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X iajs-2619 11 3 introduction introduction NOUN iajs-2619 11 4 in in ADP iajs-2619 11 5 fact fact NOUN iajs-2619 11 6 , , PUNCT iajs-2619 11 7 the the DET iajs-2619 11 8 infectious infectious ADJ iajs-2619 11 9 diseases disease NOUN iajs-2619 11 10 always always ADV iajs-2619 11 11 have have VERB iajs-2619 11 12 an an DET iajs-2619 11 13 impact impact NOUN iajs-2619 11 14 on on ADP iajs-2619 11 15 people people NOUN iajs-2619 11 16 's 's PART iajs-2619 11 17 health health NOUN iajs-2619 11 18 , , PUNCT iajs-2619 11 19 so so CCONJ iajs-2619 11 20 it it PRON iajs-2619 11 21 is be AUX iajs-2619 11 22 necessary necessary ADJ iajs-2619 11 23 to to PART iajs-2619 11 24 study study VERB iajs-2619 11 25 the the DET iajs-2619 11 26 mechanism mechanism NOUN iajs-2619 11 27 by by ADP iajs-2619 11 28 which which PRON iajs-2619 11 29 the the DET iajs-2619 11 30 disease disease NOUN iajs-2619 11 31 spread spread VERB iajs-2619 11 32 and and CCONJ iajs-2619 11 33 the the DET iajs-2619 11 34 conditions condition NOUN iajs-2619 11 35 of of ADP iajs-2619 11 36 minor minor ADJ iajs-2619 11 37 and and CCONJ iajs-2619 11 38 major major ADJ iajs-2619 11 39 infections infection NOUN iajs-2619 11 40 and and CCONJ iajs-2619 11 41 learn learn VERB iajs-2619 11 42 how how SCONJ iajs-2619 11 43 to to PART iajs-2619 11 44 control control VERB iajs-2619 11 45 diseases disease NOUN iajs-2619 11 46 . . 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PUNCT iajs-2619 15 1 in in ADP iajs-2619 15 2 the the DET iajs-2619 15 3 last last ADJ iajs-2619 15 4 decade decade NOUN iajs-2619 15 5 , , PUNCT iajs-2619 15 6 that that PRON iajs-2619 15 7 was be AUX iajs-2619 15 8 , , PUNCT iajs-2619 15 9 from from ADP iajs-2619 15 10 2010 2010 NUM iajs-2619 15 11 to to ADP iajs-2619 15 12 2020 2020 NUM iajs-2619 15 13 , , PUNCT iajs-2619 15 14 many many ADJ iajs-2619 15 15 epidemics epidemic NOUN iajs-2619 15 16 have have AUX iajs-2619 15 17 spread spread VERB iajs-2619 15 18 , , PUNCT iajs-2619 15 19 most most ADV iajs-2619 15 20 notably notably ADV iajs-2619 15 21 new new ADJ iajs-2619 15 22 types type NOUN iajs-2619 15 23 of of ADP iajs-2619 15 24 influenza influenza NOUN iajs-2619 15 25 such such ADJ iajs-2619 15 26 as as ADP iajs-2619 15 27 the the DET iajs-2619 15 28 middle middle PROPN iajs-2619 15 29 east east PROPN iajs-2619 15 30 corona corona PROPN iajs-2619 15 31 and and CCONJ iajs-2619 15 32 swine swine ADJ iajs-2619 15 33 flu flu NOUN iajs-2619 15 34 , , PUNCT iajs-2619 15 35 and and CCONJ iajs-2619 15 36 preceded precede VERB iajs-2619 15 37 them they PRON iajs-2619 15 38 in in ADP iajs-2619 15 39 the the DET iajs-2619 15 40 previous previous ADJ iajs-2619 15 41 decade decade NOUN iajs-2619 15 42 . . 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PUNCT iajs-2619 17 1 in in ADP iajs-2619 17 2 june june PROPN iajs-2619 17 3 2012 2012 NUM iajs-2619 17 4 , , PUNCT iajs-2619 17 5 a a DET iajs-2619 17 6 report report NOUN iajs-2619 17 7 was be AUX iajs-2619 17 8 published publish VERB iajs-2619 17 9 for for ADP iajs-2619 17 10 a a DET iajs-2619 17 11 study study NOUN iajs-2619 17 12 of of ADP iajs-2619 17 13 a a DET iajs-2619 17 14 group group NOUN iajs-2619 17 15 of of ADP iajs-2619 17 16 doctors doctor NOUN iajs-2619 17 17 , , PUNCT iajs-2619 17 18 researchers researcher NOUN iajs-2619 17 19 and and CCONJ iajs-2619 17 20 agencies agency NOUN iajs-2619 17 21 , , PUNCT iajs-2619 17 22 ibn ibn PROPN iajs-2619 17 23 al al PROPN iajs-2619 17 24 haitham haitham PROPN iajs-2619 17 25 journal journal PROPN iajs-2619 17 26 for for ADP iajs-2619 17 27 pure pure ADJ iajs-2619 17 28 and and CCONJ iajs-2619 17 29 applied apply VERB iajs-2619 17 30 science science NOUN iajs-2619 17 31 journal journal PROPN iajs-2619 17 32 homepage homepage NOUN iajs-2619 17 33 : : PUNCT iajs-2619 17 34 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2619 17 35 doi doi NOUN iajs-2619 17 36 : : PUNCT iajs-2619 17 37 10.30526/34.2.2619 10.30526/34.2.2619 PROPN iajs-2619 17 38 article article NOUN iajs-2619 17 39 history history NOUN iajs-2619 17 40 : : PUNCT iajs-2619 17 41 received receive VERB iajs-2619 17 42 24 24 NUM iajs-2619 17 43 , , PUNCT iajs-2619 17 44 june june PROPN iajs-2619 17 45 , , PUNCT iajs-2619 17 46 2020 2020 NUM iajs-2619 17 47 , , PUNCT iajs-2619 17 48 accepted accept VERB iajs-2619 17 49 20,july,2020 20,july,2020 NOUN iajs-2619 17 50 , , PUNCT iajs-2619 17 51 published publish VERB iajs-2619 17 52 in in ADP iajs-2619 17 53 april april PROPN iajs-2619 17 54 2021 2021 NUM iajs-2619 17 55 mailto:watheq.alasady75@gmail.com mailto:watheq.alasady75@gmail.com X iajs-2619 18 1 mailto:hassan.fadhil.r@sc.uobaghdad.edu.iq mailto:hassan.fadhil.r@sc.uobaghdad.edu.iq PROPN iajs-2619 18 2 130 130 NUM iajs-2619 18 3 ibn ibn PROPN iajs-2619 18 4 al al PROPN iajs-2619 18 5 - - PUNCT iajs-2619 18 6 haitham haitham PROPN iajs-2619 18 7 jour jour X iajs-2619 18 8 . . PROPN iajs-2619 19 1 for for ADP iajs-2619 19 2 pure pure ADJ iajs-2619 19 3 & & CCONJ iajs-2619 19 4 appl appl PROPN iajs-2619 19 5 . . PUNCT iajs-2619 20 1 sci sci PROPN iajs-2619 20 2 . . PROPN iajs-2619 21 1 34 34 NUM iajs-2619 21 2 ( ( PUNCT iajs-2619 21 3 2 2 NUM iajs-2619 21 4 ) ) PUNCT iajs-2619 21 5 2021 2021 NUM iajs-2619 22 1 announcing announce VERB iajs-2619 22 2 the the DET iajs-2619 22 3 death death NOUN iajs-2619 22 4 of of ADP iajs-2619 22 5 280,000 280,000 NUM iajs-2619 22 6 people people NOUN iajs-2619 22 7 , , PUNCT iajs-2619 22 8 while while SCONJ iajs-2619 22 9 in in ADP iajs-2619 22 10 2010 2010 NUM iajs-2619 22 11 , , PUNCT iajs-2619 22 12 the the DET iajs-2619 22 13 world world PROPN iajs-2619 22 14 health health PROPN iajs-2619 22 15 organization organization NOUN iajs-2619 22 16 announced announce VERB iajs-2619 22 17 that that SCONJ iajs-2619 22 18 18,000 18,000 NUM iajs-2619 22 19 people people NOUN iajs-2619 22 20 died die VERB iajs-2619 22 21 as as ADP iajs-2619 22 22 a a DET iajs-2619 22 23 result result NOUN iajs-2619 22 24 of of ADP iajs-2619 22 25 the the DET iajs-2619 22 26 epidemic epidemic NOUN iajs-2619 22 27 . . 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PUNCT iajs-2619 28 1 m. m. NOUN iajs-2619 28 2 al al PROPN iajs-2619 28 3 - - PUNCT iajs-2619 28 4 arydah arydah PROPN iajs-2619 28 5 , , PUNCT iajs-2619 28 6 et et NOUN iajs-2619 28 7 al al PROPN iajs-2619 29 1 [ [ X iajs-2619 29 2 3 3 NUM iajs-2619 29 3 ] ] PUNCT iajs-2619 29 4 studied study VERB iajs-2619 29 5 cholera cholera NOUN iajs-2619 29 6 disease disease NOUN iajs-2619 29 7 with with ADP iajs-2619 29 8 education education NOUN iajs-2619 29 9 and and CCONJ iajs-2619 29 10 chlorination chlorination NOUN iajs-2619 29 11 . . PUNCT iajs-2619 30 1 f.j f.j NOUN iajs-2619 30 2 . . PROPN iajs-2619 30 3 luquero luquero NOUN iajs-2619 30 4 , , PUNCT iajs-2619 30 5 et et NOUN iajs-2619 30 6 al al PROPN iajs-2619 31 1 [ [ X iajs-2619 31 2 4 4 X iajs-2619 31 3 ] ] PUNCT iajs-2619 31 4 used use VERB iajs-2619 31 5 vibrio vibrio ADJ iajs-2619 31 6 cholera cholera NOUN iajs-2619 31 7 vaccine vaccine NOUN iajs-2619 31 8 in in ADP iajs-2619 31 9 an an DET iajs-2619 31 10 outbreak outbreak NOUN iajs-2619 31 11 in in ADP iajs-2619 31 12 guinea guinea NOUN iajs-2619 31 13 . . PUNCT iajs-2619 32 1 we we PRON iajs-2619 32 2 depend depend VERB iajs-2619 32 3 here here ADV iajs-2619 32 4 on on ADP iajs-2619 32 5 one one NUM iajs-2619 32 6 from from ADP iajs-2619 32 7 the the DET iajs-2619 32 8 basic basic ADJ iajs-2619 32 9 modeling modeling NOUN iajs-2619 32 10 to to PART iajs-2619 32 11 study study VERB iajs-2619 32 12 and and CCONJ iajs-2619 32 13 analyze analyze VERB iajs-2619 32 14 the the DET iajs-2619 32 15 infection infection NOUN iajs-2619 32 16 disease disease NOUN iajs-2619 32 17 which which PRON iajs-2619 32 18 is be AUX iajs-2619 32 19 the the DET iajs-2619 32 20 sir sir NOUN iajs-2619 32 21 model model NOUN iajs-2619 32 22 of of ADP iajs-2619 32 23 kermack kermack NOUN iajs-2619 32 24 and and CCONJ iajs-2619 32 25 mckendrick mckendrick NOUN iajs-2619 32 26 [ [ X iajs-2619 32 27 5 5 NUM iajs-2619 32 28 - - SYM iajs-2619 32 29 7 7 NUM iajs-2619 32 30 . . PUNCT iajs-2619 33 1 the the DET iajs-2619 33 2 mathematical mathematical ADJ iajs-2619 33 3 modeling modeling NOUN iajs-2619 33 4 is be AUX iajs-2619 33 5 an an DET iajs-2619 33 6 important important ADJ iajs-2619 33 7 interdisciplinary interdisciplinary ADJ iajs-2619 33 8 activity activity NOUN iajs-2619 33 9 in in ADP iajs-2619 33 10 the the DET iajs-2619 33 11 study study NOUN iajs-2619 33 12 of of ADP iajs-2619 33 13 some some DET iajs-2619 33 14 aspects aspect NOUN iajs-2619 33 15 of of ADP iajs-2619 33 16 various various ADJ iajs-2619 33 17 disciplines discipline NOUN iajs-2619 33 18 . . PUNCT iajs-2619 34 1 bailey bailey PROPN iajs-2619 35 1 [ [ X iajs-2619 35 2 8 8 NUM iajs-2619 35 3 ] ] PUNCT iajs-2619 35 4 and and CCONJ iajs-2619 35 5 several several ADJ iajs-2619 35 6 authors author NOUN iajs-2619 35 7 provided provide VERB iajs-2619 35 8 many many ADJ iajs-2619 35 9 models model NOUN iajs-2619 35 10 for for ADP iajs-2619 35 11 the the DET iajs-2619 35 12 spread spread NOUN iajs-2619 35 13 and and CCONJ iajs-2619 35 14 control control NOUN iajs-2619 35 15 of of ADP iajs-2619 35 16 infectious infectious ADJ iajs-2619 35 17 diseases disease NOUN iajs-2619 35 18 [ [ X iajs-2619 35 19 9 9 NUM iajs-2619 35 20 - - SYM iajs-2619 35 21 12 12 NUM iajs-2619 35 22 ] ] PUNCT iajs-2619 35 23 . . PUNCT iajs-2619 36 1 methods method NOUN iajs-2619 36 2 used use VERB iajs-2619 36 3 to to PART iajs-2619 36 4 control control VERB iajs-2619 36 5 epidemics epidemic NOUN iajs-2619 36 6 are be AUX iajs-2619 36 7 outreach outreach NOUN iajs-2619 36 8 programs program NOUN iajs-2619 36 9 driven drive VERB iajs-2619 36 10 by by ADP iajs-2619 36 11 the the DET iajs-2619 36 12 media medium NOUN iajs-2619 36 13 that that PRON iajs-2619 36 14 can can AUX iajs-2619 36 15 modify modify VERB iajs-2619 36 16 the the DET iajs-2619 36 17 behavior behavior NOUN iajs-2619 36 18 human human ADJ iajs-2619 36 19 towards towards ADP iajs-2619 36 20 the the DET iajs-2619 36 21 disease disease NOUN iajs-2619 36 22 . . PUNCT iajs-2619 37 1 these these DET iajs-2619 37 2 awareness awareness NOUN iajs-2619 37 3 campaigns campaign NOUN iajs-2619 37 4 may may AUX iajs-2619 37 5 differ differ VERB iajs-2619 37 6 between between ADP iajs-2619 37 7 the the DET iajs-2619 37 8 various various ADJ iajs-2619 37 9 groups group NOUN iajs-2619 37 10 at at ADP iajs-2619 37 11 risk risk NOUN iajs-2619 37 12 that that PRON iajs-2619 37 13 help help VERB iajs-2619 37 14 limit limit VERB iajs-2619 37 15 the the DET iajs-2619 37 16 spread spread NOUN iajs-2619 37 17 of of ADP iajs-2619 37 18 infection infection NOUN iajs-2619 37 19 . . PUNCT iajs-2619 38 1 we we PRON iajs-2619 38 2 would would AUX iajs-2619 38 3 like like VERB iajs-2619 38 4 to to PART iajs-2619 38 5 mention mention VERB iajs-2619 38 6 those those PRON iajs-2619 38 7 who who PRON iajs-2619 38 8 have have VERB iajs-2619 38 9 a a DET iajs-2619 38 10 clear clear ADJ iajs-2619 38 11 evaluation evaluation NOUN iajs-2619 38 12 of of ADP iajs-2619 38 13 the the DET iajs-2619 38 14 impact impact NOUN iajs-2619 38 15 of of ADP iajs-2619 38 16 awareness awareness NOUN iajs-2619 38 17 programs program NOUN iajs-2619 38 18 [ [ X iajs-2619 38 19 13 13 NUM iajs-2619 38 20 - - SYM iajs-2619 38 21 15 15 NUM iajs-2619 38 22 ] ] PUNCT iajs-2619 38 23 . . PUNCT iajs-2619 39 1 there there PRON iajs-2619 39 2 are be VERB iajs-2619 39 3 studies study NOUN iajs-2619 39 4 on on ADP iajs-2619 39 5 the the DET iajs-2619 39 6 effect effect NOUN iajs-2619 39 7 of of ADP iajs-2619 39 8 media medium NOUN iajs-2619 39 9 and and CCONJ iajs-2619 39 10 the the DET iajs-2619 39 11 effectiveness effectiveness NOUN iajs-2619 39 12 of of ADP iajs-2619 39 13 using use VERB iajs-2619 39 14 face face NOUN iajs-2619 39 15 masks mask NOUN iajs-2619 39 16 to to PART iajs-2619 39 17 reduce reduce VERB iajs-2619 39 18 the the DET iajs-2619 39 19 spread spread NOUN iajs-2619 39 20 of of ADP iajs-2619 39 21 the the DET iajs-2619 39 22 influenza influenza NOUN iajs-2619 39 23 epidemic epidemic NOUN iajs-2619 39 24 [ [ X iajs-2619 39 25 16 16 NUM iajs-2619 39 26 - - SYM iajs-2619 39 27 17 17 NUM iajs-2619 39 28 ] ] PUNCT iajs-2619 39 29 . . PUNCT iajs-2619 40 1 misra misra PROPN iajs-2619 41 1 [ [ X iajs-2619 41 2 18 18 NUM iajs-2619 41 3 ] ] PUNCT iajs-2619 41 4 presented present VERB iajs-2619 41 5 a a DET iajs-2619 41 6 non non ADJ iajs-2619 41 7 - - ADJ iajs-2619 41 8 linear linear ADJ iajs-2619 41 9 mathematical mathematical ADJ iajs-2619 41 10 model model NOUN iajs-2619 41 11 that that PRON iajs-2619 41 12 evaluated evaluate VERB iajs-2619 41 13 awareness awareness NOUN iajs-2619 41 14 programs program NOUN iajs-2619 41 15 and and CCONJ iajs-2619 41 16 showed show VERB iajs-2619 41 17 their their PRON iajs-2619 41 18 control control NOUN iajs-2619 41 19 that that SCONJ iajs-2619 41 20 the the DET iajs-2619 41 21 outbreak outbreak NOUN iajs-2619 41 22 of of ADP iajs-2619 41 23 infectious infectious ADJ iajs-2619 41 24 diseases disease NOUN iajs-2619 41 25 can can AUX iajs-2619 41 26 be be AUX iajs-2619 41 27 reduced reduce VERB iajs-2619 41 28 through through ADP iajs-2619 41 29 media medium NOUN iajs-2619 41 30 coverage coverage NOUN iajs-2619 41 31 . . PUNCT iajs-2619 42 1 in in ADP iajs-2619 42 2 this this DET iajs-2619 42 3 work work NOUN iajs-2619 42 4 , , PUNCT iajs-2619 42 5 we we PRON iajs-2619 42 6 have have AUX iajs-2619 42 7 proposed propose VERB iajs-2619 42 8 the the DET iajs-2619 42 9 mathematical mathematical ADJ iajs-2619 42 10 model model NOUN iajs-2619 42 11 of of ADP iajs-2619 42 12 the the DET iajs-2619 42 13 shigella shigella PROPN iajs-2619 42 14 disease disease NOUN iajs-2619 42 15 involving involve VERB iajs-2619 42 16 media medium NOUN iajs-2619 42 17 programs program NOUN iajs-2619 42 18 effect effect VERB iajs-2619 42 19 to to PART iajs-2619 42 20 control control VERB iajs-2619 42 21 the the DET iajs-2619 42 22 spread spread NOUN iajs-2619 42 23 the the DET iajs-2619 42 24 disease disease NOUN iajs-2619 42 25 . . PUNCT iajs-2619 43 1 we we PRON iajs-2619 43 2 displayed display VERB iajs-2619 43 3 the the DET iajs-2619 43 4 full full ADJ iajs-2619 43 5 details detail NOUN iajs-2619 43 6 of of ADP iajs-2619 43 7 the the DET iajs-2619 43 8 mathematical mathematical ADJ iajs-2619 43 9 modeling modeling NOUN iajs-2619 43 10 of of ADP iajs-2619 43 11 the the DET iajs-2619 43 12 shigella shigella PROPN iajs-2619 43 13 disease disease NOUN iajs-2619 43 14 in in ADP iajs-2619 43 15 section section NOUN iajs-2619 43 16 2 2 NUM iajs-2619 43 17 . . PUNCT iajs-2619 44 1 we we PRON iajs-2619 44 2 have have AUX iajs-2619 44 3 also also ADV iajs-2619 44 4 discussed discuss VERB iajs-2619 44 5 some some DET iajs-2619 44 6 basic basic ADJ iajs-2619 44 7 properties property NOUN iajs-2619 44 8 equilibrium equilibrium NOUN iajs-2619 44 9 points point NOUN iajs-2619 44 10 in in ADP iajs-2619 44 11 section section NOUN iajs-2619 44 12 3 3 NUM iajs-2619 44 13 . . PUNCT iajs-2619 45 1 in in ADP iajs-2619 45 2 section section NOUN iajs-2619 45 3 4 4 NUM iajs-2619 45 4 the the DET iajs-2619 45 5 local local ADJ iajs-2619 45 6 stability stability NOUN iajs-2619 45 7 analysis analysis NOUN iajs-2619 45 8 was be AUX iajs-2619 45 9 studied study VERB iajs-2619 45 10 with with ADP iajs-2619 45 11 the the DET iajs-2619 45 12 support support NOUN iajs-2619 45 13 of of ADP iajs-2619 45 14 gresgorin gresgorin PROPN iajs-2619 45 15 theorem theorem PROPN iajs-2619 45 16 . . PUNCT iajs-2619 46 1 furthermore furthermore ADV iajs-2619 46 2 , , PUNCT iajs-2619 46 3 we we PRON iajs-2619 46 4 studied study VERB iajs-2619 46 5 the the DET iajs-2619 46 6 using using NOUN iajs-2619 46 7 of of ADP iajs-2619 46 8 lyapunov lyapunov ADJ iajs-2619 46 9 function function NOUN iajs-2619 46 10 to to PART iajs-2619 46 11 show show VERB iajs-2619 46 12 the the DET iajs-2619 46 13 global global ADJ iajs-2619 46 14 stability stability NOUN iajs-2619 46 15 of of ADP iajs-2619 46 16 the the DET iajs-2619 46 17 proposed propose VERB iajs-2619 46 18 model model NOUN iajs-2619 46 19 at at ADP iajs-2619 46 20 all all DET iajs-2619 46 21 equilibrium equilibrium NOUN iajs-2619 46 22 points point NOUN iajs-2619 46 23 in in ADP iajs-2619 46 24 section section NOUN iajs-2619 46 25 5 5 NUM iajs-2619 46 26 . . PUNCT iajs-2619 47 1 finally finally ADV iajs-2619 47 2 , , PUNCT iajs-2619 47 3 the the DET iajs-2619 47 4 effect effect NOUN iajs-2619 47 5 of of ADP iajs-2619 47 6 media medium NOUN iajs-2619 47 7 coverage coverage NOUN iajs-2619 47 8 have have AUX iajs-2619 47 9 been be AUX iajs-2619 47 10 done do VERB iajs-2619 47 11 to to ADP iajs-2619 47 12 awareness awareness NOUN iajs-2619 47 13 for for ADP iajs-2619 47 14 the the DET iajs-2619 47 15 shigella shigella PROPN iajs-2619 47 16 disease disease NOUN iajs-2619 47 17 and and CCONJ iajs-2619 47 18 also also ADV iajs-2619 47 19 the the DET iajs-2619 47 20 risk risk NOUN iajs-2619 47 21 of of ADP iajs-2619 47 22 direct direct ADJ iajs-2619 47 23 and and CCONJ iajs-2619 47 24 indirect indirect ADJ iajs-2619 47 25 contact contact NOUN iajs-2619 47 26 with with ADP iajs-2619 47 27 carrier carrier NOUN iajs-2619 47 28 individuals individual NOUN iajs-2619 47 29 on on ADP iajs-2619 47 30 the the DET iajs-2619 47 31 out out ADJ iajs-2619 47 32 breaking breaking NOUN iajs-2619 47 33 of of ADP iajs-2619 47 34 the the DET iajs-2619 47 35 shigella shigella PROPN iajs-2619 47 36 disease disease NOUN iajs-2619 47 37 in in ADP iajs-2619 47 38 the the DET iajs-2619 47 39 population population NOUN iajs-2619 47 40 . . PUNCT iajs-2619 48 1 this this PRON iajs-2619 48 2 was be AUX iajs-2619 48 3 done do VERB iajs-2619 48 4 through through ADP iajs-2619 48 5 a a DET iajs-2619 48 6 numerical numerical ADJ iajs-2619 48 7 simulation simulation NOUN iajs-2619 48 8 . . PUNCT iajs-2619 49 1 a a DET iajs-2619 49 2 discussion discussion NOUN iajs-2619 49 3 of of ADP iajs-2619 49 4 the the DET iajs-2619 49 5 results result NOUN iajs-2619 49 6 effects effect NOUN iajs-2619 49 7 and and CCONJ iajs-2619 49 8 limitations limitation NOUN iajs-2619 49 9 involved involve VERB iajs-2619 49 10 were be AUX iajs-2619 49 11 concluded conclude VERB iajs-2619 49 12 in in ADP iajs-2619 49 13 this this DET iajs-2619 49 14 paper paper NOUN iajs-2619 49 15 . . PUNCT iajs-2619 50 1 2 2 X iajs-2619 50 2 . . X iajs-2619 50 3 model model NOUN iajs-2619 50 4 formulation formulation NOUN iajs-2619 50 5 : : PUNCT iajs-2619 50 6 in in ADP iajs-2619 50 7 this this DET iajs-2619 50 8 section section NOUN iajs-2619 50 9 , , PUNCT iajs-2619 50 10 the the DET iajs-2619 50 11 population population NOUN iajs-2619 50 12 in in ADP iajs-2619 50 13 this this DET iajs-2619 50 14 work work NOUN iajs-2619 50 15 can can AUX iajs-2619 50 16 divided divide VERB iajs-2619 50 17 into into ADP iajs-2619 50 18 to to ADP iajs-2619 50 19 four four NUM iajs-2619 50 20 classes class NOUN iajs-2619 50 21 , , PUNCT iajs-2619 50 22 namely namely ADV iajs-2619 50 23 unaware unaware ADJ iajs-2619 50 24 susceptible susceptible ADJ iajs-2619 50 25 class class NOUN iajs-2619 50 26 denoted denote VERB iajs-2619 50 27 by by ADP iajs-2619 50 28 ( ( PUNCT iajs-2619 50 29 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 50 30 ) ) PUNCT iajs-2619 50 31 , , PUNCT iajs-2619 50 32 infected infected ADJ iajs-2619 50 33 class class NOUN iajs-2619 50 34 denoted denote VERB iajs-2619 50 35 by by ADP iajs-2619 50 36 ( ( PUNCT iajs-2619 50 37 𝐼 𝐼 PROPN iajs-2619 50 38 ) ) PUNCT iajs-2619 50 39 , , PUNCT iajs-2619 50 40 aware aware ADJ iajs-2619 50 41 susceptible susceptible ADJ iajs-2619 50 42 denoted denote VERB iajs-2619 50 43 by by ADP iajs-2619 50 44 ( ( PUNCT iajs-2619 50 45 π‘†π‘Ž π‘†π‘Ž NOUN iajs-2619 50 46 ) ) PUNCT iajs-2619 50 47 , , PUNCT iajs-2619 50 48 people people NOUN iajs-2619 50 49 carrying carry VERB iajs-2619 50 50 bacteria bacteria NOUN iajs-2619 50 51 denoted denote VERB iajs-2619 50 52 by by ADP iajs-2619 50 53 ( ( PUNCT iajs-2619 50 54 𝐡 𝐡 NOUN iajs-2619 50 55 ) ) PUNCT iajs-2619 50 56 and and CCONJ iajs-2619 50 57 the the DET iajs-2619 50 58 number number NOUN iajs-2619 50 59 of of ADP iajs-2619 50 60 media medium NOUN iajs-2619 50 61 campaigns campaign NOUN iajs-2619 50 62 in in ADP iajs-2619 50 63 that that DET iajs-2619 50 64 region region NOUN iajs-2619 50 65 at at ADP iajs-2619 50 66 time time NOUN iajs-2619 50 67 t t PROPN iajs-2619 50 68 denoted denote VERB iajs-2619 50 69 by by ADP iajs-2619 50 70 ( ( PUNCT iajs-2619 50 71 𝑀 𝑀 PROPN iajs-2619 50 72 ) ) PUNCT iajs-2619 50 73 . . PUNCT iajs-2619 51 1 the the DET iajs-2619 51 2 model model NOUN iajs-2619 51 3 is be AUX iajs-2619 51 4 given give VERB iajs-2619 51 5 by by ADP iajs-2619 51 6 the the DET iajs-2619 51 7 following follow VERB iajs-2619 51 8 differential differential ADJ iajs-2619 51 9 equation equation NOUN iajs-2619 51 10 . . PUNCT iajs-2619 52 1 131 131 NUM iajs-2619 52 2 ibn ibn PROPN iajs-2619 52 3 al al PROPN iajs-2619 52 4 - - PUNCT iajs-2619 52 5 haitham haitham PROPN iajs-2619 52 6 jour jour X iajs-2619 52 7 . . PROPN iajs-2619 52 8 for for ADP iajs-2619 52 9 pure pure ADJ iajs-2619 52 10 & & CCONJ iajs-2619 52 11 appl appl PROPN iajs-2619 52 12 . . PUNCT iajs-2619 53 1 sci sci PROPN iajs-2619 53 2 . . PROPN iajs-2619 54 1 34 34 NUM iajs-2619 54 2 ( ( PUNCT iajs-2619 54 3 2 2 NUM iajs-2619 54 4 ) ) PUNCT iajs-2619 54 5 2021 2021 NUM iajs-2619 54 6 𝑆 𝑆 PROPN iajs-2619 54 7 οΏ½ οΏ½ PROPN iajs-2619 54 8 Μ‡ Μ‡ PROPN iajs-2619 54 9 οΏ½ οΏ½ PROPN iajs-2619 54 10 = = PUNCT iajs-2619 54 11 πœƒ πœƒ PROPN iajs-2619 54 12 βˆ’ βˆ’ NOUN iajs-2619 54 13 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 54 14 βˆ’ βˆ’ PROPN iajs-2619 54 15 π‘Ž2π‘†π‘’π‘€βˆ’ π‘Ž2π‘†π‘’π‘€βˆ’ PRON iajs-2619 54 16 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 54 17 βˆ’ βˆ’ ADP iajs-2619 54 18 π‘Ž4𝑆𝑒 π‘Ž4𝑆𝑒 NOUN iajs-2619 54 19 βˆ’ βˆ’ ADP iajs-2619 54 20 πœ‡π‘†π‘’ πœ‡π‘†π‘’ PRON iajs-2619 54 21 𝐼̇ 𝐼̇ PROPN iajs-2619 54 22 = = SYM iajs-2619 54 23 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 54 24 + + CCONJ iajs-2619 54 25 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 54 26 + + NUM iajs-2619 54 27 π‘Ž4𝑆𝑒 π‘Ž4𝑆𝑒 NOUN iajs-2619 54 28 βˆ’ βˆ’ PROPN iajs-2619 54 29 ( ( PUNCT iajs-2619 54 30 πœ‡ πœ‡ ADP iajs-2619 54 31 + + X iajs-2619 54 32 𝑑 𝑑 PROPN iajs-2619 54 33 + + CCONJ iajs-2619 54 34 πœ‚)𝐼 πœ‚)𝐼 ADJ iajs-2619 54 35 𝑆 𝑆 PROPN iajs-2619 54 36 οΏ½ οΏ½ PROPN iajs-2619 54 37 Μ‡ Μ‡ VERB iajs-2619 54 38 οΏ½ οΏ½ PROPN iajs-2619 54 39 = = SYM iajs-2619 54 40 π‘Ž2𝑆𝑒𝑀 π‘Ž2𝑆𝑒𝑀 PROPN iajs-2619 54 41 βˆ’ βˆ’ PROPN iajs-2619 54 42 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž PUNCT iajs-2619 54 43 οΏ½ οΏ½ NOUN iajs-2619 54 44 Μ‡ Μ‡ VERB iajs-2619 54 45 οΏ½ οΏ½ PROPN iajs-2619 54 46 = = PUNCT iajs-2619 54 47 π‘Ÿπ΅ π‘Ÿπ΅ PROPN iajs-2619 54 48 ( ( PUNCT iajs-2619 54 49 1 1 NUM iajs-2619 54 50 βˆ’ βˆ’ PROPN iajs-2619 54 51 𝐡 𝐡 PROPN iajs-2619 54 52 𝐾 𝐾 PROPN iajs-2619 54 53 ) ) PUNCT iajs-2619 54 54 βˆ’ βˆ’ ADP iajs-2619 55 1 πœ‡π΅ πœ‡π΅ VERB iajs-2619 55 2 + + CCONJ iajs-2619 55 3 πœ–πΌ πœ–πΌ ADJ iajs-2619 55 4 οΏ½ οΏ½ NOUN iajs-2619 55 5 Μ‡ Μ‡ NOUN iajs-2619 55 6 οΏ½ οΏ½ PROPN iajs-2619 55 7 = = SYM iajs-2619 55 8 𝛼(𝑆𝑒 𝛼(𝑆𝑒 PROPN iajs-2619 55 9 + + CCONJ iajs-2619 55 10 𝐼 𝐼 PROPN iajs-2619 55 11 ) ) PUNCT iajs-2619 55 12 βˆ’ βˆ’ PROPN iajs-2619 56 1 𝛾𝑀 𝛾𝑀 ADJ iajs-2619 56 2 ( ( PUNCT iajs-2619 56 3 1 1 NUM iajs-2619 56 4 ) ) PUNCT iajs-2619 56 5 with with ADP iajs-2619 56 6 initial initial ADJ iajs-2619 56 7 conditions condition NOUN iajs-2619 56 8 𝑆𝑒(0 𝑆𝑒(0 NOUN iajs-2619 56 9 ) ) PUNCT iajs-2619 56 10 > > X iajs-2619 56 11 0 0 NUM iajs-2619 56 12 , , PUNCT iajs-2619 56 13 𝐼(0 𝐼(0 X iajs-2619 56 14 ) ) PUNCT iajs-2619 56 15 β‰₯ β‰₯ NOUN iajs-2619 56 16 0 0 NUM iajs-2619 56 17 , , PUNCT iajs-2619 56 18 π‘†π‘Ž(0 π‘†π‘Ž(0 ADJ iajs-2619 56 19 ) ) PUNCT iajs-2619 56 20 β‰₯ β‰₯ NOUN iajs-2619 56 21 0 0 NUM iajs-2619 56 22 , , PUNCT iajs-2619 56 23 𝐡(0 𝐡(0 NUM iajs-2619 56 24 ) ) PUNCT iajs-2619 56 25 β‰₯ β‰₯ NOUN iajs-2619 56 26 0,𝑀(0 0,𝑀(0 NUM iajs-2619 56 27 ) ) PUNCT iajs-2619 56 28 β‰₯ β‰₯ NOUN iajs-2619 56 29 0 0 NUM iajs-2619 56 30 . . PUNCT iajs-2619 57 1 the the DET iajs-2619 57 2 natural natural ADJ iajs-2619 57 3 birth birth NOUN iajs-2619 57 4 into into ADP iajs-2619 57 5 susceptible susceptible ADJ iajs-2619 57 6 class class NOUN iajs-2619 57 7 by by ADP iajs-2619 57 8 πœƒ πœƒ X iajs-2619 57 9 > > X iajs-2619 57 10 0 0 X iajs-2619 57 11 . . PUNCT iajs-2619 58 1 π‘Ž1 π‘Ž1 INTJ iajs-2619 58 2 > > X iajs-2619 58 3 0 0 NUM iajs-2619 58 4 is be AUX iajs-2619 58 5 the the DET iajs-2619 58 6 incidence incidence NOUN iajs-2619 58 7 of of ADP iajs-2619 58 8 the the DET iajs-2619 58 9 disease disease NOUN iajs-2619 58 10 by by ADP iajs-2619 58 11 direct direct ADJ iajs-2619 58 12 contact contact NOUN iajs-2619 58 13 with with ADP iajs-2619 58 14 the the DET iajs-2619 58 15 carriers carrier NOUN iajs-2619 58 16 of of ADP iajs-2619 58 17 the the DET iajs-2619 58 18 bacteria bacteria NOUN iajs-2619 58 19 . . PUNCT iajs-2619 59 1 π‘Ž2 π‘Ž2 NOUN iajs-2619 59 2 > > X iajs-2619 59 3 0 0 PUNCT iajs-2619 59 4 is be AUX iajs-2619 59 5 the the DET iajs-2619 59 6 awareness awareness NOUN iajs-2619 59 7 rate rate NOUN iajs-2619 59 8 . . PUNCT iajs-2619 60 1 π‘Ž3 π‘Ž3 PROPN iajs-2619 60 2 > > X iajs-2619 60 3 0 0 PUNCT iajs-2619 61 1 is be AUX iajs-2619 61 2 the the DET iajs-2619 61 3 contact contact NOUN iajs-2619 61 4 rate rate NOUN iajs-2619 61 5 between between ADP iajs-2619 61 6 susceptible susceptible ADJ iajs-2619 61 7 and and CCONJ iajs-2619 61 8 infected infected ADJ iajs-2619 61 9 . . PUNCT iajs-2619 62 1 π‘Ž4 π‘Ž4 PROPN iajs-2619 62 2 β‰₯ β‰₯ NOUN iajs-2619 62 3 0 0 NUM iajs-2619 62 4 represents represent VERB iajs-2619 62 5 the the DET iajs-2619 62 6 number number NOUN iajs-2619 62 7 of of ADP iajs-2619 62 8 cases case NOUN iajs-2619 62 9 of of ADP iajs-2619 62 10 the the DET iajs-2619 62 11 disease disease NOUN iajs-2619 62 12 due due ADP iajs-2619 62 13 to to ADP iajs-2619 62 14 the the DET iajs-2619 62 15 external external ADJ iajs-2619 62 16 sources source NOUN iajs-2619 62 17 such such ADJ iajs-2619 62 18 as as ADP iajs-2619 62 19 ( ( PUNCT iajs-2619 62 20 food food NOUN iajs-2619 62 21 , , PUNCT iajs-2619 62 22 water, water, PROPN iajs-2619 62 23 … … SYM iajs-2619 62 24 etc etc X iajs-2619 62 25 ) ) PUNCT iajs-2619 62 26 . . PUNCT iajs-2619 63 1 πœ‡ πœ‡ ADP iajs-2619 63 2 > > X iajs-2619 63 3 0 0 NUM iajs-2619 63 4 is be AUX iajs-2619 63 5 the the DET iajs-2619 63 6 natural natural ADJ iajs-2619 63 7 death death NOUN iajs-2619 63 8 rate rate NOUN iajs-2619 63 9 of of ADP iajs-2619 63 10 the the PRON iajs-2619 63 11 in in ADP iajs-2619 63 12 population population NOUN iajs-2619 63 13 . . PUNCT iajs-2619 64 1 𝑑 𝑑 PROPN iajs-2619 64 2 > > X iajs-2619 64 3 0 0 NUM iajs-2619 64 4 is be AUX iajs-2619 64 5 the the DET iajs-2619 64 6 disease disease NOUN iajs-2619 64 7 related relate VERB iajs-2619 64 8 death death NOUN iajs-2619 64 9 . . PUNCT iajs-2619 65 1 πœ‚ πœ‚ X iajs-2619 65 2 > > SYM iajs-2619 65 3 0 0 NUM iajs-2619 65 4 is be AUX iajs-2619 65 5 the the DET iajs-2619 65 6 removal removal NOUN iajs-2619 65 7 rate rate NOUN iajs-2619 65 8 . . PUNCT iajs-2619 66 1 π‘Ÿ π‘Ÿ NOUN iajs-2619 66 2 and and CCONJ iajs-2619 66 3 π‘˜ π‘˜ PROPN iajs-2619 66 4 are be AUX iajs-2619 66 5 respectively respectively ADV iajs-2619 66 6 , , PUNCT iajs-2619 66 7 the the DET iajs-2619 66 8 growth growth NOUN iajs-2619 66 9 rate rate NOUN iajs-2619 66 10 carrying carry VERB iajs-2619 66 11 capacity capacity NOUN iajs-2619 66 12 of of ADP iajs-2619 66 13 shigella shigella PROPN iajs-2619 66 14 disease disease NOUN iajs-2619 66 15 . . PUNCT iajs-2619 67 1 πœ– πœ– X iajs-2619 67 2 > > X iajs-2619 67 3 0 0 NUM iajs-2619 67 4 is be AUX iajs-2619 67 5 the the DET iajs-2619 67 6 increasing increasing NOUN iajs-2619 67 7 of of ADP iajs-2619 67 8 the the DET iajs-2619 67 9 shigella shigella PROPN iajs-2619 67 10 disease disease NOUN iajs-2619 67 11 bacteria bacteria NOUN iajs-2619 67 12 due due ADP iajs-2619 67 13 to to ADP iajs-2619 67 14 infected infected ADJ iajs-2619 67 15 class class NOUN iajs-2619 67 16 . . PUNCT iajs-2619 68 1 𝛼 𝛼 PRON iajs-2619 68 2 > > X iajs-2619 68 3 0 0 PUNCT iajs-2619 68 4 represents represent VERB iajs-2619 68 5 the the DET iajs-2619 68 6 implementation implementation NOUN iajs-2619 68 7 rate rate NOUN iajs-2619 68 8 of of ADP iajs-2619 68 9 awareness awareness NOUN iajs-2619 68 10 programs program NOUN iajs-2619 68 11 which which PRON iajs-2619 68 12 is be AUX iajs-2619 68 13 proportional proportional ADJ iajs-2619 68 14 to to ADP iajs-2619 68 15 the the DET iajs-2619 68 16 number number NOUN iajs-2619 68 17 of of ADP iajs-2619 68 18 unaware unaware ADJ iajs-2619 68 19 susceptible susceptible ADJ iajs-2619 68 20 and and CCONJ iajs-2619 68 21 infective infective ADJ iajs-2619 68 22 individuals individual NOUN iajs-2619 68 23 in in ADP iajs-2619 68 24 the the DET iajs-2619 68 25 population population NOUN iajs-2619 68 26 . . PUNCT iajs-2619 69 1 finally finally ADV iajs-2619 69 2 the the DET iajs-2619 69 3 depletion depletion NOUN iajs-2619 69 4 rate rate NOUN iajs-2619 69 5 of of ADP iajs-2619 69 6 awareness awareness NOUN iajs-2619 69 7 programs program NOUN iajs-2619 69 8 due due ADP iajs-2619 69 9 to to ADP iajs-2619 69 10 ineffectiveness ineffectiveness NOUN iajs-2619 69 11 , , PUNCT iajs-2619 69 12 social social ADJ iajs-2619 69 13 problems problem NOUN iajs-2619 69 14 in in ADP iajs-2619 69 15 the the DET iajs-2619 69 16 population population NOUN iajs-2619 69 17 and and CCONJ iajs-2619 69 18 similar similar ADJ iajs-2619 69 19 factors factor NOUN iajs-2619 69 20 is be AUX iajs-2619 69 21 represented represent VERB iajs-2619 69 22 by by ADP iajs-2619 69 23 𝛾 𝛾 X iajs-2619 69 24 > > X iajs-2619 69 25 0 0 X iajs-2619 69 26 . . PUNCT iajs-2619 69 27 theorem theorem NOUN iajs-2619 69 28 ( ( PUNCT iajs-2619 69 29 1 1 NUM iajs-2619 69 30 ): ): PUNCT iajs-2619 69 31 the the DET iajs-2619 69 32 uniformly uniformly ADV iajs-2619 69 33 bounded bound VERB iajs-2619 69 34 of of ADP iajs-2619 69 35 the the DET iajs-2619 69 36 any any DET iajs-2619 69 37 solutions solution NOUN iajs-2619 69 38 are be AUX iajs-2619 69 39 discussed discuss VERB iajs-2619 69 40 in in ADP iajs-2619 69 41 the the DET iajs-2619 69 42 following following NOUN iajs-2619 69 43 . . PUNCT iajs-2619 70 1 proof proof NOUN iajs-2619 70 2 : : PUNCT iajs-2619 70 3 let let VERB iajs-2619 70 4 ( ( PUNCT iajs-2619 70 5 𝑆𝑒(𝑑 𝑆𝑒(𝑑 NOUN iajs-2619 70 6 ) ) PUNCT iajs-2619 70 7 , , PUNCT iajs-2619 70 8 𝐼(𝑑 𝐼(𝑑 NUM iajs-2619 70 9 ) ) PUNCT iajs-2619 70 10 , , PUNCT iajs-2619 70 11 π‘†π‘Ž(𝑑 π‘†π‘Ž(𝑑 NOUN iajs-2619 70 12 ) ) PUNCT iajs-2619 70 13 , , PUNCT iajs-2619 70 14 𝐡(𝑑),𝑀(𝑑 𝐡(𝑑),𝑀(𝑑 NOUN iajs-2619 70 15 ) ) PUNCT iajs-2619 70 16 ) ) PUNCT iajs-2619 70 17 is be AUX iajs-2619 70 18 the the DET iajs-2619 70 19 solution solution NOUN iajs-2619 70 20 of of ADP iajs-2619 70 21 the the DET iajs-2619 70 22 system system NOUN iajs-2619 70 23 ( ( PUNCT iajs-2619 70 24 1 1 NUM iajs-2619 70 25 ) ) PUNCT iajs-2619 70 26 with with ADP iajs-2619 70 27 positive positive ADJ iajs-2619 70 28 initial initial ADJ iajs-2619 70 29 condition condition NOUN iajs-2619 70 30 ( ( PUNCT iajs-2619 70 31 𝑆𝑒(0 𝑆𝑒(0 PROPN iajs-2619 70 32 ) ) PUNCT iajs-2619 70 33 , , PUNCT iajs-2619 70 34 𝐼(0 𝐼(0 NOUN iajs-2619 70 35 ) ) PUNCT iajs-2619 70 36 , , PUNCT iajs-2619 70 37 π‘†π‘Ž(0 π‘†π‘Ž(0 ADJ iajs-2619 70 38 ) ) PUNCT iajs-2619 70 39 , , PUNCT iajs-2619 70 40 𝐡(0),𝑀(0 𝐡(0),𝑀(0 PROPN iajs-2619 70 41 ) ) PUNCT iajs-2619 70 42 ) ) PUNCT iajs-2619 70 43 which which PRON iajs-2619 70 44 defines define VERB iajs-2619 70 45 the the DET iajs-2619 70 46 function function NOUN iajs-2619 70 47 𝑁(𝑑 𝑁(𝑑 NOUN iajs-2619 70 48 ) ) PUNCT iajs-2619 70 49 = = SYM iajs-2619 70 50 𝑆𝑒(𝑑 𝑆𝑒(𝑑 NOUN iajs-2619 70 51 ) ) PUNCT iajs-2619 70 52 + + NUM iajs-2619 70 53 𝐼(𝑑 𝐼(𝑑 NUM iajs-2619 70 54 ) ) PUNCT iajs-2619 71 1 + + NUM iajs-2619 71 2 π‘†π‘Ž(𝑑 π‘†π‘Ž(𝑑 NOUN iajs-2619 71 3 ) ) PUNCT iajs-2619 72 1 + + CCONJ iajs-2619 72 2 𝐡(𝑑 𝐡(𝑑 X iajs-2619 72 3 ) ) PUNCT iajs-2619 72 4 then then ADV iajs-2619 72 5 take take VERB iajs-2619 72 6 the the DET iajs-2619 72 7 time time NOUN iajs-2619 72 8 derivative derivative NOUN iajs-2619 72 9 of of ADP iajs-2619 72 10 𝑁(𝑑 𝑁(𝑑 NUM iajs-2619 72 11 ) ) PUNCT iajs-2619 72 12 along along ADP iajs-2619 72 13 the the DET iajs-2619 72 14 solution solution NOUN iajs-2619 72 15 of of ADP iajs-2619 72 16 the the DET iajs-2619 72 17 system system NOUN iajs-2619 72 18 ( ( PUNCT iajs-2619 72 19 1);this 1);this PRON iajs-2619 72 20 gives give VERB iajs-2619 72 21 𝑑𝑁 𝑑𝑁 NOUN iajs-2619 72 22 𝑑𝑑 𝑑𝑑 ADP iajs-2619 72 23 = = PUNCT iajs-2619 72 24 Σ© Σ© PROPN iajs-2619 72 25 + + X iajs-2619 72 26 π‘Ÿπ΅ π‘Ÿπ΅ NOUN iajs-2619 72 27 ( ( PUNCT iajs-2619 72 28 1 1 NUM iajs-2619 72 29 βˆ’ βˆ’ PROPN iajs-2619 72 30 𝐡 𝐡 NOUN iajs-2619 72 31 𝑁 𝑁 PROPN iajs-2619 72 32 ) ) PUNCT iajs-2619 72 33 βˆ’ βˆ’ PROPN iajs-2619 72 34 π‘ž(𝑆𝑒 π‘ž(𝑆𝑒 X iajs-2619 73 1 + + CCONJ iajs-2619 73 2 𝐼 𝐼 PROPN iajs-2619 73 3 + + CCONJ iajs-2619 73 4 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 73 5 + + CCONJ iajs-2619 73 6 𝐡 𝐡 PROPN iajs-2619 73 7 ) ) PUNCT iajs-2619 73 8 𝑑𝑁 𝑑𝑁 NOUN iajs-2619 73 9 𝑑𝑑 𝑑𝑑 ADP iajs-2619 73 10 ≀ ≀ NUM iajs-2619 73 11 𝐻 𝐻 PROPN iajs-2619 73 12 βˆ’ βˆ’ PROPN iajs-2619 73 13 π‘žπ‘ π‘žπ‘ PROPN iajs-2619 73 14 ; ; PUNCT iajs-2619 73 15 𝐻 𝐻 PROPN iajs-2619 73 16 = = PUNCT iajs-2619 73 17 π‘ŸπΎ π‘ŸπΎ ADJ iajs-2619 73 18 4 4 NUM iajs-2619 73 19 + + SYM iajs-2619 73 20 Σ© Σ© NOUN iajs-2619 73 21 , , PUNCT iajs-2619 73 22 π‘ž π‘ž X iajs-2619 73 23 = = PROPN iajs-2619 73 24 π‘šπ‘–π‘› π‘šπ‘–π‘› PROPN iajs-2619 73 25 { { PUNCT iajs-2619 73 26 πœ‡ πœ‡ X iajs-2619 73 27 , , PUNCT iajs-2619 73 28 πœ‡ πœ‡ ADP iajs-2619 73 29 βˆ’ βˆ’ PROPN iajs-2619 73 30 πœ– πœ– PROPN iajs-2619 73 31 } } PUNCT iajs-2619 73 32 𝑑𝑁 𝑑𝑁 NOUN iajs-2619 73 33 𝑑𝑑 𝑑𝑑 ADP iajs-2619 73 34 + + CCONJ iajs-2619 73 35 π‘žπ‘ π‘žπ‘ PROPN iajs-2619 73 36 ≀ ≀ NOUN iajs-2619 73 37 𝐻 𝐻 PROPN iajs-2619 73 38 clearly clearly ADV iajs-2619 73 39 , , PUNCT iajs-2619 73 40 by by ADP iajs-2619 73 41 solving solve VERB iajs-2619 73 42 the the DET iajs-2619 73 43 above above ADJ iajs-2619 73 44 equation equation NOUN iajs-2619 73 45 , , PUNCT iajs-2619 73 46 we we PRON iajs-2619 73 47 obtain obtain VERB iajs-2619 73 48 𝑁(𝑑 𝑁(𝑑 NUM iajs-2619 73 49 ) ) PUNCT iajs-2619 73 50 ≀ ≀ NOUN iajs-2619 74 1 𝐻 𝐻 PROPN iajs-2619 74 2 π‘ž π‘ž X iajs-2619 74 3 + + CCONJ iajs-2619 74 4 ( ( PUNCT iajs-2619 74 5 𝑁0 𝑁0 ADJ iajs-2619 74 6 βˆ’ βˆ’ X iajs-2619 74 7 𝐻 𝐻 PROPN iajs-2619 74 8 π‘ž π‘ž NOUN iajs-2619 74 9 ) ) PUNCT iajs-2619 74 10 π‘’βˆ’π‘žπ‘‘ π‘’βˆ’π‘žπ‘‘ PROPN iajs-2619 74 11 therefore therefore ADV iajs-2619 74 12 , , PUNCT iajs-2619 74 13 𝑁(𝑑 𝑁(𝑑 NOUN iajs-2619 74 14 ) ) PUNCT iajs-2619 74 15 ≀ ≀ NOUN iajs-2619 75 1 𝐻 𝐻 PROPN iajs-2619 75 2 π‘ž π‘ž NOUN iajs-2619 75 3 , , PUNCT iajs-2619 75 4 π‘Žπ‘  π‘Žπ‘  PROPN iajs-2619 75 5 𝑑 𝑑 PROPN iajs-2619 75 6 β†’ β†’ SYM iajs-2619 75 7 ∞ ∞ X iajs-2619 75 8 𝑑𝑀 𝑑𝑀 X iajs-2619 75 9 𝑑𝑑 𝑑𝑑 ADP iajs-2619 75 10 = = PUNCT iajs-2619 75 11 𝛼 𝛼 PROPN iajs-2619 75 12 ( ( PUNCT iajs-2619 75 13 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 75 14 + + CCONJ iajs-2619 75 15 𝐼 𝐼 PROPN iajs-2619 75 16 ) ) PUNCT iajs-2619 75 17 βˆ’ βˆ’ PROPN iajs-2619 76 1 𝛾𝑀 𝛾𝑀 ADJ iajs-2619 76 2 𝑑𝑀 𝑑𝑀 PROPN iajs-2619 76 3 𝑑𝑑 𝑑𝑑 ADP iajs-2619 76 4 ≀ ≀ PROPN iajs-2619 76 5 οΏ½ οΏ½ PROPN iajs-2619 76 6 Μƒ Μƒ PROPN iajs-2619 76 7 οΏ½ οΏ½ PROPN iajs-2619 76 8 βˆ’ βˆ’ NOUN iajs-2619 76 9 𝛾𝑀 𝛾𝑀 PROPN iajs-2619 76 10 ; ; PUNCT iajs-2619 76 11 οΏ½ οΏ½ PROPN iajs-2619 76 12 Μƒ Μƒ NOUN iajs-2619 76 13 οΏ½ οΏ½ NOUN iajs-2619 76 14 = = SYM iajs-2619 76 15 𝛼 𝛼 PROPN iajs-2619 76 16 ( ( PUNCT iajs-2619 76 17 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 76 18 + + CCONJ iajs-2619 76 19 𝐼 𝐼 PROPN iajs-2619 76 20 ) ) PUNCT iajs-2619 76 21 by by ADP iajs-2619 76 22 a a DET iajs-2619 76 23 similar similar ADJ iajs-2619 76 24 way way NOUN iajs-2619 76 25 we we PRON iajs-2619 76 26 get get VERB iajs-2619 76 27 : : PUNCT iajs-2619 76 28 132 132 NUM iajs-2619 76 29 ibn ibn PROPN iajs-2619 76 30 al al PROPN iajs-2619 76 31 - - PUNCT iajs-2619 76 32 haitham haitham PROPN iajs-2619 76 33 jour jour X iajs-2619 76 34 . . PROPN iajs-2619 77 1 for for ADP iajs-2619 77 2 pure pure ADJ iajs-2619 77 3 & & CCONJ iajs-2619 77 4 appl appl PROPN iajs-2619 77 5 . . PUNCT iajs-2619 78 1 sci sci PROPN iajs-2619 78 2 . . PROPN iajs-2619 79 1 34 34 NUM iajs-2619 79 2 ( ( PUNCT iajs-2619 79 3 2 2 NUM iajs-2619 79 4 ) ) PUNCT iajs-2619 79 5 2021 2021 NUM iajs-2619 79 6 𝑀(𝑑 𝑀(𝑑 NUM iajs-2619 79 7 ) ) PUNCT iajs-2619 79 8 ≀ ≀ NOUN iajs-2619 79 9 𝛼𝐻 𝛼𝐻 VERB iajs-2619 79 10 π›Ύπ‘ž π›Ύπ‘ž ADJ iajs-2619 79 11 , , PUNCT iajs-2619 79 12 π‘Žπ‘  π‘Žπ‘  PROPN iajs-2619 79 13 𝑑 𝑑 PROPN iajs-2619 79 14 β†’ β†’ SYM iajs-2619 79 15 ∞ ∞ PROPN iajs-2619 79 16 we we PRON iajs-2619 79 17 obtain obtain VERB iajs-2619 79 18 that that SCONJ iajs-2619 79 19 , , PUNCT iajs-2619 79 20 the the DET iajs-2619 79 21 solution solution NOUN iajs-2619 79 22 of of ADP iajs-2619 79 23 system system NOUN iajs-2619 79 24 ( ( PUNCT iajs-2619 79 25 1 1 X iajs-2619 79 26 ) ) PUNCT iajs-2619 79 27 is be AUX iajs-2619 79 28 confined confine VERB iajs-2619 79 29 in in ADP iajs-2619 79 30 the the DET iajs-2619 79 31 following follow VERB iajs-2619 79 32 region region NOUN iajs-2619 79 33 Ο‰ Ο‰ PROPN iajs-2619 79 34 = = SYM iajs-2619 79 35 { { PUNCT iajs-2619 79 36 ( ( PUNCT iajs-2619 79 37 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 79 38 , , PUNCT iajs-2619 79 39 𝐼 𝐼 PROPN iajs-2619 79 40 , , PUNCT iajs-2619 79 41 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 79 42 , , PUNCT iajs-2619 79 43 𝐡,𝑀 𝐡,𝑀 VERB iajs-2619 79 44 ) ) PUNCT iajs-2619 79 45 ∈ ∈ PROPN iajs-2619 79 46 𝑅+ 𝑅+ NUM iajs-2619 79 47 5 5 NUM iajs-2619 79 48 : : PUNCT iajs-2619 79 49 𝑁 𝑁 PROPN iajs-2619 79 50 ≀ ≀ NOUN iajs-2619 79 51 𝐻 𝐻 PROPN iajs-2619 79 52 π‘ž π‘ž NOUN iajs-2619 79 53 , , PUNCT iajs-2619 79 54 0 0 NUM iajs-2619 79 55 ≀ ≀ NUM iajs-2619 79 56 𝑀 𝑀 PROPN iajs-2619 79 57 ≀ ≀ PROPN iajs-2619 79 58 𝛼𝐻 𝛼𝐻 VERB iajs-2619 79 59 π›Ύπ‘ž π›Ύπ‘ž ADJ iajs-2619 79 60 } } PUNCT iajs-2619 79 61 thus thus ADV iajs-2619 79 62 , , PUNCT iajs-2619 79 63 these these DET iajs-2619 79 64 solutions solution NOUN iajs-2619 79 65 are be AUX iajs-2619 79 66 uniformly uniformly ADV iajs-2619 79 67 bounded bound VERB iajs-2619 79 68 and and CCONJ iajs-2619 79 69 the the DET iajs-2619 79 70 proof proof NOUN iajs-2619 79 71 is be AUX iajs-2619 79 72 complete complete ADJ iajs-2619 79 73 . . PUNCT iajs-2619 80 1 ∎ ∎ NOUN iajs-2619 80 2 3 3 X iajs-2619 80 3 . . PUNCT iajs-2619 81 1 the the DET iajs-2619 81 2 number number NOUN iajs-2619 81 3 of of ADP iajs-2619 81 4 equilibrium equilibrium NOUN iajs-2619 81 5 points point NOUN iajs-2619 81 6 : : PUNCT iajs-2619 81 7 it it PRON iajs-2619 81 8 is be AUX iajs-2619 81 9 easy easy ADJ iajs-2619 81 10 that that SCONJ iajs-2619 81 11 aware aware ADJ iajs-2619 81 12 susceptible susceptible ADJ iajs-2619 81 13 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 81 14 is be AUX iajs-2619 81 15 related relate VERB iajs-2619 81 16 with with ADP iajs-2619 81 17 variable variable ADJ iajs-2619 81 18 𝑆𝑒(𝑑 𝑆𝑒(𝑑 NOUN iajs-2619 81 19 ) ) PUNCT iajs-2619 81 20 and and CCONJ iajs-2619 81 21 𝑀(𝑑 𝑀(𝑑 NUM iajs-2619 81 22 ) ) PUNCT iajs-2619 81 23 only only ADV iajs-2619 81 24 . . PUNCT iajs-2619 82 1 hence hence ADV iajs-2619 82 2 for for ADP iajs-2619 82 3 fixed fix VERB iajs-2619 82 4 values value NOUN iajs-2619 82 5 of of ADP iajs-2619 82 6 , , PUNCT iajs-2619 82 7 𝑆𝑒(𝑑 𝑆𝑒(𝑑 NOUN iajs-2619 82 8 ) ) PUNCT iajs-2619 82 9 and and CCONJ iajs-2619 82 10 𝑀(𝑑 𝑀(𝑑 NUM iajs-2619 82 11 ) ) PUNCT iajs-2619 82 12 , , PUNCT iajs-2619 82 13 the the DET iajs-2619 82 14 calculate calculate NOUN iajs-2619 82 15 value value NOUN iajs-2619 82 16 of of ADP iajs-2619 82 17 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 82 18 can can AUX iajs-2619 82 19 be be AUX iajs-2619 82 20 found find VERB iajs-2619 82 21 simply simply ADV iajs-2619 82 22 by by ADP iajs-2619 82 23 solving solve VERB iajs-2619 82 24 the the DET iajs-2619 82 25 system system NOUN iajs-2619 82 26 ( ( PUNCT iajs-2619 82 27 1 1 NUM iajs-2619 82 28 ) ) PUNCT iajs-2619 82 29 . . PUNCT iajs-2619 83 1 in in ADP iajs-2619 83 2 fact fact NOUN iajs-2619 83 3 , , PUNCT iajs-2619 83 4 we we PRON iajs-2619 83 5 can can AUX iajs-2619 83 6 determine determine VERB iajs-2619 83 7 the the DET iajs-2619 83 8 value value NOUN iajs-2619 83 9 of of ADP iajs-2619 83 10 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 83 11 by by ADP iajs-2619 83 12 the the DET iajs-2619 83 13 following follow VERB iajs-2619 83 14 equation equation NOUN iajs-2619 83 15 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 83 16 = = SYM iajs-2619 83 17 π‘Ž2 π‘Ž2 PROPN iajs-2619 83 18 οΏ½ οΏ½ PROPN iajs-2619 83 19 Μƒ Μƒ PROPN iajs-2619 83 20 οΏ½ οΏ½ NOUN iajs-2619 83 21 𝑒 𝑒 NOUN iajs-2619 83 22 οΏ½ οΏ½ PROPN iajs-2619 83 23 Μƒ Μƒ PROPN iajs-2619 83 24 οΏ½ οΏ½ PROPN iajs-2619 83 25 πœ‡ πœ‡ X iajs-2619 83 26 ( ( PUNCT iajs-2619 83 27 2 2 NUM iajs-2619 83 28 ) ) PUNCT iajs-2619 83 29 consequently consequently ADV iajs-2619 83 30 , , PUNCT iajs-2619 83 31 we we PRON iajs-2619 83 32 can can AUX iajs-2619 83 33 reduce reduce VERB iajs-2619 83 34 system system NOUN iajs-2619 83 35 ( ( PUNCT iajs-2619 83 36 1 1 NUM iajs-2619 83 37 ) ) PUNCT iajs-2619 83 38 and and CCONJ iajs-2619 83 39 rewrite rewrite VERB iajs-2619 83 40 it it PRON iajs-2619 83 41 to to ADP iajs-2619 83 42 the the DET iajs-2619 83 43 following follow VERB iajs-2619 83 44 system system NOUN iajs-2619 83 45 𝑆 𝑆 PROPN iajs-2619 83 46 οΏ½ οΏ½ PROPN iajs-2619 83 47 Μ‡ Μ‡ PROPN iajs-2619 84 1 οΏ½ οΏ½ PROPN iajs-2619 84 2 = = PUNCT iajs-2619 84 3 πœƒ πœƒ PROPN iajs-2619 84 4 βˆ’ βˆ’ NOUN iajs-2619 84 5 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 84 6 βˆ’ βˆ’ NOUN iajs-2619 84 7 π‘Ž2𝑆𝑒𝑀 π‘Ž2𝑆𝑒𝑀 NOUN iajs-2619 84 8 βˆ’ βˆ’ PROPN iajs-2619 84 9 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 84 10 βˆ’ βˆ’ NOUN iajs-2619 84 11 π‘Ž4𝑆𝑒 π‘Ž4𝑆𝑒 NOUN iajs-2619 84 12 βˆ’ βˆ’ ADP iajs-2619 84 13 πœ‡π‘†π‘’ πœ‡π‘†π‘’ PRON iajs-2619 84 14 𝐼̇ 𝐼̇ PROPN iajs-2619 84 15 = = SYM iajs-2619 84 16 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 84 17 + + CCONJ iajs-2619 84 18 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 84 19 + + NUM iajs-2619 84 20 π‘Ž4𝑆𝑒 π‘Ž4𝑆𝑒 NOUN iajs-2619 84 21 βˆ’ βˆ’ PROPN iajs-2619 84 22 ( ( PUNCT iajs-2619 84 23 πœ‡ πœ‡ ADP iajs-2619 84 24 + + X iajs-2619 84 25 𝑑 𝑑 PROPN iajs-2619 84 26 + + CCONJ iajs-2619 84 27 πœ‚)𝐼 πœ‚)𝐼 ADJ iajs-2619 84 28 οΏ½ οΏ½ PROPN iajs-2619 84 29 Μ‡ Μ‡ VERB iajs-2619 84 30 οΏ½ οΏ½ PROPN iajs-2619 84 31 = = PUNCT iajs-2619 84 32 π‘Ÿπ΅ π‘Ÿπ΅ PROPN iajs-2619 84 33 ( ( PUNCT iajs-2619 84 34 1 1 NUM iajs-2619 84 35 βˆ’ βˆ’ PROPN iajs-2619 84 36 𝐡 𝐡 PROPN iajs-2619 84 37 𝐾 𝐾 PROPN iajs-2619 84 38 ) ) PUNCT iajs-2619 84 39 βˆ’ βˆ’ ADP iajs-2619 85 1 πœ‡π΅ πœ‡π΅ VERB iajs-2619 85 2 + + CCONJ iajs-2619 85 3 πœ–πΌ πœ–πΌ ADJ iajs-2619 85 4 οΏ½ οΏ½ NOUN iajs-2619 85 5 Μ‡ Μ‡ NOUN iajs-2619 85 6 οΏ½ οΏ½ PROPN iajs-2619 85 7 = = SYM iajs-2619 85 8 𝛼 𝛼 PROPN iajs-2619 85 9 ( ( PUNCT iajs-2619 85 10 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 85 11 + + CCONJ iajs-2619 85 12 𝐼 𝐼 PROPN iajs-2619 85 13 ) ) PUNCT iajs-2619 85 14 – – PUNCT iajs-2619 85 15 𝛾𝑀 𝛾𝑀 ADJ iajs-2619 85 16 ( ( PUNCT iajs-2619 85 17 3 3 NUM iajs-2619 85 18 ) ) PUNCT iajs-2619 85 19 clearly clearly ADV iajs-2619 85 20 , , PUNCT iajs-2619 85 21 there there PRON iajs-2619 85 22 are be VERB iajs-2619 85 23 only only ADV iajs-2619 85 24 two two NUM iajs-2619 85 25 equilibrium equilibrium NOUN iajs-2619 85 26 points point NOUN iajs-2619 85 27 of of ADP iajs-2619 85 28 system system NOUN iajs-2619 85 29 ( ( PUNCT iajs-2619 85 30 3 3 NUM iajs-2619 85 31 ) ) PUNCT iajs-2619 85 32 under under ADP iajs-2619 85 33 the the DET iajs-2619 85 34 following following ADJ iajs-2619 85 35 conditions condition NOUN iajs-2619 85 36 : : PUNCT iajs-2619 85 37 ο‚· ο‚· NOUN iajs-2619 85 38 the the DET iajs-2619 85 39 first first ADJ iajs-2619 85 40 equilibrium equilibrium NOUN iajs-2619 85 41 point point NOUN iajs-2619 85 42 exists exist VERB iajs-2619 85 43 when when SCONJ iajs-2619 85 44 𝐼 𝐼 PROPN iajs-2619 85 45 = = SYM iajs-2619 85 46 0 0 PUNCT iajs-2619 85 47 ( ( PUNCT iajs-2619 85 48 when when SCONJ iajs-2619 85 49 π‘Ž4 π‘Ž4 PROPN iajs-2619 85 50 = = SYM iajs-2619 85 51 0 0 NUM iajs-2619 85 52 ) ) PUNCT iajs-2619 85 53 and and CCONJ iajs-2619 85 54 𝐡 𝐡 NOUN iajs-2619 85 55 = = SYM iajs-2619 85 56 0 0 NUM iajs-2619 85 57 , , PUNCT iajs-2619 85 58 and and CCONJ iajs-2619 85 59 is be AUX iajs-2619 85 60 called call VERB iajs-2619 85 61 disease disease NOUN iajs-2619 85 62 free free ADJ iajs-2619 85 63 equilibrium equilibrium NOUN iajs-2619 85 64 point point NOUN iajs-2619 85 65 which which PRON iajs-2619 85 66 is be AUX iajs-2619 85 67 denoted denote VERB iajs-2619 85 68 by by ADP iajs-2619 85 69 𝐸0 𝐸0 NOUN iajs-2619 85 70 = = SYM iajs-2619 85 71 ( ( PUNCT iajs-2619 85 72 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 85 73 ^ ^ PUNCT iajs-2619 85 74 , , PUNCT iajs-2619 85 75 0 0 NUM iajs-2619 85 76 , , PUNCT iajs-2619 85 77 0 0 NUM iajs-2619 85 78 , , PUNCT iajs-2619 85 79 𝑀^ 𝑀^ PROPN iajs-2619 85 80 ) ) PUNCT iajs-2619 85 81 , , PUNCT iajs-2619 85 82 where where SCONJ iajs-2619 85 83 𝑀^ 𝑀^ ADP iajs-2619 85 84 = = SYM iajs-2619 85 85 𝛼 𝛼 NOUN iajs-2619 85 86 𝛾 𝛾 NOUN iajs-2619 85 87 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 85 88 ^ ^ PUNCT iajs-2619 85 89 ( ( PUNCT iajs-2619 85 90 4 4 X iajs-2619 85 91 ) ) PUNCT iajs-2619 85 92 while while SCONJ iajs-2619 85 93 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 85 94 is be AUX iajs-2619 85 95 a a DET iajs-2619 85 96 positive positive ADJ iajs-2619 85 97 real real ADJ iajs-2619 85 98 root root NOUN iajs-2619 85 99 of of ADP iajs-2619 85 100 the the DET iajs-2619 85 101 following follow VERB iajs-2619 85 102 quadratic quadratic ADJ iajs-2619 85 103 equation equation NOUN iajs-2619 85 104 𝐴1𝑆𝑒 𝐴1𝑆𝑒 VERB iajs-2619 85 105 2 2 NUM iajs-2619 85 106 + + CCONJ iajs-2619 85 107 𝐴2𝑆𝑒 𝐴2𝑆𝑒 PROPN iajs-2619 85 108 + + CCONJ iajs-2619 85 109 𝐴3 𝐴3 PROPN iajs-2619 85 110 = = SYM iajs-2619 85 111 0 0 NUM iajs-2619 85 112 ( ( PUNCT iajs-2619 85 113 5a 5a NUM iajs-2619 85 114 ) ) PUNCT iajs-2619 85 115 here here ADV iajs-2619 85 116 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 85 117 = = PUNCT iajs-2619 85 118 βˆ’(𝐴2+√𝐴2 βˆ’(𝐴2+√𝐴2 PROPN iajs-2619 85 119 2βˆ’4𝐴1𝐴3 2βˆ’4𝐴1𝐴3 PROPN iajs-2619 85 120 ) ) PUNCT iajs-2619 85 121 2𝐴1 2𝐴1 NUM iajs-2619 85 122 ( ( PUNCT iajs-2619 85 123 5b 5b NUM iajs-2619 85 124 ) ) PUNCT iajs-2619 85 125 𝐴1 𝐴1 PROPN iajs-2619 85 126 = = PUNCT iajs-2619 86 1 βˆ’π‘Ž2𝛼 βˆ’π‘Ž2𝛼 PROPN iajs-2619 86 2 𝛾 𝛾 ADP iajs-2619 86 3 𝐴2 𝐴2 NOUN iajs-2619 86 4 = = SYM iajs-2619 86 5 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-2619 86 6 𝐴3 𝐴3 PROPN iajs-2619 86 7 = = SYM iajs-2619 86 8 πœƒ πœƒ PROPN iajs-2619 86 9 ο‚· ο‚· NOUN iajs-2619 86 10 the the DET iajs-2619 86 11 endemic endemic ADJ iajs-2619 86 12 equilibrium equilibrium NOUN iajs-2619 86 13 point point NOUN iajs-2619 86 14 , , PUNCT iajs-2619 86 15 denoted denote VERB iajs-2619 86 16 by by ADP iajs-2619 86 17 𝐸1 𝐸1 NOUN iajs-2619 86 18 = = SYM iajs-2619 86 19 ( ( PUNCT iajs-2619 86 20 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 86 21 βˆ— βˆ— NOUN iajs-2619 86 22 , , PUNCT iajs-2619 86 23 πΌβˆ— πΌβˆ— PROPN iajs-2619 86 24 , , PUNCT iajs-2619 86 25 π΅βˆ— π΅βˆ— NUM iajs-2619 86 26 , , PUNCT iajs-2619 86 27 π‘€βˆ— π‘€βˆ— NUM iajs-2619 86 28 ) ) PUNCT iajs-2619 86 29 where where SCONJ iajs-2619 86 30 πΌβˆ— πΌβˆ— NOUN iajs-2619 86 31 = = SYM iajs-2619 86 32 π΅βˆ— π΅βˆ— PROPN iajs-2619 86 33 ( ( PUNCT iajs-2619 86 34 πœ‡π‘˜+π‘Ÿπ΅βˆ—βˆ’π‘Ÿπ‘˜ πœ‡π‘˜+π‘Ÿπ΅βˆ—βˆ’π‘Ÿπ‘˜ PROPN iajs-2619 86 35 ) ) PUNCT iajs-2619 86 36 π‘˜πœ– π‘˜πœ– INTJ iajs-2619 86 37 ( ( PUNCT iajs-2619 86 38 6a 6a NOUN iajs-2619 86 39 ) ) PUNCT iajs-2619 86 40 π‘€βˆ— π‘€βˆ— NOUN iajs-2619 86 41 = = SYM iajs-2619 86 42 𝛼(π‘˜πœ–π‘†π‘’ 𝛼(π‘˜πœ–π‘†π‘’ PROPN iajs-2619 86 43 βˆ—+(πœ‡βˆ’π‘Ÿ)π‘˜π΅βˆ—+π‘Ÿπ΅βˆ— βˆ—+(πœ‡βˆ’π‘Ÿ)π‘˜π΅βˆ—+π‘Ÿπ΅βˆ— PROPN iajs-2619 86 44 2 2 NUM iajs-2619 86 45 ) ) PUNCT iajs-2619 86 46 π‘˜πœ–π›Ύ π‘˜πœ–π›Ύ NOUN iajs-2619 86 47 ( ( PUNCT iajs-2619 86 48 6b 6b NUM iajs-2619 86 49 ) ) PUNCT iajs-2619 86 50 133 133 NUM iajs-2619 86 51 ibn ibn PROPN iajs-2619 86 52 al al PROPN iajs-2619 86 53 - - PUNCT iajs-2619 86 54 haitham haitham PROPN iajs-2619 86 55 jour jour X iajs-2619 86 56 . . PROPN iajs-2619 87 1 for for ADP iajs-2619 87 2 pure pure ADJ iajs-2619 87 3 & & CCONJ iajs-2619 87 4 appl appl PROPN iajs-2619 87 5 . . PUNCT iajs-2619 88 1 sci sci PROPN iajs-2619 88 2 . . PROPN iajs-2619 89 1 34 34 NUM iajs-2619 89 2 ( ( PUNCT iajs-2619 89 3 2 2 NUM iajs-2619 89 4 ) ) PUNCT iajs-2619 89 5 2021 2021 NUM iajs-2619 89 6 while while SCONJ iajs-2619 89 7 ( ( PUNCT iajs-2619 89 8 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 89 9 βˆ— βˆ— NOUN iajs-2619 89 10 , , PUNCT iajs-2619 89 11 π΅βˆ— π΅βˆ— NUM iajs-2619 89 12 ) ) PUNCT iajs-2619 89 13 represents represent VERB iajs-2619 89 14 a a DET iajs-2619 89 15 positive positive ADJ iajs-2619 89 16 intersection intersection NOUN iajs-2619 89 17 point point NOUN iajs-2619 89 18 of of ADP iajs-2619 89 19 the the DET iajs-2619 89 20 following follow VERB iajs-2619 89 21 two two NUM iajs-2619 89 22 isoclines isocline NOUN iajs-2619 89 23 : : PUNCT iajs-2619 89 24 𝑓(𝑆𝑒 𝑓(𝑆𝑒 NUM iajs-2619 89 25 , , PUNCT iajs-2619 89 26 𝐡 𝐡 NOUN iajs-2619 89 27 ) ) PUNCT iajs-2619 89 28 = = PUNCT iajs-2619 90 1 π‘š1𝑆𝑒𝐡 π‘š1𝑆𝑒𝐡 NUM iajs-2619 90 2 2 2 NUM iajs-2619 90 3 + + CCONJ iajs-2619 90 4 π‘š2𝐡 π‘š2𝐡 NOUN iajs-2619 90 5 2 2 NUM iajs-2619 90 6 + + NUM iajs-2619 90 7 π‘š3𝑆𝑒𝐡 π‘š3𝑆𝑒𝐡 X iajs-2619 90 8 + + CCONJ iajs-2619 90 9 π‘š4𝑆𝑒 π‘š4𝑆𝑒 NOUN iajs-2619 90 10 + + NUM iajs-2619 90 11 π‘š5𝐡 π‘š5𝐡 PRON iajs-2619 90 12 = = SYM iajs-2619 90 13 0 0 NUM iajs-2619 90 14 ( ( PUNCT iajs-2619 90 15 7 7 NUM iajs-2619 90 16 ) ) PUNCT iajs-2619 90 17 𝑔(𝑆𝑒 𝑔(𝑆𝑒 NOUN iajs-2619 90 18 , , PUNCT iajs-2619 90 19 𝐡 𝐡 NOUN iajs-2619 90 20 ) ) PUNCT iajs-2619 90 21 = = SYM iajs-2619 91 1 𝑛1𝑆𝑒𝐡 𝑛1𝑆𝑒𝐡 PUNCT iajs-2619 91 2 2 2 NUM iajs-2619 91 3 + + NUM iajs-2619 91 4 𝑛2𝑆𝑒 𝑛2𝑆𝑒 PROPN iajs-2619 91 5 2 2 NUM iajs-2619 91 6 + + CCONJ iajs-2619 91 7 𝑛3𝑆𝑒𝐡 𝑛3𝑆𝑒𝐡 DET iajs-2619 91 8 + + ADJ iajs-2619 91 9 𝑛4𝑆𝑒 𝑛4𝑆𝑒 NOUN iajs-2619 91 10 + + NUM iajs-2619 91 11 𝑛5 𝑛5 NOUN iajs-2619 91 12 = = SYM iajs-2619 91 13 0 0 PUNCT iajs-2619 91 14 ( ( PUNCT iajs-2619 91 15 8) 8) NUM iajs-2619 91 16 here here ADV iajs-2619 91 17 , , PUNCT iajs-2619 91 18 π‘š1 π‘š1 NOUN iajs-2619 91 19 = = SYM iajs-2619 91 20 π‘Ž3π‘Ÿ π‘Ž3π‘Ÿ PROPN iajs-2619 91 21 , , PUNCT iajs-2619 91 22 π‘š2 π‘š2 NOUN iajs-2619 91 23 = = PUNCT iajs-2619 91 24 βˆ’(πœ‡ βˆ’(πœ‡ PROPN iajs-2619 91 25 + + CCONJ iajs-2619 91 26 𝑑 𝑑 X iajs-2619 91 27 + + ADJ iajs-2619 91 28 Ξ·),m3 Ξ·),m3 ADV iajs-2619 91 29 = = SYM iajs-2619 91 30 π‘Ž1π‘˜πœ– π‘Ž1π‘˜πœ– PROPN iajs-2619 91 31 + + PUNCT iajs-2619 91 32 π‘Ž3(πœ‡ π‘Ž3(πœ‡ NUM iajs-2619 91 33 βˆ’ βˆ’ PROPN iajs-2619 91 34 π‘Ÿ)π‘˜ π‘Ÿ)π‘˜ ADJ iajs-2619 91 35 π‘š4 π‘š4 PROPN iajs-2619 91 36 = = SYM iajs-2619 91 37 π‘Ž4π‘˜πœ– π‘Ž4π‘˜πœ– X iajs-2619 91 38 , , PUNCT iajs-2619 91 39 π‘š5 π‘š5 NOUN iajs-2619 91 40 = = SYM iajs-2619 91 41 βˆ’(πœ‡ βˆ’(πœ‡ NOUN iajs-2619 91 42 + + CCONJ iajs-2619 91 43 𝑑 𝑑 PROPN iajs-2619 91 44 + + X iajs-2619 91 45 πœ‚)(πœ‡ πœ‚)(πœ‡ ADP iajs-2619 91 46 βˆ’ βˆ’ NUM iajs-2619 91 47 π‘Ÿ π‘Ÿ X iajs-2619 91 48 ) ) PUNCT iajs-2619 91 49 n1 n1 PROPN iajs-2619 91 50 = = SYM iajs-2619 91 51 βˆ’ βˆ’ PROPN iajs-2619 91 52 ( ( PUNCT iajs-2619 91 53 π‘Ž2r π‘Ž2r NOUN iajs-2619 91 54 + + CCONJ iajs-2619 91 55 π‘Ž3π›Ύπ‘Ÿ π‘Ž3π›Ύπ‘Ÿ X iajs-2619 91 56 ) ) PUNCT iajs-2619 91 57 , , PUNCT iajs-2619 91 58 n2 n2 NOUN iajs-2619 91 59 = = PROPN iajs-2619 91 60 βˆ’π‘Ž2π›Όπ‘˜πœ– βˆ’π‘Ž2π›Όπ‘˜πœ– NOUN iajs-2619 91 61 n3 n3 NOUN iajs-2619 91 62 = = SYM iajs-2619 91 63 βˆ’ βˆ’ PROPN iajs-2619 92 1 [ [ PUNCT iajs-2619 92 2 π‘Ž1π‘˜Ο΅π›Ύ π‘Ž1π‘˜Ο΅π›Ύ X iajs-2619 92 3 + + CCONJ iajs-2619 92 4 π‘Ž2π‘˜(πœ‡ π‘Ž2π‘˜(πœ‡ PROPN iajs-2619 92 5 βˆ’ βˆ’ PROPN iajs-2619 92 6 π‘Ÿ π‘Ÿ NOUN iajs-2619 92 7 ) ) PUNCT iajs-2619 92 8 + + CCONJ iajs-2619 92 9 π‘Ž3π›Ύπ‘˜(πœ‡ π‘Ž3π›Ύπ‘˜(πœ‡ PROPN iajs-2619 92 10 βˆ’ βˆ’ PROPN iajs-2619 92 11 π‘Ÿ π‘Ÿ NOUN iajs-2619 92 12 ) ) PUNCT iajs-2619 92 13 ] ] PUNCT iajs-2619 93 1 n4 n4 PROPN iajs-2619 93 2 = = PUNCT iajs-2619 94 1 βˆ’π‘˜Ο΅π›Ύ(π‘Ž4 βˆ’π‘˜Ο΅π›Ύ(π‘Ž4 PROPN iajs-2619 94 2 + + PUNCT iajs-2619 94 3 πœ‡ πœ‡ X iajs-2619 94 4 ) ) PUNCT iajs-2619 94 5 , , PUNCT iajs-2619 94 6 n5 n5 PROPN iajs-2619 94 7 = = PUNCT iajs-2619 94 8 π‘˜Ο΅π›Ύπœƒ π‘˜Ο΅π›Ύπœƒ ADJ iajs-2619 94 9 clearly clearly ADV iajs-2619 94 10 as as SCONJ iajs-2619 94 11 𝐡 𝐡 NOUN iajs-2619 94 12 β†’ β†’ SYM iajs-2619 94 13 0 0 NUM iajs-2619 94 14 the the DET iajs-2619 94 15 first first ADJ iajs-2619 94 16 isoclines isocline NOUN iajs-2619 94 17 ( ( PUNCT iajs-2619 94 18 7 7 X iajs-2619 94 19 ) ) PUNCT iajs-2619 94 20 intersects intersect VERB iajs-2619 94 21 the the DET iajs-2619 94 22 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 94 23 βˆ’ βˆ’ PROPN iajs-2619 94 24 axis axis NOUN iajs-2619 94 25 at at ADP iajs-2619 94 26 zero zero NUM iajs-2619 94 27 however however ADV iajs-2619 94 28 when when SCONJ iajs-2619 94 29 𝐡 𝐡 PROPN iajs-2619 94 30 β†’ β†’ SYM iajs-2619 94 31 0 0 NUM iajs-2619 94 32 the the DET iajs-2619 94 33 second second ADJ iajs-2619 94 34 isoclines isocline NOUN iajs-2619 94 35 ( ( PUNCT iajs-2619 94 36 8) 8) NUM iajs-2619 94 37 will will AUX iajs-2619 94 38 intersect intersect VERB iajs-2619 94 39 the the DET iajs-2619 94 40 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 94 41 βˆ’ βˆ’ PROPN iajs-2619 94 42 axis axis NOUN iajs-2619 94 43 at at ADP iajs-2619 94 44 a a DET iajs-2619 94 45 unique unique ADJ iajs-2619 94 46 positive positive ADJ iajs-2619 94 47 point point NOUN iajs-2619 94 48 , , PUNCT iajs-2619 94 49 say say VERB iajs-2619 94 50 𝑆𝑒1 𝑆𝑒1 ADV iajs-2619 94 51 consequently consequently ADV iajs-2619 94 52 , , PUNCT iajs-2619 94 53 these these DET iajs-2619 94 54 two two NUM iajs-2619 94 55 isoclines isocline NOUN iajs-2619 94 56 ( ( PUNCT iajs-2619 94 57 7 7 NUM iajs-2619 94 58 ) ) PUNCT iajs-2619 94 59 and and CCONJ iajs-2619 94 60 ( ( PUNCT iajs-2619 94 61 8) 8) NUM iajs-2619 94 62 have have VERB iajs-2619 94 63 an an DET iajs-2619 94 64 intersection intersection NOUN iajs-2619 94 65 point point NOUN iajs-2619 94 66 in in ADP iajs-2619 94 67 the the DET iajs-2619 94 68 interior interior NOUN iajs-2619 94 69 of of ADP iajs-2619 94 70 the the DET iajs-2619 94 71 positive positive ADJ iajs-2619 94 72 quadrant quadrant NOUN iajs-2619 94 73 of of ADP iajs-2619 94 74 𝑆𝑒𝐡 𝑆𝑒𝐡 X iajs-2619 94 75 – – PUNCT iajs-2619 94 76 plane plane NOUN iajs-2619 94 77 , , PUNCT iajs-2619 94 78 namely namely ADV iajs-2619 94 79 ( ( PUNCT iajs-2619 94 80 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 94 81 βˆ— βˆ— NOUN iajs-2619 94 82 , , PUNCT iajs-2619 94 83 π΅βˆ— π΅βˆ— NUM iajs-2619 94 84 ) ) PUNCT iajs-2619 94 85 , , PUNCT iajs-2619 94 86 provided provide VERB iajs-2619 94 87 that that SCONJ iajs-2619 94 88 the the DET iajs-2619 94 89 following follow VERB iajs-2619 94 90 conditions condition NOUN iajs-2619 94 91 are be AUX iajs-2619 94 92 satisfied satisfied ADJ iajs-2619 94 93 πœ•π‘“ πœ•π‘“ ADJ iajs-2619 94 94 𝑑𝑆𝑒 𝑑𝑆𝑒 NOUN iajs-2619 94 95 > > SYM iajs-2619 94 96 0 0 NUM iajs-2619 94 97 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2619 94 98 πœ•π‘“ πœ•π‘“ NOUN iajs-2619 94 99 𝑑𝐡 𝑑𝐡 NOUN iajs-2619 94 100 < < X iajs-2619 94 101 0 0 NUM iajs-2619 94 102 π‘œπ‘Ÿ π‘œπ‘Ÿ PRON iajs-2619 94 103 πœ•π‘“ πœ•π‘“ NOUN iajs-2619 94 104 𝑑𝑆𝑒 𝑑𝑆𝑒 NOUN iajs-2619 94 105 < < X iajs-2619 94 106 0 0 NUM iajs-2619 94 107 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2619 94 108 πœ•π‘“ πœ•π‘“ ADJ iajs-2619 94 109 𝑑𝐡 𝑑𝐡 NOUN iajs-2619 94 110 > > X iajs-2619 94 111 0 0 NUM iajs-2619 94 112 } } PUNCT iajs-2619 94 113 ( ( PUNCT iajs-2619 94 114 9a 9a NOUN iajs-2619 94 115 ) ) PUNCT iajs-2619 94 116 πœ•π‘” πœ•π‘” PROPN iajs-2619 94 117 𝑑𝑆𝑒 𝑑𝑆𝑒 PROPN iajs-2619 94 118 > > X iajs-2619 94 119 0 0 NUM iajs-2619 94 120 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-2619 94 121 πœ•π‘” πœ•π‘” PROPN iajs-2619 94 122 𝑑𝐡 𝑑𝐡 VERB iajs-2619 94 123 > > X iajs-2619 94 124 0 0 PUNCT iajs-2619 95 1 π‘œπ‘Ÿ π‘œπ‘Ÿ PRON iajs-2619 95 2 πœ•π‘” πœ•π‘” PROPN iajs-2619 95 3 𝑑𝑆𝑒 𝑑𝑆𝑒 PROPN iajs-2619 95 4 < < X iajs-2619 95 5 0 0 NUM iajs-2619 95 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-2619 95 7 πœ•π‘” πœ•π‘” PROPN iajs-2619 95 8 𝑑𝐡 𝑑𝐡 VERB iajs-2619 95 9 < < X iajs-2619 95 10 0 0 NUM iajs-2619 95 11 } } PUNCT iajs-2619 95 12 ( ( PUNCT iajs-2619 95 13 9b 9b NUM iajs-2619 95 14 ) ) PUNCT iajs-2619 95 15 therefore therefore ADV iajs-2619 95 16 , , PUNCT iajs-2619 95 17 we we PRON iajs-2619 95 18 have have VERB iajs-2619 95 19 the the DET iajs-2619 95 20 endemic endemic ADJ iajs-2619 95 21 equilibrium equilibrium NOUN iajs-2619 95 22 point point NOUN iajs-2619 95 23 𝐸1 𝐸1 NOUN iajs-2619 95 24 = = PUNCT iajs-2619 95 25 ( ( PUNCT iajs-2619 95 26 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 95 27 βˆ— βˆ— NOUN iajs-2619 95 28 , , PUNCT iajs-2619 95 29 πΌβˆ— πΌβˆ— NOUN iajs-2619 95 30 , , PUNCT iajs-2619 95 31 π΅βˆ— π΅βˆ— NUM iajs-2619 95 32 , , PUNCT iajs-2619 95 33 π‘€βˆ— π‘€βˆ— NUM iajs-2619 95 34 ) ) PUNCT iajs-2619 95 35 if if SCONJ iajs-2619 95 36 the the DET iajs-2619 95 37 above above ADJ iajs-2619 95 38 conditions condition NOUN iajs-2619 95 39 ( ( PUNCT iajs-2619 95 40 9a)-(9b 9a)-(9b X iajs-2619 95 41 ) ) PUNCT iajs-2619 95 42 hold hold NOUN iajs-2619 95 43 and and CCONJ iajs-2619 95 44 the the DET iajs-2619 95 45 following follow VERB iajs-2619 95 46 condition condition NOUN iajs-2619 95 47 is be AUX iajs-2619 95 48 satisfied satisfied ADJ iajs-2619 95 49 too too ADV iajs-2619 95 50 π‘Ÿπ‘˜ π‘Ÿπ‘˜ ADP iajs-2619 95 51 < < X iajs-2619 95 52 πœ‡π‘˜ πœ‡π‘˜ PROPN iajs-2619 95 53 + + CCONJ iajs-2619 96 1 π‘Ÿπ΅βˆ— π‘Ÿπ΅βˆ— ADJ iajs-2619 96 2 ( ( PUNCT iajs-2619 96 3 10 10 NUM iajs-2619 96 4 ) ) PUNCT iajs-2619 96 5 4 4 NUM iajs-2619 96 6 . . PUNCT iajs-2619 96 7 local local ADJ iajs-2619 96 8 stability stability NOUN iajs-2619 96 9 analysis analysis NOUN iajs-2619 96 10 : : PUNCT iajs-2619 96 11 in in ADP iajs-2619 96 12 this this DET iajs-2619 96 13 section section NOUN iajs-2619 96 14 , , PUNCT iajs-2619 96 15 the the DET iajs-2619 96 16 local local ADJ iajs-2619 96 17 stability stability NOUN iajs-2619 96 18 conditions condition NOUN iajs-2619 96 19 of of ADP iajs-2619 96 20 system system NOUN iajs-2619 96 21 ( ( PUNCT iajs-2619 96 22 3 3 NUM iajs-2619 96 23 ) ) PUNCT iajs-2619 96 24 near near ADP iajs-2619 96 25 𝐸𝑖 𝐸𝑖 PROPN iajs-2619 96 26 , , PUNCT iajs-2619 96 27 𝑖 𝑖 SYM iajs-2619 96 28 = = SYM iajs-2619 96 29 0,1 0,1 NUM iajs-2619 96 30 are be AUX iajs-2619 96 31 established establish VERB iajs-2619 96 32 in in ADP iajs-2619 96 33 the the DET iajs-2619 96 34 following follow VERB iajs-2619 96 35 theorems theorem NOUN iajs-2619 96 36 . . PUNCT iajs-2619 97 1 theorem theorem NOUN iajs-2619 97 2 ( ( PUNCT iajs-2619 97 3 2 2 NUM iajs-2619 97 4 ): ): PUNCT iajs-2619 97 5 if if SCONJ iajs-2619 97 6 the the DET iajs-2619 97 7 𝐸0 𝐸0 ADJ iajs-2619 97 8 point point NOUN iajs-2619 97 9 exists exist VERB iajs-2619 97 10 , , PUNCT iajs-2619 97 11 it it PRON iajs-2619 97 12 is be AUX iajs-2619 97 13 locally locally ADV iajs-2619 97 14 asymptotically asymptotically ADV iajs-2619 97 15 stable stable ADJ iajs-2619 97 16 if if SCONJ iajs-2619 97 17 the the DET iajs-2619 97 18 following follow VERB iajs-2619 97 19 conditions condition NOUN iajs-2619 97 20 are be AUX iajs-2619 97 21 held hold VERB iajs-2619 97 22 π‘Ÿ π‘Ÿ X iajs-2619 97 23 < < X iajs-2619 97 24 πœ‡ πœ‡ X iajs-2619 97 25 ( ( PUNCT iajs-2619 97 26 11a 11a NOUN iajs-2619 97 27 ) ) PUNCT iajs-2619 97 28 𝛼 𝛼 X iajs-2619 97 29 < < X iajs-2619 97 30 π‘Ž2𝑀 π‘Ž2𝑀 NOUN iajs-2619 97 31 ^ ^ PUNCT iajs-2619 98 1 + + CCONJ iajs-2619 98 2 πœ‡ πœ‡ X iajs-2619 98 3 ( ( PUNCT iajs-2619 98 4 11b 11b NOUN iajs-2619 98 5 ) ) PUNCT iajs-2619 98 6 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 98 7 ^ ^ PUNCT iajs-2619 98 8 < < X iajs-2619 98 9 π‘šπ‘–π‘› π‘šπ‘–π‘› PROPN iajs-2619 98 10 { { PUNCT iajs-2619 98 11 ΞΌ+d+Ξ·βˆ’(πœ–+𝛼 ΞΌ+d+Ξ·βˆ’(πœ–+𝛼 PROPN iajs-2619 98 12 ) ) PUNCT iajs-2619 98 13 2π‘Ž3 2π‘Ž3 NUM iajs-2619 98 14 , , PUNCT iajs-2619 98 15 πœ‡βˆ’π‘Ÿ πœ‡βˆ’π‘Ÿ PROPN iajs-2619 98 16 2π‘Ž1 2π‘Ž1 NUM iajs-2619 98 17 , , PUNCT iajs-2619 98 18 𝛾 𝛾 ADP iajs-2619 98 19 π‘Ž2 π‘Ž2 NOUN iajs-2619 98 20 } } PUNCT iajs-2619 98 21 ( ( PUNCT iajs-2619 98 22 11c 11c NOUN iajs-2619 98 23 ) ) PUNCT iajs-2619 98 24 134 134 NUM iajs-2619 98 25 ibn ibn PROPN iajs-2619 98 26 al al PROPN iajs-2619 98 27 - - PUNCT iajs-2619 98 28 haitham haitham PROPN iajs-2619 98 29 jour jour X iajs-2619 98 30 . . PROPN iajs-2619 99 1 for for ADP iajs-2619 99 2 pure pure ADJ iajs-2619 99 3 & & CCONJ iajs-2619 99 4 appl appl PROPN iajs-2619 99 5 . . PUNCT iajs-2619 100 1 sci sci PROPN iajs-2619 100 2 . . PROPN iajs-2619 101 1 34 34 NUM iajs-2619 101 2 ( ( PUNCT iajs-2619 101 3 2 2 NUM iajs-2619 101 4 ) ) PUNCT iajs-2619 101 5 2021 2021 NUM iajs-2619 101 6 proof proof NOUN iajs-2619 101 7 : : PUNCT iajs-2619 101 8 clearly clearly ADV iajs-2619 101 9 the the DET iajs-2619 101 10 jacobian jacobian ADJ iajs-2619 101 11 matrix matrix NOUN iajs-2619 101 12 of of ADP iajs-2619 101 13 system system NOUN iajs-2619 101 14 ( ( PUNCT iajs-2619 101 15 3 3 NUM iajs-2619 101 16 ) ) PUNCT iajs-2619 101 17 at at ADP iajs-2619 101 18 𝐸0 𝐸0 PROPN iajs-2619 101 19 can can AUX iajs-2619 101 20 be be AUX iajs-2619 101 21 written write VERB iajs-2619 101 22 as as ADP iajs-2619 101 23 𝐽(𝐸0 𝐽(𝐸0 NOUN iajs-2619 101 24 ) ) PUNCT iajs-2619 101 25 = = SYM iajs-2619 102 1 ( ( PUNCT iajs-2619 102 2 𝑀𝑖𝑗)4π‘₯4 𝑀𝑖𝑗)4π‘₯4 ADP iajs-2619 102 3 where where SCONJ iajs-2619 102 4 = = PRON iajs-2619 102 5 ( ( PUNCT iajs-2619 102 6 βˆ’π‘Ž2𝑀 βˆ’π‘Ž2𝑀 NOUN iajs-2619 102 7 ^ ^ PUNCT iajs-2619 102 8 βˆ’ βˆ’ NOUN iajs-2619 102 9 πœ‡ πœ‡ X iajs-2619 102 10 βˆ’π‘Ž3𝑆𝑒 βˆ’π‘Ž3𝑆𝑒 X iajs-2619 102 11 ^ ^ PUNCT iajs-2619 102 12 βˆ’π‘Ž1𝑆𝑒 βˆ’π‘Ž1𝑆𝑒 PROPN iajs-2619 102 13 ^ ^ PUNCT iajs-2619 102 14 βˆ’π‘Ž2𝑆𝑒 βˆ’π‘Ž2𝑆𝑒 X iajs-2619 102 15 ^ ^ SYM iajs-2619 102 16 0 0 NUM iajs-2619 102 17 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 102 18 ^ ^ PUNCT iajs-2619 102 19 βˆ’ βˆ’ ADP iajs-2619 102 20 πœ‡ πœ‡ ADP iajs-2619 102 21 βˆ’ βˆ’ PROPN iajs-2619 102 22 𝑑 𝑑 PROPN iajs-2619 102 23 βˆ’ βˆ’ PROPN iajs-2619 102 24 Ξ· Ξ· X iajs-2619 102 25 π‘Ž1su π‘Ž1su PROPN iajs-2619 102 26 ^ ^ SYM iajs-2619 102 27 0 0 NUM iajs-2619 102 28 0 0 NUM iajs-2619 102 29 πœ– πœ– X iajs-2619 102 30 π‘Ÿ π‘Ÿ NOUN iajs-2619 102 31 βˆ’ βˆ’ ADP iajs-2619 102 32 πœ‡ πœ‡ ADP iajs-2619 102 33 0 0 NUM iajs-2619 102 34 𝛼 𝛼 NOUN iajs-2619 102 35 𝛼 𝛼 NOUN iajs-2619 102 36 0 0 NUM iajs-2619 102 37 βˆ’π›Ύ βˆ’π›Ύ ADJ iajs-2619 102 38 ) ) PUNCT iajs-2619 102 39 now now ADV iajs-2619 102 40 , , PUNCT iajs-2619 102 41 by by ADP iajs-2619 102 42 applying apply VERB iajs-2619 102 43 the the DET iajs-2619 102 44 condition condition NOUN iajs-2619 102 45 in in ADP iajs-2619 102 46 the the DET iajs-2619 102 47 gersgorin gersgorin NOUN iajs-2619 102 48 theorem theorem NOUN iajs-2619 102 49 [ [ X iajs-2619 102 50 19 19 NUM iajs-2619 102 51 ] ] PUNCT iajs-2619 102 52 . . PUNCT iajs-2619 103 1 |𝑀𝑖𝑖| |𝑀𝑖𝑖| NOUN iajs-2619 103 2 > > PUNCT iajs-2619 103 3 βˆ‘ βˆ‘ PUNCT iajs-2619 103 4 |𝑀𝑖𝑗| |𝑀𝑖𝑗| PROPN iajs-2619 103 5 4 4 NUM iajs-2619 103 6 𝑖=1 𝑖=1 PUNCT iajs-2619 103 7 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2619 103 8 ( ( PUNCT iajs-2619 103 9 12 12 X iajs-2619 103 10 ) ) PUNCT iajs-2619 103 11 we we PRON iajs-2619 103 12 get get VERB iajs-2619 103 13 all all DET iajs-2619 103 14 the the DET iajs-2619 103 15 eigenvalues eigenvalue NOUN iajs-2619 103 16 of of ADP iajs-2619 103 17 above above ADP iajs-2619 103 18 jacobian jacobian PROPN iajs-2619 103 19 exist exist VERB iajs-2619 103 20 in in ADP iajs-2619 103 21 the the DET iajs-2619 103 22 region region NOUN iajs-2619 104 1 𝛺 𝛺 PROPN iajs-2619 104 2 = = SYM iajs-2619 104 3 βˆͺ βˆͺ NOUN iajs-2619 104 4 { { PUNCT iajs-2619 104 5 π‘’βˆ— π‘’βˆ— NOUN iajs-2619 104 6 ∈ ∈ PROPN iajs-2619 104 7 𝐢 𝐢 PROPN iajs-2619 104 8 : : PUNCT iajs-2619 104 9 |π‘’βˆ— |π‘’βˆ— PROPN iajs-2619 104 10 βˆ’ βˆ’ PROPN iajs-2619 104 11 𝑀𝑖𝑖| 𝑀𝑖𝑖| VERB iajs-2619 104 12 < < X iajs-2619 104 13 βˆ‘|𝑀𝑖𝑗| βˆ‘|𝑀𝑖𝑗| PROPN iajs-2619 104 14 4 4 NUM iajs-2619 104 15 𝑖=1 𝑖=1 PUNCT iajs-2619 104 16 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2619 104 17 } } PUNCT iajs-2619 104 18 then then ADV iajs-2619 104 19 all all DET iajs-2619 104 20 the the DET iajs-2619 104 21 eigenvalues eigenvalue NOUN iajs-2619 104 22 of of ADP iajs-2619 104 23 𝐽(𝐸0 𝐽(𝐸0 NOUN iajs-2619 104 24 ) ) PUNCT iajs-2619 104 25 exist exist VERB iajs-2619 104 26 in in ADP iajs-2619 104 27 the the DET iajs-2619 104 28 disc disc NOUN iajs-2619 104 29 centered center VERB iajs-2619 104 30 at at ADP iajs-2619 104 31 𝑀𝑖𝑖. 𝑀𝑖𝑖. X iajs-2619 104 32 thus thus ADV iajs-2619 104 33 if if SCONJ iajs-2619 104 34 the the DET iajs-2619 104 35 diagonal diagonal ADJ iajs-2619 104 36 elements element NOUN iajs-2619 104 37 are be AUX iajs-2619 104 38 negative negative ADJ iajs-2619 104 39 and and CCONJ iajs-2619 104 40 the the DET iajs-2619 104 41 conditions condition NOUN iajs-2619 104 42 ( ( PUNCT iajs-2619 104 43 11a 11a NOUN iajs-2619 104 44 ) ) PUNCT iajs-2619 104 45 and and CCONJ iajs-2619 104 46 ( ( PUNCT iajs-2619 104 47 11c 11c NOUN iajs-2619 104 48 ) ) PUNCT iajs-2619 104 49 are be AUX iajs-2619 104 50 held hold VERB iajs-2619 104 51 , , PUNCT iajs-2619 104 52 all all DET iajs-2619 104 53 the the DET iajs-2619 104 54 eigenvalues eigenvalue NOUN iajs-2619 104 55 will will AUX iajs-2619 104 56 exist exist VERB iajs-2619 104 57 in in ADP iajs-2619 104 58 the the DET iajs-2619 104 59 left left ADJ iajs-2619 104 60 half half NOUN iajs-2619 104 61 plane plane NOUN iajs-2619 104 62 and and CCONJ iajs-2619 104 63 the the DET iajs-2619 104 64 𝐸0 𝐸0 NOUN iajs-2619 104 65 is be AUX iajs-2619 104 66 locally locally ADV iajs-2619 104 67 asymptotically asymptotically ADV iajs-2619 104 68 stable stable ADJ iajs-2619 104 69 ∎ ∎ PROPN iajs-2619 104 70 theorem theorem NOUN iajs-2619 104 71 ( ( PUNCT iajs-2619 104 72 3 3 NUM iajs-2619 104 73 ): ): PUNCT iajs-2619 104 74 the the DET iajs-2619 104 75 𝐸1 𝐸1 NOUN iajs-2619 104 76 = = PUNCT iajs-2619 104 77 ( ( PUNCT iajs-2619 104 78 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 104 79 βˆ— βˆ— NOUN iajs-2619 104 80 , , PUNCT iajs-2619 104 81 πΌβˆ— πΌβˆ— PROPN iajs-2619 104 82 , , PUNCT iajs-2619 104 83 π΅βˆ—,π‘€βˆ— π΅βˆ—,π‘€βˆ— PROPN iajs-2619 104 84 ) ) PUNCT iajs-2619 104 85 of of ADP iajs-2619 104 86 system system NOUN iajs-2619 104 87 ( ( PUNCT iajs-2619 104 88 3 3 X iajs-2619 104 89 ) ) PUNCT iajs-2619 104 90 is be AUX iajs-2619 104 91 locally locally ADV iajs-2619 104 92 asymptotically asymptotically ADV iajs-2619 104 93 stable stable ADJ iajs-2619 104 94 under under ADP iajs-2619 104 95 the the DET iajs-2619 104 96 following follow VERB iajs-2619 104 97 conditions condition NOUN iajs-2619 104 98 held hold VERB iajs-2619 104 99 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 104 100 βˆ— βˆ— NOUN iajs-2619 104 101 < < X iajs-2619 104 102 π‘šπ‘–π‘› π‘šπ‘–π‘› NOUN iajs-2619 104 103 { { PUNCT iajs-2619 104 104 ΞΌ+d+Ξ·βˆ’(πœ–+𝛼 ΞΌ+d+Ξ·βˆ’(πœ–+𝛼 PROPN iajs-2619 104 105 ) ) PUNCT iajs-2619 104 106 2π‘Ž3 2π‘Ž3 NUM iajs-2619 104 107 , , PUNCT iajs-2619 104 108 2π‘Ÿπ΅βˆ—+π‘˜(πœ‡βˆ’π‘Ÿ 2π‘Ÿπ΅βˆ—+π‘˜(πœ‡βˆ’π‘Ÿ NUM iajs-2619 104 109 ) ) PUNCT iajs-2619 104 110 2π‘Ž1π‘˜ 2π‘Ž1π‘˜ NOUN iajs-2619 104 111 , , PUNCT iajs-2619 104 112 𝛾 𝛾 ADP iajs-2619 104 113 π‘Ž2 π‘Ž2 NOUN iajs-2619 104 114 } } PUNCT iajs-2619 104 115 ( ( PUNCT iajs-2619 104 116 13a 13a NOUN iajs-2619 104 117 ) ) PUNCT iajs-2619 104 118 𝛼 𝛼 PROPN iajs-2619 104 119 < < X iajs-2619 104 120 π‘Ž2𝑀 π‘Ž2𝑀 X iajs-2619 104 121 βˆ— βˆ— NOUN iajs-2619 104 122 + + X iajs-2619 104 123 πœ‡ πœ‡ X iajs-2619 104 124 ( ( PUNCT iajs-2619 104 125 13b 13b NOUN iajs-2619 104 126 ) ) PUNCT iajs-2619 104 127 proof proof NOUN iajs-2619 104 128 : : PUNCT iajs-2619 104 129 it it PRON iajs-2619 104 130 is be AUX iajs-2619 104 131 easy easy ADJ iajs-2619 104 132 from from ADP iajs-2619 104 133 jacobian jacobian ADJ iajs-2619 104 134 matrix matrix NOUN iajs-2619 104 135 of of ADP iajs-2619 104 136 system system NOUN iajs-2619 104 137 ( ( PUNCT iajs-2619 104 138 3 3 NUM iajs-2619 104 139 ) ) PUNCT iajs-2619 104 140 at at ADP iajs-2619 104 141 𝐸1 𝐸1 NOUN iajs-2619 104 142 = = PUNCT iajs-2619 104 143 ( ( PUNCT iajs-2619 104 144 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 104 145 βˆ— βˆ— NOUN iajs-2619 104 146 , , PUNCT iajs-2619 104 147 πΌβˆ— πΌβˆ— PROPN iajs-2619 104 148 , , PUNCT iajs-2619 104 149 π΅βˆ—,π‘€βˆ— π΅βˆ—,π‘€βˆ— PROPN iajs-2619 104 150 ) ) PUNCT iajs-2619 104 151 that that PRON iajs-2619 104 152 can can AUX iajs-2619 104 153 be be AUX iajs-2619 104 154 written write VERB iajs-2619 104 155 as as ADP iajs-2619 104 156 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-2619 104 157 ) ) PUNCT iajs-2619 104 158 = = NOUN iajs-2619 104 159 ( ( PUNCT iajs-2619 104 160 𝑑𝑖𝑗)4𝑋4 𝑑𝑖𝑗)4𝑋4 ADV iajs-2619 104 161 here here ADV iajs-2619 104 162 , , PUNCT iajs-2619 104 163 𝑑11 𝑑11 NOUN iajs-2619 104 164 = = SYM iajs-2619 104 165 βˆ’π‘Ž1𝐡 βˆ’π‘Ž1𝐡 PROPN iajs-2619 104 166 βˆ— βˆ— VERB iajs-2619 104 167 βˆ’ βˆ’ NOUN iajs-2619 104 168 π‘Ž2𝑀 π‘Ž2𝑀 ADP iajs-2619 104 169 βˆ— βˆ— NOUN iajs-2619 105 1 βˆ’ βˆ’ PROPN iajs-2619 106 1 π‘Ž3𝐼 π‘Ž3𝐼 NUM iajs-2619 106 2 βˆ— βˆ— NOUN iajs-2619 106 3 βˆ’ βˆ’ PROPN iajs-2619 106 4 π‘Ž4 π‘Ž4 PROPN iajs-2619 106 5 βˆ’ βˆ’ PROPN iajs-2619 106 6 πœ‡ πœ‡ ADP iajs-2619 106 7 , , PUNCT iajs-2619 106 8 𝑑12 𝑑12 PROPN iajs-2619 106 9 = = SYM iajs-2619 106 10 βˆ’π‘Ž3𝑆𝑒 βˆ’π‘Ž3𝑆𝑒 PROPN iajs-2619 106 11 βˆ— βˆ— NOUN iajs-2619 106 12 , , PUNCT iajs-2619 106 13 𝑑13 𝑑13 PROPN iajs-2619 106 14 = = SYM iajs-2619 106 15 βˆ’π‘Ž1𝑆𝑒 βˆ’π‘Ž1𝑆𝑒 NOUN iajs-2619 106 16 βˆ— βˆ— NOUN iajs-2619 106 17 𝑑14 𝑑14 NOUN iajs-2619 106 18 = = SYM iajs-2619 106 19 βˆ’π‘Ž2𝑆𝑒 βˆ’π‘Ž2𝑆𝑒 X iajs-2619 106 20 βˆ— βˆ— NOUN iajs-2619 106 21 , , PUNCT iajs-2619 106 22 𝑑21 𝑑21 PROPN iajs-2619 106 23 = = PUNCT iajs-2619 107 1 π‘Ž1𝐡 π‘Ž1𝐡 NOUN iajs-2619 107 2 βˆ— βˆ— NOUN iajs-2619 107 3 + + CCONJ iajs-2619 107 4 π‘Ž3𝐼 π‘Ž3𝐼 NUM iajs-2619 107 5 βˆ— βˆ— NOUN iajs-2619 107 6 + + CCONJ iajs-2619 107 7 π‘Ž4 π‘Ž4 PROPN iajs-2619 107 8 , , PUNCT iajs-2619 107 9 𝑑22 𝑑22 PROPN iajs-2619 107 10 = = SYM iajs-2619 107 11 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 107 12 βˆ— βˆ— NOUN iajs-2619 107 13 βˆ’ βˆ’ PROPN iajs-2619 107 14 ( ( PUNCT iajs-2619 107 15 πœ‡ πœ‡ ADP iajs-2619 107 16 + + X iajs-2619 107 17 𝑑 𝑑 NOUN iajs-2619 107 18 ) ) PUNCT iajs-2619 107 19 βˆ’ βˆ’ PROPN iajs-2619 107 20 Ξ· Ξ· PROPN iajs-2619 107 21 d23 d23 PROPN iajs-2619 107 22 = = SYM iajs-2619 107 23 π‘Ž1su π‘Ž1su PROPN iajs-2619 107 24 βˆ— βˆ— PROPN iajs-2619 107 25 , , PUNCT iajs-2619 107 26 𝑑32 𝑑32 NOUN iajs-2619 107 27 = = SYM iajs-2619 107 28 πœ– πœ– PROPN iajs-2619 107 29 , , PUNCT iajs-2619 107 30 𝑑33 𝑑33 NOUN iajs-2619 107 31 = = SYM iajs-2619 107 32 π‘˜π‘Ÿ π‘˜π‘Ÿ PROPN iajs-2619 107 33 βˆ’ βˆ’ PROPN iajs-2619 107 34 2π‘Ÿπ΅βˆ— 2π‘Ÿπ΅βˆ— NUM iajs-2619 107 35 βˆ’ βˆ’ NOUN iajs-2619 107 36 π‘˜πœ‡ π‘˜πœ‡ VERB iajs-2619 107 37 π‘˜ π‘˜ PRON iajs-2619 107 38 , , PUNCT iajs-2619 107 39 𝑑41 𝑑41 NOUN iajs-2619 107 40 = = SYM iajs-2619 107 41 𝛼 𝛼 NOUN iajs-2619 107 42 , , PUNCT iajs-2619 107 43 𝑑42 𝑑42 PROPN iajs-2619 107 44 = = SYM iajs-2619 107 45 𝛼 𝛼 NOUN iajs-2619 107 46 𝑑44 𝑑44 NOUN iajs-2619 107 47 = = SYM iajs-2619 107 48 βˆ’π›Ύ βˆ’π›Ύ ADJ iajs-2619 107 49 , , PUNCT iajs-2619 107 50 𝑑31 𝑑31 NOUN iajs-2619 107 51 = = SYM iajs-2619 107 52 d24 d24 ADJ iajs-2619 107 53 = = SYM iajs-2619 107 54 𝑑34 𝑑34 NOUN iajs-2619 107 55 = = NOUN iajs-2619 107 56 𝑑43 𝑑43 NOUN iajs-2619 107 57 = = NOUN iajs-2619 107 58 0 0 PUNCT iajs-2619 107 59 now now ADV iajs-2619 107 60 , , PUNCT iajs-2619 107 61 by by ADP iajs-2619 107 62 applying apply VERB iajs-2619 107 63 the the DET iajs-2619 107 64 condition condition NOUN iajs-2619 107 65 in in ADP iajs-2619 107 66 the the DET iajs-2619 107 67 gersgorin gersgorin NOUN iajs-2619 107 68 theorem theorem NOUN iajs-2619 107 69 [ [ X iajs-2619 107 70 19 19 NUM iajs-2619 107 71 ] ] PUNCT iajs-2619 107 72 . . PUNCT iajs-2619 108 1 |𝑑𝑖𝑖| |𝑑𝑖𝑖| NOUN iajs-2619 108 2 > > PUNCT iajs-2619 108 3 βˆ‘ βˆ‘ PUNCT iajs-2619 108 4 |𝑑𝑖𝑗| |𝑑𝑖𝑗| PROPN iajs-2619 108 5 4 4 NUM iajs-2619 108 6 𝑖=1 𝑖=1 NUM iajs-2619 108 7 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2619 108 8 ( ( PUNCT iajs-2619 108 9 14 14 NUM iajs-2619 108 10 ) ) PUNCT iajs-2619 108 11 we we PRON iajs-2619 108 12 get get VERB iajs-2619 108 13 all all DET iajs-2619 108 14 the the DET iajs-2619 108 15 eigenvalues eigenvalue NOUN iajs-2619 108 16 of of ADP iajs-2619 108 17 above above ADP iajs-2619 108 18 jacobian jacobian PROPN iajs-2619 108 19 exist exist VERB iajs-2619 108 20 in in ADP iajs-2619 108 21 the the DET iajs-2619 108 22 region region NOUN iajs-2619 108 23 135 135 NUM iajs-2619 108 24 ibn ibn PROPN iajs-2619 108 25 al al PROPN iajs-2619 108 26 - - PUNCT iajs-2619 108 27 haitham haitham PROPN iajs-2619 108 28 jour jour X iajs-2619 108 29 . . PROPN iajs-2619 109 1 for for ADP iajs-2619 109 2 pure pure ADJ iajs-2619 109 3 & & CCONJ iajs-2619 109 4 appl appl PROPN iajs-2619 109 5 . . PUNCT iajs-2619 110 1 sci sci PROPN iajs-2619 110 2 . . PROPN iajs-2619 111 1 34 34 NUM iajs-2619 111 2 ( ( PUNCT iajs-2619 111 3 2 2 NUM iajs-2619 111 4 ) ) PUNCT iajs-2619 111 5 2021 2021 NUM iajs-2619 111 6 𝛺 𝛺 NOUN iajs-2619 111 7 = = NOUN iajs-2619 111 8 βˆͺ βˆͺ NOUN iajs-2619 111 9 { { PUNCT iajs-2619 111 10 π‘’βˆ— π‘’βˆ— NOUN iajs-2619 111 11 ∈ ∈ PROPN iajs-2619 111 12 𝐢 𝐢 PROPN iajs-2619 111 13 : : PUNCT iajs-2619 111 14 |π‘’βˆ— |π‘’βˆ— PROPN iajs-2619 111 15 βˆ’ βˆ’ PROPN iajs-2619 111 16 𝑑𝑖𝑖| 𝑑𝑖𝑖| VERB iajs-2619 111 17 < < X iajs-2619 111 18 βˆ‘|𝑑𝑖𝑗| βˆ‘|𝑑𝑖𝑗| PROPN iajs-2619 111 19 4 4 NUM iajs-2619 111 20 𝑖=1 𝑖=1 PUNCT iajs-2619 111 21 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2619 111 22 } } PUNCT iajs-2619 111 23 then then ADV iajs-2619 111 24 all all DET iajs-2619 111 25 the the DET iajs-2619 111 26 eigenvalues eigenvalue NOUN iajs-2619 111 27 of of ADP iajs-2619 111 28 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-2619 111 29 ) ) PUNCT iajs-2619 111 30 exist exist VERB iajs-2619 111 31 in in ADP iajs-2619 111 32 the the DET iajs-2619 111 33 disc disc NOUN iajs-2619 111 34 centered center VERB iajs-2619 111 35 at at ADP iajs-2619 111 36 𝑑𝑖𝑖 𝑑𝑖𝑖 PROPN iajs-2619 111 37 . . PUNCT iajs-2619 112 1 thus thus ADV iajs-2619 112 2 if if SCONJ iajs-2619 112 3 the the DET iajs-2619 112 4 diagonal diagonal ADJ iajs-2619 112 5 elements element NOUN iajs-2619 112 6 are be AUX iajs-2619 112 7 negative negative ADJ iajs-2619 112 8 and and CCONJ iajs-2619 112 9 the the DET iajs-2619 112 10 conditions condition NOUN iajs-2619 112 11 ( ( PUNCT iajs-2619 112 12 13a 13a NUM iajs-2619 112 13 ) ) PUNCT iajs-2619 112 14 and and CCONJ iajs-2619 112 15 ( ( PUNCT iajs-2619 112 16 13b 13b NOUN iajs-2619 112 17 ) ) PUNCT iajs-2619 112 18 are be AUX iajs-2619 112 19 held hold VERB iajs-2619 112 20 , , PUNCT iajs-2619 112 21 all all DET iajs-2619 112 22 the the DET iajs-2619 112 23 eigenvalues eigenvalue NOUN iajs-2619 112 24 will will AUX iajs-2619 112 25 exist exist VERB iajs-2619 112 26 in in ADP iajs-2619 112 27 the the DET iajs-2619 112 28 left left ADJ iajs-2619 112 29 half half NOUN iajs-2619 112 30 plane plane NOUN iajs-2619 112 31 and and CCONJ iajs-2619 112 32 the the DET iajs-2619 112 33 𝐸1 𝐸1 NOUN iajs-2619 112 34 is be AUX iajs-2619 112 35 locally locally ADV iajs-2619 112 36 asymptotically asymptotically ADV iajs-2619 112 37 stable stable ADJ iajs-2619 112 38 . . PUNCT iajs-2619 113 1 ∎ ∎ ADV iajs-2619 113 2 5 5 NUM iajs-2619 113 3 . . PUNCT iajs-2619 113 4 global global ADJ iajs-2619 113 5 stability stability NOUN iajs-2619 113 6 analysis analysis NOUN iajs-2619 113 7 : : PUNCT iajs-2619 113 8 in in ADP iajs-2619 113 9 this this DET iajs-2619 113 10 section section NOUN iajs-2619 113 11 , , PUNCT iajs-2619 113 12 we we PRON iajs-2619 113 13 discuss discuss VERB iajs-2619 113 14 the the DET iajs-2619 113 15 global global ADJ iajs-2619 113 16 stability stability NOUN iajs-2619 113 17 conditions condition NOUN iajs-2619 113 18 and and CCONJ iajs-2619 113 19 determine determine VERB iajs-2619 113 20 the the DET iajs-2619 113 21 basin basin NOUN iajs-2619 113 22 of of ADP iajs-2619 113 23 attraction attraction NOUN iajs-2619 113 24 of of ADP iajs-2619 113 25 these these DET iajs-2619 113 26 equilibrium equilibrium NOUN iajs-2619 113 27 points point NOUN iajs-2619 113 28 of of ADP iajs-2619 113 29 system system NOUN iajs-2619 113 30 ( ( PUNCT iajs-2619 113 31 3 3 NUM iajs-2619 113 32 ) ) PUNCT iajs-2619 113 33 that that PRON iajs-2619 113 34 is be AUX iajs-2619 113 35 presented present VERB iajs-2619 113 36 as as SCONJ iajs-2619 113 37 shown show VERB iajs-2619 113 38 in in ADP iajs-2619 113 39 the the DET iajs-2619 113 40 following follow VERB iajs-2619 113 41 theorems theorem NOUN iajs-2619 113 42 . . PUNCT iajs-2619 114 1 theorem theorem NOUN iajs-2619 114 2 ( ( PUNCT iajs-2619 114 3 4 4 NUM iajs-2619 114 4 ): ): PUNCT iajs-2619 114 5 the the DET iajs-2619 114 6 𝐸0 𝐸0 NOUN iajs-2619 114 7 of of ADP iajs-2619 114 8 system system NOUN iajs-2619 114 9 ( ( PUNCT iajs-2619 114 10 3 3 X iajs-2619 114 11 ) ) PUNCT iajs-2619 114 12 is be AUX iajs-2619 114 13 globally globally ADV iajs-2619 114 14 asymptotically asymptotically ADV iajs-2619 114 15 in in ADP iajs-2619 114 16 the the DET iajs-2619 114 17 sub sub NOUN iajs-2619 114 18 region region NOUN iajs-2619 114 19 of of ADP iajs-2619 114 20 𝑅+ 𝑅+ PROPN iajs-2619 114 21 4 4 NUM iajs-2619 114 22 under under ADP iajs-2619 114 23 the the DET iajs-2619 114 24 following follow VERB iajs-2619 114 25 sufficient sufficient ADJ iajs-2619 114 26 conditions condition NOUN iajs-2619 114 27 ( ( PUNCT iajs-2619 114 28 π‘Ž2 π‘Ž2 NOUN iajs-2619 114 29 βˆ’ βˆ’ PROPN iajs-2619 114 30 𝛼 𝛼 PROPN iajs-2619 114 31 𝑀 𝑀 PROPN iajs-2619 114 32 ) ) PUNCT iajs-2619 114 33 2 2 NUM iajs-2619 114 34 < < X iajs-2619 114 35 4 4 NUM iajs-2619 114 36 ( ( PUNCT iajs-2619 114 37 π‘Ž2𝑀 π‘Ž2𝑀 ADJ iajs-2619 114 38 ^+πœ‡ ^+πœ‡ NOUN iajs-2619 114 39 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 114 40 ) ) PUNCT iajs-2619 114 41 ( ( PUNCT iajs-2619 114 42 𝛾 𝛾 PROPN iajs-2619 114 43 𝑀 𝑀 PROPN iajs-2619 114 44 ) ) PUNCT iajs-2619 114 45 ( ( PUNCT iajs-2619 114 46 15a 15a NOUN iajs-2619 114 47 ) ) PUNCT iajs-2619 114 48 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 114 49 ^ ^ PUNCT iajs-2619 114 50 < < X iajs-2619 114 51 π‘šπ‘–π‘› π‘šπ‘–π‘› PROPN iajs-2619 114 52 { { PUNCT iajs-2619 114 53 πœ‡βˆ’π‘Ÿ πœ‡βˆ’π‘Ÿ PROPN iajs-2619 114 54 π‘Ž1 π‘Ž1 INTJ iajs-2619 114 55 , , PUNCT iajs-2619 114 56 ( ( PUNCT iajs-2619 114 57 πœ‡+𝑑+πœ‚βˆ’π›Όβˆ’πœ–)𝑀+𝛼𝑀^ πœ‡+𝑑+πœ‚βˆ’π›Όβˆ’πœ–)𝑀+𝛼𝑀^ PROPN iajs-2619 114 58 π‘Ž3𝑀 π‘Ž3𝑀 PROPN iajs-2619 114 59 } } PUNCT iajs-2619 114 60 ( ( PUNCT iajs-2619 114 61 15b 15b NOUN iajs-2619 114 62 ) ) PUNCT iajs-2619 114 63 proof proof NOUN iajs-2619 114 64 : : PUNCT iajs-2619 114 65 consider consider VERB iajs-2619 114 66 the the DET iajs-2619 114 67 following follow VERB iajs-2619 114 68 positive positive ADJ iajs-2619 114 69 definite definite ADJ iajs-2619 114 70 function function NOUN iajs-2619 114 71 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 NOUN iajs-2619 114 72 , , PUNCT iajs-2619 114 73 𝐼 𝐼 PROPN iajs-2619 114 74 , , PUNCT iajs-2619 114 75 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 114 76 ) ) PUNCT iajs-2619 114 77 = = SYM iajs-2619 115 1 ( ( PUNCT iajs-2619 115 2 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 115 3 βˆ’ βˆ’ NOUN iajs-2619 115 4 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 115 5 ^ ^ PUNCT iajs-2619 115 6 βˆ’ βˆ’ PROPN iajs-2619 115 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 115 8 ^ ^ PUNCT iajs-2619 115 9 ln ln ADJ iajs-2619 115 10 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 115 11 𝑆𝑒^ 𝑆𝑒^ PROPN iajs-2619 115 12 ) ) PUNCT iajs-2619 116 1 + + CCONJ iajs-2619 116 2 𝐼 𝐼 PROPN iajs-2619 116 3 + + CCONJ iajs-2619 116 4 𝐡 𝐡 NOUN iajs-2619 116 5 + + CCONJ iajs-2619 116 6 ( ( PUNCT iajs-2619 116 7 𝑀 𝑀 PROPN iajs-2619 116 8 βˆ’π‘€^ βˆ’π‘€^ VERB iajs-2619 116 9 βˆ’π‘€^ βˆ’π‘€^ PROPN iajs-2619 116 10 ln ln PROPN iajs-2619 116 11 𝑀 𝑀 PROPN iajs-2619 116 12 𝑀^ 𝑀^ PUNCT iajs-2619 116 13 ) ) PUNCT iajs-2619 117 1 it it PRON iajs-2619 117 2 is be AUX iajs-2619 117 3 easy easy ADJ iajs-2619 117 4 to to PART iajs-2619 117 5 see see VERB iajs-2619 117 6 that that SCONJ iajs-2619 117 7 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 NOUN iajs-2619 117 8 , , PUNCT iajs-2619 117 9 𝐼 𝐼 PROPN iajs-2619 117 10 , , PUNCT iajs-2619 117 11 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 117 12 ) ) PUNCT iajs-2619 117 13 ∈ ∈ PROPN iajs-2619 117 14 𝐢 𝐢 PROPN iajs-2619 117 15 1(𝑅+ 1(𝑅+ PROPN iajs-2619 117 16 4 4 NUM iajs-2619 117 17 , , PUNCT iajs-2619 117 18 𝑅 𝑅 NOUN iajs-2619 117 19 ) ) PUNCT iajs-2619 117 20 and and CCONJ iajs-2619 117 21 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 NOUN iajs-2619 117 22 ^ ^ SYM iajs-2619 117 23 , , PUNCT iajs-2619 117 24 0,0,𝑀^ 0,0,𝑀^ NUM iajs-2619 117 25 ) ) PUNCT iajs-2619 117 26 = = SYM iajs-2619 117 27 0 0 NUM iajs-2619 117 28 , , PUNCT iajs-2619 117 29 while while SCONJ iajs-2619 117 30 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 NOUN iajs-2619 117 31 , , PUNCT iajs-2619 117 32 𝐼 𝐼 NOUN iajs-2619 117 33 , , PUNCT iajs-2619 117 34 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 117 35 ) ) PUNCT iajs-2619 117 36 > > X iajs-2619 117 37 0 0 NUM iajs-2619 117 38 βˆ€(𝑆𝑒 βˆ€(𝑆𝑒 NOUN iajs-2619 117 39 , , PUNCT iajs-2619 117 40 𝐼 𝐼 PROPN iajs-2619 117 41 , , PUNCT iajs-2619 117 42 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 117 43 ) ) PUNCT iajs-2619 117 44 ∈ ∈ PROPN iajs-2619 117 45 𝑅+ 𝑅+ NUM iajs-2619 117 46 4 4 NUM iajs-2619 117 47 and and CCONJ iajs-2619 117 48 ( ( PUNCT iajs-2619 117 49 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 117 50 , , PUNCT iajs-2619 117 51 𝐼 𝐼 PROPN iajs-2619 117 52 , , PUNCT iajs-2619 117 53 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 117 54 ) ) PUNCT iajs-2619 117 55 β‰  β‰  PROPN iajs-2619 117 56 ( ( PUNCT iajs-2619 117 57 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 117 58 ^ ^ NUM iajs-2619 117 59 , , PUNCT iajs-2619 117 60 0,0,𝑀^ 0,0,𝑀^ NUM iajs-2619 117 61 ) ) PUNCT iajs-2619 117 62 . . PUNCT iajs-2619 118 1 𝑑𝑉0 𝑑𝑉0 ADP iajs-2619 118 2 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 3 = = SYM iajs-2619 118 4 ( ( PUNCT iajs-2619 118 5 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 6 βˆ’ βˆ’ NOUN iajs-2619 118 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 8 ^ ^ PUNCT iajs-2619 118 9 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 10 ) ) PUNCT iajs-2619 118 11 𝑑𝑆𝑒 𝑑𝑆𝑒 NOUN iajs-2619 118 12 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 13 + + NOUN iajs-2619 118 14 𝑑𝐼 𝑑𝐼 ADV iajs-2619 118 15 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 16 + + PRON iajs-2619 118 17 𝑑𝐡 𝑑𝐡 PROPN iajs-2619 118 18 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 19 + + PRON iajs-2619 118 20 ( ( PUNCT iajs-2619 118 21 𝑀 𝑀 PROPN iajs-2619 118 22 βˆ’π‘€^ βˆ’π‘€^ PROPN iajs-2619 118 23 𝑀 𝑀 PROPN iajs-2619 118 24 ) ) PUNCT iajs-2619 118 25 𝑑𝑀 𝑑𝑀 NOUN iajs-2619 118 26 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 27 𝑑𝑉0 𝑑𝑉0 ADP iajs-2619 118 28 𝑑𝑑 𝑑𝑑 ADP iajs-2619 118 29 = = PUNCT iajs-2619 118 30 ( ( PUNCT iajs-2619 118 31 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 32 βˆ’ βˆ’ NOUN iajs-2619 118 33 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 34 ^ ^ PUNCT iajs-2619 118 35 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 118 36 ) ) PUNCT iajs-2619 119 1 [ [ X iajs-2619 119 2 πœƒ πœƒ X iajs-2619 119 3 βˆ’ βˆ’ X iajs-2619 119 4 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 119 5 βˆ’ βˆ’ NOUN iajs-2619 119 6 π‘Ž2𝑆𝑒𝑀 π‘Ž2𝑆𝑒𝑀 NOUN iajs-2619 119 7 βˆ’ βˆ’ PROPN iajs-2619 119 8 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 119 9 βˆ’ βˆ’ NOUN iajs-2619 119 10 π‘Ž4𝑆𝑒 π‘Ž4𝑆𝑒 NOUN iajs-2619 119 11 βˆ’ βˆ’ PROPN iajs-2619 119 12 πœ‡π‘†π‘’ πœ‡π‘†π‘’ X iajs-2619 119 13 ] ] X iajs-2619 120 1 + + PROPN iajs-2619 120 2 [ [ X iajs-2619 120 3 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 X iajs-2619 120 4 + + ADJ iajs-2619 120 5 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 120 6 βˆ’ βˆ’ PROPN iajs-2619 120 7 ( ( PUNCT iajs-2619 120 8 πœ‡ πœ‡ X iajs-2619 120 9 + + X iajs-2619 120 10 𝑑 𝑑 X iajs-2619 120 11 + + X iajs-2619 120 12 πœ‚)𝐼 πœ‚)𝐼 ADJ iajs-2619 120 13 ] ] X iajs-2619 120 14 + + CCONJ iajs-2619 121 1 [ [ X iajs-2619 121 2 π‘Ÿπ΅ π‘Ÿπ΅ NOUN iajs-2619 121 3 ( ( PUNCT iajs-2619 121 4 1 1 NUM iajs-2619 121 5 βˆ’ βˆ’ PROPN iajs-2619 121 6 𝐡 𝐡 PROPN iajs-2619 121 7 π‘˜ π‘˜ NOUN iajs-2619 121 8 ) ) PUNCT iajs-2619 121 9 βˆ’ βˆ’ ADP iajs-2619 121 10 πœ‡π΅ πœ‡π΅ VERB iajs-2619 121 11 + + X iajs-2619 121 12 πœ–πΌ πœ–πΌ NOUN iajs-2619 121 13 ] ] X iajs-2619 121 14 + + CCONJ iajs-2619 121 15 ( ( PUNCT iajs-2619 121 16 𝑀 𝑀 PROPN iajs-2619 121 17 βˆ’π‘€^ βˆ’π‘€^ PROPN iajs-2619 121 18 𝑀 𝑀 PROPN iajs-2619 121 19 ) ) PUNCT iajs-2619 122 1 [ [ X iajs-2619 122 2 𝛼 𝛼 X iajs-2619 122 3 ( ( PUNCT iajs-2619 122 4 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 122 5 + + CCONJ iajs-2619 122 6 𝐼 𝐼 PROPN iajs-2619 122 7 ) ) PUNCT iajs-2619 122 8 – – PUNCT iajs-2619 122 9 𝛾𝑀 𝛾𝑀 ADJ iajs-2619 122 10 ] ] X iajs-2619 122 11 furthermore furthermore ADV iajs-2619 122 12 by by ADP iajs-2619 122 13 taking take VERB iajs-2619 122 14 the the DET iajs-2619 122 15 derivative derivative NOUN iajs-2619 122 16 and and CCONJ iajs-2619 122 17 simplifying simplify VERB iajs-2619 122 18 the the DET iajs-2619 122 19 resulting result VERB iajs-2619 122 20 terms term NOUN iajs-2619 122 21 , , PUNCT iajs-2619 122 22 we we PRON iajs-2619 122 23 obtain obtain VERB iajs-2619 122 24 that that SCONJ iajs-2619 122 25 𝑑𝑉0 𝑑𝑉0 ADP iajs-2619 122 26 𝑑𝑑 𝑑𝑑 ADP iajs-2619 122 27 = = SYM iajs-2619 122 28 βˆ’ βˆ’ PROPN iajs-2619 123 1 [ [ X iajs-2619 123 2 ( ( PUNCT iajs-2619 123 3 π‘Ž2𝑀 π‘Ž2𝑀 NOUN iajs-2619 123 4 ^ ^ PUNCT iajs-2619 123 5 + + CCONJ iajs-2619 123 6 πœ‡ πœ‡ ADP iajs-2619 123 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 123 8 ) ) PUNCT iajs-2619 123 9 ( ( PUNCT iajs-2619 123 10 𝑆𝑒 𝑆𝑒 ADP iajs-2619 123 11 βˆ’π‘†π‘’ βˆ’π‘†π‘’ NOUN iajs-2619 123 12 ^)2 ^)2 PUNCT iajs-2619 123 13 + + CCONJ iajs-2619 123 14 ( ( PUNCT iajs-2619 123 15 π‘Ž2 π‘Ž2 NOUN iajs-2619 123 16 βˆ’ βˆ’ PROPN iajs-2619 123 17 𝛼 𝛼 PROPN iajs-2619 123 18 𝑀 𝑀 PROPN iajs-2619 123 19 ) ) PUNCT iajs-2619 123 20 ( ( PUNCT iajs-2619 123 21 𝑀 𝑀 PROPN iajs-2619 123 22 βˆ’π‘€^)(𝑆𝑒 βˆ’π‘€^)(𝑆𝑒 AUX iajs-2619 123 23 βˆ’ βˆ’ PROPN iajs-2619 123 24 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 123 25 ^ ^ PUNCT iajs-2619 123 26 ) ) PUNCT iajs-2619 124 1 + + CCONJ iajs-2619 124 2 𝛾 𝛾 ADP iajs-2619 124 3 𝑀 𝑀 PROPN iajs-2619 124 4 ( ( PUNCT iajs-2619 124 5 π‘€βˆ’π‘€^)2 π‘€βˆ’π‘€^)2 NOUN iajs-2619 124 6 ] ] X iajs-2619 124 7 βˆ’ βˆ’ PROPN iajs-2619 124 8 ( ( PUNCT iajs-2619 124 9 πœ‡ πœ‡ X iajs-2619 124 10 βˆ’ βˆ’ PROPN iajs-2619 124 11 ( ( PUNCT iajs-2619 124 12 π‘Ÿ π‘Ÿ NOUN iajs-2619 124 13 + + CCONJ iajs-2619 124 14 π‘Ž1𝑆𝑒 π‘Ž1𝑆𝑒 PRON iajs-2619 124 15 ^))𝐡 ^))𝐡 ADV iajs-2619 124 16 βˆ’ βˆ’ PROPN iajs-2619 125 1 [ [ X iajs-2619 125 2 ( ( PUNCT iajs-2619 125 3 πœ‡ πœ‡ X iajs-2619 125 4 + + X iajs-2619 125 5 𝑑 𝑑 PROPN iajs-2619 125 6 + + ADJ iajs-2619 125 7 πœ‚ πœ‚ X iajs-2619 125 8 ) ) PUNCT iajs-2619 125 9 + + SYM iajs-2619 125 10 𝛼𝑀^ 𝛼𝑀^ NUM iajs-2619 125 11 𝑀 𝑀 PROPN iajs-2619 125 12 βˆ’ βˆ’ PROPN iajs-2619 125 13 ( ( PUNCT iajs-2619 125 14 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 125 15 ^+∈ ^+∈ VERB iajs-2619 125 16 + + NOUN iajs-2619 125 17 𝛼 𝛼 NOUN iajs-2619 125 18 ) ) PUNCT iajs-2619 125 19 ] ] PUNCT iajs-2619 126 1 𝐼 𝐼 ADP iajs-2619 126 2 136 136 NUM iajs-2619 126 3 ibn ibn PROPN iajs-2619 126 4 al al PROPN iajs-2619 126 5 - - PUNCT iajs-2619 126 6 haitham haitham PROPN iajs-2619 126 7 jour jour X iajs-2619 126 8 . . PROPN iajs-2619 127 1 for for ADP iajs-2619 127 2 pure pure ADJ iajs-2619 127 3 & & CCONJ iajs-2619 127 4 appl appl PROPN iajs-2619 127 5 . . PUNCT iajs-2619 128 1 sci sci PROPN iajs-2619 128 2 . . PROPN iajs-2619 129 1 34 34 NUM iajs-2619 129 2 ( ( PUNCT iajs-2619 129 3 2 2 NUM iajs-2619 129 4 ) ) PUNCT iajs-2619 129 5 2021 2021 NUM iajs-2619 130 1 by by ADP iajs-2619 130 2 using use VERB iajs-2619 130 3 the the DET iajs-2619 130 4 above above ADJ iajs-2619 130 5 condition condition NOUN iajs-2619 130 6 , , PUNCT iajs-2619 130 7 we we PRON iajs-2619 130 8 obtain obtain VERB iajs-2619 130 9 that that SCONJ iajs-2619 130 10 𝑑𝑉0 𝑑𝑉0 ADP iajs-2619 130 11 𝑑𝑑 𝑑𝑑 ADP iajs-2619 130 12 ≀ ≀ NOUN iajs-2619 130 13 βˆ’[√ βˆ’[√ NOUN iajs-2619 130 14 π‘Ž2𝑀^ π‘Ž2𝑀^ PROPN iajs-2619 130 15 + + PUNCT iajs-2619 130 16 πœ‡ πœ‡ X iajs-2619 130 17 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 130 18 ( ( PUNCT iajs-2619 130 19 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 130 20 βˆ’ βˆ’ NOUN iajs-2619 130 21 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 130 22 ^ ^ PUNCT iajs-2619 130 23 ) ) PUNCT iajs-2619 131 1 + + CCONJ iajs-2619 131 2 √ √ NUM iajs-2619 131 3 𝛾 𝛾 ADP iajs-2619 131 4 𝑀 𝑀 PROPN iajs-2619 131 5 ( ( PUNCT iajs-2619 131 6 𝑀 𝑀 PROPN iajs-2619 131 7 βˆ’π‘€^ βˆ’π‘€^ ADV iajs-2619 131 8 ) ) PUNCT iajs-2619 131 9 ] ] PUNCT iajs-2619 131 10 2 2 NUM iajs-2619 131 11 βˆ’ βˆ’ NOUN iajs-2619 131 12 ( ( PUNCT iajs-2619 131 13 πœ‡ πœ‡ X iajs-2619 131 14 βˆ’ βˆ’ PROPN iajs-2619 131 15 ( ( PUNCT iajs-2619 131 16 π‘Ÿ π‘Ÿ NOUN iajs-2619 131 17 + + CCONJ iajs-2619 131 18 π‘Ž1𝑆𝑒 π‘Ž1𝑆𝑒 PRON iajs-2619 131 19 ^))𝐡 ^))𝐡 ADV iajs-2619 131 20 βˆ’ βˆ’ PROPN iajs-2619 132 1 [ [ X iajs-2619 132 2 ( ( PUNCT iajs-2619 132 3 πœ‡ πœ‡ X iajs-2619 132 4 + + X iajs-2619 132 5 𝑑 𝑑 PROPN iajs-2619 132 6 + + ADJ iajs-2619 132 7 πœ‚ πœ‚ X iajs-2619 132 8 ) ) PUNCT iajs-2619 132 9 + + SYM iajs-2619 132 10 𝛼𝑀^ 𝛼𝑀^ NUM iajs-2619 132 11 𝑀 𝑀 PROPN iajs-2619 132 12 βˆ’ βˆ’ PROPN iajs-2619 132 13 ( ( PUNCT iajs-2619 132 14 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 132 15 ^+∈ ^+∈ VERB iajs-2619 132 16 + + NOUN iajs-2619 132 17 𝛼 𝛼 NOUN iajs-2619 132 18 ) ) PUNCT iajs-2619 132 19 ] ] PUNCT iajs-2619 133 1 𝐼 𝐼 ADP iajs-2619 133 2 clearly clearly ADV iajs-2619 133 3 , , PUNCT iajs-2619 133 4 𝑉0Μ‡ 𝑉0Μ‡ PROPN iajs-2619 133 5 = = NOUN iajs-2619 133 6 0 0 NUM iajs-2619 133 7 at at ADP iajs-2619 133 8 𝐸0 𝐸0 ADJ iajs-2619 133 9 = = SYM iajs-2619 133 10 ( ( PUNCT iajs-2619 133 11 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 133 12 ^ ^ SYM iajs-2619 133 13 , , PUNCT iajs-2619 133 14 0,0,𝑀^ 0,0,𝑀^ NUM iajs-2619 133 15 ) ) PUNCT iajs-2619 133 16 , , PUNCT iajs-2619 133 17 moreover moreover ADV iajs-2619 133 18 𝑉0Μ‡ 𝑉0Μ‡ VERB iajs-2619 133 19 < < X iajs-2619 133 20 0 0 X iajs-2619 133 21 otherwise otherwise ADV iajs-2619 133 22 . . PUNCT iajs-2619 134 1 hence hence ADV iajs-2619 134 2 𝑉0Μ‡ 𝑉0Μ‡ PROPN iajs-2619 134 3 is be AUX iajs-2619 134 4 negative negative ADJ iajs-2619 134 5 definite definite ADJ iajs-2619 134 6 and and CCONJ iajs-2619 134 7 then then ADV iajs-2619 134 8 the the DET iajs-2619 134 9 solution solution NOUN iajs-2619 134 10 starting start VERB iajs-2619 134 11 from from ADP iajs-2619 134 12 any any DET iajs-2619 134 13 initial initial ADJ iajs-2619 134 14 point point NOUN iajs-2619 134 15 satisfies satisfy VERB iajs-2619 134 16 the the DET iajs-2619 134 17 conditions condition NOUN iajs-2619 134 18 ( ( PUNCT iajs-2619 134 19 15a 15a NOUN iajs-2619 134 20 ) ) PUNCT iajs-2619 134 21 and and CCONJ iajs-2619 134 22 ( ( PUNCT iajs-2619 134 23 15b 15b NOUN iajs-2619 134 24 ) ) PUNCT iajs-2619 134 25 , , PUNCT iajs-2619 134 26 will will AUX iajs-2619 134 27 converge converge VERB iajs-2619 134 28 to to ADP iajs-2619 134 29 𝐸0 𝐸0 ADJ iajs-2619 134 30 point point NOUN iajs-2619 134 31 . . PUNCT iajs-2619 135 1 ∎ ∎ PROPN iajs-2619 135 2 theorem theorem ADJ iajs-2619 135 3 ( ( PUNCT iajs-2619 135 4 5 5 NUM iajs-2619 135 5 ): ): PUNCT iajs-2619 135 6 the the DET iajs-2619 135 7 𝐸1 𝐸1 NOUN iajs-2619 135 8 of of ADP iajs-2619 135 9 system system NOUN iajs-2619 135 10 ( ( PUNCT iajs-2619 135 11 3 3 X iajs-2619 135 12 ) ) PUNCT iajs-2619 135 13 is be AUX iajs-2619 135 14 globally globally ADV iajs-2619 135 15 asymptotically asymptotically ADV iajs-2619 135 16 stable stable ADJ iajs-2619 135 17 under under ADP iajs-2619 135 18 the the DET iajs-2619 135 19 following follow VERB iajs-2619 135 20 conditions condition NOUN iajs-2619 135 21 held hold VERB iajs-2619 135 22 . . PUNCT iajs-2619 136 1 𝑏12 𝑏12 VERB iajs-2619 136 2 2 2 NUM iajs-2619 136 3 < < SYM iajs-2619 136 4 4 4 NUM iajs-2619 136 5 9 9 NUM iajs-2619 136 6 𝑏11𝑏22 𝑏11𝑏22 NOUN iajs-2619 136 7 ( ( PUNCT iajs-2619 136 8 16a 16a NOUN iajs-2619 136 9 ) ) PUNCT iajs-2619 136 10 𝑏13 𝑏13 NOUN iajs-2619 136 11 2 2 NUM iajs-2619 136 12 < < SYM iajs-2619 136 13 4 4 NUM iajs-2619 136 14 6 6 NUM iajs-2619 136 15 𝑏11𝑏33 𝑏11𝑏33 NOUN iajs-2619 136 16 ( ( PUNCT iajs-2619 136 17 16b 16b X iajs-2619 136 18 ) ) PUNCT iajs-2619 136 19 𝑏14 𝑏14 NOUN iajs-2619 136 20 2 2 NUM iajs-2619 136 21 < < SYM iajs-2619 136 22 4 4 NUM iajs-2619 136 23 6 6 NUM iajs-2619 136 24 𝑏11𝑏44 𝑏11𝑏44 NOUN iajs-2619 136 25 ( ( PUNCT iajs-2619 136 26 16c 16c NOUN iajs-2619 136 27 ) ) PUNCT iajs-2619 136 28 𝑏23 𝑏23 NOUN iajs-2619 136 29 2 2 NUM iajs-2619 136 30 < < SYM iajs-2619 136 31 4 4 NUM iajs-2619 136 32 6 6 NUM iajs-2619 136 33 𝑏22𝑏33 𝑏22𝑏33 NOUN iajs-2619 136 34 ( ( PUNCT iajs-2619 136 35 16d 16d NOUN iajs-2619 136 36 ) ) PUNCT iajs-2619 136 37 𝑏24 𝑏24 NOUN iajs-2619 136 38 2 2 NUM iajs-2619 136 39 < < SYM iajs-2619 136 40 4 4 NUM iajs-2619 136 41 6 6 NUM iajs-2619 136 42 𝑏22𝑏44 𝑏22𝑏44 NOUN iajs-2619 136 43 ( ( PUNCT iajs-2619 136 44 16e 16e NUM iajs-2619 136 45 ) ) PUNCT iajs-2619 137 1 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 137 2 < < X iajs-2619 137 3 πœ‡ πœ‡ X iajs-2619 137 4 + + PROPN iajs-2619 137 5 𝑑 𝑑 PROPN iajs-2619 137 6 + + ADP iajs-2619 137 7 πœ‚ πœ‚ X iajs-2619 137 8 ( ( PUNCT iajs-2619 137 9 16f 16f NUM iajs-2619 137 10 ) ) PUNCT iajs-2619 137 11 π‘Ÿ π‘Ÿ NOUN iajs-2619 137 12 < < X iajs-2619 137 13 π‘Ÿ π‘Ÿ X iajs-2619 137 14 π‘˜ π‘˜ X iajs-2619 137 15 ( ( PUNCT iajs-2619 137 16 𝐡 𝐡 NOUN iajs-2619 137 17 + + CCONJ iajs-2619 137 18 π΅βˆ— π΅βˆ— NUM iajs-2619 137 19 ) ) PUNCT iajs-2619 137 20 + + CCONJ iajs-2619 137 21 πœ‡ πœ‡ X iajs-2619 137 22 ( ( PUNCT iajs-2619 137 23 16 16 NUM iajs-2619 137 24 g g NOUN iajs-2619 137 25 ) ) PUNCT iajs-2619 137 26 proof proof NOUN iajs-2619 137 27 : : PUNCT iajs-2619 137 28 consider consider VERB iajs-2619 137 29 the the DET iajs-2619 137 30 following follow VERB iajs-2619 137 31 function function NOUN iajs-2619 137 32 𝑉1(𝑆𝑒 𝑉1(𝑆𝑒 PROPN iajs-2619 137 33 , , PUNCT iajs-2619 137 34 𝐼 𝐼 PROPN iajs-2619 137 35 , , PUNCT iajs-2619 137 36 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 137 37 ) ) PUNCT iajs-2619 137 38 = = SYM iajs-2619 137 39 1 1 NUM iajs-2619 137 40 2 2 NUM iajs-2619 137 41 ( ( PUNCT iajs-2619 137 42 π‘†π‘’βˆ’π‘†π‘’ π‘†π‘’βˆ’π‘†π‘’ INTJ iajs-2619 137 43 βˆ—)2 βˆ—)2 ADJ iajs-2619 137 44 + + CCONJ iajs-2619 137 45 1 1 NUM iajs-2619 137 46 2 2 NUM iajs-2619 137 47 ( ( PUNCT iajs-2619 137 48 πΌβˆ’πΌβˆ—)2 πΌβˆ’πΌβˆ—)2 PROPN iajs-2619 137 49 + + NUM iajs-2619 137 50 1 1 NUM iajs-2619 137 51 2 2 NUM iajs-2619 137 52 ( ( PUNCT iajs-2619 137 53 π΅βˆ’π΅βˆ—)2 π΅βˆ’π΅βˆ—)2 PROPN iajs-2619 137 54 + + NUM iajs-2619 137 55 1 1 NUM iajs-2619 137 56 2 2 NUM iajs-2619 137 57 ( ( PUNCT iajs-2619 137 58 π‘€βˆ’π‘€βˆ—)2 π‘€βˆ’π‘€βˆ—)2 VERB iajs-2619 137 59 it it PRON iajs-2619 137 60 is be AUX iajs-2619 137 61 easy easy ADJ iajs-2619 137 62 to to PART iajs-2619 137 63 see see VERB iajs-2619 137 64 that that DET iajs-2619 137 65 𝑉1 𝑉1 NOUN iajs-2619 137 66 = = SYM iajs-2619 137 67 ( ( PUNCT iajs-2619 137 68 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 137 69 , , PUNCT iajs-2619 137 70 𝐼 𝐼 PROPN iajs-2619 137 71 , , PUNCT iajs-2619 137 72 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 137 73 ) ) PUNCT iajs-2619 137 74 ∈ ∈ PROPN iajs-2619 137 75 𝐢 𝐢 PROPN iajs-2619 137 76 1(𝑅+ 1(𝑅+ PROPN iajs-2619 137 77 4 4 NUM iajs-2619 137 78 , , PUNCT iajs-2619 137 79 𝑅 𝑅 NOUN iajs-2619 137 80 ) ) PUNCT iajs-2619 137 81 in in ADP iajs-2619 137 82 addition addition NOUN iajs-2619 137 83 𝑉1(𝑆𝑒 𝑉1(𝑆𝑒 PROPN iajs-2619 137 84 βˆ— βˆ— NOUN iajs-2619 137 85 , , PUNCT iajs-2619 137 86 πΌβˆ— πΌβˆ— PROPN iajs-2619 137 87 , , PUNCT iajs-2619 137 88 π΅βˆ— π΅βˆ— NUM iajs-2619 137 89 , , PUNCT iajs-2619 137 90 π‘€βˆ— π‘€βˆ— NUM iajs-2619 137 91 ) ) PUNCT iajs-2619 138 1 = = SYM iajs-2619 138 2 0 0 NUM iajs-2619 138 3 while while SCONJ iajs-2619 138 4 𝑉1(𝑆𝑒 𝑉1(𝑆𝑒 PROPN iajs-2619 138 5 , , PUNCT iajs-2619 138 6 𝐼 𝐼 PROPN iajs-2619 138 7 , , PUNCT iajs-2619 138 8 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 138 9 ) ) PUNCT iajs-2619 138 10 > > SYM iajs-2619 138 11 0 0 NUM iajs-2619 138 12 βˆ€ βˆ€ X iajs-2619 138 13 ( ( PUNCT iajs-2619 138 14 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 138 15 , , PUNCT iajs-2619 138 16 𝐼 𝐼 PROPN iajs-2619 138 17 , , PUNCT iajs-2619 138 18 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 138 19 ) ) PUNCT iajs-2619 138 20 ∈ ∈ PROPN iajs-2619 138 21 𝑅+ 𝑅+ NUM iajs-2619 138 22 4 4 NUM iajs-2619 138 23 and and CCONJ iajs-2619 138 24 ( ( PUNCT iajs-2619 138 25 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 138 26 , , PUNCT iajs-2619 138 27 𝐼 𝐼 PROPN iajs-2619 138 28 , , PUNCT iajs-2619 138 29 𝐡,𝑀 𝐡,𝑀 NOUN iajs-2619 138 30 ) ) PUNCT iajs-2619 138 31 β‰  β‰  PROPN iajs-2619 138 32 ( ( PUNCT iajs-2619 138 33 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 138 34 βˆ— βˆ— NOUN iajs-2619 138 35 , , PUNCT iajs-2619 138 36 πΌβˆ— πΌβˆ— PROPN iajs-2619 138 37 , , PUNCT iajs-2619 138 38 π΅βˆ— π΅βˆ— NUM iajs-2619 138 39 , , PUNCT iajs-2619 138 40 π‘€βˆ— π‘€βˆ— NUM iajs-2619 138 41 ) ) PUNCT iajs-2619 138 42 . . PUNCT iajs-2619 139 1 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2619 139 2 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 3 = = SYM iajs-2619 139 4 ( ( PUNCT iajs-2619 139 5 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 139 6 βˆ’ βˆ’ NOUN iajs-2619 139 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 139 8 βˆ— βˆ— NOUN iajs-2619 139 9 ) ) PUNCT iajs-2619 139 10 𝑑𝑆𝑒 𝑑𝑆𝑒 NOUN iajs-2619 139 11 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 12 + + CCONJ iajs-2619 139 13 ( ( PUNCT iajs-2619 139 14 𝐼 𝐼 PROPN iajs-2619 139 15 βˆ’ βˆ’ PROPN iajs-2619 139 16 πΌβˆ— πΌβˆ— NOUN iajs-2619 139 17 ) ) PUNCT iajs-2619 139 18 𝑑𝐼 𝑑𝐼 NOUN iajs-2619 139 19 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 20 + + NOUN iajs-2619 139 21 ( ( PUNCT iajs-2619 139 22 𝐡 𝐡 NOUN iajs-2619 139 23 βˆ’ βˆ’ PROPN iajs-2619 139 24 π΅βˆ— π΅βˆ— NUM iajs-2619 139 25 ) ) PUNCT iajs-2619 139 26 𝑑𝐡 𝑑𝐡 VERB iajs-2619 139 27 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 28 + + CCONJ iajs-2619 139 29 ( ( PUNCT iajs-2619 139 30 𝑀 𝑀 PROPN iajs-2619 139 31 βˆ’π‘€βˆ— βˆ’π‘€βˆ— PROPN iajs-2619 139 32 ) ) PUNCT iajs-2619 139 33 𝑑𝑀 𝑑𝑀 PROPN iajs-2619 139 34 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 35 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2619 139 36 𝑑𝑑 𝑑𝑑 ADP iajs-2619 139 37 = = PUNCT iajs-2619 139 38 ( ( PUNCT iajs-2619 139 39 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 139 40 βˆ’ βˆ’ PROPN iajs-2619 139 41 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 139 42 βˆ—)[πœƒ βˆ—)[πœƒ ADJ iajs-2619 139 43 βˆ’ βˆ’ PRON iajs-2619 139 44 π‘Ž1𝑆𝑒𝐡 π‘Ž1𝑆𝑒𝐡 NUM iajs-2619 139 45 βˆ’ βˆ’ PROPN iajs-2619 139 46 π‘Ž2π‘†π‘’π‘€βˆ’ π‘Ž2π‘†π‘’π‘€βˆ’ PRON iajs-2619 139 47 π‘Ž3𝑆𝑒𝐼 π‘Ž3𝑆𝑒𝐼 NOUN iajs-2619 139 48 βˆ’ βˆ’ ADP iajs-2619 139 49 πœ‡π‘†π‘’ πœ‡π‘†π‘’ X iajs-2619 139 50 ] ] X iajs-2619 140 1 + + PROPN iajs-2619 140 2 ( ( PUNCT iajs-2619 140 3 i i PRON iajs-2619 140 4 βˆ’ βˆ’ VERB iajs-2619 141 1 iβˆ—)[π‘Ž1𝑆u𝐡 iβˆ—)[π‘Ž1𝑆u𝐡 PROPN iajs-2619 141 2 + + NUM iajs-2619 141 3 π‘Ž3𝑆u𝐼 π‘Ž3𝑆u𝐼 NUM iajs-2619 141 4 βˆ’ βˆ’ PROPN iajs-2619 141 5 ( ( PUNCT iajs-2619 141 6 ΞΌ ΞΌ NOUN iajs-2619 141 7 + + PROPN iajs-2619 142 1 d d PROPN iajs-2619 142 2 + + CCONJ iajs-2619 142 3 Ξ·)𝐼 Ξ·)𝐼 NOUN iajs-2619 142 4 ] ] X iajs-2619 143 1 + + PROPN iajs-2619 143 2 ( ( PUNCT iajs-2619 143 3 𝐡 𝐡 NOUN iajs-2619 143 4 βˆ’ βˆ’ PROPN iajs-2619 143 5 π΅βˆ— π΅βˆ— NUM iajs-2619 143 6 ) ) PUNCT iajs-2619 144 1 [ [ X iajs-2619 144 2 r𝐡 r𝐡 X iajs-2619 144 3 ( ( PUNCT iajs-2619 144 4 1 1 NUM iajs-2619 144 5 βˆ’ βˆ’ PROPN iajs-2619 144 6 𝐡 𝐡 PROPN iajs-2619 144 7 k k PROPN iajs-2619 144 8 ) ) PUNCT iajs-2619 144 9 βˆ’ βˆ’ PROPN iajs-2619 144 10 μ𝐡 μ𝐡 NOUN iajs-2619 144 11 + + CCONJ iajs-2619 144 12 ϡ𝐼 ϡ𝐼 X iajs-2619 144 13 ] ] X iajs-2619 145 1 + + PROPN iajs-2619 145 2 ( ( PUNCT iajs-2619 145 3 𝑀 𝑀 PROPN iajs-2619 145 4 βˆ’π‘€βˆ—)[Ξ± βˆ’π‘€βˆ—)[Ξ± PROPN iajs-2619 145 5 ( ( PUNCT iajs-2619 145 6 𝑆u 𝑆u PROPN iajs-2619 145 7 + + CCONJ iajs-2619 145 8 𝐼 𝐼 PROPN iajs-2619 145 9 ) ) PUNCT iajs-2619 145 10 – – PUNCT iajs-2619 145 11 γ𝑀 γ𝑀 NOUN iajs-2619 145 12 ] ] PUNCT iajs-2619 145 13 furthermore furthermore ADV iajs-2619 145 14 , , PUNCT iajs-2619 145 15 by by ADP iajs-2619 145 16 the the DET iajs-2619 145 17 derivative derivative NOUN iajs-2619 145 18 and and CCONJ iajs-2619 145 19 simplifying simplify VERB iajs-2619 145 20 the the DET iajs-2619 145 21 137 137 NUM iajs-2619 145 22 ibn ibn PROPN iajs-2619 145 23 al al PROPN iajs-2619 145 24 - - PUNCT iajs-2619 145 25 haitham haitham PROPN iajs-2619 145 26 jour jour X iajs-2619 145 27 . . PROPN iajs-2619 146 1 for for ADP iajs-2619 146 2 pure pure ADJ iajs-2619 146 3 & & CCONJ iajs-2619 146 4 appl appl PROPN iajs-2619 146 5 . . PUNCT iajs-2619 147 1 sci sci PROPN iajs-2619 147 2 . . PROPN iajs-2619 148 1 34 34 NUM iajs-2619 148 2 ( ( PUNCT iajs-2619 148 3 2 2 NUM iajs-2619 148 4 ) ) PUNCT iajs-2619 148 5 2021 2021 NUM iajs-2619 148 6 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2619 148 7 𝑑𝑑 𝑑𝑑 ADP iajs-2619 148 8 = = SYM iajs-2619 148 9 βˆ’ βˆ’ PROPN iajs-2619 148 10 [ [ PUNCT iajs-2619 148 11 𝑏11 𝑏11 PROPN iajs-2619 148 12 3 3 NUM iajs-2619 148 13 ( ( PUNCT iajs-2619 148 14 π‘†π‘’βˆ’π‘†π‘’ π‘†π‘’βˆ’π‘†π‘’ INTJ iajs-2619 148 15 βˆ—)2 βˆ—)2 PROPN iajs-2619 149 1 + + CCONJ iajs-2619 149 2 𝑏12(𝐼 𝑏12(𝐼 ADP iajs-2619 149 3 βˆ’ βˆ’ ADV iajs-2619 149 4 𝐼 𝐼 ADP iajs-2619 149 5 βˆ—)(𝑆𝑒 βˆ—)(𝑆𝑒 PRON iajs-2619 149 6 βˆ’ βˆ’ PROPN iajs-2619 149 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 149 8 βˆ— βˆ— NOUN iajs-2619 149 9 ) ) PUNCT iajs-2619 150 1 + + CCONJ iajs-2619 150 2 𝑏22 𝑏22 NOUN iajs-2619 150 3 3 3 NUM iajs-2619 150 4 ( ( PUNCT iajs-2619 150 5 πΌβˆ’πΌβˆ—)2 πΌβˆ’πΌβˆ—)2 NUM iajs-2619 150 6 ] ] X iajs-2619 150 7 βˆ’ βˆ’ PROPN iajs-2619 150 8 [ [ PUNCT iajs-2619 150 9 𝑏11 𝑏11 PROPN iajs-2619 150 10 3 3 NUM iajs-2619 150 11 ( ( PUNCT iajs-2619 150 12 π‘†π‘’βˆ’π‘†π‘’ π‘†π‘’βˆ’π‘†π‘’ INTJ iajs-2619 150 13 βˆ—)2 βˆ—)2 PROPN iajs-2619 151 1 + + CCONJ iajs-2619 151 2 𝑏13(𝐡 𝑏13(𝐡 PROPN iajs-2619 151 3 βˆ’ βˆ’ PROPN iajs-2619 151 4 𝐡 𝐡 PROPN iajs-2619 151 5 βˆ—)(𝑆𝑒 βˆ—)(𝑆𝑒 ADP iajs-2619 151 6 βˆ’ βˆ’ PROPN iajs-2619 151 7 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 151 8 βˆ— βˆ— NOUN iajs-2619 151 9 ) ) PUNCT iajs-2619 152 1 + + NUM iajs-2619 152 2 𝑏33 𝑏33 NOUN iajs-2619 152 3 2 2 NUM iajs-2619 152 4 ( ( PUNCT iajs-2619 152 5 π΅βˆ’π΅βˆ—)2 π΅βˆ’π΅βˆ—)2 PROPN iajs-2619 152 6 ] ] X iajs-2619 152 7 βˆ’ βˆ’ PROPN iajs-2619 153 1 [ [ PUNCT iajs-2619 153 2 𝑏11 𝑏11 PROPN iajs-2619 153 3 3 3 NUM iajs-2619 153 4 ( ( PUNCT iajs-2619 153 5 π‘†π‘’βˆ’π‘†π‘’ π‘†π‘’βˆ’π‘†π‘’ INTJ iajs-2619 153 6 βˆ—)2 βˆ—)2 PROPN iajs-2619 154 1 + + CCONJ iajs-2619 154 2 𝑏14(𝑀 𝑏14(𝑀 X iajs-2619 154 3 βˆ’π‘€ βˆ’π‘€ X iajs-2619 154 4 βˆ—)(𝑆𝑒 βˆ—)(𝑆𝑒 PUNCT iajs-2619 154 5 βˆ’ βˆ’ PROPN iajs-2619 154 6 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 154 7 βˆ— βˆ— NOUN iajs-2619 154 8 ) ) PUNCT iajs-2619 155 1 + + CCONJ iajs-2619 155 2 𝑏44 𝑏44 NOUN iajs-2619 155 3 2 2 NUM iajs-2619 155 4 ( ( PUNCT iajs-2619 155 5 𝑀 𝑀 PROPN iajs-2619 155 6 βˆ’π‘€βˆ—)2 βˆ’π‘€βˆ—)2 PROPN iajs-2619 155 7 ] ] X iajs-2619 155 8 βˆ’ βˆ’ PROPN iajs-2619 155 9 [ [ PUNCT iajs-2619 155 10 𝑏22 𝑏22 NOUN iajs-2619 155 11 3 3 NUM iajs-2619 155 12 ( ( PUNCT iajs-2619 155 13 𝐼 𝐼 PROPN iajs-2619 155 14 βˆ’ βˆ’ PROPN iajs-2619 155 15 πΌβˆ—)2 πΌβˆ—)2 ADV iajs-2619 155 16 βˆ’ βˆ’ PROPN iajs-2619 155 17 𝑏23(𝐡 𝑏23(𝐡 CCONJ iajs-2619 155 18 βˆ’ βˆ’ PROPN iajs-2619 155 19 𝐡 𝐡 PROPN iajs-2619 155 20 βˆ—)(𝐼 βˆ—)(𝐼 NOUN iajs-2619 155 21 βˆ’ βˆ’ PROPN iajs-2619 155 22 πΌβˆ— πΌβˆ— NOUN iajs-2619 155 23 ) ) PUNCT iajs-2619 156 1 + + NUM iajs-2619 156 2 𝑏33 𝑏33 NOUN iajs-2619 156 3 2 2 NUM iajs-2619 156 4 ( ( PUNCT iajs-2619 156 5 𝐡 𝐡 NOUN iajs-2619 156 6 βˆ’ βˆ’ PROPN iajs-2619 156 7 π΅βˆ—)2 π΅βˆ—)2 PROPN iajs-2619 156 8 ] ] X iajs-2619 156 9 βˆ’ βˆ’ PROPN iajs-2619 157 1 [ [ PUNCT iajs-2619 157 2 𝑏22 𝑏22 NOUN iajs-2619 157 3 3 3 NUM iajs-2619 157 4 ( ( PUNCT iajs-2619 157 5 𝐼 𝐼 PROPN iajs-2619 157 6 βˆ’ βˆ’ PROPN iajs-2619 157 7 πΌβˆ—)2 πΌβˆ—)2 ADV iajs-2619 157 8 βˆ’ βˆ’ PROPN iajs-2619 157 9 𝑏24(𝑀 𝑏24(𝑀 PROPN iajs-2619 157 10 βˆ’π‘€ βˆ’π‘€ PROPN iajs-2619 157 11 βˆ—)(𝐼 βˆ—)(𝐼 NOUN iajs-2619 157 12 βˆ’ βˆ’ PROPN iajs-2619 157 13 πΌβˆ— πΌβˆ— NOUN iajs-2619 157 14 ) ) PUNCT iajs-2619 158 1 + + CCONJ iajs-2619 158 2 𝑏44 𝑏44 NOUN iajs-2619 158 3 2 2 NUM iajs-2619 158 4 ( ( PUNCT iajs-2619 158 5 𝑀 𝑀 PROPN iajs-2619 158 6 βˆ’π‘€βˆ—)2 βˆ’π‘€βˆ—)2 PROPN iajs-2619 158 7 ] ] PUNCT iajs-2619 158 8 therefore therefore ADV iajs-2619 158 9 according accord VERB iajs-2619 158 10 to to ADP iajs-2619 158 11 the the DET iajs-2619 158 12 condition condition NOUN iajs-2619 158 13 ( ( PUNCT iajs-2619 158 14 16a 16a NOUN iajs-2619 158 15 ) ) PUNCT iajs-2619 158 16 and and CCONJ iajs-2619 158 17 ( ( PUNCT iajs-2619 158 18 16 16 NUM iajs-2619 158 19 g g NOUN iajs-2619 158 20 ) ) PUNCT iajs-2619 158 21 we we PRON iajs-2619 158 22 obtain obtain VERB iajs-2619 158 23 that that PRON iajs-2619 158 24 : : PUNCT iajs-2619 158 25 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2619 158 26 𝑑𝑑 𝑑𝑑 ADP iajs-2619 158 27 < < X iajs-2619 158 28 βˆ’[√ βˆ’[√ X iajs-2619 158 29 𝑏11 𝑏11 PROPN iajs-2619 158 30 3 3 NUM iajs-2619 158 31 ( ( PUNCT iajs-2619 158 32 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 158 33 βˆ’ βˆ’ NOUN iajs-2619 159 1 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 159 2 βˆ— βˆ— NOUN iajs-2619 159 3 ) ) PUNCT iajs-2619 160 1 + + CCONJ iajs-2619 160 2 √ √ ADJ iajs-2619 160 3 𝑏22 𝑏22 NOUN iajs-2619 160 4 3 3 NUM iajs-2619 160 5 ( ( PUNCT iajs-2619 160 6 𝐼 𝐼 PROPN iajs-2619 160 7 βˆ’ βˆ’ PROPN iajs-2619 160 8 πΌβˆ— πΌβˆ— NOUN iajs-2619 160 9 ) ) PUNCT iajs-2619 160 10 ] ] PUNCT iajs-2619 161 1 2 2 NUM iajs-2619 161 2 βˆ’ βˆ’ NOUN iajs-2619 162 1 [ [ X iajs-2619 162 2 √ √ X iajs-2619 162 3 𝑏11 𝑏11 ADJ iajs-2619 162 4 3 3 NUM iajs-2619 162 5 ( ( PUNCT iajs-2619 162 6 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 162 7 βˆ’ βˆ’ NOUN iajs-2619 163 1 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 163 2 βˆ— βˆ— NOUN iajs-2619 163 3 ) ) PUNCT iajs-2619 164 1 + + CCONJ iajs-2619 164 2 √ √ NUM iajs-2619 164 3 𝑏33 𝑏33 NOUN iajs-2619 164 4 2 2 NUM iajs-2619 164 5 ( ( PUNCT iajs-2619 164 6 𝐡 𝐡 NOUN iajs-2619 164 7 βˆ’ βˆ’ PROPN iajs-2619 164 8 π΅βˆ— π΅βˆ— NUM iajs-2619 164 9 ) ) PUNCT iajs-2619 164 10 ] ] PUNCT iajs-2619 164 11 2 2 NUM iajs-2619 164 12 βˆ’ βˆ’ NOUN iajs-2619 165 1 [ [ X iajs-2619 165 2 √ √ X iajs-2619 165 3 𝑏11 𝑏11 ADJ iajs-2619 165 4 3 3 NUM iajs-2619 165 5 ( ( PUNCT iajs-2619 165 6 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 165 7 βˆ’ βˆ’ NOUN iajs-2619 166 1 𝑆𝑒 𝑆𝑒 NOUN iajs-2619 166 2 βˆ— βˆ— NOUN iajs-2619 166 3 ) ) PUNCT iajs-2619 167 1 + + CCONJ iajs-2619 167 2 √ √ VERB iajs-2619 167 3 𝑏44 𝑏44 NOUN iajs-2619 167 4 2 2 NUM iajs-2619 167 5 ( ( PUNCT iajs-2619 167 6 𝑀 𝑀 PROPN iajs-2619 167 7 βˆ’π‘€βˆ— βˆ’π‘€βˆ— PROPN iajs-2619 167 8 ) ) PUNCT iajs-2619 167 9 ] ] PUNCT iajs-2619 167 10 2 2 NUM iajs-2619 167 11 βˆ’ βˆ’ NOUN iajs-2619 168 1 [ [ X iajs-2619 168 2 √ √ ADJ iajs-2619 168 3 𝑏22 𝑏22 NOUN iajs-2619 168 4 3 3 NUM iajs-2619 168 5 ( ( PUNCT iajs-2619 168 6 𝐼 𝐼 PROPN iajs-2619 168 7 βˆ’ βˆ’ PROPN iajs-2619 168 8 πΌβˆ— πΌβˆ— NOUN iajs-2619 168 9 ) ) PUNCT iajs-2619 168 10 βˆ’ βˆ’ NOUN iajs-2619 168 11 √ √ NUM iajs-2619 168 12 𝑏33 𝑏33 NOUN iajs-2619 168 13 2 2 NUM iajs-2619 168 14 ( ( PUNCT iajs-2619 168 15 𝐡 𝐡 NOUN iajs-2619 168 16 βˆ’ βˆ’ PROPN iajs-2619 168 17 π΅βˆ— π΅βˆ— NUM iajs-2619 168 18 ) ) PUNCT iajs-2619 168 19 ] ] PUNCT iajs-2619 168 20 2 2 NUM iajs-2619 168 21 βˆ’ βˆ’ NOUN iajs-2619 168 22 [ [ X iajs-2619 168 23 √ √ ADJ iajs-2619 168 24 𝑏22 𝑏22 NOUN iajs-2619 168 25 3 3 NUM iajs-2619 168 26 ( ( PUNCT iajs-2619 168 27 𝐼 𝐼 PROPN iajs-2619 168 28 βˆ’ βˆ’ PROPN iajs-2619 168 29 πΌβˆ— πΌβˆ— NOUN iajs-2619 168 30 ) ) PUNCT iajs-2619 168 31 βˆ’ βˆ’ NOUN iajs-2619 168 32 √ √ NUM iajs-2619 168 33 𝑏44 𝑏44 NOUN iajs-2619 168 34 2 2 NUM iajs-2619 168 35 ( ( PUNCT iajs-2619 168 36 𝑀 𝑀 PROPN iajs-2619 168 37 βˆ’π‘€βˆ— βˆ’π‘€βˆ— PROPN iajs-2619 168 38 ) ) PUNCT iajs-2619 168 39 ] ] PUNCT iajs-2619 168 40 2 2 NUM iajs-2619 168 41 where where SCONJ iajs-2619 168 42 𝑏11 𝑏11 ADJ iajs-2619 168 43 = = SYM iajs-2619 168 44 π‘Ž1𝐡 π‘Ž1𝐡 NOUN iajs-2619 168 45 βˆ— βˆ— NOUN iajs-2619 168 46 + + NOUN iajs-2619 168 47 π‘Ž2𝑀 π‘Ž2𝑀 ADJ iajs-2619 168 48 βˆ— βˆ— NOUN iajs-2619 168 49 + + CCONJ iajs-2619 168 50 π‘Ž3𝐼 π‘Ž3𝐼 NUM iajs-2619 168 51 βˆ— βˆ— NOUN iajs-2619 168 52 + + NUM iajs-2619 168 53 π‘Ž4 π‘Ž4 PROPN iajs-2619 168 54 + + CCONJ iajs-2619 168 55 πœ‡ πœ‡ ADP iajs-2619 168 56 , , PUNCT iajs-2619 168 57 𝑏12 𝑏12 VERB iajs-2619 168 58 = = PUNCT iajs-2619 168 59 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 X iajs-2619 168 60 βˆ’ βˆ’ NOUN iajs-2619 168 61 π‘Ž1𝐡 π‘Ž1𝐡 NOUN iajs-2619 168 62 βˆ— βˆ— NOUN iajs-2619 168 63 βˆ’ βˆ’ ADP iajs-2619 168 64 π‘Ž3𝐼 π‘Ž3𝐼 NUM iajs-2619 168 65 βˆ— βˆ— NOUN iajs-2619 168 66 βˆ’ βˆ’ NOUN iajs-2619 168 67 π‘Ž4 π‘Ž4 PROPN iajs-2619 168 68 𝑏22 𝑏22 NOUN iajs-2619 168 69 = = SYM iajs-2619 168 70 πœ‡ πœ‡ ADP iajs-2619 168 71 + + X iajs-2619 168 72 𝑑 𝑑 PROPN iajs-2619 168 73 + + ADP iajs-2619 168 74 πœ‚ πœ‚ NOUN iajs-2619 168 75 βˆ’ βˆ’ PROPN iajs-2619 168 76 π‘Ž3𝑆𝑒 π‘Ž3𝑆𝑒 NOUN iajs-2619 168 77 , , PUNCT iajs-2619 168 78 𝑏23 𝑏23 NOUN iajs-2619 168 79 = = NOUN iajs-2619 168 80 π‘Ž1𝑆𝑒 π‘Ž1𝑆𝑒 X iajs-2619 168 81 + + CCONJ iajs-2619 168 82 πœ– πœ– PROPN iajs-2619 168 83 , , PUNCT iajs-2619 168 84 𝑏13 𝑏13 NOUN iajs-2619 168 85 = = SYM iajs-2619 168 86 π‘Ž1𝑆𝑒 π‘Ž1𝑆𝑒 NOUN iajs-2619 168 87 𝑏33 𝑏33 NOUN iajs-2619 168 88 = = NOUN iajs-2619 168 89 π‘Ÿ π‘Ÿ X iajs-2619 168 90 π‘˜ π‘˜ X iajs-2619 168 91 ( ( PUNCT iajs-2619 168 92 𝐡 𝐡 NOUN iajs-2619 168 93 + + CCONJ iajs-2619 168 94 π΅βˆ— π΅βˆ— NUM iajs-2619 168 95 ) ) PUNCT iajs-2619 168 96 βˆ’ βˆ’ PROPN iajs-2619 169 1 ( ( PUNCT iajs-2619 169 2 π‘Ÿ π‘Ÿ NOUN iajs-2619 169 3 βˆ’ βˆ’ X iajs-2619 169 4 πœ‡ πœ‡ NOUN iajs-2619 169 5 ) ) PUNCT iajs-2619 169 6 , , PUNCT iajs-2619 169 7 𝑏14 𝑏14 NOUN iajs-2619 169 8 = = PUNCT iajs-2619 169 9 π‘Ž2𝑆𝑒 π‘Ž2𝑆𝑒 NOUN iajs-2619 169 10 βˆ’ βˆ’ PROPN iajs-2619 169 11 𝛼 𝛼 NOUN iajs-2619 169 12 , , PUNCT iajs-2619 169 13 𝑏24 𝑏24 NOUN iajs-2619 169 14 = = SYM iajs-2619 169 15 𝛼 𝛼 NOUN iajs-2619 169 16 , , PUNCT iajs-2619 169 17 𝑏44 𝑏44 NOUN iajs-2619 169 18 = = SYM iajs-2619 169 19 𝛾 𝛾 VERB iajs-2619 169 20 clearly clearly ADV iajs-2619 169 21 , , PUNCT iajs-2619 169 22 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2619 169 23 𝑑𝑑 𝑑𝑑 ADP iajs-2619 169 24 < < X iajs-2619 169 25 0 0 NUM iajs-2619 169 26 , , PUNCT iajs-2619 169 27 and and CCONJ iajs-2619 169 28 then then ADV iajs-2619 169 29 𝑉1 𝑉1 PROPN iajs-2619 169 30 is be AUX iajs-2619 169 31 a a DET iajs-2619 169 32 lyapunov lyapunov ADJ iajs-2619 169 33 function function NOUN iajs-2619 169 34 provided provide VERB iajs-2619 169 35 that that SCONJ iajs-2619 169 36 the the DET iajs-2619 169 37 given give VERB iajs-2619 169 38 condition condition NOUN iajs-2619 169 39 ( ( PUNCT iajs-2619 169 40 16a 16a NOUN iajs-2619 169 41 ) ) PUNCT iajs-2619 169 42 and and CCONJ iajs-2619 169 43 ( ( PUNCT iajs-2619 169 44 16 16 NUM iajs-2619 169 45 g g NOUN iajs-2619 169 46 ) ) PUNCT iajs-2619 169 47 held hold VERB iajs-2619 169 48 . . PUNCT iajs-2619 170 1 therefore therefore ADV iajs-2619 170 2 , , PUNCT iajs-2619 170 3 𝐸1 𝐸1 PROPN iajs-2619 170 4 is be AUX iajs-2619 170 5 globally globally ADV iajs-2619 170 6 asymptotically asymptotically ADV iajs-2619 170 7 stable stable ADJ iajs-2619 170 8 . . PUNCT iajs-2619 171 1 ∎ ∎ PROPN iajs-2619 171 2 6 6 NUM iajs-2619 171 3 . . PUNCT iajs-2619 171 4 numerical numerical PROPN iajs-2619 171 5 simulation simulation PROPN iajs-2619 171 6 : : PUNCT iajs-2619 171 7 in in ADP iajs-2619 171 8 this this DET iajs-2619 171 9 present present ADJ iajs-2619 171 10 section section NOUN iajs-2619 171 11 , , PUNCT iajs-2619 171 12 the the DET iajs-2619 171 13 spread spread NOUN iajs-2619 171 14 and and CCONJ iajs-2619 171 15 control control NOUN iajs-2619 171 16 of of ADP iajs-2619 171 17 shigellosis shigellosis NOUN iajs-2619 171 18 disease disease NOUN iajs-2619 171 19 are be AUX iajs-2619 171 20 investigated investigate VERB iajs-2619 171 21 by by ADP iajs-2619 171 22 numerically numerically ADV iajs-2619 171 23 simulation simulation NOUN iajs-2619 171 24 for for ADP iajs-2619 171 25 many many ADJ iajs-2619 171 26 sets set NOUN iajs-2619 171 27 of of ADP iajs-2619 171 28 initial initial ADJ iajs-2619 171 29 values value NOUN iajs-2619 171 30 and and CCONJ iajs-2619 171 31 different different ADJ iajs-2619 171 32 sets set NOUN iajs-2619 171 33 of of ADP iajs-2619 171 34 parameters parameter NOUN iajs-2619 171 35 values value NOUN iajs-2619 171 36 . . PUNCT iajs-2619 172 1 the the DET iajs-2619 172 2 objectives objective NOUN iajs-2619 172 3 of of ADP iajs-2619 172 4 this this DET iajs-2619 172 5 section section NOUN iajs-2619 172 6 are be AUX iajs-2619 172 7 determined determine VERB iajs-2619 172 8 by by ADP iajs-2619 172 9 the the DET iajs-2619 172 10 effect effect NOUN iajs-2619 172 11 of of ADP iajs-2619 172 12 contact contact NOUN iajs-2619 172 13 rate rate NOUN iajs-2619 172 14 , , PUNCT iajs-2619 172 15 media medium NOUN iajs-2619 172 16 rate rate NOUN iajs-2619 172 17 and and CCONJ iajs-2619 172 18 external external ADJ iajs-2619 172 19 sources source NOUN iajs-2619 172 20 as as ADV iajs-2619 172 21 well well ADV iajs-2619 172 22 confirm confirm VERB iajs-2619 172 23 our our PRON iajs-2619 172 24 obtained obtain VERB iajs-2619 172 25 results result NOUN iajs-2619 172 26 . . PUNCT iajs-2619 173 1 it it PRON iajs-2619 173 2 is be AUX iajs-2619 173 3 observed observe VERB iajs-2619 173 4 that that SCONJ iajs-2619 173 5 through through ADP iajs-2619 173 6 choosing choose VERB iajs-2619 173 7 the the DET iajs-2619 173 8 following follow VERB iajs-2619 173 9 data datum NOUN iajs-2619 173 10 πœƒ πœƒ X iajs-2619 173 11 = = SYM iajs-2619 173 12 1.2 1.2 NUM iajs-2619 173 13 , , PUNCT iajs-2619 173 14 π‘Ž1 π‘Ž1 NOUN iajs-2619 173 15 = = PUNCT iajs-2619 173 16 1.5 1.5 NUM iajs-2619 173 17 , , PUNCT iajs-2619 173 18 π‘Ž2 π‘Ž2 NOUN iajs-2619 173 19 = = PROPN iajs-2619 173 20 1.19 1.19 NUM iajs-2619 173 21 , , PUNCT iajs-2619 173 22 π‘Ž3 π‘Ž3 NOUN iajs-2619 173 23 = = SYM iajs-2619 173 24 0.15 0.15 NUM iajs-2619 173 25 , , PUNCT iajs-2619 173 26 π‘Ž4 π‘Ž4 PROPN iajs-2619 173 27 = = PUNCT iajs-2619 173 28 1.15 1.15 NUM iajs-2619 173 29 , , PUNCT iajs-2619 173 30 πœ‡ πœ‡ X iajs-2619 173 31 = = NOUN iajs-2619 173 32 0.5 0.5 NUM iajs-2619 173 33 𝑑 𝑑 NOUN iajs-2619 173 34 = = SYM iajs-2619 173 35 0.3 0.3 NUM iajs-2619 173 36 , , PUNCT iajs-2619 173 37 Ξ· Ξ· PROPN iajs-2619 173 38 = = PROPN iajs-2619 173 39 0.5 0.5 NUM iajs-2619 173 40 , , PUNCT iajs-2619 173 41 r r NOUN iajs-2619 173 42 = = NOUN iajs-2619 173 43 0.3 0.3 NUM iajs-2619 173 44 , , PUNCT iajs-2619 173 45 k k PROPN iajs-2619 174 1 = = PUNCT iajs-2619 174 2 0.1 0.1 NUM iajs-2619 174 3 , , PUNCT iajs-2619 174 4 Ο΅ Ο΅ X iajs-2619 174 5 = = SYM iajs-2619 174 6 0.02 0.02 NUM iajs-2619 174 7 , , PUNCT iajs-2619 174 8 Ξ± Ξ± NOUN iajs-2619 174 9 = = SYM iajs-2619 174 10 0.1 0.1 NUM iajs-2619 174 11 , , PUNCT iajs-2619 174 12 Ξ³ Ξ³ X iajs-2619 174 13 = = SYM iajs-2619 174 14 0.3 0.3 NUM iajs-2619 174 15 ( ( PUNCT iajs-2619 174 16 17 17 NUM iajs-2619 174 17 ) ) PUNCT iajs-2619 174 18 the the DET iajs-2619 174 19 dynamical dynamical ADJ iajs-2619 174 20 behaviors behavior NOUN iajs-2619 174 21 of of ADP iajs-2619 174 22 system system NOUN iajs-2619 174 23 ( ( PUNCT iajs-2619 174 24 1 1 X iajs-2619 174 25 ) ) PUNCT iajs-2619 174 26 converge converge VERB iajs-2619 174 27 to to ADP iajs-2619 174 28 the the DET iajs-2619 174 29 𝐸1 𝐸1 NOUN iajs-2619 174 30 = = SYM iajs-2619 174 31 ( ( PUNCT iajs-2619 174 32 0.54,0.53,0.46,0.03,0.35 0.54,0.53,0.46,0.03,0.35 PROPN iajs-2619 174 33 ) ) PUNCT iajs-2619 174 34 and and CCONJ iajs-2619 174 35 the the DET iajs-2619 174 36 investigation investigation NOUN iajs-2619 174 37 of of ADP iajs-2619 174 38 the the DET iajs-2619 174 39 global global ADJ iajs-2619 174 40 stability stability NOUN iajs-2619 174 41 , , PUNCT iajs-2619 174 42 are be AUX iajs-2619 174 43 shown show VERB iajs-2619 174 44 in in ADP iajs-2619 174 45 figure figure NOUN iajs-2619 174 46 1 1 NUM iajs-2619 174 47 . . PUNCT iajs-2619 175 1 starting start VERB iajs-2619 175 2 from from ADP iajs-2619 175 3 different different ADJ iajs-2619 175 4 sets set NOUN iajs-2619 175 5 of of ADP iajs-2619 175 6 initial initial ADJ iajs-2619 175 7 points point NOUN iajs-2619 175 8 . . PUNCT iajs-2619 176 1 138 138 NUM iajs-2619 176 2 ibn ibn PROPN iajs-2619 176 3 al al PROPN iajs-2619 176 4 - - PUNCT iajs-2619 176 5 haitham haitham PROPN iajs-2619 176 6 jour jour X iajs-2619 176 7 . . PROPN iajs-2619 176 8 for for ADP iajs-2619 176 9 pure pure ADJ iajs-2619 176 10 & & CCONJ iajs-2619 176 11 appl appl PROPN iajs-2619 176 12 . . PUNCT iajs-2619 177 1 sci sci PROPN iajs-2619 177 2 . . PROPN iajs-2619 178 1 34 34 NUM iajs-2619 178 2 ( ( PUNCT iajs-2619 178 3 2 2 NUM iajs-2619 178 4 ) ) PUNCT iajs-2619 178 5 2021 2021 NUM iajs-2619 178 6 figure figure NOUN iajs-2619 178 7 1 1 NUM iajs-2619 178 8 . . PUNCT iajs-2619 179 1 the the DET iajs-2619 179 2 trajectory trajectory NOUN iajs-2619 179 3 of of ADP iajs-2619 179 4 system system NOUN iajs-2619 179 5 ( ( PUNCT iajs-2619 179 6 1 1 X iajs-2619 179 7 ) ) PUNCT iajs-2619 179 8 approaches approach VERB iajs-2619 179 9 asymptotically asymptotically ADV iajs-2619 179 10 to to ADP iajs-2619 179 11 a a DET iajs-2619 179 12 globally globally ADV iajs-2619 179 13 stable stable ADJ iajs-2619 179 14 to to ADP iajs-2619 179 15 endemic endemic ADJ iajs-2619 179 16 equilibrium equilibrium NOUN iajs-2619 179 17 point point NOUN iajs-2619 179 18 of of ADP iajs-2619 179 19 system system NOUN iajs-2619 179 20 ( ( PUNCT iajs-2619 179 21 1 1 NUM iajs-2619 179 22 ) ) PUNCT iajs-2619 179 23 for for ADP iajs-2619 179 24 the the DET iajs-2619 179 25 parameter parameter NOUN iajs-2619 179 26 set set VERB iajs-2619 179 27 in in ADP iajs-2619 179 28 eq eq ADP iajs-2619 179 29 . . PUNCT iajs-2619 180 1 ( ( PUNCT iajs-2619 180 2 17 17 NUM iajs-2619 180 3 ) ) PUNCT iajs-2619 180 4 , , PUNCT iajs-2619 180 5 started start VERB iajs-2619 180 6 from from ADP iajs-2619 180 7 different different ADJ iajs-2619 180 8 sets set NOUN iajs-2619 180 9 of of ADP iajs-2619 180 10 initial initial ADJ iajs-2619 180 11 point point NOUN iajs-2619 180 12 . . PUNCT iajs-2619 181 1 ( ( PUNCT iajs-2619 181 2 a a X iajs-2619 181 3 ) ) PUNCT iajs-2619 181 4 for for ADP iajs-2619 181 5 su(t su(t NUM iajs-2619 181 6 ) ) PUNCT iajs-2619 181 7 , , PUNCT iajs-2619 181 8 ( ( PUNCT iajs-2619 181 9 b b X iajs-2619 181 10 ) ) PUNCT iajs-2619 181 11 for for ADP iajs-2619 181 12 i(t i(t NOUN iajs-2619 181 13 ) ) PUNCT iajs-2619 181 14 , , PUNCT iajs-2619 181 15 ( ( PUNCT iajs-2619 181 16 c c X iajs-2619 181 17 ) ) PUNCT iajs-2619 181 18 for for ADP iajs-2619 181 19 sa(t sa(t NUM iajs-2619 181 20 ) ) PUNCT iajs-2619 181 21 , , PUNCT iajs-2619 181 22 ( ( PUNCT iajs-2619 181 23 d d X iajs-2619 181 24 ) ) PUNCT iajs-2619 181 25 for for ADP iajs-2619 181 26 b(t b(t NOUN iajs-2619 181 27 ) ) PUNCT iajs-2619 181 28 , , PUNCT iajs-2619 181 29 ( ( PUNCT iajs-2619 181 30 e e NOUN iajs-2619 181 31 ) ) PUNCT iajs-2619 181 32 for for ADP iajs-2619 181 33 m(t m(t NOUN iajs-2619 181 34 ) ) PUNCT iajs-2619 181 35 . . PUNCT iajs-2619 182 1 clearly clearly ADV iajs-2619 182 2 figure figure VERB iajs-2619 182 3 1 1 NUM iajs-2619 182 4 . . PUNCT iajs-2619 183 1 confirms confirm VERB iajs-2619 183 2 our our PRON iajs-2619 183 3 obtained obtain VERB iajs-2619 183 4 analytical analytical ADJ iajs-2619 183 5 results result NOUN iajs-2619 183 6 regarding regard VERB iajs-2619 183 7 the the DET iajs-2619 183 8 existence existence NOUN iajs-2619 183 9 of of ADP iajs-2619 183 10 globally globally ADV iajs-2619 183 11 asymptotically asymptotically ADV iajs-2619 183 12 stable stable ADJ iajs-2619 183 13 endemic endemic ADJ iajs-2619 183 14 equilibrium equilibrium NOUN iajs-2619 183 15 point point NOUN iajs-2619 183 16 . . PUNCT iajs-2619 184 1 however however ADV iajs-2619 184 2 , , PUNCT iajs-2619 184 3 for for ADP iajs-2619 184 4 the the DET iajs-2619 184 5 same same ADJ iajs-2619 184 6 data datum NOUN iajs-2619 184 7 by by ADP iajs-2619 184 8 equation equation NOUN iajs-2619 184 9 ( ( PUNCT iajs-2619 184 10 17 17 NUM iajs-2619 184 11 ) ) PUNCT iajs-2619 184 12 with with ADP iajs-2619 184 13 π‘Ž4 π‘Ž4 PROPN iajs-2619 184 14 = = SYM iajs-2619 184 15 0 0 NUM iajs-2619 184 16 , , PUNCT iajs-2619 184 17 π‘˜ π‘˜ X iajs-2619 184 18 = = PUNCT iajs-2619 184 19 0.01 0.01 NUM iajs-2619 184 20 the the DET iajs-2619 184 21 solution solution NOUN iajs-2619 184 22 of of ADP iajs-2619 184 23 system system NOUN iajs-2619 184 24 ( ( PUNCT iajs-2619 184 25 1 1 X iajs-2619 184 26 ) ) PUNCT iajs-2619 184 27 converge converge VERB iajs-2619 184 28 to to ADP iajs-2619 184 29 the the DET iajs-2619 184 30 disease disease NOUN iajs-2619 184 31 free free ADJ iajs-2619 184 32 equilibrium equilibrium NOUN iajs-2619 184 33 point point NOUN iajs-2619 184 34 is be AUX iajs-2619 184 35 shown show VERB iajs-2619 184 36 in in ADP iajs-2619 184 37 the the DET iajs-2619 184 38 following follow VERB iajs-2619 184 39 figure figure NOUN iajs-2619 184 40 2 2 NUM iajs-2619 184 41 . . NOUN iajs-2619 184 42 139 139 NUM iajs-2619 185 1 ibn ibn PROPN iajs-2619 185 2 al al PROPN iajs-2619 185 3 - - PUNCT iajs-2619 185 4 haitham haitham PROPN iajs-2619 185 5 jour jour X iajs-2619 185 6 . . PROPN iajs-2619 185 7 for for ADP iajs-2619 185 8 pure pure ADJ iajs-2619 185 9 & & CCONJ iajs-2619 185 10 appl appl PROPN iajs-2619 185 11 . . PUNCT iajs-2619 186 1 sci sci PROPN iajs-2619 186 2 . . PROPN iajs-2619 187 1 34 34 NUM iajs-2619 187 2 ( ( PUNCT iajs-2619 187 3 2 2 NUM iajs-2619 187 4 ) ) PUNCT iajs-2619 187 5 2021 2021 NUM iajs-2619 187 6 figure figure NOUN iajs-2619 187 7 2 2 NUM iajs-2619 187 8 . . PUNCT iajs-2619 188 1 the the DET iajs-2619 188 2 trajectory trajectory NOUN iajs-2619 188 3 of of ADP iajs-2619 188 4 system system NOUN iajs-2619 188 5 ( ( PUNCT iajs-2619 188 6 1 1 X iajs-2619 188 7 ) ) PUNCT iajs-2619 188 8 approaches approach VERB iajs-2619 188 9 asymptotically asymptotically ADV iajs-2619 188 10 to to ADP iajs-2619 188 11 a a DET iajs-2619 188 12 globally globally ADV iajs-2619 188 13 stable stable ADJ iajs-2619 188 14 to to ADP iajs-2619 188 15 e0 e0 PROPN iajs-2619 188 16 = = PUNCT iajs-2619 188 17 ( ( PUNCT iajs-2619 188 18 1.21,0,1.18,0,0.40 1.21,0,1.18,0,0.40 NUM iajs-2619 188 19 ) ) PUNCT iajs-2619 188 20 , , PUNCT iajs-2619 188 21 started start VERB iajs-2619 188 22 from from ADP iajs-2619 188 23 different different ADJ iajs-2619 188 24 sets set NOUN iajs-2619 188 25 of of ADP iajs-2619 188 26 initial initial ADJ iajs-2619 188 27 point point NOUN iajs-2619 188 28 . . PUNCT iajs-2619 189 1 ( ( PUNCT iajs-2619 189 2 a a X iajs-2619 189 3 ) ) PUNCT iajs-2619 189 4 for for ADP iajs-2619 189 5 su(t su(t NUM iajs-2619 189 6 ) ) PUNCT iajs-2619 189 7 , , PUNCT iajs-2619 189 8 ( ( PUNCT iajs-2619 189 9 b b X iajs-2619 189 10 ) ) PUNCT iajs-2619 189 11 for for ADP iajs-2619 189 12 i(t i(t NOUN iajs-2619 189 13 ) ) PUNCT iajs-2619 189 14 , , PUNCT iajs-2619 189 15 ( ( PUNCT iajs-2619 189 16 c c X iajs-2619 189 17 ) ) PUNCT iajs-2619 189 18 for for ADP iajs-2619 189 19 sa(t sa(t NUM iajs-2619 189 20 ) ) PUNCT iajs-2619 189 21 , , PUNCT iajs-2619 189 22 ( ( PUNCT iajs-2619 189 23 d d X iajs-2619 189 24 ) ) PUNCT iajs-2619 189 25 for for ADP iajs-2619 189 26 b(t b(t NOUN iajs-2619 189 27 ) ) PUNCT iajs-2619 189 28 , , PUNCT iajs-2619 189 29 ( ( PUNCT iajs-2619 189 30 e e NOUN iajs-2619 189 31 ) ) PUNCT iajs-2619 189 32 for for ADP iajs-2619 189 33 m(t m(t NOUN iajs-2619 189 34 ) ) PUNCT iajs-2619 189 35 . . PUNCT iajs-2619 190 1 it it PRON iajs-2619 190 2 is be AUX iajs-2619 190 3 easy easy ADJ iajs-2619 190 4 by by ADP iajs-2619 190 5 keeping keep VERB iajs-2619 190 6 fixed fix VERB iajs-2619 190 7 the the DET iajs-2619 190 8 parameters parameter NOUN iajs-2619 190 9 values value NOUN iajs-2619 190 10 given give VERB iajs-2619 190 11 in in ADP iajs-2619 190 12 eq eq ADP iajs-2619 190 13 . . PUNCT iajs-2619 191 1 ( ( PUNCT iajs-2619 191 2 17 17 NUM iajs-2619 191 3 ) ) PUNCT iajs-2619 191 4 with with ADP iajs-2619 191 5 putting put VERB iajs-2619 191 6 πœ‡ πœ‡ ADP iajs-2619 191 7 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2619 191 8 π‘˜ π‘˜ NOUN iajs-2619 191 9 in in ADP iajs-2619 191 10 the the DET iajs-2619 191 11 range range NOUN iajs-2619 191 12 0.5 0.5 NUM iajs-2619 191 13 ≀ ≀ NOUN iajs-2619 191 14 πœ‡ πœ‡ ADP iajs-2619 191 15 ≀ ≀ NUM iajs-2619 191 16 16.8 16.8 NUM iajs-2619 191 17 , , PUNCT iajs-2619 191 18 0.001 0.001 NUM iajs-2619 191 19 ≀ ≀ NOUN iajs-2619 191 20 π‘˜ π‘˜ DET iajs-2619 191 21 ≀ ≀ NUM iajs-2619 191 22 0.01 0.01 NUM iajs-2619 191 23 and and CCONJ iajs-2619 191 24 π‘Ž4 π‘Ž4 PROPN iajs-2619 191 25 = = SYM iajs-2619 191 26 0 0 PROPN iajs-2619 191 27 , , PUNCT iajs-2619 191 28 the the DET iajs-2619 191 29 solution solution NOUN iajs-2619 191 30 of of ADP iajs-2619 191 31 the the DET iajs-2619 191 32 system system NOUN iajs-2619 191 33 ( ( PUNCT iajs-2619 191 34 1 1 X iajs-2619 191 35 ) ) PUNCT iajs-2619 191 36 converge converge VERB iajs-2619 191 37 to to ADP iajs-2619 191 38 𝐸0 𝐸0 NOUN iajs-2619 191 39 = = SYM iajs-2619 191 40 ( ( PUNCT iajs-2619 191 41 1.21,0,1.18,0,0.40 1.21,0,1.18,0,0.40 NUM iajs-2619 191 42 ) ) PUNCT iajs-2619 191 43 is be AUX iajs-2619 191 44 shown show VERB iajs-2619 191 45 in in ADP iajs-2619 191 46 the the DET iajs-2619 191 47 typical typical ADJ iajs-2619 191 48 figure figure NOUN iajs-2619 191 49 given give VERB iajs-2619 191 50 by by ADP iajs-2619 191 51 figure figure NOUN iajs-2619 191 52 3 3 NUM iajs-2619 191 53 . . PUNCT iajs-2619 191 54 below below ADV iajs-2619 191 55 . . PUNCT iajs-2619 192 1 140 140 NUM iajs-2619 192 2 ibn ibn PROPN iajs-2619 192 3 al al PROPN iajs-2619 192 4 - - PUNCT iajs-2619 192 5 haitham haitham PROPN iajs-2619 192 6 jour jour X iajs-2619 192 7 . . PROPN iajs-2619 192 8 for for ADP iajs-2619 192 9 pure pure ADJ iajs-2619 192 10 & & CCONJ iajs-2619 192 11 appl appl PROPN iajs-2619 192 12 . . PUNCT iajs-2619 193 1 sci sci PROPN iajs-2619 193 2 . . PROPN iajs-2619 194 1 34 34 NUM iajs-2619 194 2 ( ( PUNCT iajs-2619 194 3 2 2 NUM iajs-2619 194 4 ) ) PUNCT iajs-2619 194 5 2021 2021 NUM iajs-2619 194 6 figure figure NOUN iajs-2619 194 7 3 3 NUM iajs-2619 194 8 . . PUNCT iajs-2619 195 1 the the DET iajs-2619 195 2 solution solution NOUN iajs-2619 195 3 of of ADP iajs-2619 195 4 system system NOUN iajs-2619 195 5 ( ( PUNCT iajs-2619 195 6 1 1 NUM iajs-2619 195 7 ) ) PUNCT iajs-2619 195 8 for for ADP iajs-2619 195 9 the the DET iajs-2619 195 10 data datum NOUN iajs-2619 195 11 ( ( PUNCT iajs-2619 195 12 17 17 NUM iajs-2619 195 13 ) ) PUNCT iajs-2619 195 14 ( ( PUNCT iajs-2619 195 15 a a X iajs-2619 195 16 ) ) PUNCT iajs-2619 195 17 for for ADP iajs-2619 195 18 ΞΌ ΞΌ PROPN iajs-2619 195 19 = = SYM iajs-2619 195 20 0.5 0.5 NUM iajs-2619 195 21 and and CCONJ iajs-2619 195 22 k k NOUN iajs-2619 195 23 = = SYM iajs-2619 195 24 0.1 0.1 NUM iajs-2619 195 25 ( ( PUNCT iajs-2619 195 26 b b NOUN iajs-2619 195 27 ) ) PUNCT iajs-2619 195 28 for for ADP iajs-2619 195 29 ΞΌ ΞΌ PROPN iajs-2619 195 30 = = SYM iajs-2619 195 31 0.7 0.7 NUM iajs-2619 195 32 and and CCONJ iajs-2619 195 33 k k NOUN iajs-2619 195 34 = = NOUN iajs-2619 195 35 0.01 0.01 NUM iajs-2619 195 36 . . PUNCT iajs-2619 196 1 according accord VERB iajs-2619 196 2 to to PART iajs-2619 196 3 figure figure NOUN iajs-2619 196 4 3 3 NUM iajs-2619 196 5 , , PUNCT iajs-2619 196 6 it it PRON iajs-2619 196 7 is be AUX iajs-2619 196 8 clear clear ADJ iajs-2619 196 9 that that SCONJ iajs-2619 196 10 the the DET iajs-2619 196 11 dynamical dynamical ADJ iajs-2619 196 12 behavior behavior NOUN iajs-2619 196 13 of of ADP iajs-2619 196 14 system system NOUN iajs-2619 196 15 ( ( PUNCT iajs-2619 196 16 1 1 NUM iajs-2619 196 17 ) ) PUNCT iajs-2619 196 18 , , PUNCT iajs-2619 196 19 transmission transmission NOUN iajs-2619 196 20 from from ADP iajs-2619 196 21 endemic endemic ADJ iajs-2619 196 22 point point NOUN iajs-2619 196 23 to to ADP iajs-2619 196 24 disease disease NOUN iajs-2619 196 25 free free ADJ iajs-2619 196 26 point point NOUN iajs-2619 196 27 that that PRON iajs-2619 196 28 is be AUX iajs-2619 196 29 mean mean VERB iajs-2619 196 30 the the DET iajs-2619 196 31 endemic endemic ADJ iajs-2619 196 32 point point NOUN iajs-2619 196 33 became become VERB iajs-2619 196 34 unstable unstable ADJ iajs-2619 196 35 . . PUNCT iajs-2619 197 1 now now ADV iajs-2619 197 2 , , PUNCT iajs-2619 197 3 the the DET iajs-2619 197 4 dynamical dynamical ADJ iajs-2619 197 5 behavior behavior NOUN iajs-2619 197 6 of of ADP iajs-2619 197 7 system system NOUN iajs-2619 197 8 ( ( PUNCT iajs-2619 197 9 1 1 NUM iajs-2619 197 10 ) ) PUNCT iajs-2619 197 11 under under ADP iajs-2619 197 12 the the DET iajs-2619 197 13 effect effect NOUN iajs-2619 197 14 of of ADP iajs-2619 197 15 varying vary VERB iajs-2619 197 16 the the DET iajs-2619 197 17 contact contact NOUN iajs-2619 197 18 infected infect VERB iajs-2619 197 19 rate rate NOUN iajs-2619 197 20 value value NOUN iajs-2619 197 21 π‘Ž3 π‘Ž3 PROPN iajs-2619 197 22 is be AUX iajs-2619 197 23 investigated investigate VERB iajs-2619 197 24 . . PUNCT iajs-2619 198 1 system system NOUN iajs-2619 198 2 ( ( PUNCT iajs-2619 198 3 1 1 X iajs-2619 198 4 ) ) PUNCT iajs-2619 198 5 is be AUX iajs-2619 198 6 solved solve VERB iajs-2619 198 7 numerically numerically ADV iajs-2619 198 8 by by ADP iajs-2619 198 9 choosing choose VERB iajs-2619 198 10 the the DET iajs-2619 198 11 parameters parameter NOUN iajs-2619 198 12 values value NOUN iajs-2619 198 13 given give VERB iajs-2619 198 14 by by ADP iajs-2619 198 15 equation equation NOUN iajs-2619 198 16 ( ( PUNCT iajs-2619 198 17 17 17 NUM iajs-2619 198 18 ) ) PUNCT iajs-2619 198 19 with with ADP iajs-2619 198 20 π‘Ž3 π‘Ž3 PROPN iajs-2619 198 21 = = SYM iajs-2619 198 22 3.15,6.15,9.15 3.15,6.15,9.15 NUM iajs-2619 198 23 respectively respectively ADV iajs-2619 198 24 and and CCONJ iajs-2619 198 25 then then ADV iajs-2619 198 26 the the DET iajs-2619 198 27 trajectories trajectory NOUN iajs-2619 198 28 of of ADP iajs-2619 198 29 the the DET iajs-2619 198 30 system system NOUN iajs-2619 198 31 ( ( PUNCT iajs-2619 198 32 1 1 X iajs-2619 198 33 ) ) PUNCT iajs-2619 198 34 are be AUX iajs-2619 198 35 shown show VERB iajs-2619 198 36 in in ADP iajs-2619 198 37 figure figure NOUN iajs-2619 198 38 4 4 NUM iajs-2619 198 39 . . PUNCT iajs-2619 198 40 figure figure VERB iajs-2619 198 41 4 4 NUM iajs-2619 198 42 . . PUNCT iajs-2619 199 1 the the DET iajs-2619 199 2 solutions solution NOUN iajs-2619 199 3 of of ADP iajs-2619 199 4 the the DET iajs-2619 199 5 system system NOUN iajs-2619 199 6 ( ( PUNCT iajs-2619 199 7 1 1 NUM iajs-2619 199 8 ) ) PUNCT iajs-2619 199 9 ( ( PUNCT iajs-2619 199 10 a a X iajs-2619 199 11 ) ) PUNCT iajs-2619 199 12 for for ADP iajs-2619 199 13 a3 a3 NOUN iajs-2619 199 14 = = PUNCT iajs-2619 199 15 3.15 3.15 NUM iajs-2619 199 16 ( ( PUNCT iajs-2619 199 17 b b NOUN iajs-2619 199 18 ) ) PUNCT iajs-2619 199 19 for for ADP iajs-2619 199 20 a3 a3 NOUN iajs-2619 199 21 = = PUNCT iajs-2619 199 22 6.15 6.15 NUM iajs-2619 199 23 ( ( PUNCT iajs-2619 199 24 c c NOUN iajs-2619 199 25 ) ) PUNCT iajs-2619 199 26 a3 a3 NOUN iajs-2619 199 27 = = NOUN iajs-2619 199 28 9.15 9.15 NUM iajs-2619 199 29 clearly clearly ADV iajs-2619 199 30 , , PUNCT iajs-2619 199 31 from from ADP iajs-2619 199 32 figure figure NOUN iajs-2619 199 33 4 4 NUM iajs-2619 199 34 . . PUNCT iajs-2619 200 1 we we PRON iajs-2619 200 2 see see VERB iajs-2619 200 3 that that SCONJ iajs-2619 200 4 the the DET iajs-2619 200 5 solution solution NOUN iajs-2619 200 6 of of ADP iajs-2619 200 7 system system NOUN iajs-2619 200 8 ( ( PUNCT iajs-2619 200 9 1 1 X iajs-2619 200 10 ) ) PUNCT iajs-2619 200 11 is be AUX iajs-2619 200 12 still still ADV iajs-2619 200 13 a a DET iajs-2619 200 14 converge converge NOUN iajs-2619 200 15 to to ADP iajs-2619 200 16 the the DET iajs-2619 200 17 endemic endemic ADJ iajs-2619 200 18 equilibrium equilibrium NOUN iajs-2619 200 19 point point NOUN iajs-2619 200 20 . . PUNCT iajs-2619 201 1 in in ADP iajs-2619 201 2 addition addition NOUN iajs-2619 201 3 it it PRON iajs-2619 201 4 is be AUX iajs-2619 201 5 observed observe VERB iajs-2619 201 6 that that SCONJ iajs-2619 201 7 the the DET iajs-2619 201 8 number number NOUN iajs-2619 201 9 of of ADP iajs-2619 201 10 asymptomatic asymptomatic ADJ iajs-2619 201 11 susceptible susceptible ADJ iajs-2619 201 12 unaware unaware ADJ iajs-2619 201 13 and and CCONJ iajs-2619 201 14 aware aware ADJ iajs-2619 201 15 population population NOUN iajs-2619 201 16 decreases decrease VERB iajs-2619 201 17 while while SCONJ iajs-2619 201 18 the the DET iajs-2619 201 19 number number NOUN iajs-2619 201 20 of of ADP iajs-2619 201 21 infected infected ADJ iajs-2619 201 22 population population NOUN iajs-2619 201 23 increases increase NOUN iajs-2619 201 24 . . PUNCT iajs-2619 202 1 141 141 NUM iajs-2619 202 2 ibn ibn PROPN iajs-2619 202 3 al al PROPN iajs-2619 202 4 - - PUNCT iajs-2619 202 5 haitham haitham PROPN iajs-2619 202 6 jour jour X iajs-2619 202 7 . . PROPN iajs-2619 202 8 for for ADP iajs-2619 202 9 pure pure ADJ iajs-2619 202 10 & & CCONJ iajs-2619 202 11 appl appl PROPN iajs-2619 202 12 . . PUNCT iajs-2619 203 1 sci sci PROPN iajs-2619 203 2 . . PROPN iajs-2619 204 1 34 34 NUM iajs-2619 204 2 ( ( PUNCT iajs-2619 204 3 2 2 NUM iajs-2619 204 4 ) ) PUNCT iajs-2619 204 5 2021 2021 NUM iajs-2619 204 6 finally finally ADV iajs-2619 204 7 , , PUNCT iajs-2619 204 8 the the DET iajs-2619 204 9 dynamical dynamical ADJ iajs-2619 204 10 behavior behavior NOUN iajs-2619 204 11 of of ADP iajs-2619 204 12 system system NOUN iajs-2619 204 13 ( ( PUNCT iajs-2619 204 14 1 1 NUM iajs-2619 204 15 ) ) PUNCT iajs-2619 204 16 under under ADP iajs-2619 204 17 the the DET iajs-2619 204 18 effect effect NOUN iajs-2619 204 19 of of ADP iajs-2619 204 20 varying vary VERB iajs-2619 204 21 the the DET iajs-2619 204 22 incidence incidence NOUN iajs-2619 204 23 rate rate NOUN iajs-2619 204 24 of of ADP iajs-2619 204 25 the the DET iajs-2619 204 26 disease disease NOUN iajs-2619 204 27 by by ADP iajs-2619 204 28 direct direct ADJ iajs-2619 204 29 contact contact NOUN iajs-2619 204 30 with with ADP iajs-2619 204 31 the the DET iajs-2619 204 32 carriers carrier NOUN iajs-2619 204 33 of of ADP iajs-2619 204 34 the the DET iajs-2619 204 35 bacteria bacteria NOUN iajs-2619 204 36 . . PUNCT iajs-2619 205 1 so so ADV iajs-2619 205 2 , , PUNCT iajs-2619 205 3 we we PRON iajs-2619 205 4 can can AUX iajs-2619 205 5 chose choose VERB iajs-2619 205 6 the the DET iajs-2619 205 7 same same ADJ iajs-2619 205 8 set set NOUN iajs-2619 205 9 of of ADP iajs-2619 205 10 parameters parameter NOUN iajs-2619 205 11 values value NOUN iajs-2619 205 12 given give VERB iajs-2619 205 13 by by ADP iajs-2619 205 14 equation equation NOUN iajs-2619 205 15 ( ( PUNCT iajs-2619 205 16 17 17 NUM iajs-2619 205 17 ) ) PUNCT iajs-2619 205 18 with with ADP iajs-2619 205 19 π‘Ž1 π‘Ž1 NOUN iajs-2619 205 20 = = SYM iajs-2619 205 21 5.5,20.5,35.5 5.5,20.5,35.5 NUM iajs-2619 205 22 respectively respectively ADV iajs-2619 205 23 and and CCONJ iajs-2619 205 24 then then ADV iajs-2619 205 25 the the DET iajs-2619 205 26 trajectories trajectory NOUN iajs-2619 205 27 of of ADP iajs-2619 205 28 the the DET iajs-2619 205 29 system system NOUN iajs-2619 205 30 ( ( PUNCT iajs-2619 205 31 1 1 X iajs-2619 205 32 ) ) PUNCT iajs-2619 205 33 are be AUX iajs-2619 205 34 shown show VERB iajs-2619 205 35 in in ADP iajs-2619 205 36 the the DET iajs-2619 205 37 figure figure NOUN iajs-2619 205 38 5 5 NUM iajs-2619 205 39 . . PUNCT iajs-2619 205 40 figure figure NOUN iajs-2619 205 41 5 5 NUM iajs-2619 205 42 . . NOUN iajs-2619 205 43 time time NOUN iajs-2619 205 44 series series PROPN iajs-2619 205 45 of of ADP iajs-2619 205 46 solutions solution NOUN iajs-2619 205 47 of of ADP iajs-2619 205 48 the the DET iajs-2619 205 49 system system NOUN iajs-2619 205 50 ( ( PUNCT iajs-2619 205 51 1 1 NUM iajs-2619 205 52 ) ) PUNCT iajs-2619 205 53 ( ( PUNCT iajs-2619 205 54 a a X iajs-2619 205 55 ) ) PUNCT iajs-2619 205 56 for for ADP iajs-2619 205 57 a1 a1 NOUN iajs-2619 205 58 = = SYM iajs-2619 205 59 5.5 5.5 NUM iajs-2619 205 60 ( ( PUNCT iajs-2619 205 61 b b NOUN iajs-2619 205 62 ) ) PUNCT iajs-2619 205 63 for for ADP iajs-2619 205 64 a1 a1 NOUN iajs-2619 205 65 = = SYM iajs-2619 205 66 20.5 20.5 NUM iajs-2619 205 67 ( ( PUNCT iajs-2619 205 68 c c NOUN iajs-2619 205 69 ) ) PUNCT iajs-2619 205 70 a1 a1 NOUN iajs-2619 205 71 = = NOUN iajs-2619 205 72 35.5 35.5 NUM iajs-2619 205 73 clearly clearly ADV iajs-2619 205 74 , , PUNCT iajs-2619 205 75 system system NOUN iajs-2619 205 76 ( ( PUNCT iajs-2619 205 77 1 1 X iajs-2619 205 78 ) ) PUNCT iajs-2619 205 79 has have VERB iajs-2619 205 80 an an DET iajs-2619 205 81 asymptotically asymptotically ADV iajs-2619 205 82 stable stable ADJ iajs-2619 205 83 to to ADP iajs-2619 205 84 the the DET iajs-2619 205 85 endemic endemic ADJ iajs-2619 205 86 equilibrium equilibrium NOUN iajs-2619 205 87 point point NOUN iajs-2619 205 88 . . PUNCT iajs-2619 206 1 in in ADP iajs-2619 206 2 addition addition NOUN iajs-2619 206 3 it it PRON iajs-2619 206 4 is be AUX iajs-2619 206 5 observed observe VERB iajs-2619 206 6 that that SCONJ iajs-2619 206 7 there there PRON iajs-2619 206 8 is be VERB iajs-2619 206 9 a a DET iajs-2619 206 10 slight slight ADJ iajs-2619 206 11 change change NOUN iajs-2619 206 12 in in ADP iajs-2619 206 13 the the DET iajs-2619 206 14 system system NOUN iajs-2619 206 15 . . PUNCT iajs-2619 207 1 7 7 X iajs-2619 207 2 . . X iajs-2619 207 3 conclusion conclusion NOUN iajs-2619 207 4 and and CCONJ iajs-2619 207 5 discussion discussion NOUN iajs-2619 207 6 : : PUNCT iajs-2619 207 7 awareness awareness NOUN iajs-2619 207 8 programs program NOUN iajs-2619 207 9 that that PRON iajs-2619 207 10 control control VERB iajs-2619 207 11 the the DET iajs-2619 207 12 spread spread NOUN iajs-2619 207 13 of of ADP iajs-2619 207 14 the the DET iajs-2619 207 15 epidemic epidemic NOUN iajs-2619 207 16 strong strong ADJ iajs-2619 207 17 steps step NOUN iajs-2619 207 18 should should AUX iajs-2619 207 19 be be AUX iajs-2619 207 20 taken take VERB iajs-2619 207 21 regarding regard VERB iajs-2619 207 22 their their PRON iajs-2619 207 23 implementation implementation NOUN iajs-2619 207 24 of of ADP iajs-2619 207 25 diseases disease NOUN iajs-2619 207 26 such such ADJ iajs-2619 207 27 as as ADP iajs-2619 207 28 smallpox smallpox NOUN iajs-2619 207 29 , , PUNCT iajs-2619 207 30 measles measle NOUN iajs-2619 207 31 , , PUNCT iajs-2619 207 32 influenza influenza NOUN iajs-2619 207 33 , , PUNCT iajs-2619 207 34 and and CCONJ iajs-2619 207 35 others other NOUN iajs-2619 207 36 . . PUNCT iajs-2619 208 1 behavioral behavioral ADJ iajs-2619 208 2 changes change NOUN iajs-2619 208 3 caused cause VERB iajs-2619 208 4 by by ADP iajs-2619 208 5 awareness awareness NOUN iajs-2619 208 6 programs program NOUN iajs-2619 208 7 have have VERB iajs-2619 208 8 the the DET iajs-2619 208 9 ability ability NOUN iajs-2619 208 10 to to PART iajs-2619 208 11 control control VERB iajs-2619 208 12 the the DET iajs-2619 208 13 size size NOUN iajs-2619 208 14 of of ADP iajs-2619 208 15 the the DET iajs-2619 208 16 epidemic epidemic NOUN iajs-2619 208 17 and and CCONJ iajs-2619 208 18 then then ADV iajs-2619 208 19 predict predict VERB iajs-2619 208 20 the the DET iajs-2619 208 21 future future ADJ iajs-2619 208 22 course course NOUN iajs-2619 208 23 of of ADP iajs-2619 208 24 the the DET iajs-2619 208 25 outbreak outbreak NOUN iajs-2619 208 26 to to PART iajs-2619 208 27 guide guide VERB iajs-2619 208 28 public public ADJ iajs-2619 208 29 health health NOUN iajs-2619 208 30 policy policy NOUN iajs-2619 208 31 . . PUNCT iajs-2619 209 1 this this PRON iajs-2619 209 2 is be AUX iajs-2619 209 3 a a DET iajs-2619 209 4 different different ADJ iajs-2619 209 5 thing thing NOUN iajs-2619 209 6 because because SCONJ iajs-2619 209 7 outbreak outbreak NOUN iajs-2619 209 8 of of ADP iajs-2619 209 9 the the DET iajs-2619 209 10 disease disease NOUN iajs-2619 209 11 involves involve VERB iajs-2619 209 12 the the DET iajs-2619 209 13 environment environment NOUN iajs-2619 209 14 , , PUNCT iajs-2619 209 15 and and CCONJ iajs-2619 209 16 ( ( PUNCT iajs-2619 209 17 direct direct ADJ iajs-2619 209 18 and and CCONJ iajs-2619 209 19 indirect indirect ADJ iajs-2619 209 20 ) ) PUNCT iajs-2619 209 21 multiple multiple ADJ iajs-2619 209 22 ways way NOUN iajs-2619 209 23 of of ADP iajs-2619 209 24 transmission transmission NOUN iajs-2619 209 25 how how SCONJ iajs-2619 209 26 to to PART iajs-2619 209 27 cover cover VERB iajs-2619 209 28 the the DET iajs-2619 209 29 media medium NOUN iajs-2619 209 30 , , PUNCT iajs-2619 209 31 the the DET iajs-2619 209 32 awareness awareness NOUN iajs-2619 209 33 programs program NOUN iajs-2619 209 34 will will AUX iajs-2619 209 35 be be AUX iajs-2619 209 36 affected affect VERB iajs-2619 209 37 the the DET iajs-2619 209 38 dynamics dynamic NOUN iajs-2619 209 39 of of ADP iajs-2619 209 40 shigellosis shigellosis NOUN iajs-2619 209 41 . . PUNCT iajs-2619 210 1 in in ADP iajs-2619 210 2 this this DET iajs-2619 210 3 work work NOUN iajs-2619 210 4 , , PUNCT iajs-2619 210 5 we we PRON iajs-2619 210 6 proposed propose VERB iajs-2619 210 7 and and CCONJ iajs-2619 210 8 analyzed analyze VERB iajs-2619 210 9 a a DET iajs-2619 210 10 model model NOUN iajs-2619 210 11 to to PART iajs-2619 210 12 study study VERB iajs-2619 210 13 the the DET iajs-2619 210 14 effect effect NOUN iajs-2619 210 15 of of ADP iajs-2619 210 16 awareness awareness NOUN iajs-2619 210 17 programs program NOUN iajs-2619 210 18 on on ADP iajs-2619 210 19 shigellosis shigellosis NOUN iajs-2619 210 20 disease disease NOUN iajs-2619 210 21 . . PUNCT iajs-2619 211 1 the the DET iajs-2619 211 2 model model NOUN iajs-2619 211 3 included include VERB iajs-2619 211 4 four four NUM iajs-2619 211 5 ordinary ordinary ADJ iajs-2619 211 6 differential differential ADJ iajs-2619 211 7 equations equation NOUN iajs-2619 211 8 describing describe VERB iajs-2619 211 9 four four NUM iajs-2619 211 10 different different ADJ iajs-2619 211 11 class class NOUN iajs-2619 211 12 : : PUNCT iajs-2619 211 13 unaware unaware ADJ iajs-2619 211 14 susceptible susceptible ADJ iajs-2619 211 15 individuals individual NOUN iajs-2619 211 16 𝑆𝑒 𝑆𝑒 PROPN iajs-2619 211 17 , , PUNCT iajs-2619 211 18 infected infect VERB iajs-2619 211 19 individuals individual NOUN iajs-2619 211 20 𝐼 𝐼 PROPN iajs-2619 211 21 , , PUNCT iajs-2619 211 22 aware aware ADJ iajs-2619 211 23 susceptible susceptible ADJ iajs-2619 211 24 individuals individual NOUN iajs-2619 211 25 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2619 211 26 and and CCONJ iajs-2619 211 27 people people NOUN iajs-2619 211 28 carrying carry VERB iajs-2619 211 29 bacteria bacteria NOUN iajs-2619 211 30 𝐡. 𝐡. NOUN iajs-2619 211 31 system system NOUN iajs-2619 211 32 ( ( PUNCT iajs-2619 211 33 1 1 X iajs-2619 211 34 ) ) PUNCT iajs-2619 211 35 has have VERB iajs-2619 211 36 only only ADV iajs-2619 211 37 two two NUM iajs-2619 211 38 equilibrium equilibrium NOUN iajs-2619 211 39 points point NOUN iajs-2619 211 40 . . PUNCT iajs-2619 212 1 the the DET iajs-2619 212 2 conditions condition NOUN iajs-2619 212 3 for for ADP iajs-2619 212 4 existence existence NOUN iajs-2619 212 5 , , PUNCT iajs-2619 212 6 stability stability NOUN iajs-2619 212 7 for for ADP iajs-2619 212 8 each each DET iajs-2619 212 9 equilibrium equilibrium NOUN iajs-2619 212 10 points point NOUN iajs-2619 212 11 are be AUX iajs-2619 212 12 obtained obtain VERB iajs-2619 212 13 . . PUNCT iajs-2619 213 1 further far ADV iajs-2619 213 2 , , PUNCT iajs-2619 213 3 it it PRON iajs-2619 213 4 is be AUX iajs-2619 213 5 observed observe VERB iajs-2619 213 6 that that SCONJ iajs-2619 213 7 the the DET iajs-2619 213 8 disease disease NOUN iajs-2619 213 9 free free ADJ iajs-2619 213 10 equilibrium equilibrium NOUN iajs-2619 213 11 point point NOUN iajs-2619 213 12 ( ( PUNCT iajs-2619 213 13 𝐸0 𝐸0 NOUN iajs-2619 213 14 ) ) PUNCT iajs-2619 213 15 exists exist VERB iajs-2619 213 16 when when SCONJ iajs-2619 213 17 π‘Ž4 π‘Ž4 PROPN iajs-2619 213 18 = = SYM iajs-2619 213 19 0 0 PROPN iajs-2619 213 20 and and CCONJ iajs-2619 213 21 𝐼 𝐼 PROPN iajs-2619 213 22 = = PROPN iajs-2619 213 23 𝐡 𝐡 NOUN iajs-2619 213 24 = = NOUN iajs-2619 213 25 0 0 NUM iajs-2619 213 26 and and CCONJ iajs-2619 213 27 locally locally ADV iajs-2619 213 28 stable stable ADJ iajs-2619 213 29 if if SCONJ iajs-2619 213 30 the the DET iajs-2619 213 31 conditions condition NOUN iajs-2619 213 32 ( ( PUNCT iajs-2619 213 33 11a-11c 11a-11c NOUN iajs-2619 213 34 ) ) PUNCT iajs-2619 213 35 are be AUX iajs-2619 213 36 held hold VERB iajs-2619 213 37 , , PUNCT iajs-2619 213 38 and and CCONJ iajs-2619 213 39 then then ADV iajs-2619 213 40 it it PRON iajs-2619 213 41 is be AUX iajs-2619 213 42 globally globally ADV iajs-2619 213 43 stable stable ADJ iajs-2619 213 44 if if SCONJ iajs-2619 213 45 the the DET iajs-2619 213 46 conditions condition NOUN iajs-2619 213 47 ( ( PUNCT iajs-2619 213 48 15a-15b 15a-15b NUM iajs-2619 213 49 ) ) PUNCT iajs-2619 213 50 are be AUX iajs-2619 213 51 held hold VERB iajs-2619 213 52 . . PUNCT iajs-2619 214 1 the the DET iajs-2619 214 2 endemic endemic ADJ iajs-2619 214 3 equilibrium equilibrium NOUN iajs-2619 214 4 point point NOUN iajs-2619 214 5 ( ( PUNCT iajs-2619 214 6 𝐸1 𝐸1 NOUN iajs-2619 214 7 ) ) PUNCT iajs-2619 214 8 exists exist VERB iajs-2619 214 9 if if SCONJ iajs-2619 214 10 the the DET iajs-2619 214 11 conditions condition NOUN iajs-2619 214 12 142 142 NUM iajs-2619 214 13 ibn ibn PROPN iajs-2619 214 14 al al PROPN iajs-2619 214 15 - - PUNCT iajs-2619 214 16 haitham haitham PROPN iajs-2619 214 17 jour jour X iajs-2619 214 18 . . PROPN iajs-2619 214 19 for for ADP iajs-2619 214 20 pure pure ADJ iajs-2619 214 21 & & CCONJ iajs-2619 214 22 appl appl PROPN iajs-2619 214 23 . . PUNCT iajs-2619 215 1 sci sci PROPN iajs-2619 215 2 . . PROPN iajs-2619 216 1 34 34 NUM iajs-2619 216 2 ( ( PUNCT iajs-2619 216 3 2 2 NUM iajs-2619 216 4 ) ) PUNCT iajs-2619 216 5 2021 2021 NUM iajs-2619 216 6 ( ( PUNCT iajs-2619 216 7 9a-9b 9a-9b NUM iajs-2619 216 8 ) ) PUNCT iajs-2619 216 9 are be AUX iajs-2619 216 10 held hold VERB iajs-2619 216 11 and and CCONJ iajs-2619 216 12 locally locally ADV iajs-2619 216 13 stable stable ADJ iajs-2619 216 14 if if SCONJ iajs-2619 216 15 the the DET iajs-2619 216 16 conditions condition NOUN iajs-2619 216 17 ( ( PUNCT iajs-2619 216 18 13a-13b 13a-13b NUM iajs-2619 216 19 ) ) PUNCT iajs-2619 216 20 are be AUX iajs-2619 216 21 held hold VERB iajs-2619 216 22 more more ADJ iajs-2619 216 23 than than SCONJ iajs-2619 216 24 it it PRON iajs-2619 216 25 is be AUX iajs-2619 216 26 globally globally ADV iajs-2619 216 27 stable stable ADJ iajs-2619 216 28 if if SCONJ iajs-2619 216 29 the the DET iajs-2619 216 30 conditions condition NOUN iajs-2619 216 31 ( ( PUNCT iajs-2619 216 32 16a-16 16a-16 NOUN iajs-2619 216 33 g g NOUN iajs-2619 216 34 ) ) PUNCT iajs-2619 216 35 are be AUX iajs-2619 216 36 held hold VERB iajs-2619 216 37 . . PUNCT iajs-2619 217 1 finally finally ADV iajs-2619 217 2 , , PUNCT iajs-2619 217 3 to to PART iajs-2619 217 4 understand understand VERB iajs-2619 217 5 the the DET iajs-2619 217 6 effect effect NOUN iajs-2619 217 7 of of ADP iajs-2619 217 8 varying vary VERB iajs-2619 217 9 each each DET iajs-2619 217 10 parameter parameter NOUN iajs-2619 217 11 on on ADP iajs-2619 217 12 the the DET iajs-2619 217 13 global global ADJ iajs-2619 217 14 system system NOUN iajs-2619 217 15 ( ( PUNCT iajs-2619 217 16 1 1 NUM iajs-2619 217 17 ) ) PUNCT iajs-2619 217 18 and and CCONJ iajs-2619 217 19 confirm confirm VERB iajs-2619 217 20 our our PRON iajs-2619 217 21 above above ADJ iajs-2619 217 22 analytical analytical ADJ iajs-2619 217 23 results result NOUN iajs-2619 217 24 , , PUNCT iajs-2619 217 25 system system NOUN iajs-2619 217 26 ( ( PUNCT iajs-2619 217 27 1 1 X iajs-2619 217 28 ) ) PUNCT iajs-2619 217 29 has have AUX iajs-2619 217 30 been be AUX iajs-2619 217 31 solved solve VERB iajs-2619 217 32 numerically numerically ADV iajs-2619 217 33 for for ADP iajs-2619 217 34 different different ADJ iajs-2619 217 35 sets set NOUN iajs-2619 217 36 of of ADP iajs-2619 217 37 initial initial ADJ iajs-2619 217 38 points point NOUN iajs-2619 217 39 and and CCONJ iajs-2619 217 40 different different ADJ iajs-2619 217 41 sets set NOUN iajs-2619 217 42 of of ADP iajs-2619 217 43 parameter parameter NOUN iajs-2619 217 44 given give VERB iajs-2619 217 45 by by ADP iajs-2619 217 46 equation equation NOUN iajs-2619 217 47 ( ( PUNCT iajs-2619 217 48 17 17 NUM iajs-2619 217 49 ) ) PUNCT iajs-2619 217 50 , , PUNCT iajs-2619 217 51 and and CCONJ iajs-2619 217 52 the the DET iajs-2619 217 53 following follow VERB iajs-2619 217 54 observation observation NOUN iajs-2619 217 55 are be AUX iajs-2619 217 56 made make VERB iajs-2619 217 57 : : PUNCT iajs-2619 217 58 1 1 X iajs-2619 217 59 . . X iajs-2619 218 1 for for ADP iajs-2619 218 2 the the DET iajs-2619 218 3 set set NOUN iajs-2619 218 4 of of ADP iajs-2619 218 5 hypothetical hypothetical ADJ iajs-2619 218 6 parameters parameter NOUN iajs-2619 218 7 values value NOUN iajs-2619 218 8 given give VERB iajs-2619 218 9 by by ADP iajs-2619 218 10 eq eq PROPN iajs-2619 218 11 . . PUNCT iajs-2619 219 1 ( ( PUNCT iajs-2619 219 2 17 17 NUM iajs-2619 219 3 ) ) PUNCT iajs-2619 219 4 , , PUNCT iajs-2619 219 5 the the DET iajs-2619 219 6 system system NOUN iajs-2619 219 7 ( ( PUNCT iajs-2619 219 8 1 1 X iajs-2619 219 9 ) ) PUNCT iajs-2619 219 10 approached approach VERB iajs-2619 219 11 asymptotically asymptotically ADV iajs-2619 219 12 to to ADP iajs-2619 219 13 the the DET iajs-2619 219 14 global global ADJ iajs-2619 219 15 stable stable ADJ iajs-2619 219 16 endemic endemic ADJ iajs-2619 219 17 equilibrium equilibrium NOUN iajs-2619 219 18 point point NOUN iajs-2619 219 19 , , PUNCT iajs-2619 219 20 𝐸1 𝐸1 NOUN iajs-2619 219 21 . . PUNCT iajs-2619 220 1 2 2 X iajs-2619 220 2 . . X iajs-2619 220 3 equation equation NOUN iajs-2619 220 4 ( ( PUNCT iajs-2619 220 5 17 17 NUM iajs-2619 220 6 ) ) PUNCT iajs-2619 220 7 , , PUNCT iajs-2619 220 8 when when SCONJ iajs-2619 220 9 we we PRON iajs-2619 220 10 set set VERB iajs-2619 220 11 the the DET iajs-2619 220 12 following follow VERB iajs-2619 220 13 values value NOUN iajs-2619 220 14 π‘Ž4 π‘Ž4 PROPN iajs-2619 220 15 = = SYM iajs-2619 220 16 0 0 NUM iajs-2619 220 17 and and CCONJ iajs-2619 220 18 π‘˜ π‘˜ X iajs-2619 220 19 = = NOUN iajs-2619 220 20 0.01 0.01 NUM iajs-2619 220 21 the the DET iajs-2619 220 22 system system NOUN iajs-2619 220 23 ( ( PUNCT iajs-2619 220 24 1 1 X iajs-2619 220 25 ) ) PUNCT iajs-2619 220 26 converge converge VERB iajs-2619 220 27 to to ADP iajs-2619 220 28 disease disease NOUN iajs-2619 220 29 free free ADJ iajs-2619 220 30 equilibrium equilibrium NOUN iajs-2619 220 31 point point NOUN iajs-2619 220 32 𝐸0 𝐸0 NOUN iajs-2619 220 33 and and CCONJ iajs-2619 220 34 satisfied satisfy VERB iajs-2619 220 35 the the DET iajs-2619 220 36 global global ADJ iajs-2619 220 37 stability stability NOUN iajs-2619 220 38 . . PUNCT iajs-2619 221 1 3 3 X iajs-2619 221 2 . . X iajs-2619 221 3 if if SCONJ iajs-2619 221 4 the the DET iajs-2619 221 5 contact contact NOUN iajs-2619 221 6 rates rate VERB iajs-2619 221 7 π‘Ž3 π‘Ž3 PROPN iajs-2619 221 8 and and CCONJ iajs-2619 221 9 π‘Ž1 π‘Ž1 NOUN iajs-2619 221 10 increase increase NOUN iajs-2619 221 11 , , PUNCT iajs-2619 221 12 the the DET iajs-2619 221 13 solution solution NOUN iajs-2619 221 14 of of ADP iajs-2619 221 15 system system NOUN iajs-2619 221 16 ( ( PUNCT iajs-2619 221 17 1 1 X iajs-2619 221 18 ) ) PUNCT iajs-2619 221 19 still still ADV iajs-2619 221 20 converge converge VERB iajs-2619 221 21 to to ADP iajs-2619 221 22 the the DET iajs-2619 221 23 endemic endemic ADJ iajs-2619 221 24 equilibrium equilibrium NOUN iajs-2619 221 25 point point NOUN iajs-2619 221 26 . . PUNCT iajs-2619 222 1 references reference NOUN iajs-2619 222 2 1 1 NUM iajs-2619 222 3 . . PUNCT iajs-2619 223 1 hassan hassan PROPN iajs-2619 223 2 f. f. PROPN iajs-2619 223 3 ; ; PUNCT iajs-2619 223 4 ahmed ahmed PROPN iajs-2619 223 5 a. a. PROPN iajs-2619 223 6 m. m. PROPN iajs-2619 223 7 the the DET iajs-2619 223 8 dynamical dynamical ADJ iajs-2619 223 9 behaviors behavior NOUN iajs-2619 223 10 for for ADP iajs-2619 223 11 an an DET iajs-2619 223 12 epidemic epidemic NOUN iajs-2619 223 13 disease disease NOUN iajs-2619 223 14 model model NOUN iajs-2619 223 15 with with ADP iajs-2619 223 16 general general ADJ iajs-2619 223 17 recovery recovery NOUN iajs-2619 223 18 function function NOUN iajs-2619 223 19 , , PUNCT iajs-2619 223 20 sci sci PROPN iajs-2619 223 21 . . PUNCT iajs-2619 223 22 int int PROPN iajs-2619 223 23 . . PUNCT iajs-2619 224 1 ( ( PUNCT iajs-2619 224 2 lahore lahore NOUN iajs-2619 224 3 ) ) PUNCT iajs-2619 224 4 , , PUNCT iajs-2619 224 5 2017 2017 NUM iajs-2619 224 6 , , PUNCT iajs-2619 224 7 29 29 NUM iajs-2619 224 8 , , PUNCT iajs-2619 224 9 5 5 NUM iajs-2619 224 10 , , PUNCT iajs-2619 224 11 1007 1007 NUM iajs-2619 224 12 - - SYM iajs-2619 224 13 1014 1014 NUM iajs-2619 224 14 . . PUNCT iajs-2619 225 1 2 2 X iajs-2619 225 2 . . PUNCT iajs-2619 225 3 ahmed ahmed PROPN iajs-2619 225 4 a. a. PROPN iajs-2619 225 5 m. m. PROPN iajs-2619 225 6 ; ; PUNCT iajs-2619 225 7 zhou zhou PROPN iajs-2619 225 8 x. x. PROPN iajs-2619 225 9 on on ADP iajs-2619 225 10 the the DET iajs-2619 225 11 dynamical dynamical ADJ iajs-2619 225 12 behaviors behavior NOUN iajs-2619 225 13 of of ADP iajs-2619 225 14 cholera cholera NOUN iajs-2619 225 15 model model NOUN iajs-2619 225 16 with with ADP iajs-2619 225 17 holling holle VERB iajs-2619 225 18 type type NOUN iajs-2619 225 19 ӏӏ ӏӏ CCONJ iajs-2619 225 20 functional functional ADJ iajs-2619 225 21 response response NOUN iajs-2619 225 22 , , PUNCT iajs-2619 225 23 journal journal NOUN iajs-2619 225 24 of of ADP iajs-2619 225 25 al al PROPN iajs-2619 225 26 - - PUNCT iajs-2619 225 27 nahrain nahrain PROPN iajs-2619 225 28 univ univ PROPN iajs-2619 225 29 . . PROPN iajs-2619 225 30 , , PUNCT iajs-2619 225 31 2016 2016 NUM iajs-2619 225 32 , , PUNCT iajs-2619 225 33 19 19 NUM iajs-2619 225 34 , , PUNCT iajs-2619 225 35 1 1 NUM iajs-2619 225 36 , , PUNCT iajs-2619 225 37 156 156 NUM iajs-2619 225 38 - - SYM iajs-2619 225 39 167 167 NUM iajs-2619 225 40 . . PUNCT iajs-2619 226 1 3 3 X iajs-2619 226 2 . . X iajs-2619 226 3 al al PROPN iajs-2619 226 4 - - PUNCT iajs-2619 226 5 arydah arydah PROPN iajs-2619 226 6 , , PUNCT iajs-2619 226 7 m. m. NOUN iajs-2619 226 8 ; ; PUNCT iajs-2619 226 9 mwasa mwasa PROPN iajs-2619 226 10 , , PUNCT iajs-2619 226 11 a. a. PROPN iajs-2619 226 12 ; ; PUNCT iajs-2619 226 13 tchuenche tchuenche PROPN iajs-2619 226 14 , , PUNCT iajs-2619 226 15 j. j. PROPN iajs-2619 226 16 m. m. PROPN iajs-2619 226 17 ; ; PUNCT iajs-2619 226 18 smith smith PROPN iajs-2619 226 19 r. r. PROPN iajs-2619 226 20 j. j. PROPN iajs-2619 226 21 modelling model VERB iajs-2619 226 22 cholera cholera NOUN iajs-2619 226 23 disease disease NOUN iajs-2619 226 24 with with ADP iajs-2619 226 25 education education NOUN iajs-2619 226 26 and and CCONJ iajs-2619 226 27 chlorination chlorination NOUN iajs-2619 226 28 , , PUNCT iajs-2619 226 29 journal journal NOUN iajs-2619 226 30 of of ADP iajs-2619 226 31 biological biological ADJ iajs-2619 226 32 system system NOUN iajs-2619 226 33 , , PUNCT iajs-2619 226 34 2013 2013 NUM iajs-2619 226 35 , , PUNCT iajs-2619 226 36 21 21 NUM iajs-2619 226 37 , , PUNCT iajs-2619 226 38 4 4 NUM iajs-2619 226 39 , , PUNCT iajs-2619 226 40 1 1 NUM iajs-2619 226 41 - - SYM iajs-2619 226 42 20 20 NUM iajs-2619 226 43 . . PUNCT iajs-2619 227 1 4 4 X iajs-2619 227 2 . . X iajs-2619 227 3 luquero luquero PROPN iajs-2619 227 4 f. f. PROPN iajs-2619 227 5 j. j. PROPN iajs-2619 227 6 ; ; PUNCT iajs-2619 227 7 grout grout PROPN iajs-2619 227 8 l. l. PROPN iajs-2619 227 9 ; ; PUNCT iajs-2619 227 10 ciglenecki ciglenecki PROPN iajs-2619 227 11 i. i. PROPN iajs-2619 227 12 ; ; PUNCT iajs-2619 227 13 sakoba sakoba PROPN iajs-2619 227 14 k. k. PROPN iajs-2619 227 15 ; ; PUNCT iajs-2619 227 16 traore traore PROPN iajs-2619 227 17 b. b. PROPN iajs-2619 227 18 ; ; PUNCT iajs-2619 227 19 helie helie PROPN iajs-2619 227 20 , , PUNCT iajs-2619 227 21 m. m. NOUN iajs-2619 227 22 use use NOUN iajs-2619 227 23 of of ADP iajs-2619 227 24 vibrio vibrio ADJ iajs-2619 227 25 cholerae cholerae NOUN iajs-2619 227 26 vaccine vaccine NOUN iajs-2619 227 27 in in ADP iajs-2619 227 28 an an DET iajs-2619 227 29 outbreak outbreak NOUN iajs-2619 227 30 in in ADP iajs-2619 227 31 guinea guinea NOUN iajs-2619 227 32 , , PUNCT iajs-2619 227 33 the the DET iajs-2619 227 34 new new PROPN iajs-2619 227 35 england england PROPN iajs-2619 227 36 journal journal PROPN iajs-2619 227 37 of of ADP iajs-2619 227 38 medicine medicine NOUN iajs-2619 227 39 , , PUNCT iajs-2619 227 40 2014 2014 NUM iajs-2619 227 41 , , PUNCT iajs-2619 227 42 370 370 NUM iajs-2619 227 43 , , PUNCT iajs-2619 227 44 2111 2111 NUM iajs-2619 227 45 – – PUNCT iajs-2619 227 46 2120 2120 NUM iajs-2619 227 47 . . PUNCT iajs-2619 228 1 5 5 X iajs-2619 228 2 . . X iajs-2619 228 3 kermack kermack PROPN iajs-2619 228 4 w. w. PROPN iajs-2619 228 5 o. o. PROPN iajs-2619 228 6 ; ; PUNCT iajs-2619 228 7 mckendrick mckendrick PROPN iajs-2619 228 8 a. a. PROPN iajs-2619 228 9 g. g. PROPN iajs-2619 228 10 contributions contribution NOUN iajs-2619 228 11 to to ADP iajs-2619 228 12 mathematical mathematical ADJ iajs-2619 228 13 theory theory NOUN iajs-2619 228 14 of of ADP iajs-2619 228 15 epidemics epidemic NOUN iajs-2619 228 16 , , PUNCT iajs-2619 228 17 proc proc NOUN iajs-2619 228 18 . . PUNCT iajs-2619 229 1 r. r. PROPN iajs-2619 229 2 soc soc PROPN iajs-2619 229 3 . . PUNCT iajs-2619 230 1 ser ser PROPN iajs-2619 230 2 . . PUNCT iajs-2619 230 3 a. a. PROPN iajs-2619 230 4 , , PUNCT iajs-2619 230 5 1927 1927 NUM iajs-2619 230 6 , , PUNCT iajs-2619 230 7 115 115 NUM iajs-2619 230 8 , , PUNCT iajs-2619 230 9 700 700 NUM iajs-2619 230 10 - - SYM iajs-2619 230 11 721 721 NUM iajs-2619 230 12 . . PUNCT iajs-2619 231 1 6 6 NUM iajs-2619 231 2 . . X iajs-2619 231 3 kermack kermack PROPN iajs-2619 231 4 , , PUNCT iajs-2619 231 5 w. w. PROPN iajs-2619 231 6 o. o. PROPN iajs-2619 231 7 ; ; PUNCT iajs-2619 231 8 mckendrick mckendrick PROPN iajs-2619 231 9 , , PUNCT iajs-2619 231 10 a. a. NOUN iajs-2619 231 11 g. g. PROPN iajs-2619 231 12 contributions contribution NOUN iajs-2619 231 13 to to ADP iajs-2619 231 14 mathematical mathematical ADJ iajs-2619 231 15 theory theory NOUN iajs-2619 231 16 of of ADP iajs-2619 231 17 epidemics epidemic NOUN iajs-2619 231 18 , , PUNCT iajs-2619 231 19 proc proc NOUN iajs-2619 231 20 . . PUNCT iajs-2619 232 1 r. r. PROPN iajs-2619 232 2 soc soc PROPN iajs-2619 232 3 . . PUNCT iajs-2619 233 1 lond lond PROPN iajs-2619 233 2 . . PUNCT iajs-2619 234 1 a. a. PROPN iajs-2619 234 2 , , PUNCT iajs-2619 234 3 1932 1932 NUM iajs-2619 234 4 , , PUNCT iajs-2619 234 5 138 138 NUM iajs-2619 234 6 , , PUNCT iajs-2619 234 7 55 55 NUM iajs-2619 234 8 - - SYM iajs-2619 234 9 83 83 NUM iajs-2619 234 10 . . PUNCT iajs-2619 235 1 7 7 X iajs-2619 235 2 . . 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