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PUNCT iajs-2624 5 1 the the DET iajs-2624 5 2 outbreak outbreak NOUN iajs-2624 5 3 of of ADP iajs-2624 5 4 covid-19 covid-19 PROPN iajs-2624 5 5 virus virus NOUN iajs-2624 5 6 is be AUX iajs-2624 5 7 described describe VERB iajs-2624 5 8 by by ADP iajs-2624 5 9 a a DET iajs-2624 5 10 mathematical mathematical ADJ iajs-2624 5 11 model model NOUN iajs-2624 5 12 dividing divide VERB iajs-2624 5 13 the the DET iajs-2624 5 14 population population NOUN iajs-2624 5 15 into into ADP iajs-2624 5 16 four four NUM iajs-2624 5 17 classes class NOUN iajs-2624 5 18 . . PUNCT iajs-2624 6 1 the the DET iajs-2624 6 2 first first ADJ iajs-2624 6 3 class class NOUN iajs-2624 6 4 represents represent VERB iajs-2624 6 5 the the DET iajs-2624 6 6 susceptible susceptible ADJ iajs-2624 6 7 people people NOUN iajs-2624 6 8 who who PRON iajs-2624 6 9 are be AUX iajs-2624 6 10 unaware unaware ADJ iajs-2624 6 11 of of ADP iajs-2624 6 12 the the DET iajs-2624 6 13 disease disease NOUN iajs-2624 6 14 . . 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PUNCT iajs-2624 22 1 ibn ibn PROPN iajs-2624 22 2 al al PROPN iajs-2624 22 3 haitham haitham PROPN iajs-2624 22 4 journal journal PROPN iajs-2624 22 5 for for ADP iajs-2624 22 6 pure pure ADJ iajs-2624 22 7 and and CCONJ iajs-2624 22 8 applied apply VERB iajs-2624 22 9 science science NOUN iajs-2624 22 10 journal journal PROPN iajs-2624 22 11 homepage homepage NOUN iajs-2624 22 12 : : PUNCT iajs-2624 22 13 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2624 22 14 doi doi NOUN iajs-2624 22 15 : : PUNCT iajs-2624 22 16 10.30526/34.2.2624 10.30526/34.2.2624 NUM iajs-2624 22 17 article article NOUN iajs-2624 22 18 history history NOUN iajs-2624 22 19 : : PUNCT iajs-2624 22 20 received,15,june,2020 received,15,june,2020 PROPN iajs-2624 22 21 , , PUNCT iajs-2624 22 22 accepted accept VERB iajs-2624 22 23 20,julay 20,julay NUM iajs-2624 22 24 , , PUNCT iajs-2624 22 25 2020 2020 NUM iajs-2624 22 26 , , PUNCT iajs-2624 22 27 published publish VERB iajs-2624 22 28 in in ADP iajs-2624 22 29 april april PROPN iajs-2624 22 30 2021 2021 NUM iajs-2624 22 31 husseiny husseiny PROPN iajs-2624 22 32 - - PUNCT iajs-2624 22 33 hassan hassan PROPN iajs-2624 22 34 f. f. 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PROPN iajs-2624 25 3 china china PROPN iajs-2624 25 4 mailto:aamuhseen@gmail.comdepartment mailto:aamuhseen@gmail.comdepartment PROPN iajs-2624 25 5 mailto:xueyongzhou@xynu.edu.cn mailto:xueyongzhou@xynu.edu.cn PROPN iajs-2624 25 6 43 43 NUM iajs-2624 25 7 ibn ibn PROPN iajs-2624 25 8 al al PROPN iajs-2624 25 9 - - PUNCT iajs-2624 25 10 haitham haitham PROPN iajs-2624 25 11 jour jour X iajs-2624 25 12 . . PROPN iajs-2624 26 1 for for ADP iajs-2624 26 2 pure pure ADJ iajs-2624 26 3 & & CCONJ iajs-2624 26 4 appl appl PROPN iajs-2624 26 5 . . PUNCT iajs-2624 27 1 sci sci PROPN iajs-2624 27 2 . . PROPN iajs-2624 28 1 34 34 NUM iajs-2624 28 2 ( ( PUNCT iajs-2624 28 3 2 2 NUM iajs-2624 28 4 ) ) PUNCT iajs-2624 28 5 2021 2021 NUM iajs-2624 28 6 since since SCONJ iajs-2624 28 7 december december PROPN iajs-2624 28 8 2019 2019 NUM iajs-2624 28 9 , , PUNCT iajs-2624 28 10 a a DET iajs-2624 28 11 novel novel ADJ iajs-2624 28 12 corona corona NOUN iajs-2624 28 13 virus virus NOUN iajs-2624 28 14 , , PUNCT iajs-2624 28 15 named name VERB iajs-2624 28 16 ( ( PUNCT iajs-2624 28 17 covid-19 covid-19 PROPN iajs-2624 28 18 ) ) PUNCT iajs-2624 28 19 , , PUNCT iajs-2624 28 20 emerged emerge VERB iajs-2624 28 21 in in ADP iajs-2624 28 22 wuhan wuhan PROPN iajs-2624 28 23 , , PUNCT iajs-2624 28 24 china china PROPN iajs-2624 28 25 and and CCONJ iajs-2624 28 26 the the DET iajs-2624 28 27 epidemic epidemic NOUN iajs-2624 28 28 has have AUX iajs-2624 28 29 globally globally ADV iajs-2624 28 30 spread spread VERB iajs-2624 28 31 , , PUNCT iajs-2624 28 32 updated update VERB iajs-2624 28 33 on on ADP iajs-2624 28 34 january january PROPN iajs-2624 28 35 30 30 NUM iajs-2624 28 36 , , PUNCT iajs-2624 28 37 2020 2020 NUM iajs-2624 28 38 , , PUNCT iajs-2624 28 39 covid-19 covid-19 PROPN iajs-2624 28 40 was be AUX iajs-2624 28 41 declared declare VERB iajs-2624 28 42 a a DET iajs-2624 28 43 public public ADJ iajs-2624 28 44 health health NOUN iajs-2624 28 45 emergency emergency NOUN iajs-2624 28 46 of of ADP iajs-2624 28 47 international international ADJ iajs-2624 28 48 concern concern NOUN iajs-2624 28 49 . . PUNCT iajs-2624 29 1 it it PRON iajs-2624 29 2 caused cause VERB iajs-2624 29 3 more more ADJ iajs-2624 29 4 than than ADP iajs-2624 29 5 200,000 200,000 NUM iajs-2624 29 6 deaths death NOUN iajs-2624 29 7 out out ADP iajs-2624 29 8 of of ADP iajs-2624 29 9 3,177,500 3,177,500 NUM iajs-2624 29 10 cases case NOUN iajs-2624 29 11 in in ADP iajs-2624 29 12 the the DET iajs-2624 29 13 world world NOUN iajs-2624 29 14 among among ADP iajs-2624 29 15 which which PRON iajs-2624 29 16 1,434,000 1,434,000 NUM iajs-2624 29 17 in in ADP iajs-2624 29 18 europe europe PROPN iajs-2624 29 19 , , PUNCT iajs-2624 29 20 1,291,000 1,291,000 NUM iajs-2624 29 21 in in ADP iajs-2624 29 22 americas america NOUN iajs-2624 29 23 , , PUNCT iajs-2624 29 24 148,000 148,000 NUM iajs-2624 29 25 in in ADP iajs-2624 29 26 western western ADJ iajs-2624 29 27 pacific pacific ADJ iajs-2624 29 28 and and CCONJ iajs-2624 29 29 other other ADJ iajs-2624 29 30 cases case NOUN iajs-2624 29 31 in in ADP iajs-2624 29 32 asia asia PROPN iajs-2624 29 33 and and CCONJ iajs-2624 29 34 africa africa PROPN iajs-2624 30 1 [ [ X iajs-2624 30 2 5 5 NUM iajs-2624 30 3 ] ] PUNCT iajs-2624 30 4 . . PUNCT iajs-2624 31 1 the the DET iajs-2624 31 2 number number NOUN iajs-2624 31 3 of of ADP iajs-2624 31 4 the the DET iajs-2624 31 5 total total ADJ iajs-2624 31 6 cases case NOUN iajs-2624 31 7 , , PUNCT iajs-2624 31 8 total total ADJ iajs-2624 31 9 death death NOUN iajs-2624 31 10 , , PUNCT iajs-2624 31 11 new new ADJ iajs-2624 31 12 confirmed confirm VERB iajs-2624 31 13 cases case NOUN iajs-2624 31 14 and and CCONJ iajs-2624 31 15 recovery recovery NOUN iajs-2624 31 16 for for ADP iajs-2624 31 17 some some DET iajs-2624 31 18 countries country NOUN iajs-2624 31 19 in in ADP iajs-2624 31 20 the the DET iajs-2624 31 21 world world NOUN iajs-2624 31 22 are be AUX iajs-2624 31 23 shown show VERB iajs-2624 31 24 in in ADP iajs-2624 31 25 figure figure NOUN iajs-2624 31 26 1 1 NUM iajs-2624 31 27 . . PUNCT iajs-2624 31 28 figure figure NOUN iajs-2624 31 29 1 1 NUM iajs-2624 31 30 . . PUNCT iajs-2624 32 1 the the DET iajs-2624 32 2 number number NOUN iajs-2624 32 3 of of ADP iajs-2624 32 4 total total ADJ iajs-2624 32 5 cases case NOUN iajs-2624 32 6 , , PUNCT iajs-2624 32 7 total total ADJ iajs-2624 32 8 death death NOUN iajs-2624 32 9 , , PUNCT iajs-2624 32 10 new new ADJ iajs-2624 32 11 confirmed confirm VERB iajs-2624 32 12 cases case NOUN iajs-2624 32 13 and and CCONJ iajs-2624 32 14 recovery recovery NOUN iajs-2624 32 15 of of ADP iajs-2624 32 16 some some DET iajs-2624 32 17 countries country NOUN iajs-2624 32 18 in in ADP iajs-2624 32 19 world world NOUN iajs-2624 32 20 from from ADP iajs-2624 32 21 20 20 NUM iajs-2624 32 22 january january NOUN iajs-2624 32 23 to to ADP iajs-2624 32 24 05 05 NUM iajs-2624 32 25 may may PROPN iajs-2624 32 26 2020 2020 NUM iajs-2624 32 27 . . PUNCT iajs-2624 33 1 there there PRON iajs-2624 33 2 are be VERB iajs-2624 33 3 some some DET iajs-2624 33 4 factors factor NOUN iajs-2624 33 5 that that PRON iajs-2624 33 6 complicate complicate VERB iajs-2624 33 7 the the DET iajs-2624 33 8 infection infection NOUN iajs-2624 33 9 dynamics dynamic NOUN iajs-2624 33 10 of of ADP iajs-2624 33 11 novel novel ADJ iajs-2624 33 12 coronavirus coronavirus NOUN iajs-2624 33 13 and and CCONJ iajs-2624 33 14 add add VERB iajs-2624 33 15 challenges challenge NOUN iajs-2624 33 16 to to ADP iajs-2624 33 17 how how SCONJ iajs-2624 33 18 to to PART iajs-2624 33 19 control control VERB iajs-2624 33 20 the the DET iajs-2624 33 21 epidemic epidemic NOUN iajs-2624 33 22 . . PUNCT iajs-2624 34 1 first first ADV iajs-2624 34 2 , , PUNCT iajs-2624 34 3 the the DET iajs-2624 34 4 clinical clinical ADJ iajs-2624 34 5 evidence evidence NOUN iajs-2624 34 6 shows show VERB iajs-2624 34 7 that that SCONJ iajs-2624 34 8 the the DET iajs-2624 34 9 incubation incubation NOUN iajs-2624 34 10 period period NOUN iajs-2624 34 11 of of ADP iajs-2624 34 12 this this DET iajs-2624 34 13 epidemic epidemic NOUN iajs-2624 34 14 ranges range VERB iajs-2624 34 15 from from ADP iajs-2624 34 16 7 7 NUM iajs-2624 34 17 to to PART iajs-2624 34 18 14 14 NUM iajs-2624 34 19 days day NOUN iajs-2624 34 20 . . PUNCT iajs-2624 35 1 during during ADP iajs-2624 35 2 this this DET iajs-2624 35 3 period period NOUN iajs-2624 35 4 of of ADP iajs-2624 35 5 time time NOUN iajs-2624 35 6 , , PUNCT iajs-2624 35 7 infected infect VERB iajs-2624 35 8 individuals individual NOUN iajs-2624 35 9 may may AUX iajs-2624 35 10 not not PART iajs-2624 35 11 develop develop VERB iajs-2624 35 12 any any DET iajs-2624 35 13 symptoms symptom NOUN iajs-2624 35 14 and and CCONJ iajs-2624 35 15 may may AUX iajs-2624 35 16 not not PART iajs-2624 35 17 be be AUX iajs-2624 35 18 aware aware ADJ iajs-2624 35 19 of of ADP iajs-2624 35 20 their their PRON iajs-2624 35 21 infection infection NOUN iajs-2624 35 22 , , PUNCT iajs-2624 35 23 yet yet CCONJ iajs-2624 35 24 they they PRON iajs-2624 35 25 are be AUX iajs-2624 35 26 capable capable ADJ iajs-2624 35 27 of of ADP iajs-2624 35 28 transmitting transmit VERB iajs-2624 35 29 the the DET iajs-2624 35 30 disease disease NOUN iajs-2624 35 31 to to ADP iajs-2624 35 32 other other ADJ iajs-2624 35 33 people people NOUN iajs-2624 35 34 . . PUNCT iajs-2624 36 1 secondly secondly ADV iajs-2624 36 2 , , PUNCT iajs-2624 36 3 the the DET iajs-2624 36 4 virus virus NOUN iajs-2624 36 5 is be AUX iajs-2624 36 6 new new ADJ iajs-2624 36 7 and and CCONJ iajs-2624 36 8 there there PRON iajs-2624 36 9 are be VERB iajs-2624 36 10 no no DET iajs-2624 36 11 vaccines vaccine NOUN iajs-2624 36 12 currently currently ADV iajs-2624 36 13 available available ADJ iajs-2624 36 14 for for ADP iajs-2624 36 15 treatment treatment NOUN iajs-2624 36 16 . . PUNCT iajs-2624 37 1 thirdly thirdly ADV iajs-2624 37 2 , , PUNCT iajs-2624 37 3 many many ADJ iajs-2624 37 4 people people NOUN iajs-2624 37 5 do do AUX iajs-2624 37 6 not not PART iajs-2624 37 7 take take VERB iajs-2624 37 8 into into ADP iajs-2624 37 9 account account NOUN iajs-2624 37 10 prevention prevention NOUN iajs-2624 37 11 measures measure NOUN iajs-2624 37 12 . . PUNCT iajs-2624 38 1 fourthly fourthly ADV iajs-2624 38 2 , , PUNCT iajs-2624 38 3 the the DET iajs-2624 38 4 origin origin NOUN iajs-2624 38 5 of of ADP iajs-2624 38 6 the the DET iajs-2624 38 7 infection infection NOUN iajs-2624 38 8 is be AUX iajs-2624 38 9 still still ADV iajs-2624 38 10 uncertain uncertain ADJ iajs-2624 38 11 although although SCONJ iajs-2624 38 12 it it PRON iajs-2624 38 13 is be AUX iajs-2624 38 14 widely widely ADV iajs-2624 38 15 speculated speculate VERB iajs-2624 38 16 that that SCONJ iajs-2624 38 17 wild wild ADJ iajs-2624 38 18 animals animal NOUN iajs-2624 38 19 such such ADJ iajs-2624 38 20 as as ADP iajs-2624 38 21 bats bat NOUN iajs-2624 38 22 , , PUNCT iajs-2624 38 23 civets civet NOUN iajs-2624 38 24 and and CCONJ iajs-2624 38 25 minks mink NOUN iajs-2624 38 26 are be AUX iajs-2624 38 27 responsible responsible ADJ iajs-2624 38 28 for for ADP iajs-2624 38 29 starting start VERB iajs-2624 38 30 up up ADP iajs-2624 38 31 the the DET iajs-2624 38 32 epidemic epidemic NOUN iajs-2624 39 1 [ [ X iajs-2624 39 2 6 6 NUM iajs-2624 39 3 ] ] PUNCT iajs-2624 39 4 . . PUNCT iajs-2624 40 1 a a DET iajs-2624 40 2 number number NOUN iajs-2624 40 3 of of ADP iajs-2624 40 4 modeling model VERB iajs-2624 40 5 studies study NOUN iajs-2624 40 6 have have AUX iajs-2624 40 7 already already ADV iajs-2624 40 8 been be AUX iajs-2624 40 9 performed perform VERB iajs-2624 40 10 for for ADP iajs-2624 40 11 the the DET iajs-2624 40 12 covid-19 covid-19 PROPN iajs-2624 40 13 epidemic epidemic NOUN iajs-2624 40 14 . . PUNCT iajs-2624 41 1 feng feng PROPN iajs-2624 41 2 et et PROPN iajs-2624 41 3 al al PROPN iajs-2624 41 4 . . PUNCT iajs-2624 42 1 [ [ X iajs-2624 42 2 7 7 X iajs-2624 42 3 ] ] PUNCT iajs-2624 42 4 studied study VERB iajs-2624 42 5 a a DET iajs-2624 42 6 covid-19 covid-19 PROPN iajs-2624 42 7 model model NOUN iajs-2624 42 8 in in ADP iajs-2624 42 9 uk uk PROPN iajs-2624 42 10 with with ADP iajs-2624 42 11 the the DET iajs-2624 42 12 effects effect NOUN iajs-2624 42 13 of of ADP iajs-2624 42 14 media medium NOUN iajs-2624 42 15 and and CCONJ iajs-2624 42 16 quarantine.yang quarantine.yang PROPN iajs-2624 42 17 and and CCONJ iajs-2624 42 18 wang wang PROPN iajs-2624 43 1 [ [ X iajs-2624 43 2 8 8 NUM iajs-2624 43 3 ] ] PUNCT iajs-2624 43 4 have have AUX iajs-2624 43 5 proposed propose VERB iajs-2624 43 6 a a DET iajs-2624 43 7 mathematical mathematical ADJ iajs-2624 43 8 model model NOUN iajs-2624 43 9 for for ADP iajs-2624 43 10 covid19 covid19 NOUN iajs-2624 43 11 epidemic epidemic NOUN iajs-2624 43 12 in in ADP iajs-2624 43 13 wuhan wuhan PROPN iajs-2624 43 14 , , PUNCT iajs-2624 43 15 china china PROPN iajs-2624 43 16 . . PUNCT iajs-2624 44 1 wu wu PROPN iajs-2624 44 2 et et PROPN iajs-2624 44 3 al al PROPN iajs-2624 44 4 . . PUNCT iajs-2624 45 1 [ [ X iajs-2624 45 2 9 9 NUM iajs-2624 45 3 ] ] PUNCT iajs-2624 45 4 have have AUX iajs-2624 45 5 introduced introduce VERB iajs-2624 45 6 epidemic epidemic NOUN iajs-2624 45 7 model model NOUN iajs-2624 45 8 with with ADP iajs-2624 45 9 ( ( PUNCT iajs-2624 45 10 seir seir NOUN iajs-2624 45 11 ) ) PUNCT iajs-2624 45 12 type type NOUN iajs-2624 45 13 to to PART iajs-2624 45 14 study study VERB iajs-2624 45 15 the the DET iajs-2624 45 16 transmission transmission NOUN iajs-2624 45 17 dynamic dynamic ADJ iajs-2624 45 18 and and CCONJ iajs-2624 45 19 global global ADJ iajs-2624 45 20 spread spread NOUN iajs-2624 45 21 of of ADP iajs-2624 45 22 the the DET iajs-2624 45 23 disease disease NOUN iajs-2624 45 24 depending depend VERB iajs-2624 45 25 on on ADP iajs-2624 45 26 reported reported ADJ iajs-2624 45 27 data datum NOUN iajs-2624 45 28 from from ADP iajs-2624 45 29 december december PROPN iajs-2624 45 30 31 31 NUM iajs-2624 45 31 , , PUNCT iajs-2624 45 32 2019 2019 NUM iajs-2624 45 33 to to ADP iajs-2624 45 34 january january PROPN iajs-2624 45 35 28 28 NUM iajs-2624 45 36 , , PUNCT iajs-2624 45 37 2020 2020 NUM iajs-2624 45 38 . . PUNCT iajs-2624 46 1 mohsen mohsen PROPN iajs-2624 46 2 et et PROPN iajs-2624 46 3 al al PROPN iajs-2624 46 4 . . PUNCT iajs-2624 47 1 [ [ X iajs-2624 47 2 10 10 NUM iajs-2624 47 3 ] ] PUNCT iajs-2624 47 4 have have AUX iajs-2624 47 5 introduced introduce VERB iajs-2624 47 6 a a DET iajs-2624 47 7 mathematical mathematical ADJ iajs-2624 47 8 model model NOUN iajs-2624 47 9 for for ADP iajs-2624 47 10 covid-19 covid-19 PROPN iajs-2624 47 11 pandemic pandemic NOUN iajs-2624 47 12 involving involve VERB iajs-2624 47 13 the the DET iajs-2624 47 14 quarantine quarantine NOUN iajs-2624 47 15 strategy strategy NOUN iajs-2624 47 16 and and CCONJ iajs-2624 47 17 media medium NOUN iajs-2624 47 18 coverage coverage NOUN iajs-2624 47 19 effects effect NOUN iajs-2624 47 20 . . PUNCT iajs-2624 48 1 abdulkadhim abdulkadhim PROPN iajs-2624 48 2 and and CCONJ iajs-2624 48 3 alhusseiny alhusseiny NOUN iajs-2624 49 1 [ [ X iajs-2624 49 2 11 11 NUM iajs-2624 49 3 ] ] PUNCT iajs-2624 49 4 have have AUX iajs-2624 49 5 studied study VERB iajs-2624 49 6 the the DET iajs-2624 49 7 global global ADJ iajs-2624 49 8 stability stability NOUN iajs-2624 49 9 and and CCONJ iajs-2624 49 10 bifurcation bifurcation NOUN iajs-2624 49 11 of of ADP iajs-2624 49 12 a a DET iajs-2624 49 13 covid-19 covid-19 PROPN iajs-2624 49 14 virus virus NOUN iajs-2624 49 15 modeling modeling NOUN iajs-2624 49 16 with with ADP iajs-2624 49 17 possible possible ADJ iajs-2624 49 18 loss loss NOUN iajs-2624 49 19 of of ADP iajs-2624 49 20 the the DET iajs-2624 49 21 immunity immunity NOUN iajs-2624 49 22 . . PUNCT iajs-2624 50 1 kucharski kucharski VERB iajs-2624 50 2 et et PROPN iajs-2624 50 3 al al PROPN iajs-2624 50 4 . . PUNCT iajs-2624 51 1 [ [ X iajs-2624 51 2 12 12 NUM iajs-2624 51 3 ] ] PUNCT iajs-2624 51 4 have have AUX iajs-2624 51 5 discussed discuss VERB iajs-2624 51 6 early early ADJ iajs-2624 51 7 dynamics dynamic NOUN iajs-2624 51 8 of of ADP iajs-2624 51 9 transmission transmission NOUN iajs-2624 51 10 and and CCONJ iajs-2624 51 11 control control NOUN iajs-2624 51 12 of of ADP iajs-2624 51 13 covid-19 covid-19 PROPN iajs-2624 51 14 in in ADP iajs-2624 51 15 a a DET iajs-2624 51 16 mathematical mathematical ADJ iajs-2624 51 17 model model NOUN iajs-2624 51 18 . . PUNCT iajs-2624 52 1 44 44 NUM iajs-2624 53 1 ibn ibn PROPN iajs-2624 53 2 al al PROPN iajs-2624 53 3 - - PUNCT iajs-2624 53 4 haitham haitham PROPN iajs-2624 53 5 jour jour X iajs-2624 53 6 . . PROPN iajs-2624 53 7 for for ADP iajs-2624 53 8 pure pure ADJ iajs-2624 53 9 & & CCONJ iajs-2624 53 10 appl appl PROPN iajs-2624 53 11 . . PUNCT iajs-2624 54 1 sci sci PROPN iajs-2624 54 2 . . PROPN iajs-2624 55 1 34 34 NUM iajs-2624 55 2 ( ( PUNCT iajs-2624 55 3 2 2 NUM iajs-2624 55 4 ) ) PUNCT iajs-2624 55 5 2021 2021 NUM iajs-2624 55 6 in in ADP iajs-2624 55 7 this this DET iajs-2624 55 8 study study NOUN iajs-2624 55 9 , , PUNCT iajs-2624 55 10 the the DET iajs-2624 55 11 researchers researcher NOUN iajs-2624 55 12 have have AUX iajs-2624 55 13 proposed propose VERB iajs-2624 55 14 and and CCONJ iajs-2624 55 15 studied study VERB iajs-2624 55 16 the the DET iajs-2624 55 17 mathematical mathematical ADJ iajs-2624 55 18 model model NOUN iajs-2624 55 19 of of ADP iajs-2624 55 20 the the DET iajs-2624 55 21 novel novel ADJ iajs-2624 55 22 coronavirus coronavirus NOUN iajs-2624 55 23 involving involve VERB iajs-2624 55 24 awareness awareness NOUN iajs-2624 55 25 programs program NOUN iajs-2624 55 26 . . PUNCT iajs-2624 56 1 details detail NOUN iajs-2624 56 2 about about ADP iajs-2624 56 3 the the DET iajs-2624 56 4 mathematical mathematical ADJ iajs-2624 56 5 modeling modeling NOUN iajs-2624 56 6 of of ADP iajs-2624 56 7 the the DET iajs-2624 56 8 novel novel ADJ iajs-2624 56 9 coronavirus coronavirus NOUN iajs-2624 56 10 are be AUX iajs-2624 56 11 shown show VERB iajs-2624 56 12 in in ADP iajs-2624 56 13 section section NOUN iajs-2624 56 14 2 2 NUM iajs-2624 56 15 . . PUNCT iajs-2624 57 1 some some DET iajs-2624 57 2 basic basic ADJ iajs-2624 57 3 properties property NOUN iajs-2624 57 4 ( ( PUNCT iajs-2624 57 5 existence existence NOUN iajs-2624 57 6 equilibrium equilibrium NOUN iajs-2624 57 7 points point NOUN iajs-2624 57 8 and and CCONJ iajs-2624 57 9 calculated calculate VERB iajs-2624 57 10 reproduction reproduction NOUN iajs-2624 57 11 number number NOUN iajs-2624 57 12 ) ) PUNCT iajs-2624 57 13 of of ADP iajs-2624 57 14 the the DET iajs-2624 57 15 model model NOUN iajs-2624 57 16 are be AUX iajs-2624 57 17 discussed discuss VERB iajs-2624 57 18 in in ADP iajs-2624 57 19 section section NOUN iajs-2624 57 20 3 3 NUM iajs-2624 57 21 . . PUNCT iajs-2624 58 1 the the DET iajs-2624 58 2 local local ADJ iajs-2624 58 3 stability stability NOUN iajs-2624 58 4 analysis analysis NOUN iajs-2624 58 5 is be AUX iajs-2624 58 6 studied study VERB iajs-2624 58 7 by by ADP iajs-2624 58 8 using use VERB iajs-2624 58 9 gersgorin gersgorin ADJ iajs-2624 58 10 theorem theorem NOUN iajs-2624 58 11 in in ADP iajs-2624 58 12 section section NOUN iajs-2624 58 13 4 4 NUM iajs-2624 58 14 . . PUNCT iajs-2624 59 1 the the DET iajs-2624 59 2 cases case NOUN iajs-2624 59 3 of of ADP iajs-2624 59 4 the the DET iajs-2624 59 5 backward backward ADJ iajs-2624 59 6 bifurcation bifurcation NOUN iajs-2624 59 7 occurrence occurrence NOUN iajs-2624 59 8 are be AUX iajs-2624 59 9 shown show VERB iajs-2624 59 10 in in ADP iajs-2624 59 11 section section NOUN iajs-2624 59 12 5 5 NUM iajs-2624 59 13 . . PUNCT iajs-2624 59 14 by by ADP iajs-2624 59 15 using use VERB iajs-2624 59 16 lyapunov lyapunov ADJ iajs-2624 59 17 functions function NOUN iajs-2624 59 18 , , PUNCT iajs-2624 59 19 the the DET iajs-2624 59 20 researchers researcher NOUN iajs-2624 59 21 have have AUX iajs-2624 59 22 also also ADV iajs-2624 59 23 studied study VERB iajs-2624 59 24 the the DET iajs-2624 59 25 global global ADJ iajs-2624 59 26 stability stability NOUN iajs-2624 59 27 of of ADP iajs-2624 59 28 the the DET iajs-2624 59 29 proposed propose VERB iajs-2624 59 30 model model NOUN iajs-2624 59 31 at at ADP iajs-2624 59 32 all all DET iajs-2624 59 33 equilibrium equilibrium NOUN iajs-2624 59 34 points point NOUN iajs-2624 59 35 as as SCONJ iajs-2624 59 36 shown show VERB iajs-2624 59 37 in in ADP iajs-2624 59 38 section section NOUN iajs-2624 59 39 6 6 NUM iajs-2624 59 40 . . PUNCT iajs-2624 60 1 finally finally ADV iajs-2624 60 2 , , PUNCT iajs-2624 60 3 the the DET iajs-2624 60 4 effect effect NOUN iajs-2624 60 5 of of ADP iajs-2624 60 6 media medium NOUN iajs-2624 60 7 coverage coverage NOUN iajs-2624 60 8 to to PART iajs-2624 60 9 raise raise VERB iajs-2624 60 10 the the DET iajs-2624 60 11 people people NOUN iajs-2624 60 12 's 's PART iajs-2624 60 13 awareness awareness NOUN iajs-2624 60 14 to to ADP iajs-2624 60 15 the the DET iajs-2624 60 16 risk risk NOUN iajs-2624 60 17 of of ADP iajs-2624 60 18 the the DET iajs-2624 60 19 coronavirus coronavirus NOUN iajs-2624 60 20 and and CCONJ iajs-2624 60 21 the the DET iajs-2624 60 22 effect effect NOUN iajs-2624 60 23 of of ADP iajs-2624 60 24 the the DET iajs-2624 60 25 direct direct ADJ iajs-2624 60 26 contact contact NOUN iajs-2624 60 27 with with ADP iajs-2624 60 28 carrier carrier NOUN iajs-2624 60 29 individuals individual NOUN iajs-2624 60 30 on on ADP iajs-2624 60 31 breaking break VERB iajs-2624 60 32 out out ADP iajs-2624 60 33 the the DET iajs-2624 60 34 covid-19 covid-19 PROPN iajs-2624 60 35 among among ADP iajs-2624 60 36 people people NOUN iajs-2624 60 37 are be AUX iajs-2624 60 38 shown show VERB iajs-2624 60 39 by by ADP iajs-2624 60 40 numerical numerical ADJ iajs-2624 60 41 simulation simulation NOUN iajs-2624 60 42 of of ADP iajs-2624 60 43 the the DET iajs-2624 60 44 proposed propose VERB iajs-2624 60 45 model model NOUN iajs-2624 60 46 . . PUNCT iajs-2624 61 1 2 2 X iajs-2624 61 2 . . X iajs-2624 61 3 model model NOUN iajs-2624 61 4 formulation formulation NOUN iajs-2624 61 5 the the DET iajs-2624 61 6 researchers researcher NOUN iajs-2624 61 7 have have AUX iajs-2624 61 8 formulated formulate VERB iajs-2624 61 9 the the DET iajs-2624 61 10 coronavirus coronavirus NOUN iajs-2624 61 11 mathematical mathematical ADJ iajs-2624 61 12 model model NOUN iajs-2624 61 13 involving involve VERB iajs-2624 61 14 awareness awareness NOUN iajs-2624 61 15 . . PUNCT iajs-2624 62 1 it it PRON iajs-2624 62 2 is be AUX iajs-2624 62 3 assumed assume VERB iajs-2624 62 4 that that SCONJ iajs-2624 62 5 the the DET iajs-2624 62 6 total total ADJ iajs-2624 62 7 population population NOUN iajs-2624 62 8 denoted denote VERB iajs-2624 62 9 by by ADP iajs-2624 62 10 n n PRON iajs-2624 62 11 is be AUX iajs-2624 62 12 subdivided subdivide VERB iajs-2624 62 13 into into ADP iajs-2624 62 14 four four NUM iajs-2624 62 15 compartments compartment NOUN iajs-2624 62 16 : : PUNCT iajs-2624 62 17 unaware unaware ADJ iajs-2624 62 18 susceptible susceptible ADJ iajs-2624 62 19 individuals individual NOUN iajs-2624 62 20 denoted denote VERB iajs-2624 62 21 by by ADP iajs-2624 62 22 ( ( PUNCT iajs-2624 62 23 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 62 24 ) ) PUNCT iajs-2624 62 25 , , PUNCT iajs-2624 62 26 aware aware ADJ iajs-2624 62 27 susceptible susceptible ADJ iajs-2624 62 28 individuals individual NOUN iajs-2624 62 29 denoted denote VERB iajs-2624 62 30 by by ADP iajs-2624 62 31 ( ( PUNCT iajs-2624 62 32 π‘†π‘Ž π‘†π‘Ž NOUN iajs-2624 62 33 ) ) PUNCT iajs-2624 62 34 , , PUNCT iajs-2624 62 35 asymptomatic asymptomatic ADJ iajs-2624 62 36 individuals individual NOUN iajs-2624 62 37 ( ( PUNCT iajs-2624 62 38 carrier carrier NOUN iajs-2624 62 39 ) ) PUNCT iajs-2624 62 40 denoted denote VERB iajs-2624 62 41 by by ADP iajs-2624 62 42 ( ( PUNCT iajs-2624 62 43 𝐢 𝐢 PROPN iajs-2624 62 44 ) ) PUNCT iajs-2624 62 45 , , PUNCT iajs-2624 62 46 and and CCONJ iajs-2624 62 47 infected infected ADJ iajs-2624 62 48 individuals individual NOUN iajs-2624 62 49 ( ( PUNCT iajs-2624 62 50 symptomatic symptomatic ADJ iajs-2624 62 51 ) ) PUNCT iajs-2624 62 52 denoted denote VERB iajs-2624 62 53 by by ADP iajs-2624 62 54 ( ( PUNCT iajs-2624 62 55 𝐼 𝐼 PROPN iajs-2624 62 56 ) ) PUNCT iajs-2624 62 57 , , PUNCT iajs-2624 62 58 recovery recovery NOUN iajs-2624 62 59 individuals individual NOUN iajs-2624 62 60 denoted denote VERB iajs-2624 62 61 by by ADP iajs-2624 62 62 𝑅(𝑑 𝑅(𝑑 NOUN iajs-2624 62 63 ) ) PUNCT iajs-2624 62 64 . . PUNCT iajs-2624 63 1 thus thus ADV iajs-2624 63 2 , , PUNCT iajs-2624 63 3 𝑁 𝑁 PROPN iajs-2624 63 4 = = SYM iajs-2624 63 5 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 63 6 + + CCONJ iajs-2624 63 7 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 63 8 + + CCONJ iajs-2624 63 9 𝐢 𝐢 PROPN iajs-2624 63 10 + + CCONJ iajs-2624 63 11 𝐼 𝐼 PROPN iajs-2624 63 12 + + CCONJ iajs-2624 63 13 𝑅. 𝑅. ADV iajs-2624 63 14 let let NOUN iajs-2624 63 15 ( ( PUNCT iajs-2624 63 16 𝑣 𝑣 X iajs-2624 63 17 ) ) PUNCT iajs-2624 63 18 be be AUX iajs-2624 63 19 the the DET iajs-2624 63 20 coronavirus coronavirus NOUN iajs-2624 63 21 resources resource NOUN iajs-2624 63 22 in in ADP iajs-2624 63 23 ( ( PUNCT iajs-2624 63 24 host host NOUN iajs-2624 63 25 ) ) PUNCT iajs-2624 63 26 . . PUNCT iajs-2624 64 1 this this DET iajs-2624 64 2 model model NOUN iajs-2624 64 3 describes describe VERB iajs-2624 64 4 covid-19 covid-19 PROPN iajs-2624 64 5 virus virus NOUN iajs-2624 64 6 by by ADP iajs-2624 64 7 the the DET iajs-2624 64 8 following follow VERB iajs-2624 64 9 equations equation NOUN iajs-2624 64 10 : : PUNCT iajs-2624 65 1 𝑆 𝑆 PROPN iajs-2624 65 2 οΏ½ οΏ½ PROPN iajs-2624 65 3 Μ‡ Μ‡ VERB iajs-2624 65 4 οΏ½ οΏ½ PROPN iajs-2624 65 5 = = SYM iajs-2624 65 6 πœ“ πœ“ PROPN iajs-2624 65 7 βˆ’ βˆ’ PROPN iajs-2624 65 8 𝛼𝑆𝑒 𝛼𝑆𝑒 NOUN iajs-2624 65 9 βˆ’ βˆ’ PROPN iajs-2624 65 10 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 65 11 βˆ’ βˆ’ ADP iajs-2624 66 1 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ NOUN iajs-2624 66 2 βˆ’ βˆ’ PROPN iajs-2624 66 3 πœ‡π‘†π‘’ πœ‡π‘†π‘’ NUM iajs-2624 66 4 𝑆 𝑆 PROPN iajs-2624 66 5 οΏ½ οΏ½ PROPN iajs-2624 66 6 Μ‡ Μ‡ NOUN iajs-2624 66 7 οΏ½ οΏ½ PROPN iajs-2624 66 8 = = PUNCT iajs-2624 66 9 𝛼𝑆𝑒 𝛼𝑆𝑒 PROPN iajs-2624 66 10 βˆ’ βˆ’ PROPN iajs-2624 66 11 𝛽1(1 𝛽1(1 PROPN iajs-2624 66 12 βˆ’ βˆ’ PROPN iajs-2624 66 13 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ PROPN iajs-2624 66 14 βˆ’ βˆ’ PROPN iajs-2624 66 15 𝜎1(1 𝜎1(1 NOUN iajs-2624 66 16 βˆ’ βˆ’ PROPN iajs-2624 66 17 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 66 18 βˆ’ βˆ’ PROPN iajs-2624 66 19 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 66 20 οΏ½ οΏ½ NOUN iajs-2624 66 21 Μ‡ Μ‡ VERB iajs-2624 66 22 οΏ½ οΏ½ PROPN iajs-2624 66 23 = = PUNCT iajs-2624 66 24 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 66 25 + + NUM iajs-2624 66 26 𝛽1(1 𝛽1(1 PROPN iajs-2624 66 27 βˆ’ βˆ’ PROPN iajs-2624 66 28 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ NOUN iajs-2624 66 29 + + CCONJ iajs-2624 66 30 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ X iajs-2624 66 31 + + CCONJ iajs-2624 66 32 𝜎1(1 𝜎1(1 NOUN iajs-2624 66 33 βˆ’ βˆ’ PROPN iajs-2624 66 34 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 66 35 βˆ’ βˆ’ PROPN iajs-2624 66 36 ( ( PUNCT iajs-2624 66 37 πœ‡ πœ‡ X iajs-2624 66 38 + + X iajs-2624 66 39 πœƒ πœƒ NOUN iajs-2624 66 40 + + CCONJ iajs-2624 66 41 𝛾 𝛾 VERB iajs-2624 66 42 + + CCONJ iajs-2624 66 43 𝛿1)𝐢 𝛿1)𝐢 PROPN iajs-2624 66 44 𝐼̇ 𝐼̇ PROPN iajs-2624 66 45 = = SYM iajs-2624 66 46 𝛾𝐢 𝛾𝐢 NOUN iajs-2624 66 47 βˆ’ βˆ’ PROPN iajs-2624 66 48 ( ( PUNCT iajs-2624 66 49 πœ‡ πœ‡ X iajs-2624 66 50 + + X iajs-2624 66 51 πœƒ πœƒ NOUN iajs-2624 66 52 + + NUM iajs-2624 66 53 𝛿2)𝐼 𝛿2)𝐼 NOUN iajs-2624 66 54 οΏ½ οΏ½ PROPN iajs-2624 66 55 Μ‡ Μ‡ NOUN iajs-2624 66 56 οΏ½ οΏ½ PROPN iajs-2624 66 57 = = PUNCT iajs-2624 66 58 𝛿1𝐢 𝛿1𝐢 PROPN iajs-2624 67 1 + + NOUN iajs-2624 67 2 𝛿2𝐼 𝛿2𝐼 NOUN iajs-2624 67 3 βˆ’ βˆ’ NOUN iajs-2624 67 4 πœ‡π‘… πœ‡π‘… PROPN iajs-2624 67 5 οΏ½ οΏ½ NOUN iajs-2624 67 6 Μ‡ Μ‡ NOUN iajs-2624 67 7 οΏ½ οΏ½ PROPN iajs-2624 67 8 = = SYM iajs-2624 67 9 π‘Ÿπ‘£ π‘Ÿπ‘£ PROPN iajs-2624 67 10 ( ( PUNCT iajs-2624 67 11 1 1 NUM iajs-2624 67 12 βˆ’ βˆ’ PROPN iajs-2624 67 13 𝑣 𝑣 PRON iajs-2624 67 14 π‘˜ π‘˜ NOUN iajs-2624 67 15 ) ) PUNCT iajs-2624 67 16 βˆ’ βˆ’ PROPN iajs-2624 67 17 𝑑𝑣 𝑑𝑣 SYM iajs-2624 67 18 ( ( PUNCT iajs-2624 67 19 1 1 NUM iajs-2624 67 20 ) ) PUNCT iajs-2624 67 21 with with ADP iajs-2624 67 22 initial initial ADJ iajs-2624 67 23 conditions condition NOUN iajs-2624 67 24 : : PUNCT iajs-2624 67 25 𝑆𝑒 𝑆𝑒 NOUN iajs-2624 67 26 > > PUNCT iajs-2624 67 27 0 0 NUM iajs-2624 67 28 , , PUNCT iajs-2624 67 29 π‘†π‘Ž π‘†π‘Ž NOUN iajs-2624 67 30 β‰₯ β‰₯ NOUN iajs-2624 67 31 0 0 NUM iajs-2624 67 32 , , PUNCT iajs-2624 67 33 𝐢 𝐢 PROPN iajs-2624 67 34 β‰₯ β‰₯ NUM iajs-2624 67 35 0 0 NUM iajs-2624 67 36 , , PUNCT iajs-2624 67 37 𝐼 𝐼 ADP iajs-2624 67 38 β‰₯ β‰₯ NOUN iajs-2624 67 39 0 0 NUM iajs-2624 67 40 , , PUNCT iajs-2624 67 41 𝑅 𝑅 PROPN iajs-2624 67 42 β‰₯ β‰₯ NOUN iajs-2624 67 43 0 0 NUM iajs-2624 67 44 , , PUNCT iajs-2624 67 45 𝑣 𝑣 PRON iajs-2624 67 46 β‰₯ β‰₯ NOUN iajs-2624 67 47 0 0 NUM iajs-2624 67 48 . . PUNCT iajs-2624 68 1 in in ADP iajs-2624 68 2 system system NOUN iajs-2624 68 3 ( ( PUNCT iajs-2624 68 4 1 1 X iajs-2624 68 5 ) ) PUNCT iajs-2624 68 6 the the DET iajs-2624 68 7 birth birth NOUN iajs-2624 68 8 rate rate NOUN iajs-2624 68 9 of of ADP iajs-2624 68 10 individuals individual NOUN iajs-2624 68 11 is be AUX iajs-2624 68 12 given𝛹 given𝛹 PROPN iajs-2624 68 13 > > X iajs-2624 68 14 0 0 X iajs-2624 68 15 . . PUNCT iajs-2624 69 1 the the DET iajs-2624 69 2 awareness awareness NOUN iajs-2624 69 3 rate rate NOUN iajs-2624 69 4 is be AUX iajs-2624 69 5 given𝛼 given𝛼 PROPN iajs-2624 69 6 β‰₯ β‰₯ X iajs-2624 69 7 0 0 NUM iajs-2624 69 8 . . PUNCT iajs-2624 70 1 the the DET iajs-2624 70 2 parameters parameter NOUN iajs-2624 70 3 𝛽 𝛽 X iajs-2624 70 4 > > X iajs-2624 70 5 0 0 PROPN iajs-2624 70 6 , , PUNCT iajs-2624 70 7 𝛽1 𝛽1 NOUN iajs-2624 70 8 > > X iajs-2624 70 9 0 0 PROPN iajs-2624 70 10 , , PUNCT iajs-2624 70 11 𝜎 𝜎 PROPN iajs-2624 70 12 > > X iajs-2624 70 13 0 0 PUNCT iajs-2624 70 14 and and CCONJ iajs-2624 70 15 𝜎1 𝜎1 ADJ iajs-2624 70 16 > > X iajs-2624 70 17 0 0 NUM iajs-2624 70 18 respectively respectively ADV iajs-2624 70 19 measure measure VERB iajs-2624 70 20 the the DET iajs-2624 70 21 contact contact NOUN iajs-2624 70 22 rate rate NOUN iajs-2624 70 23 between between ADP iajs-2624 70 24 susceptible susceptible ADJ iajs-2624 70 25 with with ADP iajs-2624 70 26 carrier carrier NOUN iajs-2624 70 27 , , PUNCT iajs-2624 70 28 and and CCONJ iajs-2624 70 29 coronavirus coronavirus NOUN iajs-2624 70 30 with with ADP iajs-2624 70 31 the the DET iajs-2624 70 32 prevention prevention NOUN iajs-2624 70 33 of of ADP iajs-2624 70 34 disease disease NOUN iajs-2624 70 35 that that PRON iajs-2624 70 36 is be AUX iajs-2624 70 37 given give VERB iajs-2624 70 38 a a DET iajs-2624 70 39 rate rate NOUN iajs-2624 70 40 of of ADP iajs-2624 70 41 ( ( PUNCT iajs-2624 70 42 0 0 NUM iajs-2624 70 43 ≀ ≀ NUM iajs-2624 70 44 πœ– πœ– X iajs-2624 70 45 ≀ ≀ NUM iajs-2624 70 46 1 1 NUM iajs-2624 70 47 ) ) PUNCT iajs-2624 70 48 . . PUNCT iajs-2624 71 1 the the DET iajs-2624 71 2 death death NOUN iajs-2624 71 3 rate rate NOUN iajs-2624 71 4 is be AUX iajs-2624 71 5 πœ‡ πœ‡ ADP iajs-2624 71 6 > > X iajs-2624 71 7 0 0 NUM iajs-2624 71 8 . . PUNCT iajs-2624 72 1 the the DET iajs-2624 72 2 death death NOUN iajs-2624 72 3 rate rate NOUN iajs-2624 72 4 due due ADP iajs-2624 72 5 to to ADP iajs-2624 72 6 the the DET iajs-2624 72 7 disease disease NOUN iajs-2624 72 8 is be AUX iajs-2624 72 9 πœƒ πœƒ X iajs-2624 72 10 > > X iajs-2624 72 11 0 0 NUM iajs-2624 72 12 . . PUNCT iajs-2624 73 1 after after ADP iajs-2624 73 2 14th 14th ADJ iajs-2624 73 3 days day NOUN iajs-2624 73 4 , , PUNCT iajs-2624 73 5 symptoms symptom NOUN iajs-2624 73 6 of of ADP iajs-2624 73 7 the the DET iajs-2624 73 8 disease disease NOUN iajs-2624 73 9 begin begin VERB iajs-2624 73 10 to to PART iajs-2624 73 11 appear appear VERB iajs-2624 73 12 on on ADP iajs-2624 73 13 the the DET iajs-2624 73 14 carrier carrier NOUN iajs-2624 73 15 that that PRON iajs-2624 73 16 is be AUX iajs-2624 73 17 given give VERB iajs-2624 73 18 a a DET iajs-2624 73 19 rate rate NOUN iajs-2624 73 20 of of ADP iajs-2624 73 21 𝛾 𝛾 PROPN iajs-2624 73 22 > > X iajs-2624 73 23 0 0 X iajs-2624 73 24 . . PUNCT iajs-2624 74 1 the the DET iajs-2624 74 2 recovery recovery NOUN iajs-2624 74 3 rates rate NOUN iajs-2624 74 4 from from ADP iajs-2624 74 5 carrier carrier NOUN iajs-2624 74 6 and and CCONJ iajs-2624 74 7 infected infected ADJ iajs-2624 74 8 individuals individual NOUN iajs-2624 74 9 are be AUX iajs-2624 74 10 𝛿𝑖 𝛿𝑖 ADP iajs-2624 74 11 > > X iajs-2624 74 12 0 0 PUNCT iajs-2624 74 13 , , PUNCT iajs-2624 74 14 𝑖 𝑖 SYM iajs-2624 74 15 = = NOUN iajs-2624 74 16 1,2 1,2 NUM iajs-2624 74 17 respectively respectively ADV iajs-2624 74 18 . . PUNCT iajs-2624 75 1 π‘Ÿ π‘Ÿ X iajs-2624 75 2 > > X iajs-2624 75 3 0 0 NUM iajs-2624 75 4 , , PUNCT iajs-2624 75 5 π‘˜ π‘˜ X iajs-2624 75 6 > > X iajs-2624 75 7 0 0 PUNCT iajs-2624 75 8 and and CCONJ iajs-2624 75 9 𝑑 𝑑 X iajs-2624 75 10 > > X iajs-2624 75 11 0 0 PUNCT iajs-2624 75 12 respectively respectively ADV iajs-2624 75 13 , , PUNCT iajs-2624 75 14 represent represent VERB iajs-2624 75 15 the the DET iajs-2624 75 16 spread spread ADJ iajs-2624 75 17 rate rate NOUN iajs-2624 75 18 of of ADP iajs-2624 75 19 the the DET iajs-2624 75 20 virus virus NOUN iajs-2624 75 21 , , PUNCT iajs-2624 75 22 the the DET iajs-2624 75 23 carrying carry VERB iajs-2624 75 24 capacity capacity NOUN iajs-2624 75 25 of of ADP iajs-2624 75 26 coronavirus coronavirus NOUN iajs-2624 75 27 and and CCONJ iajs-2624 75 28 the the DET iajs-2624 75 29 removal removal NOUN iajs-2624 75 30 rate rate NOUN iajs-2624 75 31 of of ADP iajs-2624 75 32 virus virus NOUN iajs-2624 75 33 . . PUNCT iajs-2624 76 1 theorem theorem NOUN iajs-2624 76 2 ( ( PUNCT iajs-2624 76 3 1 1 NUM iajs-2624 76 4 ): ): PUNCT iajs-2624 76 5 all all DET iajs-2624 76 6 solutions solution NOUN iajs-2624 76 7 of of ADP iajs-2624 76 8 system system NOUN iajs-2624 76 9 ( ( PUNCT iajs-2624 76 10 1 1 X iajs-2624 76 11 ) ) PUNCT iajs-2624 76 12 which which PRON iajs-2624 76 13 initiate initiate VERB iajs-2624 76 14 in in ADP iajs-2624 76 15 𝑅+ 𝑅+ PROPN iajs-2624 76 16 5 5 NUM iajs-2624 76 17 are be AUX iajs-2624 76 18 uniformly uniformly ADV iajs-2624 76 19 bounded bound VERB iajs-2624 76 20 . . PUNCT iajs-2624 77 1 proof proof NOUN iajs-2624 77 2 : : PUNCT iajs-2624 77 3 let let VERB iajs-2624 77 4 𝑀(𝑇 𝑀(𝑇 NUM iajs-2624 77 5 ) ) PUNCT iajs-2624 78 1 = = PRON iajs-2624 78 2 ( ( PUNCT iajs-2624 78 3 𝑁(𝑑 𝑁(𝑑 NUM iajs-2624 78 4 ) ) PUNCT iajs-2624 78 5 , , PUNCT iajs-2624 78 6 𝑣(𝑑 𝑣(𝑑 NOUN iajs-2624 78 7 ) ) PUNCT iajs-2624 78 8 ) ) PUNCT iajs-2624 79 1 where where SCONJ iajs-2624 79 2 n(t n(t NOUN iajs-2624 79 3 ) ) PUNCT iajs-2624 79 4 = = PUNCT iajs-2624 79 5 su(t su(t PROPN iajs-2624 79 6 ) ) PUNCT iajs-2624 79 7 + + CCONJ iajs-2624 79 8 π‘†π‘Ž(t π‘†π‘Ž(t NOUN iajs-2624 79 9 ) ) PUNCT iajs-2624 79 10 + + NOUN iajs-2624 79 11 c(t c(t PROPN iajs-2624 79 12 ) ) PUNCT iajs-2624 79 13 , , PUNCT iajs-2624 80 1 + + NOUN iajs-2624 80 2 i(t i(t NOUN iajs-2624 80 3 ) ) PUNCT iajs-2624 80 4 + + NUM iajs-2624 80 5 r(t r(t NOUN iajs-2624 80 6 ) ) PUNCT iajs-2624 80 7 ; ; PUNCT iajs-2624 80 8 then then ADV iajs-2624 80 9 π‘‘π‘Š π‘‘π‘Š PROPN iajs-2624 80 10 𝑑𝑑 𝑑𝑑 ADP iajs-2624 80 11 = = SYM iajs-2624 80 12 ( ( PUNCT iajs-2624 80 13 𝑑𝑁 𝑑𝑁 PROPN iajs-2624 80 14 𝑑𝑑 𝑑𝑑 ADP iajs-2624 80 15 , , PUNCT iajs-2624 80 16 𝑑𝑣 𝑑𝑣 X iajs-2624 80 17 𝑑𝑑 𝑑𝑑 ADP iajs-2624 80 18 ) ) PUNCT iajs-2624 80 19 = = SYM iajs-2624 81 1 ( ( PUNCT iajs-2624 81 2 πœ“ πœ“ PROPN iajs-2624 81 3 βˆ’ βˆ’ PROPN iajs-2624 81 4 πœ‡su πœ‡su NOUN iajs-2624 81 5 βˆ’ βˆ’ PROPN iajs-2624 81 6 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 81 7 βˆ’ βˆ’ PROPN iajs-2624 81 8 ( ( PUNCT iajs-2624 81 9 πœ‡ πœ‡ X iajs-2624 81 10 + + X iajs-2624 81 11 πœƒ)𝐢 πœƒ)𝐢 ADJ iajs-2624 81 12 βˆ’ βˆ’ X iajs-2624 81 13 ( ( PUNCT iajs-2624 81 14 πœ‡ πœ‡ ADP iajs-2624 81 15 + + X iajs-2624 81 16 πœƒ)𝐼 πœƒ)𝐼 X iajs-2624 81 17 βˆ’ βˆ’ PROPN iajs-2624 81 18 πœ‡π‘… πœ‡π‘… NOUN iajs-2624 81 19 , , PUNCT iajs-2624 81 20 π‘Ÿπ‘£ π‘Ÿπ‘£ PROPN iajs-2624 81 21 ( ( PUNCT iajs-2624 81 22 1 1 NUM iajs-2624 81 23 βˆ’ βˆ’ PROPN iajs-2624 81 24 𝑣 𝑣 PRON iajs-2624 81 25 𝐾 𝐾 PROPN iajs-2624 81 26 ) ) PUNCT iajs-2624 81 27 βˆ’ βˆ’ PROPN iajs-2624 81 28 𝑑𝑣 𝑑𝑣 NOUN iajs-2624 81 29 ) ) PUNCT iajs-2624 81 30 45 45 NUM iajs-2624 81 31 ibn ibn PROPN iajs-2624 81 32 al al PROPN iajs-2624 81 33 - - PUNCT iajs-2624 81 34 haitham haitham PROPN iajs-2624 81 35 jour jour X iajs-2624 81 36 . . PROPN iajs-2624 82 1 for for ADP iajs-2624 82 2 pure pure ADJ iajs-2624 82 3 & & CCONJ iajs-2624 82 4 appl appl PROPN iajs-2624 82 5 . . PUNCT iajs-2624 83 1 sci sci PROPN iajs-2624 83 2 . . PROPN iajs-2624 84 1 34 34 NUM iajs-2624 84 2 ( ( PUNCT iajs-2624 84 3 2 2 NUM iajs-2624 84 4 ) ) PUNCT iajs-2624 84 5 2021 2021 NUM iajs-2624 84 6 we we PRON iajs-2624 84 7 know know VERB iajs-2624 84 8 that that DET iajs-2624 84 9 𝑑𝑁 𝑑𝑁 NOUN iajs-2624 84 10 𝑑𝑑 𝑑𝑑 ADP iajs-2624 84 11 ≀ ≀ NUM iajs-2624 84 12 πœ“ πœ“ ADP iajs-2624 84 13 βˆ’ βˆ’ PROPN iajs-2624 84 14 π‘žπ‘ π‘žπ‘ PROPN iajs-2624 84 15 , , PUNCT iajs-2624 84 16 where where SCONJ iajs-2624 84 17 π‘ž π‘ž PROPN iajs-2624 84 18 = = SYM iajs-2624 84 19 π‘šπ‘–π‘›{πœ‡ π‘šπ‘–π‘›{πœ‡ ADJ iajs-2624 84 20 , , PUNCT iajs-2624 84 21 πœ‡ πœ‡ ADP iajs-2624 84 22 + + PROPN iajs-2624 84 23 πœƒ πœƒ NOUN iajs-2624 84 24 , , PUNCT iajs-2624 84 25 𝑑 𝑑 NOUN iajs-2624 84 26 } } PUNCT iajs-2624 84 27 which which PRON iajs-2624 84 28 implies imply VERB iajs-2624 84 29 that that SCONJ iajs-2624 84 30 lim lim PROPN iajs-2624 84 31 π‘‘β†’βˆž π‘‘β†’βˆž NUM iajs-2624 84 32 𝑠𝑒𝑝𝑁(𝑑 𝑠𝑒𝑝𝑁(𝑑 PROPN iajs-2624 84 33 ) ) PUNCT iajs-2624 84 34 ≀ ≀ PUNCT iajs-2624 84 35 πœ“ πœ“ ADP iajs-2624 84 36 π‘ž π‘ž NOUN iajs-2624 84 37 while while SCONJ iajs-2624 84 38 , , PUNCT iajs-2624 84 39 the the DET iajs-2624 84 40 last last ADJ iajs-2624 84 41 equation equation NOUN iajs-2624 84 42 of of ADP iajs-2624 84 43 system system NOUN iajs-2624 84 44 ( ( PUNCT iajs-2624 84 45 1 1 X iajs-2624 84 46 ) ) PUNCT iajs-2624 84 47 it it PRON iajs-2624 84 48 follows follow VERB iajs-2624 84 49 that that DET iajs-2624 84 50 sup sup NOUN iajs-2624 84 51 ( ( PUNCT iajs-2624 84 52 π‘Ÿπ‘£ π‘Ÿπ‘£ PROPN iajs-2624 84 53 ( ( PUNCT iajs-2624 84 54 1 1 NUM iajs-2624 84 55 βˆ’ βˆ’ PROPN iajs-2624 84 56 𝑣 𝑣 ADP iajs-2624 84 57 π‘˜ π‘˜ PROPN iajs-2624 84 58 ) ) PUNCT iajs-2624 84 59 ) ) PUNCT iajs-2624 84 60 ≀ ≀ NUM iajs-2624 84 61 π‘Ÿπ‘˜ π‘Ÿπ‘˜ ADP iajs-2624 84 62 4 4 NUM iajs-2624 84 63 hence hence ADV iajs-2624 84 64 𝑑𝑣 𝑑𝑣 ADP iajs-2624 84 65 𝑑𝑑 𝑑𝑑 ADP iajs-2624 84 66 ≀ ≀ NUM iajs-2624 84 67 π‘Ÿπ‘˜ π‘Ÿπ‘˜ ADP iajs-2624 84 68 4 4 NUM iajs-2624 84 69 βˆ’ βˆ’ NOUN iajs-2624 84 70 𝑑𝑣 𝑑𝑣 ADP iajs-2624 84 71 so so SCONJ iajs-2624 84 72 that that SCONJ iajs-2624 84 73 lim lim PROPN iajs-2624 84 74 π‘‘β†’βˆž π‘‘β†’βˆž NUM iajs-2624 84 75 sup sup PROPN iajs-2624 84 76 𝑣(𝑑 𝑣(𝑑 NOUN iajs-2624 84 77 ) ) PUNCT iajs-2624 84 78 ≀ ≀ NUM iajs-2624 84 79 π‘Ÿπ‘˜ π‘Ÿπ‘˜ ADP iajs-2624 84 80 4𝑑 4𝑑 NOUN iajs-2624 84 81 this this PRON iajs-2624 84 82 completes complete VERB iajs-2624 84 83 the the DET iajs-2624 84 84 proof proof NOUN iajs-2624 84 85 of of ADP iajs-2624 84 86 theorem theorem ADJ iajs-2624 84 87 β–  β–  NOUN iajs-2624 85 1 3 3 X iajs-2624 85 2 . . NOUN iajs-2624 85 3 existence existence NOUN iajs-2624 85 4 of of ADP iajs-2624 85 5 equilibrium equilibrium NOUN iajs-2624 85 6 points point NOUN iajs-2624 85 7 it it PRON iajs-2624 85 8 is be AUX iajs-2624 85 9 easy easy ADJ iajs-2624 85 10 that that SCONJ iajs-2624 85 11 the the DET iajs-2624 85 12 infected infect VERB iajs-2624 85 13 and and CCONJ iajs-2624 85 14 recovery recovery NOUN iajs-2624 85 15 population population NOUN iajs-2624 85 16 i i PRON iajs-2624 85 17 and and CCONJ iajs-2624 85 18 r r NOUN iajs-2624 85 19 , , PUNCT iajs-2624 85 20 are be AUX iajs-2624 85 21 related relate VERB iajs-2624 85 22 with with ADP iajs-2624 85 23 carrier carrier NOUN iajs-2624 85 24 population population NOUN iajs-2624 85 25 only only ADV iajs-2624 85 26 . . PUNCT iajs-2624 86 1 hence hence ADV iajs-2624 86 2 for for ADP iajs-2624 86 3 fixed fix VERB iajs-2624 86 4 value value NOUN iajs-2624 86 5 of of ADP iajs-2624 86 6 c c PROPN iajs-2624 86 7 , , PUNCT iajs-2624 86 8 the the DET iajs-2624 86 9 values value NOUN iajs-2624 86 10 of of ADP iajs-2624 86 11 i i PRON iajs-2624 86 12 and and CCONJ iajs-2624 86 13 r r NOUN iajs-2624 86 14 can can AUX iajs-2624 86 15 be be AUX iajs-2624 86 16 determined determine VERB iajs-2624 86 17 directly directly ADV iajs-2624 86 18 by by ADP iajs-2624 86 19 solving solve VERB iajs-2624 86 20 the the DET iajs-2624 86 21 fourth fourth ADJ iajs-2624 86 22 equation equation NOUN iajs-2624 86 23 in in ADP iajs-2624 86 24 system system NOUN iajs-2624 86 25 ( ( PUNCT iajs-2624 86 26 1 1 NUM iajs-2624 86 27 ) ) PUNCT iajs-2624 86 28 . . PUNCT iajs-2624 87 1 then then ADV iajs-2624 87 2 , , PUNCT iajs-2624 87 3 we we PRON iajs-2624 87 4 can can AUX iajs-2624 87 5 calculate calculate VERB iajs-2624 87 6 the the DET iajs-2624 87 7 values value NOUN iajs-2624 87 8 of of ADP iajs-2624 87 9 i i PRON iajs-2624 87 10 and and CCONJ iajs-2624 87 11 r r VERB iajs-2624 87 12 the the DET iajs-2624 87 13 following follow VERB iajs-2624 87 14 form form NOUN iajs-2624 87 15 𝐼 𝐼 NOUN iajs-2624 87 16 = = SYM iajs-2624 87 17 π›ΎπΆβˆ— π›ΎπΆβˆ— VERB iajs-2624 87 18 πœ‡+πœƒ πœ‡+πœƒ X iajs-2624 87 19 ( ( PUNCT iajs-2624 87 20 2a 2a NUM iajs-2624 87 21 ) ) PUNCT iajs-2624 87 22 𝑅 𝑅 PROPN iajs-2624 87 23 = = SYM iajs-2624 87 24 𝛿1πΆβˆ—+𝛿2πΌβˆ— 𝛿1πΆβˆ—+𝛿2πΌβˆ— PROPN iajs-2624 87 25 πœ‡ πœ‡ X iajs-2624 87 26 ( ( PUNCT iajs-2624 87 27 2b 2b NOUN iajs-2624 87 28 ) ) PUNCT iajs-2624 87 29 consequently consequently ADV iajs-2624 87 30 , , PUNCT iajs-2624 87 31 for for ADP iajs-2624 87 32 simplifying simplifying NOUN iajs-2624 87 33 , , PUNCT iajs-2624 87 34 system system NOUN iajs-2624 87 35 ( ( PUNCT iajs-2624 87 36 1 1 X iajs-2624 87 37 ) ) PUNCT iajs-2624 87 38 can can AUX iajs-2624 87 39 be be AUX iajs-2624 87 40 reduced reduce VERB iajs-2624 87 41 to to ADP iajs-2624 87 42 the the DET iajs-2624 87 43 following follow VERB iajs-2624 87 44 system system NOUN iajs-2624 87 45 , , PUNCT iajs-2624 87 46 in in ADP iajs-2624 87 47 which which PRON iajs-2624 87 48 we we PRON iajs-2624 87 49 can can AUX iajs-2624 87 50 determine determine VERB iajs-2624 87 51 the the DET iajs-2624 87 52 values value NOUN iajs-2624 87 53 of of ADP iajs-2624 87 54 i i PRON iajs-2624 87 55 and and CCONJ iajs-2624 87 56 r r NOUN iajs-2624 87 57 , , PUNCT iajs-2624 87 58 by by ADP iajs-2624 87 59 solving solve VERB iajs-2624 87 60 it it PRON iajs-2624 87 61 , , PUNCT iajs-2624 87 62 𝑆 𝑆 PROPN iajs-2624 87 63 οΏ½ οΏ½ PROPN iajs-2624 87 64 Μ‡ Μ‡ VERB iajs-2624 87 65 οΏ½ οΏ½ PROPN iajs-2624 87 66 = = SYM iajs-2624 87 67 πœ“ πœ“ PROPN iajs-2624 87 68 βˆ’ βˆ’ PROPN iajs-2624 87 69 𝛼𝑆𝑒 𝛼𝑆𝑒 NOUN iajs-2624 87 70 βˆ’ βˆ’ PROPN iajs-2624 87 71 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 87 72 βˆ’ βˆ’ ADP iajs-2624 88 1 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ NOUN iajs-2624 88 2 βˆ’ βˆ’ PROPN iajs-2624 88 3 πœ‡π‘†π‘’ πœ‡π‘†π‘’ NUM iajs-2624 88 4 𝑆 𝑆 PROPN iajs-2624 88 5 οΏ½ οΏ½ PROPN iajs-2624 88 6 Μ‡ Μ‡ NOUN iajs-2624 88 7 οΏ½ οΏ½ PROPN iajs-2624 88 8 = = PUNCT iajs-2624 88 9 𝛼𝑆𝑒 𝛼𝑆𝑒 PROPN iajs-2624 88 10 βˆ’ βˆ’ PROPN iajs-2624 88 11 𝛽1(1 𝛽1(1 PROPN iajs-2624 88 12 βˆ’ βˆ’ PROPN iajs-2624 88 13 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ PROPN iajs-2624 88 14 βˆ’ βˆ’ PROPN iajs-2624 88 15 𝜎1(1 𝜎1(1 NOUN iajs-2624 88 16 βˆ’ βˆ’ PROPN iajs-2624 88 17 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 88 18 βˆ’ βˆ’ PROPN iajs-2624 88 19 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 88 20 οΏ½ οΏ½ NOUN iajs-2624 88 21 Μ‡ Μ‡ VERB iajs-2624 88 22 οΏ½ οΏ½ PROPN iajs-2624 88 23 = = PUNCT iajs-2624 88 24 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 88 25 + + NUM iajs-2624 88 26 𝛽1(1 𝛽1(1 PROPN iajs-2624 88 27 βˆ’ βˆ’ PROPN iajs-2624 88 28 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ NOUN iajs-2624 88 29 + + CCONJ iajs-2624 88 30 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ X iajs-2624 88 31 + + CCONJ iajs-2624 88 32 𝜎1(1 𝜎1(1 NOUN iajs-2624 88 33 βˆ’ βˆ’ PROPN iajs-2624 88 34 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 88 35 βˆ’ βˆ’ PROPN iajs-2624 88 36 ( ( PUNCT iajs-2624 88 37 πœ‡ πœ‡ X iajs-2624 88 38 + + X iajs-2624 88 39 πœƒ πœƒ NOUN iajs-2624 88 40 + + CCONJ iajs-2624 88 41 𝛾 𝛾 VERB iajs-2624 88 42 + + CCONJ iajs-2624 88 43 𝛿1)𝐢 𝛿1)𝐢 PROPN iajs-2624 88 44 οΏ½ οΏ½ PROPN iajs-2624 88 45 Μ‡ Μ‡ VERB iajs-2624 88 46 οΏ½ οΏ½ PROPN iajs-2624 88 47 = = SYM iajs-2624 88 48 π‘Ÿπ‘£ π‘Ÿπ‘£ PROPN iajs-2624 88 49 ( ( PUNCT iajs-2624 88 50 1 1 NUM iajs-2624 88 51 βˆ’ βˆ’ PROPN iajs-2624 88 52 𝑣 𝑣 PRON iajs-2624 88 53 π‘˜ π‘˜ NOUN iajs-2624 88 54 ) ) PUNCT iajs-2624 88 55 βˆ’ βˆ’ PROPN iajs-2624 88 56 𝑑𝑣 𝑑𝑣 ADP iajs-2624 88 57 ( ( PUNCT iajs-2624 88 58 3 3 NUM iajs-2624 88 59 ) ) PUNCT iajs-2624 88 60 now now ADV iajs-2624 88 61 , , PUNCT iajs-2624 88 62 we we PRON iajs-2624 88 63 can can AUX iajs-2624 88 64 compute compute VERB iajs-2624 88 65 the the DET iajs-2624 88 66 reproduction reproduction NOUN iajs-2624 88 67 number number NOUN iajs-2624 88 68 for for ADP iajs-2624 88 69 the the DET iajs-2624 88 70 given give VERB iajs-2624 88 71 system system NOUN iajs-2624 88 72 ( ( PUNCT iajs-2624 88 73 3 3 NUM iajs-2624 88 74 ) ) PUNCT iajs-2624 88 75 that that PRON iajs-2624 88 76 is be AUX iajs-2624 88 77 denoted denote VERB iajs-2624 88 78 by by ADP iajs-2624 88 79 β„›0 β„›0 PROPN iajs-2624 88 80 , , PUNCT iajs-2624 88 81 such such ADJ iajs-2624 88 82 that that SCONJ iajs-2624 88 83 β„›0 β„›0 NOUN iajs-2624 88 84 = = SYM iajs-2624 88 85 πœŽπœ“ πœŽπœ“ NUM iajs-2624 88 86 πœ‡(πœ‡+πœƒ+𝛾+𝛿1 πœ‡(πœ‡+πœƒ+𝛾+𝛿1 PROPN iajs-2624 88 87 ) ) PUNCT iajs-2624 88 88 ( ( PUNCT iajs-2624 88 89 4 4 NUM iajs-2624 88 90 ) ) PUNCT iajs-2624 88 91 therefore therefore ADV iajs-2624 88 92 , , PUNCT iajs-2624 88 93 system system NOUN iajs-2624 88 94 ( ( PUNCT iajs-2624 88 95 3 3 X iajs-2624 88 96 ) ) PUNCT iajs-2624 88 97 has have VERB iajs-2624 88 98 at at ADP iajs-2624 88 99 most most ADV iajs-2624 88 100 two two NUM iajs-2624 88 101 biologically biologically ADV iajs-2624 88 102 feasible feasible ADJ iajs-2624 88 103 points point NOUN iajs-2624 88 104 , , PUNCT iajs-2624 88 105 namely namely ADV iajs-2624 88 106 , , PUNCT iajs-2624 88 107 𝐸𝑖 𝐸𝑖 PROPN iajs-2624 88 108 = = SYM iajs-2624 88 109 ( ( PUNCT iajs-2624 88 110 𝑆𝑒𝑖 𝑆𝑒𝑖 PROPN iajs-2624 88 111 , , PUNCT iajs-2624 88 112 π‘†π‘Žπ‘– π‘†π‘Žπ‘– PROPN iajs-2624 88 113 , , PUNCT iajs-2624 88 114 𝐢𝑖 𝐢𝑖 PROPN iajs-2624 88 115 , , PUNCT iajs-2624 88 116 𝑣𝑖 𝑣𝑖 NOUN iajs-2624 88 117 ) ) PUNCT iajs-2624 88 118 , , PUNCT iajs-2624 88 119 𝑖 𝑖 PROPN iajs-2624 88 120 = = SYM iajs-2624 88 121 0 0 NUM iajs-2624 88 122 , , PUNCT iajs-2624 88 123 1 1 NUM iajs-2624 88 124 . . PUNCT iajs-2624 89 1 the the DET iajs-2624 89 2 existence existence NOUN iajs-2624 89 3 conditions condition NOUN iajs-2624 89 4 for for ADP iajs-2624 89 5 each each PRON iajs-2624 89 6 of of ADP iajs-2624 89 7 these these DET iajs-2624 89 8 equilibrium equilibrium NOUN iajs-2624 89 9 points point NOUN iajs-2624 89 10 are be AUX iajs-2624 89 11 discussed discuss VERB iajs-2624 89 12 in in ADP iajs-2624 89 13 the the DET iajs-2624 89 14 following following NOUN iajs-2624 89 15 : : PUNCT iajs-2624 89 16 ο‚· ο‚· NOUN iajs-2624 89 17 in in ADP iajs-2624 89 18 the the DET iajs-2624 89 19 absence absence NOUN iajs-2624 89 20 of of ADP iajs-2624 89 21 covid-19 covid-19 PROPN iajs-2624 89 22 virus virus NOUN iajs-2624 89 23 , , PUNCT iajs-2624 89 24 that that PRON iajs-2624 89 25 is be AUX iajs-2624 89 26 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 89 27 = = PUNCT iajs-2624 89 28 𝐢 𝐢 PROPN iajs-2624 89 29 = = PUNCT iajs-2624 89 30 𝑣 𝑣 NOUN iajs-2624 89 31 = = SYM iajs-2624 89 32 0 0 PROPN iajs-2624 89 33 . . PUNCT iajs-2624 90 1 then then ADV iajs-2624 90 2 , , PUNCT iajs-2624 90 3 system system NOUN iajs-2624 90 4 ( ( PUNCT iajs-2624 90 5 3 3 X iajs-2624 90 6 ) ) PUNCT iajs-2624 90 7 has have VERB iajs-2624 90 8 a a DET iajs-2624 90 9 unique unique ADJ iajs-2624 90 10 positive positive ADJ iajs-2624 90 11 equilibrium equilibrium NOUN iajs-2624 90 12 point point NOUN iajs-2624 90 13 , , PUNCT iajs-2624 90 14 namely namely ADV iajs-2624 90 15 covid-19 covid-19 PROPN iajs-2624 90 16 free free ADJ iajs-2624 90 17 equilibrium equilibrium NOUN iajs-2624 90 18 point point NOUN iajs-2624 90 19 , , PUNCT iajs-2624 90 20 which which PRON iajs-2624 90 21 is be AUX iajs-2624 90 22 denoted denote VERB iajs-2624 90 23 by by ADP iajs-2624 90 24 𝐸0 𝐸0 ADJ iajs-2624 90 25 = = SYM iajs-2624 90 26 ( ( PUNCT iajs-2624 90 27 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 90 28 , , PUNCT iajs-2624 90 29 0,0,0 0,0,0 NOUN iajs-2624 90 30 ) ) PUNCT iajs-2624 90 31 where where SCONJ iajs-2624 90 32 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 90 33 = = PUNCT iajs-2624 90 34 πœ“ πœ“ X iajs-2624 90 35 πœ‡ πœ‡ X iajs-2624 90 36 ( ( PUNCT iajs-2624 90 37 5a 5a NUM iajs-2624 90 38 ) ) PUNCT iajs-2624 90 39 provided provide VERB iajs-2624 90 40 that that SCONJ iajs-2624 90 41 the the DET iajs-2624 90 42 following follow VERB iajs-2624 90 43 condition condition NOUN iajs-2624 90 44 holds hold VERB iajs-2624 90 45 𝛼 𝛼 X iajs-2624 90 46 = = SYM iajs-2624 90 47 0 0 NUM iajs-2624 90 48 ( ( PUNCT iajs-2624 90 49 5b 5b NUM iajs-2624 90 50 ) ) PUNCT iajs-2624 90 51 46 46 NUM iajs-2624 90 52 ibn ibn PROPN iajs-2624 90 53 al al PROPN iajs-2624 90 54 - - PUNCT iajs-2624 90 55 haitham haitham PROPN iajs-2624 90 56 jour jour X iajs-2624 90 57 . . PROPN iajs-2624 91 1 for for ADP iajs-2624 91 2 pure pure ADJ iajs-2624 91 3 & & CCONJ iajs-2624 91 4 appl appl PROPN iajs-2624 91 5 . . PUNCT iajs-2624 92 1 sci sci PROPN iajs-2624 92 2 . . PROPN iajs-2624 93 1 34 34 NUM iajs-2624 93 2 ( ( PUNCT iajs-2624 93 3 2 2 NUM iajs-2624 93 4 ) ) PUNCT iajs-2624 93 5 2021 2021 NUM iajs-2624 93 6 ο‚· ο‚· NOUN iajs-2624 93 7 the the DET iajs-2624 93 8 endemic endemic ADJ iajs-2624 93 9 equilibrium equilibrium NOUN iajs-2624 93 10 point point NOUN iajs-2624 93 11 or or CCONJ iajs-2624 93 12 covid-19 covid-19 PROPN iajs-2624 93 13 equilibrium equilibrium NOUN iajs-2624 93 14 point point NOUN iajs-2624 93 15 , , PUNCT iajs-2624 93 16 which which PRON iajs-2624 93 17 is be AUX iajs-2624 93 18 denoted denote VERB iajs-2624 93 19 by by ADP iajs-2624 93 20 𝐸1 𝐸1 NOUN iajs-2624 93 21 = = PUNCT iajs-2624 93 22 ( ( PUNCT iajs-2624 93 23 𝑆𝑒1 𝑆𝑒1 ADJ iajs-2624 93 24 , , PUNCT iajs-2624 93 25 π‘†π‘Ž1 π‘†π‘Ž1 ADJ iajs-2624 93 26 , , PUNCT iajs-2624 93 27 𝐢1 𝐢1 PRON iajs-2624 93 28 , , PUNCT iajs-2624 93 29 𝑣1 𝑣1 NOUN iajs-2624 93 30 ) ) PUNCT iajs-2624 93 31 , , PUNCT iajs-2624 93 32 where where SCONJ iajs-2624 93 33 𝑆𝑒1 𝑆𝑒1 ADJ iajs-2624 93 34 = = PUNCT iajs-2624 93 35 π‘ŸΟˆ π‘ŸΟˆ ADP iajs-2624 93 36 π‘Ÿπ›Ό+𝛽(π‘Ÿβˆ’π‘‘)π‘˜+π‘Ÿπœ‡+π‘ŸπœŽπΆ1 π‘Ÿπ›Ό+𝛽(π‘Ÿβˆ’π‘‘)π‘˜+π‘Ÿπœ‡+π‘ŸπœŽπΆ1 NUM iajs-2624 93 37 ; ; PUNCT iajs-2624 93 38 π‘†π‘Ž1 π‘†π‘Ž1 ADV iajs-2624 94 1 = = PUNCT iajs-2624 94 2 π›Όπ‘Ÿ2ψ π›Όπ‘Ÿ2ψ PROPN iajs-2624 94 3 𝐴1 𝐴1 PROPN iajs-2624 94 4 𝐢1 𝐢1 PROPN iajs-2624 94 5 2+𝐴2𝐢1+𝐴3 2+𝐴2𝐢1+𝐴3 NUM iajs-2624 94 6 ; ; PUNCT iajs-2624 94 7 𝑣1 𝑣1 NOUN iajs-2624 94 8 = = SYM iajs-2624 94 9 ( ( PUNCT iajs-2624 94 10 π‘Ÿβˆ’π‘‘)π‘˜ π‘Ÿβˆ’π‘‘)π‘˜ PROPN iajs-2624 94 11 π‘Ÿ π‘Ÿ X iajs-2624 94 12 ( ( PUNCT iajs-2624 94 13 6a 6a NOUN iajs-2624 94 14 ) ) PUNCT iajs-2624 94 15 here here ADV iajs-2624 94 16 𝐴1 𝐴1 PROPN iajs-2624 94 17 = = SYM iajs-2624 94 18 𝜎1 𝜎1 PROPN iajs-2624 94 19 ( ( PUNCT iajs-2624 94 20 1 1 NUM iajs-2624 94 21 βˆ’ βˆ’ NOUN iajs-2624 94 22 πœ–)π‘Ÿ πœ–)π‘Ÿ NOUN iajs-2624 94 23 2𝜎 2𝜎 NUM iajs-2624 94 24 𝐴2 𝐴2 NOUN iajs-2624 94 25 = = SYM iajs-2624 94 26 𝛽1(1 𝛽1(1 PROPN iajs-2624 94 27 βˆ’ βˆ’ PROPN iajs-2624 94 28 πœ–)(π‘Ÿ πœ–)(π‘Ÿ NOUN iajs-2624 94 29 βˆ’ βˆ’ PROPN iajs-2624 95 1 𝑑)π‘˜π‘ŸπœŽ 𝑑)π‘˜π‘ŸπœŽ PROPN iajs-2624 95 2 + + CCONJ iajs-2624 95 3 𝜎1 𝜎1 PROPN iajs-2624 95 4 ( ( PUNCT iajs-2624 95 5 1 1 NUM iajs-2624 95 6 βˆ’ βˆ’ PROPN iajs-2624 95 7 πœ–)π‘Ÿ[π‘Ÿπ›Ό πœ–)π‘Ÿ[π‘Ÿπ›Ό PROPN iajs-2624 95 8 + + CCONJ iajs-2624 95 9 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 95 10 βˆ’ βˆ’ NOUN iajs-2624 95 11 𝑑)π‘˜ 𝑑)π‘˜ NOUN iajs-2624 95 12 + + CCONJ iajs-2624 95 13 π‘Ÿπœ‡ π‘Ÿπœ‡ X iajs-2624 95 14 ] ] X iajs-2624 95 15 + + NUM iajs-2624 95 16 π‘Ÿ π‘Ÿ SYM iajs-2624 95 17 2 2 NUM iajs-2624 95 18 πœ‡πœŽ πœ‡πœŽ X iajs-2624 95 19 𝐴3 𝐴3 PROPN iajs-2624 95 20 = = PUNCT iajs-2624 96 1 [ [ X iajs-2624 96 2 𝛽1(1 𝛽1(1 ADP iajs-2624 96 3 βˆ’ βˆ’ PROPN iajs-2624 96 4 πœ–)(π‘Ÿ πœ–)(π‘Ÿ NOUN iajs-2624 96 5 βˆ’ βˆ’ PRON iajs-2624 96 6 𝑑)π‘˜ 𝑑)π‘˜ NOUN iajs-2624 97 1 + + CCONJ iajs-2624 97 2 π‘Ÿπœ‡][π‘Ÿπ›Ό π‘Ÿπœ‡][π‘Ÿπ›Ό PROPN iajs-2624 97 3 + + SYM iajs-2624 97 4 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 97 5 βˆ’ βˆ’ NOUN iajs-2624 97 6 𝑑)π‘˜ 𝑑)π‘˜ NOUN iajs-2624 97 7 + + CCONJ iajs-2624 97 8 π‘Ÿπœ‡ π‘Ÿπœ‡ X iajs-2624 97 9 ] ] PUNCT iajs-2624 97 10 exists exist VERB iajs-2624 97 11 under under ADP iajs-2624 97 12 the the DET iajs-2624 97 13 condition condition NOUN iajs-2624 97 14 π‘Ÿ π‘Ÿ X iajs-2624 97 15 > > X iajs-2624 97 16 𝑑 𝑑 PROPN iajs-2624 97 17 ( ( PUNCT iajs-2624 97 18 6b 6b NOUN iajs-2624 97 19 ) ) PUNCT iajs-2624 97 20 now now ADV iajs-2624 97 21 , , PUNCT iajs-2624 97 22 substituting substitute VERB iajs-2624 97 23 the the DET iajs-2624 97 24 equation equation NOUN iajs-2624 97 25 ( ( PUNCT iajs-2624 97 26 6a 6a NOUN iajs-2624 97 27 ) ) PUNCT iajs-2624 97 28 in in ADP iajs-2624 97 29 3rd 3rd ADJ iajs-2624 97 30 equation equation NOUN iajs-2624 97 31 of of ADP iajs-2624 97 32 system system NOUN iajs-2624 97 33 ( ( PUNCT iajs-2624 97 34 3 3 NUM iajs-2624 97 35 ) ) PUNCT iajs-2624 97 36 , , PUNCT iajs-2624 97 37 and and CCONJ iajs-2624 97 38 simplifying simplify VERB iajs-2624 97 39 that that DET iajs-2624 97 40 result result NOUN iajs-2624 97 41 , , PUNCT iajs-2624 97 42 we we PRON iajs-2624 97 43 get get VERB iajs-2624 97 44 𝐷1𝐢 𝐷1𝐢 ADP iajs-2624 97 45 4 4 NUM iajs-2624 97 46 + + CCONJ iajs-2624 97 47 𝐷2𝐢 𝐷2𝐢 NOUN iajs-2624 97 48 3 3 NUM iajs-2624 97 49 + + CCONJ iajs-2624 97 50 𝐷3𝐢 𝐷3𝐢 NOUN iajs-2624 97 51 2 2 NUM iajs-2624 97 52 + + NUM iajs-2624 97 53 𝐷4𝐢 𝐷4𝐢 X iajs-2624 97 54 + + CCONJ iajs-2624 97 55 𝐷5 𝐷5 X iajs-2624 97 56 = = SYM iajs-2624 97 57 0 0 NUM iajs-2624 97 58 ( ( PUNCT iajs-2624 97 59 7a 7a NUM iajs-2624 97 60 ) ) PUNCT iajs-2624 97 61 where where SCONJ iajs-2624 97 62 𝐷1 𝐷1 NOUN iajs-2624 97 63 = = PUNCT iajs-2624 97 64 βˆ’(𝛾 βˆ’(𝛾 PROPN iajs-2624 97 65 + + NOUN iajs-2624 97 66 πœƒ πœƒ NOUN iajs-2624 97 67 + + X iajs-2624 97 68 πœ‡ πœ‡ ADP iajs-2624 97 69 + + ADJ iajs-2624 97 70 𝛿1)π‘Ÿ 𝛿1)π‘Ÿ ADJ iajs-2624 97 71 2𝜎𝐴1 2𝜎𝐴1 PROPN iajs-2624 97 72 < < X iajs-2624 97 73 0 0 NUM iajs-2624 97 74 𝐷2 𝐷2 NOUN iajs-2624 97 75 = = SYM iajs-2624 97 76 πœŽπ‘Ÿ2πœ“π΄1 πœŽπ‘Ÿ2πœ“π΄1 NUM iajs-2624 97 77 βˆ’ βˆ’ NUM iajs-2624 97 78 ( ( PUNCT iajs-2624 97 79 𝛾 𝛾 ADP iajs-2624 97 80 + + NOUN iajs-2624 97 81 πœƒ πœƒ NOUN iajs-2624 97 82 + + X iajs-2624 97 83 πœ‡ πœ‡ ADP iajs-2624 97 84 + + ADJ iajs-2624 97 85 𝛿1)(π‘Ÿ 𝛿1)(π‘Ÿ PROPN iajs-2624 97 86 2𝛼 2𝛼 PROPN iajs-2624 97 87 + + SYM iajs-2624 97 88 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 98 1 βˆ’ βˆ’ X iajs-2624 98 2 𝑑)π‘˜π‘Ÿ 𝑑)π‘˜π‘Ÿ PROPN iajs-2624 98 3 + + NUM iajs-2624 98 4 π‘Ÿ2πœ‡)𝐴1 π‘Ÿ2πœ‡)𝐴1 NOUN iajs-2624 98 5 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-2624 98 6 + + CCONJ iajs-2624 98 7 πœƒ πœƒ NOUN iajs-2624 99 1 + + X iajs-2624 99 2 πœ‡ πœ‡ ADP iajs-2624 99 3 + + ADJ iajs-2624 99 4 𝛿1)π‘Ÿ 𝛿1)π‘Ÿ ADJ iajs-2624 99 5 2𝜎𝐴2 2𝜎𝐴2 PROPN iajs-2624 99 6 𝐷3 𝐷3 NOUN iajs-2624 99 7 = = SYM iajs-2624 100 1 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 100 2 βˆ’ βˆ’ NOUN iajs-2624 100 3 𝑑)π‘˜π‘Ÿπœ“π΄1 𝑑)π‘˜π‘Ÿπœ“π΄1 PUNCT iajs-2624 100 4 + + CCONJ iajs-2624 100 5 πœŽπ‘Ÿ πœŽπ‘Ÿ PROPN iajs-2624 100 6 2πœ“π΄2 2πœ“π΄2 NUM iajs-2624 100 7 + + NUM iajs-2624 100 8 𝜎1(1 𝜎1(1 NOUN iajs-2624 100 9 βˆ’ βˆ’ PROPN iajs-2624 100 10 πœ–)π›Όπ‘Ÿ πœ–)π›Όπ‘Ÿ PROPN iajs-2624 100 11 4πœ“πœŽ 4πœ“πœŽ NOUN iajs-2624 100 12 βˆ’[𝛾 βˆ’[𝛾 NOUN iajs-2624 100 13 + + CCONJ iajs-2624 100 14 πœƒ πœƒ NOUN iajs-2624 100 15 + + X iajs-2624 100 16 πœ‡ πœ‡ ADP iajs-2624 100 17 βˆ’ βˆ’ PROPN iajs-2624 100 18 ( ( PUNCT iajs-2624 100 19 𝛾 𝛾 PROPN iajs-2624 100 20 + + NOUN iajs-2624 100 21 πœƒ πœƒ NOUN iajs-2624 100 22 + + X iajs-2624 100 23 πœ‡ πœ‡ ADP iajs-2624 100 24 + + ADJ iajs-2624 100 25 𝛿1)π‘Ÿ 𝛿1)π‘Ÿ ADJ iajs-2624 100 26 2𝜎𝐴2](π‘Ÿ 2𝜎𝐴2](π‘Ÿ NOUN iajs-2624 100 27 2 2 NUM iajs-2624 100 28 𝛼 𝛼 NOUN iajs-2624 100 29 + + NOUN iajs-2624 100 30 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 100 31 βˆ’ βˆ’ NOUN iajs-2624 101 1 𝑑)π‘˜π‘Ÿ 𝑑)π‘˜π‘Ÿ PROPN iajs-2624 101 2 + + NUM iajs-2624 101 3 π‘Ÿ2πœ‡)𝐴2 π‘Ÿ2πœ‡)𝐴2 NOUN iajs-2624 101 4 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-2624 101 5 + + CCONJ iajs-2624 101 6 πœƒ πœƒ NOUN iajs-2624 101 7 + + X iajs-2624 101 8 πœ‡ πœ‡ X iajs-2624 101 9 + + ADJ iajs-2624 101 10 𝛿1)π‘Ÿ 𝛿1)π‘Ÿ ADJ iajs-2624 101 11 2𝜎𝐴3 2𝜎𝐴3 PROPN iajs-2624 101 12 𝐷4 𝐷4 NOUN iajs-2624 101 13 = = SYM iajs-2624 101 14 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 101 15 βˆ’ βˆ’ NUM iajs-2624 101 16 𝑑)π‘˜π‘Ÿπœ“π΄2 𝑑)π‘˜π‘Ÿπœ“π΄2 ADJ iajs-2624 101 17 + + CCONJ iajs-2624 101 18 𝛽1(1 𝛽1(1 PROPN iajs-2624 101 19 βˆ’ βˆ’ PROPN iajs-2624 101 20 πœ–)(π‘Ÿ πœ–)(π‘Ÿ NOUN iajs-2624 101 21 βˆ’ βˆ’ PROPN iajs-2624 101 22 𝑑)π‘˜π›Όπ‘Ÿ 𝑑)π‘˜π›Όπ‘Ÿ PROPN iajs-2624 101 23 3πœ“πœŽ 3πœ“πœŽ PROPN iajs-2624 101 24 + + CCONJ iajs-2624 101 25 πœŽπ‘Ÿ2πœ“π΄3 πœŽπ‘Ÿ2πœ“π΄3 NOUN iajs-2624 101 26 + + ADJ iajs-2624 101 27 𝜎1(1 𝜎1(1 NOUN iajs-2624 101 28 βˆ’ βˆ’ PROPN iajs-2624 101 29 πœ–)π›Όπ‘Ÿ πœ–)π›Όπ‘Ÿ PROPN iajs-2624 101 30 3πœ“(π‘Ÿπ›Ό 3πœ“(π‘Ÿπ›Ό NUM iajs-2624 101 31 + + NUM iajs-2624 101 32 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 102 1 βˆ’ βˆ’ PROPN iajs-2624 102 2 𝑑)π‘˜π‘Ÿ 𝑑)π‘˜π‘Ÿ PROPN iajs-2624 102 3 + + NUM iajs-2624 102 4 πœ‡π‘Ÿ πœ‡π‘Ÿ NOUN iajs-2624 102 5 ) ) PUNCT iajs-2624 102 6 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-2624 102 7 + + NOUN iajs-2624 102 8 πœƒ πœƒ NOUN iajs-2624 102 9 + + X iajs-2624 102 10 πœ‡ πœ‡ ADP iajs-2624 102 11 + + ADJ iajs-2624 102 12 𝛿1)[π‘Ÿ 𝛿1)[π‘Ÿ ADJ iajs-2624 102 13 2𝛼 2𝛼 NUM iajs-2624 102 14 + + SYM iajs-2624 102 15 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 102 16 βˆ’ βˆ’ PROPN iajs-2624 103 1 𝑑)π‘˜π‘Ÿ 𝑑)π‘˜π‘Ÿ PROPN iajs-2624 103 2 + + NUM iajs-2624 103 3 π‘Ÿ2πœ‡ π‘Ÿ2πœ‡ PROPN iajs-2624 103 4 ] ] X iajs-2624 103 5 𝐷5 𝐷5 PROPN iajs-2624 103 6 = = SYM iajs-2624 103 7 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 103 8 βˆ’ βˆ’ NUM iajs-2624 103 9 𝑑)π‘˜π‘Ÿπœ“π΄3 𝑑)π‘˜π‘Ÿπœ“π΄3 NOUN iajs-2624 103 10 + + CCONJ iajs-2624 103 11 𝛽1π‘˜π›Όπ‘Ÿ 𝛽1π‘˜π›Όπ‘Ÿ NUM iajs-2624 103 12 2πœ“(1 2πœ“(1 NUM iajs-2624 103 13 βˆ’ βˆ’ ADP iajs-2624 103 14 πœ–)(π‘Ÿ πœ–)(π‘Ÿ PUNCT iajs-2624 103 15 βˆ’ βˆ’ ADP iajs-2624 103 16 𝑑)[π‘Ÿπ›Ό 𝑑)[π‘Ÿπ›Ό PROPN iajs-2624 103 17 + + NUM iajs-2624 103 18 𝛽(π‘Ÿ 𝛽(π‘Ÿ NOUN iajs-2624 103 19 βˆ’ βˆ’ NOUN iajs-2624 103 20 𝑑)π‘˜ 𝑑)π‘˜ NOUN iajs-2624 103 21 + + CCONJ iajs-2624 103 22 π‘Ÿπœ‡ π‘Ÿπœ‡ X iajs-2624 103 23 ] ] PUNCT iajs-2624 103 24 > > X iajs-2624 103 25 0 0 PUNCT iajs-2624 104 1 obviously obviously ADV iajs-2624 104 2 , , PUNCT iajs-2624 104 3 the the DET iajs-2624 104 4 endemic endemic ADJ iajs-2624 104 5 equilibrium equilibrium NOUN iajs-2624 104 6 point point NOUN iajs-2624 104 7 exists exist VERB iajs-2624 104 8 unique unique ADJ iajs-2624 104 9 if if SCONJ iajs-2624 104 10 and and CCONJ iajs-2624 104 11 only only ADV iajs-2624 104 12 if if SCONJ iajs-2624 104 13 the the DET iajs-2624 104 14 following follow VERB iajs-2624 104 15 condition condition NOUN iajs-2624 104 16 holds hold VERB iajs-2624 104 17 𝐷3 𝐷3 PROPN iajs-2624 104 18 > > X iajs-2624 104 19 0 0 NUM iajs-2624 104 20 , , PUNCT iajs-2624 104 21 𝐷4 𝐷4 NOUN iajs-2624 104 22 > > X iajs-2624 104 23 0 0 PUNCT iajs-2624 105 1 ( ( PUNCT iajs-2624 105 2 7b 7b NUM iajs-2624 105 3 ) ) PUNCT iajs-2624 105 4 or or CCONJ iajs-2624 105 5 𝐷2 𝐷2 NOUN iajs-2624 105 6 < < X iajs-2624 105 7 0 0 PROPN iajs-2624 105 8 , , PUNCT iajs-2624 105 9 𝐷3 𝐷3 PROPN iajs-2624 105 10 < < X iajs-2624 105 11 0 0 PUNCT iajs-2624 105 12 ( ( PUNCT iajs-2624 105 13 7c 7c NUM iajs-2624 105 14 ) ) PUNCT iajs-2624 105 15 4 4 NUM iajs-2624 105 16 . . PUNCT iajs-2624 105 17 local local ADJ iajs-2624 105 18 stability stability NOUN iajs-2624 105 19 analysis analysis NOUN iajs-2624 105 20 in in ADP iajs-2624 105 21 this this DET iajs-2624 105 22 section section NOUN iajs-2624 105 23 , , PUNCT iajs-2624 105 24 the the DET iajs-2624 105 25 local local ADJ iajs-2624 105 26 stability stability NOUN iajs-2624 105 27 analysis analysis NOUN iajs-2624 105 28 of of ADP iajs-2624 105 29 the the DET iajs-2624 105 30 all all DET iajs-2624 105 31 equilibrium equilibrium NOUN iajs-2624 105 32 points point NOUN iajs-2624 106 1 𝐸𝑖 𝐸𝑖 PROPN iajs-2624 106 2 , , PUNCT iajs-2624 106 3 𝑖 𝑖 NOUN iajs-2624 106 4 = = SYM iajs-2624 106 5 0,1 0,1 NUM iajs-2624 106 6 of of ADP iajs-2624 106 7 system system NOUN iajs-2624 106 8 ( ( PUNCT iajs-2624 106 9 3 3 X iajs-2624 106 10 ) ) PUNCT iajs-2624 106 11 studied study VERB iajs-2624 106 12 as as SCONJ iajs-2624 106 13 shown show VERB iajs-2624 106 14 in in ADP iajs-2624 106 15 the the DET iajs-2624 106 16 following follow VERB iajs-2624 106 17 theorems theorem NOUN iajs-2624 106 18 . . PUNCT iajs-2624 107 1 theorem theorem NOUN iajs-2624 107 2 ( ( PUNCT iajs-2624 107 3 2 2 NUM iajs-2624 107 4 ): ): PUNCT iajs-2624 107 5 the the DET iajs-2624 107 6 covid-19 covid-19 PROPN iajs-2624 107 7 free free ADJ iajs-2624 107 8 equilibrium equilibrium NOUN iajs-2624 107 9 point point NOUN iajs-2624 107 10 𝐸0 𝐸0 NOUN iajs-2624 107 11 of of ADP iajs-2624 107 12 the the DET iajs-2624 107 13 system system NOUN iajs-2624 107 14 ( ( PUNCT iajs-2624 107 15 3 3 X iajs-2624 107 16 ) ) PUNCT iajs-2624 107 17 is be AUX iajs-2624 107 18 locally locally ADV iajs-2624 107 19 asymptotically asymptotically ADV iajs-2624 107 20 if if SCONJ iajs-2624 107 21 the the DET iajs-2624 107 22 following follow VERB iajs-2624 107 23 condition condition NOUN iajs-2624 107 24 is be AUX iajs-2624 107 25 satisfied satisfied ADJ iajs-2624 107 26 β„›0 β„›0 NOUN iajs-2624 107 27 < < X iajs-2624 107 28 1 1 NUM iajs-2624 107 29 ( ( PUNCT iajs-2624 107 30 8a 8a NUM iajs-2624 107 31 ) ) PUNCT iajs-2624 107 32 π‘Ÿ π‘Ÿ NOUN iajs-2624 107 33 < < X iajs-2624 107 34 𝑑 𝑑 X iajs-2624 107 35 ( ( PUNCT iajs-2624 107 36 8b 8b NUM iajs-2624 107 37 ) ) PUNCT iajs-2624 107 38 proof proof NOUN iajs-2624 107 39 : : PUNCT iajs-2624 107 40 the the DET iajs-2624 107 41 jacobian jacobian ADJ iajs-2624 107 42 matrix matrix NOUN iajs-2624 107 43 of of ADP iajs-2624 107 44 system system NOUN iajs-2624 107 45 ( ( PUNCT iajs-2624 107 46 3 3 NUM iajs-2624 107 47 ) ) PUNCT iajs-2624 107 48 at at ADP iajs-2624 107 49 𝐸0 𝐸0 PROPN iajs-2624 107 50 can can AUX iajs-2624 107 51 be be AUX iajs-2624 107 52 written write VERB iajs-2624 107 53 as as ADP iajs-2624 107 54 47 47 NUM iajs-2624 107 55 ibn ibn PROPN iajs-2624 107 56 al al PROPN iajs-2624 107 57 - - PUNCT iajs-2624 107 58 haitham haitham PROPN iajs-2624 107 59 jour jour X iajs-2624 107 60 . . PROPN iajs-2624 107 61 for for ADP iajs-2624 107 62 pure pure ADJ iajs-2624 107 63 & & CCONJ iajs-2624 107 64 appl appl PROPN iajs-2624 107 65 . . PUNCT iajs-2624 108 1 sci sci PROPN iajs-2624 108 2 . . PROPN iajs-2624 109 1 34 34 NUM iajs-2624 109 2 ( ( PUNCT iajs-2624 109 3 2 2 NUM iajs-2624 109 4 ) ) PUNCT iajs-2624 109 5 2021 2021 NUM iajs-2624 109 6 𝐽(𝑆𝑒0 𝐽(𝑆𝑒0 PROPN iajs-2624 109 7 , , PUNCT iajs-2624 109 8 0,0,0 0,0,0 NUM iajs-2624 109 9 ) ) PUNCT iajs-2624 110 1 = = SYM iajs-2624 110 2 ( ( PUNCT iajs-2624 110 3 βˆ’πœ‡ βˆ’πœ‡ NUM iajs-2624 110 4 0 0 NUM iajs-2624 110 5 βˆ’πœŽπ‘†π‘’0 βˆ’πœŽπ‘†π‘’0 VERB iajs-2624 110 6 βˆ’π›½π‘†π‘’0 βˆ’π›½π‘†π‘’0 NOUN iajs-2624 110 7 0 0 PUNCT iajs-2624 111 1 βˆ’πœ‡ βˆ’πœ‡ NUM iajs-2624 111 2 0 0 NUM iajs-2624 111 3 0 0 NUM iajs-2624 111 4 0 0 NUM iajs-2624 111 5 0 0 NUM iajs-2624 111 6 πœŽπ‘†π‘’0 πœŽπ‘†π‘’0 PROPN iajs-2624 111 7 βˆ’ βˆ’ PROPN iajs-2624 111 8 ( ( PUNCT iajs-2624 111 9 𝛾 𝛾 ADP iajs-2624 111 10 + + NOUN iajs-2624 111 11 πœƒ πœƒ NOUN iajs-2624 111 12 + + NUM iajs-2624 111 13 πœ‡ πœ‡ X iajs-2624 111 14 + + X iajs-2624 111 15 𝛿1 𝛿1 NOUN iajs-2624 111 16 ) ) PUNCT iajs-2624 111 17 𝛽𝑆𝑒0 𝛽𝑆𝑒0 PROPN iajs-2624 111 18 0 0 NUM iajs-2624 111 19 0 0 NUM iajs-2624 111 20 0 0 NUM iajs-2624 112 1 π‘Ÿ π‘Ÿ NOUN iajs-2624 112 2 βˆ’ βˆ’ PROPN iajs-2624 112 3 𝑑 𝑑 NOUN iajs-2624 112 4 ) ) PUNCT iajs-2624 112 5 thus thus ADV iajs-2624 112 6 , , PUNCT iajs-2624 112 7 the the DET iajs-2624 112 8 eigenvalues eigenvalue NOUN iajs-2624 112 9 of of ADP iajs-2624 112 10 𝐽(𝐸0 𝐽(𝐸0 NOUN iajs-2624 112 11 ) ) PUNCT iajs-2624 112 12 are be AUX iajs-2624 112 13 given give VERB iajs-2624 112 14 by by ADP iajs-2624 112 15 πœ†1 πœ†1 NOUN iajs-2624 112 16 = = PUNCT iajs-2624 112 17 πœ†2 πœ†2 NOUN iajs-2624 112 18 = = PUNCT iajs-2624 112 19 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-2624 112 20 < < X iajs-2624 112 21 0 0 NUM iajs-2624 112 22 πœ†3 πœ†3 NOUN iajs-2624 112 23 = = SYM iajs-2624 112 24 πœŽπ‘†π‘’0 πœŽπ‘†π‘’0 PROPN iajs-2624 112 25 βˆ’ βˆ’ PROPN iajs-2624 112 26 ( ( PUNCT iajs-2624 112 27 𝛾 𝛾 ADP iajs-2624 112 28 + + NOUN iajs-2624 112 29 πœƒ πœƒ NOUN iajs-2624 112 30 + + NUM iajs-2624 112 31 πœ‡ πœ‡ X iajs-2624 112 32 + + X iajs-2624 112 33 𝛿1 𝛿1 ADJ iajs-2624 112 34 ) ) PUNCT iajs-2624 112 35 πœ†4 πœ†4 NOUN iajs-2624 112 36 = = PUNCT iajs-2624 113 1 π‘Ÿ π‘Ÿ NOUN iajs-2624 114 1 βˆ’ βˆ’ NOUN iajs-2624 115 1 𝑑 𝑑 PRON iajs-2624 116 1 it it PRON iajs-2624 116 2 is be AUX iajs-2624 116 3 easy easy ADJ iajs-2624 116 4 from from ADP iajs-2624 116 5 above above ADV iajs-2624 116 6 to to PART iajs-2624 116 7 verify verify VERB iajs-2624 116 8 that that SCONJ iajs-2624 116 9 conditions condition NOUN iajs-2624 116 10 ( ( PUNCT iajs-2624 116 11 8a 8a NUM iajs-2624 116 12 ) ) PUNCT iajs-2624 116 13 and and CCONJ iajs-2624 116 14 ( ( PUNCT iajs-2624 116 15 8b 8b NUM iajs-2624 116 16 ) ) PUNCT iajs-2624 116 17 guarantee guarantee VERB iajs-2624 116 18 that that SCONJ iajs-2624 116 19 the the DET iajs-2624 116 20 eigenvalue eigenvalue PROPN iajs-2624 116 21 πœ†3 πœ†3 NOUN iajs-2624 116 22 and and CCONJ iajs-2624 116 23 πœ†4 πœ†4 NOUN iajs-2624 116 24 are be AUX iajs-2624 116 25 negative negative ADJ iajs-2624 116 26 respectively respectively ADV iajs-2624 116 27 . . PUNCT iajs-2624 117 1 then then ADV iajs-2624 117 2 , , PUNCT iajs-2624 117 3 𝐸0 𝐸0 PROPN iajs-2624 117 4 is be AUX iajs-2624 117 5 locally locally ADV iajs-2624 117 6 asymptotically asymptotically ADV iajs-2624 117 7 stable stable ADJ iajs-2624 117 8 . . PUNCT iajs-2624 118 1 however however ADV iajs-2624 118 2 , , PUNCT iajs-2624 118 3 it it PRON iajs-2624 118 4 is be AUX iajs-2624 118 5 a a DET iajs-2624 118 6 saddle saddle NOUN iajs-2624 118 7 point point NOUN iajs-2624 118 8 otherwise otherwise ADV iajs-2624 118 9 . . PUNCT iajs-2624 119 1 β–  β–  PUNCT iajs-2624 119 2 theorem theorem ADJ iajs-2624 119 3 ( ( PUNCT iajs-2624 119 4 3 3 NUM iajs-2624 119 5 ): ): PUNCT iajs-2624 119 6 the the DET iajs-2624 119 7 endemic endemic ADJ iajs-2624 119 8 equilibrium equilibrium NOUN iajs-2624 119 9 point point NOUN iajs-2624 119 10 𝐸1 𝐸1 NOUN iajs-2624 119 11 of of ADP iajs-2624 119 12 the the DET iajs-2624 119 13 system system NOUN iajs-2624 119 14 ( ( PUNCT iajs-2624 119 15 3 3 X iajs-2624 119 16 ) ) PUNCT iajs-2624 119 17 is be AUX iajs-2624 119 18 locally locally ADV iajs-2624 119 19 asymptotically asymptotically ADV iajs-2624 119 20 if if SCONJ iajs-2624 119 21 the the DET iajs-2624 119 22 following follow VERB iajs-2624 119 23 conditions condition NOUN iajs-2624 119 24 are be AUX iajs-2624 119 25 hold hold VERB iajs-2624 119 26 2πœŽπ‘†π‘’1 2πœŽπ‘†π‘’1 NUM iajs-2624 119 27 + + CCONJ iajs-2624 119 28 2𝜎1(1 2𝜎1(1 NUM iajs-2624 119 29 βˆ’ βˆ’ ADP iajs-2624 119 30 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 119 31 < < X iajs-2624 119 32 Ξ³ Ξ³ X iajs-2624 119 33 + + NOUN iajs-2624 119 34 ΞΈ ΞΈ PROPN iajs-2624 119 35 + + CCONJ iajs-2624 120 1 ΞΌ ΞΌ PROPN iajs-2624 120 2 + + CCONJ iajs-2624 120 3 𝛿1 𝛿1 NOUN iajs-2624 120 4 ( ( PUNCT iajs-2624 120 5 9a 9a NOUN iajs-2624 120 6 ) ) PUNCT iajs-2624 120 7 π‘˜ π‘˜ X iajs-2624 121 1 [ [ X iajs-2624 121 2 2𝛽𝑆𝑒1 2𝛽𝑆𝑒1 NUM iajs-2624 121 3 + + NUM iajs-2624 121 4 2𝛽1(1 2𝛽1(1 NUM iajs-2624 121 5 βˆ’ βˆ’ X iajs-2624 121 6 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 121 7 + + CCONJ iajs-2624 121 8 π‘Ÿ π‘Ÿ NOUN iajs-2624 121 9 ] ] X iajs-2624 121 10 < < X iajs-2624 121 11 2π‘Ÿπ‘£1 2π‘Ÿπ‘£1 NUM iajs-2624 121 12 + + CCONJ iajs-2624 121 13 π‘˜π‘‘ π‘˜π‘‘ INTJ iajs-2624 121 14 ( ( PUNCT iajs-2624 121 15 9b 9b NOUN iajs-2624 121 16 ) ) PUNCT iajs-2624 121 17 proof proof NOUN iajs-2624 121 18 : : PUNCT iajs-2624 121 19 the the DET iajs-2624 121 20 jacobian jacobian ADJ iajs-2624 121 21 matrix matrix NOUN iajs-2624 121 22 of of ADP iajs-2624 121 23 system system NOUN iajs-2624 121 24 ( ( PUNCT iajs-2624 121 25 3 3 NUM iajs-2624 121 26 ) ) PUNCT iajs-2624 121 27 at at ADP iajs-2624 121 28 𝐸1 𝐸1 NOUN iajs-2624 121 29 can can AUX iajs-2624 121 30 be be AUX iajs-2624 121 31 written write VERB iajs-2624 121 32 as as ADP iajs-2624 121 33 𝐽(𝑆𝑒1 𝐽(𝑆𝑒1 PROPN iajs-2624 121 34 , , PUNCT iajs-2624 121 35 π‘†π‘Ž1 π‘†π‘Ž1 ADV iajs-2624 121 36 , , PUNCT iajs-2624 121 37 𝐢1 𝐢1 PRON iajs-2624 121 38 , , PUNCT iajs-2624 121 39 𝑣1 𝑣1 NOUN iajs-2624 121 40 ) ) PUNCT iajs-2624 121 41 = = PUNCT iajs-2624 122 1 ( ( PUNCT iajs-2624 122 2 𝑏11 𝑏11 PROPN iajs-2624 122 3 0 0 NUM iajs-2624 122 4 𝑏13 𝑏13 NOUN iajs-2624 122 5 𝑏14 𝑏14 PROPN iajs-2624 122 6 𝑏21 𝑏21 NOUN iajs-2624 122 7 𝑏22 𝑏22 PROPN iajs-2624 122 8 𝑏23 𝑏23 PROPN iajs-2624 122 9 𝑏24 𝑏24 PROPN iajs-2624 122 10 𝑏31 𝑏31 NOUN iajs-2624 122 11 𝑏32 𝑏32 NOUN iajs-2624 122 12 𝑏33 𝑏33 NOUN iajs-2624 122 13 𝑏34 𝑏34 NOUN iajs-2624 122 14 0 0 NUM iajs-2624 122 15 0 0 NUM iajs-2624 122 16 0 0 NUM iajs-2624 122 17 𝑏44 𝑏44 NOUN iajs-2624 122 18 ) ) PUNCT iajs-2624 122 19 here here ADV iajs-2624 122 20 𝑏11 𝑏11 PROPN iajs-2624 122 21 = = SYM iajs-2624 122 22 βˆ’(𝛼 βˆ’(𝛼 NOUN iajs-2624 122 23 + + CCONJ iajs-2624 122 24 𝛽𝑣1 𝛽𝑣1 ADJ iajs-2624 123 1 + + CCONJ iajs-2624 123 2 𝜎𝐢1 𝜎𝐢1 NUM iajs-2624 123 3 + + CCONJ iajs-2624 123 4 πœ‡ πœ‡ X iajs-2624 123 5 ) ) PUNCT iajs-2624 123 6 ; ; PUNCT iajs-2624 123 7 𝑏13 𝑏13 NOUN iajs-2624 123 8 = = SYM iajs-2624 123 9 βˆ’πœŽπ‘†π‘’1 βˆ’πœŽπ‘†π‘’1 X iajs-2624 123 10 ; ; PUNCT iajs-2624 123 11 𝑏14 𝑏14 PROPN iajs-2624 123 12 = = SYM iajs-2624 123 13 βˆ’π›½π‘†π‘’1 βˆ’π›½π‘†π‘’1 NOUN iajs-2624 123 14 ; ; PUNCT iajs-2624 123 15 𝑏21 𝑏21 PROPN iajs-2624 123 16 = = PROPN iajs-2624 123 17 𝛼 𝛼 NOUN iajs-2624 123 18 ; ; PUNCT iajs-2624 123 19 𝑏22 𝑏22 NOUN iajs-2624 123 20 = = SYM iajs-2624 123 21 βˆ’[𝛽1(1 βˆ’[𝛽1(1 PROPN iajs-2624 124 1 βˆ’ βˆ’ PROPN iajs-2624 124 2 πœ–)𝑣1 πœ–)𝑣1 PROPN iajs-2624 124 3 + + NUM iajs-2624 124 4 𝜎1(1 𝜎1(1 PROPN iajs-2624 124 5 βˆ’ βˆ’ PROPN iajs-2624 124 6 πœ–)𝐢1 πœ–)𝐢1 NOUN iajs-2624 124 7 + + CCONJ iajs-2624 124 8 πœ‡ πœ‡ X iajs-2624 124 9 ] ] PUNCT iajs-2624 124 10 ; ; PUNCT iajs-2624 124 11 𝑏23 𝑏23 NOUN iajs-2624 124 12 = = SYM iajs-2624 124 13 βˆ’πœŽ1 βˆ’πœŽ1 PROPN iajs-2624 124 14 ( ( PUNCT iajs-2624 124 15 1 1 NUM iajs-2624 124 16 βˆ’ βˆ’ PROPN iajs-2624 124 17 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 124 18 ; ; PUNCT iajs-2624 124 19 𝑏24 𝑏24 NUM iajs-2624 124 20 = = SYM iajs-2624 124 21 βˆ’π›½1(1 βˆ’π›½1(1 SYM iajs-2624 124 22 βˆ’ βˆ’ X iajs-2624 124 23 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 124 24 𝑏31 𝑏31 NOUN iajs-2624 124 25 = = SYM iajs-2624 125 1 𝛽𝑣1 𝛽𝑣1 NOUN iajs-2624 125 2 + + CCONJ iajs-2624 125 3 𝜎𝐢1 𝜎𝐢1 NUM iajs-2624 125 4 ; ; PUNCT iajs-2624 125 5 𝑏32 𝑏32 NOUN iajs-2624 125 6 = = SYM iajs-2624 125 7 𝛽1(1 𝛽1(1 PROPN iajs-2624 125 8 βˆ’ βˆ’ PROPN iajs-2624 125 9 πœ–)𝑣1 πœ–)𝑣1 PROPN iajs-2624 125 10 + + CCONJ iajs-2624 125 11 𝜎1(1 𝜎1(1 PROPN iajs-2624 125 12 βˆ’ βˆ’ PROPN iajs-2624 125 13 πœ–)𝐢1 πœ–)𝐢1 NOUN iajs-2624 125 14 ; ; PUNCT iajs-2624 125 15 𝑏33 𝑏33 NOUN iajs-2624 125 16 = = SYM iajs-2624 125 17 πœŽπ‘†π‘’1 πœŽπ‘†π‘’1 PROPN iajs-2624 125 18 + + CCONJ iajs-2624 125 19 𝜎1(1 𝜎1(1 PROPN iajs-2624 125 20 βˆ’ βˆ’ PROPN iajs-2624 125 21 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 125 22 βˆ’ βˆ’ PROPN iajs-2624 125 23 ( ( PUNCT iajs-2624 125 24 𝛾 𝛾 PROPN iajs-2624 125 25 + + NOUN iajs-2624 125 26 πœƒ πœƒ NOUN iajs-2624 125 27 + + NUM iajs-2624 125 28 πœ‡ πœ‡ X iajs-2624 125 29 + + X iajs-2624 125 30 𝛿1 𝛿1 NOUN iajs-2624 125 31 ) ) PUNCT iajs-2624 125 32 ; ; PUNCT iajs-2624 125 33 𝑏34 𝑏34 NOUN iajs-2624 125 34 = = PUNCT iajs-2624 125 35 𝛽𝑆𝑒1 𝛽𝑆𝑒1 PROPN iajs-2624 125 36 + + CCONJ iajs-2624 125 37 𝛽1(1 𝛽1(1 PROPN iajs-2624 125 38 βˆ’ βˆ’ PROPN iajs-2624 125 39 πœ–)π‘†π‘Ž1 πœ–)π‘†π‘Ž1 PROPN iajs-2624 125 40 𝑏44 𝑏44 NOUN iajs-2624 125 41 = = SYM iajs-2624 125 42 βˆ’2π‘Ÿπ‘£1 βˆ’2π‘Ÿπ‘£1 NOUN iajs-2624 126 1 π‘˜ π‘˜ X iajs-2624 127 1 + + CCONJ iajs-2624 127 2 ( ( PUNCT iajs-2624 127 3 π‘Ÿ π‘Ÿ NOUN iajs-2624 127 4 βˆ’ βˆ’ PROPN iajs-2624 127 5 𝑑 𝑑 NOUN iajs-2624 127 6 ) ) PUNCT iajs-2624 127 7 ; ; PUNCT iajs-2624 127 8 𝑏12 𝑏12 VERB iajs-2624 127 9 = = NOUN iajs-2624 127 10 𝑏41 𝑏41 NOUN iajs-2624 127 11 = = NOUN iajs-2624 127 12 𝑏42 𝑏42 NOUN iajs-2624 127 13 = = PROPN iajs-2624 127 14 𝑏43 𝑏43 NOUN iajs-2624 127 15 = = NOUN iajs-2624 127 16 0 0 NUM iajs-2624 127 17 now now ADV iajs-2624 127 18 , , PUNCT iajs-2624 127 19 according accord VERB iajs-2624 127 20 to to ADP iajs-2624 127 21 gersgorin gersgorin ADJ iajs-2624 127 22 theorem theorem ADJ iajs-2624 127 23 [ [ X iajs-2624 127 24 13 13 NUM iajs-2624 127 25 ] ] PUNCT iajs-2624 127 26 , , PUNCT iajs-2624 127 27 if if SCONJ iajs-2624 127 28 the the DET iajs-2624 127 29 following follow VERB iajs-2624 127 30 condition condition NOUN iajs-2624 127 31 holds hold VERB iajs-2624 127 32 : : PUNCT iajs-2624 127 33 |𝑏𝑖𝑖| |𝑏𝑖𝑖| NOUN iajs-2624 127 34 > > PUNCT iajs-2624 127 35 βˆ‘|𝑏𝑖𝑗| βˆ‘|𝑏𝑖𝑗| PROPN iajs-2624 127 36 4 4 NUM iajs-2624 127 37 𝑖=1 𝑖=1 PUNCT iajs-2624 127 38 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2624 127 39 = = PUNCT iajs-2624 128 1 𝑃𝑖 𝑃𝑖 SCONJ iajs-2624 128 2 then then ADV iajs-2624 128 3 all all DET iajs-2624 128 4 eigenvalues eigenvalue NOUN iajs-2624 128 5 of of ADP iajs-2624 128 6 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-2624 128 7 ) ) PUNCT iajs-2624 128 8 exist exist VERB iajs-2624 128 9 in in ADP iajs-2624 128 10 the the DET iajs-2624 128 11 region region NOUN iajs-2624 128 12 : : PUNCT iajs-2624 129 1 Ο‰ Ο‰ NOUN iajs-2624 129 2 = = NOUN iajs-2624 129 3 βˆͺ βˆͺ X iajs-2624 129 4 { { PUNCT iajs-2624 129 5 π‘ˆβˆ—πœ– π‘ˆβˆ—πœ– PROPN iajs-2624 129 6 𝐢 𝐢 PROPN iajs-2624 129 7 ∢ ∢ NOUN iajs-2624 129 8 |π‘ˆβˆ— |π‘ˆβˆ— PUNCT iajs-2624 129 9 βˆ’ βˆ’ ADP iajs-2624 129 10 𝑏𝑖𝑖| 𝑏𝑖𝑖| VERB iajs-2624 129 11 < < X iajs-2624 129 12 βˆ‘|𝑏𝑖𝑗| βˆ‘|𝑏𝑖𝑗| PROPN iajs-2624 129 13 4 4 NUM iajs-2624 129 14 𝑖=1 𝑖=1 PUNCT iajs-2624 129 15 𝑖≠𝑗 𝑖≠𝑗 PUNCT iajs-2624 129 16 } } PUNCT iajs-2624 129 17 then then ADV iajs-2624 129 18 , , PUNCT iajs-2624 129 19 all all DET iajs-2624 129 20 the the DET iajs-2624 129 21 eigenvalues eigenvalue NOUN iajs-2624 129 22 of of ADP iajs-2624 129 23 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-2624 129 24 ) ) PUNCT iajs-2624 129 25 exist exist VERB iajs-2624 129 26 in in ADP iajs-2624 129 27 the the DET iajs-2624 129 28 disc disc NOUN iajs-2624 129 29 centered center VERB iajs-2624 129 30 at at ADP iajs-2624 129 31 𝑏𝑖𝑖 𝑏𝑖𝑖 NOUN iajs-2624 129 32 with with ADP iajs-2624 129 33 radius radius NOUN iajs-2624 129 34 𝑃𝑖. 𝑃𝑖. PROPN iajs-2624 129 35 thus thus ADV iajs-2624 129 36 if if SCONJ iajs-2624 129 37 the the DET iajs-2624 129 38 diagonal diagonal ADJ iajs-2624 129 39 elements element NOUN iajs-2624 129 40 are be AUX iajs-2624 129 41 negative negative ADJ iajs-2624 129 42 and and CCONJ iajs-2624 129 43 the the DET iajs-2624 129 44 condition condition NOUN iajs-2624 129 45 ( ( PUNCT iajs-2624 129 46 9a 9a NOUN iajs-2624 129 47 ) ) PUNCT iajs-2624 129 48 holds hold NOUN iajs-2624 129 49 , , PUNCT iajs-2624 129 50 all all DET iajs-2624 129 51 the the DET iajs-2624 129 52 eigenvalues eigenvalue NOUN iajs-2624 129 53 will will AUX iajs-2624 129 54 be be AUX iajs-2624 129 55 exist exist VERB iajs-2624 129 56 in in ADP iajs-2624 129 57 the the DET iajs-2624 129 58 left left ADJ iajs-2624 129 59 half half NOUN iajs-2624 129 60 plane plane NOUN iajs-2624 129 61 and and CCONJ iajs-2624 129 62 the the DET iajs-2624 129 63 𝐸1 𝐸1 NOUN iajs-2624 129 64 of of ADP iajs-2624 129 65 system system NOUN iajs-2624 129 66 ( ( PUNCT iajs-2624 129 67 3 3 X iajs-2624 129 68 ) ) PUNCT iajs-2624 129 69 is be AUX iajs-2624 129 70 locally locally ADV iajs-2624 129 71 asymptotically asymptotically ADV iajs-2624 129 72 stabile stabile ADJ iajs-2624 129 73 . . PUNCT iajs-2624 130 1 clearly clearly ADV iajs-2624 130 2 conditions condition NOUN iajs-2624 130 3 ( ( PUNCT iajs-2624 130 4 9a)-(9b 9a)-(9b X iajs-2624 130 5 ) ) PUNCT iajs-2624 130 6 guarantee guarantee VERB iajs-2624 130 7 the the DET iajs-2624 130 8 existence existence NOUN iajs-2624 130 9 of of ADP iajs-2624 130 10 all all DET iajs-2624 130 11 eigenvalues eigenvalue NOUN iajs-2624 130 12 in in ADP iajs-2624 130 13 the the DET iajs-2624 130 14 left left ADJ iajs-2624 130 15 half half NOUN iajs-2624 130 16 plane plane NOUN iajs-2624 130 17 and and CCONJ iajs-2624 130 18 the the DET iajs-2624 130 19 proof proof NOUN iajs-2624 130 20 follows follow VERB iajs-2624 130 21 . . PUNCT iajs-2624 131 1 β–  β–  PUNCT iajs-2624 131 2 48 48 NUM iajs-2624 131 3 ibn ibn PROPN iajs-2624 131 4 al al PROPN iajs-2624 131 5 - - PUNCT iajs-2624 131 6 haitham haitham PROPN iajs-2624 131 7 jour jour X iajs-2624 131 8 . . PROPN iajs-2624 132 1 for for ADP iajs-2624 132 2 pure pure ADJ iajs-2624 132 3 & & CCONJ iajs-2624 132 4 appl appl PROPN iajs-2624 132 5 . . PUNCT iajs-2624 133 1 sci sci PROPN iajs-2624 133 2 . . PROPN iajs-2624 134 1 34 34 NUM iajs-2624 134 2 ( ( PUNCT iajs-2624 134 3 2 2 NUM iajs-2624 134 4 ) ) PUNCT iajs-2624 134 5 2021 2021 NUM iajs-2624 134 6 5 5 NUM iajs-2624 134 7 . . PUNCT iajs-2624 134 8 global global ADJ iajs-2624 134 9 stability stability NOUN iajs-2624 134 10 analysis analysis NOUN iajs-2624 134 11 the the DET iajs-2624 134 12 propose propose NOUN iajs-2624 134 13 of of ADP iajs-2624 134 14 this this DET iajs-2624 134 15 section section NOUN iajs-2624 134 16 is be AUX iajs-2624 134 17 to to PART iajs-2624 134 18 investigate investigate VERB iajs-2624 134 19 the the DET iajs-2624 134 20 global global ADJ iajs-2624 134 21 stability stability NOUN iajs-2624 134 22 of of ADP iajs-2624 134 23 covid-19 covid-19 PROPN iajs-2624 134 24 virus virus NOUN iajs-2624 134 25 mathematical mathematical ADJ iajs-2624 134 26 model model NOUN iajs-2624 134 27 given give VERB iajs-2624 134 28 by by ADP iajs-2624 134 29 equations equation NOUN iajs-2624 134 30 in in ADP iajs-2624 134 31 system system NOUN iajs-2624 134 32 ( ( PUNCT iajs-2624 134 33 3 3 NUM iajs-2624 134 34 ) ) PUNCT iajs-2624 134 35 near near ADP iajs-2624 134 36 covid-19 covid-19 PROPN iajs-2624 134 37 free free ADJ iajs-2624 134 38 and and CCONJ iajs-2624 134 39 covid-19 covid-19 PROPN iajs-2624 134 40 points point NOUN iajs-2624 134 41 respectively respectively ADV iajs-2624 134 42 given give VERB iajs-2624 134 43 by by ADP iajs-2624 134 44 𝐸𝑖 𝐸𝑖 NOUN iajs-2624 134 45 , , PUNCT iajs-2624 134 46 𝑖 𝑖 NOUN iajs-2624 134 47 = = SYM iajs-2624 134 48 0,1 0,1 NUM iajs-2624 134 49 . . PUNCT iajs-2624 135 1 we we PRON iajs-2624 135 2 obtain obtain VERB iajs-2624 135 3 the the DET iajs-2624 135 4 result result NOUN iajs-2624 135 5 in in ADP iajs-2624 135 6 the the DET iajs-2624 135 7 following follow VERB iajs-2624 135 8 theorems theorem NOUN iajs-2624 135 9 theorem theorem NOUN iajs-2624 135 10 ( ( PUNCT iajs-2624 135 11 4 4 NUM iajs-2624 135 12 ): ): PUNCT iajs-2624 135 13 𝐸0 𝐸0 PROPN iajs-2624 135 14 is be AUX iajs-2624 135 15 globally globally ADV iajs-2624 135 16 asymptotically asymptotically ADV iajs-2624 135 17 stable stable ADJ iajs-2624 135 18 provided provide VERB iajs-2624 135 19 that that SCONJ iajs-2624 135 20 the the DET iajs-2624 135 21 following follow VERB iajs-2624 135 22 conditions condition NOUN iajs-2624 135 23 hold hold VERB iajs-2624 135 24 : : PUNCT iajs-2624 135 25 β„›0 β„›0 PROPN iajs-2624 135 26 < < X iajs-2624 135 27 1 1 NUM iajs-2624 135 28 ( ( PUNCT iajs-2624 135 29 10a 10a NUM iajs-2624 135 30 ) ) PUNCT iajs-2624 135 31 𝛽𝑆𝑒0 𝛽𝑆𝑒0 PROPN iajs-2624 135 32 < < X iajs-2624 135 33 𝑑 𝑑 PROPN iajs-2624 135 34 βˆ’ βˆ’ PROPN iajs-2624 135 35 π‘Ÿ π‘Ÿ NOUN iajs-2624 135 36 ( ( PUNCT iajs-2624 135 37 10b 10b NOUN iajs-2624 135 38 ) ) PUNCT iajs-2624 135 39 proof proof NOUN iajs-2624 135 40 : : PUNCT iajs-2624 135 41 consider consider VERB iajs-2624 135 42 the the DET iajs-2624 135 43 following follow VERB iajs-2624 135 44 function function NOUN iajs-2624 135 45 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 NOUN iajs-2624 135 46 , , PUNCT iajs-2624 135 47 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 135 48 , , PUNCT iajs-2624 135 49 𝐢 𝐢 PROPN iajs-2624 135 50 , , PUNCT iajs-2624 135 51 𝑣 𝑣 NOUN iajs-2624 135 52 ) ) PUNCT iajs-2624 135 53 = = SYM iajs-2624 135 54 ( ( PUNCT iajs-2624 135 55 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 135 56 βˆ’ βˆ’ PROPN iajs-2624 135 57 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 135 58 βˆ’ βˆ’ PROPN iajs-2624 135 59 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 135 60 ln ln ADJ iajs-2624 135 61 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 135 62 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 135 63 ) ) PUNCT iajs-2624 136 1 + + CCONJ iajs-2624 136 2 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 136 3 + + CCONJ iajs-2624 136 4 𝐢 𝐢 PROPN iajs-2624 136 5 + + CCONJ iajs-2624 136 6 𝑣 𝑣 ADV iajs-2624 136 7 clearly clearly ADV iajs-2624 136 8 , , PUNCT iajs-2624 136 9 𝑉0 𝑉0 ADV iajs-2624 136 10 : : PUNCT iajs-2624 136 11 𝑅+ 𝑅+ PROPN iajs-2624 136 12 4 4 NUM iajs-2624 136 13 β†’ β†’ SYM iajs-2624 136 14 𝑅 𝑅 PROPN iajs-2624 136 15 is be AUX iajs-2624 136 16 a a DET iajs-2624 136 17 continuously continuously ADV iajs-2624 136 18 differentiable differentiable ADJ iajs-2624 136 19 function function NOUN iajs-2624 136 20 such such ADJ iajs-2624 136 21 that that DET iajs-2624 136 22 𝑉0(𝑆𝑒0 𝑉0(𝑆𝑒0 NOUN iajs-2624 136 23 , , PUNCT iajs-2624 136 24 0 0 NUM iajs-2624 136 25 , , PUNCT iajs-2624 136 26 0,0 0,0 NUM iajs-2624 136 27 ) ) PUNCT iajs-2624 136 28 = = SYM iajs-2624 136 29 0 0 NUM iajs-2624 136 30 and and CCONJ iajs-2624 136 31 𝑉0(𝑆𝑒 𝑉0(𝑆𝑒 PROPN iajs-2624 136 32 , , PUNCT iajs-2624 136 33 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 136 34 , , PUNCT iajs-2624 136 35 𝐢 𝐢 PROPN iajs-2624 136 36 , , PUNCT iajs-2624 136 37 𝑣 𝑣 NOUN iajs-2624 136 38 ) ) PUNCT iajs-2624 136 39 > > X iajs-2624 136 40 0 0 NUM iajs-2624 136 41 , , PUNCT iajs-2624 136 42 βˆ€ βˆ€ X iajs-2624 136 43 ( ( PUNCT iajs-2624 136 44 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 136 45 , , PUNCT iajs-2624 136 46 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 136 47 , , PUNCT iajs-2624 136 48 𝐢 𝐢 PROPN iajs-2624 136 49 , , PUNCT iajs-2624 136 50 𝑣 𝑣 NOUN iajs-2624 136 51 ) ) PUNCT iajs-2624 136 52 β‰  β‰  PROPN iajs-2624 136 53 ( ( PUNCT iajs-2624 136 54 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 136 55 , , PUNCT iajs-2624 136 56 0 0 NUM iajs-2624 136 57 , , PUNCT iajs-2624 136 58 0,0 0,0 NOUN iajs-2624 136 59 ) ) PUNCT iajs-2624 136 60 . . PUNCT iajs-2624 137 1 further far ADV iajs-2624 137 2 , , PUNCT iajs-2624 137 3 we we PRON iajs-2624 137 4 have have VERB iajs-2624 137 5 𝑑𝑉0 𝑑𝑉0 ADP iajs-2624 137 6 𝑑𝑑 𝑑𝑑 ADP iajs-2624 137 7 = = SYM iajs-2624 137 8 ( ( PUNCT iajs-2624 137 9 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 137 10 βˆ’ βˆ’ PROPN iajs-2624 137 11 𝑆𝑒0 𝑆𝑒0 VERB iajs-2624 137 12 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 137 13 ) ) PUNCT iajs-2624 138 1 [ [ X iajs-2624 138 2 ψ ψ X iajs-2624 138 3 βˆ’ βˆ’ PROPN iajs-2624 138 4 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 138 5 βˆ’ βˆ’ ADP iajs-2624 138 6 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ ADJ iajs-2624 138 7 βˆ’ βˆ’ NOUN iajs-2624 138 8 πœ‡π‘†π‘’ πœ‡π‘†π‘’ NOUN iajs-2624 138 9 ] ] X iajs-2624 139 1 + + CCONJ iajs-2624 140 1 [ [ X iajs-2624 140 2 βˆ’π›½1(1 βˆ’π›½1(1 NUM iajs-2624 140 3 βˆ’ βˆ’ PRON iajs-2624 140 4 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ PROPN iajs-2624 140 5 βˆ’ βˆ’ PROPN iajs-2624 140 6 𝜎1(1 𝜎1(1 NOUN iajs-2624 140 7 βˆ’ βˆ’ PROPN iajs-2624 140 8 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 140 9 βˆ’ βˆ’ PROPN iajs-2624 140 10 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 140 11 ] ] X iajs-2624 141 1 + + PROPN iajs-2624 141 2 [ [ X iajs-2624 141 3 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 141 4 + + CCONJ iajs-2624 141 5 𝛽1(1 𝛽1(1 PROPN iajs-2624 141 6 βˆ’ βˆ’ PROPN iajs-2624 141 7 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ NOUN iajs-2624 141 8 + + CCONJ iajs-2624 141 9 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ X iajs-2624 141 10 + + CCONJ iajs-2624 141 11 𝜎1(1 𝜎1(1 NOUN iajs-2624 141 12 βˆ’ βˆ’ PROPN iajs-2624 141 13 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 141 14 βˆ’ βˆ’ PROPN iajs-2624 141 15 ( ( PUNCT iajs-2624 141 16 𝛾 𝛾 ADP iajs-2624 141 17 + + NOUN iajs-2624 141 18 πœƒ πœƒ NOUN iajs-2624 141 19 + + X iajs-2624 141 20 πœ‡ πœ‡ X iajs-2624 141 21 + + X iajs-2624 141 22 𝛿1)𝐢 𝛿1)𝐢 ADJ iajs-2624 141 23 ] ] X iajs-2624 141 24 + + CCONJ iajs-2624 141 25 [ [ X iajs-2624 141 26 ( ( PUNCT iajs-2624 141 27 π‘Ÿ π‘Ÿ NOUN iajs-2624 141 28 βˆ’ βˆ’ NOUN iajs-2624 141 29 𝑑)𝑣 𝑑)𝑣 PUNCT iajs-2624 141 30 βˆ’ βˆ’ PROPN iajs-2624 142 1 π‘Ÿ π‘Ÿ PRON iajs-2624 142 2 π‘˜ π‘˜ X iajs-2624 142 3 𝑣2 𝑣2 PROPN iajs-2624 142 4 ] ] PUNCT iajs-2624 142 5 now now ADV iajs-2624 142 6 , , PUNCT iajs-2624 142 7 by by ADP iajs-2624 142 8 doing do VERB iajs-2624 142 9 some some DET iajs-2624 142 10 algebraic algebraic ADJ iajs-2624 142 11 manipulation manipulation NOUN iajs-2624 142 12 and and CCONJ iajs-2624 142 13 using use VERB iajs-2624 142 14 the the DET iajs-2624 142 15 conditions condition NOUN iajs-2624 142 16 ( ( PUNCT iajs-2624 142 17 8) 8) NUM iajs-2624 142 18 , , PUNCT iajs-2624 142 19 ( ( PUNCT iajs-2624 142 20 10a 10a NUM iajs-2624 142 21 ) ) PUNCT iajs-2624 142 22 and and CCONJ iajs-2624 142 23 ( ( PUNCT iajs-2624 142 24 10b 10b NOUN iajs-2624 142 25 ) ) PUNCT iajs-2624 142 26 , , PUNCT iajs-2624 142 27 we we PRON iajs-2624 142 28 get get VERB iajs-2624 142 29 𝑑𝑉0 𝑑𝑉0 ADP iajs-2624 142 30 𝑑𝑑 𝑑𝑑 ADP iajs-2624 142 31 ≀ ≀ NOUN iajs-2624 143 1 βˆ’πœ‡ βˆ’πœ‡ NUM iajs-2624 143 2 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 143 3 ( ( PUNCT iajs-2624 143 4 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 143 5 βˆ’ βˆ’ NOUN iajs-2624 143 6 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 143 7 ) ) PUNCT iajs-2624 143 8 2 2 NUM iajs-2624 143 9 βˆ’ βˆ’ NOUN iajs-2624 143 10 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 143 11 βˆ’ βˆ’ PROPN iajs-2624 144 1 [ [ X iajs-2624 144 2 ( ( PUNCT iajs-2624 144 3 𝛾 𝛾 NOUN iajs-2624 144 4 + + NOUN iajs-2624 144 5 πœƒ πœƒ NOUN iajs-2624 144 6 + + NUM iajs-2624 144 7 πœ‡ πœ‡ X iajs-2624 144 8 + + X iajs-2624 144 9 𝛿1 𝛿1 ADJ iajs-2624 144 10 ) ) PUNCT iajs-2624 144 11 βˆ’ βˆ’ PROPN iajs-2624 144 12 πœŽπ‘†π‘’0]𝐢 πœŽπ‘†π‘’0]𝐢 NOUN iajs-2624 144 13 βˆ’ βˆ’ NOUN iajs-2624 144 14 π‘Ÿ π‘Ÿ NOUN iajs-2624 144 15 π‘˜ π‘˜ X iajs-2624 144 16 𝑣2 𝑣2 PRON iajs-2624 144 17 βˆ’ βˆ’ PROPN iajs-2624 145 1 [ [ X iajs-2624 145 2 ( ( PUNCT iajs-2624 145 3 𝑑 𝑑 PROPN iajs-2624 145 4 βˆ’ βˆ’ PROPN iajs-2624 145 5 π‘Ÿ π‘Ÿ NOUN iajs-2624 145 6 ) ) PUNCT iajs-2624 145 7 βˆ’ βˆ’ PROPN iajs-2624 145 8 𝛽𝑆𝑒0]𝑣 𝛽𝑆𝑒0]𝑣 X iajs-2624 145 9 obviously obviously ADV iajs-2624 145 10 , , PUNCT iajs-2624 145 11 οΏ½ οΏ½ PROPN iajs-2624 145 12 Μ‡ Μ‡ PROPN iajs-2624 145 13 οΏ½ οΏ½ PROPN iajs-2624 145 14 0 0 NUM iajs-2624 145 15 = = SYM iajs-2624 145 16 0 0 NUM iajs-2624 145 17 at at ADP iajs-2624 145 18 𝐸0 𝐸0 ADJ iajs-2624 145 19 = = SYM iajs-2624 145 20 ( ( PUNCT iajs-2624 145 21 𝑆𝑒0 𝑆𝑒0 NOUN iajs-2624 145 22 , , PUNCT iajs-2624 145 23 0,0,0 0,0,0 NOUN iajs-2624 145 24 ) ) PUNCT iajs-2624 145 25 , , PUNCT iajs-2624 145 26 moreover moreover PROPN iajs-2624 145 27 οΏ½ οΏ½ PROPN iajs-2624 145 28 Μ‡ Μ‡ NOUN iajs-2624 145 29 οΏ½ οΏ½ PROPN iajs-2624 145 30 0 0 NUM iajs-2624 145 31 < < X iajs-2624 145 32 0 0 X iajs-2624 146 1 otherwise otherwise ADV iajs-2624 146 2 . . PUNCT iajs-2624 147 1 hence hence ADV iajs-2624 147 2 , , PUNCT iajs-2624 147 3 οΏ½ οΏ½ PROPN iajs-2624 147 4 Μ‡ Μ‡ PROPN iajs-2624 147 5 οΏ½ οΏ½ PROPN iajs-2624 147 6 0 0 NUM iajs-2624 147 7 is be AUX iajs-2624 147 8 negative negative ADJ iajs-2624 147 9 definite definite ADJ iajs-2624 147 10 and and CCONJ iajs-2624 147 11 then then ADV iajs-2624 147 12 the the DET iajs-2624 147 13 solution solution NOUN iajs-2624 147 14 starting start VERB iajs-2624 147 15 from from ADP iajs-2624 147 16 any any DET iajs-2624 147 17 initial initial ADJ iajs-2624 147 18 point point NOUN iajs-2624 147 19 that that PRON iajs-2624 147 20 satisfies satisfy VERB iajs-2624 147 21 the the DET iajs-2624 147 22 conditions condition NOUN iajs-2624 147 23 ( ( PUNCT iajs-2624 147 24 8b 8b NUM iajs-2624 147 25 ) ) PUNCT iajs-2624 147 26 with with ADP iajs-2624 147 27 ( ( PUNCT iajs-2624 147 28 10a 10a NOUN iajs-2624 147 29 ) ) PUNCT iajs-2624 147 30 and and CCONJ iajs-2624 147 31 ( ( PUNCT iajs-2624 147 32 10b 10b NOUN iajs-2624 147 33 ) ) PUNCT iajs-2624 147 34 will will AUX iajs-2624 147 35 approach approach VERB iajs-2624 147 36 asymptotically asymptotically ADV iajs-2624 147 37 to to ADP iajs-2624 147 38 covid-19 covid-19 PROPN iajs-2624 147 39 free free ADJ iajs-2624 147 40 equilibrium equilibrium NOUN iajs-2624 147 41 point point NOUN iajs-2624 147 42 . . PUNCT iajs-2624 148 1 hence hence ADV iajs-2624 148 2 , , PUNCT iajs-2624 148 3 the the DET iajs-2624 148 4 proof proof NOUN iajs-2624 148 5 is be AUX iajs-2624 148 6 complete complete ADJ iajs-2624 148 7 . . PUNCT iajs-2624 149 1 β–  β–  PUNCT iajs-2624 149 2 theorem theorem ADJ iajs-2624 149 3 ( ( PUNCT iajs-2624 149 4 5 5 NUM iajs-2624 149 5 ): ): PUNCT iajs-2624 149 6 𝐸1 𝐸1 NOUN iajs-2624 149 7 is be AUX iajs-2624 149 8 global global ADJ iajs-2624 149 9 asymptotically asymptotically ADV iajs-2624 149 10 stable stable ADJ iajs-2624 149 11 if if SCONJ iajs-2624 149 12 β„›0 β„›0 PROPN iajs-2624 149 13 > > X iajs-2624 149 14 1 1 X iajs-2624 149 15 . . X iajs-2624 150 1 proof proof NOUN iajs-2624 150 2 : : PUNCT iajs-2624 150 3 for for ADP iajs-2624 150 4 the the DET iajs-2624 150 5 covid-19 covid-19 PROPN iajs-2624 150 6 equilibrium equilibrium NOUN iajs-2624 150 7 𝐸1 𝐸1 NOUN iajs-2624 150 8 = = PUNCT iajs-2624 150 9 ( ( PUNCT iajs-2624 150 10 𝑆𝑒1 𝑆𝑒1 ADJ iajs-2624 150 11 , , PUNCT iajs-2624 150 12 π‘†π‘Ž1 π‘†π‘Ž1 ADJ iajs-2624 150 13 , , PUNCT iajs-2624 150 14 𝐢1 𝐢1 PRON iajs-2624 150 15 , , PUNCT iajs-2624 150 16 𝑣1 𝑣1 NOUN iajs-2624 150 17 ) ) PUNCT iajs-2624 150 18 ; ; PUNCT iajs-2624 150 19 𝑆𝑒1 𝑆𝑒1 X iajs-2624 150 20 , , PUNCT iajs-2624 150 21 π‘†π‘Ž1 π‘†π‘Ž1 ADV iajs-2624 150 22 , , PUNCT iajs-2624 150 23 𝐢1 𝐢1 PRON iajs-2624 150 24 and and CCONJ iajs-2624 150 25 𝑣1 𝑣1 NOUN iajs-2624 150 26 satisfies satisfie NOUN iajs-2624 150 27 equations equation NOUN iajs-2624 150 28 πœ“ πœ“ ADP iajs-2624 150 29 βˆ’ βˆ’ PROPN iajs-2624 150 30 𝛼𝑆𝑒 𝛼𝑆𝑒 NOUN iajs-2624 150 31 βˆ’ βˆ’ PROPN iajs-2624 150 32 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 150 33 βˆ’ βˆ’ ADP iajs-2624 151 1 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ NOUN iajs-2624 151 2 βˆ’ βˆ’ NUM iajs-2624 151 3 πœ‡π‘†π‘’ πœ‡π‘†π‘’ NOUN iajs-2624 151 4 = = SYM iajs-2624 151 5 0 0 NUM iajs-2624 151 6 𝛼𝑆𝑒 𝛼𝑆𝑒 NOUN iajs-2624 151 7 βˆ’ βˆ’ PROPN iajs-2624 151 8 𝛽1(1 𝛽1(1 PROPN iajs-2624 151 9 βˆ’ βˆ’ PROPN iajs-2624 151 10 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ PROPN iajs-2624 151 11 βˆ’ βˆ’ PROPN iajs-2624 151 12 𝜎1(1 𝜎1(1 NOUN iajs-2624 151 13 βˆ’ βˆ’ PROPN iajs-2624 151 14 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 151 15 βˆ’ βˆ’ PROPN iajs-2624 151 16 πœ‡π‘†π‘Ž πœ‡π‘†π‘Ž NOUN iajs-2624 151 17 = = SYM iajs-2624 151 18 0 0 PUNCT iajs-2624 151 19 𝛽𝑆𝑒𝑣 𝛽𝑆𝑒𝑣 PROPN iajs-2624 151 20 + + CCONJ iajs-2624 151 21 𝛽1(1 𝛽1(1 PROPN iajs-2624 151 22 βˆ’ βˆ’ PROPN iajs-2624 151 23 πœ–)π‘†π‘Žπ‘£ πœ–)π‘†π‘Žπ‘£ NOUN iajs-2624 152 1 + + CCONJ iajs-2624 152 2 πœŽπ‘†π‘’πΆ πœŽπ‘†π‘’πΆ X iajs-2624 152 3 + + CCONJ iajs-2624 152 4 𝜎1(1 𝜎1(1 NOUN iajs-2624 152 5 βˆ’ βˆ’ PROPN iajs-2624 152 6 πœ–)π‘†π‘ŽπΆ πœ–)π‘†π‘ŽπΆ PRON iajs-2624 152 7 βˆ’ βˆ’ PROPN iajs-2624 152 8 ( ( PUNCT iajs-2624 152 9 πœ‡ πœ‡ X iajs-2624 152 10 + + X iajs-2624 152 11 πœƒ πœƒ NOUN iajs-2624 152 12 + + CCONJ iajs-2624 152 13 𝛾 𝛾 VERB iajs-2624 152 14 + + CCONJ iajs-2624 152 15 𝛿1)𝐢 𝛿1)𝐢 ADJ iajs-2624 152 16 = = SYM iajs-2624 152 17 0 0 NUM iajs-2624 152 18 π‘Ÿπ‘£ π‘Ÿπ‘£ PROPN iajs-2624 152 19 ( ( PUNCT iajs-2624 152 20 1 1 NUM iajs-2624 152 21 βˆ’ βˆ’ PROPN iajs-2624 152 22 𝑣 𝑣 PRON iajs-2624 152 23 π‘˜ π‘˜ NOUN iajs-2624 152 24 ) ) PUNCT iajs-2624 152 25 βˆ’ βˆ’ PROPN iajs-2624 152 26 𝑑𝑣 𝑑𝑣 ADP iajs-2624 152 27 = = SYM iajs-2624 152 28 0 0 NUM iajs-2624 152 29 ( ( PUNCT iajs-2624 152 30 11 11 NUM iajs-2624 152 31 ) ) PUNCT iajs-2624 152 32 by by ADP iajs-2624 152 33 applying apply VERB iajs-2624 152 34 ( ( PUNCT iajs-2624 152 35 11 11 NUM iajs-2624 152 36 ) ) PUNCT iajs-2624 152 37 and and CCONJ iajs-2624 152 38 putting put VERB iajs-2624 152 39 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 152 40 𝑆𝑒1 𝑆𝑒1 NOUN iajs-2624 152 41 = = SYM iajs-2624 152 42 π‘₯ π‘₯ INTJ iajs-2624 152 43 , , PUNCT iajs-2624 152 44 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 152 45 π‘†π‘Ž1 π‘†π‘Ž1 NOUN iajs-2624 152 46 = = SYM iajs-2624 152 47 𝑦 𝑦 NOUN iajs-2624 152 48 , , PUNCT iajs-2624 152 49 𝐢 𝐢 PROPN iajs-2624 152 50 𝐢1 𝐢1 PROPN iajs-2624 152 51 = = SYM iajs-2624 152 52 𝑧 𝑧 NOUN iajs-2624 152 53 , , PUNCT iajs-2624 152 54 𝑣 𝑣 PRON iajs-2624 152 55 𝑣1 𝑣1 NOUN iajs-2624 152 56 = = PROPN iajs-2624 152 57 𝑒 𝑒 PROPN iajs-2624 152 58 49 49 NUM iajs-2624 152 59 ibn ibn PROPN iajs-2624 152 60 al al PROPN iajs-2624 152 61 - - PUNCT iajs-2624 152 62 haitham haitham PROPN iajs-2624 152 63 jour jour X iajs-2624 152 64 . . PROPN iajs-2624 153 1 for for ADP iajs-2624 153 2 pure pure ADJ iajs-2624 153 3 & & CCONJ iajs-2624 153 4 appl appl PROPN iajs-2624 153 5 . . PUNCT iajs-2624 154 1 sci sci PROPN iajs-2624 154 2 . . PROPN iajs-2624 155 1 34 34 NUM iajs-2624 155 2 ( ( PUNCT iajs-2624 155 3 2 2 NUM iajs-2624 155 4 ) ) PUNCT iajs-2624 155 5 2021 2021 NUM iajs-2624 155 6 we we PRON iajs-2624 155 7 have have VERB iajs-2624 155 8 π‘₯β€² π‘₯β€² VERB iajs-2624 156 1 = = PUNCT iajs-2624 156 2 π‘₯ π‘₯ PROPN iajs-2624 156 3 [ [ PUNCT iajs-2624 156 4 πœ“ πœ“ X iajs-2624 156 5 𝑆𝑒1 𝑆𝑒1 X iajs-2624 156 6 ( ( PUNCT iajs-2624 156 7 1 1 NUM iajs-2624 156 8 π‘₯ π‘₯ PRON iajs-2624 156 9 βˆ’ βˆ’ NUM iajs-2624 156 10 1 1 NUM iajs-2624 156 11 ) ) PUNCT iajs-2624 157 1 βˆ’ βˆ’ PROPN iajs-2624 157 2 𝛽𝑣1(𝑒 𝛽𝑣1(𝑒 NOUN iajs-2624 157 3 βˆ’ βˆ’ NOUN iajs-2624 157 4 1 1 NUM iajs-2624 157 5 ) ) PUNCT iajs-2624 157 6 βˆ’ βˆ’ NOUN iajs-2624 157 7 𝜎𝐢1(𝑧 𝜎𝐢1(𝑧 VERB iajs-2624 157 8 βˆ’ βˆ’ NOUN iajs-2624 157 9 1 1 NUM iajs-2624 157 10 ) ) PUNCT iajs-2624 157 11 ] ] PUNCT iajs-2624 157 12 𝑦′ 𝑦′ X iajs-2624 157 13 = = SYM iajs-2624 157 14 𝑦 𝑦 SYM iajs-2624 158 1 [ [ X iajs-2624 158 2 𝛼 𝛼 X iajs-2624 158 3 𝑆𝑒1 𝑆𝑒1 ADJ iajs-2624 158 4 π‘ π‘Ž1 π‘ π‘Ž1 INTJ iajs-2624 158 5 ( ( PUNCT iajs-2624 158 6 π‘₯ π‘₯ NOUN iajs-2624 158 7 βˆ’ βˆ’ PROPN iajs-2624 158 8 1 1 NUM iajs-2624 158 9 ) ) PUNCT iajs-2624 158 10 βˆ’ βˆ’ PROPN iajs-2624 158 11 𝛽1(1 𝛽1(1 PROPN iajs-2624 158 12 βˆ’ βˆ’ PROPN iajs-2624 158 13 πœ–)𝑣1(𝑒 πœ–)𝑣1(𝑒 NUM iajs-2624 158 14 βˆ’ βˆ’ PROPN iajs-2624 158 15 1 1 NUM iajs-2624 158 16 ) ) PUNCT iajs-2624 158 17 βˆ’ βˆ’ PROPN iajs-2624 158 18 𝜎1(1 𝜎1(1 PROPN iajs-2624 158 19 βˆ’ βˆ’ PROPN iajs-2624 158 20 πœ–)𝐢1(𝑧 πœ–)𝐢1(𝑧 NUM iajs-2624 158 21 βˆ’ βˆ’ PROPN iajs-2624 158 22 1 1 NUM iajs-2624 158 23 ) ) PUNCT iajs-2624 158 24 ] ] PUNCT iajs-2624 158 25 𝑧′ 𝑧′ PUNCT iajs-2624 159 1 = = SYM iajs-2624 159 2 𝑧 𝑧 X iajs-2624 159 3 [ [ PUNCT iajs-2624 159 4 𝛽𝑆𝑒1𝑣1 𝛽𝑆𝑒1𝑣1 ADJ iajs-2624 159 5 𝐢1 𝐢1 X iajs-2624 159 6 ( ( PUNCT iajs-2624 159 7 π‘₯𝑒 π‘₯𝑒 INTJ iajs-2624 159 8 𝑧 𝑧 PRON iajs-2624 159 9 βˆ’ βˆ’ PROPN iajs-2624 159 10 1 1 NUM iajs-2624 159 11 ) ) PUNCT iajs-2624 159 12 + + CCONJ iajs-2624 159 13 𝛽1(1 𝛽1(1 ADJ iajs-2624 159 14 βˆ’ βˆ’ PROPN iajs-2624 159 15 πœ–)π‘†π‘Ž1𝑣1 πœ–)π‘†π‘Ž1𝑣1 NOUN iajs-2624 159 16 𝑐1 𝑐1 NOUN iajs-2624 159 17 ( ( PUNCT iajs-2624 159 18 𝑦𝑒 𝑦𝑒 NOUN iajs-2624 159 19 𝑧 𝑧 PRON iajs-2624 159 20 βˆ’ βˆ’ PROPN iajs-2624 159 21 1 1 NUM iajs-2624 159 22 ) ) PUNCT iajs-2624 159 23 + + CCONJ iajs-2624 159 24 πœŽπ‘†π‘’1(π‘₯ πœŽπ‘†π‘’1(π‘₯ ADJ iajs-2624 159 25 βˆ’ βˆ’ NOUN iajs-2624 159 26 1 1 X iajs-2624 159 27 ) ) PUNCT iajs-2624 159 28 + + CCONJ iajs-2624 159 29 𝜎1(1 𝜎1(1 NOUN iajs-2624 159 30 βˆ’ βˆ’ PROPN iajs-2624 159 31 πœ–)(𝑦 πœ–)(𝑦 NOUN iajs-2624 159 32 βˆ’ βˆ’ PROPN iajs-2624 159 33 1 1 NUM iajs-2624 159 34 ) ) PUNCT iajs-2624 159 35 ] ] PUNCT iajs-2624 159 36 𝑒′ 𝑒′ ADP iajs-2624 159 37 = = SYM iajs-2624 159 38 𝑒 𝑒 X iajs-2624 159 39 [ [ PUNCT iajs-2624 159 40 βˆ’π‘Ÿπ‘£1 βˆ’π‘Ÿπ‘£1 ADP iajs-2624 159 41 π‘˜ π‘˜ X iajs-2624 159 42 ( ( PUNCT iajs-2624 159 43 𝑧 𝑧 PROPN iajs-2624 159 44 βˆ’ βˆ’ PROPN iajs-2624 159 45 1 1 NUM iajs-2624 159 46 ) ) PUNCT iajs-2624 159 47 ] ] PUNCT iajs-2624 159 48 ( ( PUNCT iajs-2624 159 49 12 12 NUM iajs-2624 159 50 ) ) PUNCT iajs-2624 159 51 define define VERB iajs-2624 159 52 the the DET iajs-2624 159 53 lyapunov lyapunov ADJ iajs-2624 159 54 function function NOUN iajs-2624 159 55 𝑉1 𝑉1 NOUN iajs-2624 159 56 = = SYM iajs-2624 159 57 𝑆𝑒1(π‘₯ 𝑆𝑒1(π‘₯ ADP iajs-2624 159 58 βˆ’ βˆ’ PROPN iajs-2624 159 59 1 1 NUM iajs-2624 159 60 βˆ’ βˆ’ PROPN iajs-2624 159 61 ln ln NOUN iajs-2624 159 62 π‘₯ π‘₯ NOUN iajs-2624 159 63 ) ) PUNCT iajs-2624 159 64 + + CCONJ iajs-2624 159 65 π‘†π‘Ž1(𝑦 π‘†π‘Ž1(𝑦 PROPN iajs-2624 159 66 βˆ’ βˆ’ PROPN iajs-2624 159 67 1 1 NUM iajs-2624 159 68 βˆ’ βˆ’ PROPN iajs-2624 159 69 ln ln PROPN iajs-2624 159 70 𝑦 𝑦 NOUN iajs-2624 159 71 ) ) PUNCT iajs-2624 159 72 + + ADJ iajs-2624 159 73 𝐢1(𝑧 𝐢1(𝑧 VERB iajs-2624 159 74 βˆ’ βˆ’ NUM iajs-2624 159 75 1 1 NUM iajs-2624 159 76 βˆ’ βˆ’ PROPN iajs-2624 159 77 ln ln NOUN iajs-2624 159 78 𝑧 𝑧 NOUN iajs-2624 159 79 ) ) PUNCT iajs-2624 159 80 + + CCONJ iajs-2624 160 1 𝑣1(𝑒 𝑣1(𝑒 NUM iajs-2624 160 2 βˆ’ βˆ’ NUM iajs-2624 160 3 1 1 NUM iajs-2624 160 4 βˆ’ βˆ’ PROPN iajs-2624 160 5 ln ln ADJ iajs-2624 160 6 𝑒 𝑒 NOUN iajs-2624 160 7 ) ) PUNCT iajs-2624 160 8 ( ( PUNCT iajs-2624 160 9 13 13 NUM iajs-2624 160 10 ) ) PUNCT iajs-2624 160 11 the the DET iajs-2624 160 12 derivative derivative NOUN iajs-2624 160 13 of of ADP iajs-2624 160 14 𝑉1 𝑉1 NOUN iajs-2624 160 15 is be AUX iajs-2624 160 16 given give VERB iajs-2624 160 17 by by ADP iajs-2624 160 18 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2624 160 19 𝑑𝑑 𝑑𝑑 ADP iajs-2624 160 20 = = PUNCT iajs-2624 160 21 𝑆𝑒1 𝑆𝑒1 ADJ iajs-2624 160 22 π‘₯βˆ’1 π‘₯βˆ’1 NOUN iajs-2624 160 23 π‘₯ π‘₯ NOUN iajs-2624 160 24 π‘₯β€² π‘₯β€² PUNCT iajs-2624 161 1 + + CCONJ iajs-2624 161 2 π‘†π‘Ž1 π‘†π‘Ž1 ADJ iajs-2624 161 3 π‘¦βˆ’1 π‘¦βˆ’1 PROPN iajs-2624 161 4 𝑦 𝑦 NOUN iajs-2624 161 5 𝑦′ 𝑦′ X iajs-2624 161 6 + + CCONJ iajs-2624 161 7 𝐢1 𝐢1 PROPN iajs-2624 161 8 π‘§βˆ’1 π‘§βˆ’1 PROPN iajs-2624 161 9 𝑧 𝑧 DET iajs-2624 161 10 𝑧′ 𝑧′ NUM iajs-2624 161 11 + + CCONJ iajs-2624 161 12 𝑣1 𝑣1 PROPN iajs-2624 161 13 π‘’βˆ’1 π‘’βˆ’1 PROPN iajs-2624 161 14 𝑒 𝑒 PROPN iajs-2624 161 15 𝑣′ 𝑣′ PROPN iajs-2624 161 16 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2624 161 17 𝑑𝑑 𝑑𝑑 ADP iajs-2624 161 18 = = SYM iajs-2624 161 19 ( ( PUNCT iajs-2624 161 20 π‘₯ π‘₯ NOUN iajs-2624 161 21 βˆ’ βˆ’ NOUN iajs-2624 161 22 1 1 NUM iajs-2624 161 23 ) ) PUNCT iajs-2624 162 1 [ [ X iajs-2624 162 2 πœ“ πœ“ X iajs-2624 162 3 ( ( PUNCT iajs-2624 162 4 1 1 NUM iajs-2624 162 5 π‘₯ π‘₯ PRON iajs-2624 162 6 βˆ’ βˆ’ NUM iajs-2624 162 7 1 1 NUM iajs-2624 162 8 ) ) PUNCT iajs-2624 162 9 βˆ’ βˆ’ ADP iajs-2624 162 10 𝛽𝑆𝑒1𝑣1(𝑒 𝛽𝑆𝑒1𝑣1(𝑒 NUM iajs-2624 162 11 βˆ’ βˆ’ NUM iajs-2624 162 12 1 1 NUM iajs-2624 162 13 ) ) PUNCT iajs-2624 162 14 βˆ’ βˆ’ PROPN iajs-2624 163 1 πœŽπ‘†π‘’1𝐢1(𝑧 πœŽπ‘†π‘’1𝐢1(𝑧 VERB iajs-2624 163 2 βˆ’ βˆ’ NOUN iajs-2624 163 3 1 1 NUM iajs-2624 163 4 ) ) PUNCT iajs-2624 163 5 ] ] PUNCT iajs-2624 164 1 + + ADJ iajs-2624 164 2 ( ( PUNCT iajs-2624 164 3 𝑦 𝑦 NOUN iajs-2624 164 4 βˆ’ βˆ’ PROPN iajs-2624 164 5 1)[𝛼𝑆𝑒1(π‘₯ 1)[𝛼𝑆𝑒1(π‘₯ NUM iajs-2624 164 6 βˆ’ βˆ’ NOUN iajs-2624 164 7 1 1 NUM iajs-2624 164 8 ) ) PUNCT iajs-2624 164 9 βˆ’ βˆ’ PROPN iajs-2624 164 10 𝛽1(1 𝛽1(1 PROPN iajs-2624 164 11 βˆ’ βˆ’ PROPN iajs-2624 164 12 πœ–)π‘†π‘Ž1𝑣1(𝑒 πœ–)π‘†π‘Ž1𝑣1(𝑒 PROPN iajs-2624 164 13 βˆ’ βˆ’ NOUN iajs-2624 164 14 1 1 NUM iajs-2624 164 15 ) ) PUNCT iajs-2624 164 16 βˆ’ βˆ’ PROPN iajs-2624 164 17 𝜎1(1 𝜎1(1 NOUN iajs-2624 164 18 βˆ’ βˆ’ NOUN iajs-2624 164 19 πœ–)π‘†π‘Ž1𝐢1(𝑧 πœ–)π‘†π‘Ž1𝐢1(𝑧 NOUN iajs-2624 164 20 βˆ’ βˆ’ PROPN iajs-2624 164 21 1 1 NUM iajs-2624 164 22 ) ) PUNCT iajs-2624 164 23 ] ] PUNCT iajs-2624 165 1 + + PROPN iajs-2624 165 2 ( ( PUNCT iajs-2624 165 3 𝑧 𝑧 DET iajs-2624 165 4 βˆ’ βˆ’ PROPN iajs-2624 165 5 1 1 NUM iajs-2624 165 6 ) ) PUNCT iajs-2624 165 7 [ [ X iajs-2624 165 8 𝛽𝑆𝑒1𝑣1 𝛽𝑆𝑒1𝑣1 NOUN iajs-2624 165 9 ( ( PUNCT iajs-2624 165 10 π‘₯𝑒 π‘₯𝑒 ADV iajs-2624 165 11 𝑧 𝑧 PRON iajs-2624 165 12 βˆ’ βˆ’ PROPN iajs-2624 165 13 1 1 NUM iajs-2624 165 14 ) ) PUNCT iajs-2624 165 15 + + CCONJ iajs-2624 165 16 𝛽1(1 𝛽1(1 ADJ iajs-2624 165 17 βˆ’ βˆ’ PROPN iajs-2624 165 18 πœ–)π‘†π‘Ž1𝑣1 πœ–)π‘†π‘Ž1𝑣1 NOUN iajs-2624 165 19 ( ( PUNCT iajs-2624 165 20 𝑦𝑒 𝑦𝑒 NOUN iajs-2624 165 21 𝑧 𝑧 PRON iajs-2624 165 22 βˆ’ βˆ’ NUM iajs-2624 165 23 1 1 NUM iajs-2624 165 24 ) ) PUNCT iajs-2624 165 25 + + NOUN iajs-2624 165 26 πœŽπ‘†π‘’1𝐢1(π‘₯ πœŽπ‘†π‘’1𝐢1(π‘₯ NUM iajs-2624 165 27 βˆ’ βˆ’ NOUN iajs-2624 165 28 1 1 NUM iajs-2624 165 29 ) ) PUNCT iajs-2624 165 30 + + CCONJ iajs-2624 165 31 𝜎1(1 𝜎1(1 PROPN iajs-2624 165 32 βˆ’ βˆ’ PROPN iajs-2624 165 33 πœ–)𝐢1(𝑦 πœ–)𝐢1(𝑦 PROPN iajs-2624 165 34 βˆ’ βˆ’ NOUN iajs-2624 165 35 1 1 NUM iajs-2624 165 36 ) ) PUNCT iajs-2624 165 37 ] ] PUNCT iajs-2624 166 1 + + CCONJ iajs-2624 166 2 ( ( PUNCT iajs-2624 166 3 𝑒 𝑒 PRON iajs-2624 166 4 βˆ’ βˆ’ PROPN iajs-2624 166 5 1 1 NUM iajs-2624 166 6 ) ) PUNCT iajs-2624 166 7 [ [ PUNCT iajs-2624 166 8 βˆ’π‘Ÿπ‘£1 βˆ’π‘Ÿπ‘£1 NOUN iajs-2624 166 9 2 2 NUM iajs-2624 166 10 π‘˜ π‘˜ PRON iajs-2624 166 11 ( ( PUNCT iajs-2624 166 12 𝑧 𝑧 PROPN iajs-2624 166 13 βˆ’ βˆ’ PROPN iajs-2624 166 14 1 1 NUM iajs-2624 166 15 ) ) PUNCT iajs-2624 166 16 ] ] PUNCT iajs-2624 166 17 furthermore furthermore ADV iajs-2624 166 18 , , PUNCT iajs-2624 166 19 by by ADP iajs-2624 166 20 simplifying simplify VERB iajs-2624 166 21 the the DET iajs-2624 166 22 resulting result VERB iajs-2624 166 23 terms term NOUN iajs-2624 166 24 , , PUNCT iajs-2624 166 25 we we PRON iajs-2624 166 26 get get VERB iajs-2624 166 27 that that PRON iajs-2624 166 28 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2624 166 29 𝑑𝑑 𝑑𝑑 ADP iajs-2624 166 30 = = SYM iajs-2624 166 31 πœ“(2 πœ“(2 PROPN iajs-2624 166 32 βˆ’ βˆ’ ADP iajs-2624 166 33 π‘₯ π‘₯ INTJ iajs-2624 167 1 βˆ’ βˆ’ PROPN iajs-2624 167 2 1 1 NUM iajs-2624 167 3 π‘₯ π‘₯ NOUN iajs-2624 167 4 ) ) PUNCT iajs-2624 168 1 + + PUNCT iajs-2624 168 2 𝛽𝑆𝑒1𝑣1 𝛽𝑆𝑒1𝑣1 NOUN iajs-2624 168 3 ( ( PUNCT iajs-2624 168 4 π‘₯ π‘₯ PROPN iajs-2624 168 5 + + NUM iajs-2624 168 6 𝑒 𝑒 PROPN iajs-2624 168 7 βˆ’ βˆ’ NOUN iajs-2624 168 8 𝑧 𝑧 PRON iajs-2624 168 9 βˆ’ βˆ’ NOUN iajs-2624 168 10 π‘₯𝑒 π‘₯𝑒 NOUN iajs-2624 168 11 𝑧 𝑧 NOUN iajs-2624 168 12 ) ) PUNCT iajs-2624 169 1 + + CCONJ iajs-2624 169 2 𝛼𝑆𝑒1(π‘₯𝑦 𝛼𝑆𝑒1(π‘₯𝑦 PROPN iajs-2624 169 3 βˆ’ βˆ’ NOUN iajs-2624 169 4 π‘₯ π‘₯ PRON iajs-2624 169 5 βˆ’ βˆ’ PROPN iajs-2624 169 6 𝑦 𝑦 NOUN iajs-2624 169 7 + + NOUN iajs-2624 169 8 1 1 NUM iajs-2624 169 9 ) ) PUNCT iajs-2624 169 10 + + ADJ iajs-2624 169 11 𝛽1(1 𝛽1(1 ADJ iajs-2624 169 12 βˆ’ βˆ’ PROPN iajs-2624 169 13 πœ–)π‘†π‘Ž1𝑣1 πœ–)π‘†π‘Ž1𝑣1 NOUN iajs-2624 169 14 ( ( PUNCT iajs-2624 169 15 𝑦 𝑦 NOUN iajs-2624 169 16 + + X iajs-2624 169 17 𝑒 𝑒 NOUN iajs-2624 169 18 βˆ’ βˆ’ NOUN iajs-2624 169 19 𝑧 𝑧 PRON iajs-2624 169 20 βˆ’ βˆ’ PROPN iajs-2624 169 21 𝑦𝑒 𝑦𝑒 NOUN iajs-2624 169 22 𝑧 𝑧 PROPN iajs-2624 169 23 ) ) PUNCT iajs-2624 169 24 βˆ’ βˆ’ ADP iajs-2624 169 25 π‘Ÿπ‘£1 π‘Ÿπ‘£1 NOUN iajs-2624 169 26 2 2 NUM iajs-2624 169 27 π‘˜ π‘˜ NOUN iajs-2624 169 28 ( ( PUNCT iajs-2624 169 29 𝑧𝑒 𝑧𝑒 ADP iajs-2624 169 30 βˆ’ βˆ’ PROPN iajs-2624 169 31 𝑧 𝑧 PRON iajs-2624 169 32 βˆ’ βˆ’ PUNCT iajs-2624 169 33 𝑒 𝑒 PART iajs-2624 169 34 + + NOUN iajs-2624 169 35 1 1 NUM iajs-2624 169 36 ) ) PUNCT iajs-2624 169 37 since since SCONJ iajs-2624 169 38 the the DET iajs-2624 169 39 arithmetical arithmetical ADJ iajs-2624 169 40 mean mean NOUN iajs-2624 169 41 is be AUX iajs-2624 169 42 greater great ADJ iajs-2624 169 43 than than ADP iajs-2624 169 44 , , PUNCT iajs-2624 169 45 or or CCONJ iajs-2624 169 46 equal equal ADJ iajs-2624 169 47 to to ADP iajs-2624 169 48 the the DET iajs-2624 169 49 geometrical geometrical ADJ iajs-2624 169 50 mean mean NOUN iajs-2624 169 51 , , PUNCT iajs-2624 169 52 then then ADV iajs-2624 169 53 , , PUNCT iajs-2624 169 54 2 2 NUM iajs-2624 169 55 βˆ’ βˆ’ NOUN iajs-2624 169 56 π‘₯ π‘₯ PRON iajs-2624 169 57 βˆ’ βˆ’ NUM iajs-2624 169 58 1 1 NUM iajs-2624 169 59 π‘₯ π‘₯ NOUN iajs-2624 169 60 ≀ ≀ NUM iajs-2624 169 61 0 0 NUM iajs-2624 169 62 for for ADP iajs-2624 169 63 π‘₯ π‘₯ PROPN iajs-2624 169 64 > > SYM iajs-2624 169 65 0 0 NUM iajs-2624 169 66 and and CCONJ iajs-2624 169 67 2 2 NUM iajs-2624 169 68 βˆ’ βˆ’ NOUN iajs-2624 169 69 π‘₯ π‘₯ PRON iajs-2624 169 70 βˆ’ βˆ’ NUM iajs-2624 169 71 1 1 NUM iajs-2624 169 72 π‘₯ π‘₯ NOUN iajs-2624 169 73 = = SYM iajs-2624 169 74 0 0 PUNCT iajs-2624 169 75 if if SCONJ iajs-2624 169 76 and and CCONJ iajs-2624 169 77 only only ADV iajs-2624 169 78 if if SCONJ iajs-2624 169 79 π‘₯ π‘₯ PROPN iajs-2624 169 80 = = SYM iajs-2624 169 81 1 1 NUM iajs-2624 169 82 ; ; PUNCT iajs-2624 169 83 π‘₯ π‘₯ SYM iajs-2624 169 84 + + NUM iajs-2624 169 85 𝑒 𝑒 PROPN iajs-2624 169 86 βˆ’ βˆ’ NOUN iajs-2624 169 87 𝑧 𝑧 PRON iajs-2624 169 88 βˆ’ βˆ’ PROPN iajs-2624 169 89 π‘₯𝑒 π‘₯𝑒 INTJ iajs-2624 169 90 𝑧 𝑧 DET iajs-2624 169 91 ≀ ≀ NUM iajs-2624 169 92 0 0 NUM iajs-2624 169 93 for for ADP iajs-2624 169 94 π‘₯ π‘₯ PROPN iajs-2624 169 95 , , PUNCT iajs-2624 169 96 𝑧 𝑧 PROPN iajs-2624 169 97 , , PUNCT iajs-2624 169 98 𝑒 𝑒 X iajs-2624 169 99 > > X iajs-2624 169 100 0 0 PUNCT iajs-2624 170 1 and and CCONJ iajs-2624 170 2 π‘₯ π‘₯ PRON iajs-2624 170 3 + + NUM iajs-2624 170 4 𝑒 𝑒 PROPN iajs-2624 170 5 βˆ’ βˆ’ NOUN iajs-2624 170 6 𝑧 𝑧 PRON iajs-2624 170 7 βˆ’ βˆ’ NOUN iajs-2624 170 8 π‘₯𝑒 π‘₯𝑒 NOUN iajs-2624 170 9 𝑧 𝑧 NOUN iajs-2624 170 10 = = NOUN iajs-2624 170 11 0 0 PUNCT iajs-2624 170 12 if if SCONJ iajs-2624 170 13 and and CCONJ iajs-2624 170 14 only only ADV iajs-2624 170 15 if if SCONJ iajs-2624 170 16 π‘₯ π‘₯ PROPN iajs-2624 170 17 = = SYM iajs-2624 170 18 1 1 NUM iajs-2624 170 19 , , PUNCT iajs-2624 171 1 𝑧 𝑧 NOUN iajs-2624 171 2 = = X iajs-2624 171 3 𝑒 𝑒 NOUN iajs-2624 171 4 ; ; PUNCT iajs-2624 171 5 π‘₯𝑦 π‘₯𝑦 ADP iajs-2624 171 6 βˆ’ βˆ’ ADP iajs-2624 171 7 π‘₯ π‘₯ PRON iajs-2624 171 8 βˆ’ βˆ’ PROPN iajs-2624 171 9 𝑦 𝑦 SYM iajs-2624 171 10 + + CCONJ iajs-2624 171 11 1 1 NUM iajs-2624 171 12 ≀ ≀ NUM iajs-2624 171 13 0 0 NUM iajs-2624 171 14 for for ADP iajs-2624 171 15 π‘₯ π‘₯ PROPN iajs-2624 171 16 , , PUNCT iajs-2624 171 17 𝑦 𝑦 NOUN iajs-2624 171 18 > > X iajs-2624 171 19 0 0 PUNCT iajs-2624 171 20 and and CCONJ iajs-2624 171 21 π‘₯𝑦 π‘₯𝑦 ADP iajs-2624 171 22 βˆ’ βˆ’ ADP iajs-2624 171 23 π‘₯ π‘₯ PRON iajs-2624 171 24 βˆ’ βˆ’ PROPN iajs-2624 171 25 𝑦 𝑦 SYM iajs-2624 171 26 + + CCONJ iajs-2624 171 27 1 1 NUM iajs-2624 171 28 = = SYM iajs-2624 171 29 0 0 PUNCT iajs-2624 172 1 if if SCONJ iajs-2624 172 2 and and CCONJ iajs-2624 172 3 only only ADV iajs-2624 172 4 if if SCONJ iajs-2624 172 5 π‘₯ π‘₯ DET iajs-2624 172 6 = = SYM iajs-2624 172 7 𝑦 𝑦 NOUN iajs-2624 172 8 = = SYM iajs-2624 172 9 1 1 NUM iajs-2624 172 10 ; ; PUNCT iajs-2624 172 11 𝑦 𝑦 X iajs-2624 172 12 + + SYM iajs-2624 172 13 𝑒 𝑒 PROPN iajs-2624 172 14 βˆ’ βˆ’ NOUN iajs-2624 172 15 𝑧 𝑧 PRON iajs-2624 172 16 βˆ’ βˆ’ PROPN iajs-2624 172 17 𝑦𝑒 𝑦𝑒 NOUN iajs-2624 172 18 𝑧 𝑧 DET iajs-2624 172 19 ≀ ≀ NUM iajs-2624 172 20 0 0 NUM iajs-2624 172 21 for for ADP iajs-2624 172 22 𝑦 𝑦 PROPN iajs-2624 172 23 , , PUNCT iajs-2624 172 24 𝑧 𝑧 PROPN iajs-2624 172 25 , , PUNCT iajs-2624 172 26 𝑒 𝑒 X iajs-2624 172 27 > > X iajs-2624 172 28 0 0 NUM iajs-2624 172 29 and and CCONJ iajs-2624 172 30 𝑦 𝑦 PRON iajs-2624 172 31 + + CCONJ iajs-2624 172 32 𝑒 𝑒 PROPN iajs-2624 172 33 βˆ’ βˆ’ NOUN iajs-2624 172 34 𝑧 𝑧 PRON iajs-2624 172 35 βˆ’ βˆ’ PROPN iajs-2624 172 36 𝑦𝑒 𝑦𝑒 NOUN iajs-2624 172 37 𝑧 𝑧 NOUN iajs-2624 172 38 = = NOUN iajs-2624 172 39 0 0 PUNCT iajs-2624 172 40 if if SCONJ iajs-2624 172 41 and and CCONJ iajs-2624 172 42 only only ADV iajs-2624 172 43 if if SCONJ iajs-2624 172 44 𝑦 𝑦 NOUN iajs-2624 172 45 = = SYM iajs-2624 172 46 1 1 NUM iajs-2624 172 47 , , PUNCT iajs-2624 172 48 𝑧 𝑧 NOUN iajs-2624 172 49 = = X iajs-2624 172 50 𝑒 𝑒 NOUN iajs-2624 172 51 ; ; PUNCT iajs-2624 172 52 𝑧𝑒 𝑧𝑒 ADP iajs-2624 172 53 βˆ’ βˆ’ PROPN iajs-2624 172 54 𝑧 𝑧 PRON iajs-2624 172 55 βˆ’ βˆ’ PUNCT iajs-2624 172 56 𝑒 𝑒 NOUN iajs-2624 172 57 + + NOUN iajs-2624 172 58 1 1 NUM iajs-2624 172 59 ≀ ≀ NUM iajs-2624 172 60 0 0 NUM iajs-2624 172 61 for for ADP iajs-2624 172 62 𝑧 𝑧 PROPN iajs-2624 172 63 , , PUNCT iajs-2624 172 64 𝑒 𝑒 X iajs-2624 172 65 > > X iajs-2624 172 66 0 0 PUNCT iajs-2624 172 67 and and CCONJ iajs-2624 172 68 𝑧𝑒 𝑧𝑒 ADP iajs-2624 172 69 βˆ’ βˆ’ PROPN iajs-2624 172 70 𝑧 𝑧 PRON iajs-2624 172 71 βˆ’ βˆ’ PUNCT iajs-2624 172 72 𝑒 𝑒 PART iajs-2624 172 73 + + NOUN iajs-2624 172 74 1 1 NUM iajs-2624 172 75 = = SYM iajs-2624 172 76 0 0 PUNCT iajs-2624 172 77 if if SCONJ iajs-2624 172 78 and and CCONJ iajs-2624 172 79 only only ADV iajs-2624 172 80 if if SCONJ iajs-2624 172 81 𝑧 𝑧 ADJ iajs-2624 172 82 = = VERB iajs-2624 172 83 𝑒 𝑒 NOUN iajs-2624 172 84 = = SYM iajs-2624 172 85 1 1 X iajs-2624 172 86 . . PUNCT iajs-2624 172 87 therefore therefore ADV iajs-2624 172 88 , , PUNCT iajs-2624 172 89 𝑉1 𝑉1 NOUN iajs-2624 172 90 β€² β€² NOUN iajs-2624 172 91 ≀ ≀ NUM iajs-2624 172 92 0 0 NUM iajs-2624 173 1 for for ADP iajs-2624 173 2 π‘₯ π‘₯ PROPN iajs-2624 173 3 , , PUNCT iajs-2624 173 4 𝑦 𝑦 PROPN iajs-2624 173 5 , , PUNCT iajs-2624 173 6 𝑧 𝑧 PRON iajs-2624 173 7 , , PUNCT iajs-2624 173 8 𝑒 𝑒 X iajs-2624 173 9 > > X iajs-2624 173 10 0 0 NUM iajs-2624 173 11 and and CCONJ iajs-2624 173 12 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2624 173 13 𝑑𝑑⁄ 𝑑𝑑⁄ PROPN iajs-2624 173 14 = = PROPN iajs-2624 173 15 0 0 PUNCT iajs-2624 173 16 if if SCONJ iajs-2624 173 17 and and CCONJ iajs-2624 173 18 only only ADV iajs-2624 173 19 if if SCONJ iajs-2624 173 20 π‘₯ π‘₯ PRON iajs-2624 173 21 = = SYM iajs-2624 173 22 𝑦 𝑦 NOUN iajs-2624 173 23 = = SYM iajs-2624 173 24 1 1 NUM iajs-2624 173 25 , , PUNCT iajs-2624 173 26 𝑧 𝑧 NOUN iajs-2624 173 27 = = X iajs-2624 173 28 𝑒 𝑒 X iajs-2624 173 29 = = SYM iajs-2624 173 30 1 1 NUM iajs-2624 173 31 , , PUNCT iajs-2624 173 32 the the DET iajs-2624 173 33 maximum maximum ADJ iajs-2624 173 34 invariant invariant ADJ iajs-2624 173 35 set set NOUN iajs-2624 173 36 of of ADP iajs-2624 173 37 system system NOUN iajs-2624 173 38 ( ( PUNCT iajs-2624 173 39 3 3 NUM iajs-2624 173 40 ) ) PUNCT iajs-2624 173 41 on on ADP iajs-2624 173 42 the the DET iajs-2624 173 43 set set NOUN iajs-2624 173 44 { { PUNCT iajs-2624 173 45 ( ( PUNCT iajs-2624 173 46 π‘₯ π‘₯ PROPN iajs-2624 173 47 , , PUNCT iajs-2624 173 48 𝑦 𝑦 NOUN iajs-2624 173 49 , , PUNCT iajs-2624 173 50 𝑧 𝑧 PRON iajs-2624 173 51 , , PUNCT iajs-2624 173 52 𝑒 𝑒 X iajs-2624 173 53 ): ): PUNCT iajs-2624 173 54 𝑑𝑉1 𝑑𝑉1 PROPN iajs-2624 173 55 𝑑𝑑⁄ 𝑑𝑑⁄ PROPN iajs-2624 173 56 = = PROPN iajs-2624 173 57 0 0 NUM iajs-2624 173 58 } } PUNCT iajs-2624 173 59 is be AUX iajs-2624 173 60 the the DET iajs-2624 173 61 singleton singleton NOUN iajs-2624 173 62 ( ( PUNCT iajs-2624 173 63 1,1,1,1 1,1,1,1 NUM iajs-2624 173 64 ) ) PUNCT iajs-2624 173 65 . . PUNCT iajs-2624 174 1 thus thus ADV iajs-2624 174 2 , , PUNCT iajs-2624 174 3 for for ADP iajs-2624 174 4 system system NOUN iajs-2624 174 5 ( ( PUNCT iajs-2624 174 6 3 3 NUM iajs-2624 174 7 ) ) PUNCT iajs-2624 174 8 , , PUNCT iajs-2624 174 9 the the DET iajs-2624 174 10 covid-19 covid-19 PROPN iajs-2624 174 11 equilibrium equilibrium NOUN iajs-2624 174 12 point point NOUN iajs-2624 174 13 𝐸1 𝐸1 NOUN iajs-2624 174 14 is be AUX iajs-2624 174 15 globally globally ADV iajs-2624 174 16 asymptotically asymptotically ADV iajs-2624 174 17 stable stable ADJ iajs-2624 174 18 if if SCONJ iajs-2624 174 19 β„›0 β„›0 PROPN iajs-2624 174 20 > > X iajs-2624 174 21 1 1 NUM iajs-2624 174 22 by by ADP iajs-2624 174 23 lasalle lasalle PROPN iajs-2624 174 24 principle principle NOUN iajs-2624 174 25 [ [ X iajs-2624 174 26 14 14 NUM iajs-2624 174 27 ] ] PUNCT iajs-2624 174 28 . . PUNCT iajs-2624 175 1 50 50 NUM iajs-2624 175 2 ibn ibn PROPN iajs-2624 175 3 al al PROPN iajs-2624 175 4 - - PUNCT iajs-2624 175 5 haitham haitham PROPN iajs-2624 175 6 jour jour X iajs-2624 175 7 . . PROPN iajs-2624 175 8 for for ADP iajs-2624 175 9 pure pure ADJ iajs-2624 175 10 & & CCONJ iajs-2624 175 11 appl appl PROPN iajs-2624 175 12 . . PUNCT iajs-2624 176 1 sci sci PROPN iajs-2624 176 2 . . PROPN iajs-2624 177 1 34 34 NUM iajs-2624 177 2 ( ( PUNCT iajs-2624 177 3 2 2 NUM iajs-2624 177 4 ) ) PUNCT iajs-2624 177 5 2021 2021 NUM iajs-2624 177 6 6 6 NUM iajs-2624 177 7 . . PUNCT iajs-2624 177 8 numerical numerical PROPN iajs-2624 177 9 simulation simulation PROPN iajs-2624 177 10 in in ADP iajs-2624 177 11 this this DET iajs-2624 177 12 section section NOUN iajs-2624 177 13 , , PUNCT iajs-2624 177 14 we we PRON iajs-2624 177 15 present present VERB iajs-2624 177 16 the the DET iajs-2624 177 17 numerical numerical PROPN iajs-2624 177 18 simulation simulation PROPN iajs-2624 177 19 results result NOUN iajs-2624 177 20 and and CCONJ iajs-2624 177 21 effect effect NOUN iajs-2624 177 22 of of ADP iajs-2624 177 23 the the DET iajs-2624 177 24 parameters parameter NOUN iajs-2624 177 25 of of ADP iajs-2624 177 26 the the DET iajs-2624 177 27 analytic analytic ADJ iajs-2624 177 28 results result NOUN iajs-2624 177 29 of of ADP iajs-2624 177 30 system system NOUN iajs-2624 177 31 ( ( PUNCT iajs-2624 177 32 3 3 NUM iajs-2624 177 33 ) ) PUNCT iajs-2624 177 34 . . PUNCT iajs-2624 178 1 using use VERB iajs-2624 178 2 matlab matlab PROPN iajs-2624 178 3 version version PROPN iajs-2624 178 4 8 8 NUM iajs-2624 178 5 and and CCONJ iajs-2624 178 6 c++ c++ NOUN iajs-2624 178 7 software software NOUN iajs-2624 178 8 , , PUNCT iajs-2624 178 9 we we PRON iajs-2624 178 10 begin begin VERB iajs-2624 178 11 the the DET iajs-2624 178 12 simulation simulation NOUN iajs-2624 178 13 with with ADP iajs-2624 178 14 a a DET iajs-2624 178 15 set set VERB iajs-2624 178 16 outbreak outbreak NOUN iajs-2624 178 17 data datum NOUN iajs-2624 178 18 published publish VERB iajs-2624 178 19 daily daily ADV iajs-2624 178 20 by by ADP iajs-2624 178 21 who who PRON iajs-2624 178 22 and and CCONJ iajs-2624 178 23 other other ADJ iajs-2624 178 24 sources source NOUN iajs-2624 178 25 [ [ X iajs-2624 178 26 8 8 NUM iajs-2624 178 27 , , PUNCT iajs-2624 178 28 15 15 NUM iajs-2624 178 29 - - SYM iajs-2624 178 30 20 20 NUM iajs-2624 178 31 ] ] PUNCT iajs-2624 178 32 , , PUNCT iajs-2624 178 33 as as SCONJ iajs-2624 178 34 shown show VERB iajs-2624 178 35 in in ADP iajs-2624 178 36 table table NOUN iajs-2624 178 37 1 1 NUM iajs-2624 178 38 which which PRON iajs-2624 178 39 are be AUX iajs-2624 178 40 described describe VERB iajs-2624 178 41 in in ADP iajs-2624 178 42 section section NOUN iajs-2624 178 43 2 2 NUM iajs-2624 178 44 . . PUNCT iajs-2624 178 45 table table NOUN iajs-2624 178 46 1 1 NUM iajs-2624 178 47 : : PUNCT iajs-2624 178 48 definitions definition NOUN iajs-2624 178 49 and and CCONJ iajs-2624 178 50 values value NOUN iajs-2624 178 51 of of ADP iajs-2624 178 52 parameters parameter NOUN iajs-2624 178 53 in in ADP iajs-2624 178 54 system system NOUN iajs-2624 178 55 ( ( PUNCT iajs-2624 178 56 3 3 X iajs-2624 178 57 ) ) PUNCT iajs-2624 178 58 parameter parameter NOUN iajs-2624 178 59 definition definition NOUN iajs-2624 178 60 estimated estimate VERB iajs-2624 178 61 mean mean ADJ iajs-2624 178 62 values value NOUN iajs-2624 178 63 reference reference VERB iajs-2624 178 64 πœ“ πœ“ ADP iajs-2624 178 65 𝛼 𝛼 VERB iajs-2624 178 66 𝛽 𝛽 PROPN iajs-2624 178 67 𝛽1 𝛽1 NOUN iajs-2624 178 68 𝜎 𝜎 NOUN iajs-2624 178 69 𝜎1 𝜎1 NOUN iajs-2624 178 70 πœ‡ πœ‡ ADP iajs-2624 178 71 𝛾 𝛾 NOUN iajs-2624 178 72 πœ– πœ– X iajs-2624 178 73 πœƒ πœƒ DET iajs-2624 178 74 π‘Ÿ π‘Ÿ PRON iajs-2624 178 75 π‘˜ π‘˜ NOUN iajs-2624 178 76 𝑑 𝑑 PROPN iajs-2624 178 77 𝛿1 𝛿1 ADJ iajs-2624 178 78 birth birth NOUN iajs-2624 178 79 rate rate NOUN iajs-2624 178 80 awareness awareness NOUN iajs-2624 178 81 rate rate NOUN iajs-2624 178 82 contact contact NOUN iajs-2624 178 83 rate rate NOUN iajs-2624 178 84 between between ADP iajs-2624 178 85 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 178 86 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 178 87 𝑣 𝑣 DET iajs-2624 178 88 contact contact NOUN iajs-2624 178 89 rate rate NOUN iajs-2624 178 90 between between ADP iajs-2624 178 91 π‘†π‘Ž π‘†π‘Ž NOUN iajs-2624 178 92 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2624 178 93 𝑣 𝑣 DET iajs-2624 178 94 contact contact NOUN iajs-2624 178 95 rate rate NOUN iajs-2624 178 96 between between ADP iajs-2624 178 97 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 178 98 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 178 99 𝐢 𝐢 PROPN iajs-2624 178 100 contact contact NOUN iajs-2624 178 101 rate rate NOUN iajs-2624 178 102 between between ADP iajs-2624 178 103 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 178 104 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 178 105 𝐢 𝐢 PROPN iajs-2624 178 106 natural natural ADJ iajs-2624 178 107 death death NOUN iajs-2624 178 108 rate rate NOUN iajs-2624 178 109 transmission transmission NOUN iajs-2624 178 110 asymptomatic asymptomatic NOUN iajs-2624 178 111 to to ADP iajs-2624 178 112 symptomatic symptomatic ADJ iajs-2624 178 113 prevention prevention NOUN iajs-2624 178 114 rate rate NOUN iajs-2624 178 115 death death NOUN iajs-2624 178 116 rate rate NOUN iajs-2624 178 117 due due ADP iajs-2624 178 118 to to ADP iajs-2624 178 119 the the DET iajs-2624 178 120 disease disease NOUN iajs-2624 178 121 spread spread VERB iajs-2624 178 122 rate rate NOUN iajs-2624 178 123 of of ADP iajs-2624 178 124 the the DET iajs-2624 178 125 virus virus NOUN iajs-2624 178 126 carrying carry VERB iajs-2624 178 127 capacity capacity NOUN iajs-2624 178 128 in in ADP iajs-2624 178 129 environment environment NOUN iajs-2624 178 130 removal removal NOUN iajs-2624 178 131 rate rate NOUN iajs-2624 178 132 of of ADP iajs-2624 178 133 virus virus NOUN iajs-2624 178 134 recovery recovery NOUN iajs-2624 178 135 rate rate NOUN iajs-2624 178 136 from from ADP iajs-2624 178 137 carrier carrier NOUN iajs-2624 178 138 271.23 271.23 NUM iajs-2624 178 139 π‘π‘’π‘Ÿ π‘π‘’π‘Ÿ PROPN iajs-2624 178 140 π‘‘π‘Žπ‘¦ π‘‘π‘Žπ‘¦ NOUN iajs-2624 178 141 𝛼 𝛼 PROPN iajs-2624 178 142 β‰₯ β‰₯ PROPN iajs-2624 178 143 0 0 NUM iajs-2624 178 144 π‘π‘’π‘Ÿ π‘π‘’π‘Ÿ PROPN iajs-2624 178 145 π‘‘π‘Žπ‘¦ π‘‘π‘Žπ‘¦ NOUN iajs-2624 178 146 1.555 1.555 NUM iajs-2624 178 147 Γ— Γ— NOUN iajs-2624 178 148 10βˆ’8 10βˆ’8 NUM iajs-2624 178 149 /π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ /π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ PUNCT iajs-2624 178 150 3.1 3.1 NUM iajs-2624 178 151 Γ— Γ— NOUN iajs-2624 178 152 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ NUM iajs-2624 178 153 1.555 1.555 NUM iajs-2624 178 154 Γ— Γ— PROPN iajs-2624 178 155 10βˆ’8/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 10βˆ’8/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ NUM iajs-2624 178 156 3.1 3.1 NUM iajs-2624 178 157 Γ— Γ— PROPN iajs-2624 178 158 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ NUM iajs-2624 178 159 3.01 3.01 NUM iajs-2624 178 160 Γ— Γ— NOUN iajs-2624 178 161 10βˆ’4/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 10βˆ’4/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ NUM iajs-2624 178 162 7 7 NUM iajs-2624 178 163 π‘‘π‘Žπ‘¦π‘  π‘‘π‘Žπ‘¦π‘  NOUN iajs-2624 178 164 0.03/π‘π‘’π‘Ÿπ‘ π‘œπ‘› 0.03/π‘π‘’π‘Ÿπ‘ π‘œπ‘› NOUN iajs-2624 178 165 0.01 0.01 NUM iajs-2624 178 166 π‘π‘’π‘Ÿ π‘π‘’π‘Ÿ NOUN iajs-2624 178 167 π‘‘π‘Žπ‘¦ π‘‘π‘Žπ‘¦ NOUN iajs-2624 178 168 2 2 NUM iajs-2624 178 169 0.5 0.5 NUM iajs-2624 178 170 3 3 NUM iajs-2624 178 171 π‘π‘’π‘Ÿ π‘π‘’π‘Ÿ NOUN iajs-2624 178 172 π‘‘π‘Žπ‘¦ π‘‘π‘Žπ‘¦ NOUN iajs-2624 178 173 0.03 0.03 NUM iajs-2624 178 174 π‘π‘’π‘Ÿπ‘‘π‘Žπ‘¦ π‘π‘’π‘Ÿπ‘‘π‘Žπ‘¦ NOUN iajs-2624 178 175 [ [ X iajs-2624 178 176 8 8 NUM iajs-2624 178 177 ] ] X iajs-2624 179 1 [ [ X iajs-2624 179 2 8,19 8,19 X iajs-2624 179 3 ] ] X iajs-2624 180 1 [ [ X iajs-2624 180 2 8,19 8,19 X iajs-2624 180 3 ] ] X iajs-2624 181 1 [ [ X iajs-2624 181 2 8,19 8,19 X iajs-2624 181 3 ] ] X iajs-2624 181 4 [ [ X iajs-2624 181 5 8,19 8,19 X iajs-2624 181 6 ] ] X iajs-2624 181 7 [ [ X iajs-2624 181 8 18 18 NUM iajs-2624 181 9 ] ] X iajs-2624 181 10 [ [ X iajs-2624 181 11 8 8 NUM iajs-2624 181 12 ] ] X iajs-2624 181 13 [ [ X iajs-2624 181 14 20 20 NUM iajs-2624 181 15 ] ] PUNCT iajs-2624 181 16 clearly clearly ADV iajs-2624 181 17 , , PUNCT iajs-2624 181 18 the the DET iajs-2624 181 19 initial initial ADJ iajs-2624 181 20 point point NOUN iajs-2624 181 21 for for ADP iajs-2624 181 22 𝐼(0 𝐼(0 NOUN iajs-2624 181 23 ) ) PUNCT iajs-2624 181 24 = = SYM iajs-2624 181 25 25 25 NUM iajs-2624 181 26 , , PUNCT iajs-2624 181 27 we we PRON iajs-2624 181 28 assume assume VERB iajs-2624 181 29 that that SCONJ iajs-2624 181 30 the the DET iajs-2624 181 31 𝐢(0 𝐢(0 NOUN iajs-2624 181 32 ) ) PUNCT iajs-2624 181 33 = = SYM iajs-2624 181 34 10 10 NUM iajs-2624 181 35 , , PUNCT iajs-2624 181 36 π‘†π‘Ž π‘†π‘Ž PROPN iajs-2624 181 37 = = NOUN iajs-2624 181 38 10 10 NUM iajs-2624 181 39 , , PUNCT iajs-2624 181 40 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 181 41 = = SYM iajs-2624 181 42 15 15 NUM iajs-2624 181 43 and and CCONJ iajs-2624 181 44 𝑣 𝑣 X iajs-2624 181 45 = = NOUN iajs-2624 181 46 20 20 NUM iajs-2624 181 47 . . PUNCT iajs-2624 182 1 using use VERB iajs-2624 182 2 this this DET iajs-2624 182 3 initial initial ADJ iajs-2624 182 4 point point NOUN iajs-2624 182 5 and and CCONJ iajs-2624 182 6 simulated simulate VERB iajs-2624 182 7 the the DET iajs-2624 182 8 system system NOUN iajs-2624 182 9 ( ( PUNCT iajs-2624 182 10 1 1 NUM iajs-2624 182 11 ) ) PUNCT iajs-2624 182 12 , , PUNCT iajs-2624 182 13 we we PRON iajs-2624 182 14 get get VERB iajs-2624 182 15 the the DET iajs-2624 182 16 values value NOUN iajs-2624 182 17 of of ADP iajs-2624 182 18 parameters parameter NOUN iajs-2624 182 19 shown show VERB iajs-2624 182 20 in in ADP iajs-2624 182 21 table table NOUN iajs-2624 182 22 1 1 NUM iajs-2624 182 23 , , PUNCT iajs-2624 182 24 whereas whereas SCONJ iajs-2624 182 25 the the DET iajs-2624 182 26 basic basic ADJ iajs-2624 182 27 reproduction reproduction NOUN iajs-2624 182 28 number number NOUN iajs-2624 182 29 is be AUX iajs-2624 182 30 estimated estimate VERB iajs-2624 182 31 β„›0 β„›0 ADJ iajs-2624 182 32 = = NOUN iajs-2624 182 33 0.914 0.914 NUM iajs-2624 182 34 < < X iajs-2624 182 35 1 1 NUM iajs-2624 182 36 with with ADP iajs-2624 182 37 condition condition NOUN iajs-2624 182 38 ( ( PUNCT iajs-2624 182 39 8b 8b NUM iajs-2624 182 40 ) ) PUNCT iajs-2624 182 41 , , PUNCT iajs-2624 182 42 we we PRON iajs-2624 182 43 obtain obtain VERB iajs-2624 182 44 the the DET iajs-2624 182 45 dynamical dynamical ADJ iajs-2624 182 46 behavior behavior NOUN iajs-2624 182 47 of of ADP iajs-2624 182 48 system system NOUN iajs-2624 182 49 ( ( PUNCT iajs-2624 182 50 3 3 X iajs-2624 182 51 ) ) PUNCT iajs-2624 182 52 converge converge VERB iajs-2624 182 53 to to ADP iajs-2624 182 54 covid-19 covid-19 PROPN iajs-2624 182 55 free free ADJ iajs-2624 182 56 equilibrium equilibrium NOUN iajs-2624 182 57 point point NOUN iajs-2624 182 58 𝐸0 𝐸0 NOUN iajs-2624 182 59 = = SYM iajs-2624 182 60 ( ( PUNCT iajs-2624 182 61 901096,0,0,0,0 901096,0,0,0,0 NOUN iajs-2624 182 62 ) ) PUNCT iajs-2624 182 63 , , PUNCT iajs-2624 182 64 illustrating illustrate VERB iajs-2624 182 65 its its PRON iajs-2624 182 66 asymptotical asymptotical ADJ iajs-2624 182 67 stability stability NOUN iajs-2624 182 68 that that PRON iajs-2624 182 69 is be AUX iajs-2624 182 70 stated state VERB iajs-2624 182 71 in in ADP iajs-2624 182 72 theorem theorem NOUN iajs-2624 182 73 ( ( PUNCT iajs-2624 182 74 2 2 NUM iajs-2624 182 75 ) ) PUNCT iajs-2624 182 76 . . PUNCT iajs-2624 183 1 this this PRON iajs-2624 183 2 is be AUX iajs-2624 183 3 shown show VERB iajs-2624 183 4 in in ADP iajs-2624 183 5 figure figure NOUN iajs-2624 183 6 2 2 NUM iajs-2624 183 7 . . NUM iajs-2624 183 8 51 51 NUM iajs-2624 183 9 ibn ibn PROPN iajs-2624 183 10 al al PROPN iajs-2624 183 11 - - PUNCT iajs-2624 183 12 haitham haitham PROPN iajs-2624 183 13 jour jour X iajs-2624 183 14 . . PROPN iajs-2624 184 1 for for ADP iajs-2624 184 2 pure pure ADJ iajs-2624 184 3 & & CCONJ iajs-2624 184 4 appl appl PROPN iajs-2624 184 5 . . PUNCT iajs-2624 185 1 sci sci PROPN iajs-2624 185 2 . . PROPN iajs-2624 186 1 34 34 NUM iajs-2624 186 2 ( ( PUNCT iajs-2624 186 3 2 2 NUM iajs-2624 186 4 ) ) PUNCT iajs-2624 186 5 2021 2021 NUM iajs-2624 186 6 figure figure NOUN iajs-2624 186 7 2 2 NUM iajs-2624 186 8 : : PUNCT iajs-2624 186 9 a a DET iajs-2624 186 10 simulation simulation NOUN iajs-2624 186 11 result result VERB iajs-2624 186 12 for for ADP iajs-2624 186 13 the the DET iajs-2624 186 14 dynamics dynamic NOUN iajs-2624 186 15 of of ADP iajs-2624 186 16 coronavirus coronavirus NOUN iajs-2624 186 17 model model NOUN iajs-2624 186 18 using use VERB iajs-2624 186 19 the the DET iajs-2624 186 20 parameters parameter NOUN iajs-2624 186 21 from from ADP iajs-2624 186 22 table table NOUN iajs-2624 186 23 1 1 NUM iajs-2624 186 24 . . PUNCT iajs-2624 187 1 the the DET iajs-2624 187 2 reproduction reproduction NOUN iajs-2624 187 3 number number NOUN iajs-2624 187 4 β„›0 β„›0 NOUN iajs-2624 187 5 = = NOUN iajs-2624 187 6 0.07 0.07 NUM iajs-2624 187 7 and and CCONJ iajs-2624 187 8 the the DET iajs-2624 187 9 behavior behavior NOUN iajs-2624 187 10 coronavirus coronavirus NOUN iajs-2624 187 11 converge converge VERB iajs-2624 187 12 to to ADP iajs-2624 187 13 covid-19 covid-19 PROPN iajs-2624 187 14 free free ADJ iajs-2624 187 15 equilibrium equilibrium NOUN iajs-2624 187 16 point point NOUN iajs-2624 187 17 . . PUNCT iajs-2624 188 1 however however ADV iajs-2624 188 2 , , PUNCT iajs-2624 188 3 for for ADP iajs-2624 188 4 the the DET iajs-2624 188 5 same same ADJ iajs-2624 188 6 data datum NOUN iajs-2624 188 7 given give VERB iajs-2624 188 8 by by ADP iajs-2624 188 9 table table NOUN iajs-2624 188 10 1 1 NUM iajs-2624 188 11 . . PUNCT iajs-2624 189 1 with with ADP iajs-2624 189 2 𝑑 𝑑 PROPN iajs-2624 189 3 = = SYM iajs-2624 189 4 1 1 NUM iajs-2624 189 5 , , PUNCT iajs-2624 189 6 ( ( PUNCT iajs-2624 189 7 i.e. i.e. X iajs-2624 189 8 the the DET iajs-2624 189 9 condition condition NOUN iajs-2624 189 10 6b 6b NOUN iajs-2624 189 11 is be AUX iajs-2624 189 12 holds hold NOUN iajs-2624 189 13 ) ) PUNCT iajs-2624 189 14 , , PUNCT iajs-2624 189 15 and and CCONJ iajs-2624 189 16 𝜎 𝜎 X iajs-2624 189 17 = = NOUN iajs-2624 189 18 1.555 1.555 NUM iajs-2624 189 19 Γ— Γ— NOUN iajs-2624 189 20 10βˆ’5 10βˆ’5 NUM iajs-2624 189 21 ( ( PUNCT iajs-2624 189 22 i.e. i.e. X iajs-2624 189 23 the the DET iajs-2624 189 24 contact contact NOUN iajs-2624 189 25 rate rate NOUN iajs-2624 189 26 between between ADP iajs-2624 189 27 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 189 28 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 189 29 𝐢 𝐢 PROPN iajs-2624 189 30 will will AUX iajs-2624 189 31 increase increase VERB iajs-2624 189 32 ) ) PUNCT iajs-2624 189 33 . . PUNCT iajs-2624 190 1 it it PRON iajs-2624 190 2 is be AUX iajs-2624 190 3 observed observe VERB iajs-2624 190 4 that that SCONJ iajs-2624 190 5 , , PUNCT iajs-2624 190 6 although although SCONJ iajs-2624 190 7 system system NOUN iajs-2624 190 8 ( ( PUNCT iajs-2624 190 9 3 3 X iajs-2624 190 10 ) ) PUNCT iajs-2624 190 11 is be AUX iajs-2624 190 12 solved solve VERB iajs-2624 190 13 with with ADP iajs-2624 190 14 the the DET iajs-2624 190 15 same same ADJ iajs-2624 190 16 initial initial ADJ iajs-2624 190 17 point point NOUN iajs-2624 190 18 , , PUNCT iajs-2624 190 19 the the DET iajs-2624 190 20 dynamical dynamical ADJ iajs-2624 190 21 behavior behavior NOUN iajs-2624 190 22 of of ADP iajs-2624 190 23 system system NOUN iajs-2624 190 24 ( ( PUNCT iajs-2624 190 25 3 3 X iajs-2624 190 26 ) ) PUNCT iajs-2624 190 27 converge converge VERB iajs-2624 190 28 to to ADP iajs-2624 190 29 covid-19 covid-19 PROPN iajs-2624 190 30 equilibrium equilibrium NOUN iajs-2624 190 31 point point NOUN iajs-2624 190 32 𝐸1 𝐸1 NOUN iajs-2624 190 33 = = PUNCT iajs-2624 190 34 ( ( PUNCT iajs-2624 190 35 979,2734,1752,24299,0,25 979,2734,1752,24299,0,25 NUM iajs-2624 190 36 ) ) PUNCT iajs-2624 190 37 , , PUNCT iajs-2624 190 38 illustrating illustrate VERB iajs-2624 190 39 its its PRON iajs-2624 190 40 asymptotical asymptotical ADJ iajs-2624 190 41 stability stability NOUN iajs-2624 190 42 that that PRON iajs-2624 190 43 is be AUX iajs-2624 190 44 stated state VERB iajs-2624 190 45 in in ADP iajs-2624 190 46 theorem theorem NOUN iajs-2624 190 47 ( ( PUNCT iajs-2624 190 48 3 3 NUM iajs-2624 190 49 ) ) PUNCT iajs-2624 190 50 , , PUNCT iajs-2624 190 51 and and CCONJ iajs-2624 190 52 reproduction reproduction NOUN iajs-2624 190 53 number number NOUN iajs-2624 190 54 is be AUX iajs-2624 190 55 estimated estimate VERB iajs-2624 190 56 β„›0 β„›0 ADJ iajs-2624 190 57 = = NOUN iajs-2624 190 58 7.77 7.77 NUM iajs-2624 190 59 > > SYM iajs-2624 191 1 1 1 NUM iajs-2624 191 2 . . PUNCT iajs-2624 192 1 this this PRON iajs-2624 192 2 is be AUX iajs-2624 192 3 shown show VERB iajs-2624 192 4 in in ADP iajs-2624 192 5 figure figure NOUN iajs-2624 192 6 3 3 NUM iajs-2624 192 7 . . PROPN iajs-2624 192 8 52 52 NUM iajs-2624 192 9 ibn ibn PROPN iajs-2624 192 10 al al PROPN iajs-2624 192 11 - - PUNCT iajs-2624 192 12 haitham haitham PROPN iajs-2624 192 13 jour jour X iajs-2624 192 14 . . PROPN iajs-2624 193 1 for for ADP iajs-2624 193 2 pure pure ADJ iajs-2624 193 3 & & CCONJ iajs-2624 193 4 appl appl PROPN iajs-2624 193 5 . . PUNCT iajs-2624 194 1 sci sci PROPN iajs-2624 194 2 . . PROPN iajs-2624 195 1 34 34 NUM iajs-2624 195 2 ( ( PUNCT iajs-2624 195 3 2 2 NUM iajs-2624 195 4 ) ) PUNCT iajs-2624 195 5 2021 2021 NUM iajs-2624 195 6 figure figure NOUN iajs-2624 195 7 3 3 NUM iajs-2624 195 8 : : PUNCT iajs-2624 195 9 a a DET iajs-2624 195 10 simulation simulation NOUN iajs-2624 195 11 result result VERB iajs-2624 195 12 for for ADP iajs-2624 195 13 the the DET iajs-2624 195 14 dynamics dynamic NOUN iajs-2624 195 15 of of ADP iajs-2624 195 16 coronavirus coronavirus NOUN iajs-2624 195 17 model model NOUN iajs-2624 195 18 using use VERB iajs-2624 195 19 the the DET iajs-2624 195 20 parameters parameter NOUN iajs-2624 195 21 from from ADP iajs-2624 195 22 table table NOUN iajs-2624 195 23 1 1 NUM iajs-2624 195 24 with with ADP iajs-2624 195 25 𝑑 𝑑 PROPN iajs-2624 195 26 = = SYM iajs-2624 195 27 1 1 NUM iajs-2624 195 28 and and CCONJ iajs-2624 195 29 𝜎 𝜎 PRON iajs-2624 195 30 = = NOUN iajs-2624 195 31 1.555 1.555 NUM iajs-2624 195 32 Γ— Γ— NOUN iajs-2624 195 33 10βˆ’5 10βˆ’5 NUM iajs-2624 195 34 . . PUNCT iajs-2624 196 1 the the DET iajs-2624 196 2 reproduction reproduction NOUN iajs-2624 196 3 number number NOUN iajs-2624 196 4 β„›0 β„›0 NOUN iajs-2624 196 5 = = NOUN iajs-2624 196 6 7.77 7.77 NUM iajs-2624 196 7 and and CCONJ iajs-2624 196 8 the the DET iajs-2624 196 9 behavior behavior NOUN iajs-2624 196 10 of of ADP iajs-2624 196 11 coronavirus coronavirus NOUN iajs-2624 196 12 converge converge VERB iajs-2624 196 13 to to ADP iajs-2624 196 14 covid-19 covid-19 PROPN iajs-2624 196 15 equilibrium equilibrium NOUN iajs-2624 196 16 point point NOUN iajs-2624 196 17 . . PUNCT iajs-2624 197 1 similarly similarly ADV iajs-2624 197 2 , , PUNCT iajs-2624 197 3 if if SCONJ iajs-2624 197 4 the the DET iajs-2624 197 5 increase increase NOUN iajs-2624 197 6 effect effect NOUN iajs-2624 197 7 of of ADP iajs-2624 197 8 the the DET iajs-2624 197 9 contact contact NOUN iajs-2624 197 10 rate rate NOUN iajs-2624 197 11 between between ADP iajs-2624 197 12 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 197 13 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 197 14 𝑣 𝑣 X iajs-2624 197 15 ( ( PUNCT iajs-2624 197 16 𝛽 𝛽 NOUN iajs-2624 197 17 ) ) PUNCT iajs-2624 197 18 , , PUNCT iajs-2624 197 19 as as ADV iajs-2624 197 20 well well ADV iajs-2624 197 21 as as ADP iajs-2624 197 22 used use VERB iajs-2624 197 23 the the DET iajs-2624 197 24 values value NOUN iajs-2624 197 25 of of ADP iajs-2624 197 26 parameters parameter NOUN iajs-2624 197 27 shown show VERB iajs-2624 197 28 in in ADP iajs-2624 197 29 table table NOUN iajs-2624 197 30 1 1 NUM iajs-2624 197 31 with with ADP iajs-2624 197 32 𝑑 𝑑 PROPN iajs-2624 197 33 = = SYM iajs-2624 197 34 1 1 NUM iajs-2624 197 35 , , PUNCT iajs-2624 197 36 ( ( PUNCT iajs-2624 197 37 i.e. i.e. X iajs-2624 197 38 the the DET iajs-2624 197 39 condition condition NOUN iajs-2624 197 40 6b 6b NOUN iajs-2624 197 41 is be AUX iajs-2624 197 42 holds hold NOUN iajs-2624 197 43 ) ) PUNCT iajs-2624 197 44 , , PUNCT iajs-2624 197 45 and and CCONJ iajs-2624 197 46 𝛽 𝛽 NOUN iajs-2624 197 47 = = NOUN iajs-2624 197 48 1.555 1.555 NUM iajs-2624 197 49 Γ— Γ— PROPN iajs-2624 197 50 10βˆ’4 10βˆ’4 NUM iajs-2624 197 51 ( ( PUNCT iajs-2624 197 52 i.e. i.e. X iajs-2624 197 53 the the DET iajs-2624 197 54 contact contact NOUN iajs-2624 197 55 rate rate NOUN iajs-2624 197 56 between between ADP iajs-2624 197 57 𝑆𝑒 𝑆𝑒 PROPN iajs-2624 197 58 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2624 197 59 𝑣 𝑣 NOUN iajs-2624 197 60 will will AUX iajs-2624 197 61 increase increase VERB iajs-2624 197 62 ) ) PUNCT iajs-2624 197 63 . . PUNCT iajs-2624 198 1 , , PUNCT iajs-2624 198 2 we we PRON iajs-2624 198 3 obtain obtain VERB iajs-2624 198 4 the the DET iajs-2624 198 5 dynamical dynamical ADJ iajs-2624 198 6 behavior behavior NOUN iajs-2624 198 7 of of ADP iajs-2624 198 8 system system NOUN iajs-2624 198 9 ( ( PUNCT iajs-2624 198 10 3 3 X iajs-2624 198 11 ) ) PUNCT iajs-2624 198 12 still still ADV iajs-2624 198 13 converge converge VERB iajs-2624 198 14 to to ADP iajs-2624 198 15 covid-19 covid-19 PROPN iajs-2624 198 16 equilibrium equilibrium NOUN iajs-2624 198 17 point point NOUN iajs-2624 198 18 𝐸1 𝐸1 NOUN iajs-2624 198 19 = = SYM iajs-2624 198 20 ( ( PUNCT iajs-2624 198 21 ( ( PUNCT iajs-2624 198 22 61636,170389,25,334,0.25 61636,170389,25,334,0.25 NOUN iajs-2624 198 23 ) ) PUNCT iajs-2624 198 24 ) ) PUNCT iajs-2624 198 25 , , PUNCT iajs-2624 198 26 illustrating illustrate VERB iajs-2624 198 27 its its PRON iajs-2624 198 28 asymptotical asymptotical ADJ iajs-2624 198 29 stability stability NOUN iajs-2624 198 30 that that PRON iajs-2624 198 31 is be AUX iajs-2624 198 32 stated state VERB iajs-2624 198 33 in in ADP iajs-2624 198 34 theorem theorem NOUN iajs-2624 198 35 ( ( PUNCT iajs-2624 198 36 3 3 NUM iajs-2624 198 37 ) ) PUNCT iajs-2624 198 38 . . PUNCT iajs-2624 199 1 this this PRON iajs-2624 199 2 is be AUX iajs-2624 199 3 shown show VERB iajs-2624 199 4 in in ADP iajs-2624 199 5 figure figure NOUN iajs-2624 199 6 4 4 NUM iajs-2624 199 7 . . X iajs-2624 199 8 53 53 NUM iajs-2624 199 9 ibn ibn PROPN iajs-2624 199 10 al al PROPN iajs-2624 199 11 - - PUNCT iajs-2624 199 12 haitham haitham PROPN iajs-2624 199 13 jour jour X iajs-2624 199 14 . . PROPN iajs-2624 200 1 for for ADP iajs-2624 200 2 pure pure ADJ iajs-2624 200 3 & & CCONJ iajs-2624 200 4 appl appl PROPN iajs-2624 200 5 . . PUNCT iajs-2624 201 1 sci sci PROPN iajs-2624 201 2 . . PROPN iajs-2624 202 1 34 34 NUM iajs-2624 202 2 ( ( PUNCT iajs-2624 202 3 2 2 NUM iajs-2624 202 4 ) ) PUNCT iajs-2624 202 5 2021 2021 NUM iajs-2624 202 6 figure figure NOUN iajs-2624 202 7 4 4 NUM iajs-2624 202 8 : : PUNCT iajs-2624 202 9 a a DET iajs-2624 202 10 simulation simulation NOUN iajs-2624 202 11 result result VERB iajs-2624 202 12 for for ADP iajs-2624 202 13 the the DET iajs-2624 202 14 dynamics dynamic NOUN iajs-2624 202 15 of of ADP iajs-2624 202 16 coronavirus coronavirus NOUN iajs-2624 202 17 model model NOUN iajs-2624 202 18 using use VERB iajs-2624 202 19 the the DET iajs-2624 202 20 parameters parameter NOUN iajs-2624 202 21 from from ADP iajs-2624 202 22 table table NOUN iajs-2624 202 23 1 1 NUM iajs-2624 202 24 with with ADP iajs-2624 202 25 𝑑 𝑑 PROPN iajs-2624 202 26 = = SYM iajs-2624 202 27 1 1 NUM iajs-2624 202 28 and and CCONJ iajs-2624 202 29 𝛽 𝛽 NOUN iajs-2624 202 30 = = NOUN iajs-2624 202 31 1.555 1.555 NUM iajs-2624 202 32 Γ— Γ— PROPN iajs-2624 202 33 10βˆ’4 10βˆ’4 NUM iajs-2624 202 34 . . PUNCT iajs-2624 203 1 the the DET iajs-2624 203 2 dynamical dynamical ADJ iajs-2624 203 3 behavior behavior NOUN iajs-2624 203 4 of of ADP iajs-2624 203 5 coronavirus coronavirus NOUN iajs-2624 203 6 model model NOUN iajs-2624 203 7 converge converge VERB iajs-2624 203 8 to to ADP iajs-2624 203 9 covid-19 covid-19 PROPN iajs-2624 203 10 equilibrium equilibrium NOUN iajs-2624 203 11 point point NOUN iajs-2624 203 12 . . PUNCT iajs-2624 204 1 7 7 X iajs-2624 204 2 . . X iajs-2624 204 3 discussions discussion NOUN iajs-2624 204 4 and and CCONJ iajs-2624 204 5 conclusions conclusion NOUN iajs-2624 204 6 in in ADP iajs-2624 204 7 this this DET iajs-2624 204 8 manuscript manuscript NOUN iajs-2624 204 9 , , PUNCT iajs-2624 204 10 a a DET iajs-2624 204 11 coronavirus coronavirus NOUN iajs-2624 204 12 model model NOUN iajs-2624 204 13 is be AUX iajs-2624 204 14 proposed propose VERB iajs-2624 204 15 and and CCONJ iajs-2624 204 16 discussed discuss VERB iajs-2624 204 17 . . PUNCT iajs-2624 205 1 this this DET iajs-2624 205 2 model model NOUN iajs-2624 205 3 has have VERB iajs-2624 205 4 two two NUM iajs-2624 205 5 feasible feasible ADJ iajs-2624 205 6 equilibrium equilibrium NOUN iajs-2624 205 7 points point NOUN iajs-2624 205 8 which which PRON iajs-2624 205 9 are be AUX iajs-2624 205 10 covid-19 covid-19 PROPN iajs-2624 205 11 free free ADJ iajs-2624 205 12 equilibrium equilibrium NOUN iajs-2624 205 13 point point NOUN iajs-2624 205 14 and and CCONJ iajs-2624 205 15 epidemic epidemic NOUN iajs-2624 205 16 equilibrium equilibrium NOUN iajs-2624 205 17 point point NOUN iajs-2624 205 18 . . PUNCT iajs-2624 206 1 the the DET iajs-2624 206 2 results result NOUN iajs-2624 206 3 of of ADP iajs-2624 206 4 the the DET iajs-2624 206 5 proposed propose VERB iajs-2624 206 6 work work NOUN iajs-2624 206 7 may may AUX iajs-2624 206 8 be be AUX iajs-2624 206 9 used use VERB iajs-2624 206 10 to to PART iajs-2624 206 11 show show VERB iajs-2624 206 12 and and CCONJ iajs-2624 206 13 understand understand VERB iajs-2624 206 14 the the DET iajs-2624 206 15 effect effect NOUN iajs-2624 206 16 of of ADP iajs-2624 206 17 awareness awareness NOUN iajs-2624 206 18 rate rate NOUN iajs-2624 206 19 , , PUNCT iajs-2624 206 20 prevention prevention NOUN iajs-2624 206 21 rate rate NOUN iajs-2624 206 22 of of ADP iajs-2624 206 23 disease disease NOUN iajs-2624 206 24 and and CCONJ iajs-2624 206 25 contact contact NOUN iajs-2624 206 26 rate rate NOUN iajs-2624 206 27 between between ADP iajs-2624 206 28 asymptomatic asymptomatic ADJ iajs-2624 206 29 people people NOUN iajs-2624 206 30 and and CCONJ iajs-2624 206 31 other other ADJ iajs-2624 206 32 . . PUNCT iajs-2624 207 1 the the DET iajs-2624 207 2 stability stability NOUN iajs-2624 207 3 ( ( PUNCT iajs-2624 207 4 local local ADJ iajs-2624 207 5 and and CCONJ iajs-2624 207 6 global global ADJ iajs-2624 207 7 ) ) PUNCT iajs-2624 207 8 of of ADP iajs-2624 207 9 the the DET iajs-2624 207 10 model model NOUN iajs-2624 207 11 has have AUX iajs-2624 207 12 been be AUX iajs-2624 207 13 analyzed analyze VERB iajs-2624 207 14 at at ADP iajs-2624 207 15 all all DET iajs-2624 207 16 equilibrium equilibrium NOUN iajs-2624 207 17 points point NOUN iajs-2624 207 18 . . PUNCT iajs-2624 208 1 it it PRON iajs-2624 208 2 has have AUX iajs-2624 208 3 been be AUX iajs-2624 208 4 obtained obtain VERB iajs-2624 208 5 that that SCONJ iajs-2624 208 6 covid-19 covid-19 PROPN iajs-2624 208 7 free free ADJ iajs-2624 208 8 is be AUX iajs-2624 208 9 locally locally ADV iajs-2624 208 10 asymptotically asymptotically ADV iajs-2624 208 11 stable stable ADJ iajs-2624 208 12 if if SCONJ iajs-2624 208 13 the the DET iajs-2624 208 14 conditions condition NOUN iajs-2624 208 15 ( ( PUNCT iajs-2624 208 16 8a 8a NUM iajs-2624 208 17 ) ) PUNCT iajs-2624 208 18 and and CCONJ iajs-2624 208 19 ( ( PUNCT iajs-2624 208 20 8b 8b NUM iajs-2624 208 21 ) ) PUNCT iajs-2624 208 22 are be AUX iajs-2624 208 23 hold hold VERB iajs-2624 208 24 and and CCONJ iajs-2624 208 25 it it PRON iajs-2624 208 26 is be AUX iajs-2624 208 27 global global ADJ iajs-2624 208 28 stable stable ADJ iajs-2624 208 29 under under ADP iajs-2624 208 30 the the DET iajs-2624 208 31 conditions condition NOUN iajs-2624 208 32 ( ( PUNCT iajs-2624 208 33 10a 10a NUM iajs-2624 208 34 ) ) PUNCT iajs-2624 208 35 and and CCONJ iajs-2624 208 36 ( ( PUNCT iajs-2624 208 37 10b 10b NOUN iajs-2624 208 38 ) ) PUNCT iajs-2624 208 39 are be AUX iajs-2624 208 40 hold hold ADJ iajs-2624 208 41 . . PUNCT iajs-2624 209 1 while while SCONJ iajs-2624 209 2 , , PUNCT iajs-2624 209 3 the the DET iajs-2624 209 4 epidemic epidemic NOUN iajs-2624 209 5 point point NOUN iajs-2624 209 6 it it PRON iajs-2624 209 7 has have AUX iajs-2624 209 8 been be AUX iajs-2624 209 9 obtained obtain VERB iajs-2624 209 10 under under ADP iajs-2624 209 11 certain certain ADJ iajs-2624 209 12 conditions condition NOUN iajs-2624 209 13 for for ADP iajs-2624 209 14 locally locally ADV iajs-2624 209 15 as as ADV iajs-2624 209 16 well well ADV iajs-2624 209 17 as as ADP iajs-2624 209 18 globally globally ADV iajs-2624 209 19 asymptotically asymptotically ADV iajs-2624 209 20 stable stable ADJ iajs-2624 209 21 under under ADP iajs-2624 209 22 the the DET iajs-2624 209 23 conditions condition NOUN iajs-2624 209 24 ( ( PUNCT iajs-2624 209 25 9a-9b 9a-9b NUM iajs-2624 209 26 ) ) PUNCT iajs-2624 209 27 and and CCONJ iajs-2624 209 28 ( ( PUNCT iajs-2624 209 29 11a-11 11a-11 NUM iajs-2624 209 30 g g NOUN iajs-2624 209 31 ) ) PUNCT iajs-2624 209 32 are be AUX iajs-2624 209 33 hold hold VERB iajs-2624 209 34 respectively respectively ADV iajs-2624 209 35 . . PUNCT iajs-2624 210 1 to to PART iajs-2624 210 2 validate validate VERB iajs-2624 210 3 the the DET iajs-2624 210 4 analytical analytical ADJ iajs-2624 210 5 results result NOUN iajs-2624 210 6 , , PUNCT iajs-2624 210 7 we we PRON iajs-2624 210 8 have have AUX iajs-2624 210 9 executed execute VERB iajs-2624 210 10 the the DET iajs-2624 210 11 numerical numerical ADJ iajs-2624 210 12 simulations simulation NOUN iajs-2624 210 13 for for ADP iajs-2624 210 14 investigating investigate VERB iajs-2624 210 15 the the DET iajs-2624 210 16 dynamics dynamic NOUN iajs-2624 210 17 of of ADP iajs-2624 210 18 a a DET iajs-2624 210 19 coronavirus coronavirus NOUN iajs-2624 210 20 mathematical mathematical ADJ iajs-2624 210 21 model model NOUN iajs-2624 210 22 . . PUNCT iajs-2624 211 1 we we PRON iajs-2624 211 2 conclude conclude VERB iajs-2624 211 3 if if SCONJ iajs-2624 211 4 the the DET iajs-2624 211 5 removal removal NOUN iajs-2624 211 6 rate rate NOUN iajs-2624 211 7 of of ADP iajs-2624 211 8 virus virus NOUN iajs-2624 211 9 is be AUX iajs-2624 211 10 increased increase VERB iajs-2624 211 11 through through ADP iajs-2624 211 12 used use VERB iajs-2624 211 13 the the DET iajs-2624 211 14 sterilization sterilization NOUN iajs-2624 211 15 materials material NOUN iajs-2624 211 16 could could AUX iajs-2624 211 17 us we PRON iajs-2624 211 18 to to PART iajs-2624 211 19 decrease decrease VERB iajs-2624 211 20 virus virus NOUN iajs-2624 211 21 rate rate NOUN iajs-2624 211 22 and and CCONJ iajs-2624 211 23 hence hence ADV iajs-2624 211 24 , , PUNCT iajs-2624 211 25 we we PRON iajs-2624 211 26 get get VERB iajs-2624 211 27 the the DET iajs-2624 211 28 basic basic ADJ iajs-2624 211 29 reproduction reproduction NOUN iajs-2624 211 30 number number NOUN iajs-2624 211 31 less less ADJ iajs-2624 211 32 than than ADP iajs-2624 211 33 one one NUM iajs-2624 211 34 ( ( PUNCT iajs-2624 211 35 i.e. i.e. X iajs-2624 211 36 satisfied satisfied ADJ iajs-2624 211 37 the the DET iajs-2624 211 38 condition condition NOUN iajs-2624 211 39 10a 10a NOUN iajs-2624 211 40 and and CCONJ iajs-2624 211 41 10b 10b NOUN iajs-2624 211 42 ) ) PUNCT iajs-2624 211 43 . . PUNCT iajs-2624 212 1 also also ADV iajs-2624 212 2 caution caution NOUN iajs-2624 212 3 should should AUX iajs-2624 212 4 be be AUX iajs-2624 212 5 exercised exercise VERB iajs-2624 212 6 against against ADP iajs-2624 212 7 asymptomatic asymptomatic ADJ iajs-2624 212 8 people people NOUN iajs-2624 212 9 who who PRON iajs-2624 212 10 are be AUX iajs-2624 212 11 the the DET iajs-2624 212 12 source source NOUN iajs-2624 212 13 of of ADP iajs-2624 212 14 the the DET iajs-2624 212 15 risk risk NOUN iajs-2624 212 16 , , PUNCT iajs-2624 212 17 so so CCONJ iajs-2624 212 18 direct direct ADJ iajs-2624 212 19 contact contact NOUN iajs-2624 212 20 should should AUX iajs-2624 212 21 be be AUX iajs-2624 212 22 reduced reduce VERB iajs-2624 212 23 with with ADP iajs-2624 212 24 them they PRON iajs-2624 212 25 . . PUNCT iajs-2624 213 1 finally finally ADV iajs-2624 213 2 , , PUNCT iajs-2624 213 3 a a DET iajs-2624 213 4 rapid rapid ADJ iajs-2624 213 5 health health NOUN iajs-2624 213 6 examination examination NOUN iajs-2624 213 7 to to PART iajs-2624 213 8 detect detect VERB iajs-2624 213 9 the the DET iajs-2624 213 10 infected infected NOUN iajs-2624 213 11 is be AUX iajs-2624 213 12 one one NUM iajs-2624 213 13 of of ADP iajs-2624 213 14 the the DET iajs-2624 213 15 important important ADJ iajs-2624 213 16 factors factor NOUN iajs-2624 213 17 in in ADP iajs-2624 213 18 the the DET iajs-2624 213 19 control control NOUN iajs-2624 213 20 to to PART iajs-2624 213 21 spread spread VERB iajs-2624 213 22 of of ADP iajs-2624 213 23 the the DET iajs-2624 213 24 epidemic epidemic NOUN iajs-2624 213 25 . . PUNCT iajs-2624 214 1 54 54 NUM iajs-2624 214 2 ibn ibn PROPN iajs-2624 214 3 al al PROPN iajs-2624 214 4 - - PUNCT iajs-2624 214 5 haitham haitham PROPN iajs-2624 214 6 jour jour X iajs-2624 214 7 . . PROPN iajs-2624 214 8 for for ADP iajs-2624 214 9 pure pure ADJ iajs-2624 214 10 & & CCONJ iajs-2624 214 11 appl appl PROPN iajs-2624 214 12 . . PUNCT iajs-2624 215 1 sci sci PROPN iajs-2624 215 2 . . 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