Double Compartment CA Simulation of Drug Treatments Inhibiting HIV Growth and Replication at Various Stages of Life Cycle

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1 Double Comparmen CA Simulaion of Drug Treamens Inhibiing HIV Growh and Repliaion a Various Sages of ife Cyle Sompop Moonhai, and Yongwimon enbury Absra Alhough here is no ure for AIDS a his ime, inense researh effors have yielded several reamens ha may be relied upon o delay HIV progression and improve he qualiy of life of hose who have beome sympomai. Human immunodefiieny virus (HIV) infeion ypially follows a hree phase paern; he primary response phase, he linial laeny phase, and he final phase of onse of aquired immunodefiieny syndrome (AIDS). In order o es he effiieny of differen proools in drug herapy for HIV paiens, i is imporan o have a realisi model whih reliably simulaes he ourse of he infeion whih exhibis wo drasially differen ime sales, days and deades. The lassial ordinary or parial differenial equaions have been found o be inadequae in oping wih suh exreme spread in ime sales. In his paper, we employ a wo-omparmen Cellular Auomaa (CA) model o sudy he dynamis of drug herapy of HIV infeion. The levels of healhy an infeed CD+T ells are raked in boh he lymph node and peripheral blood omparmens oupled and updaed simulaneously wih eah ime sep. The viral loads in he wo omparmens are also updaed hrough a sysem of differene equaions. The ell updae rules in he CA model are modified o simulae he impas of herapeui measures where various ypes of anireroviral drugs are applied o inhibi he growh and iaion of HIV a various sages of is life yle. y adjusing he rules o updae he ells in he CA laie, i beomes possible o sudy he effiaies of differen reamen sraegies or drugs of hoie, as well as he reperussion of drug resisane over ime. Keywords human immunodefiieny virus, Cellular Auomaa (CA) model, drug herapy, drug resisane. I. INTRODUCTION UMAN immunodefiieny virus (HIV) is a member of H he rerovirus family ha auses a ondiion alled aquired immunodefiieny syndrome (AIDS) [, ] in whih he human s immune sysem begins o fail. One his sage in he progression of he disease is reahed, he infeed person beomes susepible o life-hreaening opporunisi Manusrip reeived Marh 5, : Revised version reeived Marh 5,. This work was suppored by he Cenre of Exellene in Mahemais, CHE, Thailand. S. Moonhai is wih he Deparmen of Mahemais, Fauly of Siene, Chiangmai Universiy and he Cenre of Exellene in Mahemais, CHE, 3 Si Ayuhaya Road, angkok, Thailand ( umah@gmail.om). Y. enbury is wih he Deparmen of Mahemais, Fauly of Siene, Mahidol Universiy and he Cenre of Exellene in Mahemais, CHE, 3 Si Ayuhaya Road, angkok, Thailand (orresponding auhor, phone: - -5; fax: --533; sylb@mahidol.a.h). infeions. Wihin he infeed person s bodily fluids, HIV is presen as boh free virus pariles and virus wihin infeed immune ells. Sine he Human Immunodefiieny Virus auses a lehal disease wih suh an insidious ime ourse, he sudy of he dynamis and pharmaokineis of he virus-immune sysem in he human body is neessary in order o disover a proper herapeui sraegy for HIV infeion and disover how he disease migh be onrolled. Developmen of reliable models o simulae he ourse of HIV infeion an be an imporan and ruial omponen in he pharmaologial researh for effeive HIV reamens. Reenly, i has been disovered ha ellular CA modelling ehnique offers a more suiable ool for he sudy of he progress of he disease sine i an aommodae he drasially differen ime sales exhibied in he enire ourse of HIV infeion. Classial aemps a modelling based on sysems of ordinary differenial equaions (ODES) and parial differenial equaions (DES) have been found inadequae for he ask sine he developmen of he disease ypially exhibis hree phases of infeion, ha is, an aue phase (measured in days), a hroni phase (measured in weeks), and developmen o full blown AIDS (measured in years). In reen years, a few ellular CA models have been developed o model HIV infeion in he lymph node [3-5]. In, Sloo e al. repored on a non-uniform CA model whih sudies he dynamis of drug herapy in HIV infeion []. Their model was employed o simulae four phases of infeion dynamis; aue, hroni, drug reamen response, and onse of AIDS. Their resuls indiaed ha boh simulaions (wih or wihou reamen) evolved oward o same seady sae. Three differen drug herapies were invesigaed, mono-herapy, ombined drug herapy, and highly aive anireroviral reamen (HAART). More reenly, Shi e al. developed a CA model o sudy he effe of drug reamen of HIV, inorporaing he virus iaion yle and he role of viral load and laenly infeed ells in susaining HIV infeion []. Drug reamen ombinaions wih reverse ransripase inhibiors and proease inhibiors are simulaed wih various drug effiaies. Reverse ransripase inhibiors (RTIs) are a lass of anireroviral drug used o rea HIV infeion. When HIV infes a ell, reverse ransripase opies he viral single Issue 3, Volume 5, 3

2 sranded RNA genome ino a double-sranded viral DNA. The viral DNA is hen inegraed ino he hos hromosomal DNA. This allows ransripion and ranslaion o our in he hos s ells in order o reprodue he virus. The role of RTIs is o blok reverse ransripase's enzymai funion and preven ompleion of synhesis of he double-sranded viral DNA. HIV is hus prevened from muliplying. Anireroviral drugs inhibi he growh and iaion of HIV a various sages of is life yle. RTIs ome in hree forms. The firs is he nuleoide analog reverse ransripase inhibiors (NRTIs), someimes alled nukes, whih work by bloking an enzyme (reverse ransripase) ruial o he produion and iaion of HIV. The seond form is he nuleoside analog reverse ransripase inhibiors (NNRTIs), someimes referred o as Non-Nuleoside Analogs or "nonnukes", whih operaes by prevening he enzyme from onvering RNA o DNA, and prevens he genei maerial of he HIV virus from being inorporaed ino he healhy genei maerial of he ell. Thus, hey render i impossible for he ell o produe a new virus. Finally proease inhibiors (Is) work by inerfering wih he enzyme HIV proease. HIV iaion is hus inerruped a a laer sage in is life yle, ausing HIV pariles in he body o beome sruurally disorganized and noninfeious []. Mos CA models so far only onsidered he dynamis in he lymph node. However, mos linial indiaions of progression are based on blood daa, beause hese daa are mos easily obained. Sine viral populaion irulaes beween he lymph node and plasma omparmens, viral load in boh omparmens are imporan for he desripion of he dynamis of HIV infeion. In our earlier paper [7], he CA rules based on hose uilized by Sanos and Couinho in heir CA model were modified o onsru a double laied CA model o invesigae he dynamis of HIV infeion in boh he lymph node and blood omparmens while he viral loads in he wo omparmens are oninuously updaed hroughou he simulaion. The model also akes ino aoun a delay in he ransformaion of a newly infeed CD+ T ell ha is free o spread he infeion, ino a final saged infeed ell. In his paper, differen forms of drug herapy are simulaed by inorporaing is effes in he ell updae rules of he double omparmen CA model we developed in [7]. Firs, we invesigae he aion of mediaion by drugs in he group of proease inhibiors whih ause some HIV pariles o beome sruurally disorganized and non-infeious. We hen uilize a modified se of ell updae rules o simulae he aion of drugs in he group whih work by bloking reverse ransripase s enzymai funion ourring in he hos s ells so ha viral DNA synhesis is bloked. y appropriaely adjusing he rules o updae he ells in he wo CA laies, he laie represening he lymph node issue and ha for he peripheral blood omparmen, i beomes possible o sudy he effiaies of differen reamen opions or drugs of hoie, as well as he reperussion of drug resisane over ime. II. CA MODE AND SIMUATIONS Here, we employ a CA model whih is defined on wo oupled square laies of sizes. The Moore neighborhood is adoped o define he rules. The saes of he ells in eah of he laies are updaed a eah ime sep in parallel aording o he rules, wih eah ime sep orresponding o one week. Eah sie on he laie is oupied by a ell whih is assigned one of he five saes ha desribe he possible saes in whih hose ells may be found: non-aivaed ells, aive healhy ells (represening CD+ T- ells whih are he main arge of he HIV), infeed A ells (orresponding o infeed ells ha are free o spread he infeion), infeed A ells (infeed ells in he final sage before dying due o he aion of he immune sysem) or dead ells (infeed ells killed by he immune response). In simulaing he CA model of HIV infeion in wo oupled omparmens, for eah omparmen, he simulaion seps sar wih N non-aivaed or non-proliferaing ells, H healhy aive ells, and a small fraion HIV of infeed A ells (A), suh ha A HIV H, disribued randomly. These numbers depend on he iniial viral load V. A eah ime sep, all ells are updaed using he rules desribed below. The effiay of drug use in he proease inhibior group is refleed in he probabiliy appearing below in Rule for he updaes of healhy ells. I redues he hane ha a healhy ell is infeed by he virus ino an A infeed ell. The definiions and values of all he parameers and probabiliies used in hese rules are given in Tables -. The updaing rules are as follows. Rule : Updaes of non-proliferaing ells. (a) If a non-proliferaing ell has non-assigned slos as neighbors, i may beome an aive healhy ell a he probabiliy op, aouning for opporunisi infeion, or i remains he same a he probabiliy - op. (b) If i has a neighbor whih is A - or A - infeed, i beomes an aive healhy ell, by whih he body ries o figh he infeion. Rule : Updae of healhy ells. (a) An aive healhy ell ges infeed by oming in ona wih a virus a he probabiliy av ( ) f( V ) ( ) ( e ). () v d v d v (b) If i has a leas one infeed A neighbor, i beomes an infeed A ell a he probabiliy d v ( ) r( ). () Table. Model parameers in he CA model in he lymph node omparmen. Value Symbol Definiion aie size 5 Number of non-aivaed or 5, non-proliferaing ells a N Issue 3, Volume 5, 33

3 H Number of healhy aive ells a robabiliy or perenage of iniial infeed ells op robabiliy for a nonproliferaing ell o be aed wih an aive healhy ell V Consan in probabiliy for a healhy ell o ome in ona wih a virus A Consan in probabiliy in Eq. () r Consan in probabiliy in Eq. () r Consan in probabiliy in Eq. (3) Time delay for an infeed A ell o beome an infeed A ell robabiliy for a healhy ell o be aed wih an infeed A ell HIV infe robabiliy for a deah ell o be aed wih a healhy ell nona R robabiliy for a deah ell o be aed wih non-aivaed ells Number of infeed A ells in a ell neighbourhood o indue a healhy ell o beome an infeed A ell, () If i has no infeed A neighbor, bu has a leas R ( < R < ) infeed A neighbors, i beomes an infeed A ell a he probabiliy ( d) r r ( v ). (3) (d) Oherwise, i remains a healhy ell a he probabiliy v where v and is a drug effeiveness,. Rule 3: Updae of infeed A ells. An infeed A ell beomes an infeed A ell afer ime seps. Thus, infeed A ells beome infeed A ells a differen ime wih a delay of. Rule : Updae of infeed A ells. Infeed A ells beome dead ells, orresponding o he depleion of infeed ells by he immune response. Rule 5: Updaes of dead ells (a) Dead ells an be aed by healhy ells wih he probabiliy ( infe ) in he nex sep, or by an infeed A ell wih he probabiliy infe. Oherwise, i remains a dead ell a probabiliy. (b) Afer sep (a), dead ells an be aed by an inaivaed ell wih probabiliy. Oherwise, i remains a dead ell a he probabiliy in. in Table. Model parameers in he CA model in he blood omparmen. Symbol Definiion Value aie size N Number of non-aivaed or, non-proliferaing ells a H Number of healhy ells a 5, HIV robabiliy or perenage of iniial infeed ells.5 op robabiliy for a nonproliferaing. ell o be aed wih an aive healhy ell V Consan in probabiliy for a healhy ell o ome in ona wih a virus A Consan in probabiliy in Eq. () r Consan in probabiliy in Eq. () r Consan in probabiliy in Eq. (3) Time delay for an infeed A ell o beome an infeed A ell robabiliy for a healhy ell infe o be aed wih an infeed A ell robabiliy for a deah ell o be aed wih a healhy ell nona R robabiliy for a deah ell o be aed wih a nonaivaed ell Number of infeed A ells in he neighborhood of a ell o indue a healhy ell o beome an infeed A ell [3].99 [3].9 The same rules are applied o updae he ells in he laie for he peripheral blood omparmen. III. VIRA OAD DETERMINATIONS The viral load influenes he dynamis of he healhy and infeed ells hrough he probabiliy v. Following wha has Issue 3, Volume 5, 3

4 been done in our earlier work [7], afer all of he five ell saes are updaed in he wo laies, he virus presene in eah omparmen is alulaed using Eq. ()-() and he following differene equaions whih represen he evoluion of viral load in he lymph node omparmen (wih V = V ) and peripheral blood omparmen (wih V = V ) a ime. In he lymph node omparmen V V ps I ( V V ) H V V () H where I = virus-produing infeed ells = A A V e V V In he blood omparmen V V ps I ( V V ) H V V (5) H = A A V e V V I = virus-produing infeed ells = week (ime sep) As in [7], A and A are he numbers a ime of A and A infeed ells in he lymph node, respeively, while A and A are he orresponding amouns in he blood omparmen. H and H are he numbers of aive healhy ells in he respeive omparmens a ime, p is he average viral produion rae per infeed ell, e represens he irulaion of virus beween he wo omparmens, and is he deah rae of free virus. The values of he parameers appearing in Eq. -5 used in our simulaions are given in Table 3. The value of drug effiay probabiliy is varied in he simulaion shown in Figures -. The healhy ells, he infeed A ells, he infeed A ells, and he viral load in he lymph node are shown in Figures -3, omparing noreamen ourse of infeion o effes of reamens a various drug effiaies. The levels of hese ell ypes in he peripheral blood are shown in Figures -. We observe in hese simulaion samples ha he appliaion of ani-viral drugs exer a marked effe only briefly immediaely afer he beginning of drug use. The drug seems o evenually beome less effeive in he long run, however high is effiay is, even hough is use has no been erminaed, as long as % effiay an no be assured ( d ). Our simulaions indiae ha he virus will be able o adjus quie quikly o he drugs aemp a bloking is infeion of he healhy ells whose level of healhy ells sill drops o a low level, and hose of infeed ells and viral load sill rise o unhealhily high levels. However, hese levels do sabilize o seady sae levels, whih ould be onrollable. In onras, he paien free of herapy shows a oninued drop in he level of healhy ells, and non-sabilizing rise in he levels of infeed ells and viral load in he blood omparmen. IV. DRUG EFFICACIES WITH RESISTANCE To invesigae he possible long erm onsequenes of drug resisane a differen drug effiaies, we ae, in he updaing rule, whih inorporaes he effe of drug herapy in fighing HIV infeion, by ( T d ) d e wih, Td 3, and.. This means ha as ime progresses, he given drug loses is abiliy o figh he virus in an exponenial fashion. Table 3. Model parameers in viral load simulaion. Symbol V V p S S Definiion lasma virus onen-raion a Virus onenraion in he lymph node a Average virion produion rae per infeed ell Saling faor in he lymph node Saling faor in he blood Clearane rae of free virus H in he lymph node Clearane rae of free virus H in he blood Value [] (an vary) [9] /H, / H.. Free virus deah rae.3 [9] e Cirulaion fraion of virus beween lymph node and blood. [] Saling faor: lymph node blood Saling faor: lymph node blood 7 [] 5 [] Issue 3, Volume 5, 35

5 Healhy ells in he peripheral blood Healhy ells in he lymph node Infeed A ells in he lymph node x x =.5 =. =.7 =. Figure =.5 =. =.7 =. 3 5 Figure =.5 =. =.7 =. 3 5 Figure 3 In Figures 7-9, we see he omparison of effes of drug reamen a differen resisane values of. The drug effiay wears off wih ime and he healhy ells sele o a lower seady sae level han he ase where drug resisane is no under display. The A infeed ells and viral load are seen o sele o higher seady sae levels han hose in he ase where drug resisane is no inorporaed. The urves are averages over simulaions. No signifian differene has been observed in he urves when averaged over a higher number of simulaions. Infeed A ells in he lymph node Infeed A ells in he peripheral blood.5 x =. =.7 =. = Figure Viral load in he peripheral blood =.5 =. =.7 =. 3 5 x 5 Figure 5 =.5 =. =.7 =. 3 5 Figure Issue 3, Volume 5, 3

6 Healhy ells in he lymph node Infeed A ells in he lymph node Viral load in he peripheral blood x 5 =. =.7 =. = x =. =.7 =. = x 5 =. =.7 =. =.9 Figure 7 Figure Figure 9 V. ACTION OF NRTIS/NNRTIS To simulae he effiay of reverse ransripase inhibiors in he form of NRTIs or NNRTIs, we adjus he ell updae rules as follows. Rule : Updaes of non-proliferaing ells. (a) If a non-proliferaing ell has non-assigned slos as neighbors, i may beome an aive healhy ell a he probabiliy op, aouning for opporunisi infeion, or i remains he same a he probabiliy - op. (b) If i has a neighbor whih is A - or A - infeed, i beomes an aive healhy ell, by whih he body ries o figh he infeion. Rule : Updae of healhy ells. (a) A healhy ell ges infeed by oming in ona wih a virus a he probabiliy av v v f ( V ) ( v e ). () (b) If i has a leas one infeed A neighbor, i beomes an infeed A ell a he probabiliy r( v ). (7) () If i has no infeed A neighbor, bu has a leas R ( < R < ) infeed A neighbors, i beomes an infeed A ell a he probabiliy r r ( ). () v (d) Oherwise, i remains a healhy ell a he probabiliy v where v. (e) Afer sep (d), an infeed A ell remains an infeed A ell a he probabiliy d.oherwise, i beomes a healhy ell. (f) Afer sep (e), a healhy ell an be aed by an infeed A ell wih probabiliy. Oherwise, i remains a healhy ell a he probabiliy in he nex ime sep, where is probabiliy ha reverse ransripion is no ompleed and is a drug effiay of reverse ransripase inhibior,. Rule 3: Updae of infeed A ells. An infeed A ell beomes an infeed A ell afer ime seps. Thus, infeed A ells beome infeed A ells a differen ime wih a delay of. Rule : Updae of infeed A ells. Infeed A ells beome dead ells in he nex ime sep, orresponding o he depleion of infeed ells by he immune response. Rule 5: Updaes of dead ells (a) Dead ells an be aed by healhy ells wih he probabiliy ( ) infe Issue 3, Volume 5, 37

7 in he nex sep, or by an infeed A ell wih he probabiliy. Oherwise, i remains a dead ell a probabiliy infe. (b) Afer sep (a), dead ells an be aed by an inaivaed ell wih probabiliy in. Oherwise, i remains a dead ell a he probabiliy. in The parameri values used in our simulaions in his ase are given in Tables -5. Table. Model parameers in he CA model in he lymph node omparmen. Symbol Value in lymph node Value in blood 5 N 5,, H, 3, HIV.5 [].5 [] op.. v.5 a 5 r.997 r.997 [] infe nona 5 [].99 [].9 R [] [] 5 [].99 [].9 [] d Figures - show he simulaed resuls in he lymph node and blood omparmens when is se equal o.99, while is varied from.5 o.9. d Figures 3- show he simulaion resuls when is fixed d a.9, while is varied beween.5 and.99. The urves have been obained from averaging over simulaions. No observable differene was found when averaged over higher number of simulaions. Table 5. Model parameers in viral load simulaion. Symbol V V Definiion lasma virus onenraion a Virus onenraion in he lymph node a p Average virion produion rae per infeed ell S S H H Saling faor in he lymph node Saling faor in he blood Clearane rae of free virus in he lymph node Value [] (an vary) [9] /H / H. Clearane rae of free virus in he blood Deah rae of free virus.3 [9] e Cirulaion fraion of. virus beween lymph [] node and blood Saling faor: lymph node blood Saling faor: lymph node blood. 7 [] 5 [] Sine he simulaed urves in he lymph node and he peripheral blood omparmens are qualiaively similar, we only show he simulaed urves in he lymph node for some quaniies, and in he blood omparmen for ohers. We observe in hese figures ha when he effiaies and are suffiienly high, his ype of drug reamen affes a d lear improvemen in he ourse of infeion: he levels of viral load and infeed ells drops drasially and remain signifianly lower han hose of he no-reamen ase, while he level of healhy ells rises o reah and remain a a level signifianly higher han he no-reamen ase. Thus, our simulaions appear o indiae ha his form of drug hoie an be expeed o be more effeive han he use of proease inhibiors simulaed in he previous seion. Issue 3, Volume 5, 3

8 x x Infeed A ells in he lymph node =.5 =. =.7 =. =.9 Infeed A ells in he lymph node =.5 =. =.7 =. =.9 =.99 Healhy ells in he lymph node x 5 Figure =.5 =. =.7 =. =.9 Healhy ells in he lymph node x 5 Figure 3 =.5 =. =.7 =. =.9 = Figure x Figure Viral load in he peripheral blood x =.5 =. =.7 =. =.9 Viral load in he lymph node =.5 =. =.7 =. =.9 = Figure 5 Figure Issue 3, Volume 5, 39

9 Infeed A ells in he peripheral blood 3.5 x =.5 =. =.7 =. =.9 = VI. CONCUSION The simulaions of our CA model allow us o invesigae he effes of drug effiaies and drug resisane on he developmen of HIV infeion. Effes of differen ypes of drugs argeing differen infeion mehanisms, suh as he use of drug reamen ombinaions wih reverse ransripase inhibiors and proease inhibiors may be simulaed subje o various drug effiaies. Our sudy indiaes ha he drugs whih ause he virus o beome sruurally disorganized and hene less infeious lead o beer improvemen in he lymph node han in he peripheral blood omparmen. On he oher hand, drugs whih work by bloking he enzyme reverse ransripase ruial for he produion and iaion of he virus lead o more observable improvemens in he blood omparmen han in he lymph node. Moreover, he laer drug ype seems o be more effiien in fighing he infeion. Healhy ells in he peripheral blood Viral load in he peripheral blood x =.5 =. =.7 =. =.9 = x =.5 =. =.7 =. =.9 =.99 Figure Figure Figure I would seem ha our resuls differ qualiaively from hose of Sloo e al. [] in ha he non-reaed and reaed urves simulaed wih our model do no onverge o he same seady sae. The same an be observed in he ase where drug resisane is assumed o ake plae. In our opinion, our resul is more reasonable as i suggess ha even hough drug reamen does no effeively annihilae he infeion alogeher, and even wih he ensuing drug resisane, he paiens benefi from he reamen o a erain exen over no being given any reamen a all. Of ourse, he appliaions of drug reamen a differen poins in ime in a paien ourse of HIV infeion is expeed o lead o differen ouomes. Simulaions an be furher arried ou o deermine he opimal poin in ime during he ourse of infeion a whih drug reamen should be sared. Effes of differen ypes of drugs argeing differen infeion mehanisms, suh as he use of drug reamen ombinaions wih reverse ransripase inhibiors and proease inhibiors may be simulaed subje o various drug effiaies. The ouomes of hese differen drug hoies or ombinaions ould be ompared in order for proper prognosis and deisions an be made by he physiians on he bes ourse of aion o be aken for heir paiens under heir are. ACKNOWEDGMENT Appreiaion is exended oward he Cenre of Exellene in Mahemais, CHE, Thailand, for finanial suppor. REFERENCES [] R. A. Weiss, How does HIV auses AIDS?, Siene, Vol., 993, pp [] D. C. Douek, M. Roederer, R. A. Koup, Emerging Coneps in he Immunophahogenesis of AIDS, Ann. Rev. Med., Vol, 9, pp. 7-. [3] V. Shi, A. Tridane, Y. Kuang, A Viral oad-ased Cellular Auomaa Approah o Modelling Hiv Dynamis and Drug Treamen, Journal of Theoreial iology, Vol.53, No.,, pp []. Sloo, F. Chen, C. ouher, Cellular Auomaa Model of Drug Therapy for HIV Infeion, eure Noes in Compuer Siene, Vol. 93,, pp. 93. Issue 3, Volume 5,

10 R.M. Zorzenon dos Sanos, S. Couinho, Dynamis of HIV Infeion: A Cellular Auomaa Approah, hysis Review eers, Vol.7, No.,, pp. S. [] M. andrisina, A. Fabiano, S. Alamura, C. agalà, A. isazzi, A. Cassano, C. Spadafora, F. Giorgino, C. arone, M. Cignarelli, Reverse Transripase Inhibiors Down-Regulae Cell roliferaion in Viro and in Vivo and Resore Thyroropin Signaling and Iodine Upake in Human Thyroid Anaplasi Carinoma, The Journal of Clinial Endorinology & Meabolism, Vol. 9, No., pp [7] S. Moonhai, Y. enbury, W. Triampo, Muliple aied Cellular Auomaa: HIV Dynamis in Coupled ymph Node and eripheral lood Comparmens, he nd WSEAS Conferene on Applied Compuer Siene, Iwae, Japan, Oober -,. [] S. H. ajaria, G. Webb, M. Cloyd, D. Kirshner, Dynamis of naive and memory CD+ T lymphoyes in HIV- disease progression, J Aquir. Immune Defi. Syndr.,, Vol. 3, pp. -5. [9].W. Nelson, J.D. Murray, A.S. erelson, A model of HIV- pahogenesis ha inludes an inraellular delay, Mah. iosi.,, Vol. 3, pp. 5. [] D. J. Sekel, C. E. arker, M. A. Nowak, A model of lymphoye reirulaion, Immunol. Today, Vol., 997, pp. -. Issue 3, Volume 5,