id sid tid token lemma pos cana-5899 1 1 communications communication NOUN cana-5899 1 2 on on ADP cana-5899 1 3 applied apply VERB cana-5899 1 4 nonlinear nonlinear ADJ cana-5899 1 5 analysis analysis NOUN cana-5899 1 6 issn issn NOUN cana-5899 1 7 : : PUNCT cana-5899 1 8 1074 1074 NUM cana-5899 1 9 - - PUNCT cana-5899 1 10 133x 133x NUM cana-5899 1 11 vol vol NOUN cana-5899 1 12 31 31 NUM cana-5899 1 13 no no NOUN cana-5899 1 14 . . PUNCT cana-5899 2 1 8s 8s PROPN cana-5899 2 2 ( ( PUNCT cana-5899 2 3 2024 2024 NUM cana-5899 2 4 ) ) PUNCT cana-5899 2 5 1084 1084 NUM cana-5899 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 2 7 dynamics dynamic NOUN cana-5899 2 8 of of ADP cana-5899 2 9 an an DET cana-5899 2 10 infectious infectious ADJ cana-5899 2 11 disease disease NOUN cana-5899 2 12 model model NOUN cana-5899 2 13 with with ADP cana-5899 2 14 transmission transmission NOUN cana-5899 2 15 delay delay NOUN cana-5899 2 16 from from ADP cana-5899 2 17 recovery recovery NOUN cana-5899 2 18 to to ADP cana-5899 2 19 susceptibility susceptibility NOUN cana-5899 2 20 1*g 1*g PROPN cana-5899 2 21 . . PUNCT cana-5899 3 1 shailaja shailaja PROPN cana-5899 3 2 , , PUNCT cana-5899 3 3 2dr 2dr NOUN cana-5899 3 4 . . PUNCT cana-5899 4 1 m. m. NOUN cana-5899 4 2 a. a. PROPN cana-5899 4 3 srinivas srinivas PROPN cana-5899 4 4 1 1 NUM cana-5899 4 5 , , PUNCT cana-5899 4 6 * * PUNCT cana-5899 4 7 research research NOUN cana-5899 4 8 scholar scholar NOUN cana-5899 4 9 , , PUNCT cana-5899 4 10 ph.d ph.d PROPN cana-5899 4 11 . . PUNCT cana-5899 5 1 in in ADP cana-5899 5 2 mathematics mathematics PROPN cana-5899 5 3 , , PUNCT cana-5899 5 4 jntuh jntuh PROPN cana-5899 5 5 , , PUNCT cana-5899 5 6 hyderabad hyderabad PROPN cana-5899 5 7 , , PUNCT cana-5899 5 8 telangana telangana PROPN cana-5899 5 9 , , PUNCT cana-5899 5 10 india india PROPN cana-5899 5 11 . . PROPN cana-5899 5 12 2 2 NUM cana-5899 5 13 emeritus emeritus ADJ cana-5899 5 14 professor professor NOUN cana-5899 5 15 , , PUNCT cana-5899 5 16 department department NOUN cana-5899 5 17 of of ADP cana-5899 5 18 mathematics mathematic NOUN cana-5899 5 19 , , PUNCT cana-5899 5 20 jntuh jntuh PROPN cana-5899 5 21 , , PUNCT cana-5899 5 22 hyderabad hyderabad PROPN cana-5899 5 23 , , PUNCT cana-5899 5 24 telangana telangana PROPN cana-5899 5 25 , , PUNCT cana-5899 5 26 india india PROPN cana-5899 5 27 . . PUNCT cana-5899 6 1 * * PUNCT cana-5899 6 2 corresponding correspond VERB cana-5899 6 3 author author NOUN cana-5899 6 4 : : PUNCT cana-5899 6 5 g. g. PROPN cana-5899 6 6 shailaja shailaja PROPN cana-5899 6 7 article article NOUN cana-5899 6 8 history history NOUN cana-5899 6 9 : : PUNCT cana-5899 6 10 received receive VERB cana-5899 6 11 : : PUNCT cana-5899 6 12 21 21 NUM cana-5899 6 13 - - SYM cana-5899 6 14 10 10 NUM cana-5899 6 15 - - PUNCT cana-5899 6 16 2024 2024 NUM cana-5899 6 17 revised revise VERB cana-5899 6 18 : : PUNCT cana-5899 6 19 04 04 NUM cana-5899 6 20 - - SYM cana-5899 6 21 12 12 NUM cana-5899 6 22 - - PUNCT cana-5899 6 23 2024 2024 NUM cana-5899 6 24 accepted accept VERB cana-5899 6 25 : : PUNCT cana-5899 6 26 28 28 NUM cana-5899 6 27 - - SYM cana-5899 6 28 12 12 NUM cana-5899 6 29 - - PUNCT cana-5899 6 30 2024 2024 NUM cana-5899 6 31 abstract abstract NOUN cana-5899 6 32 : : PUNCT cana-5899 6 33 in in ADP cana-5899 6 34 this this DET cana-5899 6 35 study study NOUN cana-5899 6 36 , , PUNCT cana-5899 6 37 an an DET cana-5899 6 38 extended extended ADJ cana-5899 6 39 sir sir NOUN cana-5899 6 40 model model NOUN cana-5899 6 41 with with ADP cana-5899 6 42 vaccination vaccination NOUN cana-5899 6 43 and and CCONJ cana-5899 6 44 treatment treatment NOUN cana-5899 6 45 is be AUX cana-5899 6 46 considered consider VERB cana-5899 6 47 by by ADP cana-5899 6 48 introducing introduce VERB cana-5899 6 49 a a DET cana-5899 6 50 delay delay NOUN cana-5899 6 51 in in ADP cana-5899 6 52 the the DET cana-5899 6 53 transition transition NOUN cana-5899 6 54 from from ADP cana-5899 6 55 the the DET cana-5899 6 56 recovered recovered ADJ cana-5899 6 57 class class NOUN cana-5899 6 58 back back ADV cana-5899 6 59 to to ADP cana-5899 6 60 the the DET cana-5899 6 61 susceptible susceptible ADJ cana-5899 6 62 class class NOUN cana-5899 6 63 . . PUNCT cana-5899 7 1 this this DET cana-5899 7 2 delay delay NOUN cana-5899 7 3 reflects reflect VERB cana-5899 7 4 the the DET cana-5899 7 5 time time NOUN cana-5899 7 6 it it PRON cana-5899 7 7 takes take VERB cana-5899 7 8 for for SCONJ cana-5899 7 9 individuals individual NOUN cana-5899 7 10 to to PART cana-5899 7 11 lose lose VERB cana-5899 7 12 immunity immunity NOUN cana-5899 7 13 after after ADP cana-5899 7 14 recovery recovery NOUN cana-5899 7 15 or or CCONJ cana-5899 7 16 vaccination vaccination NOUN cana-5899 7 17 . . PUNCT cana-5899 8 1 to to PART cana-5899 8 2 understand understand VERB cana-5899 8 3 the the DET cana-5899 8 4 dynamics dynamic NOUN cana-5899 8 5 of of ADP cana-5899 8 6 the the DET cana-5899 8 7 disease disease NOUN cana-5899 8 8 , , PUNCT cana-5899 8 9 both both CCONJ cana-5899 8 10 local local ADJ cana-5899 8 11 and and CCONJ cana-5899 8 12 global global ADJ cana-5899 8 13 stability stability NOUN cana-5899 8 14 of of ADP cana-5899 8 15 the the DET cana-5899 8 16 equilibrium equilibrium NOUN cana-5899 8 17 points point NOUN cana-5899 8 18 are be AUX cana-5899 8 19 analyzed analyze VERB cana-5899 8 20 . . PUNCT cana-5899 9 1 local local ADJ cana-5899 9 2 stability stability NOUN cana-5899 9 3 conditions condition NOUN cana-5899 9 4 are be AUX cana-5899 9 5 derived derive VERB cana-5899 9 6 using use VERB cana-5899 9 7 the the DET cana-5899 9 8 basic basic ADJ cana-5899 9 9 reproduction reproduction NOUN cana-5899 9 10 number number NOUN cana-5899 9 11 and and CCONJ cana-5899 9 12 characteristic characteristic ADJ cana-5899 9 13 equations equation NOUN cana-5899 9 14 , , PUNCT cana-5899 9 15 while while SCONJ cana-5899 9 16 global global ADJ cana-5899 9 17 stability stability NOUN cana-5899 9 18 is be AUX cana-5899 9 19 established establish VERB cana-5899 9 20 through through ADP cana-5899 9 21 lyapunov lyapunov PROPN cana-5899 9 22 ’s ’s PART cana-5899 9 23 method method NOUN cana-5899 9 24 . . PUNCT cana-5899 10 1 a a DET cana-5899 10 2 numerical numerical ADJ cana-5899 10 3 example example NOUN cana-5899 10 4 is be AUX cana-5899 10 5 included include VERB cana-5899 10 6 to to PART cana-5899 10 7 demonstrate demonstrate VERB cana-5899 10 8 how how SCONJ cana-5899 10 9 different different ADJ cana-5899 10 10 values value NOUN cana-5899 10 11 of of ADP cana-5899 10 12 the the DET cana-5899 10 13 delay delay NOUN cana-5899 10 14 affect affect VERB cana-5899 10 15 the the DET cana-5899 10 16 system system NOUN cana-5899 10 17 . . PUNCT cana-5899 11 1 keywords keyword NOUN cana-5899 11 2 : : PUNCT cana-5899 11 3 characteristic characteristic ADJ cana-5899 11 4 , , PUNCT cana-5899 11 5 equations equation NOUN cana-5899 11 6 , , PUNCT cana-5899 11 7 equilibrium equilibrium NOUN cana-5899 11 8 , , PUNCT cana-5899 11 9 demonstrate demonstrate VERB cana-5899 11 10 1 1 NUM cana-5899 11 11 . . PUNCT cana-5899 12 1 introduction introduction NOUN cana-5899 12 2 infectious infectious ADJ cana-5899 12 3 diseases disease NOUN cana-5899 12 4 have have AUX cana-5899 12 5 always always ADV cana-5899 12 6 been be AUX cana-5899 12 7 a a DET cana-5899 12 8 major major ADJ cana-5899 12 9 concern concern NOUN cana-5899 12 10 for for ADP cana-5899 12 11 human human ADJ cana-5899 12 12 health health NOUN cana-5899 12 13 , , PUNCT cana-5899 12 14 especially especially ADV cana-5899 12 15 because because SCONJ cana-5899 12 16 of of ADP cana-5899 12 17 how how SCONJ cana-5899 12 18 quickly quickly ADV cana-5899 12 19 they they PRON cana-5899 12 20 can can AUX cana-5899 12 21 spread spread VERB cana-5899 12 22 and and CCONJ cana-5899 12 23 reappear reappear VERB cana-5899 12 24 , , PUNCT cana-5899 12 25 often often ADV cana-5899 12 26 when when SCONJ cana-5899 12 27 we we PRON cana-5899 12 28 least least ADV cana-5899 12 29 expect expect VERB cana-5899 12 30 it it PRON cana-5899 12 31 . . PUNCT cana-5899 13 1 over over ADP cana-5899 13 2 the the DET cana-5899 13 3 years year NOUN cana-5899 13 4 , , PUNCT cana-5899 13 5 researchers researcher NOUN cana-5899 13 6 and and CCONJ cana-5899 13 7 scientists scientist NOUN cana-5899 13 8 have have AUX cana-5899 13 9 worked work VERB cana-5899 13 10 hard hard ADV cana-5899 13 11 to to PART cana-5899 13 12 understand understand VERB cana-5899 13 13 how how SCONJ cana-5899 13 14 these these DET cana-5899 13 15 diseases disease NOUN cana-5899 13 16 behave behave VERB cana-5899 13 17 and and CCONJ cana-5899 13 18 how how SCONJ cana-5899 13 19 we we PRON cana-5899 13 20 might might AUX cana-5899 13 21 stop stop VERB cana-5899 13 22 them they PRON cana-5899 13 23 [ [ X cana-5899 13 24 10],[11 10],[11 NUM cana-5899 13 25 ] ] PUNCT cana-5899 13 26 . . PUNCT cana-5899 14 1 one one NUM cana-5899 14 2 of of ADP cana-5899 14 3 the the DET cana-5899 14 4 most most ADV cana-5899 14 5 useful useful ADJ cana-5899 14 6 tools tool NOUN cana-5899 14 7 in in ADP cana-5899 14 8 this this DET cana-5899 14 9 effort effort NOUN cana-5899 14 10 has have AUX cana-5899 14 11 been be AUX cana-5899 14 12 mathematical mathematical ADJ cana-5899 14 13 modelling modelling NOUN cana-5899 14 14 , , PUNCT cana-5899 14 15 which which PRON cana-5899 14 16 helps help VERB cana-5899 14 17 us we PRON cana-5899 14 18 see see VERB cana-5899 14 19 the the DET cana-5899 14 20 bigger big ADJ cana-5899 14 21 picture picture NOUN cana-5899 14 22 by by ADP cana-5899 14 23 using use VERB cana-5899 14 24 equations equation NOUN cana-5899 14 25 to to PART cana-5899 14 26 describe describe VERB cana-5899 14 27 how how SCONJ cana-5899 14 28 diseases disease NOUN cana-5899 14 29 move move VERB cana-5899 14 30 through through ADP cana-5899 14 31 a a DET cana-5899 14 32 population population NOUN cana-5899 15 1 [ [ X cana-5899 15 2 6],[7],[8],[9 6],[7],[8],[9 NOUN cana-5899 15 3 ] ] X cana-5899 15 4 . . PUNCT cana-5899 16 1 most most ADJ cana-5899 16 2 models model NOUN cana-5899 16 3 divide divide VERB cana-5899 16 4 the the DET cana-5899 16 5 population population NOUN cana-5899 16 6 into into ADP cana-5899 16 7 groups group NOUN cana-5899 16 8 like like ADP cana-5899 16 9 those those PRON cana-5899 16 10 who who PRON cana-5899 16 11 are be AUX cana-5899 16 12 susceptible susceptible ADJ cana-5899 16 13 to to ADP cana-5899 16 14 infection infection NOUN cana-5899 16 15 , , PUNCT cana-5899 16 16 those those PRON cana-5899 16 17 who who PRON cana-5899 16 18 are be AUX cana-5899 16 19 currently currently ADV cana-5899 16 20 infected infect VERB cana-5899 16 21 , , PUNCT cana-5899 16 22 and and CCONJ cana-5899 16 23 those those PRON cana-5899 16 24 who who PRON cana-5899 16 25 have have AUX cana-5899 16 26 recovered recover VERB cana-5899 16 27 . . PUNCT cana-5899 17 1 these these DET cana-5899 17 2 simple simple ADJ cana-5899 17 3 models model NOUN cana-5899 17 4 have have AUX cana-5899 17 5 been be AUX cana-5899 17 6 very very ADV cana-5899 17 7 helpful helpful ADJ cana-5899 17 8 in in ADP cana-5899 17 9 studying study VERB cana-5899 17 10 a a DET cana-5899 17 11 range range NOUN cana-5899 17 12 of of ADP cana-5899 17 13 diseases disease NOUN cana-5899 17 14 . . PUNCT cana-5899 18 1 but but CCONJ cana-5899 18 2 real real ADJ cana-5899 18 3 life life NOUN cana-5899 18 4 is be AUX cana-5899 18 5 often often ADV cana-5899 18 6 more more ADV cana-5899 18 7 complicated complicated ADJ cana-5899 18 8 . . PUNCT cana-5899 19 1 one one NUM cana-5899 19 2 important important ADJ cana-5899 19 3 factor factor NOUN cana-5899 19 4 that that PRON cana-5899 19 5 's be AUX cana-5899 19 6 sometimes sometimes ADV cana-5899 19 7 left leave VERB cana-5899 19 8 out out ADP cana-5899 19 9 is be AUX cana-5899 19 10 the the DET cana-5899 19 11 time time NOUN cana-5899 19 12 it it PRON cana-5899 19 13 takes take VERB cana-5899 19 14 for for ADP cana-5899 19 15 someone someone PRON cana-5899 19 16 who who PRON cana-5899 19 17 has have AUX cana-5899 19 18 recovered recover VERB cana-5899 19 19 from from ADP cana-5899 19 20 an an DET cana-5899 19 21 illness illness NOUN cana-5899 19 22 , , PUNCT cana-5899 19 23 especially especially ADV cana-5899 19 24 someone someone PRON cana-5899 19 25 who who PRON cana-5899 19 26 has have AUX cana-5899 19 27 n't not PART cana-5899 19 28 been be AUX cana-5899 19 29 vaccinated vaccinate VERB cana-5899 19 30 , , PUNCT cana-5899 19 31 to to PART cana-5899 19 32 become become VERB cana-5899 19 33 vulnerable vulnerable ADJ cana-5899 19 34 again again ADV cana-5899 19 35 . . PUNCT cana-5899 20 1 in in ADP cana-5899 20 2 many many ADJ cana-5899 20 3 infections infection NOUN cana-5899 20 4 , , PUNCT cana-5899 20 5 recovery recovery NOUN cana-5899 20 6 does do AUX cana-5899 20 7 n't not PART cana-5899 20 8 mean mean VERB cana-5899 20 9 someone someone PRON cana-5899 20 10 is be AUX cana-5899 20 11 safe safe ADJ cana-5899 20 12 forever forever ADV cana-5899 20 13 . . PUNCT cana-5899 21 1 their their PRON cana-5899 21 2 immunity immunity NOUN cana-5899 21 3 might might AUX cana-5899 21 4 only only ADV cana-5899 21 5 last last VERB cana-5899 21 6 a a DET cana-5899 21 7 few few ADJ cana-5899 21 8 months month NOUN cana-5899 21 9 or or CCONJ cana-5899 21 10 years year NOUN cana-5899 21 11 , , PUNCT cana-5899 21 12 and and CCONJ cana-5899 21 13 after after ADP cana-5899 21 14 that that PRON cana-5899 21 15 , , PUNCT cana-5899 21 16 they they PRON cana-5899 21 17 can can AUX cana-5899 21 18 get get VERB cana-5899 21 19 sick sick ADJ cana-5899 21 20 again again ADV cana-5899 21 21 . . PUNCT cana-5899 22 1 this this DET cana-5899 22 2 delay delay NOUN cana-5899 22 3 between between ADP cana-5899 22 4 recovering recover VERB cana-5899 22 5 and and CCONJ cana-5899 22 6 becoming become VERB cana-5899 22 7 susceptible susceptible ADJ cana-5899 22 8 once once ADV cana-5899 22 9 more more ADJ cana-5899 22 10 can can AUX cana-5899 22 11 have have VERB cana-5899 22 12 a a DET cana-5899 22 13 big big ADJ cana-5899 22 14 impact impact NOUN cana-5899 22 15 on on ADP cana-5899 22 16 how how SCONJ cana-5899 22 17 a a DET cana-5899 22 18 disease disease NOUN cana-5899 22 19 spreads spread VERB cana-5899 22 20 , , PUNCT cana-5899 22 21 especially especially ADV cana-5899 22 22 in in ADP cana-5899 22 23 communities community NOUN cana-5899 22 24 where where SCONJ cana-5899 22 25 not not PART cana-5899 22 26 everyone everyone PRON cana-5899 22 27 gets get VERB cana-5899 22 28 vaccinated vaccinate VERB cana-5899 22 29 or or CCONJ cana-5899 22 30 where where SCONJ cana-5899 22 31 immunity immunity NOUN cana-5899 22 32 fades fade VERB cana-5899 22 33 over over ADP cana-5899 22 34 time time NOUN cana-5899 22 35 . . PUNCT cana-5899 23 1 by by ADP cana-5899 23 2 including include VERB cana-5899 23 3 this this DET cana-5899 23 4 delay delay NOUN cana-5899 23 5 in in ADP cana-5899 23 6 our our PRON cana-5899 23 7 models model NOUN cana-5899 23 8 , , PUNCT cana-5899 23 9 we we PRON cana-5899 23 10 get get VERB cana-5899 23 11 a a DET cana-5899 23 12 much much ADV cana-5899 23 13 clearer clear ADJ cana-5899 23 14 understanding understanding NOUN cana-5899 23 15 of of ADP cana-5899 23 16 how how SCONJ cana-5899 23 17 diseases disease NOUN cana-5899 23 18 can can AUX cana-5899 23 19 re re VERB cana-5899 23 20 - - VERB cana-5899 23 21 emerge emerge VERB cana-5899 23 22 , , PUNCT cana-5899 23 23 even even ADV cana-5899 23 24 after after SCONJ cana-5899 23 25 they they PRON cana-5899 23 26 seem seem VERB cana-5899 23 27 to to PART cana-5899 23 28 be be AUX cana-5899 23 29 controlled control VERB cana-5899 23 30 . . PUNCT cana-5899 24 1 it it PRON cana-5899 24 2 also also ADV cana-5899 24 3 helps help VERB cana-5899 24 4 us we PRON cana-5899 24 5 make make VERB cana-5899 24 6 better well ADJ cana-5899 24 7 decisions decision NOUN cana-5899 24 8 about about ADP cana-5899 24 9 when when SCONJ cana-5899 24 10 and and CCONJ cana-5899 24 11 how how SCONJ cana-5899 24 12 to to PART cana-5899 24 13 use use VERB cana-5899 24 14 tools tool NOUN cana-5899 24 15 like like ADP cana-5899 24 16 vaccination vaccination NOUN cana-5899 24 17 and and CCONJ cana-5899 24 18 treatment treatment NOUN cana-5899 24 19 . . PUNCT cana-5899 25 1 for for ADP cana-5899 25 2 example example NOUN cana-5899 25 3 , , PUNCT cana-5899 25 4 if if SCONJ cana-5899 25 5 we we PRON cana-5899 25 6 know know VERB cana-5899 25 7 there there PRON cana-5899 25 8 's be VERB cana-5899 25 9 a a DET cana-5899 25 10 time time NOUN cana-5899 25 11 lag lag NOUN cana-5899 25 12 before before SCONJ cana-5899 25 13 people people NOUN cana-5899 25 14 become become VERB cana-5899 25 15 susceptible susceptible ADJ cana-5899 25 16 again again ADV cana-5899 25 17 , , PUNCT cana-5899 25 18 we we PRON cana-5899 25 19 can can AUX cana-5899 25 20 plan plan VERB cana-5899 25 21 booster booster NOUN cana-5899 25 22 vaccinations vaccination NOUN cana-5899 25 23 or or CCONJ cana-5899 25 24 public public ADJ cana-5899 25 25 health health NOUN cana-5899 25 26 responses response NOUN cana-5899 25 27 more more ADV cana-5899 25 28 effectively effectively ADV cana-5899 25 29 . . PUNCT cana-5899 26 1 communications communication NOUN cana-5899 26 2 on on ADP cana-5899 26 3 applied apply VERB cana-5899 26 4 nonlinear nonlinear ADJ cana-5899 26 5 analysis analysis NOUN cana-5899 26 6 issn issn NOUN cana-5899 26 7 : : PUNCT cana-5899 26 8 1074 1074 NUM cana-5899 26 9 - - PUNCT cana-5899 26 10 133x 133x NUM cana-5899 26 11 vol vol NOUN cana-5899 26 12 31 31 NUM cana-5899 26 13 no no NOUN cana-5899 26 14 . . PUNCT cana-5899 27 1 8s 8s PROPN cana-5899 27 2 ( ( PUNCT cana-5899 27 3 2024 2024 NUM cana-5899 27 4 ) ) PUNCT cana-5899 27 5 1085 1085 NUM cana-5899 27 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 28 1 this this DET cana-5899 28 2 study study NOUN cana-5899 28 3 looks look VERB cana-5899 28 4 at at ADP cana-5899 28 5 what what PRON cana-5899 28 6 happens happen VERB cana-5899 28 7 when when SCONJ cana-5899 28 8 we we PRON cana-5899 28 9 account account VERB cana-5899 28 10 for for ADP cana-5899 28 11 that that DET cana-5899 28 12 delay delay NOUN cana-5899 28 13 , , PUNCT cana-5899 28 14 when when SCONJ cana-5899 28 15 recovered recover VERB cana-5899 28 16 individuals individual NOUN cana-5899 28 17 slowly slowly ADV cana-5899 28 18 lose lose VERB cana-5899 28 19 their their PRON cana-5899 28 20 immunity immunity NOUN cana-5899 28 21 and and CCONJ cana-5899 28 22 rejoin rejoin VERB cana-5899 28 23 the the DET cana-5899 28 24 group group NOUN cana-5899 28 25 of of ADP cana-5899 28 26 people people NOUN cana-5899 28 27 who who PRON cana-5899 28 28 can can AUX cana-5899 28 29 catch catch VERB cana-5899 28 30 the the DET cana-5899 28 31 disease disease NOUN cana-5899 28 32 . . PUNCT cana-5899 29 1 our our PRON cana-5899 29 2 goal goal NOUN cana-5899 29 3 is be AUX cana-5899 29 4 to to PART cana-5899 29 5 use use VERB cana-5899 29 6 this this DET cana-5899 29 7 more more ADV cana-5899 29 8 realistic realistic ADJ cana-5899 29 9 approach approach NOUN cana-5899 29 10 to to PART cana-5899 29 11 better well ADV cana-5899 29 12 understand understand VERB cana-5899 29 13 the the DET cana-5899 29 14 ups up NOUN cana-5899 29 15 and and CCONJ cana-5899 29 16 downs down NOUN cana-5899 29 17 of of ADP cana-5899 29 18 disease disease NOUN cana-5899 29 19 outbreaks outbreak NOUN cana-5899 29 20 , , PUNCT cana-5899 29 21 and and CCONJ cana-5899 29 22 to to PART cana-5899 29 23 figure figure VERB cana-5899 29 24 out out ADP cana-5899 29 25 how how SCONJ cana-5899 29 26 we we PRON cana-5899 29 27 can can AUX cana-5899 29 28 respond respond VERB cana-5899 29 29 in in ADP cana-5899 29 30 smarter smart ADJ cana-5899 29 31 , , PUNCT cana-5899 29 32 more more ADV cana-5899 29 33 timely timely ADJ cana-5899 29 34 ways way NOUN cana-5899 29 35 . . PUNCT cana-5899 30 1 the the DET cana-5899 30 2 sir sir PROPN cana-5899 30 3 model model NOUN cana-5899 30 4 with with ADP cana-5899 30 5 treatment treatment NOUN cana-5899 30 6 and and CCONJ cana-5899 30 7 vaccinations vaccination NOUN cana-5899 30 8 , , PUNCT cana-5899 30 9 which which PRON cana-5899 30 10 was be AUX cana-5899 30 11 introduced introduce VERB cana-5899 30 12 in in ADP cana-5899 30 13 [ [ X cana-5899 30 14 1 1 NUM cana-5899 30 15 ] ] PUNCT cana-5899 30 16 , , PUNCT cana-5899 30 17 is be AUX cana-5899 30 18 taken take VERB cana-5899 30 19 into into ADP cana-5899 30 20 consideration consideration NOUN cana-5899 30 21 . . PUNCT cana-5899 31 1 in in ADP cana-5899 31 2 [ [ X cana-5899 31 3 3 3 NUM cana-5899 31 4 ] ] PUNCT cana-5899 31 5 , , PUNCT cana-5899 31 6 various various ADJ cana-5899 31 7 global global ADJ cana-5899 31 8 stability stability NOUN cana-5899 31 9 conditions condition NOUN cana-5899 31 10 for for ADP cana-5899 31 11 the the DET cana-5899 31 12 considered consider VERB cana-5899 31 13 model model NOUN cana-5899 31 14 are be AUX cana-5899 31 15 examined examine VERB cana-5899 31 16 . . PUNCT cana-5899 32 1 further far ADV cana-5899 32 2 , , PUNCT cana-5899 32 3 the the DET cana-5899 32 4 influence influence NOUN cana-5899 32 5 of of ADP cana-5899 32 6 time time NOUN cana-5899 32 7 - - PUNCT cana-5899 32 8 varying vary VERB cana-5899 32 9 rates rate NOUN cana-5899 32 10 of of ADP cana-5899 32 11 vaccination vaccination NOUN cana-5899 32 12 and and CCONJ cana-5899 32 13 treatment treatment NOUN cana-5899 32 14 is be AUX cana-5899 32 15 investigated investigate VERB cana-5899 32 16 in in ADP cana-5899 32 17 [ [ X cana-5899 32 18 2 2 NUM cana-5899 32 19 ] ] PUNCT cana-5899 32 20 . . PUNCT cana-5899 33 1 the the DET cana-5899 33 2 effects effect NOUN cana-5899 33 3 of of ADP cana-5899 33 4 the the DET cana-5899 33 5 vaccine vaccine NOUN cana-5899 33 6 delay delay NOUN cana-5899 33 7 are be AUX cana-5899 33 8 highlighted highlight VERB cana-5899 33 9 in in ADP cana-5899 33 10 [ [ X cana-5899 33 11 4 4 NUM cana-5899 33 12 ] ] PUNCT cana-5899 33 13 . . PUNCT cana-5899 34 1 in in ADP cana-5899 34 2 this this DET cana-5899 34 3 article article NOUN cana-5899 34 4 , , PUNCT cana-5899 34 5 the the DET cana-5899 34 6 delay delay NOUN cana-5899 34 7 in in ADP cana-5899 34 8 recovering recover VERB cana-5899 34 9 individuals individual NOUN cana-5899 34 10 becoming become VERB cana-5899 34 11 susceptible susceptible ADJ cana-5899 34 12 again again ADV cana-5899 34 13 is be AUX cana-5899 34 14 introduced introduce VERB cana-5899 34 15 , , PUNCT cana-5899 34 16 and and CCONJ cana-5899 34 17 its its PRON cana-5899 34 18 dynamics dynamic NOUN cana-5899 34 19 on on ADP cana-5899 34 20 the the DET cana-5899 34 21 disease disease NOUN cana-5899 34 22 spread spread NOUN cana-5899 34 23 are be AUX cana-5899 34 24 studied study VERB cana-5899 34 25 . . PUNCT cana-5899 35 1 this this DET cana-5899 35 2 document document NOUN cana-5899 35 3 has have VERB cana-5899 35 4 the the DET cana-5899 35 5 following follow VERB cana-5899 35 6 structure structure NOUN cana-5899 35 7 . . PUNCT cana-5899 36 1 the the DET cana-5899 36 2 model model NOUN cana-5899 36 3 is be AUX cana-5899 36 4 presented present VERB cana-5899 36 5 , , PUNCT cana-5899 36 6 and and CCONJ cana-5899 36 7 basic basic ADJ cana-5899 36 8 properties property NOUN cana-5899 36 9 are be AUX cana-5899 36 10 examined examine VERB cana-5899 36 11 in in ADP cana-5899 36 12 section section NOUN cana-5899 36 13 2 2 NUM cana-5899 36 14 . . PUNCT cana-5899 36 15 section section NOUN cana-5899 36 16 3 3 NUM cana-5899 36 17 looks look VERB cana-5899 36 18 at at ADP cana-5899 36 19 the the DET cana-5899 36 20 model model NOUN cana-5899 36 21 's 's PART cana-5899 36 22 stability stability NOUN cana-5899 36 23 both both CCONJ cana-5899 36 24 locally locally ADV cana-5899 36 25 and and CCONJ cana-5899 36 26 globally globally ADV cana-5899 36 27 . . PUNCT cana-5899 37 1 in in ADP cana-5899 37 2 section section NOUN cana-5899 37 3 4 4 NUM cana-5899 37 4 , , PUNCT cana-5899 37 5 the the DET cana-5899 37 6 numerical numerical ADJ cana-5899 37 7 examples example NOUN cana-5899 37 8 are be AUX cana-5899 37 9 discussed discuss VERB cana-5899 37 10 . . PUNCT cana-5899 38 1 in in ADP cana-5899 38 2 section section NOUN cana-5899 38 3 5 5 NUM cana-5899 38 4 , , PUNCT cana-5899 38 5 the the DET cana-5899 38 6 conclusion conclusion NOUN cana-5899 38 7 is be AUX cana-5899 38 8 provided provide VERB cana-5899 38 9 . . PUNCT cana-5899 39 1 2 2 X cana-5899 39 2 . . X cana-5899 39 3 model model NOUN cana-5899 39 4 description description NOUN cana-5899 39 5 a a DET cana-5899 39 6 three three NUM cana-5899 39 7 - - PUNCT cana-5899 39 8 compartmental compartmental ADJ cana-5899 39 9 , , PUNCT cana-5899 39 10 susceptible susceptible ADJ cana-5899 39 11 - - PUNCT cana-5899 39 12 infected infect VERB cana-5899 39 13 - - PUNCT cana-5899 39 14 recovered recover VERB cana-5899 39 15 ( ( PUNCT cana-5899 39 16 sir sir NOUN cana-5899 39 17 ) ) PUNCT cana-5899 39 18 model model NOUN cana-5899 39 19 is be AUX cana-5899 39 20 considered consider VERB cana-5899 39 21 . . PUNCT cana-5899 40 1 ′𝑒′ ′𝑒′ PROPN cana-5899 40 2 denotes denote VERB cana-5899 40 3 susceptible susceptible ADJ cana-5899 40 4 population population NOUN cana-5899 40 5 , , PUNCT cana-5899 40 6 ′𝑣′ ′𝑣′ PRON cana-5899 40 7 denotes denote VERB cana-5899 40 8 the the DET cana-5899 40 9 infected infected ADJ cana-5899 40 10 population population NOUN cana-5899 40 11 and and CCONJ cana-5899 40 12 ′𝑀′ ′𝑀′ NOUN cana-5899 40 13 denotes denote VERB cana-5899 40 14 the the DET cana-5899 40 15 recovered recovered ADJ cana-5899 40 16 population population NOUN cana-5899 40 17 . . PUNCT cana-5899 41 1 the the DET cana-5899 41 2 dynamics dynamic NOUN cana-5899 41 3 of of ADP cana-5899 41 4 the the DET cana-5899 41 5 model model NOUN cana-5899 41 6 are be AUX cana-5899 41 7 described describe VERB cana-5899 41 8 by by ADP cana-5899 41 9 the the DET cana-5899 41 10 following follow VERB cana-5899 41 11 differential differential ADJ cana-5899 41 12 equations equation NOUN cana-5899 41 13 𝑒′ 𝑒′ ADP cana-5899 41 14 = = PUNCT cana-5899 41 15 π‘Ž π‘Ž DET cana-5899 41 16 βˆ’ βˆ’ NOUN cana-5899 41 17 𝑏𝑓(𝑒 𝑏𝑓(𝑒 ADV cana-5899 41 18 , , PUNCT cana-5899 41 19 𝑣 𝑣 X cana-5899 41 20 ) ) PUNCT cana-5899 41 21 βˆ’ βˆ’ PROPN cana-5899 41 22 𝑑𝑒 𝑑𝑒 INTJ cana-5899 42 1 βˆ’ βˆ’ NOUN cana-5899 42 2 𝑐𝑉(𝑒 𝑐𝑉(𝑒 NUM cana-5899 42 3 ) ) PUNCT cana-5899 43 1 + + CCONJ cana-5899 43 2 𝛼𝑀(𝑑 𝛼𝑀(𝑑 ADP cana-5899 43 3 βˆ’ βˆ’ PRON cana-5899 43 4 πœ‡ πœ‡ NOUN cana-5899 43 5 ) ) PUNCT cana-5899 43 6 𝑣′ 𝑣′ NUM cana-5899 43 7 = = PUNCT cana-5899 44 1 𝑏1𝑓(𝑒(t 𝑏1𝑓(𝑒(t PROPN cana-5899 44 2 βˆ’ βˆ’ PROPN cana-5899 44 3 Ο„ Ο„ PROPN cana-5899 44 4 ) ) PUNCT cana-5899 44 5 , , PUNCT cana-5899 44 6 𝑣 𝑣 X cana-5899 44 7 ) ) PUNCT cana-5899 44 8 βˆ’ βˆ’ PROPN cana-5899 45 1 π‘Ÿπ‘ƒ(𝑣 π‘Ÿπ‘ƒ(𝑣 NUM cana-5899 45 2 ) ) PUNCT cana-5899 46 1 βˆ’ βˆ’ PROPN cana-5899 47 1 𝑑1𝑣 𝑑1𝑣 PROPN cana-5899 47 2 𝑀′ 𝑀′ X cana-5899 47 3 = = PUNCT cana-5899 47 4 π‘Ÿπ‘ƒ(𝑣(𝑑 π‘Ÿπ‘ƒ(𝑣(𝑑 ADJ cana-5899 47 5 βˆ’ βˆ’ PROPN cana-5899 47 6 Ξ΄ Ξ΄ PROPN cana-5899 47 7 ) ) PUNCT cana-5899 47 8 ) ) PUNCT cana-5899 48 1 βˆ’ βˆ’ PROPN cana-5899 48 2 𝛼𝑀 𝛼𝑀 NOUN cana-5899 48 3 ( ( PUNCT cana-5899 48 4 1 1 NUM cana-5899 48 5 ) ) PUNCT cana-5899 48 6 the the DET cana-5899 48 7 susceptible susceptible ADJ cana-5899 48 8 population population NOUN cana-5899 48 9 growth growth NOUN cana-5899 48 10 rate rate NOUN cana-5899 48 11 is be AUX cana-5899 48 12 represented represent VERB cana-5899 48 13 by by ADP cana-5899 48 14 𝒂 𝒂 NOUN cana-5899 48 15 , , PUNCT cana-5899 48 16 and and CCONJ cana-5899 48 17 the the DET cana-5899 48 18 non non ADJ cana-5899 48 19 - - ADJ cana-5899 48 20 linear linear ADJ cana-5899 48 21 infection infection NOUN cana-5899 48 22 function function NOUN cana-5899 48 23 that that PRON cana-5899 48 24 illustrates illustrate VERB cana-5899 48 25 how how SCONJ cana-5899 48 26 susceptibles susceptible NOUN cana-5899 48 27 become become VERB cana-5899 48 28 infected infected ADJ cana-5899 48 29 is be AUX cana-5899 48 30 indicated indicate VERB cana-5899 48 31 by by ADP cana-5899 48 32 𝒇(𝒖 𝒇(𝒖 PROPN cana-5899 48 33 , , PUNCT cana-5899 48 34 𝒗 𝒗 PROPN cana-5899 48 35 ) ) PUNCT cana-5899 48 36 . . PUNCT cana-5899 49 1 𝒃 𝒃 PROPN cana-5899 49 2 > > X cana-5899 49 3 𝟎 𝟎 PROPN cana-5899 49 4 indicates indicate VERB cana-5899 49 5 that that SCONJ cana-5899 49 6 susceptible susceptible ADJ cana-5899 49 7 and and CCONJ cana-5899 49 8 infected infected ADJ cana-5899 49 9 people people NOUN cana-5899 49 10 interact interact VERB cana-5899 49 11 , , PUNCT cana-5899 49 12 and and CCONJ cana-5899 49 13 𝒅 𝒅 PRON cana-5899 49 14 is be AUX cana-5899 49 15 the the DET cana-5899 49 16 rate rate NOUN cana-5899 49 17 at at ADP cana-5899 49 18 which which PRON cana-5899 49 19 susceptible susceptible ADJ cana-5899 49 20 people people NOUN cana-5899 49 21 who who PRON cana-5899 49 22 are be AUX cana-5899 49 23 inherently inherently ADV cana-5899 49 24 immune immune ADJ cana-5899 49 25 to to ADP cana-5899 49 26 the the DET cana-5899 49 27 illness illness NOUN cana-5899 49 28 and and CCONJ cana-5899 49 29 do do AUX cana-5899 49 30 not not PART cana-5899 49 31 become become VERB cana-5899 49 32 infected infect VERB cana-5899 49 33 are be AUX cana-5899 49 34 removed remove VERB cana-5899 49 35 from from ADP cana-5899 49 36 the the DET cana-5899 49 37 system system NOUN cana-5899 49 38 . . PUNCT cana-5899 50 1 the the DET cana-5899 50 2 vaccine vaccine NOUN cana-5899 50 3 function function VERB cana-5899 50 4 𝑽(𝒖 𝑽(𝒖 NUM cana-5899 50 5 ) ) PUNCT cana-5899 50 6 is be AUX cana-5899 50 7 the the DET cana-5899 50 8 vaccination vaccination NOUN cana-5899 50 9 function function NOUN cana-5899 50 10 which which PRON cana-5899 50 11 depends depend VERB cana-5899 50 12 on on ADP cana-5899 50 13 the the DET cana-5899 50 14 susceptible susceptible ADJ cana-5899 50 15 population population NOUN cana-5899 50 16 , , PUNCT cana-5899 50 17 the the DET cana-5899 50 18 successful successful ADJ cana-5899 50 19 vaccination vaccination NOUN cana-5899 50 20 rate rate NOUN cana-5899 50 21 is be AUX cana-5899 50 22 𝒄 𝒄 NOUN cana-5899 50 23 > > X cana-5899 50 24 𝟎 𝟎 PROPN cana-5899 50 25 , , PUNCT cana-5899 50 26 and and CCONJ cana-5899 50 27 the the DET cana-5899 50 28 rate rate NOUN cana-5899 50 29 at at ADP cana-5899 50 30 which which PRON cana-5899 50 31 susceptible susceptible ADJ cana-5899 50 32 people people NOUN cana-5899 50 33 become become VERB cana-5899 50 34 infected infected ADJ cana-5899 50 35 is be AUX cana-5899 50 36 𝑏1 𝑏1 ADJ cana-5899 50 37 ( ( PUNCT cana-5899 50 38 < < X cana-5899 50 39 𝑏 𝑏 NOUN cana-5899 50 40 ) ) PUNCT cana-5899 50 41 . . PUNCT cana-5899 51 1 the the DET cana-5899 51 2 rate rate NOUN cana-5899 51 3 at at ADP cana-5899 51 4 which which PRON cana-5899 51 5 an an DET cana-5899 51 6 unvaccinated unvaccinated ADJ cana-5899 51 7 or or CCONJ cana-5899 51 8 less less ADJ cana-5899 51 9 immunity immunity NOUN cana-5899 51 10 person person NOUN cana-5899 51 11 who who PRON cana-5899 51 12 has have AUX cana-5899 51 13 recovered recover VERB cana-5899 51 14 may may AUX cana-5899 51 15 be be AUX cana-5899 51 16 exposed expose VERB cana-5899 51 17 to to ADP cana-5899 51 18 infection infection NOUN cana-5899 51 19 again again ADV cana-5899 51 20 and and CCONJ cana-5899 51 21 become become VERB cana-5899 51 22 susceptible susceptible ADJ cana-5899 51 23 again again ADV cana-5899 51 24 is be AUX cana-5899 51 25 indicated indicate VERB cana-5899 51 26 by by ADP cana-5899 51 27 the the DET cana-5899 51 28 parameter parameter NOUN cana-5899 51 29 𝜢 𝜢 AUX cana-5899 51 30 > > X cana-5899 51 31 𝟎. 𝟎. PUNCT cana-5899 51 32 𝑷(𝒗 𝑷(𝒗 CCONJ cana-5899 51 33 ) ) PUNCT cana-5899 51 34 is be AUX cana-5899 51 35 the the DET cana-5899 51 36 recovery recovery NOUN cana-5899 51 37 ( ( PUNCT cana-5899 51 38 by by ADP cana-5899 51 39 treatment treatment NOUN cana-5899 51 40 ) ) PUNCT cana-5899 51 41 function function NOUN cana-5899 51 42 of of ADP cana-5899 51 43 the the DET cana-5899 51 44 infected infected NOUN cana-5899 51 45 , , PUNCT cana-5899 51 46 𝒓 𝒓 PROPN cana-5899 51 47 > > X cana-5899 51 48 𝟎 𝟎 PROPN cana-5899 51 49 indicates indicate VERB cana-5899 51 50 the the DET cana-5899 51 51 rate rate NOUN cana-5899 51 52 of of ADP cana-5899 51 53 recovery recovery NOUN cana-5899 51 54 , , PUNCT cana-5899 51 55 and and CCONJ cana-5899 51 56 π’…πŸ π’…πŸ NOUN cana-5899 51 57 is be AUX cana-5899 51 58 the the DET cana-5899 51 59 infection infection NOUN cana-5899 51 60 - - PUNCT cana-5899 51 61 related relate VERB cana-5899 51 62 mortality mortality NOUN cana-5899 51 63 rate rate NOUN cana-5899 51 64 . . PUNCT cana-5899 52 1 time time NOUN cana-5899 52 2 delay delay NOUN cana-5899 52 3 𝛕 𝛕 X cana-5899 52 4 > > X cana-5899 52 5 0 0 NUM cana-5899 52 6 indicates indicate VERB cana-5899 52 7 the the DET cana-5899 52 8 time time NOUN cana-5899 52 9 it it PRON cana-5899 52 10 takes take VERB cana-5899 52 11 for for ADP cana-5899 52 12 susceptible susceptible ADJ cana-5899 52 13 𝑒 𝑒 NOUN cana-5899 52 14 to to PART cana-5899 52 15 become become AUX cana-5899 52 16 infected infected ADJ cana-5899 52 17 when when SCONJ cana-5899 52 18 it it PRON cana-5899 52 19 comes come VERB cana-5899 52 20 in in ADP cana-5899 52 21 contact contact NOUN cana-5899 52 22 with with ADP cana-5899 52 23 𝑣. 𝑣. NOUN cana-5899 52 24 time time NOUN cana-5899 52 25 delay delay VERB cana-5899 52 26 𝛅 𝛅 PROPN cana-5899 52 27 > > X cana-5899 52 28 0 0 NUM cana-5899 52 29 is be AUX cana-5899 52 30 the the DET cana-5899 52 31 time time NOUN cana-5899 52 32 taken take VERB cana-5899 52 33 by by ADP cana-5899 52 34 a a DET cana-5899 52 35 person person NOUN cana-5899 52 36 to to PART cana-5899 52 37 get get AUX cana-5899 52 38 recovered recover VERB cana-5899 52 39 by by ADP cana-5899 52 40 the the DET cana-5899 52 41 treatment treatment NOUN cana-5899 52 42 . . PUNCT cana-5899 53 1 we we PRON cana-5899 53 2 introduce introduce VERB cana-5899 53 3 a a DET cana-5899 53 4 time time NOUN cana-5899 53 5 delay delay NOUN cana-5899 53 6 𝝁 𝝁 PRON cana-5899 53 7 > > PUNCT cana-5899 53 8 𝟎. 𝟎. X cana-5899 53 9 it it PRON cana-5899 53 10 is be AUX cana-5899 53 11 the the DET cana-5899 53 12 time time NOUN cana-5899 53 13 taken take VERB cana-5899 53 14 by by ADP cana-5899 53 15 the the DET cana-5899 53 16 recovered recovered ADJ cana-5899 53 17 individual individual NOUN cana-5899 53 18 to to PART cana-5899 53 19 become become VERB cana-5899 53 20 susceptible susceptible ADJ cana-5899 53 21 again again ADV cana-5899 53 22 due due ADP cana-5899 53 23 to to ADP cana-5899 53 24 a a DET cana-5899 53 25 lack lack NOUN cana-5899 53 26 of of ADP cana-5899 53 27 immunity immunity NOUN cana-5899 53 28 or or CCONJ cana-5899 53 29 because because SCONJ cana-5899 53 30 of of ADP cana-5899 53 31 no no DET cana-5899 53 32 vaccination vaccination NOUN cana-5899 53 33 . . PUNCT cana-5899 54 1 the the DET cana-5899 54 2 following follow VERB cana-5899 54 3 are be AUX cana-5899 54 4 the the DET cana-5899 54 5 assumptions assumption NOUN cana-5899 54 6 for for ADP cana-5899 54 7 the the DET cana-5899 54 8 infection infection NOUN cana-5899 54 9 function function NOUN cana-5899 54 10 , , PUNCT cana-5899 54 11 the the DET cana-5899 54 12 recovery recovery NOUN cana-5899 54 13 function function NOUN cana-5899 54 14 and and CCONJ cana-5899 54 15 the the DET cana-5899 54 16 vaccination vaccination NOUN cana-5899 54 17 function function NOUN cana-5899 54 18 [ [ X cana-5899 54 19 1 1 NUM cana-5899 54 20 ] ] PUNCT cana-5899 54 21 . . PUNCT cana-5899 55 1 ( ( PUNCT cana-5899 55 2 i i NOUN cana-5899 55 3 ) ) PUNCT cana-5899 55 4 . . PUNCT cana-5899 56 1 𝑓(𝑒 𝑓(𝑒 PROPN cana-5899 56 2 , , PUNCT cana-5899 56 3 𝑣 𝑣 NOUN cana-5899 56 4 ) ) PUNCT cana-5899 56 5 β‰₯ β‰₯ NOUN cana-5899 56 6 0 0 NUM cana-5899 56 7 , , PUNCT cana-5899 56 8 βˆ€π‘’ βˆ€π‘’ DET cana-5899 56 9 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5899 56 10 βˆ€π‘£ βˆ€π‘£ PROPN cana-5899 56 11 communications communication NOUN cana-5899 56 12 on on ADP cana-5899 56 13 applied apply VERB cana-5899 56 14 nonlinear nonlinear ADJ cana-5899 56 15 analysis analysis NOUN cana-5899 56 16 issn issn NOUN cana-5899 56 17 : : PUNCT cana-5899 56 18 1074 1074 NUM cana-5899 56 19 - - PUNCT cana-5899 56 20 133x 133x NUM cana-5899 56 21 vol vol NOUN cana-5899 56 22 31 31 NUM cana-5899 56 23 no no NOUN cana-5899 56 24 . . PUNCT cana-5899 57 1 8s 8s PROPN cana-5899 57 2 ( ( PUNCT cana-5899 57 3 2024 2024 NUM cana-5899 57 4 ) ) PUNCT cana-5899 57 5 1086 1086 NUM cana-5899 57 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 57 7 ( ( PUNCT cana-5899 57 8 ii ii NOUN cana-5899 57 9 ) ) PUNCT cana-5899 57 10 . . PUNCT cana-5899 58 1 𝑓(𝑒 𝑓(𝑒 PROPN cana-5899 58 2 , , PUNCT cana-5899 58 3 0 0 NUM cana-5899 58 4 ) ) PUNCT cana-5899 58 5 β‰₯ β‰₯ NOUN cana-5899 58 6 0 0 NUM cana-5899 58 7 , , PUNCT cana-5899 58 8 βˆ€π‘’ βˆ€π‘’ DET cana-5899 58 9 ( ( PUNCT cana-5899 58 10 iii iii NOUN cana-5899 58 11 ) ) PUNCT cana-5899 58 12 . . PUNCT cana-5899 59 1 𝑓(0 𝑓(0 NOUN cana-5899 59 2 , , PUNCT cana-5899 59 3 𝑣 𝑣 NOUN cana-5899 59 4 ) ) PUNCT cana-5899 59 5 β‰₯ β‰₯ NOUN cana-5899 59 6 0 0 NUM cana-5899 59 7 , , PUNCT cana-5899 59 8 βˆ€π‘£ βˆ€π‘£ PROPN cana-5899 59 9 ( ( PUNCT cana-5899 59 10 iv iv NOUN cana-5899 59 11 ) ) PUNCT cana-5899 59 12 . . PUNCT cana-5899 60 1 𝑓(0,0 𝑓(0,0 NOUN cana-5899 60 2 ) ) PUNCT cana-5899 60 3 = = SYM cana-5899 60 4 0 0 NUM cana-5899 60 5 , , PUNCT cana-5899 60 6 ( ( PUNCT cana-5899 60 7 v v NOUN cana-5899 60 8 ) ) PUNCT cana-5899 60 9 . . PUNCT cana-5899 61 1 𝑃(𝑣 𝑃(𝑣 VERB cana-5899 61 2 ) ) PUNCT cana-5899 61 3 β‰₯ β‰₯ NOUN cana-5899 61 4 0 0 NUM cana-5899 61 5 , , PUNCT cana-5899 61 6 βˆ€π‘£ βˆ€π‘£ PROPN cana-5899 61 7 ( ( PUNCT cana-5899 61 8 vi vi NOUN cana-5899 61 9 ) ) PUNCT cana-5899 61 10 . . PUNCT cana-5899 62 1 𝑃(0 𝑃(0 NUM cana-5899 62 2 ) ) PUNCT cana-5899 62 3 = = SYM cana-5899 62 4 0 0 NUM cana-5899 62 5 , , PUNCT cana-5899 62 6 ( ( PUNCT cana-5899 62 7 vii vii PROPN cana-5899 62 8 ) ) PUNCT cana-5899 62 9 . . PUNCT cana-5899 63 1 𝑉(𝑒 𝑉(𝑒 NUM cana-5899 63 2 ) ) PUNCT cana-5899 63 3 β‰₯ β‰₯ NOUN cana-5899 63 4 0 0 NUM cana-5899 63 5 , , PUNCT cana-5899 63 6 βˆ€π‘’ βˆ€π‘’ PRON cana-5899 63 7 ( ( PUNCT cana-5899 63 8 viii viii NOUN cana-5899 63 9 ) ) PUNCT cana-5899 63 10 𝑉(0 𝑉(0 NUM cana-5899 63 11 ) ) PUNCT cana-5899 64 1 = = SYM cana-5899 64 2 0 0 X cana-5899 64 3 . . PUNCT cana-5899 64 4 ( ( PUNCT cana-5899 64 5 ii ii NOUN cana-5899 64 6 ) ) PUNCT cana-5899 64 7 . . PUNCT cana-5899 65 1 𝑓(0,0 𝑓(0,0 NOUN cana-5899 65 2 ) ) PUNCT cana-5899 65 3 = = SYM cana-5899 65 4 0 0 X cana-5899 65 5 . . PUNCT cana-5899 66 1 by by ADP cana-5899 66 2 following follow VERB cana-5899 66 3 the the DET cana-5899 66 4 procedure procedure NOUN cana-5899 66 5 as as ADP cana-5899 66 6 in in ADP cana-5899 66 7 [ [ X cana-5899 66 8 4 4 NUM cana-5899 66 9 ] ] PUNCT cana-5899 66 10 , , PUNCT cana-5899 66 11 we we PRON cana-5899 66 12 can can AUX cana-5899 66 13 prove prove VERB cana-5899 66 14 the the DET cana-5899 66 15 result result NOUN cana-5899 66 16 on on ADP cana-5899 66 17 positivity positivity NOUN cana-5899 66 18 and and CCONJ cana-5899 66 19 boundedness boundedness NOUN cana-5899 66 20 of of ADP cana-5899 66 21 the the DET cana-5899 66 22 solutions solution NOUN cana-5899 66 23 of of ADP cana-5899 66 24 the the DET cana-5899 66 25 model model NOUN cana-5899 66 26 ( ( PUNCT cana-5899 66 27 1 1 NUM cana-5899 66 28 ) ) PUNCT cana-5899 66 29 and and CCONJ cana-5899 66 30 state state VERB cana-5899 66 31 the the DET cana-5899 66 32 result result NOUN cana-5899 66 33 as as SCONJ cana-5899 66 34 theorem theorem ADJ cana-5899 66 35 2.1 2.1 NUM cana-5899 66 36 : : PUNCT cana-5899 66 37 all all DET cana-5899 66 38 solutions solution NOUN cana-5899 66 39 of of ADP cana-5899 66 40 the the DET cana-5899 66 41 system system NOUN cana-5899 66 42 ( ( PUNCT cana-5899 66 43 1 1 NUM cana-5899 66 44 ) ) PUNCT cana-5899 66 45 with with ADP cana-5899 66 46 nonnegative nonnegative ADJ cana-5899 66 47 initial initial ADJ cana-5899 66 48 conditions condition NOUN cana-5899 66 49 are be AUX cana-5899 66 50 nonnegative nonnegative ADJ cana-5899 66 51 and and CCONJ cana-5899 66 52 bounded bound VERB cana-5899 66 53 [ [ X cana-5899 66 54 4 4 NUM cana-5899 66 55 ] ] PUNCT cana-5899 66 56 . . PUNCT cana-5899 67 1 in in ADP cana-5899 67 2 the the DET cana-5899 67 3 next next ADJ cana-5899 67 4 section section NOUN cana-5899 67 5 , , PUNCT cana-5899 67 6 we we PRON cana-5899 67 7 will will AUX cana-5899 67 8 examine examine VERB cana-5899 67 9 the the DET cana-5899 67 10 local local ADJ cana-5899 67 11 and and CCONJ cana-5899 67 12 global global ADJ cana-5899 67 13 stability stability NOUN cana-5899 67 14 of of ADP cana-5899 67 15 the the DET cana-5899 67 16 model model NOUN cana-5899 67 17 ( ( PUNCT cana-5899 67 18 1 1 NUM cana-5899 67 19 ) ) PUNCT cana-5899 67 20 at at ADP cana-5899 67 21 equilibria equilibrium NOUN cana-5899 67 22 3 3 X cana-5899 67 23 . . X cana-5899 67 24 equilibria equilibrium NOUN cana-5899 67 25 and and CCONJ cana-5899 67 26 stability stability NOUN cana-5899 67 27 3.1 3.1 NUM cana-5899 67 28 equilibrium equilibrium NOUN cana-5899 67 29 in in ADP cana-5899 67 30 studying study VERB cana-5899 67 31 how how SCONJ cana-5899 67 32 diseases disease NOUN cana-5899 67 33 spread spread VERB cana-5899 67 34 and and CCONJ cana-5899 67 35 persist persist VERB cana-5899 67 36 within within ADP cana-5899 67 37 a a DET cana-5899 67 38 population population NOUN cana-5899 67 39 through through ADP cana-5899 67 40 mathematical mathematical ADJ cana-5899 67 41 models model NOUN cana-5899 67 42 like like ADP cana-5899 67 43 model model NOUN cana-5899 67 44 ( ( PUNCT cana-5899 67 45 1 1 NUM cana-5899 67 46 ) ) PUNCT cana-5899 67 47 , , PUNCT cana-5899 67 48 we we PRON cana-5899 67 49 typically typically ADV cana-5899 67 50 identify identify VERB cana-5899 67 51 two two NUM cana-5899 67 52 possible possible ADJ cana-5899 67 53 long long ADJ cana-5899 67 54 - - PUNCT cana-5899 67 55 term term NOUN cana-5899 67 56 scenarios scenario NOUN cana-5899 67 57 , , PUNCT cana-5899 67 58 known know VERB cana-5899 67 59 as as ADP cana-5899 67 60 equilibrium equilibrium NOUN cana-5899 67 61 points point NOUN cana-5899 67 62 . . PUNCT cana-5899 68 1 these these PRON cana-5899 68 2 describe describe VERB cana-5899 68 3 the the DET cana-5899 68 4 steady steady ADJ cana-5899 68 5 states state NOUN cana-5899 68 6 a a DET cana-5899 68 7 population population NOUN cana-5899 68 8 may may AUX cana-5899 68 9 reach reach VERB cana-5899 68 10 over over ADP cana-5899 68 11 time time NOUN cana-5899 68 12 in in ADP cana-5899 68 13 response response NOUN cana-5899 68 14 to to ADP cana-5899 68 15 the the DET cana-5899 68 16 infection infection NOUN cana-5899 68 17 dynamics dynamic NOUN cana-5899 68 18 . . PUNCT cana-5899 69 1 for for ADP cana-5899 69 2 such such ADJ cana-5899 69 3 models model NOUN cana-5899 69 4 , , PUNCT cana-5899 69 5 the the DET cana-5899 69 6 two two NUM cana-5899 69 7 main main ADJ cana-5899 69 8 types type NOUN cana-5899 69 9 of of ADP cana-5899 69 10 equilibrium equilibrium NOUN cana-5899 69 11 are be AUX cana-5899 69 12 the the DET cana-5899 69 13 disease disease NOUN cana-5899 69 14 - - PUNCT cana-5899 69 15 free free ADJ cana-5899 69 16 equilibrium equilibrium NOUN cana-5899 69 17 and and CCONJ cana-5899 69 18 the the DET cana-5899 69 19 endemic endemic ADJ cana-5899 69 20 equilibrium equilibrium NOUN cana-5899 69 21 . . PUNCT cana-5899 70 1 the the DET cana-5899 70 2 disease disease NOUN cana-5899 70 3 - - PUNCT cana-5899 70 4 free free ADJ cana-5899 70 5 equilibrium equilibrium NOUN cana-5899 70 6 represents represent VERB cana-5899 70 7 a a DET cana-5899 70 8 state state NOUN cana-5899 70 9 in in ADP cana-5899 70 10 which which PRON cana-5899 70 11 the the DET cana-5899 70 12 infection infection NOUN cana-5899 70 13 has have AUX cana-5899 70 14 been be AUX cana-5899 70 15 eliminated eliminate VERB cana-5899 70 16 from from ADP cana-5899 70 17 the the DET cana-5899 70 18 population population NOUN cana-5899 70 19 . . PUNCT cana-5899 71 1 in in ADP cana-5899 71 2 this this DET cana-5899 71 3 case case NOUN cana-5899 71 4 , , PUNCT cana-5899 71 5 there there PRON cana-5899 71 6 are be VERB cana-5899 71 7 no no DET cana-5899 71 8 infected infected ADJ cana-5899 71 9 or or CCONJ cana-5899 71 10 recovered recover VERB cana-5899 71 11 individuals individual NOUN cana-5899 71 12 . . PUNCT cana-5899 72 1 it it PRON cana-5899 72 2 is be AUX cana-5899 72 3 expressed express VERB cana-5899 72 4 as as ADP cana-5899 72 5 ( ( PUNCT cana-5899 72 6 π‘’βˆ— π‘’βˆ— NOUN cana-5899 72 7 , , PUNCT cana-5899 72 8 0,0 0,0 NOUN cana-5899 72 9 ) ) PUNCT cana-5899 72 10 , , PUNCT cana-5899 72 11 where where SCONJ cana-5899 72 12 π‘’βˆ— π‘’βˆ— VERB cana-5899 72 13 > > SYM cana-5899 72 14 0 0 NUM cana-5899 72 15 denotes denote NOUN cana-5899 72 16 a a DET cana-5899 72 17 healthy healthy ADJ cana-5899 72 18 population population NOUN cana-5899 72 19 of of ADP cana-5899 72 20 susceptible susceptible ADJ cana-5899 72 21 individuals individual NOUN cana-5899 72 22 with with ADP cana-5899 72 23 no no DET cana-5899 72 24 presence presence NOUN cana-5899 72 25 of of ADP cana-5899 72 26 disease disease NOUN cana-5899 72 27 . . PUNCT cana-5899 73 1 in in ADP cana-5899 73 2 contrast contrast NOUN cana-5899 73 3 , , PUNCT cana-5899 73 4 the the DET cana-5899 73 5 endemic endemic ADJ cana-5899 73 6 equilibrium equilibrium NOUN cana-5899 73 7 reflects reflect VERB cana-5899 73 8 a a DET cana-5899 73 9 scenario scenario NOUN cana-5899 73 10 in in ADP cana-5899 73 11 which which PRON cana-5899 73 12 the the DET cana-5899 73 13 disease disease NOUN cana-5899 73 14 continues continue VERB cana-5899 73 15 to to PART cana-5899 73 16 exist exist VERB cana-5899 73 17 within within ADP cana-5899 73 18 the the DET cana-5899 73 19 population population NOUN cana-5899 73 20 over over ADP cana-5899 73 21 the the DET cana-5899 73 22 long long ADJ cana-5899 73 23 term term NOUN cana-5899 73 24 . . PUNCT cana-5899 74 1 here here ADV cana-5899 74 2 , , PUNCT cana-5899 74 3 all all DET cana-5899 74 4 population population NOUN cana-5899 74 5 groups group NOUN cana-5899 74 6 β€” β€” PUNCT cana-5899 74 7 susceptible susceptible ADJ cana-5899 74 8 , , PUNCT cana-5899 74 9 infected infected ADJ cana-5899 74 10 , , PUNCT cana-5899 74 11 and and CCONJ cana-5899 74 12 recovered recover VERB cana-5899 74 13 or or CCONJ cana-5899 74 14 immune immune ADJ cana-5899 74 15 β€” β€” PUNCT cana-5899 74 16 maintain maintain VERB cana-5899 74 17 positive positive ADJ cana-5899 74 18 values value NOUN cana-5899 74 19 . . PUNCT cana-5899 75 1 this this DET cana-5899 75 2 equilibrium equilibrium NOUN cana-5899 75 3 is be AUX cana-5899 75 4 denoted denote VERB cana-5899 75 5 by by ADP cana-5899 75 6 ( ( PUNCT cana-5899 75 7 π‘’βˆ— π‘’βˆ— PROPN cana-5899 75 8 , , PUNCT cana-5899 75 9 π‘£βˆ— π‘£βˆ— ADJ cana-5899 75 10 , , PUNCT cana-5899 75 11 π‘€βˆ— π‘€βˆ— PROPN cana-5899 75 12 ) ) PUNCT cana-5899 75 13 , , PUNCT cana-5899 75 14 where where SCONJ cana-5899 75 15 each each DET cana-5899 75 16 component component NOUN cana-5899 75 17 is be AUX cana-5899 75 18 greater great ADJ cana-5899 75 19 than than ADP cana-5899 75 20 zero zero NUM cana-5899 75 21 . . PUNCT cana-5899 76 1 it it PRON cana-5899 76 2 implies imply VERB cana-5899 76 3 that that SCONJ cana-5899 76 4 while while SCONJ cana-5899 76 5 some some DET cana-5899 76 6 communications communication NOUN cana-5899 76 7 on on ADP cana-5899 76 8 applied apply VERB cana-5899 76 9 nonlinear nonlinear ADJ cana-5899 76 10 analysis analysis NOUN cana-5899 76 11 issn issn NOUN cana-5899 76 12 : : PUNCT cana-5899 76 13 1074 1074 NUM cana-5899 76 14 - - PUNCT cana-5899 76 15 133x 133x NUM cana-5899 76 16 vol vol NOUN cana-5899 76 17 31 31 NUM cana-5899 76 18 no no NOUN cana-5899 76 19 . . PUNCT cana-5899 77 1 8s 8s PROPN cana-5899 77 2 ( ( PUNCT cana-5899 77 3 2024 2024 NUM cana-5899 77 4 ) ) PUNCT cana-5899 77 5 1087 1087 NUM cana-5899 77 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 77 7 individuals individual NOUN cana-5899 77 8 remain remain VERB cana-5899 77 9 uninfected uninfected ADJ cana-5899 77 10 , , PUNCT cana-5899 77 11 a a DET cana-5899 77 12 consistent consistent ADJ cana-5899 77 13 number number NOUN cana-5899 77 14 of of ADP cana-5899 77 15 people people NOUN cana-5899 77 16 are be AUX cana-5899 77 17 carrying carry VERB cana-5899 77 18 or or CCONJ cana-5899 77 19 recovering recover VERB cana-5899 77 20 from from ADP cana-5899 77 21 the the DET cana-5899 77 22 disease disease NOUN cana-5899 77 23 . . PUNCT cana-5899 78 1 understanding understand VERB cana-5899 78 2 the the DET cana-5899 78 3 stability stability NOUN cana-5899 78 4 of of ADP cana-5899 78 5 the the DET cana-5899 78 6 model model NOUN cana-5899 78 7 at at ADP cana-5899 78 8 equilibrium equilibrium NOUN cana-5899 78 9 points point NOUN cana-5899 78 10 is be AUX cana-5899 78 11 crucial crucial ADJ cana-5899 78 12 to to ADP cana-5899 78 13 determining determine VERB cana-5899 78 14 whether whether SCONJ cana-5899 78 15 an an DET cana-5899 78 16 infection infection NOUN cana-5899 78 17 will will AUX cana-5899 78 18 disappear disappear VERB cana-5899 78 19 or or CCONJ cana-5899 78 20 persist persist VERB cana-5899 78 21 as as ADP cana-5899 78 22 a a DET cana-5899 78 23 permanent permanent ADJ cana-5899 78 24 feature feature NOUN cana-5899 78 25 of of ADP cana-5899 78 26 the the DET cana-5899 78 27 population population NOUN cana-5899 78 28 's 's PART cana-5899 78 29 health health NOUN cana-5899 78 30 environment environment NOUN cana-5899 78 31 . . PUNCT cana-5899 79 1 they they PRON cana-5899 79 2 are be AUX cana-5899 79 3 also also ADV cana-5899 79 4 essential essential ADJ cana-5899 79 5 in in ADP cana-5899 79 6 assessing assess VERB cana-5899 79 7 public public ADJ cana-5899 79 8 health health NOUN cana-5899 79 9 initiatives initiative NOUN cana-5899 79 10 and and CCONJ cana-5899 79 11 establishing establish VERB cana-5899 79 12 acceptable acceptable ADJ cana-5899 79 13 levels level NOUN cana-5899 79 14 of of ADP cana-5899 79 15 disease disease NOUN cana-5899 79 16 control control NOUN cana-5899 79 17 and and CCONJ cana-5899 79 18 elimination elimination NOUN cana-5899 79 19 . . PUNCT cana-5899 80 1 3.2 3.2 NUM cana-5899 80 2 basic basic ADJ cana-5899 80 3 reproduction reproduction NOUN cana-5899 80 4 number number NOUN cana-5899 80 5 the the DET cana-5899 80 6 number number NOUN cana-5899 80 7 of of ADP cana-5899 80 8 secondary secondary ADJ cana-5899 80 9 infections infection NOUN cana-5899 80 10 caused cause VERB cana-5899 80 11 by by ADP cana-5899 80 12 an an DET cana-5899 80 13 infectious infectious ADJ cana-5899 80 14 individual individual NOUN cana-5899 80 15 moving move VERB cana-5899 80 16 through through ADP cana-5899 80 17 a a DET cana-5899 80 18 vulnerable vulnerable ADJ cana-5899 80 19 community community NOUN cana-5899 80 20 is be AUX cana-5899 80 21 determined determine VERB cana-5899 80 22 by by ADP cana-5899 80 23 the the DET cana-5899 80 24 basic basic ADJ cana-5899 80 25 reproduction reproduction NOUN cana-5899 80 26 number number NOUN cana-5899 80 27 ( ( PUNCT cana-5899 80 28 brn brn PROPN cana-5899 80 29 ) ) PUNCT cana-5899 80 30 , , PUNCT cana-5899 80 31 a a DET cana-5899 80 32 threshold threshold NOUN cana-5899 80 33 in in ADP cana-5899 80 34 epidemiology epidemiology NOUN cana-5899 80 35 . . PUNCT cana-5899 81 1 it it PRON cana-5899 81 2 is be AUX cana-5899 81 3 represented represent VERB cana-5899 81 4 by by ADP cana-5899 81 5 𝑅0 𝑅0 PROPN cana-5899 81 6 . . PUNCT cana-5899 82 1 for for ADP cana-5899 82 2 model model NOUN cana-5899 82 3 ( ( PUNCT cana-5899 82 4 1 1 X cana-5899 82 5 ) ) PUNCT cana-5899 82 6 using use VERB cana-5899 82 7 the the DET cana-5899 82 8 next next ADJ cana-5899 82 9 - - PUNCT cana-5899 82 10 generation generation NOUN cana-5899 82 11 matrix matrix NOUN cana-5899 82 12 method method NOUN cana-5899 82 13 , , PUNCT cana-5899 82 14 we we PRON cana-5899 82 15 get get VERB cana-5899 82 16 the the DET cana-5899 82 17 brn brn NOUN cana-5899 82 18 as as ADP cana-5899 82 19 𝑅0 𝑅0 NOUN cana-5899 82 20 = = SYM cana-5899 82 21 𝑏1𝑓𝑣(π‘’βˆ—,0 𝑏1𝑓𝑣(π‘’βˆ—,0 PROPN cana-5899 82 22 ) ) PUNCT cana-5899 82 23 π‘Ÿπ‘ƒβ€²(0)+𝑑1 π‘Ÿπ‘ƒβ€²(0)+𝑑1 NUM cana-5899 82 24 . . PUNCT cana-5899 83 1 ( ( PUNCT cana-5899 83 2 one one PRON cana-5899 83 3 can can AUX cana-5899 83 4 refer refer VERB cana-5899 83 5 to to ADP cana-5899 83 6 [ [ X cana-5899 83 7 4 4 X cana-5899 83 8 ] ] PUNCT cana-5899 83 9 for for ADP cana-5899 83 10 the the DET cana-5899 83 11 derivation derivation NOUN cana-5899 83 12 of of ADP cana-5899 83 13 𝑅0 𝑅0 NOUN cana-5899 83 14 ) ) PUNCT cana-5899 83 15 3.3 3.3 NUM cana-5899 83 16 local local ADJ cana-5899 83 17 stability stability NOUN cana-5899 83 18 studying study VERB cana-5899 83 19 the the DET cana-5899 83 20 local local ADJ cana-5899 83 21 stability stability NOUN cana-5899 83 22 of of ADP cana-5899 83 23 the the DET cana-5899 83 24 equilibrium equilibrium NOUN cana-5899 83 25 points point NOUN cana-5899 83 26 in in ADP cana-5899 83 27 the the DET cana-5899 83 28 model model NOUN cana-5899 83 29 is be AUX cana-5899 83 30 crucial crucial ADJ cana-5899 83 31 to to ADP cana-5899 83 32 comprehending comprehend VERB cana-5899 83 33 how how SCONJ cana-5899 83 34 a a DET cana-5899 83 35 disease disease NOUN cana-5899 83 36 develops develop VERB cana-5899 83 37 or or CCONJ cana-5899 83 38 disappears disappear VERB cana-5899 83 39 over over ADP cana-5899 83 40 time time NOUN cana-5899 83 41 [ [ X cana-5899 83 42 14 14 NUM cana-5899 83 43 ] ] PUNCT cana-5899 83 44 , , PUNCT cana-5899 83 45 [ [ X cana-5899 83 46 17 17 NUM cana-5899 83 47 ] ] PUNCT cana-5899 83 48 . . PUNCT cana-5899 84 1 by by ADP cana-5899 84 2 examining examine VERB cana-5899 84 3 local local ADJ cana-5899 84 4 stability stability NOUN cana-5899 84 5 around around ADP cana-5899 84 6 equilibrium equilibrium NOUN cana-5899 84 7 points point NOUN cana-5899 84 8 , , PUNCT cana-5899 84 9 we we PRON cana-5899 84 10 may may AUX cana-5899 84 11 observe observe VERB cana-5899 84 12 how how SCONJ cana-5899 84 13 the the DET cana-5899 84 14 system system NOUN cana-5899 84 15 responds respond VERB cana-5899 84 16 to to ADP cana-5899 84 17 minor minor ADJ cana-5899 84 18 changes change NOUN cana-5899 84 19 . . PUNCT cana-5899 85 1 a a DET cana-5899 85 2 point point NOUN cana-5899 85 3 is be AUX cana-5899 85 4 considered consider VERB cana-5899 85 5 stable stable ADJ cana-5899 85 6 if if SCONJ cana-5899 85 7 the the DET cana-5899 85 8 system system NOUN cana-5899 85 9 regains regain VERB cana-5899 85 10 equilibrium equilibrium NOUN cana-5899 85 11 following follow VERB cana-5899 85 12 a a DET cana-5899 85 13 minor minor ADJ cana-5899 85 14 disruption disruption NOUN cana-5899 85 15 ; ; PUNCT cana-5899 85 16 if if SCONJ cana-5899 85 17 not not PART cana-5899 85 18 , , PUNCT cana-5899 85 19 it it PRON cana-5899 85 20 may may AUX cana-5899 85 21 lead lead VERB cana-5899 85 22 to to PART cana-5899 85 23 further far ADV cana-5899 85 24 spread spread VERB cana-5899 85 25 or or CCONJ cana-5899 85 26 decline decline NOUN cana-5899 85 27 of of ADP cana-5899 85 28 the the DET cana-5899 85 29 infection infection NOUN cana-5899 85 30 . . PUNCT cana-5899 86 1 first first ADV cana-5899 86 2 , , PUNCT cana-5899 86 3 we we PRON cana-5899 86 4 state state VERB cana-5899 86 5 the the DET cana-5899 86 6 conditions condition NOUN cana-5899 86 7 for for ADP cana-5899 86 8 local local ADJ cana-5899 86 9 stability stability NOUN cana-5899 86 10 at at ADP cana-5899 86 11 the the DET cana-5899 86 12 disease disease NOUN cana-5899 86 13 - - PUNCT cana-5899 86 14 free free ADJ cana-5899 86 15 equilibrium equilibrium NOUN cana-5899 86 16 . . PUNCT cana-5899 87 1 theorem theorem VERB cana-5899 87 2 3.1 3.1 NUM cana-5899 87 3 : : PUNCT cana-5899 87 4 system system NOUN cana-5899 87 5 ( ( PUNCT cana-5899 87 6 1 1 X cana-5899 87 7 ) ) PUNCT cana-5899 87 8 will will AUX cana-5899 87 9 be be AUX cana-5899 87 10 unstable unstable ADJ cana-5899 87 11 at at ADP cana-5899 87 12 disease disease NOUN cana-5899 87 13 - - PUNCT cana-5899 87 14 free free ADJ cana-5899 87 15 equilibrium equilibrium NOUN cana-5899 87 16 for for ADP cana-5899 87 17 𝑅0 𝑅0 PROPN cana-5899 87 18 > > SYM cana-5899 87 19 1 1 NUM cana-5899 87 20 and and CCONJ cana-5899 87 21 will will AUX cana-5899 87 22 be be AUX cana-5899 87 23 locally locally ADV cana-5899 87 24 stable stable ADJ cana-5899 87 25 if if SCONJ cana-5899 87 26 ( ( PUNCT cana-5899 87 27 i i NOUN cana-5899 87 28 ) ) PUNCT cana-5899 87 29 𝑅0 𝑅0 VERB cana-5899 87 30 < < X cana-5899 87 31 1 1 NUM cana-5899 87 32 , , PUNCT cana-5899 87 33 ( ( PUNCT cana-5899 87 34 ii)π›Όπ‘’βˆ’πœ†πœ‡ ii)π›Όπ‘’βˆ’πœ†πœ‡ ADP cana-5899 87 35 < < X cana-5899 87 36 𝑏𝑓𝑒(π‘’βˆ— 𝑏𝑓𝑒(π‘’βˆ— NOUN cana-5899 87 37 , , PUNCT cana-5899 87 38 0 0 NUM cana-5899 87 39 ) ) PUNCT cana-5899 88 1 + + CCONJ cana-5899 88 2 𝑑 𝑑 PROPN cana-5899 88 3 + + ADJ cana-5899 88 4 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— NUM cana-5899 88 5 ) ) PUNCT cana-5899 88 6 , , PUNCT cana-5899 88 7 ( ( PUNCT cana-5899 88 8 iii)π‘Ÿπ‘’βˆ’πœ†π›Ώ iii)π‘Ÿπ‘’βˆ’πœ†π›Ώ NOUN cana-5899 88 9 πœ•π‘ƒ πœ•π‘ƒ SYM cana-5899 88 10 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 88 11 ( ( PUNCT cana-5899 88 12 0 0 NUM cana-5899 88 13 ) ) PUNCT cana-5899 88 14 < < X cana-5899 88 15 𝛼 𝛼 X cana-5899 88 16 proof proof NOUN cana-5899 88 17 : : PUNCT cana-5899 88 18 the the DET cana-5899 88 19 characteristic characteristic ADJ cana-5899 88 20 equation equation NOUN cana-5899 88 21 for for ADP cana-5899 88 22 the the DET cana-5899 88 23 system system NOUN cana-5899 88 24 ( ( PUNCT cana-5899 88 25 1 1 X cana-5899 88 26 ) ) PUNCT cana-5899 88 27 at at ADP cana-5899 88 28 ( ( PUNCT cana-5899 88 29 π‘’βˆ— π‘’βˆ— NOUN cana-5899 88 30 , , PUNCT cana-5899 88 31 0,0 0,0 NOUN cana-5899 88 32 ) ) PUNCT cana-5899 88 33 is be AUX cana-5899 88 34 given give VERB cana-5899 88 35 by by ADP cana-5899 88 36 [ [ PUNCT cana-5899 88 37 πœ† πœ† DET cana-5899 88 38 βˆ’ βˆ’ PROPN cana-5899 88 39 𝐴1 𝐴1 PROPN cana-5899 88 40 𝑄1 𝑄1 VERB cana-5899 88 41 0 0 NUM cana-5899 88 42 0 0 NUM cana-5899 89 1 πœ† πœ† DET cana-5899 89 2 βˆ’ βˆ’ PROPN cana-5899 89 3 𝐡1 𝐡1 NOUN cana-5899 89 4 0 0 NUM cana-5899 89 5 0 0 NUM cana-5899 89 6 0 0 NUM cana-5899 90 1 πœ† πœ† DET cana-5899 90 2 βˆ’ βˆ’ PROPN cana-5899 90 3 𝐢1 𝐢1 NOUN cana-5899 90 4 ] ] PUNCT cana-5899 91 1 = = SYM cana-5899 91 2 0 0 NUM cana-5899 91 3 𝐴1 𝐴1 PROPN cana-5899 91 4 = = SYM cana-5899 92 1 βˆ’π‘π‘“π‘’(π‘’βˆ— βˆ’π‘π‘“π‘’(π‘’βˆ— PROPN cana-5899 92 2 , , PUNCT cana-5899 92 3 0 0 NUM cana-5899 92 4 ) ) PUNCT cana-5899 92 5 βˆ’ βˆ’ PROPN cana-5899 92 6 𝑑 𝑑 PROPN cana-5899 92 7 βˆ’ βˆ’ PROPN cana-5899 92 8 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— PROPN cana-5899 92 9 ) ) PUNCT cana-5899 93 1 + + NUM cana-5899 93 2 π›Όπ‘’βˆ’πœ†πœ‡ π›Όπ‘’βˆ’πœ†πœ‡ X cana-5899 93 3 𝑄1 𝑄1 NOUN cana-5899 93 4 = = SYM cana-5899 93 5 βˆ’π‘π‘“π‘£(π‘’βˆ— βˆ’π‘π‘“π‘£(π‘’βˆ— PROPN cana-5899 93 6 , , PUNCT cana-5899 93 7 0 0 NUM cana-5899 93 8 ) ) PUNCT cana-5899 93 9 𝐡1 𝐡1 NOUN cana-5899 93 10 = = SYM cana-5899 93 11 𝑏1𝑓𝑣(π‘’βˆ— 𝑏1𝑓𝑣(π‘’βˆ— PROPN cana-5899 93 12 , , PUNCT cana-5899 93 13 0 0 NUM cana-5899 93 14 ) ) PUNCT cana-5899 93 15 βˆ’ βˆ’ PROPN cana-5899 93 16 π‘Ÿ π‘Ÿ NOUN cana-5899 93 17 𝑃′(0 𝑃′(0 NOUN cana-5899 93 18 ) ) PUNCT cana-5899 93 19 βˆ’ βˆ’ PROPN cana-5899 93 20 𝑑1 𝑑1 NOUN cana-5899 93 21 + + CCONJ cana-5899 93 22 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ NOUN cana-5899 93 23 ( ( PUNCT cana-5899 93 24 π‘’βˆ— π‘’βˆ— NOUN cana-5899 93 25 , , PUNCT cana-5899 93 26 0 0 NUM cana-5899 93 27 ) ) PUNCT cana-5899 93 28 𝐢1 𝐢1 NOUN cana-5899 93 29 = = PUNCT cana-5899 93 30 βˆ’π›Ό βˆ’π›Ό PROPN cana-5899 93 31 + + CCONJ cana-5899 93 32 π‘Ÿπ‘’βˆ’πœ†π›Ώ π‘Ÿπ‘’βˆ’πœ†π›Ώ PROPN cana-5899 93 33 πœ•π‘ƒ πœ•π‘ƒ NUM cana-5899 93 34 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 93 35 ( ( PUNCT cana-5899 93 36 0 0 NUM cana-5899 93 37 ) ) PUNCT cana-5899 93 38 . . PUNCT cana-5899 94 1 clearly clearly ADV cana-5899 94 2 𝐴1 𝐴1 PROPN cana-5899 94 3 , , PUNCT cana-5899 94 4 𝐡1 𝐡1 NOUN cana-5899 94 5 and and CCONJ cana-5899 94 6 𝐢1are 𝐢1are VERB cana-5899 94 7 the the DET cana-5899 94 8 eigenvalues eigenvalue NOUN cana-5899 94 9 . . PUNCT cana-5899 95 1 communications communication NOUN cana-5899 95 2 on on ADP cana-5899 95 3 applied apply VERB cana-5899 95 4 nonlinear nonlinear ADJ cana-5899 95 5 analysis analysis NOUN cana-5899 95 6 issn issn NOUN cana-5899 95 7 : : PUNCT cana-5899 95 8 1074 1074 NUM cana-5899 95 9 - - PUNCT cana-5899 95 10 133x 133x NUM cana-5899 95 11 vol vol NOUN cana-5899 95 12 31 31 NUM cana-5899 95 13 no no NOUN cana-5899 95 14 . . PUNCT cana-5899 96 1 8s 8s PROPN cana-5899 96 2 ( ( PUNCT cana-5899 96 3 2024 2024 NUM cana-5899 96 4 ) ) PUNCT cana-5899 96 5 1088 1088 NUM cana-5899 96 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 96 7 a a DET cana-5899 96 8 system system NOUN cana-5899 96 9 of of ADP cana-5899 96 10 the the DET cana-5899 96 11 form form NOUN cana-5899 96 12 ( ( PUNCT cana-5899 96 13 1 1 X cana-5899 96 14 ) ) PUNCT cana-5899 96 15 is be AUX cana-5899 96 16 unstable unstable ADJ cana-5899 96 17 if if SCONJ cana-5899 96 18 any any PRON cana-5899 96 19 of of ADP cana-5899 96 20 its its PRON cana-5899 96 21 eigenvalues eigenvalue NOUN cana-5899 96 22 have have VERB cana-5899 96 23 a a DET cana-5899 96 24 positive positive ADJ cana-5899 96 25 real real ADJ cana-5899 96 26 part part NOUN cana-5899 96 27 , , PUNCT cana-5899 96 28 and and CCONJ cana-5899 96 29 it it PRON cana-5899 96 30 is be AUX cana-5899 96 31 stable stable ADJ cana-5899 96 32 if if SCONJ cana-5899 96 33 all all DET cana-5899 96 34 the the DET cana-5899 96 35 eigenvalues eigenvalue NOUN cana-5899 96 36 are be AUX cana-5899 96 37 negative negative ADJ cana-5899 96 38 . . PUNCT cana-5899 97 1 if if SCONJ cana-5899 97 2 𝐡1 𝐡1 PROPN cana-5899 97 3 > > X cana-5899 97 4 0 0 NUM cana-5899 97 5 , , PUNCT cana-5899 97 6 then then ADV cana-5899 97 7 clearly clearly ADV cana-5899 97 8 𝑅0 𝑅0 VERB cana-5899 97 9 > > X cana-5899 97 10 1.thus 1.thus NUM cana-5899 97 11 , , PUNCT cana-5899 97 12 if if SCONJ cana-5899 97 13 𝑅0 𝑅0 PROPN cana-5899 97 14 > > X cana-5899 97 15 1 1 NUM cana-5899 97 16 then then ADV cana-5899 97 17 the the DET cana-5899 97 18 system system NOUN cana-5899 97 19 is be AUX cana-5899 97 20 unstable unstable ADJ cana-5899 97 21 . . PUNCT cana-5899 98 1 for for SCONJ cana-5899 98 2 the the DET cana-5899 98 3 system system NOUN cana-5899 98 4 to to PART cana-5899 98 5 be be AUX cana-5899 98 6 stable stable ADJ cana-5899 98 7 𝐴1 𝐴1 PROPN cana-5899 98 8 < < X cana-5899 98 9 0 0 PROPN cana-5899 98 10 , , PUNCT cana-5899 98 11 𝐡1 𝐡1 NOUN cana-5899 98 12 < < X cana-5899 98 13 0 0 NUM cana-5899 98 14 , , PUNCT cana-5899 98 15 and and CCONJ cana-5899 98 16 𝐢1 𝐢1 PRON cana-5899 98 17 < < X cana-5899 98 18 0 0 NUM cana-5899 98 19 . . PUNCT cana-5899 99 1 𝐴1 𝐴1 PROPN cana-5899 99 2 < < X cana-5899 99 3 0 0 PUNCT cana-5899 99 4 ⟹ ⟹ NUM cana-5899 99 5 π›Όπ‘’βˆ’πœ†πœ‡ π›Όπ‘’βˆ’πœ†πœ‡ NOUN cana-5899 99 6 < < X cana-5899 99 7 𝑏𝑓𝑒(π‘’βˆ— 𝑏𝑓𝑒(π‘’βˆ— NUM cana-5899 99 8 , , PUNCT cana-5899 99 9 0 0 NUM cana-5899 99 10 ) ) PUNCT cana-5899 100 1 + + CCONJ cana-5899 100 2 𝑑 𝑑 PROPN cana-5899 100 3 + + ADJ cana-5899 100 4 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— NUM cana-5899 100 5 ) ) PUNCT cana-5899 100 6 , , PUNCT cana-5899 100 7 𝐡1 𝐡1 NOUN cana-5899 100 8 < < X cana-5899 100 9 0 0 NUM cana-5899 100 10 ⟹ ⟹ NUM cana-5899 100 11 𝑅0 𝑅0 NOUN cana-5899 100 12 < < X cana-5899 100 13 1 1 NUM cana-5899 100 14 , , PUNCT cana-5899 100 15 𝐢1 𝐢1 X cana-5899 100 16 < < X cana-5899 100 17 0 0 NUM cana-5899 100 18 ⟹ ⟹ NUM cana-5899 100 19 π‘Ÿπ‘’βˆ’πœ†π›Ώ π‘Ÿπ‘’βˆ’πœ†π›Ώ NUM cana-5899 100 20 πœ•π‘ƒ πœ•π‘ƒ SYM cana-5899 100 21 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 100 22 ( ( PUNCT cana-5899 100 23 0 0 NUM cana-5899 100 24 ) ) PUNCT cana-5899 100 25 < < X cana-5899 100 26 𝛼. 𝛼. NOUN cana-5899 100 27 thus thus ADV cana-5899 100 28 proved prove VERB cana-5899 100 29 . . PUNCT cana-5899 101 1 next next ADV cana-5899 101 2 , , PUNCT cana-5899 101 3 we we PRON cana-5899 101 4 state state VERB cana-5899 101 5 the the DET cana-5899 101 6 conditions condition NOUN cana-5899 101 7 for for ADP cana-5899 101 8 local local ADJ cana-5899 101 9 stability stability NOUN cana-5899 101 10 at at ADP cana-5899 101 11 the the DET cana-5899 101 12 endemic endemic ADJ cana-5899 101 13 equilibrium equilibrium NOUN cana-5899 101 14 . . PUNCT cana-5899 102 1 theorem theorem VERB cana-5899 102 2 3.2 3.2 NUM cana-5899 102 3 : : PUNCT cana-5899 102 4 system system NOUN cana-5899 102 5 ( ( PUNCT cana-5899 102 6 1 1 X cana-5899 102 7 ) ) PUNCT cana-5899 102 8 is be AUX cana-5899 102 9 locally locally ADV cana-5899 102 10 stable stable ADJ cana-5899 102 11 at at ADP cana-5899 102 12 the the DET cana-5899 102 13 endemic endemic ADJ cana-5899 102 14 equilibrium equilibrium NOUN cana-5899 102 15 point point NOUN cana-5899 102 16 ( ( PUNCT cana-5899 102 17 π‘’βˆ— π‘’βˆ— NOUN cana-5899 102 18 , , PUNCT cana-5899 102 19 π‘£βˆ— π‘£βˆ— ADJ cana-5899 102 20 , , PUNCT cana-5899 102 21 π‘€βˆ— π‘€βˆ— PROPN cana-5899 102 22 ) ) PUNCT cana-5899 102 23 , , PUNCT cana-5899 102 24 if if SCONJ cana-5899 102 25 ( ( PUNCT cana-5899 102 26 i i NOUN cana-5899 102 27 ) ) PUNCT cana-5899 102 28 π›Όπ‘’βˆ’πœ†πœ‡ π›Όπ‘’βˆ’πœ†πœ‡ NOUN cana-5899 102 29 < < X cana-5899 102 30 𝑏𝑓𝑒(π‘’βˆ— 𝑏𝑓𝑒(π‘’βˆ— PROPN cana-5899 102 31 , , PUNCT cana-5899 102 32 π‘£βˆ— π‘£βˆ— ADJ cana-5899 102 33 ) ) PUNCT cana-5899 103 1 + + CCONJ cana-5899 103 2 𝑑 𝑑 PROPN cana-5899 103 3 + + ADJ cana-5899 103 4 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— NUM cana-5899 103 5 ) ) PUNCT cana-5899 103 6 , , PUNCT cana-5899 103 7 ( ( PUNCT cana-5899 103 8 ii ii NOUN cana-5899 103 9 ) ) PUNCT cana-5899 103 10 𝑏1𝑓𝑣(π‘’βˆ— 𝑏1𝑓𝑣(π‘’βˆ— PROPN cana-5899 103 11 , , PUNCT cana-5899 103 12 π‘£βˆ— π‘£βˆ— ADJ cana-5899 103 13 ) ) PUNCT cana-5899 104 1 + + CCONJ cana-5899 104 2 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ NOUN cana-5899 104 3 ( ( PUNCT cana-5899 104 4 π‘’βˆ— π‘’βˆ— NOUN cana-5899 104 5 , , PUNCT cana-5899 104 6 π‘£βˆ— π‘£βˆ— ADJ cana-5899 104 7 ) ) PUNCT cana-5899 105 1 < < X cana-5899 105 2 π‘Ÿ π‘Ÿ X cana-5899 105 3 𝑃′(π‘£βˆ— 𝑃′(π‘£βˆ— NOUN cana-5899 105 4 ) ) PUNCT cana-5899 105 5 + + NUM cana-5899 105 6 𝑑1 𝑑1 NOUN cana-5899 105 7 , , PUNCT cana-5899 105 8 ( ( PUNCT cana-5899 105 9 iii iii X cana-5899 105 10 ) ) PUNCT cana-5899 105 11 π‘Ÿπ‘’βˆ’πœ†π›Ώ π‘Ÿπ‘’βˆ’πœ†π›Ώ NOUN cana-5899 105 12 πœ•π‘ƒ πœ•π‘ƒ SYM cana-5899 105 13 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 105 14 ( ( PUNCT cana-5899 105 15 π‘£βˆ— π‘£βˆ— ADJ cana-5899 105 16 ) ) PUNCT cana-5899 105 17 < < X cana-5899 105 18 𝛼. 𝛼. NOUN cana-5899 105 19 proof proof NOUN cana-5899 105 20 : : PUNCT cana-5899 105 21 the the DET cana-5899 105 22 characteristic characteristic ADJ cana-5899 105 23 equation equation NOUN cana-5899 105 24 for for ADP cana-5899 105 25 the the DET cana-5899 105 26 system system NOUN cana-5899 105 27 ( ( PUNCT cana-5899 105 28 1 1 X cana-5899 105 29 ) ) PUNCT cana-5899 105 30 at at ADP cana-5899 105 31 ( ( PUNCT cana-5899 105 32 π‘’βˆ— π‘’βˆ— NOUN cana-5899 105 33 , , PUNCT cana-5899 105 34 π‘£βˆ— π‘£βˆ— ADJ cana-5899 105 35 , , PUNCT cana-5899 105 36 π‘€βˆ— π‘€βˆ— PROPN cana-5899 105 37 ) ) PUNCT cana-5899 105 38 is be AUX cana-5899 105 39 given give VERB cana-5899 105 40 by by ADP cana-5899 105 41 [ [ PUNCT cana-5899 105 42 πœ† πœ† PRON cana-5899 105 43 βˆ’ βˆ’ PROPN cana-5899 105 44 𝐴2 𝐴2 PROPN cana-5899 105 45 𝑄1 𝑄1 NOUN cana-5899 105 46 0 0 NUM cana-5899 105 47 0 0 NUM cana-5899 106 1 πœ† πœ† DET cana-5899 106 2 βˆ’ βˆ’ NOUN cana-5899 106 3 𝐡2 𝐡2 NOUN cana-5899 106 4 0 0 NUM cana-5899 106 5 0 0 NUM cana-5899 106 6 0 0 NUM cana-5899 107 1 πœ† πœ† DET cana-5899 107 2 βˆ’ βˆ’ PROPN cana-5899 107 3 𝐢2 𝐢2 INTJ cana-5899 107 4 ] ] PUNCT cana-5899 107 5 = = SYM cana-5899 107 6 0 0 NUM cana-5899 107 7 𝐴2 𝐴2 PROPN cana-5899 107 8 = = SYM cana-5899 107 9 βˆ’π‘π‘“π‘’(π‘’βˆ— βˆ’π‘π‘“π‘’(π‘’βˆ— PROPN cana-5899 107 10 , , PUNCT cana-5899 107 11 π‘£βˆ— π‘£βˆ— ADJ cana-5899 107 12 ) ) PUNCT cana-5899 107 13 βˆ’ βˆ’ PROPN cana-5899 108 1 𝑑 𝑑 PROPN cana-5899 108 2 βˆ’ βˆ’ PROPN cana-5899 108 3 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— PROPN cana-5899 108 4 ) ) PUNCT cana-5899 109 1 + + CCONJ cana-5899 109 2 π›Όπ‘’βˆ’πœ†πœ‡ π›Όπ‘’βˆ’πœ†πœ‡ X cana-5899 109 3 𝑄2 𝑄2 X cana-5899 109 4 = = SYM cana-5899 109 5 βˆ’π‘π‘“π‘£(π‘’βˆ— βˆ’π‘π‘“π‘£(π‘’βˆ— PROPN cana-5899 109 6 , , PUNCT cana-5899 109 7 π‘£βˆ— π‘£βˆ— NUM cana-5899 109 8 ) ) PUNCT cana-5899 109 9 𝐡2 𝐡2 NOUN cana-5899 109 10 = = SYM cana-5899 109 11 𝑏1𝑓𝑣(π‘’βˆ— 𝑏1𝑓𝑣(π‘’βˆ— PROPN cana-5899 109 12 , , PUNCT cana-5899 109 13 π‘£βˆ— π‘£βˆ— ADJ cana-5899 109 14 ) ) PUNCT cana-5899 109 15 βˆ’ βˆ’ ADP cana-5899 109 16 π‘Ÿ π‘Ÿ SYM cana-5899 109 17 𝑃′(π‘£βˆ— 𝑃′(π‘£βˆ— NOUN cana-5899 109 18 ) ) PUNCT cana-5899 109 19 βˆ’ βˆ’ NOUN cana-5899 109 20 𝑑1 𝑑1 NOUN cana-5899 109 21 + + CCONJ cana-5899 109 22 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ NOUN cana-5899 109 23 ( ( PUNCT cana-5899 109 24 π‘’βˆ— π‘’βˆ— NOUN cana-5899 109 25 , , PUNCT cana-5899 109 26 π‘£βˆ— π‘£βˆ— ADJ cana-5899 109 27 ) ) PUNCT cana-5899 109 28 𝐢2 𝐢2 PUNCT cana-5899 110 1 = = PUNCT cana-5899 110 2 βˆ’π›Ό βˆ’π›Ό PROPN cana-5899 110 3 + + CCONJ cana-5899 110 4 π‘Ÿπ‘’βˆ’πœ†π›Ώ π‘Ÿπ‘’βˆ’πœ†π›Ώ PROPN cana-5899 110 5 πœ•π‘ƒ πœ•π‘ƒ NUM cana-5899 110 6 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 110 7 ( ( PUNCT cana-5899 110 8 π‘£βˆ— π‘£βˆ— ADJ cana-5899 110 9 ) ) PUNCT cana-5899 110 10 . . PUNCT cana-5899 111 1 clearly clearly ADV cana-5899 111 2 𝐴2 𝐴2 PROPN cana-5899 111 3 , , PUNCT cana-5899 111 4 𝐡2 𝐡2 NOUN cana-5899 111 5 and and CCONJ cana-5899 111 6 𝐢2are 𝐢2are PROPN cana-5899 111 7 the the DET cana-5899 111 8 eigenvalues eigenvalue NOUN cana-5899 111 9 . . PUNCT cana-5899 112 1 𝐴2 𝐴2 PROPN cana-5899 112 2 < < X cana-5899 112 3 0 0 NUM cana-5899 112 4 ⟹ ⟹ NUM cana-5899 112 5 π›Όπ‘’βˆ’πœ†πœ‡ π›Όπ‘’βˆ’πœ†πœ‡ NOUN cana-5899 112 6 < < X cana-5899 112 7 𝑏𝑓𝑒(π‘’βˆ— 𝑏𝑓𝑒(π‘’βˆ— PROPN cana-5899 112 8 , , PUNCT cana-5899 112 9 π‘£βˆ— π‘£βˆ— ADJ cana-5899 112 10 ) ) PUNCT cana-5899 113 1 + + CCONJ cana-5899 113 2 𝑑 𝑑 PROPN cana-5899 113 3 + + X cana-5899 113 4 𝑐𝑉′(π‘’βˆ— 𝑐𝑉′(π‘’βˆ— NUM cana-5899 113 5 ) ) PUNCT cana-5899 113 6 , , PUNCT cana-5899 113 7 𝐡2 𝐡2 NOUN cana-5899 113 8 < < X cana-5899 113 9 0 0 PROPN cana-5899 113 10 ⟹ ⟹ NUM cana-5899 113 11 𝑏1𝑓𝑣(π‘’βˆ— 𝑏1𝑓𝑣(π‘’βˆ— PROPN cana-5899 113 12 , , PUNCT cana-5899 113 13 π‘£βˆ— π‘£βˆ— ADJ cana-5899 113 14 ) ) PUNCT cana-5899 114 1 + + CCONJ cana-5899 114 2 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ 𝑏1π‘’βˆ’πœ†πœπ‘“π‘’πœ NOUN cana-5899 114 3 ( ( PUNCT cana-5899 114 4 π‘’βˆ— π‘’βˆ— NOUN cana-5899 114 5 , , PUNCT cana-5899 114 6 π‘£βˆ— π‘£βˆ— ADJ cana-5899 114 7 ) ) PUNCT cana-5899 115 1 < < X cana-5899 115 2 π‘Ÿ π‘Ÿ X cana-5899 115 3 𝑃′(π‘£βˆ— 𝑃′(π‘£βˆ— NOUN cana-5899 115 4 ) ) PUNCT cana-5899 115 5 + + NUM cana-5899 115 6 𝑑1 𝑑1 NOUN cana-5899 115 7 , , PUNCT cana-5899 115 8 𝐢2 𝐢2 X cana-5899 115 9 < < X cana-5899 115 10 0 0 NUM cana-5899 115 11 ⟹ ⟹ NUM cana-5899 115 12 π‘Ÿπ‘’βˆ’πœ†π›Ώ π‘Ÿπ‘’βˆ’πœ†π›Ώ NUM cana-5899 115 13 πœ•π‘ƒ πœ•π‘ƒ SYM cana-5899 115 14 πœ•π‘£π›Ώ πœ•π‘£π›Ώ NOUN cana-5899 115 15 ( ( PUNCT cana-5899 115 16 π‘£βˆ— π‘£βˆ— ADJ cana-5899 115 17 ) ) PUNCT cana-5899 115 18 < < X cana-5899 115 19 𝛼. 𝛼. NOUN cana-5899 115 20 for for ADP cana-5899 115 21 the the DET cana-5899 115 22 above above ADJ cana-5899 115 23 3 3 NUM cana-5899 115 24 conditions condition NOUN cana-5899 115 25 , , PUNCT cana-5899 115 26 the the DET cana-5899 115 27 eigenvalues eigenvalue NOUN cana-5899 115 28 of of ADP cana-5899 115 29 the the DET cana-5899 115 30 system system NOUN cana-5899 115 31 will will AUX cana-5899 115 32 be be AUX cana-5899 115 33 negative negative ADJ cana-5899 115 34 , , PUNCT cana-5899 115 35 which which PRON cana-5899 115 36 implies imply VERB cana-5899 115 37 the the DET cana-5899 115 38 system system NOUN cana-5899 115 39 is be AUX cana-5899 115 40 locally locally ADV cana-5899 115 41 stable stable ADJ cana-5899 115 42 next next ADV cana-5899 115 43 , , PUNCT cana-5899 115 44 we we PRON cana-5899 115 45 examine examine VERB cana-5899 115 46 the the DET cana-5899 115 47 conditions condition NOUN cana-5899 115 48 for for ADP cana-5899 115 49 global global ADJ cana-5899 115 50 stability stability NOUN cana-5899 115 51 of of ADP cana-5899 115 52 the the DET cana-5899 115 53 model model NOUN cana-5899 115 54 . . PUNCT cana-5899 116 1 3.4 3.4 NUM cana-5899 116 2 . . PUNCT cana-5899 116 3 global global ADJ cana-5899 116 4 stability stability NOUN cana-5899 116 5 at at ADP cana-5899 116 6 endemic endemic ADJ cana-5899 116 7 equilibria equilibrium NOUN cana-5899 116 8 in in ADP cana-5899 116 9 models model NOUN cana-5899 116 10 of of ADP cana-5899 116 11 type type NOUN cana-5899 116 12 ( ( PUNCT cana-5899 116 13 1 1 NUM cana-5899 116 14 ) ) PUNCT cana-5899 116 15 that that PRON cana-5899 116 16 describe describe VERB cana-5899 116 17 the the DET cana-5899 116 18 spread spread NOUN cana-5899 116 19 of of ADP cana-5899 116 20 infectious infectious ADJ cana-5899 116 21 diseases disease NOUN cana-5899 116 22 , , PUNCT cana-5899 116 23 where where SCONJ cana-5899 116 24 the the DET cana-5899 116 25 disease disease NOUN cana-5899 116 26 may may AUX cana-5899 116 27 remain remain VERB cana-5899 116 28 present present ADJ cana-5899 116 29 in in ADP cana-5899 116 30 the the DET cana-5899 116 31 population population NOUN cana-5899 116 32 over over ADP cana-5899 116 33 time time NOUN cana-5899 117 1 , , PUNCT cana-5899 117 2 it it PRON cana-5899 117 3 's be AUX cana-5899 117 4 important important ADJ cana-5899 117 5 to to PART cana-5899 117 6 understand understand VERB cana-5899 117 7 how how SCONJ cana-5899 117 8 the the DET cana-5899 117 9 system system NOUN cana-5899 117 10 behaves behave VERB cana-5899 117 11 in in ADP cana-5899 117 12 the the DET cana-5899 117 13 long long ADJ cana-5899 117 14 run run NOUN cana-5899 117 15 . . PUNCT cana-5899 118 1 this this PRON cana-5899 118 2 is be AUX cana-5899 118 3 where where SCONJ cana-5899 118 4 the the DET cana-5899 118 5 idea idea NOUN cana-5899 118 6 of of ADP cana-5899 118 7 global global ADJ cana-5899 118 8 stability stability NOUN cana-5899 118 9 becomes become VERB cana-5899 118 10 useful useful ADJ cana-5899 118 11 [ [ X cana-5899 118 12 5 5 NUM cana-5899 118 13 ] ] PUNCT cana-5899 118 14 . . PUNCT cana-5899 119 1 if if SCONJ cana-5899 119 2 an an DET cana-5899 119 3 equilibrium equilibrium NOUN cana-5899 119 4 point point NOUN cana-5899 119 5 is be AUX cana-5899 119 6 said say VERB cana-5899 119 7 to to PART cana-5899 119 8 be be AUX cana-5899 119 9 globally globally ADV cana-5899 119 10 stable stable ADJ cana-5899 119 11 , , PUNCT cana-5899 119 12 it it PRON cana-5899 119 13 means mean VERB cana-5899 119 14 that that SCONJ cana-5899 119 15 no no ADV cana-5899 119 16 matter matter ADV cana-5899 119 17 how how SCONJ cana-5899 119 18 the the DET cana-5899 119 19 disease disease NOUN cana-5899 119 20 starts start VERB cana-5899 119 21 β€” β€” PUNCT cana-5899 119 22 whether whether SCONJ cana-5899 119 23 the the DET cana-5899 119 24 number number NOUN cana-5899 119 25 of of ADP cana-5899 119 26 infected infected ADJ cana-5899 119 27 people people NOUN cana-5899 119 28 is be AUX cana-5899 119 29 high high ADJ cana-5899 119 30 or or CCONJ cana-5899 119 31 low low ADJ cana-5899 119 32 β€” β€” PUNCT cana-5899 119 33 the the DET cana-5899 119 34 system system NOUN cana-5899 119 35 will will AUX cana-5899 119 36 eventually eventually ADV cana-5899 119 37 move move VERB cana-5899 119 38 toward toward ADP cana-5899 119 39 and and CCONJ cana-5899 119 40 settle settle VERB cana-5899 119 41 at at ADP cana-5899 119 42 that that DET cana-5899 119 43 same same ADJ cana-5899 119 44 point point NOUN cana-5899 119 45 . . PUNCT cana-5899 120 1 when when SCONJ cana-5899 120 2 this this DET cana-5899 120 3 point point NOUN cana-5899 120 4 represents represent VERB cana-5899 120 5 an an DET cana-5899 120 6 endemic endemic ADJ cana-5899 120 7 communications communication NOUN cana-5899 120 8 on on ADP cana-5899 120 9 applied apply VERB cana-5899 120 10 nonlinear nonlinear ADJ cana-5899 120 11 analysis analysis NOUN cana-5899 120 12 issn issn NOUN cana-5899 120 13 : : PUNCT cana-5899 120 14 1074 1074 NUM cana-5899 120 15 - - PUNCT cana-5899 120 16 133x 133x NUM cana-5899 120 17 vol vol NOUN cana-5899 120 18 31 31 NUM cana-5899 120 19 no no NOUN cana-5899 120 20 . . PUNCT cana-5899 121 1 8s 8s PROPN cana-5899 121 2 ( ( PUNCT cana-5899 121 3 2024 2024 NUM cana-5899 121 4 ) ) PUNCT cana-5899 121 5 1089 1089 NUM cana-5899 121 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 121 7 equilibrium equilibrium NOUN cana-5899 121 8 , , PUNCT cana-5899 121 9 it it PRON cana-5899 121 10 tells tell VERB cana-5899 121 11 us we PRON cana-5899 121 12 that that SCONJ cana-5899 121 13 the the DET cana-5899 121 14 disease disease NOUN cana-5899 121 15 will will AUX cana-5899 121 16 continue continue VERB cana-5899 121 17 to to PART cana-5899 121 18 exist exist VERB cana-5899 121 19 in in ADP cana-5899 121 20 the the DET cana-5899 121 21 population population NOUN cana-5899 121 22 at at ADP cana-5899 121 23 a a DET cana-5899 121 24 steady steady ADJ cana-5899 121 25 level level NOUN cana-5899 121 26 over over ADP cana-5899 121 27 time time NOUN cana-5899 121 28 . . PUNCT cana-5899 122 1 even even ADV cana-5899 122 2 if if SCONJ cana-5899 122 3 there there PRON cana-5899 122 4 are be VERB cana-5899 122 5 temporary temporary ADJ cana-5899 122 6 changes change NOUN cana-5899 122 7 , , PUNCT cana-5899 122 8 like like ADP cana-5899 122 9 sudden sudden ADJ cana-5899 122 10 increases increase NOUN cana-5899 122 11 or or CCONJ cana-5899 122 12 decreases decrease NOUN cana-5899 122 13 in in ADP cana-5899 122 14 cases case NOUN cana-5899 122 15 , , PUNCT cana-5899 122 16 the the DET cana-5899 122 17 system system NOUN cana-5899 122 18 will will AUX cana-5899 122 19 return return VERB cana-5899 122 20 to to ADP cana-5899 122 21 this this DET cana-5899 122 22 balance balance NOUN cana-5899 122 23 in in ADP cana-5899 122 24 the the DET cana-5899 122 25 long long ADJ cana-5899 122 26 run run NOUN cana-5899 122 27 . . PUNCT cana-5899 123 1 this this DET cana-5899 123 2 kind kind NOUN cana-5899 123 3 of of ADP cana-5899 123 4 analysis analysis NOUN cana-5899 123 5 helps help VERB cana-5899 123 6 researchers researcher NOUN cana-5899 123 7 understand understand VERB cana-5899 123 8 whether whether SCONJ cana-5899 123 9 the the DET cana-5899 123 10 disease disease NOUN cana-5899 123 11 will will AUX cana-5899 123 12 die die VERB cana-5899 123 13 out out ADP cana-5899 123 14 or or CCONJ cana-5899 123 15 continue continue VERB cana-5899 123 16 spreading spread VERB cana-5899 123 17 . . PUNCT cana-5899 124 1 we we PRON cana-5899 124 2 use use VERB cana-5899 124 3 the the DET cana-5899 124 4 approach approach NOUN cana-5899 124 5 of of ADP cana-5899 124 6 lyapunov lyapunov ADJ cana-5899 124 7 stability stability NOUN cana-5899 124 8 to to PART cana-5899 124 9 examine examine VERB cana-5899 124 10 the the DET cana-5899 124 11 conditions condition NOUN cana-5899 124 12 for for ADP cana-5899 124 13 the the DET cana-5899 124 14 model model NOUN cana-5899 124 15 ( ( PUNCT cana-5899 124 16 1 1 NUM cana-5899 124 17 ) ) PUNCT cana-5899 124 18 to to PART cana-5899 124 19 be be AUX cana-5899 124 20 globally globally ADV cana-5899 124 21 stable stable ADJ cana-5899 124 22 . . PUNCT cana-5899 125 1 as as SCONJ cana-5899 125 2 the the DET cana-5899 125 3 equilibrium equilibrium NOUN cana-5899 125 4 point point NOUN cana-5899 125 5 ( ( PUNCT cana-5899 125 6 π‘’βˆ— π‘’βˆ— NOUN cana-5899 125 7 , , PUNCT cana-5899 125 8 π‘£βˆ— π‘£βˆ— ADJ cana-5899 125 9 , , PUNCT cana-5899 125 10 π‘€βˆ— π‘€βˆ— PROPN cana-5899 125 11 ) ) PUNCT cana-5899 125 12 is be AUX cana-5899 125 13 the the DET cana-5899 125 14 solution solution NOUN cana-5899 125 15 of of ADP cana-5899 125 16 the the DET cana-5899 125 17 system system NOUN cana-5899 125 18 ( ( PUNCT cana-5899 125 19 1 1 NUM cana-5899 125 20 ) ) PUNCT cana-5899 125 21 . . PUNCT cana-5899 126 1 we we PRON cana-5899 126 2 can can AUX cana-5899 126 3 rewrite rewrite VERB cana-5899 126 4 the the DET cana-5899 126 5 system system NOUN cana-5899 126 6 as as ADP cana-5899 126 7 ( ( PUNCT cana-5899 126 8 𝑒 𝑒 PROPN cana-5899 126 9 βˆ’ βˆ’ NOUN cana-5899 126 10 π‘’βˆ—)β€² π‘’βˆ—)β€² X cana-5899 126 11 = = SYM cana-5899 126 12 βˆ’π‘(𝑓(𝑒 βˆ’π‘(𝑓(𝑒 PROPN cana-5899 126 13 , , PUNCT cana-5899 126 14 𝑣 𝑣 NOUN cana-5899 126 15 ) ) PUNCT cana-5899 126 16 βˆ’ βˆ’ ADP cana-5899 126 17 𝑓(π‘’βˆ— 𝑓(π‘’βˆ— NOUN cana-5899 126 18 , , PUNCT cana-5899 126 19 π‘£βˆ— π‘£βˆ— NUM cana-5899 126 20 ) ) PUNCT cana-5899 126 21 ) ) PUNCT cana-5899 127 1 βˆ’ βˆ’ ADP cana-5899 127 2 𝑑(𝑒 𝑑(𝑒 ADJ cana-5899 127 3 βˆ’ βˆ’ PROPN cana-5899 127 4 π‘’βˆ— π‘’βˆ— NOUN cana-5899 127 5 ) ) PUNCT cana-5899 127 6 βˆ’ βˆ’ ADP cana-5899 127 7 𝑐(𝑉(𝑒 𝑐(𝑉(𝑒 ADJ cana-5899 127 8 ) ) PUNCT cana-5899 128 1 βˆ’ βˆ’ PROPN cana-5899 128 2 𝑉(π‘’βˆ— 𝑉(π‘’βˆ— PROPN cana-5899 128 3 ) ) PUNCT cana-5899 128 4 ) ) PUNCT cana-5899 129 1 + + CCONJ cana-5899 129 2 𝛼(𝑀(𝑑 𝛼(𝑀(𝑑 NOUN cana-5899 129 3 βˆ’ βˆ’ NOUN cana-5899 129 4 πœ‡ πœ‡ NOUN cana-5899 129 5 ) ) PUNCT cana-5899 129 6 βˆ’ βˆ’ PROPN cana-5899 129 7 π‘€βˆ— π‘€βˆ— PROPN cana-5899 129 8 ) ) PUNCT cana-5899 129 9 ( ( PUNCT cana-5899 129 10 𝑣 𝑣 DET cana-5899 129 11 βˆ’ βˆ’ X cana-5899 129 12 π‘£βˆ—)β€² π‘£βˆ—)β€² PROPN cana-5899 130 1 = = PUNCT cana-5899 130 2 𝑏1(𝑓(𝑒(t 𝑏1(𝑓(𝑒(t NUM cana-5899 130 3 βˆ’ βˆ’ PROPN cana-5899 130 4 Ο„ Ο„ PROPN cana-5899 130 5 ) ) PUNCT cana-5899 130 6 , , PUNCT cana-5899 130 7 𝑣 𝑣 X cana-5899 130 8 ) ) PUNCT cana-5899 130 9 βˆ’ βˆ’ ADP cana-5899 130 10 𝑓(π‘’βˆ— 𝑓(π‘’βˆ— NOUN cana-5899 130 11 , , PUNCT cana-5899 130 12 π‘£βˆ— π‘£βˆ— NUM cana-5899 130 13 ) ) PUNCT cana-5899 130 14 ) ) PUNCT cana-5899 131 1 βˆ’ βˆ’ ADP cana-5899 131 2 π‘Ÿ(𝑃(𝑣 π‘Ÿ(𝑃(𝑣 NOUN cana-5899 131 3 ) ) PUNCT cana-5899 131 4 βˆ’ βˆ’ PROPN cana-5899 131 5 𝑃 𝑃 PROPN cana-5899 131 6 ( ( PUNCT cana-5899 131 7 π‘£βˆ— π‘£βˆ— ADJ cana-5899 131 8 ) ) PUNCT cana-5899 131 9 ) ) PUNCT cana-5899 132 1 βˆ’ βˆ’ PROPN cana-5899 133 1 𝑑1(𝑣 𝑑1(𝑣 CCONJ cana-5899 134 1 βˆ’ βˆ’ PROPN cana-5899 134 2 π‘£βˆ— π‘£βˆ— ADJ cana-5899 134 3 ) ) PUNCT cana-5899 134 4 ( ( PUNCT cana-5899 134 5 𝑀 𝑀 PART cana-5899 134 6 βˆ’ βˆ’ NOUN cana-5899 134 7 π‘€βˆ—)β€² π‘€βˆ—)β€² X cana-5899 134 8 = = SYM cana-5899 134 9 π‘Ÿ(𝑃(𝑣(𝑑 π‘Ÿ(𝑃(𝑣(𝑑 NUM cana-5899 134 10 βˆ’ βˆ’ PROPN cana-5899 134 11 Ξ΄ Ξ΄ PROPN cana-5899 134 12 ) ) PUNCT cana-5899 134 13 ) ) PUNCT cana-5899 135 1 βˆ’ βˆ’ ADP cana-5899 135 2 𝑃 𝑃 PROPN cana-5899 135 3 ( ( PUNCT cana-5899 135 4 π‘£βˆ— π‘£βˆ— ADJ cana-5899 135 5 ) ) PUNCT cana-5899 135 6 ) ) PUNCT cana-5899 136 1 βˆ’ βˆ’ ADP cana-5899 136 2 𝛼(𝑀 𝛼(𝑀 ADP cana-5899 136 3 βˆ’ βˆ’ PROPN cana-5899 136 4 π‘€βˆ— π‘€βˆ— NOUN cana-5899 136 5 ) ) PUNCT cana-5899 136 6 ( ( PUNCT cana-5899 136 7 2 2 X cana-5899 136 8 ) ) PUNCT cana-5899 136 9 to to PART cana-5899 136 10 prove prove VERB cana-5899 136 11 the the DET cana-5899 136 12 result result NOUN cana-5899 136 13 of of ADP cana-5899 136 14 global global ADJ cana-5899 136 15 stability stability NOUN cana-5899 136 16 , , PUNCT cana-5899 136 17 we we PRON cana-5899 136 18 assume assume VERB cana-5899 136 19 the the DET cana-5899 136 20 following follow VERB cana-5899 136 21 lipschitz lipschitz NOUN cana-5899 136 22 conditions condition NOUN cana-5899 136 23 on on ADP cana-5899 136 24 the the DET cana-5899 136 25 functions function NOUN cana-5899 136 26 𝑓 𝑓 PRON cana-5899 136 27 , , PUNCT cana-5899 136 28 𝑉 𝑉 PROPN cana-5899 136 29 and and CCONJ cana-5899 136 30 𝑃. 𝑃. PROPN cana-5899 136 31 𝐾1|𝑒 𝐾1|𝑒 NOUN cana-5899 136 32 βˆ’ βˆ’ NUM cana-5899 136 33 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 136 34 + + CCONJ cana-5899 137 1 𝐾2|𝑣 𝐾2|𝑣 ADP cana-5899 137 2 βˆ’ βˆ’ NOUN cana-5899 137 3 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 137 4 ≀ ≀ NOUN cana-5899 137 5 |𝑓(𝑒 |𝑓(𝑒 PROPN cana-5899 137 6 , , PUNCT cana-5899 137 7 𝑣 𝑣 NOUN cana-5899 137 8 ) ) PUNCT cana-5899 137 9 βˆ’ βˆ’ ADP cana-5899 137 10 𝑓(π‘’βˆ— 𝑓(π‘’βˆ— NOUN cana-5899 137 11 , , PUNCT cana-5899 137 12 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 137 13 ≀ ≀ NUM cana-5899 137 14 𝐾3|𝑒 𝐾3|𝑒 NOUN cana-5899 137 15 βˆ’ βˆ’ ADP cana-5899 137 16 π‘’βˆ—| π‘’βˆ—| X cana-5899 137 17 + + CCONJ cana-5899 137 18 𝐾4|𝑣 𝐾4|𝑣 PROPN cana-5899 137 19 βˆ’ βˆ’ NOUN cana-5899 137 20 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 137 21 𝑀1|𝑒 𝑀1|𝑒 NOUN cana-5899 137 22 βˆ’ βˆ’ ADP cana-5899 137 23 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 137 24 ≀ ≀ NOUN cana-5899 137 25 |𝑉(𝑒 |𝑉(𝑒 NOUN cana-5899 137 26 ) ) PUNCT cana-5899 138 1 βˆ’ βˆ’ PROPN cana-5899 138 2 𝑉(π‘’βˆ—)| 𝑉(π‘’βˆ—)| PROPN cana-5899 138 3 ≀ ≀ PROPN cana-5899 138 4 𝑀2|𝑒 𝑀2|𝑒 PROPN cana-5899 138 5 βˆ’ βˆ’ PROPN cana-5899 138 6 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 138 7 𝑁1|𝑣 𝑁1|𝑣 VERB cana-5899 138 8 βˆ’ βˆ’ PROPN cana-5899 138 9 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 138 10 ≀ ≀ NUM cana-5899 138 11 |𝑃(𝑣 |𝑃(𝑣 NOUN cana-5899 138 12 ) ) PUNCT cana-5899 138 13 βˆ’ βˆ’ ADP cana-5899 139 1 𝑃 𝑃 PROPN cana-5899 139 2 ( ( PUNCT cana-5899 139 3 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 139 4 ≀ ≀ NUM cana-5899 140 1 𝑁2|𝑣 𝑁2|𝑣 NUM cana-5899 140 2 βˆ’ βˆ’ NOUN cana-5899 140 3 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 140 4 ( ( PUNCT cana-5899 140 5 3 3 X cana-5899 140 6 ) ) PUNCT cana-5899 140 7 we we PRON cana-5899 140 8 state state NOUN cana-5899 140 9 theorem theorem VERB cana-5899 140 10 3.3 3.3 NUM cana-5899 140 11 . . PUNCT cana-5899 141 1 the the DET cana-5899 141 2 system system NOUN cana-5899 141 3 ( ( PUNCT cana-5899 141 4 1 1 X cana-5899 141 5 ) ) PUNCT cana-5899 141 6 is be AUX cana-5899 141 7 globally globally ADV cana-5899 141 8 stable stable ADJ cana-5899 141 9 at at ADP cana-5899 141 10 the the DET cana-5899 141 11 equilibrium equilibrium NOUN cana-5899 141 12 point point NOUN cana-5899 141 13 if if SCONJ cana-5899 141 14 the the DET cana-5899 141 15 functions function NOUN cana-5899 141 16 of of ADP cana-5899 141 17 system system NOUN cana-5899 141 18 ( ( PUNCT cana-5899 141 19 1 1 X cana-5899 141 20 ) ) PUNCT cana-5899 141 21 satisfy satisfy NOUN cana-5899 141 22 lipschitz lipschitz NOUN cana-5899 141 23 conditions condition NOUN cana-5899 141 24 ( ( PUNCT cana-5899 141 25 3 3 NUM cana-5899 141 26 ) ) PUNCT cana-5899 141 27 and and CCONJ cana-5899 141 28 the the DET cana-5899 141 29 parameters parameter NOUN cana-5899 141 30 of of ADP cana-5899 141 31 the the DET cana-5899 141 32 system system NOUN cana-5899 141 33 satisfy satisfy VERB cana-5899 141 34 𝑏𝐾1 𝑏𝐾1 ADV cana-5899 141 35 + + CCONJ cana-5899 141 36 𝑑 𝑑 PROPN cana-5899 141 37 + + X cana-5899 141 38 𝑐𝑀1 𝑐𝑀1 X cana-5899 141 39 > > SYM cana-5899 141 40 𝑏1𝐾3 𝑏1𝐾3 PUNCT cana-5899 141 41 and and CCONJ cana-5899 141 42 𝑏𝐾2 𝑏𝐾2 PROPN cana-5899 141 43 + + NUM cana-5899 141 44 𝑑1 𝑑1 NOUN cana-5899 141 45 + + CCONJ cana-5899 141 46 π‘Ÿπ‘1 π‘Ÿπ‘1 X cana-5899 141 47 > > SYM cana-5899 141 48 𝑏1𝐾4 𝑏1𝐾4 PROPN cana-5899 141 49 + + NUM cana-5899 142 1 π‘Ÿπ‘2 π‘Ÿπ‘2 PROPN cana-5899 142 2 . . PUNCT cana-5899 142 3 proof proof NOUN cana-5899 142 4 : : PUNCT cana-5899 142 5 let let VERB cana-5899 142 6 the the DET cana-5899 142 7 lyapunov lyapunov ADJ cana-5899 142 8 function function NOUN cana-5899 142 9 be be AUX cana-5899 142 10 𝐿 𝐿 PROPN cana-5899 142 11 = = PUNCT cana-5899 142 12 |𝑒 |𝑒 ADJ cana-5899 142 13 βˆ’ βˆ’ NUM cana-5899 142 14 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 142 15 + + CCONJ cana-5899 142 16 |𝑣 |𝑣 NOUN cana-5899 142 17 βˆ’ βˆ’ NOUN cana-5899 142 18 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 142 19 + + CCONJ cana-5899 142 20 |𝑀 |𝑀 X cana-5899 142 21 βˆ’ βˆ’ PROPN cana-5899 142 22 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 142 23 + + CCONJ cana-5899 143 1 𝛼 𝛼 PRON cana-5899 143 2 ∫ ∫ PROPN cana-5899 143 3 |𝑀(𝑠 |𝑀(𝑠 PROPN cana-5899 143 4 ) ) PUNCT cana-5899 143 5 βˆ’ βˆ’ PROPN cana-5899 143 6 π‘€βˆ—| π‘€βˆ—| ADP cana-5899 143 7 𝑑 𝑑 PROPN cana-5899 143 8 π‘‘βˆ’πœ‡ π‘‘βˆ’πœ‡ PROPN cana-5899 143 9 𝑑𝑠 𝑑𝑠 X cana-5899 143 10 + + SYM cana-5899 143 11 𝑏1𝐾3 𝑏1𝐾3 PROPN cana-5899 143 12 ∫ ∫ PROPN cana-5899 143 13 |𝑒(𝑠 |𝑒(𝑠 PROPN cana-5899 143 14 ) ) PUNCT cana-5899 143 15 βˆ’ βˆ’ ADP cana-5899 143 16 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 144 1 𝑑 𝑑 PROPN cana-5899 144 2 π‘‘βˆ’πœ π‘‘βˆ’πœ PRON cana-5899 144 3 𝑑𝑠 𝑑𝑠 X cana-5899 144 4 + + X cana-5899 144 5 π‘Ÿπ‘1 π‘Ÿπ‘1 PROPN cana-5899 144 6 ∫ ∫ PROPN cana-5899 144 7 |𝑣(𝑠 |𝑣(𝑠 PROPN cana-5899 144 8 ) ) PUNCT cana-5899 144 9 βˆ’ βˆ’ PROPN cana-5899 144 10 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 144 11 𝑑 𝑑 PROPN cana-5899 144 12 π‘‘βˆ’π›Ώ π‘‘βˆ’π›Ώ PROPN cana-5899 144 13 . . PUNCT cana-5899 145 1 then then ADV cana-5899 145 2 the the DET cana-5899 145 3 dini dini NOUN cana-5899 145 4 derivative derivative NOUN cana-5899 145 5 along along ADP cana-5899 145 6 the the DET cana-5899 145 7 solutions solution NOUN cana-5899 145 8 of of ADP cana-5899 145 9 ( ( PUNCT cana-5899 145 10 1 1 NUM cana-5899 145 11 ) ) PUNCT cana-5899 145 12 equations equation NOUN cana-5899 145 13 ( ( PUNCT cana-5899 145 14 2 2 X cana-5899 145 15 ) ) PUNCT cana-5899 145 16 are be AUX cana-5899 145 17 𝐷+𝐿 𝐷+𝐿 X cana-5899 146 1 ≀ ≀ ADV cana-5899 147 1 βˆ’π‘|𝑓(𝑒 βˆ’π‘|𝑓(𝑒 ADV cana-5899 147 2 , , PUNCT cana-5899 147 3 𝑣 𝑣 NOUN cana-5899 147 4 ) ) PUNCT cana-5899 147 5 βˆ’ βˆ’ ADP cana-5899 147 6 𝑓(π‘’βˆ— 𝑓(π‘’βˆ— NOUN cana-5899 147 7 , , PUNCT cana-5899 147 8 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 147 9 βˆ’ βˆ’ PROPN cana-5899 147 10 𝑑|𝑒 𝑑|𝑒 NOUN cana-5899 148 1 βˆ’ βˆ’ NOUN cana-5899 148 2 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 148 3 βˆ’ βˆ’ NOUN cana-5899 148 4 𝑐|𝑉(𝑒 𝑐|𝑉(𝑒 NOUN cana-5899 148 5 ) ) PUNCT cana-5899 148 6 βˆ’ βˆ’ PROPN cana-5899 149 1 𝑉(π‘’βˆ—)| 𝑉(π‘’βˆ—)| PROPN cana-5899 149 2 + + CCONJ cana-5899 149 3 𝛼|𝑀(𝑑 𝛼|𝑀(𝑑 NOUN cana-5899 149 4 βˆ’ βˆ’ NOUN cana-5899 149 5 πœ‡ πœ‡ NOUN cana-5899 149 6 ) ) PUNCT cana-5899 149 7 βˆ’ βˆ’ NOUN cana-5899 149 8 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 149 9 + + CCONJ cana-5899 149 10 𝑏1|𝑓(𝑒(t 𝑏1|𝑓(𝑒(t VERB cana-5899 149 11 βˆ’ βˆ’ PROPN cana-5899 149 12 Ο„ Ο„ X cana-5899 149 13 ) ) PUNCT cana-5899 149 14 , , PUNCT cana-5899 149 15 𝑣 𝑣 X cana-5899 149 16 ) ) PUNCT cana-5899 149 17 βˆ’ βˆ’ ADP cana-5899 149 18 𝑓(π‘’βˆ— 𝑓(π‘’βˆ— NOUN cana-5899 149 19 , , PUNCT cana-5899 149 20 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 149 21 βˆ’ βˆ’ NUM cana-5899 149 22 π‘Ÿ|𝑃(𝑣 π‘Ÿ|𝑃(𝑣 SYM cana-5899 149 23 ) ) PUNCT cana-5899 149 24 βˆ’ βˆ’ PROPN cana-5899 150 1 𝑃 𝑃 PROPN cana-5899 150 2 ( ( PUNCT cana-5899 150 3 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 150 4 βˆ’ βˆ’ PROPN cana-5899 150 5 𝑑1|𝑣 𝑑1|𝑣 VERB cana-5899 150 6 βˆ’ βˆ’ PROPN cana-5899 150 7 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 150 8 + + PUNCT cana-5899 151 1 π‘Ÿ|𝑃(𝑣(𝑑 π‘Ÿ|𝑃(𝑣(𝑑 NUM cana-5899 151 2 βˆ’ βˆ’ PROPN cana-5899 151 3 Ξ΄ Ξ΄ NOUN cana-5899 151 4 ) ) PUNCT cana-5899 151 5 ) ) PUNCT cana-5899 152 1 βˆ’ βˆ’ PROPN cana-5899 152 2 𝑃 𝑃 PROPN cana-5899 152 3 ( ( PUNCT cana-5899 152 4 π‘£βˆ—)| π‘£βˆ—)| PROPN cana-5899 152 5 βˆ’ βˆ’ PROPN cana-5899 152 6 𝛼|𝑀 𝛼|𝑀 NOUN cana-5899 152 7 βˆ’ βˆ’ ADP cana-5899 152 8 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 152 9 + + CCONJ cana-5899 152 10 𝛼|𝑀 𝛼|𝑀 NOUN cana-5899 152 11 βˆ’ βˆ’ NOUN cana-5899 152 12 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 152 13 βˆ’ βˆ’ PROPN cana-5899 152 14 𝛼|𝑀(𝑑 𝛼|𝑀(𝑑 PROPN cana-5899 152 15 βˆ’ βˆ’ NOUN cana-5899 152 16 πœ‡ πœ‡ NOUN cana-5899 152 17 ) ) PUNCT cana-5899 152 18 βˆ’ βˆ’ NOUN cana-5899 152 19 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 152 20 + + CCONJ cana-5899 152 21 𝑏1𝐾3|𝑒 𝑏1𝐾3|𝑒 ADJ cana-5899 152 22 βˆ’ βˆ’ PROPN cana-5899 152 23 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 152 24 βˆ’ βˆ’ PROPN cana-5899 152 25 𝑏1𝐾3|𝑒(t 𝑏1𝐾3|𝑒(t PROPN cana-5899 152 26 βˆ’ βˆ’ PROPN cana-5899 152 27 Ο„ Ο„ PROPN cana-5899 152 28 ) ) PUNCT cana-5899 152 29 βˆ’ βˆ’ ADP cana-5899 152 30 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 152 31 + + CCONJ cana-5899 152 32 π‘Ÿπ‘1|𝑣 π‘Ÿπ‘1|𝑣 PROPN cana-5899 152 33 βˆ’ βˆ’ PROPN cana-5899 152 34 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 152 35 βˆ’ βˆ’ NUM cana-5899 152 36 π‘Ÿπ‘1|𝑣(𝑑 π‘Ÿπ‘1|𝑣(𝑑 NUM cana-5899 152 37 βˆ’ βˆ’ PROPN cana-5899 152 38 Ξ΄ Ξ΄ PROPN cana-5899 152 39 ) ) PUNCT cana-5899 152 40 βˆ’ βˆ’ NOUN cana-5899 152 41 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 152 42 communications communication NOUN cana-5899 152 43 on on ADP cana-5899 152 44 applied apply VERB cana-5899 152 45 nonlinear nonlinear ADJ cana-5899 152 46 analysis analysis NOUN cana-5899 152 47 issn issn NOUN cana-5899 152 48 : : PUNCT cana-5899 152 49 1074 1074 NUM cana-5899 152 50 - - PUNCT cana-5899 152 51 133x 133x NUM cana-5899 152 52 vol vol NOUN cana-5899 152 53 31 31 NUM cana-5899 152 54 no no NOUN cana-5899 152 55 . . PUNCT cana-5899 153 1 8s 8s PROPN cana-5899 153 2 ( ( PUNCT cana-5899 153 3 2024 2024 NUM cana-5899 153 4 ) ) PUNCT cana-5899 153 5 1090 1090 NUM cana-5899 153 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 153 7 using use VERB cana-5899 153 8 conditions condition NOUN cana-5899 153 9 ( ( PUNCT cana-5899 153 10 3 3 NUM cana-5899 153 11 ) ) PUNCT cana-5899 153 12 , , PUNCT cana-5899 153 13 we we PRON cana-5899 153 14 get get VERB cana-5899 153 15 𝐷+ 𝐷+ VERB cana-5899 153 16 ≀ ≀ NUM cana-5899 153 17 βˆ’π‘(𝐾1|𝑒 βˆ’π‘(𝐾1|𝑒 ADP cana-5899 153 18 βˆ’ βˆ’ NUM cana-5899 153 19 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 153 20 + + CCONJ cana-5899 154 1 𝐾2|𝑣 𝐾2|𝑣 ADP cana-5899 154 2 βˆ’ βˆ’ NOUN cana-5899 154 3 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 154 4 ) ) PUNCT cana-5899 154 5 βˆ’ βˆ’ NOUN cana-5899 154 6 𝑑|𝑒 𝑑|𝑒 NOUN cana-5899 155 1 βˆ’ βˆ’ ADP cana-5899 155 2 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 155 3 βˆ’ βˆ’ PROPN cana-5899 155 4 𝑐𝑀1|𝑒 𝑐𝑀1|𝑒 PROPN cana-5899 155 5 βˆ’ βˆ’ PROPN cana-5899 155 6 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 155 7 + + CCONJ cana-5899 155 8 𝛼|𝑀(𝑑 𝛼|𝑀(𝑑 NUM cana-5899 155 9 βˆ’ βˆ’ NOUN cana-5899 155 10 πœ‡ πœ‡ NOUN cana-5899 155 11 ) ) PUNCT cana-5899 155 12 βˆ’ βˆ’ NOUN cana-5899 155 13 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 155 14 + + CCONJ cana-5899 155 15 𝑏1(𝐾3|𝑒(t 𝑏1(𝐾3|𝑒(t PROPN cana-5899 155 16 βˆ’ βˆ’ PROPN cana-5899 155 17 Ο„ Ο„ PROPN cana-5899 155 18 ) ) PUNCT cana-5899 155 19 βˆ’ βˆ’ ADP cana-5899 155 20 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 155 21 + + CCONJ cana-5899 155 22 𝐾4|𝑣 𝐾4|𝑣 PROPN cana-5899 155 23 βˆ’ βˆ’ NUM cana-5899 155 24 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 155 25 ) ) PUNCT cana-5899 155 26 βˆ’ βˆ’ PROPN cana-5899 156 1 π‘Ÿπ‘1|𝑣 π‘Ÿπ‘1|𝑣 PROPN cana-5899 156 2 βˆ’ βˆ’ PROPN cana-5899 156 3 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 156 4 βˆ’ βˆ’ NOUN cana-5899 156 5 𝑑1|𝑣 𝑑1|𝑣 NOUN cana-5899 156 6 βˆ’ βˆ’ PROPN cana-5899 156 7 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 156 8 + + CCONJ cana-5899 156 9 π‘Ÿπ‘2|𝑣(𝑑 π‘Ÿπ‘2|𝑣(𝑑 NUM cana-5899 156 10 βˆ’ βˆ’ PROPN cana-5899 156 11 Ξ΄ Ξ΄ PROPN cana-5899 156 12 ) ) PUNCT cana-5899 156 13 βˆ’ βˆ’ PROPN cana-5899 156 14 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 156 15 βˆ’ βˆ’ PRON cana-5899 156 16 𝛼|𝑀 𝛼|𝑀 NOUN cana-5899 156 17 βˆ’ βˆ’ ADP cana-5899 156 18 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 156 19 + + CCONJ cana-5899 156 20 𝛼|𝑀 𝛼|𝑀 NOUN cana-5899 156 21 βˆ’ βˆ’ NOUN cana-5899 156 22 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 156 23 βˆ’ βˆ’ PROPN cana-5899 156 24 𝛼|𝑀(𝑑 𝛼|𝑀(𝑑 PROPN cana-5899 156 25 βˆ’ βˆ’ NOUN cana-5899 156 26 πœ‡ πœ‡ NOUN cana-5899 156 27 ) ) PUNCT cana-5899 156 28 βˆ’ βˆ’ NOUN cana-5899 156 29 π‘€βˆ—| π‘€βˆ—| NOUN cana-5899 156 30 + + CCONJ cana-5899 156 31 𝑏1𝐾3|𝑒 𝑏1𝐾3|𝑒 ADJ cana-5899 156 32 βˆ’ βˆ’ PROPN cana-5899 156 33 π‘’βˆ—| π‘’βˆ—| NOUN cana-5899 156 34 βˆ’ βˆ’ PROPN cana-5899 156 35 𝑏1𝐾3|𝑒(t 𝑏1𝐾3|𝑒(t PROPN cana-5899 156 36 βˆ’ βˆ’ PROPN cana-5899 156 37 Ο„ Ο„ PROPN cana-5899 156 38 ) ) PUNCT cana-5899 156 39 βˆ’ βˆ’ ADP cana-5899 156 40 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 156 41 + + CCONJ cana-5899 156 42 π‘Ÿπ‘2|𝑣 π‘Ÿπ‘2|𝑣 NUM cana-5899 156 43 βˆ’ βˆ’ NOUN cana-5899 156 44 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 156 45 βˆ’ βˆ’ PROPN cana-5899 156 46 π‘Ÿπ‘2|𝑣(𝑑 π‘Ÿπ‘2|𝑣(𝑑 NUM cana-5899 156 47 βˆ’ βˆ’ PROPN cana-5899 156 48 Ξ΄ Ξ΄ PROPN cana-5899 156 49 ) ) PUNCT cana-5899 156 50 βˆ’ βˆ’ PROPN cana-5899 156 51 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 156 52 𝐷+ 𝐷+ VERB cana-5899 156 53 ≀ ≀ NOUN cana-5899 156 54 ( ( PUNCT cana-5899 156 55 βˆ’π‘πΎ1 βˆ’π‘πΎ1 ADV cana-5899 156 56 βˆ’ βˆ’ PROPN cana-5899 156 57 𝑑 𝑑 NOUN cana-5899 156 58 βˆ’ βˆ’ PROPN cana-5899 156 59 𝑐𝑀1 𝑐𝑀1 NUM cana-5899 156 60 + + CCONJ cana-5899 156 61 𝑏1𝐾3)|𝑒 𝑏1𝐾3)|𝑒 PROPN cana-5899 156 62 βˆ’ βˆ’ PROPN cana-5899 156 63 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 156 64 + + CCONJ cana-5899 156 65 ( ( PUNCT cana-5899 156 66 βˆ’π‘πΎ2 βˆ’π‘πΎ2 PROPN cana-5899 156 67 + + CCONJ cana-5899 156 68 𝑏1𝐾4 𝑏1𝐾4 PROPN cana-5899 156 69 βˆ’ βˆ’ PROPN cana-5899 156 70 𝑑1 𝑑1 NOUN cana-5899 156 71 βˆ’ βˆ’ PROPN cana-5899 156 72 π‘Ÿπ‘1 π‘Ÿπ‘1 PROPN cana-5899 156 73 + + CCONJ cana-5899 156 74 π‘Ÿπ‘2)|𝑣 π‘Ÿπ‘2)|𝑣 PROPN cana-5899 156 75 βˆ’ βˆ’ PROPN cana-5899 156 76 π‘£βˆ—| π‘£βˆ—| NOUN cana-5899 156 77 𝐷+ 𝐷+ VERB cana-5899 156 78 ≀ ≀ NUM cana-5899 156 79 βˆ’(𝑏𝐾1 βˆ’(𝑏𝐾1 ADV cana-5899 156 80 + + CCONJ cana-5899 156 81 𝑑 𝑑 PRON cana-5899 156 82 + + NOUN cana-5899 156 83 𝑐𝑀1 𝑐𝑀1 VERB cana-5899 156 84 βˆ’ βˆ’ NOUN cana-5899 157 1 𝑏1𝐾3)|𝑒 𝑏1𝐾3)|𝑒 PROPN cana-5899 157 2 βˆ’ βˆ’ PROPN cana-5899 157 3 π‘’βˆ—| π‘’βˆ—| PROPN cana-5899 157 4 βˆ’ βˆ’ PROPN cana-5899 157 5 ( ( PUNCT cana-5899 157 6 𝑏𝐾2 𝑏𝐾2 PROPN cana-5899 157 7 βˆ’ βˆ’ PROPN cana-5899 157 8 𝑏1𝐾4 𝑏1𝐾4 PROPN cana-5899 157 9 + + NUM cana-5899 157 10 𝑑1 𝑑1 NOUN cana-5899 157 11 + + CCONJ cana-5899 157 12 π‘Ÿπ‘1 π‘Ÿπ‘1 PROPN cana-5899 157 13 βˆ’ βˆ’ NOUN cana-5899 157 14 π‘Ÿπ‘2)|𝑣 π‘Ÿπ‘2)|𝑣 PROPN cana-5899 157 15 βˆ’ βˆ’ PROPN cana-5899 157 16 π‘£βˆ—| π‘£βˆ—| NUM cana-5899 157 17 by by ADP cana-5899 157 18 our our PRON cana-5899 157 19 assumptions assumption NOUN cana-5899 157 20 on on ADP cana-5899 157 21 the the DET cana-5899 157 22 parameter parameter NOUN cana-5899 157 23 𝐷+ 𝐷+ VERB cana-5899 157 24 ≀ ≀ NOUN cana-5899 157 25 0 0 NUM cana-5899 157 26 . . PUNCT cana-5899 158 1 hence hence ADV cana-5899 158 2 by by ADP cana-5899 158 3 lyapunov lyapunov PROPN cana-5899 158 4 theory theory NOUN cana-5899 158 5 𝑒 𝑒 PART cana-5899 158 6 ⟢ ⟢ NOUN cana-5899 158 7 π‘’βˆ— π‘’βˆ— NOUN cana-5899 158 8 , , PUNCT cana-5899 158 9 𝑣 𝑣 PRON cana-5899 158 10 ⟢ ⟢ NOUN cana-5899 158 11 π‘£βˆ—and π‘£βˆ—and ADP cana-5899 158 12 𝑀 𝑀 SYM cana-5899 158 13 ⟢ ⟢ NOUN cana-5899 158 14 π‘€βˆ—. π‘€βˆ—. X cana-5899 158 15 therefore therefore ADV cana-5899 158 16 , , PUNCT cana-5899 158 17 the the DET cana-5899 158 18 system system NOUN cana-5899 158 19 ( ( PUNCT cana-5899 158 20 1 1 X cana-5899 158 21 ) ) PUNCT cana-5899 158 22 is be AUX cana-5899 158 23 globally globally ADV cana-5899 158 24 stable stable ADJ cana-5899 158 25 at at ADP cana-5899 158 26 equilibria equilibrium NOUN cana-5899 158 27 ( ( PUNCT cana-5899 158 28 π‘’βˆ— π‘’βˆ— NOUN cana-5899 158 29 , , PUNCT cana-5899 158 30 π‘£βˆ— π‘£βˆ— ADJ cana-5899 158 31 , , PUNCT cana-5899 158 32 π‘€βˆ— π‘€βˆ— PROPN cana-5899 158 33 ) ) PUNCT cana-5899 158 34 . . PUNCT cana-5899 159 1 remark remark VERB cana-5899 159 2 3.1 3.1 NUM cana-5899 159 3 : : PUNCT cana-5899 159 4 it it PRON cana-5899 159 5 is be AUX cana-5899 159 6 clear clear ADJ cana-5899 159 7 from from ADP cana-5899 159 8 the the DET cana-5899 159 9 previous previous ADJ cana-5899 159 10 results result NOUN cana-5899 159 11 that that SCONJ cana-5899 159 12 the the DET cana-5899 159 13 system system NOUN cana-5899 159 14 can can AUX cana-5899 159 15 still still ADV cana-5899 159 16 achieve achieve VERB cana-5899 159 17 both both CCONJ cana-5899 159 18 local local ADJ cana-5899 159 19 and and CCONJ cana-5899 159 20 global global ADJ cana-5899 159 21 stability stability NOUN cana-5899 159 22 even even ADV cana-5899 159 23 in in ADP cana-5899 159 24 the the DET cana-5899 159 25 presence presence NOUN cana-5899 159 26 of of ADP cana-5899 159 27 delays delay NOUN cana-5899 159 28 . . PUNCT cana-5899 160 1 this this DET cana-5899 160 2 outcome outcome NOUN cana-5899 160 3 depends depend VERB cana-5899 160 4 on on ADP cana-5899 160 5 how how SCONJ cana-5899 160 6 the the DET cana-5899 160 7 parameters parameter NOUN cana-5899 160 8 are be AUX cana-5899 160 9 chosen choose VERB cana-5899 160 10 and and CCONJ cana-5899 160 11 how how SCONJ cana-5899 160 12 the the DET cana-5899 160 13 nonlinear nonlinear ADJ cana-5899 160 14 terms term NOUN cana-5899 160 15 are be AUX cana-5899 160 16 structured structure VERB cana-5899 160 17 . . PUNCT cana-5899 161 1 by by ADP cana-5899 161 2 appropriately appropriately ADV cana-5899 161 3 choosing choose VERB cana-5899 161 4 parameter parameter NOUN cana-5899 161 5 values value NOUN cana-5899 161 6 and and CCONJ cana-5899 161 7 ensuring ensure VERB cana-5899 161 8 that that SCONJ cana-5899 161 9 the the DET cana-5899 161 10 functional functional ADJ cana-5899 161 11 forms form NOUN cana-5899 161 12 of of ADP cana-5899 161 13 the the DET cana-5899 161 14 nonlinear nonlinear ADJ cana-5899 161 15 terms term NOUN cana-5899 161 16 satisfy satisfy VERB cana-5899 161 17 specific specific ADJ cana-5899 161 18 constraints constraint NOUN cana-5899 161 19 , , PUNCT cana-5899 161 20 the the DET cana-5899 161 21 stability stability NOUN cana-5899 161 22 of of ADP cana-5899 161 23 the the DET cana-5899 161 24 system system NOUN cana-5899 161 25 can can AUX cana-5899 161 26 still still ADV cana-5899 161 27 be be AUX cana-5899 161 28 preserved preserve VERB cana-5899 161 29 . . PUNCT cana-5899 162 1 this this PRON cana-5899 162 2 shows show VERB cana-5899 162 3 that that SCONJ cana-5899 162 4 delays delay NOUN cana-5899 162 5 do do AUX cana-5899 162 6 not not PART cana-5899 162 7 necessarily necessarily ADV cana-5899 162 8 destabilize destabilize VERB cana-5899 162 9 the the DET cana-5899 162 10 system system NOUN cana-5899 162 11 if if SCONJ cana-5899 162 12 the the DET cana-5899 162 13 model model NOUN cana-5899 162 14 is be AUX cana-5899 162 15 constructed construct VERB cana-5899 162 16 carefully carefully ADV cana-5899 162 17 . . PUNCT cana-5899 163 1 in in ADP cana-5899 163 2 the the DET cana-5899 163 3 next next ADJ cana-5899 163 4 section section NOUN cana-5899 163 5 numerical numerical ADJ cana-5899 163 6 examples example NOUN cana-5899 163 7 are be AUX cana-5899 163 8 demonstrated demonstrate VERB cana-5899 163 9 to to PART cana-5899 163 10 show show VERB cana-5899 163 11 the the DET cana-5899 163 12 behavior behavior NOUN cana-5899 163 13 of of ADP cana-5899 163 14 the the DET cana-5899 163 15 model model NOUN cana-5899 163 16 when when SCONJ cana-5899 163 17 the the DET cana-5899 163 18 delay delay NOUN cana-5899 163 19 vary vary VERB cana-5899 163 20 . . PUNCT cana-5899 164 1 4 4 X cana-5899 164 2 . . X cana-5899 164 3 numerical numerical ADJ cana-5899 164 4 examples example NOUN cana-5899 164 5 in in ADP cana-5899 164 6 this this DET cana-5899 164 7 section section NOUN cana-5899 164 8 , , PUNCT cana-5899 164 9 we we PRON cana-5899 164 10 present present VERB cana-5899 164 11 several several ADJ cana-5899 164 12 examples example NOUN cana-5899 164 13 of of ADP cana-5899 164 14 system system NOUN cana-5899 164 15 ( ( PUNCT cana-5899 164 16 1 1 NUM cana-5899 164 17 ) ) PUNCT cana-5899 164 18 by by ADP cana-5899 164 19 fixing fix VERB cana-5899 164 20 the the DET cana-5899 164 21 parameters parameter NOUN cana-5899 164 22 , , PUNCT cana-5899 164 23 functional functional ADJ cana-5899 164 24 forms form NOUN cana-5899 164 25 , , PUNCT cana-5899 164 26 and and CCONJ cana-5899 164 27 the the DET cana-5899 164 28 delays delay NOUN cana-5899 164 29 Ο„ Ο„ PROPN cana-5899 164 30 and and CCONJ cana-5899 164 31 Ξ΄ Ξ΄ PROPN cana-5899 164 32 . . PUNCT cana-5899 165 1 the the DET cana-5899 165 2 delay delay NOUN cana-5899 165 3 ΞΌ ΞΌ PROPN cana-5899 165 4 is be AUX cana-5899 165 5 varied varied ADJ cana-5899 165 6 to to PART cana-5899 165 7 examine examine VERB cana-5899 165 8 how how SCONJ cana-5899 165 9 it it PRON cana-5899 165 10 affects affect VERB cana-5899 165 11 the the DET cana-5899 165 12 behaviour behaviour NOUN cana-5899 165 13 of of ADP cana-5899 165 14 the the DET cana-5899 165 15 model model NOUN cana-5899 165 16 . . PUNCT cana-5899 166 1 the the DET cana-5899 166 2 delay delay NOUN cana-5899 166 3 differential differential ADJ cana-5899 166 4 equations equation NOUN cana-5899 166 5 are be AUX cana-5899 166 6 solved solve VERB cana-5899 166 7 using use VERB cana-5899 166 8 matlab matlab PROPN cana-5899 166 9 ’s ’s PART cana-5899 166 10 dde23 dde23 PROPN cana-5899 166 11 solver solver NOUN cana-5899 166 12 , , PUNCT cana-5899 166 13 and and CCONJ cana-5899 166 14 the the DET cana-5899 166 15 corresponding corresponding ADJ cana-5899 166 16 solutions solution NOUN cana-5899 166 17 are be AUX cana-5899 166 18 plotted plot VERB cana-5899 166 19 to to PART cana-5899 166 20 illustrate illustrate VERB cana-5899 166 21 the the DET cana-5899 166 22 dynamics dynamic NOUN cana-5899 166 23 . . PUNCT cana-5899 167 1 consider consider VERB cana-5899 167 2 the the DET cana-5899 167 3 system system NOUN cana-5899 167 4 𝑒′ 𝑒′ ADP cana-5899 167 5 = = NUM cana-5899 167 6 10 10 NUM cana-5899 167 7 βˆ’ βˆ’ NOUN cana-5899 167 8 3𝑓(𝑒 3𝑓(𝑒 NUM cana-5899 167 9 , , PUNCT cana-5899 167 10 𝑣 𝑣 NOUN cana-5899 167 11 ) ) PUNCT cana-5899 167 12 βˆ’ βˆ’ NOUN cana-5899 167 13 𝑒 𝑒 PART cana-5899 167 14 βˆ’ βˆ’ PROPN cana-5899 167 15 2𝑉(𝑒 2𝑉(𝑒 NUM cana-5899 167 16 ) ) PUNCT cana-5899 168 1 + + NUM cana-5899 168 2 3𝑀(𝑑 3𝑀(𝑑 ADJ cana-5899 168 3 βˆ’ βˆ’ NOUN cana-5899 168 4 πœ‡ πœ‡ NOUN cana-5899 168 5 ) ) PUNCT cana-5899 168 6 𝑣′ 𝑣′ NUM cana-5899 169 1 = = SYM cana-5899 169 2 3𝑓(𝑒(t 3𝑓(𝑒(t NUM cana-5899 169 3 βˆ’ βˆ’ PROPN cana-5899 169 4 Ο„ Ο„ PROPN cana-5899 169 5 ) ) PUNCT cana-5899 169 6 , , PUNCT cana-5899 169 7 𝑣 𝑣 X cana-5899 169 8 ) ) PUNCT cana-5899 169 9 βˆ’ βˆ’ ADP cana-5899 169 10 1.5𝑃(𝑣 1.5𝑃(𝑣 NUM cana-5899 169 11 ) ) PUNCT cana-5899 170 1 βˆ’ βˆ’ PROPN cana-5899 170 2 3.5𝑣 3.5𝑣 NUM cana-5899 170 3 𝑀′ 𝑀′ NOUN cana-5899 170 4 = = SYM cana-5899 171 1 1.5𝑃(𝑣(𝑑 1.5𝑃(𝑣(𝑑 NUM cana-5899 171 2 βˆ’ βˆ’ PROPN cana-5899 171 3 Ξ΄ Ξ΄ PROPN cana-5899 171 4 ) ) PUNCT cana-5899 171 5 ) ) PUNCT cana-5899 172 1 βˆ’ βˆ’ PROPN cana-5899 172 2 3𝑀 3𝑀 NUM cana-5899 172 3 ( ( PUNCT cana-5899 172 4 4 4 NUM cana-5899 172 5 ) ) PUNCT cana-5899 172 6 ( ( PUNCT cana-5899 172 7 i i NOUN cana-5899 172 8 ) ) PUNCT cana-5899 172 9 letting let VERB cana-5899 172 10 the the DET cana-5899 172 11 functional functional ADJ cana-5899 172 12 values value NOUN cana-5899 172 13 to to PART cana-5899 172 14 be be AUX cana-5899 172 15 𝑓(𝑒 𝑓(𝑒 PROPN cana-5899 172 16 , , PUNCT cana-5899 172 17 𝑣 𝑣 X cana-5899 172 18 ) ) PUNCT cana-5899 172 19 = = SYM cana-5899 172 20 𝑒𝑣 𝑒𝑣 X cana-5899 172 21 , , PUNCT cana-5899 172 22 𝑉(𝑒 𝑉(𝑒 NUM cana-5899 172 23 ) ) PUNCT cana-5899 172 24 = = SYM cana-5899 172 25 𝑒 𝑒 NOUN cana-5899 172 26 , , PUNCT cana-5899 172 27 𝑃(𝑣 𝑃(𝑣 X cana-5899 172 28 ) ) PUNCT cana-5899 172 29 = = SYM cana-5899 172 30 𝑣 𝑣 NOUN cana-5899 172 31 and and CCONJ cana-5899 172 32 fixing fix VERB cana-5899 172 33 the the DET cana-5899 172 34 delays delay NOUN cana-5899 172 35 Ο„ Ο„ PROPN cana-5899 173 1 = = NOUN cana-5899 173 2 0.1 0.1 NUM cana-5899 173 3 and and CCONJ cana-5899 173 4 Ξ΄ Ξ΄ X cana-5899 173 5 = = NOUN cana-5899 173 6 0.1 0.1 NUM cana-5899 173 7 the the DET cana-5899 173 8 system system NOUN cana-5899 173 9 ( ( PUNCT cana-5899 173 10 4 4 X cana-5899 173 11 ) ) PUNCT cana-5899 173 12 satisfies satisfy VERB cana-5899 173 13 the the DET cana-5899 173 14 constrains constrain NOUN cana-5899 173 15 of of ADP cana-5899 173 16 theorem theorem ADJ cana-5899 173 17 3.3 3.3 NUM cana-5899 173 18 . . PUNCT cana-5899 174 1 therefore therefore ADV cana-5899 174 2 , , PUNCT cana-5899 174 3 the the DET cana-5899 174 4 system system NOUN cana-5899 174 5 is be AUX cana-5899 174 6 stable stable ADJ cana-5899 174 7 globally globally ADV cana-5899 174 8 at at ADP cana-5899 174 9 equilibrium equilibrium NOUN cana-5899 174 10 point point NOUN cana-5899 174 11 ( ( PUNCT cana-5899 174 12 1.6 1.6 NUM cana-5899 174 13 . . NOUN cana-5899 174 14 2 2 NUM cana-5899 174 15 , , PUNCT cana-5899 174 16 1 1 NUM cana-5899 174 17 ) ) PUNCT cana-5899 174 18 . . PUNCT cana-5899 175 1 the the DET cana-5899 175 2 behaviour behaviour NOUN cana-5899 175 3 of of ADP cana-5899 175 4 the the DET cana-5899 175 5 solution solution NOUN cana-5899 175 6 of of ADP cana-5899 175 7 the the DET cana-5899 175 8 system system NOUN cana-5899 175 9 ( ( PUNCT cana-5899 175 10 4 4 NUM cana-5899 175 11 ) ) PUNCT cana-5899 175 12 for for ADP cana-5899 175 13 various various ADJ cana-5899 175 14 values value NOUN cana-5899 175 15 delay delay VERB cana-5899 175 16 πœ‡ πœ‡ PRON cana-5899 175 17 can can AUX cana-5899 175 18 be be AUX cana-5899 175 19 seen see VERB cana-5899 175 20 in in ADP cana-5899 175 21 figure figure NOUN cana-5899 175 22 1 1 NUM cana-5899 175 23 . . PUNCT cana-5899 176 1 communications communication NOUN cana-5899 176 2 on on ADP cana-5899 176 3 applied apply VERB cana-5899 176 4 nonlinear nonlinear ADJ cana-5899 176 5 analysis analysis NOUN cana-5899 176 6 issn issn NOUN cana-5899 176 7 : : PUNCT cana-5899 176 8 1074 1074 NUM cana-5899 176 9 - - PUNCT cana-5899 176 10 133x 133x NUM cana-5899 176 11 vol vol NOUN cana-5899 176 12 31 31 NUM cana-5899 176 13 no no NOUN cana-5899 176 14 . . PUNCT cana-5899 177 1 8s 8s PROPN cana-5899 177 2 ( ( PUNCT cana-5899 177 3 2024 2024 NUM cana-5899 177 4 ) ) PUNCT cana-5899 177 5 1091 1091 NUM cana-5899 177 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 177 7 figure figure NOUN cana-5899 177 8 1 1 NUM cana-5899 177 9 : : PUNCT cana-5899 177 10 solution solution NOUN cana-5899 177 11 profile profile NOUN cana-5899 177 12 of of ADP cana-5899 177 13 system system NOUN cana-5899 177 14 ( ( PUNCT cana-5899 177 15 1 1 NUM cana-5899 177 16 ) ) PUNCT cana-5899 177 17 for for ADP cana-5899 177 18 various various ADJ cana-5899 177 19 values value NOUN cana-5899 177 20 of of ADP cana-5899 177 21 the the DET cana-5899 177 22 delay delay NOUN cana-5899 177 23 𝝁 𝝁 PRON cana-5899 177 24 , , PUNCT cana-5899 177 25 where where SCONJ cana-5899 177 26 the the DET cana-5899 177 27 functional functional ADJ cana-5899 177 28 values value NOUN cana-5899 177 29 are be AUX cana-5899 177 30 𝒇(𝒖 𝒇(𝒖 PROPN cana-5899 177 31 , , PUNCT cana-5899 177 32 𝒗 𝒗 ADJ cana-5899 177 33 ) ) PUNCT cana-5899 177 34 = = SYM cana-5899 177 35 𝒖𝒗 𝒖𝒗 NOUN cana-5899 177 36 , , PUNCT cana-5899 177 37 𝑽(𝒖 𝑽(𝒖 NUM cana-5899 177 38 ) ) PUNCT cana-5899 177 39 = = SYM cana-5899 177 40 𝒖 𝒖 X cana-5899 177 41 , , PUNCT cana-5899 177 42 𝑷(𝒗 𝑷(𝒗 NUM cana-5899 177 43 ) ) PUNCT cana-5899 177 44 = = SYM cana-5899 177 45 𝒗 𝒗 PROPN cana-5899 177 46 and and CCONJ cana-5899 177 47 fixed fix VERB cana-5899 177 48 delays delay NOUN cana-5899 177 49 𝛕 𝛕 X cana-5899 177 50 = = SYM cana-5899 177 51 0.1 0.1 NUM cana-5899 177 52 and and CCONJ cana-5899 177 53 𝛅 𝛅 X cana-5899 177 54 = = SYM cana-5899 177 55 0.1 0.1 NUM cana-5899 177 56 . . PUNCT cana-5899 178 1 ( ( PUNCT cana-5899 178 2 i i NOUN cana-5899 178 3 ) ) PUNCT cana-5899 178 4 for for ADP cana-5899 178 5 the the DET cana-5899 178 6 functional functional ADJ cana-5899 178 7 values value NOUN cana-5899 178 8 𝑓(𝑒 𝑓(𝑒 ADV cana-5899 178 9 , , PUNCT cana-5899 178 10 𝑣 𝑣 X cana-5899 178 11 ) ) PUNCT cana-5899 178 12 = = SYM cana-5899 178 13 𝑒𝑣 𝑒𝑣 X cana-5899 178 14 , , PUNCT cana-5899 178 15 𝑉(𝑒 𝑉(𝑒 NOUN cana-5899 178 16 ) ) PUNCT cana-5899 178 17 = = PUNCT cana-5899 178 18 𝑒 𝑒 PROPN cana-5899 178 19 𝑒+1 𝑒+1 NUM cana-5899 178 20 , , PUNCT cana-5899 178 21 𝑃(𝑣 𝑃(𝑣 ADP cana-5899 178 22 ) ) PUNCT cana-5899 178 23 = = SYM cana-5899 178 24 tanh tanh NOUN cana-5899 178 25 ( ( PUNCT cana-5899 178 26 𝑣 𝑣 NOUN cana-5899 178 27 ) ) PUNCT cana-5899 178 28 and and CCONJ cana-5899 178 29 fixing fix VERB cana-5899 178 30 the the DET cana-5899 178 31 delays delay NOUN cana-5899 178 32 Ο„ Ο„ X cana-5899 179 1 = = NOUN cana-5899 179 2 0.13 0.13 NUM cana-5899 179 3 and and CCONJ cana-5899 179 4 Ξ΄ Ξ΄ PROPN cana-5899 179 5 = = NOUN cana-5899 179 6 0.13 0.13 NUM cana-5899 179 7 the the DET cana-5899 179 8 system system NOUN cana-5899 179 9 ( ( PUNCT cana-5899 179 10 4 4 X cana-5899 179 11 ) ) PUNCT cana-5899 179 12 satisfies satisfy VERB cana-5899 179 13 global global ADJ cana-5899 179 14 stability stability NOUN cana-5899 179 15 conditions condition NOUN cana-5899 179 16 of of ADP cana-5899 179 17 theorem theorem ADJ cana-5899 179 18 3.3 3.3 NUM cana-5899 179 19 . . PUNCT cana-5899 180 1 thus thus ADV cana-5899 180 2 the the DET cana-5899 180 3 solutions solution NOUN cana-5899 180 4 will will AUX cana-5899 180 5 reach reach VERB cana-5899 180 6 equilibrium equilibrium NOUN cana-5899 180 7 point point NOUN cana-5899 180 8 ( ( PUNCT cana-5899 180 9 1.4 1.4 NUM cana-5899 180 10 , , PUNCT cana-5899 180 11 2.9 2.9 NUM cana-5899 180 12 , , PUNCT cana-5899 180 13 0.5 0.5 NUM cana-5899 180 14 ) ) PUNCT cana-5899 180 15 . . PUNCT cana-5899 181 1 the the DET cana-5899 181 2 behaviour behaviour NOUN cana-5899 181 3 of of ADP cana-5899 181 4 the the DET cana-5899 181 5 solution solution NOUN cana-5899 181 6 of of ADP cana-5899 181 7 the the DET cana-5899 181 8 system system NOUN cana-5899 181 9 ( ( PUNCT cana-5899 181 10 4 4 NUM cana-5899 181 11 ) ) PUNCT cana-5899 181 12 for for ADP cana-5899 181 13 different different ADJ cana-5899 181 14 values value NOUN cana-5899 181 15 of of ADP cana-5899 181 16 the the DET cana-5899 181 17 delay delay NOUN cana-5899 181 18 πœ‡ πœ‡ PRON cana-5899 181 19 can can AUX cana-5899 181 20 be be AUX cana-5899 181 21 seen see VERB cana-5899 181 22 in in ADP cana-5899 181 23 figure figure NOUN cana-5899 181 24 2 2 NUM cana-5899 181 25 . . PUNCT cana-5899 181 26 figure figure NOUN cana-5899 181 27 2 2 NUM cana-5899 181 28 : : PUNCT cana-5899 181 29 solution solution NOUN cana-5899 181 30 profile profile NOUN cana-5899 181 31 of of ADP cana-5899 181 32 system system NOUN cana-5899 181 33 ( ( PUNCT cana-5899 181 34 1 1 NUM cana-5899 181 35 ) ) PUNCT cana-5899 181 36 for for ADP cana-5899 181 37 various various ADJ cana-5899 181 38 values value NOUN cana-5899 181 39 of of ADP cana-5899 181 40 the the DET cana-5899 181 41 delay delay NOUN cana-5899 181 42 ΞΌ ΞΌ PROPN cana-5899 181 43 , , PUNCT cana-5899 181 44 where where SCONJ cana-5899 181 45 the the DET cana-5899 181 46 functional functional ADJ cana-5899 181 47 values value NOUN cana-5899 181 48 are be AUX cana-5899 181 49 𝒇(𝒖 𝒇(𝒖 PROPN cana-5899 181 50 , , PUNCT cana-5899 181 51 𝒗 𝒗 ADJ cana-5899 181 52 ) ) PUNCT cana-5899 181 53 = = SYM cana-5899 181 54 𝒖𝒗 𝒖𝒗 NOUN cana-5899 181 55 , , PUNCT cana-5899 181 56 v(u)= v(u)= PROPN cana-5899 181 57 𝒖 𝒖 X cana-5899 181 58 𝒖+𝟏 𝒖+𝟏 X cana-5899 181 59 , , PUNCT cana-5899 181 60 𝑷(𝒗 𝑷(𝒗 NUM cana-5899 181 61 ) ) PUNCT cana-5899 181 62 = = PUNCT cana-5899 181 63 𝒕𝒂𝒉(𝒗 𝒕𝒂𝒉(𝒗 X cana-5899 181 64 ) ) PUNCT cana-5899 181 65 and and CCONJ cana-5899 181 66 fixed fix VERB cana-5899 181 67 delays delay NOUN cana-5899 181 68 Ο„ Ο„ X cana-5899 182 1 = = NOUN cana-5899 182 2 0.13 0.13 NUM cana-5899 182 3 and and CCONJ cana-5899 182 4 Ξ΄ Ξ΄ NOUN cana-5899 182 5 = = NOUN cana-5899 182 6 0.13 0.13 NUM cana-5899 182 7 . . PUNCT cana-5899 183 1 communications communication NOUN cana-5899 183 2 on on ADP cana-5899 183 3 applied apply VERB cana-5899 183 4 nonlinear nonlinear ADJ cana-5899 183 5 analysis analysis NOUN cana-5899 183 6 issn issn NOUN cana-5899 183 7 : : PUNCT cana-5899 183 8 1074 1074 NUM cana-5899 183 9 - - PUNCT cana-5899 183 10 133x 133x NUM cana-5899 183 11 vol vol NOUN cana-5899 183 12 31 31 NUM cana-5899 183 13 no no NOUN cana-5899 183 14 . . PUNCT cana-5899 184 1 8s 8s PROPN cana-5899 184 2 ( ( PUNCT cana-5899 184 3 2024 2024 NUM cana-5899 184 4 ) ) PUNCT cana-5899 184 5 1092 1092 NUM cana-5899 185 1 https://internationalpubls.com https://internationalpubls.com X cana-5899 185 2 remark remark VERB cana-5899 185 3 4.1 4.1 NUM cana-5899 185 4 : : PUNCT cana-5899 185 5 from from ADP cana-5899 185 6 figures figure NOUN cana-5899 185 7 1 1 NUM cana-5899 185 8 and and CCONJ cana-5899 185 9 2 2 NUM cana-5899 185 10 , , PUNCT cana-5899 185 11 we we PRON cana-5899 185 12 can can AUX cana-5899 185 13 see see VERB cana-5899 185 14 that that SCONJ cana-5899 185 15 when when SCONJ cana-5899 185 16 the the DET cana-5899 185 17 delay delay NOUN cana-5899 185 18 ΞΌ\muΞΌ ΞΌ\muΞΌ AUX cana-5899 185 19 becomes become VERB cana-5899 185 20 larger large ADJ cana-5899 185 21 , , PUNCT cana-5899 185 22 the the DET cana-5899 185 23 system system NOUN cana-5899 185 24 takes take VERB cana-5899 185 25 longer long ADV cana-5899 185 26 to to PART cana-5899 185 27 settle settle VERB cana-5899 185 28 down down ADP cana-5899 185 29 to to ADP cana-5899 185 30 a a DET cana-5899 185 31 steady steady ADJ cana-5899 185 32 state state NOUN cana-5899 185 33 . . PUNCT cana-5899 186 1 this this PRON cana-5899 186 2 means mean VERB cana-5899 186 3 the the DET cana-5899 186 4 presence presence NOUN cana-5899 186 5 of of ADP cana-5899 186 6 delay delay NOUN cana-5899 186 7 slows slow VERB cana-5899 186 8 down down ADP cana-5899 186 9 how how SCONJ cana-5899 186 10 quickly quickly ADV cana-5899 186 11 the the DET cana-5899 186 12 system system NOUN cana-5899 186 13 reaches reach VERB cana-5899 186 14 stability stability NOUN cana-5899 186 15 . . PUNCT cana-5899 187 1 however however ADV cana-5899 187 2 , , PUNCT cana-5899 187 3 since since SCONJ cana-5899 187 4 the the DET cana-5899 187 5 equations equation NOUN cana-5899 187 6 and and CCONJ cana-5899 187 7 parameters parameter NOUN cana-5899 187 8 used use VERB cana-5899 187 9 in in ADP cana-5899 187 10 the the DET cana-5899 187 11 model model NOUN cana-5899 187 12 are be AUX cana-5899 187 13 chosen choose VERB cana-5899 187 14 carefully carefully ADV cana-5899 187 15 to to PART cana-5899 187 16 meet meet VERB cana-5899 187 17 specific specific ADJ cana-5899 187 18 conditions condition NOUN cana-5899 187 19 , , PUNCT cana-5899 187 20 the the DET cana-5899 187 21 system system NOUN cana-5899 187 22 still still ADV cana-5899 187 23 behaves behave VERB cana-5899 187 24 in in ADP cana-5899 187 25 a a DET cana-5899 187 26 controlled control VERB cana-5899 187 27 manner manner NOUN cana-5899 187 28 . . PUNCT cana-5899 188 1 even even ADV cana-5899 188 2 with with ADP cana-5899 188 3 increasing increase VERB cana-5899 188 4 delay delay NOUN cana-5899 188 5 , , PUNCT cana-5899 188 6 the the DET cana-5899 188 7 system system NOUN cana-5899 188 8 does do AUX cana-5899 188 9 n’t not PART cana-5899 188 10 drift drift VERB cana-5899 188 11 too too ADV cana-5899 188 12 far far ADV cana-5899 188 13 or or CCONJ cana-5899 188 14 become become VERB cana-5899 188 15 unstable unstable ADJ cana-5899 188 16 . . PUNCT cana-5899 189 1 it it PRON cana-5899 189 2 may may AUX cana-5899 189 3 take take VERB cana-5899 189 4 more more ADJ cana-5899 189 5 time time NOUN cana-5899 189 6 , , PUNCT cana-5899 189 7 but but CCONJ cana-5899 189 8 it it PRON cana-5899 189 9 eventually eventually ADV cana-5899 189 10 moves move VERB cana-5899 189 11 toward toward ADP cana-5899 189 12 the the DET cana-5899 189 13 stable stable ADJ cana-5899 189 14 point point NOUN cana-5899 189 15 without without ADP cana-5899 189 16 showing show VERB cana-5899 189 17 large large ADJ cana-5899 189 18 fluctuations fluctuation NOUN cana-5899 189 19 or or CCONJ cana-5899 189 20 unexpected unexpected ADJ cana-5899 189 21 changes change NOUN cana-5899 189 22 . . PUNCT cana-5899 190 1 this this PRON cana-5899 190 2 shows show VERB cana-5899 190 3 that that SCONJ cana-5899 190 4 the the DET cana-5899 190 5 model model NOUN cana-5899 190 6 is be AUX cana-5899 190 7 stable stable ADJ cana-5899 190 8 and and CCONJ cana-5899 190 9 can can AUX cana-5899 190 10 handle handle VERB cana-5899 190 11 delays delay NOUN cana-5899 190 12 up up ADP cana-5899 190 13 to to ADP cana-5899 190 14 a a DET cana-5899 190 15 certain certain ADJ cana-5899 190 16 extent extent NOUN cana-5899 190 17 without without ADP cana-5899 190 18 losing lose VERB cana-5899 190 19 its its PRON cana-5899 190 20 overall overall ADJ cana-5899 190 21 balance balance NOUN cana-5899 190 22 . . PUNCT cana-5899 191 1 5 5 X cana-5899 191 2 . . X cana-5899 191 3 discussion discussion NOUN cana-5899 191 4 this this DET cana-5899 191 5 study study NOUN cana-5899 191 6 highlights highlight VERB cana-5899 191 7 the the DET cana-5899 191 8 effect effect NOUN cana-5899 191 9 of of ADP cana-5899 191 10 immunity immunity NOUN cana-5899 191 11 loss loss NOUN cana-5899 191 12 over over ADP cana-5899 191 13 time time NOUN cana-5899 191 14 on on ADP cana-5899 191 15 the the DET cana-5899 191 16 long long ADJ cana-5899 191 17 - - PUNCT cana-5899 191 18 term term NOUN cana-5899 191 19 behaviour behaviour NOUN cana-5899 191 20 of of ADP cana-5899 191 21 an an DET cana-5899 191 22 sir sir NOUN cana-5899 191 23 model model NOUN cana-5899 191 24 that that PRON cana-5899 191 25 includes include VERB cana-5899 191 26 both both DET cana-5899 191 27 vaccination vaccination NOUN cana-5899 191 28 and and CCONJ cana-5899 191 29 treatment treatment NOUN cana-5899 191 30 . . PUNCT cana-5899 192 1 by by ADP cana-5899 192 2 introducing introduce VERB cana-5899 192 3 a a DET cana-5899 192 4 delay delay NOUN cana-5899 192 5 in in ADP cana-5899 192 6 the the DET cana-5899 192 7 return return NOUN cana-5899 192 8 of of ADP cana-5899 192 9 recovered recover VERB cana-5899 192 10 individuals individual NOUN cana-5899 192 11 to to ADP cana-5899 192 12 the the DET cana-5899 192 13 susceptible susceptible ADJ cana-5899 192 14 group group NOUN cana-5899 192 15 , , PUNCT cana-5899 192 16 the the DET cana-5899 192 17 model model NOUN cana-5899 192 18 reflects reflect VERB cana-5899 192 19 a a DET cana-5899 192 20 more more ADV cana-5899 192 21 realistic realistic ADJ cana-5899 192 22 scenario scenario NOUN cana-5899 192 23 where where SCONJ cana-5899 192 24 immunity immunity NOUN cana-5899 192 25 is be AUX cana-5899 192 26 not not PART cana-5899 192 27 permanent permanent ADJ cana-5899 192 28 . . PUNCT cana-5899 193 1 the the DET cana-5899 193 2 analysis analysis NOUN cana-5899 193 3 shows show VERB cana-5899 193 4 that that SCONJ cana-5899 193 5 , , PUNCT cana-5899 193 6 despite despite SCONJ cana-5899 193 7 this this DET cana-5899 193 8 delay delay NOUN cana-5899 193 9 , , PUNCT cana-5899 193 10 the the DET cana-5899 193 11 system system NOUN cana-5899 193 12 can can AUX cana-5899 193 13 remain remain VERB cana-5899 193 14 stable stable ADJ cana-5899 193 15 under under ADP cana-5899 193 16 the the DET cana-5899 193 17 right right ADJ cana-5899 193 18 conditions condition NOUN cana-5899 193 19 . . PUNCT cana-5899 194 1 both both DET cana-5899 194 2 local local ADJ cana-5899 194 3 and and CCONJ cana-5899 194 4 global global ADJ cana-5899 194 5 stability stability NOUN cana-5899 194 6 are be AUX cana-5899 194 7 preserved preserve VERB cana-5899 194 8 when when SCONJ cana-5899 194 9 the the DET cana-5899 194 10 parameters parameter NOUN cana-5899 194 11 are be AUX cana-5899 194 12 chosen choose VERB cana-5899 194 13 carefully carefully ADV cana-5899 194 14 . . PUNCT cana-5899 195 1 numerical numerical PROPN cana-5899 195 2 results result NOUN cana-5899 195 3 support support VERB cana-5899 195 4 this this PRON cana-5899 195 5 by by ADP cana-5899 195 6 showing show VERB cana-5899 195 7 that that SCONJ cana-5899 195 8 while while SCONJ cana-5899 195 9 the the DET cana-5899 195 10 delay delay NOUN cana-5899 195 11 may may AUX cana-5899 195 12 slow slow VERB cana-5899 195 13 down down ADP cana-5899 195 14 the the DET cana-5899 195 15 path path NOUN cana-5899 195 16 to to ADP cana-5899 195 17 equilibrium equilibrium NOUN cana-5899 195 18 , , PUNCT cana-5899 195 19 it it PRON cana-5899 195 20 does do AUX cana-5899 195 21 not not PART cana-5899 195 22 disturb disturb VERB cana-5899 195 23 the the DET cana-5899 195 24 system system NOUN cana-5899 195 25 ’s ’s PART cana-5899 195 26 overall overall ADJ cana-5899 195 27 stability stability NOUN cana-5899 195 28 . . PUNCT cana-5899 196 1 these these DET cana-5899 196 2 results result NOUN cana-5899 196 3 underline underline VERB cana-5899 196 4 the the DET cana-5899 196 5 importance importance NOUN cana-5899 196 6 of of ADP cana-5899 196 7 accounting account VERB cana-5899 196 8 for for ADP cana-5899 196 9 delays delay NOUN cana-5899 196 10 when when SCONJ cana-5899 196 11 modeling modeling NOUN cana-5899 196 12 diseases disease NOUN cana-5899 196 13 where where SCONJ cana-5899 196 14 immunity immunity NOUN cana-5899 196 15 may may AUX cana-5899 196 16 fade fade VERB cana-5899 196 17 or or CCONJ cana-5899 196 18 vaccination vaccination NOUN cana-5899 196 19 is be AUX cana-5899 196 20 not not PART cana-5899 196 21 fully fully ADV cana-5899 196 22 effective effective ADJ cana-5899 196 23 . . PUNCT cana-5899 197 1 with with ADP cana-5899 197 2 appropriate appropriate ADJ cana-5899 197 3 assumptions assumption NOUN cana-5899 197 4 , , PUNCT cana-5899 197 5 such such ADJ cana-5899 197 6 models model NOUN cana-5899 197 7 can can AUX cana-5899 197 8 still still ADV cana-5899 197 9 offer offer VERB cana-5899 197 10 dependable dependable ADJ cana-5899 197 11 insights insight NOUN cana-5899 197 12 into into ADP cana-5899 197 13 how how SCONJ cana-5899 197 14 a a DET cana-5899 197 15 disease disease NOUN cana-5899 197 16 might might AUX cana-5899 197 17 behave behave VERB cana-5899 197 18 over over ADP cana-5899 197 19 time time NOUN cana-5899 197 20 . . PUNCT cana-5899 198 1 references reference NOUN cana-5899 198 2 [ [ X cana-5899 198 3 1 1 X cana-5899 198 4 ] ] PUNCT cana-5899 198 5 raja raja PROPN cana-5899 198 6 sekhara sekhara PROPN cana-5899 198 7 rao rao PROPN cana-5899 198 8 , , PUNCT cana-5899 198 9 p. p. PROPN cana-5899 198 10 , , PUNCT cana-5899 198 11 & & CCONJ cana-5899 198 12 naresh naresh PROPN cana-5899 198 13 kumar kumar PROPN cana-5899 198 14 , , PUNCT cana-5899 198 15 m. m. NOUN cana-5899 198 16 ( ( PUNCT cana-5899 198 17 2015 2015 NUM cana-5899 198 18 ) ) PUNCT cana-5899 198 19 . . PUNCT cana-5899 199 1 a a DET cana-5899 199 2 dynamic dynamic ADJ cana-5899 199 3 model model NOUN cana-5899 199 4 for for ADP cana-5899 199 5 infectious infectious ADJ cana-5899 199 6 diseases disease NOUN cana-5899 199 7 : : PUNCT cana-5899 199 8 the the DET cana-5899 199 9 role role NOUN cana-5899 199 10 of of ADP cana-5899 199 11 vaccination vaccination NOUN cana-5899 199 12 and and CCONJ cana-5899 199 13 treatment treatment NOUN cana-5899 199 14 . . PUNCT cana-5899 200 1 chaos chaos NOUN cana-5899 200 2 , , PUNCT cana-5899 200 3 solitons soliton NOUN cana-5899 200 4 , , PUNCT cana-5899 200 5 and and CCONJ cana-5899 200 6 fractals fractal NOUN cana-5899 200 7 , , PUNCT cana-5899 200 8 75 75 NUM cana-5899 200 9 , , PUNCT cana-5899 200 10 34–49 34–49 NUM cana-5899 200 11 . . PUNCT cana-5899 201 1 https://doi.org/10.1016/j.chaos.2015.02.004 https://doi.org/10.1016/j.chaos.2015.02.004 PROPN cana-5899 201 2 [ [ X cana-5899 201 3 2 2 NUM cana-5899 201 4 ] ] X cana-5899 201 5 g. g. PROPN cana-5899 201 6 shailaja shailaja PROPN cana-5899 201 7 , , PUNCT cana-5899 201 8 dr dr PROPN cana-5899 201 9 . . PROPN cana-5899 201 10 m.a m.a PROPN cana-5899 201 11 . . PROPN cana-5899 201 12 srinivas srinivas PROPN cana-5899 201 13 . . PUNCT cana-5899 202 1 ( ( PUNCT cana-5899 202 2 2024 2024 NUM cana-5899 202 3 ) ) PUNCT cana-5899 202 4 . . PUNCT cana-5899 203 1 control control NOUN cana-5899 203 2 of of ADP cana-5899 203 3 disease disease NOUN cana-5899 203 4 dynamics dynamic NOUN cana-5899 203 5 under under ADP cana-5899 203 6 time time NOUN cana-5899 203 7 varying vary VERB cana-5899 203 8 vaccination vaccination NOUN cana-5899 203 9 and and CCONJ cana-5899 203 10 treatment treatment NOUN cana-5899 203 11 rates rate NOUN cana-5899 203 12 . . PUNCT cana-5899 204 1 journal journal NOUN cana-5899 204 2 of of ADP cana-5899 204 3 computational computational ADJ cana-5899 204 4 analysis analysis NOUN cana-5899 204 5 and and CCONJ cana-5899 204 6 applications application NOUN cana-5899 204 7 ( ( PUNCT cana-5899 204 8 jocaaa jocaaa NOUN cana-5899 204 9 ) ) PUNCT cana-5899 204 10 , , PUNCT cana-5899 204 11 33(2),963–971 33(2),963–971 PROPN cana-5899 204 12 . . PUNCT cana-5899 204 13 https://eudoxuspress.com/index.php/pub/article/view/1586 https://eudoxuspress.com/index.php/pub/article/view/1586 X cana-5899 205 1 [ [ X cana-5899 205 2 3 3 X cana-5899 205 3 ] ] X cana-5899 205 4 g. g. PROPN cana-5899 205 5 shailaja shailaja PROPN cana-5899 205 6 , , PUNCT cana-5899 205 7 dr dr PROPN cana-5899 205 8 . . PROPN cana-5899 205 9 m.a m.a PROPN cana-5899 205 10 . . PROPN cana-5899 205 11 srinivas srinivas PROPN cana-5899 205 12 . . PUNCT cana-5899 206 1 ( ( PUNCT cana-5899 206 2 2024 2024 NUM cana-5899 206 3 ) ) PUNCT cana-5899 206 4 . . PUNCT cana-5899 207 1 global global ADJ cana-5899 207 2 stability stability NOUN cana-5899 207 3 of of ADP cana-5899 207 4 an an DET cana-5899 207 5 sir sir NOUN cana-5899 207 6 model model NOUN cana-5899 207 7 characterized characterize VERB cana-5899 207 8 by by ADP cana-5899 207 9 vaccination vaccination NOUN cana-5899 207 10 and and CCONJ cana-5899 207 11 treatment treatment NOUN cana-5899 207 12 . . PUNCT cana-5899 208 1 journal journal PROPN cana-5899 208 2 of of ADP cana-5899 208 3 nonlinear nonlinear ADJ cana-5899 208 4 modeling modeling NOUN cana-5899 208 5 and and CCONJ cana-5899 208 6 analysis analysis NOUN cana-5899 208 7 , , PUNCT cana-5899 208 8 6(4 6(4 NUM cana-5899 208 9 ) ) PUNCT cana-5899 208 10 , , PUNCT cana-5899 208 11 10461063 10461063 NUM cana-5899 208 12 . . PUNCT cana-5899 209 1 https://doi.org/10.12150/jnma.2024.1046 https://doi.org/10.12150/jnma.2024.1046 PROPN cana-5899 210 1 [ [ X cana-5899 210 2 4 4 NUM cana-5899 210 3 ] ] X cana-5899 210 4 pathak pathak PROPN cana-5899 210 5 , , PUNCT cana-5899 210 6 s. s. PROPN cana-5899 210 7 , , PUNCT cana-5899 210 8 kota kota PROPN cana-5899 210 9 , , PUNCT cana-5899 210 10 v.r v.r PROPN cana-5899 210 11 . . PROPN cana-5899 210 12 ( ( PUNCT cana-5899 210 13 2025 2025 NUM cana-5899 210 14 ) ) PUNCT cana-5899 210 15 . . PUNCT cana-5899 211 1 an an DET cana-5899 211 2 influential influential ADJ cana-5899 211 3 study study NOUN cana-5899 211 4 of of ADP cana-5899 211 5 a a DET cana-5899 211 6 time time NOUN cana-5899 211 7 - - PUNCT cana-5899 211 8 delayed delay VERB cana-5899 211 9 epidemic epidemic NOUN cana-5899 211 10 model model NOUN cana-5899 211 11 incorporating incorporate VERB cana-5899 211 12 vaccination vaccination NOUN cana-5899 211 13 and and CCONJ cana-5899 211 14 treatment treatment NOUN cana-5899 211 15 interventions intervention NOUN cana-5899 211 16 . . PUNCT cana-5899 212 1 adv adv PROPN cana-5899 212 2 cont cont PROPN cana-5899 212 3 discr discr PROPN cana-5899 212 4 mod mod PROPN cana-5899 212 5 2025 2025 NUM cana-5899 212 6 , , PUNCT cana-5899 212 7 57 57 NUM cana-5899 212 8 ( ( PUNCT cana-5899 212 9 2025 2025 NUM cana-5899 212 10 ) ) PUNCT cana-5899 212 11 . . PUNCT cana-5899 213 1 https://doi.org/10.1186/s13662-025-03920-0 https://doi.org/10.1186/s13662-025-03920-0 NUM cana-5899 214 1 [ [ X cana-5899 214 2 5 5 NUM cana-5899 214 3 ] ] PUNCT cana-5899 214 4 monalisa monalisa PROPN cana-5899 214 5 anand anand PROPN cana-5899 214 6 , , PUNCT cana-5899 214 7 p. p. PROPN cana-5899 214 8 danumjaya danumjaya PROPN cana-5899 214 9 , , PUNCT cana-5899 214 10 p. p. PROPN cana-5899 214 11 raja raja PROPN cana-5899 214 12 sekhara sekhara PROPN cana-5899 214 13 rao rao PROPN cana-5899 214 14 . . PUNCT cana-5899 215 1 a a DET cana-5899 215 2 nonlinear nonlinear ADJ cana-5899 215 3 mathematical mathematical ADJ cana-5899 215 4 model model NOUN cana-5899 215 5 on on ADP cana-5899 215 6 the the DET cana-5899 215 7 covid-19 covid-19 PROPN cana-5899 215 8 transmission transmission NOUN cana-5899 215 9 pattern pattern NOUN cana-5899 215 10 among among ADP cana-5899 215 11 diabetic diabetic ADJ cana-5899 215 12 and and CCONJ cana-5899 215 13 non non ADJ cana-5899 215 14 - - ADJ cana-5899 215 15 diabetic diabetic ADJ cana-5899 215 16 population population NOUN cana-5899 215 17 . . PUNCT cana-5899 216 1 mathematics mathematic NOUN cana-5899 216 2 and and CCONJ cana-5899 216 3 computers computer NOUN cana-5899 216 4 in in ADP cana-5899 216 5 simulation simulation NOUN cana-5899 216 6 210 210 NUM cana-5899 216 7 , , PUNCT cana-5899 216 8 346 346 NUM cana-5899 216 9 - - SYM cana-5899 216 10 369(2023 369(2023 NUM cana-5899 216 11 ) ) PUNCT cana-5899 216 12 . . PUNCT cana-5899 217 1 https://doi.org/10.1016/j.matcom.2023.03.016 https://doi.org/10.1016/j.matcom.2023.03.016 PRON cana-5899 217 2 [ [ PUNCT cana-5899 217 3 6 6 NUM cana-5899 217 4 ] ] X cana-5899 217 5 diana diana PROPN cana-5899 217 6 , , PUNCT cana-5899 217 7 b. b. PROPN cana-5899 217 8 , , PUNCT cana-5899 217 9 shirali shirali PROPN cana-5899 217 10 , , PUNCT cana-5899 217 11 k. k. PROPN cana-5899 217 12 , , PUNCT cana-5899 217 13 ardak ardak PROPN cana-5899 217 14 , , PUNCT cana-5899 217 15 k.mathematical k.mathematical ADJ cana-5899 217 16 modeling modeling NOUN cana-5899 217 17 of of ADP cana-5899 217 18 infectious infectious ADJ cana-5899 217 19 diseases disease NOUN cana-5899 217 20 and and CCONJ cana-5899 217 21 the the DET cana-5899 217 22 impact impact NOUN cana-5899 217 23 of of ADP cana-5899 217 24 vaccination vaccination NOUN cana-5899 217 25 strategies strategy NOUN cana-5899 217 26 , , PUNCT cana-5899 217 27 math math PROPN cana-5899 217 28 biosci biosci PROPN cana-5899 217 29 eng eng PROPN cana-5899 217 30 . . PROPN cana-5899 218 1 2024 2024 NUM cana-5899 218 2 sep sep PROPN cana-5899 218 3 19;21(9):7103 19;21(9):7103 NUM cana-5899 218 4 - - SYM cana-5899 218 5 7123 7123 NUM cana-5899 218 6 : : PUNCT cana-5899 218 7 doi:10.3934 doi:10.3934 NOUN cana-5899 218 8 / / SYM cana-5899 218 9 mbe.2024314 mbe.2024314 PROPN cana-5899 218 10 https://doi.org/10.1016/j.chaos.2015.02.004 https://doi.org/10.1016/j.chaos.2015.02.004 PROPN cana-5899 218 11 https://eudoxuspress.com/index.php/pub/article/view/1586 https://eudoxuspress.com/index.php/pub/article/view/1586 X cana-5899 218 12 https://doi.org/10.12150/jnma.2024.1046 https://doi.org/10.12150/jnma.2024.1046 PROPN cana-5899 218 13 https://doi.org/10.1186/s13662-025-03920-0 https://doi.org/10.1186/s13662-025-03920-0 NUM cana-5899 218 14 https://doi.org/10.1016/j.matcom.2023.03.016 https://doi.org/10.1016/j.matcom.2023.03.016 ADJ cana-5899 218 15 communications communication NOUN cana-5899 218 16 on on ADP cana-5899 218 17 applied apply VERB cana-5899 218 18 nonlinear nonlinear ADJ cana-5899 218 19 analysis analysis NOUN cana-5899 218 20 issn issn NOUN cana-5899 218 21 : : PUNCT cana-5899 218 22 1074 1074 NUM cana-5899 218 23 - - PUNCT cana-5899 218 24 133x 133x NUM cana-5899 218 25 vol vol NOUN cana-5899 218 26 31 31 NUM cana-5899 218 27 no no NOUN cana-5899 218 28 . . PUNCT cana-5899 219 1 8s 8s PROPN cana-5899 219 2 ( ( PUNCT cana-5899 219 3 2024 2024 NUM cana-5899 219 4 ) ) PUNCT cana-5899 219 5 1093 1093 NUM cana-5899 219 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 220 1 [ [ X cana-5899 220 2 7 7 NUM cana-5899 220 3 ] ] PUNCT cana-5899 220 4 a. a. NOUN cana-5899 220 5 huppert huppert PROPN cana-5899 220 6 , , PUNCT cana-5899 220 7 g. g. PROPN cana-5899 220 8 katriel katriel PROPN cana-5899 220 9 - - PUNCT cana-5899 220 10 mathematical mathematical ADJ cana-5899 220 11 modelling modelling NOUN cana-5899 220 12 and and CCONJ cana-5899 220 13 prediction prediction NOUN cana-5899 220 14 in in ADP cana-5899 220 15 infectious infectious ADJ cana-5899 220 16 disease disease NOUN cana-5899 220 17 epidemiology epidemiology NOUN cana-5899 220 18 . . PUNCT cana-5899 221 1 https://doi.org/10.1111/1469-0691.12308 https://doi.org/10.1111/1469-0691.12308 PROPN cana-5899 222 1 [ [ X cana-5899 222 2 8 8 NUM cana-5899 222 3 ] ] X cana-5899 222 4 sousa sousa NOUN cana-5899 222 5 , , PUNCT cana-5899 222 6 g.a g.a PROPN cana-5899 222 7 . . PROPN cana-5899 222 8 , , PUNCT cana-5899 222 9 souza souza PROPN cana-5899 222 10 , , PUNCT cana-5899 222 11 d.l.m d.l.m NOUN cana-5899 222 12 . . PROPN cana-5899 222 13 , , PUNCT cana-5899 222 14 gabrick gabrick PROPN cana-5899 222 15 , , PUNCT cana-5899 222 16 e.c e.c PROPN cana-5899 222 17 . . PROPN cana-5899 222 18 et et PROPN cana-5899 222 19 al al PROPN cana-5899 222 20 . . PROPN cana-5899 222 21 continuous continuous ADJ cana-5899 222 22 and and CCONJ cana-5899 222 23 discrete discrete ADJ cana-5899 222 24 compartmental compartmental NOUN cana-5899 222 25 models model NOUN cana-5899 222 26 for for ADP cana-5899 222 27 infectious infectious ADJ cana-5899 222 28 disease disease NOUN cana-5899 222 29 . . PUNCT cana-5899 223 1 braz braz PROPN cana-5899 223 2 j j PROPN cana-5899 223 3 phys phy NOUN cana-5899 223 4 55 55 NUM cana-5899 223 5 , , PUNCT cana-5899 223 6 137 137 NUM cana-5899 223 7 ( ( PUNCT cana-5899 223 8 2025 2025 NUM cana-5899 223 9 ) ) PUNCT cana-5899 223 10 . . PUNCT cana-5899 224 1 [ [ X cana-5899 224 2 9 9 NUM cana-5899 224 3 ] ] PUNCT cana-5899 224 4 adebayo adebayo PROPN cana-5899 224 5 , , PUNCT cana-5899 224 6 m. m. NOUN cana-5899 224 7 deepfakes deepfake NOUN cana-5899 224 8 and and CCONJ cana-5899 224 9 data datum NOUN cana-5899 224 10 privacy privacy NOUN cana-5899 224 11 : : PUNCT cana-5899 224 12 navigating navigate VERB cana-5899 224 13 the the DET cana-5899 224 14 risks risk NOUN cana-5899 224 15 in in ADP cana-5899 224 16 the the DET cana-5899 224 17 age age NOUN cana-5899 224 18 of of ADP cana-5899 224 19 ai ai PROPN cana-5899 224 20 . . PROPN cana-5899 224 21 ndpc ndpc PROPN cana-5899 224 22 – – PUNCT cana-5899 224 23 , , PUNCT cana-5899 224 24 106 106 NUM cana-5899 224 25 . . PUNCT cana-5899 225 1 [ [ X cana-5899 225 2 10 10 NUM cana-5899 225 3 ] ] X cana-5899 225 4 g. g. PROPN cana-5899 225 5 zaman zaman PROPN cana-5899 225 6 , , PUNCT cana-5899 225 7 i.h i.h PROPN cana-5899 225 8 . . PROPN cana-5899 225 9 jung jung PROPN cana-5899 225 10 , , PUNCT cana-5899 225 11 d.f.m d.f.m PROPN cana-5899 225 12 . . PROPN cana-5899 225 13 torres torre NOUN cana-5899 225 14 and and CCONJ cana-5899 225 15 a. a. NOUN cana-5899 225 16 zeb zeb NUM cana-5899 225 17 -mathematical -mathematical ADJ cana-5899 225 18 modeling modeling NOUN cana-5899 225 19 and and CCONJ cana-5899 225 20 control control NOUN cana-5899 225 21 of of ADP cana-5899 225 22 infectious infectious ADJ cana-5899 225 23 diseases disease NOUN cana-5899 225 24 , , PUNCT cana-5899 225 25 computational computational ADJ cana-5899 225 26 and and CCONJ cana-5899 225 27 mathematical mathematical ADJ cana-5899 225 28 methods method NOUN cana-5899 225 29 in in ADP cana-5899 225 30 medicine medicine NOUN cana-5899 225 31 , , PUNCT cana-5899 225 32 vol vol NOUN cana-5899 225 33 . . PROPN cana-5899 225 34 2017 2017 NUM cana-5899 225 35 , , PUNCT cana-5899 225 36 pp pp ADJ cana-5899 225 37 . . PUNCT cana-5899 226 1 1 1 NUM cana-5899 226 2 - - SYM cana-5899 226 3 1 1 NUM cana-5899 226 4 , , PUNCT cana-5899 226 5 12 12 NUM cana-5899 226 6 2017 2017 NUM cana-5899 226 7 . . PUNCT cana-5899 227 1 doi:10.1155/2017/7149154 doi:10.1155/2017/7149154 NOUN cana-5899 227 2 [ [ PUNCT cana-5899 227 3 11 11 NUM cana-5899 227 4 ] ] X cana-5899 227 5 simran simran PROPN cana-5899 227 6 ghosh ghosh PROPN cana-5899 227 7 , , PUNCT cana-5899 227 8 vitaly vitaly PROPN cana-5899 227 9 volpert volpert PROPN cana-5899 227 10 , , PUNCT cana-5899 227 11 malay malay PROPN cana-5899 227 12 banerjee banerjee PROPN cana-5899 227 13 an an DET cana-5899 227 14 epidemic epidemic NOUN cana-5899 227 15 model model NOUN cana-5899 227 16 with with ADP cana-5899 227 17 time time NOUN cana-5899 227 18 delay delay NOUN cana-5899 227 19 determined determine VERB cana-5899 227 20 by by ADP cana-5899 227 21 the the DET cana-5899 227 22 disease disease NOUN cana-5899 227 23 duration duration NOUN cana-5899 227 24 . . PUNCT cana-5899 228 1 mathematics mathematic NOUN cana-5899 228 2 2022 2022 NUM cana-5899 228 3 , , PUNCT cana-5899 228 4 10(15 10(15 NUM cana-5899 228 5 ) ) PUNCT cana-5899 228 6 , , PUNCT cana-5899 228 7 2561 2561 NUM cana-5899 228 8 ; ; PUNCT cana-5899 228 9 https://doi.org/10.3390/math10152561 https://doi.org/10.3390/math10152561 NOUN cana-5899 229 1 [ [ X cana-5899 229 2 12 12 NUM cana-5899 229 3 ] ] X cana-5899 229 4 zhang zhang PROPN cana-5899 229 5 et et PROPN cana-5899 229 6 al al PROPN cana-5899 229 7 . . PROPN cana-5899 229 8 ( ( PUNCT cana-5899 229 9 2024 2024 NUM cana-5899 229 10 ) ) PUNCT cana-5899 229 11 – – PUNCT cana-5899 229 12 dynamics dynamic NOUN cana-5899 229 13 of of ADP cana-5899 229 14 infectious infectious ADJ cana-5899 229 15 diseases disease NOUN cana-5899 229 16 incorporating incorporate VERB cana-5899 229 17 a a DET cana-5899 229 18 testing testing NOUN cana-5899 229 19 compartment compartment NOUN cana-5899 229 20 ( ( PUNCT cana-5899 229 21 mdpi mdpi ADJ cana-5899 229 22 math math NOUN cana-5899 229 23 ) ) PUNCT cana-5899 229 24 mathematics mathematic NOUN cana-5899 229 25 2024 2024 NUM cana-5899 229 26 , , PUNCT cana-5899 229 27 12(12 12(12 NUM cana-5899 229 28 ) ) PUNCT cana-5899 229 29 , , PUNCT cana-5899 229 30 1797 1797 NUM cana-5899 229 31 . . PUNCT cana-5899 230 1 [ [ X cana-5899 230 2 13 13 NUM cana-5899 230 3 ] ] PUNCT cana-5899 230 4 cangiotti cangiotti PROPN cana-5899 230 5 et et PROPN cana-5899 230 6 β€― β€― SPACE cana-5899 230 7 al al PROPN cana-5899 230 8 . . PROPN cana-5899 231 1 ( ( PUNCT cana-5899 231 2 2023 2023 NUM cana-5899 231 3 ) ) PUNCT cana-5899 231 4 – – PUNCT cana-5899 231 5 a a DET cana-5899 231 6 survey survey NOUN cana-5899 231 7 on on ADP cana-5899 231 8 lyapunov lyapunov ADJ cana-5899 231 9 functions function NOUN cana-5899 231 10 for for ADP cana-5899 231 11 epidemic epidemic NOUN cana-5899 231 12 compartmental compartmental NOUN cana-5899 231 13 models model NOUN cana-5899 231 14 ( ( PUNCT cana-5899 231 15 arxiv arxiv PROPN cana-5899 231 16 ) ) PUNCT cana-5899 231 17 . . PUNCT cana-5899 232 1 [ [ X cana-5899 232 2 14 14 NUM cana-5899 232 3 ] ] PUNCT cana-5899 232 4 blyuss blyuss NOUN cana-5899 232 5 & & CCONJ cana-5899 232 6 β€― β€― SPACE cana-5899 232 7 kyrychko kyrychko PROPN cana-5899 232 8 ( ( PUNCT cana-5899 232 9 2012 2012 NUM cana-5899 232 10 ) ) PUNCT cana-5899 232 11 – – PUNCT cana-5899 232 12 stability stability NOUN cana-5899 232 13 and and CCONJ cana-5899 232 14 bifurcations bifurcation NOUN cana-5899 232 15 in in ADP cana-5899 232 16 an an DET cana-5899 232 17 epidemic epidemic NOUN cana-5899 232 18 model model NOUN cana-5899 232 19 with with ADP cana-5899 232 20 varying vary VERB cana-5899 232 21 immunity immunity NOUN cana-5899 232 22 period period NOUN cana-5899 232 23 ( ( PUNCT cana-5899 232 24 arxiv arxiv PROPN cana-5899 232 25 ) ) PUNCT cana-5899 232 26 . . PUNCT cana-5899 233 1 [ [ X cana-5899 233 2 15 15 NUM cana-5899 233 3 ] ] X cana-5899 233 4 olukayode olukayode ADJ cana-5899 233 5 adebimpe adebimpe NOUN cana-5899 233 6 , , PUNCT cana-5899 233 7 kehinde kehinde ADJ cana-5899 233 8 adekunle adekunle NOUN cana-5899 233 9 bashiru bashiru NOUN cana-5899 233 10 , , PUNCT cana-5899 233 11 taiwo taiwo PROPN cana-5899 233 12 adetola adetola PROPN cana-5899 233 13 ojurongbe ojurongbe NOUN cana-5899 233 14 – – PUNCT cana-5899 233 15 stability stability NOUN cana-5899 233 16 analysis analysis NOUN cana-5899 233 17 of of ADP cana-5899 233 18 an an DET cana-5899 233 19 sir sir NOUN cana-5899 233 20 epidemic epidemic NOUN cana-5899 233 21 model model NOUN cana-5899 233 22 with with ADP cana-5899 233 23 non non ADJ cana-5899 233 24 - - ADJ cana-5899 233 25 linear linear ADJ cana-5899 233 26 incidence incidence NOUN cana-5899 233 27 rate rate NOUN cana-5899 233 28 and and CCONJ cana-5899 233 29 treatment treatment NOUN cana-5899 233 30 . . PUNCT cana-5899 234 1 published publish VERB cana-5899 234 2 by by ADP cana-5899 234 3 open open ADJ cana-5899 234 4 journal journal NOUN cana-5899 234 5 of of ADP cana-5899 234 6 modelling modelling NOUN cana-5899 234 7 and and CCONJ cana-5899 234 8 simulation simulation NOUN cana-5899 234 9 , , PUNCT cana-5899 234 10 vol.3 vol.3 PROPN cana-5899 234 11 no.3 no.3 PROPN cana-5899 234 12 , , PUNCT cana-5899 234 13 2015 2015 NUM cana-5899 234 14 . . PUNCT cana-5899 235 1 [ [ X cana-5899 235 2 16 16 NUM cana-5899 235 3 ] ] X cana-5899 235 4 p.r.s p.r.s PROPN cana-5899 235 5 . . PUNCT cana-5899 235 6 rao rao PROPN cana-5899 235 7 , , PUNCT cana-5899 235 8 k.v k.v PROPN cana-5899 235 9 . . PROPN cana-5899 235 10 ratnam ratnam PROPN cana-5899 235 11 and and CCONJ cana-5899 235 12 sita sita PROPN cana-5899 235 13 rama rama PROPN cana-5899 236 1 murthy murthy PROPN cana-5899 236 2 mushunuri mushunuri PROPN cana-5899 236 3 predictive predictive ADJ cana-5899 236 4 dynamics dynamic NOUN cana-5899 236 5 of of ADP cana-5899 236 6 infectious infectious ADJ cana-5899 236 7 diseases disease NOUN cana-5899 236 8 – – PUNCT cana-5899 236 9 a a DET cana-5899 236 10 new new ADJ cana-5899 236 11 technique technique NOUN cana-5899 236 12 . . PUNCT cana-5899 237 1 world world PROPN cana-5899 237 2 journal journal PROPN cana-5899 237 3 of of ADP cana-5899 237 4 nonlinear nonlinear ADJ cana-5899 237 5 modelling modelling NOUN cana-5899 237 6 and and CCONJ cana-5899 237 7 simulation15(2):128 simulation15(2):128 NOUN cana-5899 237 8 - - PUNCT cana-5899 237 9 139https://www.researchgate.net 139https://www.researchgate.net NOUN cana-5899 237 10 / / SYM cana-5899 237 11 publication/343058437 publication/343058437 NOUN cana-5899 237 12 [ [ X cana-5899 237 13 17 17 NUM cana-5899 237 14 ] ] PUNCT cana-5899 237 15 andres andres PROPN cana-5899 237 16 david david PROPN cana-5899 237 17 baez baez PROPN cana-5899 237 18 - - PUNCT cana-5899 237 19 sanchez sanchez PROPN cana-5899 237 20 , , PUNCT cana-5899 237 21 naro naro NOUN cana-5899 237 22 bobko bobko NOUN cana-5899 237 23 on on ADP cana-5899 237 24 equilibria equilibrium NOUN cana-5899 237 25 stability stability NOUN cana-5899 237 26 in in ADP cana-5899 237 27 an an DET cana-5899 237 28 epidemiological epidemiological ADJ cana-5899 237 29 sir sir NOUN cana-5899 237 30 model model NOUN cana-5899 237 31 with with ADP cana-5899 237 32 recovery recovery NOUN cana-5899 237 33 - - PUNCT cana-5899 237 34 dependent dependent ADJ cana-5899 237 35 infection infection NOUN cana-5899 237 36 rate rate NOUN cana-5899 237 37 . . PUNCT cana-5899 238 1 [ [ X cana-5899 238 2 18 18 NUM cana-5899 238 3 ] ] X cana-5899 238 4 k.a k.a PROPN cana-5899 238 5 . . PUNCT cana-5899 238 6 kabir kabir PROPN cana-5899 238 7 , , PUNCT cana-5899 238 8 k. k. PROPN cana-5899 238 9 kurga kurga PROPN cana-5899 238 10 and and CCONJ cana-5899 238 11 j. j. PROPN cana-5899 238 12 tanimotoanalysis tanimotoanalysis NOUN cana-5899 238 13 of of ADP cana-5899 238 14 sir sir PROPN cana-5899 238 15 epidemic epidemic NOUN cana-5899 238 16 model model NOUN cana-5899 238 17 with with ADP cana-5899 238 18 information information NOUN cana-5899 238 19 spreading spread VERB cana-5899 238 20 of of ADP cana-5899 238 21 awareness awareness NOUN cana-5899 238 22 , , PUNCT cana-5899 238 23 chaos chaos NOUN cana-5899 238 24 , , PUNCT cana-5899 238 25 solitons soliton NOUN cana-5899 238 26 & & CCONJ cana-5899 238 27 fractals fractal NOUN cana-5899 238 28 , , PUNCT cana-5899 238 29 vol vol NOUN cana-5899 238 30 . . PROPN cana-5899 238 31 119 119 NUM cana-5899 238 32 , , PUNCT cana-5899 238 33 pp pp ADJ cana-5899 238 34 . . PUNCT cana-5899 239 1 118 118 NUM cana-5899 239 2 - - SYM cana-5899 239 3 125 125 NUM cana-5899 239 4 , , PUNCT cana-5899 239 5 2019 2019 NUM cana-5899 239 6 , , PUNCT cana-5899 239 7 https://doi.org/10.1016/j.chaos2018.12.017 https://doi.org/10.1016/j.chaos2018.12.017 PRON cana-5899 239 8 [ [ X cana-5899 239 9 19 19 NUM cana-5899 239 10 ] ] X cana-5899 239 11 pathak pathak PROPN cana-5899 239 12 s s PROPN cana-5899 239 13 , , PUNCT cana-5899 239 14 shirisha shirisha VERB cana-5899 239 15 g g NOUN cana-5899 239 16 , , PUNCT cana-5899 239 17 ratnam ratnam PROPN cana-5899 239 18 k.v k.v PROPN cana-5899 239 19 -dynamical -dynamical ADJ cana-5899 239 20 behavior behavior NOUN cana-5899 239 21 of of ADP cana-5899 239 22 a a DET cana-5899 239 23 time time NOUN cana-5899 239 24 - - PUNCT cana-5899 239 25 delayed delay VERB cana-5899 239 26 infectious infectious ADJ cana-5899 239 27 disease disease NOUN cana-5899 239 28 model model NOUN cana-5899 239 29 with with ADP cana-5899 239 30 a a DET cana-5899 239 31 non non ADJ cana-5899 239 32 - - ADJ cana-5899 239 33 linear linear ADJ cana-5899 239 34 incidence incidence NOUN cana-5899 239 35 function function VERB cana-5899 239 36 under under ADP cana-5899 239 37 the the DET cana-5899 239 38 effect effect NOUN cana-5899 239 39 of of ADP cana-5899 239 40 vaccination vaccination NOUN cana-5899 239 41 and and CCONJ cana-5899 239 42 treatment(2023).https://doi.org/10.48550 treatment(2023).https://doi.org/10.48550 ADJ cana-5899 239 43 / / SYM cana-5899 239 44 arxiv.2307.00339 arxiv.2307.00339 X cana-5899 239 45 [ [ X cana-5899 239 46 20 20 NUM cana-5899 239 47 ] ] X cana-5899 239 48 elazzouzi elazzouzi PROPN cana-5899 239 49 , , PUNCT cana-5899 239 50 a. a. PROPN cana-5899 239 51 , , PUNCT cana-5899 239 52 alaoui alaoui PROPN cana-5899 239 53 , , PUNCT cana-5899 239 54 a.l a.l PROPN cana-5899 239 55 . . PROPN cana-5899 239 56 , , PUNCT cana-5899 239 57 tiloua tiloua ADV cana-5899 239 58 , , PUNCT cana-5899 239 59 m. m. NOUN cana-5899 239 60 and and CCONJ cana-5899 239 61 tridane tridane VERB cana-5899 239 62 aglobal aglobal ADJ cana-5899 239 63 stability stability NOUN cana-5899 239 64 analysis analysis NOUN cana-5899 239 65 for for ADP cana-5899 239 66 a a DET cana-5899 239 67 generalized generalize VERB cana-5899 239 68 delayed delay VERB cana-5899 239 69 sir sir NOUN cana-5899 239 70 model model NOUN cana-5899 239 71 with with ADP cana-5899 239 72 vaccination vaccination NOUN cana-5899 239 73 and and CCONJ cana-5899 239 74 treatment treatment NOUN cana-5899 239 75 . . PUNCT cana-5899 240 1 adv.differ adv.differ PROPN cana-5899 240 2 . . PUNCT cana-5899 241 1 equ.2019(1 equ.2019(1 ADJ cana-5899 241 2 ) ) PUNCT cana-5899 241 3 , , PUNCT cana-5899 241 4 532(2019 532(2019 NUM cana-5899 241 5 ) ) PUNCT cana-5899 241 6 . . PUNCT cana-5899 242 1 [ [ X cana-5899 242 2 21 21 NUM cana-5899 242 3 ] ] X cana-5899 242 4 zhai zhai PROPN cana-5899 242 5 , , PUNCT cana-5899 242 6 s. s. PROPN cana-5899 242 7 , , PUNCT cana-5899 242 8 luo luo PROPN cana-5899 242 9 , , PUNCT cana-5899 242 10 g. g. PROPN cana-5899 242 11 , , PUNCT cana-5899 242 12 huang huang PROPN cana-5899 242 13 , , PUNCT cana-5899 242 14 x. x. PROPN cana-5899 242 15 , , PUNCT cana-5899 242 16 tao tao PROPN cana-5899 242 17 , , PUNCT cana-5899 242 18 j. j. PROPN cana-5899 242 19 , , PUNCT cana-5899 242 20 zhou zhou PROPN cana-5899 242 21 , , PUNCT cana-5899 242 22 p. p. NOUN cana-5899 242 23 – – PUNCT cana-5899 242 24 vaccination vaccination NOUN cana-5899 242 25 control control NOUN cana-5899 242 26 of of ADP cana-5899 242 27 an an DET cana-5899 242 28 epidemic epidemic NOUN cana-5899 242 29 model model NOUN cana-5899 242 30 with with ADP cana-5899 242 31 time time NOUN cana-5899 242 32 delay delay NOUN cana-5899 242 33 and and CCONJ cana-5899 242 34 its its PRON cana-5899 242 35 application application NOUN cana-5899 242 36 to to ADP cana-5899 242 37 covid-19 covid-19 PROPN cana-5899 242 38 , , PUNCT cana-5899 242 39 nonlinear nonlinear ADJ cana-5899 242 40 dyn dyn NOUN cana-5899 242 41 . . PUNCT cana-5899 242 42 106(2 106(2 NUM cana-5899 242 43 ) ) PUNCT cana-5899 242 44 , , PUNCT cana-5899 242 45 12791292)2021 12791292)2021 NOUN cana-5899 242 46 ) ) PUNCT cana-5899 242 47 . . PUNCT cana-5899 243 1 [ [ X cana-5899 243 2 22 22 NUM cana-5899 243 3 ] ] X cana-5899 243 4 singh singh PROPN cana-5899 243 5 , , PUNCT cana-5899 243 6 r.p.bifurcation r.p.bifurcation NOUN cana-5899 243 7 and and CCONJ cana-5899 243 8 stability stability NOUN cana-5899 243 9 analysis analysis NOUN cana-5899 243 10 of of ADP cana-5899 243 11 delayed delay VERB cana-5899 243 12 sir sir PROPN cana-5899 243 13 model model NOUN cana-5899 243 14 . . PUNCT cana-5899 244 1 j. j. PROPN cana-5899 244 2 phys phys PROPN cana-5899 244 3 . . PUNCT cana-5899 244 4 conf conf PROPN cana-5899 244 5 . . PUNCT cana-5899 245 1 ser.2267(1 ser.2267(1 NOUN cana-5899 245 2 ) ) PUNCT cana-5899 245 3 , , PUNCT cana-5899 245 4 012011(2022 012011(2022 NOUN cana-5899 245 5 ) ) PUNCT cana-5899 245 6 . . PUNCT cana-5899 246 1 https://10.1088/1742 https://10.1088/1742 ADV cana-5899 246 2 - - PUNCT cana-5899 246 3 6595/2267/1/012011 6595/2267/1/012011 NUM cana-5899 246 4 https://doi.org/10.1111/1469-0691.12308 https://doi.org/10.1111/1469-0691.12308 NOUN cana-5899 246 5 https://doi.org/10.3390/math10152561 https://doi.org/10.3390/math10152561 PROPN cana-5899 246 6 https://www.researchgate.net/publication/343058437 https://www.researchgate.net/publication/343058437 VERB cana-5899 246 7 https://doi.org/10.1016/j.chaos2018.12.017 https://doi.org/10.1016/j.chaos2018.12.017 NUM cana-5899 246 8 https://doi.org/10.48550/arxiv.2307.00339 https://doi.org/10.48550/arxiv.2307.00339 PROPN cana-5899 246 9 https://10.0.4.64/1742 https://10.0.4.64/1742 PROPN cana-5899 246 10 - - PUNCT cana-5899 246 11 6595/2267/1/012011 6595/2267/1/012011 NUM cana-5899 246 12 communications communication NOUN cana-5899 246 13 on on ADP cana-5899 246 14 applied apply VERB cana-5899 246 15 nonlinear nonlinear ADJ cana-5899 246 16 analysis analysis NOUN cana-5899 246 17 issn issn NOUN cana-5899 246 18 : : PUNCT cana-5899 246 19 1074 1074 NUM cana-5899 246 20 - - PUNCT cana-5899 246 21 133x 133x NUM cana-5899 246 22 vol vol NOUN cana-5899 246 23 31 31 NUM cana-5899 246 24 no no NOUN cana-5899 246 25 . . PUNCT cana-5899 247 1 8s 8s PROPN cana-5899 247 2 ( ( PUNCT cana-5899 247 3 2024 2024 NUM cana-5899 247 4 ) ) PUNCT cana-5899 247 5 1094 1094 NUM cana-5899 247 6 https://internationalpubls.com https://internationalpubls.com X cana-5899 248 1 [ [ X cana-5899 248 2 23 23 NUM cana-5899 248 3 ] ] X cana-5899 248 4 gupta gupta PROPN cana-5899 248 5 , , PUNCT cana-5899 248 6 s. s. PROPN cana-5899 248 7 , , PUNCT cana-5899 248 8 bhatia bhatia PROPN cana-5899 248 9 , , PUNCT cana-5899 248 10 s.k s.k PROPN cana-5899 248 11 . . PROPN cana-5899 248 12 , , PUNCT cana-5899 248 13 arya arya PROPN cana-5899 248 14 , , PUNCT cana-5899 248 15 n. n. NOUN cana-5899 248 16 – – PUNCT cana-5899 248 17 effect effect NOUN cana-5899 248 18 of of ADP cana-5899 248 19 incubation incubation NOUN cana-5899 248 20 delay delay NOUN cana-5899 248 21 and and CCONJ cana-5899 248 22 pollution pollution NOUN cana-5899 248 23 on on ADP cana-5899 248 24 the the DET cana-5899 248 25 transmission transmission NOUN cana-5899 248 26 dynamics dynamic NOUN cana-5899 248 27 of of ADP cana-5899 248 28 infectious infectious ADJ cana-5899 248 29 disease disease NOUN cana-5899 248 30 . . PUNCT cana-5899 249 1 ann ann PROPN cana-5899 249 2 . . PROPN cana-5899 249 3 univ univ PROPN cana-5899 249 4 . . PROPN cana-5899 249 5 ferrara ferrara PROPN cana-5899 249 6 69(1),23 69(1),23 PROPN cana-5899 249 7 - - PUNCT cana-5899 249 8 47(2023 47(2023 NUM cana-5899 249 9 ) ) PUNCT cana-5899 249 10 . . PUNCT cana-5899 250 1 doi:10.1017 doi:10.1017 NOUN cana-5899 250 2 / / SYM cana-5899 250 3 s11565 s11565 PROPN cana-5899 250 4 - - PUNCT cana-5899 250 5 022 022 NUM cana-5899 250 6 - - PUNCT cana-5899 250 7 00399 00399 NUM cana-5899 250 8 - - SYM cana-5899 250 9 5 5 NUM cana-5899 250 10 [ [ SYM cana-5899 250 11 24 24 NUM cana-5899 250 12 ] ] X cana-5899 250 13 goel goel PROPN cana-5899 250 14 , , PUNCT cana-5899 250 15 k. k. PROPN cana-5899 250 16 and and CCONJ cana-5899 250 17 neelam neelam PROPN cana-5899 250 18 stability stability PROPN cana-5899 250 19 behavior behavior NOUN cana-5899 250 20 of of ADP cana-5899 250 21 a a DET cana-5899 250 22 nonlinear nonlinear ADJ cana-5899 250 23 epidemic epidemic NOUN cana-5899 250 24 transmission transmission NOUN cana-5899 250 25 model model NOUN cana-5899 250 26 with with ADP cana-5899 250 27 time time NOUN cana-5899 250 28 delay delay NOUN cana-5899 250 29 – – PUNCT cana-5899 250 30 nonlinear nonlinear ADJ cana-5899 250 31 dyn dyn NOUN cana-5899 250 32 98 98 NUM cana-5899 250 33 , , PUNCT cana-5899 250 34 1501 1501 NUM cana-5899 250 35 - - SYM cana-5899 250 36 1518(2019 1518(2019 NUM cana-5899 250 37 ) ) PUNCT cana-5899 250 38 . . PUNCT cana-5899 251 1 [ [ X cana-5899 251 2 25 25 NUM cana-5899 251 3 ] ] X cana-5899 251 4 zhen zhen PROPN cana-5899 251 5 jin jin PROPN cana-5899 251 6 , , PUNCT cana-5899 251 7 zhien zhien PROPN cana-5899 251 8 ma ma PROPN cana-5899 251 9 the the DET cana-5899 251 10 stability stability NOUN cana-5899 251 11 of of ADP cana-5899 251 12 an an DET cana-5899 251 13 sir sir NOUN cana-5899 251 14 epidemic epidemic NOUN cana-5899 251 15 model model NOUN cana-5899 251 16 with with ADP cana-5899 251 17 time time NOUN cana-5899 251 18 delays delay NOUN cana-5899 251 19 . . PUNCT cana-5899 252 1 th th X cana-5899 252 2 . . PUNCT cana-5899 253 1 biosci biosci PROPN cana-5899 253 2 . . PUNCT cana-5899 254 1 eng eng PROPN cana-5899 254 2 . . PROPN cana-5899 254 3 2006 2006 NUM cana-5899 254 4 jan;3(1):101 jan;3(1):101 PROPN cana-5899 254 5 - - PUNCT cana-5899 254 6 9 9 NUM cana-5899 254 7 . . PUNCT cana-5899 255 1 [ [ X cana-5899 255 2 26 26 NUM cana-5899 255 3 ] ] PUNCT cana-5899 255 4 wencai wencai PROPN cana-5899 255 5 zhao zhao PROPN cana-5899 255 6 , , PUNCT cana-5899 255 7 tongqian tongqian PROPN cana-5899 255 8 zhang zhang PROPN cana-5899 255 9 , , PUNCT cana-5899 255 10 zhengbo zhengbo PROPN cana-5899 255 11 chang chang PROPN cana-5899 255 12 , , PUNCT cana-5899 255 13 xinzhu xinzhu PROPN cana-5899 255 14 meng meng PROPN cana-5899 255 15 , , PUNCT cana-5899 255 16 yulin yulin PROPN cana-5899 255 17 liu liu PROPN cana-5899 255 18 – – PUNCT cana-5899 255 19 dynamical dynamical ADJ cana-5899 255 20 analysis analysis NOUN cana-5899 255 21 of of ADP cana-5899 255 22 sir sir NOUN cana-5899 255 23 epidemic epidemic NOUN cana-5899 255 24 models model NOUN cana-5899 255 25 with with ADP cana-5899 255 26 distributed distributed ADJ cana-5899 255 27 delay delay NOUN cana-5899 255 28 . . PUNCT cana-5899 256 1 journal journal NOUN cana-5899 256 2 of of ADP cana-5899 256 3 applied apply VERB cana-5899 256 4 mathematics mathematic NOUN cana-5899 256 5 . . PUNCT cana-5899 257 1 https://doi.org/10.1155/2013/154387 https://doi.org/10.1155/2013/154387 PROPN cana-5899 257 2 https://doi.org/10.1155/2013/154387 https://doi.org/10.1155/2013/154387 PROPN