Dynamical Behavior Analysis and Control of a Fractional-order Discretized Tumor Model

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1 6 International Conference on Information Engineering and Communication Technolog (IECT 6) ISBN: Dnamical Behavior Anali and Control of a Fractional-order Dicretied Tumor Model Yaling Zhang,a, Xiaodan Zhang,b, Yinghan Zhang 3,c School of Mathematic and Phic, Univerit of Science and Technolog Beijing, Beijing 83, China a hangaling8@6.com, b bkdd@63.com, c hanginghan7@6.com Keword: fractional-order, cancer model, Dicretiation, chao, feedback control Abtract. In thi wo, we contructed a new fractional-order dnamical model of tumor and appl Euler method to obtain the dicrete tem. Local tabilit of the fied point of the dicretied tem i tudied. Numerical imulation how the chaotic attractor and the richer dnamical behavior of the dicretied tem. Linear feedback control method i ued to control chao in the conidered dicretied tem. Numerical imulation reult how that the controller can control the chao effectivel.. Introduction In the pat few decade, fractional calculu ha been an active reearch field from both theoretical and practical perpective [, ]. Fractional differential euation which are generaliation of claical differential euation decribe the memor effect, and it i the major advantage over integer-order derivative [3]. Mathematical model for tumor growth have been etenivel tudied in the literature to undertand the mechanim of the dieae and to predict it future behavior [4]. Some author dicued the problem of chao and tabilit anali of ome biological model uch a cancer and tumor model, tochatic lattice ga pre predator model and man other model [5, 6]. However, the current model are all retricted to integer-order ordinar differential euation. A major difference between fractional-order model and integer order model i that fractional-order model poe memor, while the main feature of immune repone involve memor [7]. Hence, we will propoe a model of fractional-order differential euation to decribe tumor growth. Moreover, the need for a dicretiation of fractional-order model arie from the fundamental realiation that nonlinear fractional-order tem generall do not have analtic olution epreible in term of a finite repreentation of elementar function. Some eample of dnamical tem generated b piecewie contant argument have been tudied in [8, 9]. Here, we appl Euler method to obtain the dicrete verion of the tem under tud. There are man cheme to achieve chao control, uch a linear and nonlinear feedback control and active control etc. [, ]. In thi paper, linear feedback control method i ued to control chao. In thi paper, we have introduced the fractional-order tumor model and dicretied tem. The fied point and their amptotic tabilit of the dicretied tumor tem are tudied. Numerical imulation how the tem fractional parameter ha effect on the tabilit of the dicretied tem which how rich variet of dnamical behavior uch a an attractor crii and chaotic attractor. Furthermore, linear feedback control method i ued to control chao in the conidered dicretied tem. The eperiment reult how that the controller can control the chao effectivel. Finall, brief concluion are given.

2 . Fractional-order tumor model and it dicretiation Our model i propoed baed on Ref. [6]. Conider that the immune cell can die naturall and the main feature of immune repone involve memor, the natural death rate of immune cell i added to the econd euation, and a fractional-order cancer model i given a follow: dt T rt ( ) TE TH, ( i) dt k d E rte TE de, ( ii) dt T k d H H rh 3 ( ) THdH. ( iii) dt k3 () Where Tt (), Etand () Ht () repreent the number of tumor cell, effector immune cell, and health hot cell at time t, repectivel. And i the fractional order atifing. E. (i ) give the rate of change in the population of the tumor cell rate of with time t. In E. (i ), r i the growth rate of tumor cell and k i the maimum carring capacit. Tumor cell are killed b immune cell at the rate and the competition between the hot cell and the tumor cell which reult in the lo of the tumor cell i given b the term TH. E. ( ii ) decribe the change in the immune cell population with time t. The immune repone timulate b poitive nonlinear growth in the preence of tumor cell, that i rte T k, where r and k i poitive contant. The immune cell are inactivated b the tumor cell at the rate of and the die naturall at the rate d. E. ( iii ) decribe the rate of change in the population of hot cell with timet. In E. ( iii ), the hot cell grow logiticall with the growth rate of r and maimum carring capacit i 3 k 3. i the rate of the hot cell inactivated b tumor cell a well a the die naturall at the rate d. We note that all the tem parameter are poitive and non-dimenional. Net, the dicretiation proce of the fractional-order cancer tem i given a follow: Let Tt () t (), Et () t (), Ht () t (), aume that (), (), () are the initial condition of tem (),and ( t ), ( t ), ( t ), t i, i,,, here, i the tep ie. i i i i i i i B Euler method, the tem () can be dicretied a follow:

3 n n ( r n( n / k) nn nn), ( ) n n ( rnn /( k n) nn dn), ( ) n n ( r3n( n / k3) nn dn). ( ) () 3. Dnamical behavior of the dicretied fractional order tumor model 3.. Stabilit of the fied point of the dicretied tem We can obtain eight fied point of tem (): () E (,,) (trivial fied point). The fied point at the origin correpond to a ituation where there i no cell at all. () E( k,,), which mean the tumor cell grow to it maimal ie. (3) (4) r d E (,, ) (tumor free fied point), which mean that the tem i in health tage ( d r r ) r r rd k E (,, )(tumor cell and hot cell fied point) r 3 k3 r 3 k3 E (5) 4,5 (,,)(tumor cell and immune cell fied point), where r rd k r k k, r r rd k r k k. M N 3 3dk3k3 E (,, )(nontrivial fied point), where,, r (6) 6,7 r k. M krd, N M 4 kd, k 3 A tated above, in the cancer model, the three tate variable, and are nonnegative, o we are onl intereted in the poitive fied point. Net, we will dicu the tabilit at the fied point. Theorem. For tem (), the following tatement hold true: () E i untable point. () E i locall amptoticall table if and onl if r (), () kd k k () r k d. and 3 Theorem. The fied point E of tem () i locall amptoticall table if and onl if

4 () ( r3 d) () 3 3( r3 d) () d, r, r 3 d Finall, let u dicu the tabilit of the other fied point. The characteritic euation of J ( E i ) i given b: 3 K K K3. According to the Jur criterion [], the fied point are locall amptoticall table if K K K,K K K, K K K K,K K K K Numerical imulation In order to anale the tabilit of the interior fied point, we invetigate the global dnamical behavior of tem () b uing numerical imulation. We fi the parameter ( r, r, r,,,,, d, d, k, k, k ) (,.8,4.5,.5,,., 3.,,.,,,), then we var the 3 3 parameter and.two chaotic attractor at.95,.85 and. are depicted in Fig. a and b. Alo in Fig. c and d, the phae portrait of the tem () are preented for.95,.85 and., repectivel. From Fig., we notice that decreaing the parameter and increaing the fractional-order parameter tabilie the chaotic behavior of tem ().The trange attractor of the tem () i hown in Fig..e for and.which confirm the above reult; becaue it i hown that the reverible of the above reult when i mall and decreaing the fractional-order parameter detabilie the table behavior of tem (). (3) (a) (b) (c) (d) (e) Figure. Trajector of the tem () in phae pace. (a).95,., (b).85,., (c).95,., (d).85,., (e),..

5 4. Chao control of the dicretied fractional-order tumor tem In thi ection, the linear feedback control method i applied to control chao. According to the tem (), a new controlled tem i contructed a follow: n n ( r n( n / k) nn nn) p( ), ( ) n n ( rnn /( k n) nn dn) p( ), (4) ( ) n n ( r3n( n / k3) nn dn) p3( ). ( ) Wherep, p and p3 are the poitive control parameter and (,, ) i the fied point of the tem (). Obvioul, (,, ) i alo fied point of tem (4). Net, we will chooe parameter p, p and p uch that the Jur condition (3) are 3 atified. Let ( r, r, r3,,,,, d, d, k, k, k3) (,4.5,.8,.5,,., 3.,,.,,,). At the fied point E (.35,.347,), the eigenvalue are i, i, 3.43.Here 3, o E 6 (.35,.347,) i abolutel untable. Theorem3 when p p,.466 p3.33, the tem (4) at E (.35,.347,) 6 i locall amptoticall table. In order to reveal the effect of chao control, ome numerical imulation are given below. Keep the parameter ( r, r, r 3,,,,, d, d, k, k, k 3 ) (,4.5,.8,.5,,.,3.,,.,,,) unchanged. Let and.. At fied point E (.35,.347,), chooe 6 p, p., p3, we haveaa a3, aa a3, a aa 3a3, a aa a So the tem (4) i locall amptoticall table. The imulation reult i depicted 3 3. through Fig..Therefore, b adding the feedback controller, the tumor cell can be controlled to change from the untable chaotic tate to the table tate. (a).8.6 (b) Figure. Phae diagram of a controlled tem (5). (a) Three-dimenional phae diagram; (b) curve.

6 5 Concluion In thi paper, we have introduced the fractional-order tumor model and dicretied tem. We have tudied the local tabilit of the fied point of the dicretied tem. It ha been found that, the fractional parameter ha effect on the tabilit of the dicretied tem. We have alo hown that when decreaing the parameter and increaing the fractional-order parameter, the chaotic behavior of tem () will be tabilied. Meanwhile, decreaing the fractional-order parameter detabilie the table behavior of tem (). Numerical imulation have been ued to how the richer dnamic of the dicretied tem. Alo, the controlled tem of dicretied tem i contructed b appling the linear feedback method. Simulation reult how that the feedback control i ea to implement even for controlling the dicretied chaotic tem. Reference [] B. Datko, Y. Luchko, Pattern formation in fractional reaction diffuion tem with multiple homogeneou tate, Int. J. Bifurcation Chao (). [] J.J. Huo and H.Y. Zhao, The effect of vaccine on backward bifurcation in a fractional order HIV model, Nonlinear Anal. RWA. 6 (5) [3] I. Podlubn, Fractional Differential Euation, Academic Pre, New Yo, 999. [4] R. Saar, S. Banerjee. Cancer and elf-remiion and tumor tabilit, a tochatic approach. Math. Bioci. 69(5) [5] A. El-Gohar, Bukhari F. Optimal control of tochatic pre predator model. Appl. Math. Comput. 46(3) [6] M. Itik and S.P. Bank, Chao in a three-dimenional cancer model, Internat. J. Bifurc. Chao () [7] V.A. Kunetov and I.A. Makalkin, Nonlinear dnamic of immunogenic tumor: Parameter etimation and global bifurcation anali, Bull. Math. Biol. 56 (994) [8] M.U. Akhmet, Stabilit of differential euation with piecewie contant argument of generalied tpe, Nonlinear Anali 68 (8) [9] A.M.A. El-Saed, S.M. Salman, Chao and bifurcation of the Logitic dicontinuou dnamical tem with piecewie contant argument, Malaa Journal of Matematik 3 (3) 4. [] X.R. Shi, Z.L. Wang, Adaptive added-order anti nchroniation of chaotic tem with full unknown parameter, Appl. Math. Comput. 5 (9) []S.K. Agrawal, M. Srivatava, S. Da, Snchroniation between fractional-order Rabinovich Fabrikant and Lotka Volterra tem, Nonlinear Dnam. 69 () [] L. Edeltein-Kehet, Mathematical Model in Biolog. McGraw-Hill; 988.