Time delay in a basic model of the immune response

Transcription

1 Chos, Solitons nd Frctls 12 (2001) 483±489 Time dely in bsic model of the immune response N. Buric, M. Mudrinic, N. Vsovic * Deprtment of Physics, Fculty of Phrmcy, Vojvode Stepe 450, Beogrd, Yugoslvi Accepted 9 December 1999 Abstrct The e ects of time dely on the two-dimensionl system of Myer et l., which represents the bsic model of the immune response, re nlysed (cf. Myer H, Zenker KS, n der Heiden U. A bsic mthemticl model of the immune response. Chos, Solitons nd Frctls 1995;5:155±61). We studied vritions of the stbility of the xed points due to the time dely nd the possibility for the occurrence of the chotic solutions. Ó 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction Time dely in models of popultion dynmics nd in prticulr in mcroscopic models of the immune response re nturl nd common [2,3]. Complex systems with mny components nd vrious interctions, often unknown, re replced by systems with just few mesurble quntities nd ll the complexity is introduced by joint e ect of the nonlinerities nd the time dely. Most often, the nonlinerities re introduced by resonble theoreticl model of the interction between the severl components (good surveys cn be found in [4,5]). However, such pproch fils to chieve the mjor gol of reducing the number of the necessry quntities to minimum. On the other hnd, the nonlinerities cn be introduced by incorporting independently obtined models of the behviour of prticulr components of the system into the model of the whole system. Prtil spects of the system's behviour re treted s known, nd re explicitly used in constructing the model for the whole system. One of the models obtined long these lines is the Myer model of the immune response [1]. The model is described by system of just two ODEs, nd the rich behviour is mde possible by using highly nonliner functions in order to model the joint e ect of vrious processes. These functions re justi ed on the bsis of previous models of the prts of the system. We shll nlyse vrious e ects of the time dely, introduced in these functions in nturl wy. The direct motivtion for the introduction of the time dely is tht the Myer model consists of only two ODEs, nd thus, it hs only regulr solutions, such s xed points, periodic orbits nd orbits symptotic to these [6]. As Myer et l. pointed out the model cnnot describe, frequently observed, irregulr or chotic behviour. In recent pper, we hve nlyse miniml extension of the Myer model, bsed on the periodic prmetric perturbtions, which hs the chotic solutions [7]. It is the gol of this pper to present nd nlyse the e ects of the time dely just su cient to introduce the chotic behviour into the Myer model. * Corresponding uthor. E-mil ddress: mjb@rudjer..bg.c.yu (N. VsovicÂ) /00/$ - see front mtter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S ( 9 9 )

2 484 N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483± Description of the model We shll be interested in the e ects of the time dely on the dynmics in very simple model of the immune response. The bsic model [1] describes the immune response s n interction between trget popultion, denoted T, nd the most relevnt (for such trget) feture of the immune system, denoted E. The quntity E is clled the immune competence, nd refers to the concentrtion of the most relevnt immune gents, like certin ntibodies, NK cells or cytotoxic T-cells. The trget cn be some mesurble property of nything susceptible to n immune response, like micro-orgnisms, mcro-molecules (proteins, polyscchrides, lipids), or immunogenic tumour cells, etc. The clss of models, which we shll nlyse is given by the following equtions: dt dt ˆ rt kte; de dt ˆ pf T t 1 T t s T sg be t 1 b E t s E E; 2 where either or b is necessrily equl to unity. In fct, the cse with ˆ 1; b 6ˆ 1 does not show long-term irregulr behviour, so we shll present in detils the nlyses of the cse 6ˆ 1; b ˆ 1, nd the results for the other cse shll be only indicted. In the cse of no time dely the Eqs. (1) nd (2) give the model of Myer et l. The functions f nd g describe the rte of chnge of the immune competence E due to the presence of the trget (f T ), nd due to uto-ctlytic nd/or co-opertive reinforcement (g E ) (see [1]). These processes re described by the following functions: 1 f T ˆ T 4 1 T 4 ; T P 0; 3 g E ˆ E3 1 E ; E P 0: 4 3 Such choice describes, depending on the vlues of the prmeters r; k; p nd s, lrge vriety of biologiclly plusible situtions. The e ect of the immune response on the trget is describe by the term kte in the rst eqution. All the complexity of the interction between the immune system nd the trget, nd of the processes between vrious immuno-competent gents, leding to the rech vriety of possible outcomes, is contined in the nonliner functions f nd g. The liner terms rt nd E describe the verge production rte of the trget cells proportionl to T, nd the nite verge life time of the immuno-competent gents. The model with no time dely exhibits vrious stble nd unstble xed points with the corresponding symptotic orbits. There re lso lrge intervls of the prmeter vlues when the system hs got limit cycle, corresponding to periodic vritions of the immune stte. This limit cycle is born from Hopf bifurction of stble xed point. The bifurction nlyses of the model nd its prmetric perturbtions were presented in the Ref. [7]. However the two-dimensionl model hs only regulr solutions. There re severl wys of extending it so tht the new model might exhibit chotic behviour. One possibility, nlysed in [7], is to suppose periodic time dependence for the prmeters. Prmetric oscilltions coupled with the limit cycle oscilltions in the model then generte the chotic solutions. It is lso well-known tht single di erentil eqution with delyed rgument could hve very complex solutions [8,9]. Furthermore, introduction of the time dely into the functions f nd g is obviously biologiclly justi ed. 3. Liner stbility of the xed points The bsic elements of the qulittive description of dynmicl system re the xed points nd the periodic orbits. The time dely introduces importnt qulittive chnges in the dynmics of the system. We shll be concerned with the e ects introduced by the time dely upon the stble xed points nd the periodic

3 orbits (generl theory of delyed-di erentil eqution cn be found in [10]). Liner stbility of the xed point will be nlysed in this section, nd the perturbtions of the limit cycle will be nlysed numericlly in the next section. There re two xed points of the system. The rst one is in the origin nd is lwys unstble. The second one is given by: E 0 ˆ r=k d; r 4 T 0 ˆ ; ˆ d 1 p d s p d 3 1 d 3 : 5 As the time dely s is vried this xed point undergoes Hopf bifurction i.e. it is chnged from stble into n unstble xed point, nd limit cycle is born. We wnt to nd the criticl vlue of the time dely s c, when the bifurction from the stble into the unstble xed point occurs. We shll present the nlyses nd the results for the cse 6ˆ 1; b ˆ 1, since, s we shll see, only in this cse the long term irregulr solutions re possible. In order to determine stbility of the xed point we linerise the Eqs. (1) nd (2) with 6ˆ 1; b ˆ 1, in neighbourhood of E ˆ E 0 ; T ˆ T 0 to obtin the mtrix A ˆ 1 2 b 1 1 b 6 1 exp ks b 2 nd the chrcteristic eqution det A ki k 2 1 f 1 k f 2 f 2 e ks ˆ 0; 7 where nd f 1 ˆ b 2 ; f 2 ˆ 2 b 1 1 ˆ 0; 2 ˆ kt 0 ; b 1 ˆ 4pT T ; b ˆ 3sE2 0 d: E0 3 2 The chrcteristic eqution is used to determine the liner stbility of the xed point. It is trnscendentl eqution, with n in nite number of solutions k i. The xed point is unstble if there is t lest one k i with Rek i > 0. If ll k i hve k i < 0 then the xed point is stble. The bifurction occurs for those vlues of the prmeters when there is t lest one k i with Rek i ˆ 0. The existence of s c nd the number of purely imginry k i for this s c is found by treting the chrcteristic eqution s complex vrible mpping problem. Consider the trnsformtion from the k-plne to w-plne de ned by w ˆ k 2 f 1 k f 2 1 f 2 e ks ; s > 0: 10 Setting k ˆ l im this gives w ˆ l 2 m 2 f 1 l f 2 1 f 2 e ls cos ms N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483± i 2lm 8 f 1 m 1 f 2 e ls sin ms : 11 We wish to show tht there is s c such tht the imge of the imginry xes in the k-plne by the complex mp w k, goes through the origin in the w-plne. Consider the contour in the k-plne consisting of the imginry xis nd semi-circle of in nite rdius s shown schemticlly in Fig. 1. We strt by plotting the imge of this contour by the mpping (11) in the cse s ˆ 0. Time dely will modify the imge by just superposing oscilltions proportionl to s. We know tht in the cse f 1 > 0; f 2 > 0nds ˆ 0 the xed point is stble nd Rek < 0, so w s function of k does not pss through the origin in the w-plne. Actully, AE in Fig. 1, on which l ˆ 0, is mpped by

4 486 N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483±489 Fig. 1. A domin V in k-plne, bounded by the curve AEF, is mpped by the chrcteristic polynomil into A 0 E 0 F 0 in w-plne. AE is on the imginry xes. w ˆ m 2 f 2 i f 1 mš 12 onto A 0 E 0 in Fig. 1b with C 0 ˆ f 2 = i0 in the w-plne corresponding to C ˆ 0 i0 in the k-plne. The domin V is mpped into V 0. The points A; B; C; D nd E in Fig. 1 re mpped onto their primed equivlents s indicted in Tble 1. We now consider the mpping with s > 0. The line AE hs l ˆ 0 so it is mpped onto w ˆ m 2 f 2 1 f 2 cos ms i f 1 m 1 f 2 sin ms : 13 The e ect of the trigonometric terms is simply to dd oscilltions onto the curve A 0 B 0 C 0 D 0 E 0, s is shown schemticlly on Fig. 2. For s lrger thn the criticl s c, which depends on f 1 ; f 2 nd, the point B 0 is below nd the point D 0 is bove the Rew xes, s in Fig. 2b. The roots of the chrcteristic eqution Tble 1 A 1e p ip=2 A 0 1e p ip B f 2 = e ip=2 B 0 f 1 f 2 = e ip=2 C (0) p C 0 f 2 p= D f 2 = e ip=2 D 0 f 1 f 2 = e ip=2 E 1e ip=2 E 0 1e ip Fig. 2. Imges of the curve AEF from Fig. 1 by the chrcteristic eqution with: () smll s; (b) lrger s.

5 N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483± w k ˆ0 re lwys complex conjugte pirs. The number of roots with Rek > 0 is obtined by computing the chnge in the rgument of w upon trversing the boundry of V 0. In the supercriticl cse, represented in the Fig. 2b, the chnge in rg w is 4p, corresponding to two complex conjugte roots with Rek > 0. We shll not go into the nlyses of the dependence of s c on the prmeters f 1 ; f 2 nd. This could be investigted by grphicl or numericl solutions of the following system of equtions: f 1 ˆ 1 sin ms c m 1 1 = cos ms c ; 14 f 2 ˆ m = cos ms c ; 15 or nlyticlly in the cse of smll nd lrge s. Let us just mention in order tht there is the criticl s c, i.e. solution with rel m, the prmeter must be > 1=2. 4. Nonliner e ects We expect tht the nonliner terms in the two-dimensionl system, coupled with the time dely, could introduce the chotic solutions in the cse when the originl system, with no time dely, hs limit cycle. There re lrge intervls of the prmeter vlues when the instbility domin of the unstble xed point of the Myer model is bounded by the stble limit cycle. We shll now nlyse wht hppens with this limit cycle due to the time dely. In prticulr we wnt to investigte if the chotic solutions could pper by introducing the time dely in only one term of the originl model. Systemtic numericl computtions led us to conclude tht there re three qulittively di erent situtions, which could occur for the vlues of the prmeters corresponding to the existence of the limit cycle in the originl model. Two of these could occur for ny of the time-delyed models i.e. for ny vlues of nd b nd the third one is possible only in the cse when the time dely is introduced in the function f T i.e. for 6ˆ 1; b ˆ 1. In order to obtin prticulr solution of the system of delyed di erentil equtions we need n initil function de ned on the intervl t 0 s; t 0, which corresponds to reltively short intervl before the complex interctions nd processes between E nd T cells strted. In our numericl computtions we used the clss of liner functions for the initil dt, which is biologiclly plusible. Our conclusions re thus restricted to this clss of the initil functions, but they re qulittively the sme for the functions in this clss. In the rst two cses the system is eventully ttrcted towrds simple ttrctor. Depending on the vlues of the prmeters r; k; s; p nd the time dely s the ttrctor is either the stble xed point or the stble limit cycle, bounding the instbility domin of the unstble xed point. This limiting long term behviour is presided either by short period of irregulr or by regulr trnsient oscilltions. The combintion of the criticl vlues of the prmeters nd the time dely, r c ; k c ; s c ; p c nd s c, corresponding to the trnsition from the one to the other limiting behviour, cn be determined using the liner stbility nlyses of the xed point in ech cse of interest. The nlyses, using method (D-resolution) di erent from the one of the lst section is rther lengthy nd will not be given here [11]. These two types of behviour re the only possible types in the cse the time dely is introduced in the function g E ( ˆ 1; b 6ˆ 1) or in the rst term of the eqution for T. T components of the typicl orbits re illustrted in Fig. 3 nd b. Fourier mplitude spectrum (Fig. 4), nd the lrgest Lypunov exponents lso correspond to the regulr ttrctors. Very lrge time dely leds to unbounded solutions. No solution with long term chotic behviour is found. The chotic solutions re found only in the cse of the time dely in the function f T. A typicl chotic orbit in the phse spce is represented in Fig. 5, together with its T component nd the Fourier mplitude spectrum in Fig. 6 nd b. For xed vlues of the prmeters r; k; s; p, dmitting limit cycle in the originl model (s ˆ 0), nd for xed ˆ 0:75; b ˆ 1, incresing the vlue of the time dely s chnges the limiting set of typicl orbit from circle to multiply periodic orbit, for even lrger s, the orbit becomes chotic. The

6 488 N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483±489 Fig. 3. T component of typicl orbits ttrcted towrds: () the stble xed point; (b) the stble limit circle. Fig. 4. Amplitude Fourier spectrum of the T component of the orbit in Fig. 3b. Fig. 5. Typicl chotic orbit (k ˆ 1, r ˆ 0:5, s ˆ 1:75, p ˆ 0:35, ˆ 0:75, s ˆ 20). Fig. 6. () T component. (b) Its mplitude spectrum of the chotic orbit in Fig. 5.

7 N. Buric et l. / Chos, Solitons nd Frctls 12 (2001) 483± chotic solutions pper for the vlues of s which re close to the period of the limit cycle of the system with no dely. There is smll intervl (bout 10% of the time dely) of the time dely s for which the orbits re chotic. Then follows smll intervl of s for which periodic limiting behviour is restored. Lrger time delys led gin to the chotic solutions, which continue to pper for ll lrge s. Similr chotic solutions re found for other vlues of lying pproximtely in the intervl 0:65; 0:8 nd b ˆ 1. No chotic solutions re found for the vlues of r; k; s; p which corresponding to the stble xed point in the originl s ˆ 0 model nd for ny s, (or nd b) lthough, s shown in the lst section, the time dely cn destbilise the xed point. To summrise, extensive numericl computtions show tht: For ny vlues of the prmeters r; s; p; d, for which there is limit circle in the model with no time dely, there re su ciently lrge time dely s nd n intervl of 6ˆ 1ndb ˆ 1, such tht the system is chotic. The cse ˆ 1; b < 1 hs no chotic solution for ny vlue of the prmeters nd the time dely. 5. Conclusion The model nlysed in this pper is typicl exmple of n pproch to the modelling of the immune system. The pproch is bsed on the guiding ide tht the model should stisfy the bsic requirement, one should need s little s possible of the input dt in order to be ble to describe mny typicl situtions occurring in the rel system. The Myer model without the time dely contins only two chrcteristic vribles. All vriety of di erent dynmicl behviour is mde possible by the two nonliner functions, introduced on the bses of n independent nlyses. However, the model is to simple to be ble to reproduce (t lest qulittively) the irregulr oscilltions of the stte of the immune system, which re observed quite often. Wht re possible miniml dditions which would mke model with irregulr behviour? We hve prtilly nswered this question by nlysing clss of systems with the time-delyed rguments, introducing the time dely in ech nd only one of the terms in the Myer model. The min result of our nlyses is tht the irregulr behviour cn be introduced by the time dely in the function which describes the incresed production of the immuno competent cells due to the presence of the trget cells. Time dely in ny other single term of the equtions produces stbilistion of the trnsient irregulr behviour onto the simple ttrctor i.e. the xed point or the periodic orbit. This conclusion seems to be generlly vlid for ll tested initil dt functions. References [1] Myer M, Zenker KS, n der Heiden U. A bsic mthemticl model of the immune response. Chos, Solitons & Frctls 1995;5:155±61. [2] Murry JD. Mthemticl biology. Berlin Heidelberg: Springer; [3] McDonld N. Biologicl dely systems. Cmbridge: Cmbridge University Press; [4] Romnovski Y, Stepnov NV, Cernevski DS. Mthemticl biophysics. Moskow: Nuk; [5] Perelson AS, Theoreticl immunology I, II. New York: Addison-Wesley; [6] Wiggins S. Introduction to pplied nonliner dynmicl systems nd chos. New York: Springer; [7] Buric N, Vsovic N. A simple model of the chotic immune response. Chos, Solitons & Frctls 1999;10:1185. [8] Mckey MC, Glss L. Oscilltion nd chos in phisiologicl control systems. Science 1977;197:287. [9] Peters H. Chotic behviour of nonliner di erentil dely equtions. Nonlin Anl 1983;7:1315. [10] Hle J, Lunel SV. Introduction to functionl di erentil equtions. New York: Springer; [11] Buric N, Mudrinic M, Vsovic N. Preprint, 1999.