Global analysis of a delay virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response

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1 Glol nlsis of del irus dnmis model wih Beddingon-DeAngelis inidene re nd CTL immune response Lish Ling Shool of Mhemis nd Phsis Uniersi of Siene nd Tehnolog Beijing Beijing Chin Yongmei Su Shool of Mhemis nd Phsis Uniersi of Siene nd Tehnolog Beijing Beijing Chin Asr In his pper n HIV- infeion model wih Beddingon-DeAngelis infeion re nd CTL immune response is inesged We derie he si reproduion numer for he irl infeion model B onsruing suile Lpuno funionls nd using LSlle inrin priniple for he del differenil equions we find when he infeion-free R R equilirium is gloll smpoill sle And if he CTL immune reproduie numer he immune-free equilirium nd he endemi equilirium re gloll smpoill sle Kewords Beddingon-DeAngelis; CTL immune response ; Lpuno funionl; LSlle inrin priniple;glol sili I INTRODUCTION In reen ers he dnmis of HIV- infeion model he een sudied due o suh models n e helpful in he onrol of endemi diseses nd proide insighs ino he dnmis of irl lod [-8] The nlsis of hese dnmi ehiors m pl signifin role in he deelopmen of eer undersnding of diseses nd rious drug herp sregies gins hem A si irl infeion model [9] hs een widel used for inesiging he dnmis of irus infeions whih hs he following forms: d ( k u where susepile ells ( re produed onsn re die densi-dependen re d nd eome infeed wih re ; infeed ells ( re produed re nd die re ; free irus priles ( re relesed from infeed ells re k nd die re u In reli Cooi T Lmphoes (CTL immune response is uniersl nd neessr o elimine or onrol he R disese fer he infeion Indeed i is elieed h CTL ells re he min hos immune for h deermines irus lod [] Therefore he dnmis of irus infeion wih CTL response hs reenl drwn muh enion of reserhers in he reled res [-6] pper [] ge he following immune model d pz ( k z z z where infeed ells re killed re immune response nd he irus-speifi CTL ells prolifered re z on wih infeed ells nd die re The riles nd oher prmeers he sme iologil menings s in he model ( z ( pz he CTL Besides he iliner inidene re nd ( he Beddingon-DeAngelis funionl response ws ofen used for irus infeion model [78] m n used in model ( Xi Wng [3] nd Youde To onsru he following model: d m n pz (3 m n k u z z z where z he he sme iologil menings s in he model ( Reenl i hs een relized h ime del should e ken ino onsiderion [9-] Beuse here m e lg eween he ime for rge ells o e oned he irus priles nd he ime for he oned ells o eome iel ffeed Th is he oning irions need ime o ener ells Then G Hung W M [3] propose he following model: The 8h Inernionl Conferene on Ssems Biolog (ISB //$3 IEEE 8 Qingdo Chin Ooer 7

2 ( ( ( d( ( ( p ( ( ( e p( ( ( ( k( u( ( As poined in [] if we ssume h immune responses n poenill deelop he ondiions we inrodue n immune response reproduion numer ke u( d m R ( k ndk ume where he se rile nd onsn he sme mening s model (3 nd represens he ime del Bsed on oe disussion we propose he following model: ( ( ( d( m( n( ( ( ( e ( p( z ( (5 m( n( ( k( u( z( ( z( z( where he se rile nd onsn he sme mening s model ( This pper is orgnized s follows In Seion II we will gie si reproduie numer nd wo equiliriums Then we proe h he hree equiliriums re gloll smpoill sle in seion III A ls his pper ends wih rief onlusion in Seion IV II BASIC REPRODUCTIVE NUMBER AND EQUILIBRIUM A dire ompuion shows h he si reproduie numer of model (5 is R e k / u( d m I shows here eiss n infeion-free equilirium E ( nd d / If R in he sene of n immune response here eiss n immune-free equilirium E ( where ue nk k ndk ume ke ( R ( k ndk ume k e ( R u( k ndk ume Noe h R so mens k ume whih n mke nd n lso e posiie wih R When R E ( z where dmu k( n mdu ( mu k du dnk k u e ( d here eiss n endemi equilirium e ( d z is equilen o z e ( d > whih n e proed ( ke u( d m > ( ( k ndk ume h is R poin Theorem 3 If E III STABILITY OF EQUILIBRIUMS R he infeion-free equilirium is glol smpoill sle for n del Proof Choosing Lpuno funion W ( ( W ( [ ( ln ] e ( e ( m k p ( ( ( e z d m( n( s follows where / d We luling he deriie of long he posiie soluions of he ssem (5 nd noe h d we oin p W ( [ ] ( e ( e ( e z( m ( k ( ( ( ( m( n( m( n( W ( ( ( [ ]( d( ] m ( m( n( ( ( e ( e ( p( z( m( n( p e [ k( u( ] e [ ( z( z( ] k ( ( ( ( m( n( m( n( The 8h Inernionl Conferene on Ssems Biolog (ISB //$3 IEEE 9 Qingdo Chin Ooer 7

3 d ( ( ( ( [ ( ( m ( m( m( n( ( ( ( ] m m( n( m( n( u e ( e z( k d( ( m( ( ( ( m m m( n( u e ( e z( k d( ( ( u e ( e z( ( ( m m k d( ( u ( ( m k e ( R ( e z( Oiousl when z R we he W ( z Therefore he infeion-free equilirium sle W ( z if nd onl if z e he lrges inrin se of ( z : from he seond equion of (5 we oin shows E M E R W for ll E is Le M hen whih so we ge he glol smpoil sili of LSlle inrine priniple If W ( z if nd onl if lrges inrin se of ( z : z R W R we oin Le M e he hen from he seond nd hird equions we oin so LSelle inrine priniple we n know gloll smpoil sili of E R n lso ensure he Theorem 3 If he immune-free equilirium poin is glol smpoill sle E R Proof Define Lpuno funion W s follows m n ( W ( e [ ( d ] [ ( ln ] ( m n ( p [ ( ln ] z( k e ( ( g( d ( m( n( where g( ln We luling he deriie of long he posiie soluions of he ssem (5 m( n W e ( [ ( ( ] [ ( ( ] mn ( p ( ( [ ( ( ] z( e k m( n( W ( ( ( ( e ln m( n( m( n( m( ( ( ( m ( n ( ( e m n ( m( n( [ ][ d( ] ( ( m( n( ( [ e ( p( z( ] p k ( [ k( u( ] [ ( z( z( ] ( ( ( ( e e m( n( m( n( ln ( ( m( n( m ( n ( ( ( Noe h d e m n u k e de ( ( ( n W ( ( ( m n ln ( ( m( n( m ( n ( ( ( m( n ( ( [3 ( m n ( m( n( m n ( ( ( m( n ] [ ] ( m( n( pz( ( de ( ( ( n m( n [ ( ( m n ( m n m( n ( ( m n ln ] ( ( m n ( m( n( ( ( m n ( ln ] [ ( m( n( ( ( m( n( m( n( ln ] [ ln ] ( m( n m( n ( ( m( n m( n( [ ] m( n( m( n pz( ( The 8h Inernionl Conferene on Ssems Biolog (ISB //$3 IEEE Qingdo Chin Ooer 7

4 de ( ( ( n m( n g ( ( ( m n ( m n g ( ( m n ( ( m ( n ( ( m( n( g( g( ( m( n n( m( ( ( pz( ( R ( m( n ( m( n( Sine g ( for ll if R hen W ( z And W ( z = if nd onl if z so is gloll smpoill sle When we he E W ( z R = if nd onl if from he seond equion of (5 i is es o know he lrges inrin se of ( z : is so LSelle R W inrine priniple we n know gloll smpoil sili of E R E n lso ensure he Theorem 33 The endemi equilirium poin glol smpoill sle Proof We onsru Lpuno funion s follows m n ( W ( e [ ( d ] [ ( ln ] ( m n pz ( p z( [ ( ln ] [ ( ln ] z z z k z e ( ( ( pz g( d ( p z ( m( n( where g( ln B luling he deriie of W z long he posiie soluions of he ssem (5 ( m( n W e ( [ ( ( ] [ ( ( ] m n ( pz p z [ ( ( ] [ z( z( ] k z( e ( ( ( ( e m( n( m( n( ( pzln m ( n ( ( ( E ( ( m( n( m ( n ( ( e m n ( m( n( [ ][ d( ] ( ( ( [ e p( z( ] ( m( n( pz ( [ ( u( ] k ( k p z z( ( ][ ( z( z( ] is ( ( ( ( e e m( n( m( n( Noe h ( ( m( n( ( pzln m ( n ( ( ( d e ( p z m n u k e ( p z de ( n( ( ( W ( ( pz[ ( ( m n ( m ( n ( ] ( p z [3 m n m( n( m n ( ( ( ( m n ( ( m( n( ] ( ( m( n( ( pzln m ( n ( ( ( de ( n ( ( m( n ( [ ( ( m n m n ( pz ( ( ( ln m n ] ( [ p z m n ( ( m n ( ( m n ln ] m( n( ( m( n( m( n( m( n( ( pz[ ln ] m( n m( n ( ( pz g[ ] ( n( m( ( ( ( m( n( ( m( n de ( n ( ( m( n [ [ ] ( ( m n m n ( pz g ( ( m n ( pz g[ ] ( m ( n ( m( n( ( pz g[ ] m( n ( ( pz g[ ] ( n( m( ( ( ( m( n( ( m( n Sine g ( for ll we he h W ( z for ll z Therefore he endemi equilirium E is sle W ( z = if nd onl if The 8h Inernionl Conferene on Ssems Biolog (ISB //$3 IEEE Qingdo Chin Ooer 7

5 Le M e he lrges inrin se of ( z : R W (5 we oin endemi equilirium z z E IV hen from he forh equion of LSelle inrine priniple he is lso gloll smpoill sle CONCLUSIONS In his pper sed on Beddingon-DeAngelis infeion re nd CTL immune response we he disussed irus infeion model wih ime del The sle nlsis of he gien model is rried ou While R he infeion-free equilirium is gloll smpoill sle for n B onsruing suile Lpuno funion nd using LSlle inrine priniple we he h when he immunefree equilirium is lso gloll smpoill sle when R we lso proe he glol sili of he endemi equilirium E E E ACKNOWLEDGMENT R We would like o hnk he suppor of Nionl Nurl Siene Foundion of Chin nd he Fundmenl Reserh Funds for he Cenrl Uniersiies REFERENCES [] Perelson A Nelson P: Mhemil models of HIV dnmis in io SIAM Re (999 pp3- [] Perelson A Neumnn A Mrkowiz M Leonrd J Ho D: HIV- dnmis in io: irion lerne re infeed ells life-spn nd irl generion ime Siene 7(996pp [3] Nowk MA Bnghm CRM Populion dnmis of immune response o persisen iruses Siene 7(996pp7-79 [] Cnrro AA Gleri IM Lr ML Periodi soluions nd hos in non-liner model for he deled ellulr immune response Phsi A 3(pp3- [5] Zhu H Zou X Imp of dels in ell infeion nd irus produion on HIV- dnmis Mh Med Biol 5(8pp99- [6] Li Mihel Y Shu H Glol dnmis of mhemil model for HTLV- infeion of CD T ells wih deled CTL response Nonlier Anl Rel World Appl 3(pp8-9 [7] Shi X Zhou X Song X Dnmil ehior of del irus dnmis model wih CTL immune response Nonliner Anl Rel World Appl (pp [8] Song X Wng S Zhou X Sili nd Hopf ifurion for irl infeion model wih deled non-li immune response J Appl Mh Compu33(pp5-65 [9] Nowk MA M RM Virus Dnmis New York:Oford Uniersi Press: [] Arnou R Nowk M Wodrz D: HIV- dnmis reisied: iphsi de ooi lmphoe killing Pro R So Lond B65(pp37-35 [] Culshw RV Run SG Spieri RJ:Opiml HIV remen mimising immune response J Mh Biol 8(pp55-56 [] Wng KF Wng WD Liu XN: Glol sili in irl infeion model wih li nd nonli immune response Compu Mh Appl5(6pp593-6 [3] Hung G M WB Tkeuhi Y: Glol properies for irus dnmis model wih Beddingon-DeAngelis funionl response Appl Mh Le(9pp [] Zhou XY Shi XY Zhng ZH Song XY: Dnmil ehior of irus dnmis model wih CTL immune response Appl Mh Compu3(9pp39-37 [5] Yun Z M J Tng X Glol sili of deled HIV infeion model wih nonliner inidene re Nonliner Dn 68(pp7- [6] Zhou X Shi X Zhng Z Song X Dnmil ehior of irus dnmis model wih CTL immune response Appl Mh Compu 3(9pp39-37 [7] Beddingon JR: Muul inerferene eween prsies or predors nd is effe on serhing effiien J Anim Eol(975pp33-3 [8] DeAngelis DL Goldsein RA O Neill RV:A model for rophi inerion Eolog 56(975pp88-89 [9] V Herz S Bonhoeffer R Anderson RM M MA Nowk Virl dnmis in io: limiions on esimions on inrellulr del nd irus del Pro Nl Ad Si USA 93(996pp77-75 [] P Nelson J Murr A Perelson A model of HIV- phogenesis h inludes n inrellulr del Mh Biosi 63(pp-5 [] P Nelson A Perelson Mhmil nlsis of del differenil equion models of HIV- infeion Mh Biosi79(pp73-9 [] Nowk MA M RM: Virl dnmis (Oford Uniersi Press [3] Xi Wng Youde To Xinu Song Glol sili of irus dnmis model wih Beddingon-DeAngelis inidene re nd CTL immune response Originl 66(pp85-83 [] Wodrz D Hepiis C irus dnmis nd pholog: he role of CTL nd niod response Journl of Generl Virolog(83pp73-75 The 8h Inernionl Conferene on Ssems Biolog (ISB //$3 IEEE Qingdo Chin Ooer 7