MODELING THE DYNAMICS OF A MONOCYCLIC CELL AGGREGATION SYSTEM

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1 Cybereics a Sysems Aaysis Vo 47 No MODELING THE DYNAMICS OF A MONOCYCLIC CELL AGGREGATION SYSTEM V V Akimeko a a Yu V Zagoroiy a UDC 5356 Absrac Te paper cosiers a yamic moe of moocycic ce aggregaio base o a iiia bouary-vaue probem for yperboic raspor equaios A aayic souio o e probem a coiios of is coiuous iffereiabiiy are fou Numerica cacuaios for casses a iffer i e smooess of ipu moe parameers are carrie ou for wo scearios of moocycic aggregaio of bioogica ces Keywors: bioogica sysems moocycic ce aggregaio ce esiy iiia bouary-vaue probem sysem of raspor equaios aayica souio umerica moeig INTRODUCTION Moeig of e ife cyce of a moocycic ce aggregaio as a bioogica sysem is base o iiia bouary-vaue probems for sysems of yperboic equaios a escribe raspor processes [ 6] Tese moes perai o biokieic probems a escribe e evouio of bioogica ces as eir raspor roug e sages of bir ivisio agig a yig (wi respec o age parameer [7 ] We wi cosier a maemaica moe of e evouio of a moocycic aggregaio of bioogica ces Te eory of paria iffereia equaios wi be use o erive a exac aayic souio of a sysem of raspor equaios a o aayze is smooess properies for a give cass of ipu parameers of e moe Te resus wi aow oe o suy various scearios of e eveopme of bioogica moocycic ce aggregaios wic correspo o various eviromea coiios o eveop efficie meos o moe is ype of bioogica processes a o formuae opima coro probems for e sysem Te umerica aaysis of e yamics of moocycic aggregaio carrie ou i e paper for wo iffere casses of ipu parameers wi revea e possibiiy of usig e obaie aayic epeeces i appie bioogica suies uer isufficiey smoo a priori aa o e beavior of e bioogica sysem MONOCYCLIC CELL AGGREGATION MODEL Le us irouce a fucio ( of e isribuio esiy of ces of moocycic aggregaio wi respec o bioogica age [ max ] o a ime ierva [ T] max T wi e omai of efiiio {( : max T} Cosier a moe of a bioogica sysem for wic e firs group of e moocycic ce aggregaio is ivie io auger ces (N oy oce a bioogica age as bee acieve e remaiig ces coiues exisig wiou ivisio Te age of ces from e firs group oes o excee Le us prese eir reaive age isribuio esiy by a fucio ( wi e omai of efiiio {( : T} Te esiy of e seco group of ces wose age is greaer a is efie by a fucio ( specifie i e omai {( : max T} Here a Taras Sevceko Naioa Uiversiy of Kyiv Kyiv Ukraie akvv@uicybkievua; yuzagor@ukre Trasae from Kibereika i Sisemyi Aaiz No pp Jauary February Origia arice submie Ocober //47-9 Spriger Sciece+Busiess Meia Ic 9

2 For e isribuio esiy of ces of moocycic aggregaio ( e us cosier a iiia bouary-vaue probem for a sysem of raspor equaios (of yperboic ype [ ]: s ( for ( ( ( ( ( ( ( (3 s( for max (4 ( ( ( ( (5 ( ( (6 ( ( (7 ( max were ( is e sare of ces a are ivie a e age ; s ( a s ( are e ea raes of reproucive a vegeaive ces respecivey; cos a cos are cosa veociies of ce agig (moio wi respec o e parameer ; cos is a reproucio coefficie i ce ivisio; ( a ( are e iiia age isribuios of reproucive a vegeaive ces respecivey Wiou oss of geeraiy of e probem saeme we wi assume a ere exiss a ieger N a eermies e fia ime T of moeig of e process Te foowig cosrais are impose o parameers of e moe ( (7: ~ i si ( ~ ~ ( (i T ( N (8 were ~ ~ ~ a are kow posiive cosas ANALYTIC SOLUTION OF THE INITIAL BOUNDARY-VALUE PROBLEM FOR SYSTEM ( (8 Le us cosier e foowig eorem THEOREM Le e coefficies of probem ( (8 saisfy e coiios C ([ ] C ([ T ] s s C ( ( ( ( max ( ( s s ( ( ( ( ( ( (9 ( ( ( ( ( ( s( ( ( ( s ( Te e cassica souio (7 of probem ( (8 exiss a is uique a ( C ( a ( C ( : ( ( ( for ( ( ( ( for ( ( were ( k( ( k / ( ( ( exp ( k ( (3 k ( ( k / ( k 3

3 ( s( ( ( ( ( N ; (4 ( exp s( ( for ( ( ( ( ( ( ( exp s( ( for ( (5 for k were ( k for k ( ( / (6 is e Heavisie fucio e ses ( ( { } {( ( } N are efie for eac N ( {( T } T max } a {( Proof As foows from e properies of souios of iiia bouary-vaue probems for raspor equaios (of yperboic ype [5 6] for posiive iiia coiios (3 (6 suc a ( ( ( ( (7 souios ( a (5 of probem ( (8 remai posiive everywere for a are boue o a arbirary ime ierva [ T] Le us firs prese e fucio ( from (3 as ( ( ( were ( ( ( / ( k( ( k / exp ( k ( k ( ( k / ( k ( k ( s ( k ( ( ( k ( ( ( Fi e paria erivaives of e fucio ( a iera pois of e omais a : ( ( ( ( ( ( ( ( ( ( k( ( k / ( k ( ( k ( ( k / ( k 3

4 (( ( k ( k( k( k ( k( ( k / k ( k ( ( k / k ( ( k( ( k / k ( ( k / ( k ( k ( ( k ( ( ( k / ( k ( ( k / k k Comparig ese erivaives yies e ieiy ( ( Te we ca wrie ( ( ( ( ( ( ( Represe e fucio ( from ( as ( ( ( P ( for ( ( P ( for ( were P ( ( ( We obai e expressio P ( ( P ( ( ( ( ( P ( ( ( ( ( P( ( P( ( ( s ( P ( P ( ( P( ( ( wic eas o Eq ( for e fucio P ( : P( P( s ( P ( N a ece for e fucio ( Tus ( from ( is a coiuousy iffereiabe fucio a a cassica souio of Eq ( a iera pois of e omais a Le us sow a e fucio ( from ( saisfies e iiia a bouary coiios ( a (3 Te foowig equaiies are rue for : 3

5 ( ( ( ( ( ( for ( ( ( for wic correspos o (3 uer e coiio of Teorem were ( ( ( Le us ceck e coiuiy of e fucio ( a e ierface of e omais a wic is e se of vaues of a : im / ( ( ( im im / ( ( ( im / ( ( ( / ( ( ( ( k( ( k / ( ( exp (( k k (( k / ( k ( / ( k( ( k / ( (exp (( k k (( k / ( k ( k( ( k / ( ( ( exp (( k k (( k / ( k ( k( ( k / ( ( exp (( k k ( k( k / ( / ( k( ( k / ( ( ( exp (( k k (( k / ( k ( / ( k( ( k / ( ( exp (( k k ( k( k / Herefrom i foows a e fucio ( is coiuous i provie a ( ( ( Le us cosier e vaues of e fucio ( a e bouary pois a : ( ( ( (( (( / ( k( k / exp ( k ( k ( k / ( k (( / ( (( ( k( k / exp ( k ( ; k ( k / ( k 33

6 (( / ( ( ( ( ( (( ( k k / exp (( k ( k ( ( k / ( k ( wece i foows a ( wic correspos o coiio ( Express e fucio ( ( from ( Te e coiio ( yies e cosrai ( ( Le us sow a e fucio ( is coiuousy iffereiabe wi respec o a e ierface of e omais a wic is e se of vaues of ( : im ( P ( ( ( P ( ( / ( im ( P ( ( / P ( ( ( ( Sice e fucio ( is coiuous a e pois of e sraig ie ( we ave P( P ( Te e coiio of e coiuous iffereiabiiy of ( wi respec o as e form ( ( Le us eermie e paria erivaives k ( ( ( ( ( (( k ( k k ( ( ( / k ( k( k / (( ( k ( k( k( k ( k( ( k / ( k ( ( k / ( ( k ( k( ( k / ( exp (( k ; k ( k( k / ( ( ( ( ( 34 (( k ( ( k( ( k / k ( k( k /

7 (( ( k ( k( k( k ( k( ( k / k ( k ( ( k / ( ( k ( k( ( k / ( ( exp (( k k ( k( k / Subsiuig ese expressios io e coiuous iffereiabiiy coiio yies e expressio ( k( ( k / ( exp (( k k ( k( k / ( k( ( k / ( ( ( ( exp (( k k (( k / ( k ( k( ( k / ( ( exp (( k k ( k( k / wece we obai e equaio ( ( ( ( ( ( ( Subsiuig e expressios ( s( ( ( ( ( io is equaio we obai coiio ( of Teorem ( s ( ( ( ( s( s( ( ( ( ( ( wece i foows a ( is coiuousy iffereiabe a e pois of e sraig ie Le us sow a ( is coiuousy iffereiabe wi respec o a e pois of e sraig ie : im ( P ( ( P ( ( / ( im ( P ( ( P ( ( / ( Te e coiuous iffereiabiiy of ( wi respec o a e pois of e sraig ie foows from e coiios obaie i e proof of e coiuous iffereiabiiy wi respec o : P ( P ( ( ( 35

8 Fiay i foows erefrom a ( C ( is a cassica souio of probem ( (3 Le us sow a e fucio ( is coiuous (see (5 i Represe i as Q( for ( ( Q( for ( Q( ( exp s( ( Q( ( ( ( ( (exp s( ( ( ( / Specify e erivaive of e fucio Q ( wi respec o a : Q ( ( ( s( Q ( ( ( exp s( ( Te erivaive of e fucio Q ( wi respec o a is Q Q s ( ( s( ( exp ( ( s ( ( ( ( ( ( ( ( ( ( ( ( s ( s ( ( s( ( Q ( ( ( ( ( ( ( ( ( ( ( ( ( s ( ( s ( ( Q ( Subsiue Q ( io Eq (4 a e iiia coiio (6 For max Q ( Q ( s( Q( Q ( ( exp( ( If a we obai e equaiy Te bouary coiio (5 yies e ieiy Q ( Q ( s ( Q ( ( Q( ( ( ( exp( ( ( ( we ave e ieiy 36

9 Le us eermie e ( If coiios of Teorem o e e fucio ( from (5 saisfies e coiuous iffereiabiiy coiio a e iera pois of e ses a I e bouary omai of e ierface of a a e pois e fucio ( saisfies e imi reaios im ( Q ( ( exp s ( im ( Q ( ( ( ( exp s ( I foows from e coiio ( ( ( ( a e fucio ( is coiuous a e pois Simiary coiio (5 yies e coiuiy of e fucio ( a e pois Te paria erivaive ( a e bouary pois saisfies e foowig imi reaios: im ( ( s ( s ( ( Q ( s im ( Q ( I foows from e coiios s ( exp ( s ( ( ( ( ( Q ( ( ( ( ( s ( ( s ( s ( ( ( ( ( exp s( ( ( ( ( s ( ( exp ( s ( ( ( ( ( ( ( s ( ( a e erivaive ( is coiuous a e bouary pois Simiary we obai a e paria erivaive ( is coiuous a e bouary pois Te expressios ( ( ( ( ( ( ( ( ( ( ( 37

10 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( for ( s( ( ( ( ( yie e foowig reaios: ( ( ( ( ( ( ( ( ( ( ( ( s ( ( ( Te e coiio wereby ( is coiuousy iffereiabe wi respec o i e bouary omai of e ierface of a a e pois becomes ( ( ( ( ( s( ( ( ( s ( Te paria erivaive (saisfies e imi reaios ( ( im im ( Q ( s ( exp s( ( ( Q ( ( ( ( ( ( ( s ( ( s( ( exp ( s wece i foows a e paria erivaive (is coiuous a e bouary pois uer e same coiios for wic e paria erivaive ( is coiuous Simiary we obai a e paria erivaive (is coiuous a e bouary pois Te eorem is prove Remark I e aaysis of appie bioogica sysems coiios ( a ( may appear icoveie for pracica cacuaios sice ey ivove compex a uwiey maemaica operaios If coiio ( is vioae souio ( ceases o be coiuousy iffereiabe i e omai a e ierface of e omais a a e pois a if coiio ( is vioae souio (5 is o ay more coiuousy iffereiabe i e omai a e ierface of e omais a a e pois However usig e resus of Teorem i is 38

11 case ie e aayic souio of probem ( (8 is possibe a iera pois of e omais a wic is ofe of pracica imporace a eermies e basic pricipes of e yamics of bioogica sysems Beow we wi cosier e exampes of appyig e resus of e eorem o various ypes of ipu parameers wic saisfy a o o saisfy aiioa ierface a smooess coiios ( a ( NUMERICAL ANALYSIS OF THE BEHAVIOR OF MONOCYCLIC CELL AGGREGATION SYSTEM FOR VARIOUS MODEL SCENARIOS Le us cosier e resus of compuer moeig of e yamics of moocycic aggregaio base o moe ( (8 for es parameers of e sysem Irouce a iegra fucio max H ( ( Le us eermie e foowig form for e ea fucios a e sare of iviig ces: ( u ( q exp( s ( s ( c c c were q N a For ese fucios we ave ( q ( k ( q ( k ( k Te we obai e reaio ( k ( q k ( ( k For we ave a e foowig reaios are rue: ( / A ( ( ( exp( A q ( ( ( ( ( ( ( / u k u exp k q For we ave ( ( u ( ( ( exp ( ( / A ( ( ( exp( / A ( ( ( ( 39

12 Te we obai e reaio ( u ( ( ( exp ( q u For we ave ( / ( ( ( exp( A A ( k ( k ( k( ( k / ( ( k / ( k ( ( k / ( ( ( k ( / k ( k / ( ( / ( ( Te rasformaios yie e reaio ( u ( ( ( exp q Le us ow eermie e fucio ( ( Q( for max ( Q ( for were Q( ( exp s( ( c c ( exp ( c c Q( ( ( ( ( (exp s( ( ( exp( ( c exp q c c c c ( ( ( We wi specify e foowig vaues of e parameers of moe ( (8: T max 5 q 75 8 s ( a s ( a cosier various scearios of e eveopme of e sysem Le us cosier e case wic is caracerisic of e iiia coiios of e form sou o 4 ( ( N kexp( ap bp ( M exp( c p For e coiio ( ( ( ( o be saisfie (see (7 e equaiy p p p M ( q ( N k exp( a ( b c

13 H Fig Fucio H (for e case A H Fig Fucio H (for e case B Te coiuiy coiio for e fucio ( as e form b p q ( N k a N For e fucio ( o be coiuousy iffereiabe o e bouary coiio ( becomes a p k k ( N k N Coiio ( of e coiuous iffereiabiiy of e fucio ( a e bouary becomes c p q k ( q N k p ( ap bp Te curves i Figs 4 sow e resus of umerica moeig for is sceario of e eveopme of e bioogica sysem 4

14 H Fig 3 Fucio H (for e case C H Le us cosier e case A for wic N a k If a p a bp ( q we obai o oy e coiuiy of e fucio ( bu aso e ieiy ( ( Te curve of e yamics of e iegra fucio H (for e sceario A is presee i Fig Tis case may escribe a eay sress-ess grow of e usmber of ces i a orgaism or e umber of iiviuas of a moocycic popuaio for exampe uices Le us cosier e cases B a C for wic N 5 a k Teir oy ifferece is a e coiuiy coiio is o saisfie for e fucio ( i e case B Figures a 3 sow e ifferece bewee e scearios Te cases B a C may escribe e yamics of ce aggregaio i a orgaism or e yamics of e eveopme of a moocycic popuaio of uices uer eviromea sress Le us ow cosier a iscoiuous iiia coiio of e form N ( ; ( wi Te sceario of is case is presee i Fig 4 (case D Te grap emosraes a e ce esiy fucio is sarpy wavy Tis case may escribe e yamics of ce aggregaio uer coiios of osmoo eveopme were e orma grow is impossibe because of eviromea sresses 4 Fig 4 Fucio H (for e case D

15 Tus moe ( (6 ca be use o ivesigae various scearios of e eveopme of e esiy of iiviuas of moocycic aggregaios Te souio of is sysem is presee i Eqs ( (6 wece e coiios of is membersip i e cass of coiuousy iffereiabe fucios foow Te auors are graefu o Prof A G Nakoecyi for e iscussios a vauabe feeback REFERENCES Yu V Zagoroiy Simuaio moes of pa grow a eveopme uer favorabe a agerous coiios of ifecious iseases Visyk KNU im Tarasa Sevceka No 7 5 (6 V V Akimeko a A G Nakoecyi Opima coro moes for ierregioa migraio uer socia risk Cyber Sys Aaysis 4 No (6 3 V V Akimeko Simuaio of wo-imesioa raspor processes usig oiear moooe seco-orer scemes Cyber Sys Aaysis 39 No (3 4 V V Akimeko Noiear moooe iger-orer scemes for raspor equaios Z Vyc Maem Ma Fiziki 39 No (999 5 G I Marcuk V P Dymikov a V B Zaesyi Maemaica Moes i Geopysica Hyroyamics a Numerica Meos of eir Impemeaio [i Russia] Giromeeoiza Leigra (987 6 A N Tikoov a A A Samarskii Equaios of Maemaica Pysics [i Russia] Nauka Moscow (4 7 D A Kyusi N I Lyasko a Yu N Oopcuk Maemaica moeig a opimizaio of iraumor rug raspor Cyber Sys Aaysis 43 No (7 8 R A Pouekov (e Dyamic Teory of Bioogica Popuaios [i Russia] Nauka Moscow (974 9 J H M Torey Maemaica Moes of Pa Pysioogy (Experimea Boay Aca Press (976 M M Musieko Pa Pysioogy [i Ukraiia] Fiososioser Kyiv ( 43