Applied Mathematics, 2013, 4, 16371646 Published Online December 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.412223 Open Access AM Solutions of Impulsive Diffusion and VonFoersterMakendrick Models Using the BTransform* Benjamin Oyediran Oyelami, Samson Olatunji Ale National Mathematical Centre, Abuja, Nigeria Email: boyelami2000@yahoo.com, aleola@ymail.com Received October 2, 2013; revised November 2, 2013; accepted November 9, 2013 Copyright © 2013 Benjamin Oyediran Oyelami, Samson Olatunji Ale. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we explore the possibility of using the scientific computing method to obtain the inverse BTransform of Oyelami and Ale [1]. Using some suitable conditions and the symbolic programming method in Maple 15 we obtained the asymptotic expansion for the inverse Btransform then used the residue theorem to obtain solutions of Impulsive Diffusion and VonFoersterMakendrick models. The results obtained suggest that drugs that are needed for prophylac tic or chemotherapeutic purposing the concentration must not be allowed to oscillate about the steady state. Drugs that are to be used for immunization should not oscillate at steady state in order to have long residue effect in the blood. From VonFoersterMakendrick model, we obtained the conditions for population of the specie to attain super satura tion level through the “dying effect” phenomenon ([24]). We used this phenomenon to establish that the environment cannot accommodate the population of the specie anymore which mean that a catastrophic stage t* is reached that only the fittest can survive beyond this regime (i.e. t > t*) and that there would be sharp competition for food, shelter and waste disposal etc. Keywords: BTransform; Impulsive Diffusion; VonFoersterMakendrick Models; Residue Theorem; Maple Symbolic Programme and Asymptotic Expansion 1. Introduction Impulsive differential equations (IDEs) describe proc esses which quickly change their states for short moment of times when compare to the total evolution time for the systems. The impulses may happen at fixed or nonfixed moments and the behavior of state variables describing the processes may show some “jumps”, “shocks” attri butes etc. This kind of impulsive behavior makes the IDEs not easily accessible to most existing concepts and theories in differential equations, ecology, biomathe matics, engineering and control systems ([29]). More over, impulsive moments not only depend on some im pulsive sets but also on the dynamics of the systems, this special feature gives rich perspective for investigation for many real life processes ([2,3,8,10]). Btransform is an operational calculus method de veloped by Oyelami and Ale almost a decade ago for finding closed solution forms for fixed moment im pulsive systems ([1]). Most known existing transform methods are not known to be applicable to the (IDEs). It’s interesting to say that the evolution times describing (IDEs) are intermittently interrupted by small perturb ations (impulses) at certain fixed or nonfixed moments. This peculiarity makes the solutions of IDEs to exhibit some strange behaviors. Therefore any transform method for finding solution of IDEs must take into consideration the impulsive effect of the systems. The Btransform was developed to take into consideration the impulsive effect of the system to provide a method for finding solutions for the fixed moments IDEs ([1,1113]). Btransform has been successfully applied to solve a variety of problems in Biomedicine, Physics, engineering, control theory etc (see [1,7,12]). One major problem that one encounters just like most known transform methods ([1]) is how to obtain in the inverse transform for some particular problems ([14]). In this paper we explore the possibility of introducing scientific computing method to obtain the inverse BTransform. We would make use of *Dedicated to: Professor V. Lakshimikantham who has contributed ex tensively to the development of Differential Equations, Impulsive Dif ferential Equations and Nonlinear Analysis.
B. O. OYELAMI, S. O. ALE 1638 some suitable conditions and the symbolic programming method in Maple 15 to obtain the asymptotic expansion of the inverse Btransform, then followed by the use of residue theorem to obtain solutions to Impulsive Diffu sion and VonFoersterMakendrick models. We note that, the Impulsive Diffusion model is very useful for modelling the diffusion of drug across the cells and tissues in the body and the attendant osmoregulation in the apparent volume and also modelling of environ mental degradation problems ([5,15]). More recently, stochastic diffusion models are used for analyzing be havior of financial markets. The entropy trap threatens the existence of man as a result of continuous emission of carbon dioxide into the air and water. The earth is getting more saturated carbon dioxide than the oxygen that plants absorb and release. The sea water is fast becoming acidic because of dissolution of carbon dioxide in the water and hence the marine lives are also threat ened. The diffusion model considered here or its modi fications have potential applications for studying dif fusion of carbon dioxide into the atmosphere and the sea and the long time chaotic effect can be studied from impulsive perspective. The theory of taxism, loosely speaking is the tendency of an organism to move forward or be away from some stimuli. There are many taxes, positive and negative which influence presumably direct organisms in favor able physical environments ([15]). Most notable or perhaps widely studied taxes are the chemo taxi that is response for organisms to be sensitive to chemical chan ges. If the response is measured in term of concen tration of such chemicals, the impulsive diffusion model will give a clue to study chemo taxis of the expected organisms. More research works are expected to be on taxes in cellular ecology, that is, molecular and cellular biology. The study of bacterial and viral chemotaxes from impulsive point of view will offer researchers the opportunity that has integrated approach to understand the interaction of genetics, physiology and ecology. We must emphasize that under certain conditions like rapid changes i.e. advent of war, earthquake, displace ment of persons etc., the population tends to be impulsive in nature (see [2,8,1618]). For this reason, we consider impulsive analog of the VonFoersterMakendrick model of an agedependent population in given ecosystem. The model is typical impulsive partial differential equations and has potential applications in modelling the popula tion of species stratified into age groups and epidemiol ogy of infectious diseases like malaria and HIV/AIDs. 2. BTransform Consider the impulsive differential equations ,,, 0,1,2, k k x ftx t k t xIxt (IDE) where : : nn n RR R IR R f and I are assumed to satisfy the continuity and Lipchitz conditions for the existence and uniqueness of solution of (IDE) (see [3,4]). The Btransform of the function t with impulses at fixed moments ,1,2, k tk during the evolution ary process is [1,11] 1 1 nc tq xx Bq (1) where C q and I q are the components of the Btransform and are defined as and c LI L 1 0 d,, 0,1,2, en tq c ck qxt xtttk xt L (2) And 1 1 1en k ok q tk t tt qxt Ixt xL (3) where 0,1,2,,nn ; n is the order of the transform. For sake of simplicity, we will choose . The advantage of taking 1n 1n lies in the derivation of the inverse transform. The inverse transform for components of C q and I q was obtained (see [1], Theorem 1) as follows: 1d e 2π vi sq cc vi txq xi q k x (4) 1, ok k t tt tqI t t (5) The Btransform is valid in the union of the sets 1 0 :ed , ,1,2,,1,2, n k tq n k xtRxtt m tRtkn And 0 2 :e 1, 2, n k k tq n kk ttt BIxtRI xtm n , And such that Open Access AM
B. O. OYELAMI, S. O. ALE Open Access AM 1639 Diammax, k ABxt Ixt ing effect”, “loss of autonomy” and “dying effect” are hampering the rate of its development ([2,4,9,15]). Let us return to the problem, if we slot ,Ctx into Equations (6) and (7), we get where . is the ndimensional Euclidean norm, m1 and m2 are constants that are assumed to exit and finite. d,,0,1,2, d i iik Ct kCt t tk t (10) 3. Impulsive Diffusion Problems Consider an example of application of Btransform to impulsive partial differential equations. Impulsive partial differential equations have so many interesting applica tions. We will consider a diffusion problem of a porous media of substrate whose concentration is given by at time t and a distance of x from the source. This type of problem is very useful in drug administra tion. The underlying mathematical model can be stated as follows: ,Cxt 2 2 d, d ,0,1,2, i ii iiii k Cx Cxkrxth ut x t k t (11) ki k Ct t ICt (12) 012 0; ,constants,1,2, lim k ki k tttt t i 2 2 ,, , , ii iii k Ctx Ctx kCt xD tx krx thuttt (6) Subject to given boundary conditions applying the Btransform to Equations (10)(12) and solving the equa tions we get ,,,0,1 ik ik Ct tx ICtx k 12 01 1 , ok i ii i ttti kk q Cq k C qkq tqICt (13) ,2, (7) Subject to the boundary conditions at 0,,0 i C x tx x (8) 1 deq k i t t ik c t,q q (14) , at ,0 i Ctx x L x k (9) And solving Equation (11) we get where sin cos ,d O ii i x ii i x CxAx x Gstrxsh s s (15) , i Ctx and i k is the concentration of the substance that diffuses across the apparent volume into gastrointestinal tracts at time t and the thickness of the tract being x units. i are some rate constants, Di the coefficients of diffusion. ,cos iii Gst t s . (16) i rxt h accounts for the biological gestation function lagged by a constant Ci. Now, we determine the inverse of and the constants ,1,2 i Cqi i and . Thus i B i ut is the relevant biological control function that is responsible for osmoregulation of the medium. 2 ee ,,1, ii ok kt itk io i ik k ttt Ct C k ttICti 2, (17) Using a transformation of variable, a kind of separa tion of variable per say, we can express in the form where and , i Ctx ,CtxCtCxCt Cx are purely functions of t and x respectively. This separa tion works for this type of problem, for other forms of impulsive partial differential equations, it may not work. This kind of features makes the theory of impulsive dif ferential equation to be expanding at a rapid rate of late, even though the discovery of new phenomena like “beat The values for constants i and , i B1, 2i are found to be cos 0 iiii i i tr ut A (18) 32 cos 0coscos sin iii iiiiiii i ii tr utL L uL BL (19)
B. O. OYELAMI, S. O. ALE 1640 Therefore given the require solution to the model, that is, ,Ctx ,, ee ii ok i ktt k ttt otk i Ctx C ttICt k k 2 sin cos, d , ee cos ,d o i ok o iii ii x iii i ktt k ttt ok i i iii ii i x x x Gst rxth utt C tt k Cx Gstrxth utt (20) where 22 1 ,tan i iiii i C AB B The problem exhibits oscillatory behavior at 14 π2 2 i i i k x If 0,1, 2, iii rxt hut i At steady state, when ; t 0 lim , tCtxC x where (, cos lim d. ok o sikkiii tttt x ii iii x Cx ttICt Cx + Gstrxsh +uss (21) If the system does not oscillate about the points xi as enumerated above and if it oscillates. 0 s Cx From the above derivation, drugs that are needed for prophylactic or chemotherapeutic purposes the concen tration must not be allowed to oscillate about the equilib rium at the steady state. Drugs that are to be used for immunization should not oscillate about the equilibrium at steady state in order to have long residue effect in the blood. 4. Impulsive Population Model We consider impulsive analog of the VonFoersterMak endrick model of an agedependent population in given ecosystem. The equation is given by 12 ,,, ,k NN txNFNtxt t xt (22) ,0Nx x t (23) 0 0,,, dNt xtNxt (24) , k Nxt t INxt , k (25) 012 lim 0, kk tt t t t k. where , :,NDR R CR 0,R , t,Nx is the population of a single species in given ecological setup at time t and x age group. If , k INxt if 0 substrate (biomass) is taken away and , k INxt 0 biomass is added to the environment ([3]). We must emphasize that under certain conditions like rapid changes i.e. advent of war, earthquake, displace ment of persons etc., and the population tends to be im pulsive in nature [18]. ++ : D R RR : RR Is the growth rate, F is a nonlinear function and ,1,2 ii . are some relevant rate constants. Applying Btransform to model we arrive at 0 1 0 0 0 2 12 ,, ,d e ,, , e ,,d e ,d , e 1 ,,d e ,, k ok tq cc qt Ik ttt tq c tq tq c c cc LN xtNxqN xtt LNxtNxq INxt LNxt NxttNxq xx x Nxtt Nxq xx LNxtNxtt xq N xx q NxqNxqx Gxq xq , ,. , (26) 0 11 00 11 00 ,,,, e 0,d,,d d. ,,dd. e tq tq c tq GxqtxNFNtxt Nq txtNxttt e xtNxttt 1 d Substituting the value of and then solving Equation (26) for , c Nxq , c Nxq for fixed q, we get Open Access AM
B. O. OYELAMI, S. O. ALE 1641 22 111 11 00 d, 1 , d 0,e,,d d. c c tq c Nxq aa Nxqx Gxq xaqa a Nq xtNxttt 1 , (27) Solving Equation (27) we get 2 1 0 2 1 0 1 ,0,exp 1 ,0,exp cc cc NxqNq s q NxqNq s q d d ,. (28) Therefore 1 ,, c NxqNxq Nxq It follows that 1 1 ,, e 2π ,, ok i tq cc i kk ttt Nxt Nxqq i Nxq ttINxt d , ,. (29) Let C be contour for which the integration is carried out in the complex plain. Therefore 1 ,, c Nxt Nxt Nxt (30) 0 2 1 222 111 2 1 1d ,0,exp 2π d exp , 1 expdde, k k x c Ca xx Ca a xtq k attt as Nxt Nqtqq iaq aaa stqsGx q aqa a asq INxt aq d (31) We note that the system has no solution passing through if ,tNt 122 1 ,,0,1, , ik k IN ttk CN tttC 2, where are possitive constans such that i C 2 2 22 ,, k C C tt Ntt 0. This model wills exhibits the ‘dying effects’, which mean that the solution is not continuous across some natural boundary containing the solution ([24]). To this particular model, this means that the population of the specie attains super saturation level such that after time the population starts showing some strange behavior that the population cannot be quantified. t tt The “dying effect” as tt suggests that the envi ronment cannot accommodate the population of the spe cie any more. By Darwin’s theory a catastrophic stage is reached that only the fittest can survive beyond this re gime (i.e. tt ); there expected to be sharp competition for food, shelter and waste disposal etc. It is interesting to note that dying effect does not exist if 20 ,, . We consider special cases of Equations (22)(25). Case I When if μ = const and NxtNtx. Then 2 1 2 0, , 1 c c Nqx s Nxq x x 0 d x x s (32) 2 1 :exp xq . Thus using Equations (28), (30) & (32), we find ,Nxt for this particular case. If 2 20,0,0 and xx Then 0 , x s 22 1 ,0d cc NqxN qs xx By asymptotic expansion using Maple 15 Symbolic Programming at dos command prompt: > asympt (exp (theta/q), q); 23 4 23 46 12624 120 qqqqq 5 5 1 O q > subs (theta = alpha [2]/alpha [1]*x, %); 22 3355 22 22 22 33556 111 1 1 126 120 xx xx O qqqqq > %*exp (t*q); 22 3355 22 22 22 33556 111 1 1 1e 2 6120 tq xx xx O qqqqq And the asymptotic expansion of NI(tk,q) being 2 5 62 0 35 4 345 1 1e 2 ,6e 11 1 6 120 24 tq tq k tq Ik k k tq tq tq kk k et I Ntq IOtI q tq et Iet I etI qq q where the summation over is taken over ,,0 kk . INt ttt Therefore Open Access AM
B. O. OYELAMI, S. O. ALE Open Access AM 1642 22 3355 22 22 22 3355 111 1 6 2 0 0 , 11 2π26 120 0,e d 1 e0,d 2π e 1dd , 4πk C tq c tp C C tq x kk tt C Nxt xx xx iq qq q Nq q NqqO iq sqttINt it , N , NtNKN K is the saturation (maxi mum) population the environment can support. For this case, we found that 22 2 2 2 1 ,1 , 0,d , cc x c x Nxq xxNxq K Nqx xss It follows that Case II: For μ = const and 2232 2 1 2 , 4 11 10,d 2 c t co Nxq x x xx xNq xss K xx K (33) 2232 2 1 2 , 4 11 10,d 2 c t co Nxq x x xx xNq xss K xx K (34) And is any real number. 0 If we slot Equations (33) and (34) into Equations (29) and (30) the solution to the model can be found for cases II and I respectively. qIf 1 and K is very large than 1 ,2 22 c B Nqx y A If we let Hence 2 1 , 1exp2expd0 2π as c C Ntx x Ktqq xq x 2 2 2 2 2 ,1, 0, 8 d and . c x x, BxCNq K A Dxssy B x We can approximate , c Nxq in Taylors series us ing the Maple 15 software as Since is analytic in C and by the Cauchy integration theorem it is zero. exp d C tqq 23 456 11 1 ,11 2281 57 128 256 c B Nqxyyy A yyOy The asymptotical expansion for by Dos command prompt in Maple 15 we have: , c Ntx 6 (35) > N[c](x,q):=1/2*pi/i*int(%%*N[c](0,q),q)+alpha2/2* int(exp(t*q),q)*int(phi(s), s=0…x); 22 334455 22 222 2223344 111 11 2 26 0 ,1 16 π2624 120 0, e 11 0,ddedd 4π 4π cC tq x Ctq cC C xx xxx K Nxq q xiq qqq Nq KK NqqOq qss xixit q 55 where the integration is take over a contour C containing the pole of order . By the use of Cachy Goursat theorem, the integral can be reduced to only an integration around a small circle about the pole 0q6k0q of
B. O. OYELAMI, S. O. ALE 1643 order . The value for 6ked tq k Cq q can be obtained by the use of residue theorem as 1 1 0 0 ed 1de 2πRee2π 1!d 2π 1! tq k C ktq tq k q q k q q is i kx t ik (37) Substitute the above value for gives the asymptotic approximation for and where 0,1, 2,, 6k , C Ntx ,, IC NtxN txNtx , 1 0,0,,5 6 66 1 , ! ,, k kk Ik ttk k Ntxt t k INtxtOtINt xt (38) By residue theorem which is unbounded as if and excepttx 0 0 ,exp where 1 and 0 are constants. kk k k k; Nt xttt xt k The Maple was used to find involving some special functions see the appendix and the use of the residue theorem can then be applied to complete the so lution. , c Ntq 5. Conclusions From the two models considered, the Btransform has been demonstrated to offer simple way for solving fixed moment impulsive system (see [19]) for more applica tions). More works need to be done on applications of Btransform especially taxism problems by studying the response of living organisms to stimulus. Moreover, in molecular chemistry, the use of impulsive diffusion mod el can be explored to investigate the movement of che mical substances in and out cells and tissues of organ isms. Furthermore, the VonFoersterMakendrick models can be further exploited to study epidemiology of infec tious diseases like malaria and HIV/AIDS. In view of these, it is recommended that more work should be done on the applications of Btransform couple with the use symbolic programming approach to solve problems. 6. Acknowledgements The authors are grateful to the National Mathematical Centre, Abuja, Nigeria and the Kaduna State University Kaduna, Nigeria for their supports. REFERENCES [1] B. O. Oyelami and S. O. Ale, “Btransform Method and its Applications, in obtaining Solutions of some Impul sive Models,” International Journal of Mathematical Edu cation in Science and Technology, Vol. 31, No. 4, 2000, pp. 525538. http://dx.doi.org/10.1080/002073900412633 [2] S. O. Ale and B. O. Oyelami, “Impulsive System and Applications,” International Journal of Mathematical Education in Science and Technology, Vol. 31, No. 4, 2000, pp. 539544. http://dx.doi.org/10.1080/002073900412642 [3] D. D. Bainov, V. Lakshikantham and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publication, Singapore, 1989. [4] P. S. Simeonov and D. D. Bainov, “Theory of Impulsive Differential Equations: Periodic Solutions and Applica tions,” Essex, Longman, 1993. [5] E. Beltrami, “Mathematics for Dynamic Modeling,” Academy Press, London, 1987. [6] B. O. Oyelami, “On Military Model for Impulsive Rein forcement Functions using Exclusion and Marginalization Techniques,” Nonlinear Analysis, Vol. 35, No. 8, 1999, pp. 947958. http://dx.doi.org/10.1016/S0362546X(98)00114X [7] B. O. Oyelami, S. O. Ale, P. Onumanyi and J. A. Ogidi, “BTransform Method and Application to Sickle Cell Aneamia,” 2008, pp. 202220. http://siriusc.ncat.edu/asn/ajp/allissue/ajpISOTPAND/in dex.html [8] B. O. Oyelami and S. O. Ale, “On Existence of Solution, Oscillation and NonOscillation Properties of Delay Equations Containing ‘Maximum’,” Acta Applicandae Mathematicae Journal, Vol. 109, No. 3, 2010, pp. 683 701. http://dx.doi.org/10.1007/s1044000893401 [9] P. S. Simeonov and D. D. Bainov, “Impulsive Differential Equation: Asymptotic Properties of the Solutions,” World Scientific Publication, Singapore, 1989. [10] S. G. Pandit and H. Deo Sudashiv, “Differential Systems Involving Impulsive. Lecture Notes in Maths,” Springer Verlag, BerlinHeidelbergNew York, 1982. [11] B. O. Oyelami and S. O. Ale, “BTransform and Its Ap plications to a FishHyacinth Model,” International Jour nal of Mathematical Education in Science and Technol ogy, Vol. 33, No. 4, 2002, pp. 565573. http://dx.doi.org/10.1080/00207390210131353 [12] B. O. Oyelami, S. O. Ale, P. Onumanyi and J. A. Ogidi, “Impulsive HIV1 Model in the Presence of Antiretroviral Drugs Using BTransform Method,” Proceedings of Af rican Mathematical Union, Vol. 1, No. 1, 2003, pp. 62 76. [13] B. O. Oyelami, S. O. Ale and P. Onumanyi, “Impulsive HIV Model Using BTransform,” The Proceedings of Na tional Mathematical Centre on Conference on Computa tional Mathematics, Vol. 2, No. 1, 2005, pp. 5064. http://nmcabuja.org/nmc_proceedings.html Open Access AM
B. O. OYELAMI, S. O. ALE Open Access AM 1644 [14] B. Davies, “Brian Davies Integral Transforms and Their Applications,” Springer Publisher, BerlinHeidelberg New York, 2002. [15] M. B. V. Robert, “Biology; a Functional Approval,” Nel son Butler and Tanner Ltd., Rome and London, 1971. [16] V. Lakshimikantham and Z. Dric, “Positivity and Bound edness of Solutions of Impulsive ReactionDiffusion Equations,” Journal of Computational and Applied Ma thematics, Vol. 88, No. 1, 1988, pp. 175184. http://dx.doi.org/10.1016/S03770427(97)002100 [17] L. H. Erbe, H. I. Freeman, X. Z. Lin and H. J. Wu, “Com parison Principles for Impulsive Parabolic Species Grow th,” The Journal of the Australian Mathematical Society, Series B, Vol. 32, No. 4, 1991, pp. 382400. [18] S. O. Ale and B. O. Oyelami, “On Chemotherapy of Im pulsive Models Involving Malignant Cancer Cells,” Aba cus, Journal of Mathematical Association of Nigeria, Vol. 24, No. 2, 1996, pp. 110. [19] B. O. Oyelami and S. O. Ale, “Impulsive Differential Equations and Applications to Some Models: Theory and Applications. A Monograph,” Lambert Academic Pub lisher, Saarbrücken, 2012.
B. O. OYELAMI, S. O. ALE 1645 Appendix > approx2:=asympt (1 + exp (theta/q), q); 23 45 23 456 1 approx2:22624 120 O qqqqqq > Subs (theta=−2*alpha2*x/alpha1, %); 22 33 22 33 44 55 4455 6 222242 21131 22 421 31 151 xx qqq xx O qq > %*exp (t*q); 22 33 22 33 44 55 4455 6 22 2242 21131 22421 e 31 151 tq xx x qqq xx O qqq w = K/(2*pi*mu*x) 2π w q > N(t,x):=%*int (%%*exp (t*q), q); 2 2 22 2 2 222 2 333 222 22 3 555 44 e 222 1,2 22 1,2 ,: e11 ee e 21,222 4 31 1 ee 11 246 4 15 tq tq tq tqtq tq tq xtEitq tq xtEi tq Ntx 21,2 tEitqxtEi tq tq tq xt tq tq 22 33 5 e 12 1, 2 33 1 tq Eitq t tq tq 22222 22 2 2 55 44 33 22 444 33 22 4 2 eeee e 11 1 1 21 4 1, 2 5101515151515 eee 11 24 21,2 33 33 24 1, 2 31 1 22 22 1,2 1e 21 tq tq tq tqtq tq tq tq tq OE tq tq tqtq tq xtEitq tq tq tq Eitq w xtEi tq 5 itq 2 22 2 e21,2 1 41,2π 15 tq Ei tq xt tq Eitq Kx >Subs (alpha1 = 0.001, alpha2 = 0.05, pi = 3.142, K = 1000, %); Open Access AM
B. O. OYELAMI, S. O. ALE 1646 2 2 22 22 2 22 33 22 744 33 22 e e1001, 2500021, 2 ee 1 166666.666621, 2 2 eee 11 24 0.4166666667 101,2. 33 33 tq tq tq tq tq tqtq xtEi tqxtEi tq tq xtEi tq tq tq tE tq tq tq itq 22222 55 44 3322 eeeee 111124 1, 2 51015151515 tq tq tq tq tq Ei tqx tq tq tqtq tq Open Access AM
