J. Math. Of Anali and Application [9] Bichi G. I,Gardini L, Mira C 999 Int. J. Bifurcat. Chao.9 9 [0] Bichi G. I,Gardini L, Mira C 003 Int. J. Bifurcat. Chao [] Bichi G.I,Gardini L, Mira.C. 005 Int. J. Bifurcat. Chao.5 45 [] Gu E.-G., Rung J. 005 Int. J. Bifurcat. Chao [3] Gu E.-G. 006 Int. J. Bifurcat. Chao.6 60 [4] Gu E.-G., HAO Y,-D 007 Chao. Soliton. Fract [5] Gu E.-G 007 Chao. Soliton. Fract. 3 4 [6] Gu E.-G. 007 Int. J. Bifurcat. Chao

Computers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order

Computers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order Computer and Mathematic with Application 64 (2012) 2262 2274 Content lit available at SciVere ScienceDirect Computer and Mathematic with Application journal homepage: wwweleviercom/locate/camwa Sharp algebraic