id sid tid token lemma pos cord-308115-bjyr6ehq 1 1 key key JJ cord-308115-bjyr6ehq 2 1 : : : cord-308115-bjyr6ehq 2 2 cord-308115-bjyr6ehq cord-308115-bjyr6ehq NN cord-308115-bjyr6ehq 2 3 authors author NNS cord-308115-bjyr6ehq 2 4 : : : cord-308115-bjyr6ehq 2 5 Baba Baba NNP cord-308115-bjyr6ehq 2 6 , , , cord-308115-bjyr6ehq 2 7 Isa Isa NNP cord-308115-bjyr6ehq 2 8 Abdullah Abdullah NNP cord-308115-bjyr6ehq 2 9 ; ; : cord-308115-bjyr6ehq 2 10 Nasidi Nasidi NNP cord-308115-bjyr6ehq 2 11 , , , cord-308115-bjyr6ehq 2 12 Bashir Bashir NNP cord-308115-bjyr6ehq 2 13 Ahmad Ahmad NNP cord-308115-bjyr6ehq 2 14 title title NN cord-308115-bjyr6ehq 2 15 : : : cord-308115-bjyr6ehq 2 16 Fractional Fractional NNP cord-308115-bjyr6ehq 2 17 Order Order NNP cord-308115-bjyr6ehq 2 18 Model Model NNP cord-308115-bjyr6ehq 2 19 for for IN cord-308115-bjyr6ehq 2 20 the the DT cord-308115-bjyr6ehq 2 21 Role role NN cord-308115-bjyr6ehq 2 22 of of IN cord-308115-bjyr6ehq 2 23 Mild mild JJ cord-308115-bjyr6ehq 2 24 Cases case NNS cord-308115-bjyr6ehq 2 25 in in IN cord-308115-bjyr6ehq 2 26 the the DT cord-308115-bjyr6ehq 2 27 Transmission Transmission NNP cord-308115-bjyr6ehq 2 28 of of IN cord-308115-bjyr6ehq 2 29 COVID-19 covid-19 CD cord-308115-bjyr6ehq 2 30 date date NN cord-308115-bjyr6ehq 2 31 : : : cord-308115-bjyr6ehq 2 32 2020 2020 CD cord-308115-bjyr6ehq 2 33 - - SYM cord-308115-bjyr6ehq 2 34 10 10 CD cord-308115-bjyr6ehq 2 35 - - SYM cord-308115-bjyr6ehq 2 36 20 20 CD cord-308115-bjyr6ehq 2 37 journal journal NN cord-308115-bjyr6ehq 2 38 : : : cord-308115-bjyr6ehq 3 1 Chaos Chaos NNP cord-308115-bjyr6ehq 3 2 Solitons Solitons NNPS cord-308115-bjyr6ehq 4 1 Fractals fractal NNS cord-308115-bjyr6ehq 4 2 DOI DOI NNP cord-308115-bjyr6ehq 4 3 : : : cord-308115-bjyr6ehq 4 4 10.1016 10.1016 CD cord-308115-bjyr6ehq 4 5 / / SYM cord-308115-bjyr6ehq 4 6 j.chaos.2020.110374 j.chaos.2020.110374 NNP cord-308115-bjyr6ehq 4 7 sha sha NNP cord-308115-bjyr6ehq 4 8 : : : cord-308115-bjyr6ehq 4 9 3685bbb851b235b4f5e1efc09c5401e36b13329e 3685bbb851b235b4f5e1efc09c5401e36b13329e CD cord-308115-bjyr6ehq 4 10 doc_id doc_id CD cord-308115-bjyr6ehq 4 11 : : : cord-308115-bjyr6ehq 4 12 308115 308115 CD cord-308115-bjyr6ehq 4 13 cord_uid cord_uid NNS cord-308115-bjyr6ehq 4 14 : : : cord-308115-bjyr6ehq 5 1 bjyr6ehq bjyr6ehq NNP cord-308115-bjyr6ehq 6 1 Most Most JJS cord-308115-bjyr6ehq 6 2 of of IN cord-308115-bjyr6ehq 6 3 the the DT cord-308115-bjyr6ehq 6 4 nations nation NNS cord-308115-bjyr6ehq 6 5 with with IN cord-308115-bjyr6ehq 6 6 deplorable deplorable JJ cord-308115-bjyr6ehq 6 7 health health NN cord-308115-bjyr6ehq 6 8 conditions condition NNS cord-308115-bjyr6ehq 6 9 lack lack VBP cord-308115-bjyr6ehq 6 10 rapid rapid JJ cord-308115-bjyr6ehq 6 11 COVID-19diagnostic covid-19diagnostic NN cord-308115-bjyr6ehq 6 12 test test NN cord-308115-bjyr6ehq 6 13 due due IN cord-308115-bjyr6ehq 6 14 to to IN cord-308115-bjyr6ehq 6 15 limited limited JJ cord-308115-bjyr6ehq 6 16 testing testing NN cord-308115-bjyr6ehq 6 17 kits kit NNS cord-308115-bjyr6ehq 6 18 and and CC cord-308115-bjyr6ehq 6 19 laboratories laboratory NNS cord-308115-bjyr6ehq 6 20 . . . cord-308115-bjyr6ehq 7 1 The the DT cord-308115-bjyr6ehq 7 2 un un NNP cord-308115-bjyr6ehq 7 3 - - JJ cord-308115-bjyr6ehq 7 4 diagnosticmild diagnosticmild JJ cord-308115-bjyr6ehq 7 5 cases case NNS cord-308115-bjyr6ehq 7 6 ( ( -LRB- cord-308115-bjyr6ehq 7 7 who who WP cord-308115-bjyr6ehq 7 8 show show VBP cord-308115-bjyr6ehq 7 9 no no DT cord-308115-bjyr6ehq 7 10 critical critical JJ cord-308115-bjyr6ehq 7 11 sign sign NN cord-308115-bjyr6ehq 7 12 and and CC cord-308115-bjyr6ehq 7 13 symptoms symptom NNS cord-308115-bjyr6ehq 7 14 ) ) -RRB- cord-308115-bjyr6ehq 7 15 play play VBP cord-308115-bjyr6ehq 7 16 the the DT cord-308115-bjyr6ehq 7 17 role role NN cord-308115-bjyr6ehq 7 18 as as IN cord-308115-bjyr6ehq 7 19 a a DT cord-308115-bjyr6ehq 7 20 route route NN cord-308115-bjyr6ehq 7 21 that that WDT cord-308115-bjyr6ehq 7 22 spread spread VBD cord-308115-bjyr6ehq 7 23 the the DT cord-308115-bjyr6ehq 7 24 infection infection NN cord-308115-bjyr6ehq 7 25 unknowingly unknowingly RB cord-308115-bjyr6ehq 7 26 to to IN cord-308115-bjyr6ehq 7 27 healthy healthy JJ cord-308115-bjyr6ehq 7 28 individuals individual NNS cord-308115-bjyr6ehq 7 29 . . . cord-308115-bjyr6ehq 8 1 In in IN cord-308115-bjyr6ehq 8 2 this this DT cord-308115-bjyr6ehq 8 3 paper paper NN cord-308115-bjyr6ehq 8 4 , , , cord-308115-bjyr6ehq 8 5 we -PRON- PRP cord-308115-bjyr6ehq 8 6 present present VBP cord-308115-bjyr6ehq 8 7 a a DT cord-308115-bjyr6ehq 8 8 fractional fractional JJ cord-308115-bjyr6ehq 8 9 order order NN cord-308115-bjyr6ehq 8 10 SIR SIR NNP cord-308115-bjyr6ehq 8 11 model model NN cord-308115-bjyr6ehq 8 12 incorporating incorporate VBG cord-308115-bjyr6ehq 8 13 individual individual JJ cord-308115-bjyr6ehq 8 14 with with IN cord-308115-bjyr6ehq 8 15 mild mild JJ cord-308115-bjyr6ehq 8 16 cases case NNS cord-308115-bjyr6ehq 8 17 as as IN cord-308115-bjyr6ehq 8 18 a a DT cord-308115-bjyr6ehq 8 19 compartment compartment NN cord-308115-bjyr6ehq 8 20 to to TO cord-308115-bjyr6ehq 8 21 become become VB cord-308115-bjyr6ehq 8 22 SMIR SMIR NNP cord-308115-bjyr6ehq 8 23 model model NN cord-308115-bjyr6ehq 8 24 . . . cord-308115-bjyr6ehq 9 1 The the DT cord-308115-bjyr6ehq 9 2 existence existence NN cord-308115-bjyr6ehq 9 3 of of IN cord-308115-bjyr6ehq 9 4 the the DT cord-308115-bjyr6ehq 9 5 solutions solution NNS cord-308115-bjyr6ehq 9 6 of of IN cord-308115-bjyr6ehq 9 7 the the DT cord-308115-bjyr6ehq 9 8 model model NN cord-308115-bjyr6ehq 9 9 is be VBZ cord-308115-bjyr6ehq 9 10 investigated investigate VBN cord-308115-bjyr6ehq 9 11 by by IN cord-308115-bjyr6ehq 9 12 solving solve VBG cord-308115-bjyr6ehq 9 13 the the DT cord-308115-bjyr6ehq 9 14 fractional fractional JJ cord-308115-bjyr6ehq 9 15 Gronwall Gronwall NNP cord-308115-bjyr6ehq 9 16 's 's POS cord-308115-bjyr6ehq 9 17 inequality inequality NN cord-308115-bjyr6ehq 9 18 using use VBG cord-308115-bjyr6ehq 9 19 the the DT cord-308115-bjyr6ehq 9 20 Laplace Laplace NNP cord-308115-bjyr6ehq 9 21 transform transform VB cord-308115-bjyr6ehq 9 22 approach approach NN cord-308115-bjyr6ehq 9 23 . . . cord-308115-bjyr6ehq 10 1 The the DT cord-308115-bjyr6ehq 10 2 equilibrium equilibrium NN cord-308115-bjyr6ehq 10 3 solutions solution NNS cord-308115-bjyr6ehq 10 4 ( ( -LRB- cord-308115-bjyr6ehq 10 5 DFE DFE NNP cord-308115-bjyr6ehq 10 6 & & CC cord-308115-bjyr6ehq 10 7 Endemic Endemic NNP cord-308115-bjyr6ehq 10 8 ) ) -RRB- cord-308115-bjyr6ehq 10 9 are be VBP cord-308115-bjyr6ehq 10 10 found find VBN cord-308115-bjyr6ehq 10 11 to to TO cord-308115-bjyr6ehq 10 12 be be VB cord-308115-bjyr6ehq 10 13 locally locally RB cord-308115-bjyr6ehq 10 14 asymptotically asymptotically RB cord-308115-bjyr6ehq 10 15 stable stable JJ cord-308115-bjyr6ehq 10 16 , , , cord-308115-bjyr6ehq 10 17 and and CC cord-308115-bjyr6ehq 10 18 subsequently subsequently RB cord-308115-bjyr6ehq 10 19 the the DT cord-308115-bjyr6ehq 10 20 basic basic JJ cord-308115-bjyr6ehq 10 21 reproduction reproduction NN cord-308115-bjyr6ehq 10 22 number number NN cord-308115-bjyr6ehq 10 23 is be VBZ cord-308115-bjyr6ehq 10 24 obtained obtain VBN cord-308115-bjyr6ehq 10 25 . . . cord-308115-bjyr6ehq 11 1 Also also RB cord-308115-bjyr6ehq 11 2 the the DT cord-308115-bjyr6ehq 11 3 global global JJ cord-308115-bjyr6ehq 11 4 stability stability NN cord-308115-bjyr6ehq 11 5 analysis analysis NN cord-308115-bjyr6ehq 11 6 is be VBZ cord-308115-bjyr6ehq 11 7 carried carry VBN cord-308115-bjyr6ehq 11 8 out out RP cord-308115-bjyr6ehq 11 9 by by IN cord-308115-bjyr6ehq 11 10 constructing construct VBG cord-308115-bjyr6ehq 11 11 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 11 12 function function NN cord-308115-bjyr6ehq 11 13 . . . cord-308115-bjyr6ehq 12 1 Lastly lastly RB cord-308115-bjyr6ehq 12 2 , , , cord-308115-bjyr6ehq 12 3 numerical numerical JJ cord-308115-bjyr6ehq 12 4 simulations simulation NNS cord-308115-bjyr6ehq 12 5 that that WDT cord-308115-bjyr6ehq 12 6 support support VBP cord-308115-bjyr6ehq 12 7 analytic analytic JJ cord-308115-bjyr6ehq 12 8 solution solution NN cord-308115-bjyr6ehq 12 9 follow follow VB cord-308115-bjyr6ehq 12 10 . . . cord-308115-bjyr6ehq 13 1 It -PRON- PRP cord-308115-bjyr6ehq 13 2 was be VBD cord-308115-bjyr6ehq 13 3 also also RB cord-308115-bjyr6ehq 13 4 shown show VBN cord-308115-bjyr6ehq 13 5 that that IN cord-308115-bjyr6ehq 13 6 when when WRB cord-308115-bjyr6ehq 13 7 the the DT cord-308115-bjyr6ehq 13 8 rate rate NN cord-308115-bjyr6ehq 13 9 of of IN cord-308115-bjyr6ehq 13 10 infection infection NN cord-308115-bjyr6ehq 13 11 of of IN cord-308115-bjyr6ehq 13 12 the the DT cord-308115-bjyr6ehq 13 13 mild mild JJ cord-308115-bjyr6ehq 13 14 cases case NNS cord-308115-bjyr6ehq 13 15 increases increase NNS cord-308115-bjyr6ehq 13 16 , , , cord-308115-bjyr6ehq 13 17 there there EX cord-308115-bjyr6ehq 13 18 is be VBZ cord-308115-bjyr6ehq 13 19 equivalent equivalent JJ cord-308115-bjyr6ehq 13 20 increase increase NN cord-308115-bjyr6ehq 13 21 in in IN cord-308115-bjyr6ehq 13 22 the the DT cord-308115-bjyr6ehq 13 23 overall overall JJ cord-308115-bjyr6ehq 13 24 population population NN cord-308115-bjyr6ehq 13 25 of of IN cord-308115-bjyr6ehq 13 26 infected infected JJ cord-308115-bjyr6ehq 13 27 individuals individual NNS cord-308115-bjyr6ehq 13 28 . . . cord-308115-bjyr6ehq 14 1 Hence hence RB cord-308115-bjyr6ehq 14 2 to to TO cord-308115-bjyr6ehq 14 3 curtail curtail VB cord-308115-bjyr6ehq 14 4 the the DT cord-308115-bjyr6ehq 14 5 spread spread NN cord-308115-bjyr6ehq 14 6 of of IN cord-308115-bjyr6ehq 14 7 the the DT cord-308115-bjyr6ehq 14 8 disease disease NN cord-308115-bjyr6ehq 14 9 there there EX cord-308115-bjyr6ehq 14 10 is be VBZ cord-308115-bjyr6ehq 14 11 need need NN cord-308115-bjyr6ehq 14 12 to to TO cord-308115-bjyr6ehq 14 13 take take VB cord-308115-bjyr6ehq 14 14 care care NN cord-308115-bjyr6ehq 14 15 of of IN cord-308115-bjyr6ehq 14 16 the the DT cord-308115-bjyr6ehq 14 17 Mild mild JJ cord-308115-bjyr6ehq 14 18 cases case NNS cord-308115-bjyr6ehq 14 19 as as RB cord-308115-bjyr6ehq 14 20 well well RB cord-308115-bjyr6ehq 14 21 . . . cord-308115-bjyr6ehq 15 1 The the DT cord-308115-bjyr6ehq 15 2 outbreak outbreak NN cord-308115-bjyr6ehq 15 3 of of IN cord-308115-bjyr6ehq 15 4 the the DT cord-308115-bjyr6ehq 15 5 novel novel JJ cord-308115-bjyr6ehq 15 6 strain strain NN cord-308115-bjyr6ehq 15 7 of of IN cord-308115-bjyr6ehq 15 8 Corona Corona NNP cord-308115-bjyr6ehq 15 9 viruses virus NNS cord-308115-bjyr6ehq 15 10 ( ( -LRB- cord-308115-bjyr6ehq 15 11 COVID- covid- NN cord-308115-bjyr6ehq 15 12 19 19 CD cord-308115-bjyr6ehq 15 13 ) ) -RRB- cord-308115-bjyr6ehq 15 14 started start VBD cord-308115-bjyr6ehq 15 15 late late JJ cord-308115-bjyr6ehq 15 16 December December NNP cord-308115-bjyr6ehq 15 17 in in IN cord-308115-bjyr6ehq 15 18 the the DT cord-308115-bjyr6ehq 15 19 Wuhan Wuhan NNP cord-308115-bjyr6ehq 15 20 province province NN cord-308115-bjyr6ehq 15 21 in in IN cord-308115-bjyr6ehq 15 22 China China NNP cord-308115-bjyr6ehq 15 23 [ [ -LRB- cord-308115-bjyr6ehq 15 24 1 1 CD cord-308115-bjyr6ehq 15 25 ] ] -RRB- cord-308115-bjyr6ehq 15 26 . . . cord-308115-bjyr6ehq 16 1 It -PRON- PRP cord-308115-bjyr6ehq 16 2 became become VBD cord-308115-bjyr6ehq 16 3 a a DT cord-308115-bjyr6ehq 16 4 global global JJ cord-308115-bjyr6ehq 16 5 pandemic pandemic NN cord-308115-bjyr6ehq 16 6 causing cause VBG cord-308115-bjyr6ehq 16 7 the the DT cord-308115-bjyr6ehq 16 8 devastating devastating JJ cord-308115-bjyr6ehq 16 9 impact impact NN cord-308115-bjyr6ehq 16 10 in in IN cord-308115-bjyr6ehq 16 11 terms term NNS cord-308115-bjyr6ehq 16 12 of of IN cord-308115-bjyr6ehq 16 13 morbidity morbidity NN cord-308115-bjyr6ehq 16 14 , , , cord-308115-bjyr6ehq 16 15 infections infection NNS cord-308115-bjyr6ehq 16 16 and and CC cord-308115-bjyr6ehq 16 17 fatality fatality NN cord-308115-bjyr6ehq 16 18 in in IN cord-308115-bjyr6ehq 16 19 addition addition NN cord-308115-bjyr6ehq 16 20 to to IN cord-308115-bjyr6ehq 16 21 socio socio JJ cord-308115-bjyr6ehq 16 22 - - JJ cord-308115-bjyr6ehq 16 23 economic economic JJ cord-308115-bjyr6ehq 16 24 disaster disaster NN cord-308115-bjyr6ehq 16 25 . . . cord-308115-bjyr6ehq 17 1 The the DT cord-308115-bjyr6ehq 17 2 virus virus NN cord-308115-bjyr6ehq 17 3 source source NN cord-308115-bjyr6ehq 17 4 which which WDT cord-308115-bjyr6ehq 17 5 is be VBZ cord-308115-bjyr6ehq 17 6 yet yet RB cord-308115-bjyr6ehq 17 7 to to TO cord-308115-bjyr6ehq 17 8 be be VB cord-308115-bjyr6ehq 17 9 identified identify VBN cord-308115-bjyr6ehq 17 10 is be VBZ cord-308115-bjyr6ehq 17 11 said say VBN cord-308115-bjyr6ehq 17 12 to to TO cord-308115-bjyr6ehq 17 13 have have VB cord-308115-bjyr6ehq 17 14 genetic genetic JJ cord-308115-bjyr6ehq 17 15 linkages linkage NNS cord-308115-bjyr6ehq 17 16 with with IN cord-308115-bjyr6ehq 17 17 Severe severe JJ cord-308115-bjyr6ehq 17 18 Acute acute JJ cord-308115-bjyr6ehq 17 19 Respiratory Respiratory NNP cord-308115-bjyr6ehq 17 20 Syndrome Syndrome NNP cord-308115-bjyr6ehq 17 21 ( ( -LRB- cord-308115-bjyr6ehq 17 22 SARS SARS NNP cord-308115-bjyr6ehq 17 23 - - HYPH cord-308115-bjyr6ehq 17 24 COV COV NNP cord-308115-bjyr6ehq 17 25 ) ) -RRB- cord-308115-bjyr6ehq 17 26 but but CC cord-308115-bjyr6ehq 17 27 less less RBR cord-308115-bjyr6ehq 17 28 severe severe JJ cord-308115-bjyr6ehq 17 29 than than IN cord-308115-bjyr6ehq 17 30 Middle Middle NNP cord-308115-bjyr6ehq 17 31 East East NNP cord-308115-bjyr6ehq 17 32 Respiratory Respiratory NNP cord-308115-bjyr6ehq 17 33 Syndrome Syndrome NNP cord-308115-bjyr6ehq 17 34 ( ( -LRB- cord-308115-bjyr6ehq 17 35 MERS MERS NNP cord-308115-bjyr6ehq 17 36 - - HYPH cord-308115-bjyr6ehq 17 37 COV COV NNP cord-308115-bjyr6ehq 17 38 ) ) -RRB- cord-308115-bjyr6ehq 18 1 [ [ -LRB- cord-308115-bjyr6ehq 18 2 2 2 CD cord-308115-bjyr6ehq 18 3 ] ] -RRB- cord-308115-bjyr6ehq 18 4 . . . cord-308115-bjyr6ehq 19 1 The the DT cord-308115-bjyr6ehq 19 2 virus virus NN cord-308115-bjyr6ehq 19 3 is be VBZ cord-308115-bjyr6ehq 19 4 transmitted transmit VBN cord-308115-bjyr6ehq 19 5 to to IN cord-308115-bjyr6ehq 19 6 healthy healthy JJ cord-308115-bjyr6ehq 19 7 persons person NNS cord-308115-bjyr6ehq 19 8 via via IN cord-308115-bjyr6ehq 19 9 eyes eye NNS cord-308115-bjyr6ehq 19 10 , , , cord-308115-bjyr6ehq 19 11 mouth mouth NN cord-308115-bjyr6ehq 19 12 and and CC cord-308115-bjyr6ehq 19 13 nose nose NN cord-308115-bjyr6ehq 19 14 when when WRB cord-308115-bjyr6ehq 19 15 an an DT cord-308115-bjyr6ehq 19 16 infected infected JJ cord-308115-bjyr6ehq 19 17 person person NN cord-308115-bjyr6ehq 19 18 produced produce VBD cord-308115-bjyr6ehq 19 19 respiratory respiratory JJ cord-308115-bjyr6ehq 19 20 droplets droplet NNS cord-308115-bjyr6ehq 19 21 of of IN cord-308115-bjyr6ehq 19 22 cough cough NN cord-308115-bjyr6ehq 19 23 and and CC cord-308115-bjyr6ehq 19 24 sneeze sneeze VBP cord-308115-bjyr6ehq 19 25 or or CC cord-308115-bjyr6ehq 19 26 as as IN cord-308115-bjyr6ehq 19 27 a a DT cord-308115-bjyr6ehq 19 28 result result NN cord-308115-bjyr6ehq 19 29 of of IN cord-308115-bjyr6ehq 19 30 contact contact NN cord-308115-bjyr6ehq 19 31 with with IN cord-308115-bjyr6ehq 19 32 contaminated contaminated JJ cord-308115-bjyr6ehq 19 33 surfaces surface NNS cord-308115-bjyr6ehq 19 34 . . . cord-308115-bjyr6ehq 20 1 The the DT cord-308115-bjyr6ehq 20 2 average average JJ cord-308115-bjyr6ehq 20 3 incubation incubation NN cord-308115-bjyr6ehq 20 4 period period NN cord-308115-bjyr6ehq 20 5 from from IN cord-308115-bjyr6ehq 20 6 catching catch VBG cord-308115-bjyr6ehq 20 7 the the DT cord-308115-bjyr6ehq 20 8 virus virus NN cord-308115-bjyr6ehq 20 9 to to IN cord-308115-bjyr6ehq 20 10 time time NN cord-308115-bjyr6ehq 20 11 onset onset NN cord-308115-bjyr6ehq 20 12 of of IN cord-308115-bjyr6ehq 20 13 major major JJ cord-308115-bjyr6ehq 20 14 symptoms symptom NNS cord-308115-bjyr6ehq 20 15 ( ( -LRB- cord-308115-bjyr6ehq 20 16 like like IN cord-308115-bjyr6ehq 20 17 fever fever NN cord-308115-bjyr6ehq 20 18 , , , cord-308115-bjyr6ehq 20 19 cough cough NN cord-308115-bjyr6ehq 20 20 and and CC cord-308115-bjyr6ehq 20 21 sneeze sneeze NN cord-308115-bjyr6ehq 20 22 ) ) -RRB- cord-308115-bjyr6ehq 20 23 is be VBZ cord-308115-bjyr6ehq 20 24 between between IN cord-308115-bjyr6ehq 20 25 2 2 CD cord-308115-bjyr6ehq 20 26 -14days -14days : cord-308115-bjyr6ehq 20 27 [ [ -LRB- cord-308115-bjyr6ehq 20 28 3 3 CD cord-308115-bjyr6ehq 20 29 ] ] -RRB- cord-308115-bjyr6ehq 20 30 . . . cord-308115-bjyr6ehq 21 1 As as IN cord-308115-bjyr6ehq 21 2 the the DT cord-308115-bjyr6ehq 21 3 vaccine vaccine NN cord-308115-bjyr6ehq 21 4 is be VBZ cord-308115-bjyr6ehq 21 5 not not RB cord-308115-bjyr6ehq 21 6 yet yet RB cord-308115-bjyr6ehq 21 7 found find VBN cord-308115-bjyr6ehq 21 8 , , , cord-308115-bjyr6ehq 21 9 the the DT cord-308115-bjyr6ehq 21 10 control control NN cord-308115-bjyr6ehq 21 11 measures measure NNS cord-308115-bjyr6ehq 21 12 such such JJ cord-308115-bjyr6ehq 21 13 as as IN cord-308115-bjyr6ehq 21 14 : : : cord-308115-bjyr6ehq 21 15 social social JJ cord-308115-bjyr6ehq 21 16 distancing distancing NN cord-308115-bjyr6ehq 21 17 , , , cord-308115-bjyr6ehq 21 18 quarantine quarantine NN cord-308115-bjyr6ehq 21 19 of of IN cord-308115-bjyr6ehq 21 20 suspected suspect VBN cord-308115-bjyr6ehq 21 21 case case NN cord-308115-bjyr6ehq 21 22 , , , cord-308115-bjyr6ehq 21 23 use use NN cord-308115-bjyr6ehq 21 24 of of IN cord-308115-bjyr6ehq 21 25 personal personal JJ cord-308115-bjyr6ehq 21 26 protective protective JJ cord-308115-bjyr6ehq 21 27 equipment equipment NN cord-308115-bjyr6ehq 21 28 ( ( -LRB- cord-308115-bjyr6ehq 21 29 like like UH cord-308115-bjyr6ehq 21 30 face face NN cord-308115-bjyr6ehq 21 31 Mask Mask NNP cord-308115-bjyr6ehq 21 32 , , , cord-308115-bjyr6ehq 21 33 hand hand NN cord-308115-bjyr6ehq 21 34 globes globe NNS cord-308115-bjyr6ehq 21 35 , , , cord-308115-bjyr6ehq 21 36 gown gown NN cord-308115-bjyr6ehq 21 37 ) ) -RRB- cord-308115-bjyr6ehq 21 38 , , , cord-308115-bjyr6ehq 21 39 regular regular JJ cord-308115-bjyr6ehq 21 40 hand hand NN cord-308115-bjyr6ehq 21 41 sanitation sanitation NN cord-308115-bjyr6ehq 21 42 using use VBG cord-308115-bjyr6ehq 21 43 antibacterial antibacterial JJ cord-308115-bjyr6ehq 21 44 agents agent NNS cord-308115-bjyr6ehq 21 45 ( ( -LRB- cord-308115-bjyr6ehq 21 46 like like IN cord-308115-bjyr6ehq 21 47 soaps soap NNS cord-308115-bjyr6ehq 21 48 , , , cord-308115-bjyr6ehq 21 49 sanitizer sanitizer NN cord-308115-bjyr6ehq 21 50 ) ) -RRB- cord-308115-bjyr6ehq 21 51 and and CC cord-308115-bjyr6ehq 21 52 imposing impose VBG cord-308115-bjyr6ehq 21 53 the the DT cord-308115-bjyr6ehq 21 54 lockdown lockdown NN cord-308115-bjyr6ehq 21 55 curfew(when curfew(when VBG cord-308115-bjyr6ehq 21 56 necessary necessary JJ cord-308115-bjyr6ehq 21 57 ) ) -RRB- cord-308115-bjyr6ehq 21 58 are be VBP cord-308115-bjyr6ehq 21 59 the the DT cord-308115-bjyr6ehq 21 60 effective effective JJ cord-308115-bjyr6ehq 21 61 intervention intervention NN cord-308115-bjyr6ehq 21 62 that that WDT cord-308115-bjyr6ehq 21 63 mitigate mitigate VBP cord-308115-bjyr6ehq 21 64 the the DT cord-308115-bjyr6ehq 21 65 transmission transmission NN cord-308115-bjyr6ehq 21 66 of of IN cord-308115-bjyr6ehq 21 67 the the DT cord-308115-bjyr6ehq 21 68 infection infection NN cord-308115-bjyr6ehq 21 69 . . . cord-308115-bjyr6ehq 22 1 To to TO cord-308115-bjyr6ehq 22 2 execute execute VB cord-308115-bjyr6ehq 22 3 these these DT cord-308115-bjyr6ehq 22 4 measures measure NNS cord-308115-bjyr6ehq 22 5 effectively effectively RB cord-308115-bjyr6ehq 22 6 , , , cord-308115-bjyr6ehq 22 7 there there EX cord-308115-bjyr6ehq 22 8 is be VBZ cord-308115-bjyr6ehq 22 9 need need NN cord-308115-bjyr6ehq 22 10 to to TO cord-308115-bjyr6ehq 22 11 have have VB cord-308115-bjyr6ehq 22 12 an an DT cord-308115-bjyr6ehq 22 13 in in IN cord-308115-bjyr6ehq 22 14 depth depth NN cord-308115-bjyr6ehq 22 15 study study NN cord-308115-bjyr6ehq 22 16 about about IN cord-308115-bjyr6ehq 22 17 the the DT cord-308115-bjyr6ehq 22 18 number number NN cord-308115-bjyr6ehq 22 19 of of IN cord-308115-bjyr6ehq 22 20 persons person NNS cord-308115-bjyr6ehq 22 21 that that IN cord-308115-bjyr6ehq 22 22 each each DT cord-308115-bjyr6ehq 22 23 infected infect VBN cord-308115-bjyr6ehq 22 24 individual individual NN cord-308115-bjyr6ehq 22 25 can can MD cord-308115-bjyr6ehq 22 26 infect infect VB cord-308115-bjyr6ehq 22 27 , , , cord-308115-bjyr6ehq 22 28 meanwhile meanwhile RB cord-308115-bjyr6ehq 22 29 a a DT cord-308115-bjyr6ehq 22 30 mathematical mathematical JJ cord-308115-bjyr6ehq 22 31 model model NN cord-308115-bjyr6ehq 22 32 describing describe VBG cord-308115-bjyr6ehq 22 33 the the DT cord-308115-bjyr6ehq 22 34 transmission transmission NN cord-308115-bjyr6ehq 22 35 dynamics dynamic NNS cord-308115-bjyr6ehq 22 36 of of IN cord-308115-bjyr6ehq 22 37 the the DT cord-308115-bjyr6ehq 22 38 disease disease NN cord-308115-bjyr6ehq 22 39 should should MD cord-308115-bjyr6ehq 22 40 be be VB cord-308115-bjyr6ehq 22 41 established establish VBN cord-308115-bjyr6ehq 22 42 . . . cord-308115-bjyr6ehq 23 1 In in IN cord-308115-bjyr6ehq 23 2 this this DT cord-308115-bjyr6ehq 23 3 regard regard NN cord-308115-bjyr6ehq 23 4 , , , cord-308115-bjyr6ehq 23 5 Zhao Zhao NNP cord-308115-bjyr6ehq 23 6 and and CC cord-308115-bjyr6ehq 23 7 Chen Chen NNP cord-308115-bjyr6ehq 23 8 [ [ -LRB- cord-308115-bjyr6ehq 23 9 4 4 CD cord-308115-bjyr6ehq 23 10 ] ] -RRB- cord-308115-bjyr6ehq 23 11 developed develop VBD cord-308115-bjyr6ehq 23 12 a a DT cord-308115-bjyr6ehq 23 13 Susceptible susceptible JJ cord-308115-bjyr6ehq 23 14 , , , cord-308115-bjyr6ehq 23 15 Un Un NNP cord-308115-bjyr6ehq 23 16 - - HYPH cord-308115-bjyr6ehq 23 17 quarantined quarantine VBN cord-308115-bjyr6ehq 23 18 infected infect VBN cord-308115-bjyr6ehq 23 19 , , , cord-308115-bjyr6ehq 23 20 Quarantined quarantine VBN cord-308115-bjyr6ehq 23 21 infected infect VBN cord-308115-bjyr6ehq 23 22 , , , cord-308115-bjyr6ehq 23 23 Confirmed confirm VBD cord-308115-bjyr6ehq 23 24 infected infect VBN cord-308115-bjyr6ehq 23 25 ( ( -LRB- cord-308115-bjyr6ehq 23 26 SUQC SUQC NNP cord-308115-bjyr6ehq 23 27 ) ) -RRB- cord-308115-bjyr6ehq 23 28 model model NN cord-308115-bjyr6ehq 23 29 to to TO cord-308115-bjyr6ehq 23 30 characterize characterize VB cord-308115-bjyr6ehq 23 31 the the DT cord-308115-bjyr6ehq 23 32 dynamics dynamic NNS cord-308115-bjyr6ehq 23 33 of of IN cord-308115-bjyr6ehq 23 34 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 23 35 and and CC cord-308115-bjyr6ehq 23 36 explicitly explicitly RB cord-308115-bjyr6ehq 23 37 parameterized parameterize VBD cord-308115-bjyr6ehq 23 38 the the DT cord-308115-bjyr6ehq 23 39 intervention intervention NN cord-308115-bjyr6ehq 23 40 effects effect NNS cord-308115-bjyr6ehq 23 41 of of IN cord-308115-bjyr6ehq 23 42 control control NN cord-308115-bjyr6ehq 23 43 measures measure NNS cord-308115-bjyr6ehq 23 44 . . . cord-308115-bjyr6ehq 24 1 Similarly similarly RB cord-308115-bjyr6ehq 24 2 , , , cord-308115-bjyr6ehq 24 3 Song Song NNP cord-308115-bjyr6ehq 24 4 et et FW cord-308115-bjyr6ehq 24 5 al al NNP cord-308115-bjyr6ehq 24 6 . . . cord-308115-bjyr6ehq 25 1 [ [ -LRB- cord-308115-bjyr6ehq 25 2 5 5 CD cord-308115-bjyr6ehq 25 3 ] ] -RRB- cord-308115-bjyr6ehq 25 4 developed develop VBD cord-308115-bjyr6ehq 25 5 a a DT cord-308115-bjyr6ehq 25 6 mathematical mathematical JJ cord-308115-bjyr6ehq 25 7 model model NN cord-308115-bjyr6ehq 25 8 based base VBN cord-308115-bjyr6ehq 25 9 on on IN cord-308115-bjyr6ehq 25 10 the the DT cord-308115-bjyr6ehq 25 11 epidemiology epidemiology NN cord-308115-bjyr6ehq 25 12 of of IN cord-308115-bjyr6ehq 25 13 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 25 14 , , , cord-308115-bjyr6ehq 25 15 incorporating incorporate VBG cord-308115-bjyr6ehq 25 16 the the DT cord-308115-bjyr6ehq 25 17 isolation isolation NN cord-308115-bjyr6ehq 25 18 of of IN cord-308115-bjyr6ehq 25 19 healthy healthy JJ cord-308115-bjyr6ehq 25 20 people people NNS cord-308115-bjyr6ehq 25 21 , , , cord-308115-bjyr6ehq 25 22 confirmed confirm VBD cord-308115-bjyr6ehq 25 23 cases case NNS cord-308115-bjyr6ehq 25 24 and and CC cord-308115-bjyr6ehq 25 25 close close JJ cord-308115-bjyr6ehq 25 26 contacts contact NNS cord-308115-bjyr6ehq 25 27 . . . cord-308115-bjyr6ehq 26 1 Tahir Tahir NNP cord-308115-bjyr6ehq 26 2 et et NNP cord-308115-bjyr6ehq 26 3 al al NNP cord-308115-bjyr6ehq 26 4 . . . cord-308115-bjyr6ehq 27 1 [ [ -LRB- cord-308115-bjyr6ehq 27 2 6 6 CD cord-308115-bjyr6ehq 27 3 ] ] -RRB- cord-308115-bjyr6ehq 27 4 developed develop VBD cord-308115-bjyr6ehq 27 5 a a DT cord-308115-bjyr6ehq 27 6 mathematical mathematical JJ cord-308115-bjyr6ehq 27 7 model model NN cord-308115-bjyr6ehq 27 8 ( ( -LRB- cord-308115-bjyr6ehq 27 9 for for IN cord-308115-bjyr6ehq 27 10 MERS MERS NNP cord-308115-bjyr6ehq 27 11 ) ) -RRB- cord-308115-bjyr6ehq 27 12 inform inform NN cord-308115-bjyr6ehq 27 13 of of IN cord-308115-bjyr6ehq 27 14 nonlinear nonlinear JJ cord-308115-bjyr6ehq 27 15 system system NN cord-308115-bjyr6ehq 27 16 of of IN cord-308115-bjyr6ehq 27 17 differential differential JJ cord-308115-bjyr6ehq 27 18 equations equation NNS cord-308115-bjyr6ehq 27 19 , , , cord-308115-bjyr6ehq 27 20 in in IN cord-308115-bjyr6ehq 27 21 which which WDT cord-308115-bjyr6ehq 27 22 he -PRON- PRP cord-308115-bjyr6ehq 27 23 considered consider VBD cord-308115-bjyr6ehq 27 24 a a DT cord-308115-bjyr6ehq 27 25 camel camel NN cord-308115-bjyr6ehq 27 26 to to TO cord-308115-bjyr6ehq 27 27 be be VB cord-308115-bjyr6ehq 27 28 the the DT cord-308115-bjyr6ehq 27 29 source source NN cord-308115-bjyr6ehq 27 30 of of IN cord-308115-bjyr6ehq 27 31 infection infection NN cord-308115-bjyr6ehq 27 32 that that WDT cord-308115-bjyr6ehq 27 33 spread spread VBD cord-308115-bjyr6ehq 27 34 the the DT cord-308115-bjyr6ehq 27 35 virus virus NN cord-308115-bjyr6ehq 27 36 to to IN cord-308115-bjyr6ehq 27 37 infective infective JJ cord-308115-bjyr6ehq 27 38 human human JJ cord-308115-bjyr6ehq 27 39 population population NN cord-308115-bjyr6ehq 27 40 , , , cord-308115-bjyr6ehq 27 41 then then RB cord-308115-bjyr6ehq 27 42 human human JJ cord-308115-bjyr6ehq 27 43 to to IN cord-308115-bjyr6ehq 27 44 human human JJ cord-308115-bjyr6ehq 27 45 transmission transmission NN cord-308115-bjyr6ehq 27 46 , , , cord-308115-bjyr6ehq 27 47 then then RB cord-308115-bjyr6ehq 27 48 to to IN cord-308115-bjyr6ehq 27 49 clinic clinic NN cord-308115-bjyr6ehq 27 50 center center NN cord-308115-bjyr6ehq 27 51 then then RB cord-308115-bjyr6ehq 27 52 to to TO cord-308115-bjyr6ehq 27 53 care care VB cord-308115-bjyr6ehq 27 54 center center NN cord-308115-bjyr6ehq 27 55 . . . cord-308115-bjyr6ehq 28 1 However however RB cord-308115-bjyr6ehq 28 2 , , , cord-308115-bjyr6ehq 28 3 they -PRON- PRP cord-308115-bjyr6ehq 28 4 constructed construct VBD cord-308115-bjyr6ehq 28 5 the the DT cord-308115-bjyr6ehq 28 6 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 28 7 candidate candidate NN cord-308115-bjyr6ehq 28 8 function function NN cord-308115-bjyr6ehq 28 9 to to TO cord-308115-bjyr6ehq 28 10 investigate investigate VB cord-308115-bjyr6ehq 28 11 the the DT cord-308115-bjyr6ehq 28 12 local local JJ cord-308115-bjyr6ehq 28 13 and and CC cord-308115-bjyr6ehq 28 14 global global JJ cord-308115-bjyr6ehq 28 15 stability stability NN cord-308115-bjyr6ehq 28 16 analysis analysis NN cord-308115-bjyr6ehq 28 17 of of IN cord-308115-bjyr6ehq 28 18 the the DT cord-308115-bjyr6ehq 28 19 equilibriums equilibrium NNS cord-308115-bjyr6ehq 28 20 solution solution NN cord-308115-bjyr6ehq 28 21 and and CC cord-308115-bjyr6ehq 28 22 subsequently subsequently RB cord-308115-bjyr6ehq 28 23 obtained obtain VBD cord-308115-bjyr6ehq 28 24 the the DT cord-308115-bjyr6ehq 28 25 basic basic JJ cord-308115-bjyr6ehq 28 26 reproduction reproduction NN cord-308115-bjyr6ehq 28 27 number number NN cord-308115-bjyr6ehq 28 28 or or CC cord-308115-bjyr6ehq 28 29 roughly roughly RB cord-308115-bjyr6ehq 28 30 , , , cord-308115-bjyr6ehq 28 31 a a DT cord-308115-bjyr6ehq 28 32 key key JJ cord-308115-bjyr6ehq 28 33 parameter parameter NN cord-308115-bjyr6ehq 28 34 describing describe VBG cord-308115-bjyr6ehq 28 35 transmission transmission NN cord-308115-bjyr6ehq 28 36 of of IN cord-308115-bjyr6ehq 28 37 the the DT cord-308115-bjyr6ehq 28 38 infection infection NN cord-308115-bjyr6ehq 28 39 . . . cord-308115-bjyr6ehq 29 1 Also also RB cord-308115-bjyr6ehq 29 2 , , , cord-308115-bjyr6ehq 29 3 Chen Chen NNP cord-308115-bjyr6ehq 29 4 et et NNP cord-308115-bjyr6ehq 29 5 al al NNP cord-308115-bjyr6ehq 29 6 . . . cord-308115-bjyr6ehq 30 1 [ [ -LRB- cord-308115-bjyr6ehq 30 2 7 7 CD cord-308115-bjyr6ehq 30 3 ] ] -RRB- cord-308115-bjyr6ehq 30 4 developed develop VBD cord-308115-bjyr6ehq 30 5 a a DT cord-308115-bjyr6ehq 30 6 Bats Bats NNP cord-308115-bjyr6ehq 30 7 - - HYPH cord-308115-bjyr6ehq 30 8 Hosts Hosts NNP cord-308115-bjyr6ehq 30 9 - - HYPH cord-308115-bjyr6ehq 30 10 Reservoir Reservoir NNP cord-308115-bjyr6ehq 30 11 - - HYPH cord-308115-bjyr6ehq 30 12 People People NNP cord-308115-bjyr6ehq 30 13 ( ( -LRB- cord-308115-bjyr6ehq 30 14 BHRP BHRP NNP cord-308115-bjyr6ehq 30 15 ) ) -RRB- cord-308115-bjyr6ehq 30 16 transmission transmission NN cord-308115-bjyr6ehq 30 17 network network NN cord-308115-bjyr6ehq 30 18 model model NN cord-308115-bjyr6ehq 30 19 for for IN cord-308115-bjyr6ehq 30 20 the the DT cord-308115-bjyr6ehq 30 21 potential potential JJ cord-308115-bjyr6ehq 30 22 transmission transmission NN cord-308115-bjyr6ehq 30 23 from from IN cord-308115-bjyr6ehq 30 24 the the DT cord-308115-bjyr6ehq 30 25 infection infection NN cord-308115-bjyr6ehq 30 26 source source NN cord-308115-bjyr6ehq 30 27 ( ( -LRB- cord-308115-bjyr6ehq 30 28 probably probably RB cord-308115-bjyr6ehq 30 29 bats bat NNS cord-308115-bjyr6ehq 30 30 ) ) -RRB- cord-308115-bjyr6ehq 30 31 to to IN cord-308115-bjyr6ehq 30 32 the the DT cord-308115-bjyr6ehq 30 33 human human JJ cord-308115-bjyr6ehq 30 34 infection infection NN cord-308115-bjyr6ehq 30 35 , , , cord-308115-bjyr6ehq 30 36 which which WDT cord-308115-bjyr6ehq 30 37 focuses focus VBZ cord-308115-bjyr6ehq 30 38 on on IN cord-308115-bjyr6ehq 30 39 calculating calculate VBG cord-308115-bjyr6ehq 30 40 . . . cord-308115-bjyr6ehq 31 1 To to TO cord-308115-bjyr6ehq 31 2 suit suit VB cord-308115-bjyr6ehq 31 3 Korean korean JJ cord-308115-bjyr6ehq 31 4 outbreak outbreak NN cord-308115-bjyr6ehq 31 5 , , , cord-308115-bjyr6ehq 31 6 Sunhwa Sunhwa NNP cord-308115-bjyr6ehq 31 7 and and CC cord-308115-bjyr6ehq 31 8 Moran Moran NNP cord-308115-bjyr6ehq 31 9 [ [ -LRB- cord-308115-bjyr6ehq 31 10 8 8 CD cord-308115-bjyr6ehq 31 11 ] ] -RRB- cord-308115-bjyr6ehq 31 12 established establish VBN cord-308115-bjyr6ehq 31 13 deterministic deterministic JJ cord-308115-bjyr6ehq 31 14 mathematical mathematical JJ cord-308115-bjyr6ehq 31 15 model model NN cord-308115-bjyr6ehq 31 16 ( ( -LRB- cord-308115-bjyr6ehq 31 17 in in IN cord-308115-bjyr6ehq 31 18 form form NN cord-308115-bjyr6ehq 31 19 of of IN cord-308115-bjyr6ehq 31 20 SEIHR SEIHR NNP cord-308115-bjyr6ehq 31 21 ) ) -RRB- cord-308115-bjyr6ehq 31 22 , , , cord-308115-bjyr6ehq 31 23 in in IN cord-308115-bjyr6ehq 31 24 which which WDT cord-308115-bjyr6ehq 31 25 they -PRON- PRP cord-308115-bjyr6ehq 31 26 estimated estimate VBD cord-308115-bjyr6ehq 31 27 the the DT cord-308115-bjyr6ehq 31 28 reproduction reproduction NN cord-308115-bjyr6ehq 31 29 number number NN cord-308115-bjyr6ehq 31 30 and and CC cord-308115-bjyr6ehq 31 31 assessed assess VBD cord-308115-bjyr6ehq 31 32 the the DT cord-308115-bjyr6ehq 31 33 effect effect NN cord-308115-bjyr6ehq 31 34 of of IN cord-308115-bjyr6ehq 31 35 preventive preventive JJ cord-308115-bjyr6ehq 31 36 measures measure NNS cord-308115-bjyr6ehq 31 37 . . . cord-308115-bjyr6ehq 32 1 Similarly similarly RB cord-308115-bjyr6ehq 32 2 , , , cord-308115-bjyr6ehq 32 3 Lin Lin NNP cord-308115-bjyr6ehq 32 4 et et NNP cord-308115-bjyr6ehq 32 5 al al NNP cord-308115-bjyr6ehq 32 6 . . . cord-308115-bjyr6ehq 33 1 [ [ -LRB- cord-308115-bjyr6ehq 33 2 9 9 CD cord-308115-bjyr6ehq 33 3 ] ] -RRB- cord-308115-bjyr6ehq 33 4 modeled model VBN cord-308115-bjyr6ehq 33 5 ( ( -LRB- cord-308115-bjyr6ehq 33 6 based base VBN cord-308115-bjyr6ehq 33 7 on on IN cord-308115-bjyr6ehq 33 8 SEIR SEIR NNP cord-308115-bjyr6ehq 33 9 ) ) -RRB- cord-308115-bjyr6ehq 34 1 the the DT cord-308115-bjyr6ehq 34 2 outbreak outbreak NN cord-308115-bjyr6ehq 34 3 in in IN cord-308115-bjyr6ehq 34 4 Wuhan Wuhan NNP cord-308115-bjyr6ehq 34 5 with with IN cord-308115-bjyr6ehq 34 6 individual individual JJ cord-308115-bjyr6ehq 34 7 reaction reaction NN cord-308115-bjyr6ehq 34 8 and and CC cord-308115-bjyr6ehq 34 9 governmental governmental JJ cord-308115-bjyr6ehq 34 10 action action NN cord-308115-bjyr6ehq 34 11 ( ( -LRB- cord-308115-bjyr6ehq 34 12 holiday holiday NN cord-308115-bjyr6ehq 34 13 extension extension NN cord-308115-bjyr6ehq 34 14 , , , cord-308115-bjyr6ehq 34 15 city city NN cord-308115-bjyr6ehq 34 16 lockdown lockdown NN cord-308115-bjyr6ehq 34 17 , , , cord-308115-bjyr6ehq 34 18 hospitalization hospitalization NN cord-308115-bjyr6ehq 34 19 and and CC cord-308115-bjyr6ehq 34 20 quarantine quarantine NN cord-308115-bjyr6ehq 34 21 ) ) -RRB- cord-308115-bjyr6ehq 34 22 in in IN cord-308115-bjyr6ehq 34 23 which which WDT cord-308115-bjyr6ehq 34 24 they -PRON- PRP cord-308115-bjyr6ehq 34 25 estimated estimate VBD cord-308115-bjyr6ehq 34 26 the the DT cord-308115-bjyr6ehq 34 27 preliminary preliminary JJ cord-308115-bjyr6ehq 34 28 magnitude magnitude NN cord-308115-bjyr6ehq 34 29 of of IN cord-308115-bjyr6ehq 34 30 different different JJ cord-308115-bjyr6ehq 34 31 effect effect NN cord-308115-bjyr6ehq 34 32 of of IN cord-308115-bjyr6ehq 34 33 individual individual JJ cord-308115-bjyr6ehq 34 34 reaction reaction NN cord-308115-bjyr6ehq 34 35 and and CC cord-308115-bjyr6ehq 34 36 governmental governmental JJ cord-308115-bjyr6ehq 34 37 action action NN cord-308115-bjyr6ehq 34 38 . . . cord-308115-bjyr6ehq 35 1 Also also RB cord-308115-bjyr6ehq 35 2 Yang Yang NNP cord-308115-bjyr6ehq 35 3 and and CC cord-308115-bjyr6ehq 35 4 Wang Wang NNP cord-308115-bjyr6ehq 35 5 [ [ -LRB- cord-308115-bjyr6ehq 35 6 10 10 CD cord-308115-bjyr6ehq 35 7 ] ] -RRB- cord-308115-bjyr6ehq 35 8 proposed propose VBD cord-308115-bjyr6ehq 35 9 a a DT cord-308115-bjyr6ehq 35 10 mathematical mathematical JJ cord-308115-bjyr6ehq 35 11 model model NN cord-308115-bjyr6ehq 35 12 to to TO cord-308115-bjyr6ehq 35 13 investigate investigate VB cord-308115-bjyr6ehq 35 14 the the DT cord-308115-bjyr6ehq 35 15 current current JJ cord-308115-bjyr6ehq 35 16 outbreak outbreak NN cord-308115-bjyr6ehq 35 17 of of IN cord-308115-bjyr6ehq 35 18 the the DT cord-308115-bjyr6ehq 35 19 coronavirus coronavirus NN cord-308115-bjyr6ehq 35 20 disease disease NN cord-308115-bjyr6ehq 35 21 ( ( -LRB- cord-308115-bjyr6ehq 35 22 COVID- covid- NN cord-308115-bjyr6ehq 35 23 19 19 CD cord-308115-bjyr6ehq 35 24 ) ) -RRB- cord-308115-bjyr6ehq 35 25 in in IN cord-308115-bjyr6ehq 35 26 Wuhan Wuhan NNP cord-308115-bjyr6ehq 35 27 , , , cord-308115-bjyr6ehq 35 28 China China NNP cord-308115-bjyr6ehq 35 29 . . . cord-308115-bjyr6ehq 36 1 The the DT cord-308115-bjyr6ehq 36 2 model model NN cord-308115-bjyr6ehq 36 3 described describe VBD cord-308115-bjyr6ehq 36 4 the the DT cord-308115-bjyr6ehq 36 5 multiple multiple JJ cord-308115-bjyr6ehq 36 6 transmission transmission NN cord-308115-bjyr6ehq 36 7 pathways pathway NNS cord-308115-bjyr6ehq 36 8 in in IN cord-308115-bjyr6ehq 36 9 the the DT cord-308115-bjyr6ehq 36 10 infection infection NN cord-308115-bjyr6ehq 36 11 dynamics dynamic NNS cord-308115-bjyr6ehq 36 12 , , , cord-308115-bjyr6ehq 36 13 and and CC cord-308115-bjyr6ehq 36 14 emphasized emphasize VBD cord-308115-bjyr6ehq 36 15 the the DT cord-308115-bjyr6ehq 36 16 role role NN cord-308115-bjyr6ehq 36 17 of of IN cord-308115-bjyr6ehq 36 18 environmental environmental JJ cord-308115-bjyr6ehq 36 19 reservoir reservoir NN cord-308115-bjyr6ehq 36 20 in in IN cord-308115-bjyr6ehq 36 21 the the DT cord-308115-bjyr6ehq 36 22 transmission transmission NN cord-308115-bjyr6ehq 36 23 and and CC cord-308115-bjyr6ehq 36 24 spread spread NN cord-308115-bjyr6ehq 36 25 of of IN cord-308115-bjyr6ehq 36 26 the the DT cord-308115-bjyr6ehq 36 27 disease disease NN cord-308115-bjyr6ehq 36 28 . . . cord-308115-bjyr6ehq 37 1 However however RB cord-308115-bjyr6ehq 37 2 , , , cord-308115-bjyr6ehq 37 3 the the DT cord-308115-bjyr6ehq 37 4 model model NN cord-308115-bjyr6ehq 37 5 employed employ VBD cord-308115-bjyr6ehq 37 6 non non JJ cord-308115-bjyr6ehq 37 7 - - JJ cord-308115-bjyr6ehq 37 8 constant constant JJ cord-308115-bjyr6ehq 37 9 transmission transmission NN cord-308115-bjyr6ehq 37 10 rates rate NNS cord-308115-bjyr6ehq 37 11 which which WDT cord-308115-bjyr6ehq 37 12 change change VBP cord-308115-bjyr6ehq 37 13 with with IN cord-308115-bjyr6ehq 37 14 the the DT cord-308115-bjyr6ehq 37 15 epidemiological epidemiological JJ cord-308115-bjyr6ehq 37 16 status status NN cord-308115-bjyr6ehq 37 17 and and CC cord-308115-bjyr6ehq 37 18 environmental environmental JJ cord-308115-bjyr6ehq 37 19 conditions condition NNS cord-308115-bjyr6ehq 37 20 and and CC cord-308115-bjyr6ehq 37 21 which which WDT cord-308115-bjyr6ehq 37 22 reflect reflect VBP cord-308115-bjyr6ehq 37 23 the the DT cord-308115-bjyr6ehq 37 24 impact impact NN cord-308115-bjyr6ehq 37 25 of of IN cord-308115-bjyr6ehq 37 26 the the DT cord-308115-bjyr6ehq 37 27 ongoing ongoing JJ cord-308115-bjyr6ehq 37 28 disease disease NN cord-308115-bjyr6ehq 37 29 control control NN cord-308115-bjyr6ehq 37 30 measures measure NNS cord-308115-bjyr6ehq 37 31 . . . cord-308115-bjyr6ehq 38 1 Nonlocality Nonlocality NNP cord-308115-bjyr6ehq 38 2 is be VBZ cord-308115-bjyr6ehq 38 3 one one CD cord-308115-bjyr6ehq 38 4 of of IN cord-308115-bjyr6ehq 38 5 the the DT cord-308115-bjyr6ehq 38 6 main main JJ cord-308115-bjyr6ehq 38 7 drivers driver NNS cord-308115-bjyr6ehq 38 8 of of IN cord-308115-bjyr6ehq 38 9 interest interest NN cord-308115-bjyr6ehq 38 10 in in IN cord-308115-bjyr6ehq 38 11 fractional fractional JJ cord-308115-bjyr6ehq 38 12 calculus calculus NN cord-308115-bjyr6ehq 38 13 applications application NNS cord-308115-bjyr6ehq 38 14 . . . cord-308115-bjyr6ehq 39 1 There there EX cord-308115-bjyr6ehq 39 2 are be VBP cord-308115-bjyr6ehq 39 3 interesting interesting JJ cord-308115-bjyr6ehq 39 4 phenomena phenomenon NNS cord-308115-bjyr6ehq 39 5 that that WDT cord-308115-bjyr6ehq 39 6 have have VBP cord-308115-bjyr6ehq 39 7 what what WP cord-308115-bjyr6ehq 39 8 are be VBP cord-308115-bjyr6ehq 39 9 called call VBN cord-308115-bjyr6ehq 39 10 memory memory NN cord-308115-bjyr6ehq 39 11 effects effect NNS cord-308115-bjyr6ehq 39 12 , , , cord-308115-bjyr6ehq 39 13 meaning mean VBG cord-308115-bjyr6ehq 39 14 their -PRON- PRP$ cord-308115-bjyr6ehq 39 15 state state NN cord-308115-bjyr6ehq 39 16 does do VBZ cord-308115-bjyr6ehq 39 17 not not RB cord-308115-bjyr6ehq 39 18 depend depend VB cord-308115-bjyr6ehq 39 19 solely solely RB cord-308115-bjyr6ehq 39 20 on on IN cord-308115-bjyr6ehq 39 21 time time NN cord-308115-bjyr6ehq 39 22 and and CC cord-308115-bjyr6ehq 39 23 position position NN cord-308115-bjyr6ehq 39 24 but but CC cord-308115-bjyr6ehq 39 25 also also RB cord-308115-bjyr6ehq 39 26 on on IN cord-308115-bjyr6ehq 39 27 previous previous JJ cord-308115-bjyr6ehq 39 28 state state NN cord-308115-bjyr6ehq 39 29 . . . cord-308115-bjyr6ehq 40 1 Such such JJ cord-308115-bjyr6ehq 40 2 system system NN cord-308115-bjyr6ehq 40 3 can can MD cord-308115-bjyr6ehq 40 4 be be VB cord-308115-bjyr6ehq 40 5 very very RB cord-308115-bjyr6ehq 40 6 difficult difficult JJ cord-308115-bjyr6ehq 40 7 to to TO cord-308115-bjyr6ehq 40 8 model model VB cord-308115-bjyr6ehq 40 9 and and CC cord-308115-bjyr6ehq 40 10 analyze analyze VB cord-308115-bjyr6ehq 40 11 with with IN cord-308115-bjyr6ehq 40 12 classical classical JJ cord-308115-bjyr6ehq 40 13 differential differential JJ cord-308115-bjyr6ehq 40 14 equations equation NNS cord-308115-bjyr6ehq 40 15 , , , cord-308115-bjyr6ehq 40 16 but but CC cord-308115-bjyr6ehq 40 17 nonlocality nonlocality NN cord-308115-bjyr6ehq 40 18 gives give VBZ cord-308115-bjyr6ehq 40 19 fractional fractional JJ cord-308115-bjyr6ehq 40 20 derivative derivative JJ cord-308115-bjyr6ehq 40 21 built build VBN cord-308115-bjyr6ehq 40 22 - - HYPH cord-308115-bjyr6ehq 40 23 in in RP cord-308115-bjyr6ehq 40 24 ability ability NN cord-308115-bjyr6ehq 40 25 to to TO cord-308115-bjyr6ehq 40 26 incorporate incorporate VB cord-308115-bjyr6ehq 40 27 memory memory NN cord-308115-bjyr6ehq 40 28 effects effect NNS cord-308115-bjyr6ehq 40 29 [ [ -LRB- cord-308115-bjyr6ehq 40 30 11 11 CD cord-308115-bjyr6ehq 40 31 ] ] -RRB- cord-308115-bjyr6ehq 40 32 .Fractional .Fractional . cord-308115-bjyr6ehq 40 33 differential differential JJ cord-308115-bjyr6ehq 40 34 equations equation NNS cord-308115-bjyr6ehq 40 35 appear appear VBP cord-308115-bjyr6ehq 40 36 naturally naturally RB cord-308115-bjyr6ehq 40 37 in in IN cord-308115-bjyr6ehq 40 38 numerous numerous JJ cord-308115-bjyr6ehq 40 39 fields field NNS cord-308115-bjyr6ehq 40 40 of of IN cord-308115-bjyr6ehq 40 41 study study NN cord-308115-bjyr6ehq 40 42 including include VBG cord-308115-bjyr6ehq 40 43 physics physics NN cord-308115-bjyr6ehq 40 44 , , , cord-308115-bjyr6ehq 40 45 polymer polymer NN cord-308115-bjyr6ehq 40 46 rheology rheology NN cord-308115-bjyr6ehq 40 47 , , , cord-308115-bjyr6ehq 40 48 regular regular JJ cord-308115-bjyr6ehq 40 49 variation variation NN cord-308115-bjyr6ehq 40 50 in in IN cord-308115-bjyr6ehq 40 51 thermodynamics thermodynamic NNS cord-308115-bjyr6ehq 40 52 , , , cord-308115-bjyr6ehq 40 53 biophysics biophysic NNS cord-308115-bjyr6ehq 40 54 , , , cord-308115-bjyr6ehq 40 55 blood blood NN cord-308115-bjyr6ehq 40 56 flow flow NN cord-308115-bjyr6ehq 40 57 phenomena phenomenon NNS cord-308115-bjyr6ehq 40 58 , , , cord-308115-bjyr6ehq 40 59 aerodynamics aerodynamic NNS cord-308115-bjyr6ehq 40 60 , , , cord-308115-bjyr6ehq 40 61 electrodynamics electrodynamic NNS cord-308115-bjyr6ehq 40 62 of of IN cord-308115-bjyr6ehq 40 63 complex complex JJ cord-308115-bjyr6ehq 40 64 medium medium NN cord-308115-bjyr6ehq 40 65 , , , cord-308115-bjyr6ehq 40 66 viscoelasticity viscoelasticity NN cord-308115-bjyr6ehq 40 67 , , , cord-308115-bjyr6ehq 40 68 capacitor capacitor NN cord-308115-bjyr6ehq 40 69 theory theory NN cord-308115-bjyr6ehq 40 70 , , , cord-308115-bjyr6ehq 40 71 electrical electrical JJ cord-308115-bjyr6ehq 40 72 circuits circuit NNS cord-308115-bjyr6ehq 40 73 , , , cord-308115-bjyr6ehq 40 74 electron electron NN cord-308115-bjyr6ehq 40 75 - - HYPH cord-308115-bjyr6ehq 40 76 analytical analytical JJ cord-308115-bjyr6ehq 40 77 chemistry chemistry NN cord-308115-bjyr6ehq 40 78 , , , cord-308115-bjyr6ehq 40 79 biology biology NN cord-308115-bjyr6ehq 40 80 , , , cord-308115-bjyr6ehq 40 81 control control NN cord-308115-bjyr6ehq 40 82 theory theory NN cord-308115-bjyr6ehq 40 83 , , , cord-308115-bjyr6ehq 40 84 and and CC cord-308115-bjyr6ehq 40 85 fitting fitting NN cord-308115-bjyr6ehq 40 86 of of IN cord-308115-bjyr6ehq 40 87 experimental experimental JJ cord-308115-bjyr6ehq 40 88 data datum NNS cord-308115-bjyr6ehq 40 89 [ [ -LRB- cord-308115-bjyr6ehq 40 90 12 12 CD cord-308115-bjyr6ehq 40 91 -14 -14 NN cord-308115-bjyr6ehq 40 92 , , , cord-308115-bjyr6ehq 40 93 23 23 CD cord-308115-bjyr6ehq 40 94 -26 -26 CD cord-308115-bjyr6ehq 40 95 ] ] -RRB- cord-308115-bjyr6ehq 40 96 . . . cord-308115-bjyr6ehq 41 1 Recently recently RB cord-308115-bjyr6ehq 41 2 there there EX cord-308115-bjyr6ehq 41 3 are be VBP cord-308115-bjyr6ehq 41 4 many many JJ cord-308115-bjyr6ehq 41 5 studies study NNS cord-308115-bjyr6ehq 41 6 on on IN cord-308115-bjyr6ehq 41 7 epidemiological epidemiological JJ cord-308115-bjyr6ehq 41 8 disease disease NN cord-308115-bjyr6ehq 41 9 modeling modeling NN cord-308115-bjyr6ehq 41 10 using use VBG cord-308115-bjyr6ehq 41 11 fractional fractional JJ cord-308115-bjyr6ehq 41 12 order order NN cord-308115-bjyr6ehq 41 13 differential differential NN cord-308115-bjyr6ehq 41 14 equations equation NNS cord-308115-bjyr6ehq 41 15 [ [ -LRB- cord-308115-bjyr6ehq 41 16 27 27 CD cord-308115-bjyr6ehq 41 17 -33 -33 NNS cord-308115-bjyr6ehq 41 18 ] ] -RRB- cord-308115-bjyr6ehq 41 19 . . . cord-308115-bjyr6ehq 42 1 The the DT cord-308115-bjyr6ehq 42 2 Rieman Rieman NNP cord-308115-bjyr6ehq 42 3 - - HYPH cord-308115-bjyr6ehq 42 4 Liouville Liouville NNP cord-308115-bjyr6ehq 42 5 fractional fractional JJ cord-308115-bjyr6ehq 42 6 derivative derivative NN cord-308115-bjyr6ehq 42 7 is be VBZ cord-308115-bjyr6ehq 42 8 mostly mostly RB cord-308115-bjyr6ehq 42 9 used use VBN cord-308115-bjyr6ehq 42 10 by by IN cord-308115-bjyr6ehq 42 11 mathematicians mathematician NNS cord-308115-bjyr6ehq 42 12 , , , cord-308115-bjyr6ehq 42 13 but but CC cord-308115-bjyr6ehq 42 14 this this DT cord-308115-bjyr6ehq 42 15 approach approach NN cord-308115-bjyr6ehq 42 16 is be VBZ cord-308115-bjyr6ehq 42 17 not not RB cord-308115-bjyr6ehq 42 18 suitable suitable JJ cord-308115-bjyr6ehq 42 19 for for IN cord-308115-bjyr6ehq 42 20 real real JJ cord-308115-bjyr6ehq 42 21 world world NN cord-308115-bjyr6ehq 42 22 physical physical JJ cord-308115-bjyr6ehq 42 23 problems problem NNS cord-308115-bjyr6ehq 42 24 since since IN cord-308115-bjyr6ehq 42 25 it -PRON- PRP cord-308115-bjyr6ehq 42 26 requires require VBZ cord-308115-bjyr6ehq 42 27 the the DT cord-308115-bjyr6ehq 42 28 definition definition NN cord-308115-bjyr6ehq 42 29 of of IN cord-308115-bjyr6ehq 42 30 fractional fractional JJ cord-308115-bjyr6ehq 42 31 order order NN cord-308115-bjyr6ehq 42 32 initial initial JJ cord-308115-bjyr6ehq 42 33 conditions condition NNS cord-308115-bjyr6ehq 42 34 , , , cord-308115-bjyr6ehq 42 35 which which WDT cord-308115-bjyr6ehq 42 36 have have VBP cord-308115-bjyr6ehq 42 37 no no DT cord-308115-bjyr6ehq 42 38 physically physically RB cord-308115-bjyr6ehq 42 39 meaningful meaningful JJ cord-308115-bjyr6ehq 42 40 explanation explanation NN cord-308115-bjyr6ehq 42 41 yet yet RB cord-308115-bjyr6ehq 42 42 . . . cord-308115-bjyr6ehq 43 1 Caputo Caputo NNP cord-308115-bjyr6ehq 43 2 introduced introduce VBD cord-308115-bjyr6ehq 43 3 an an DT cord-308115-bjyr6ehq 43 4 alternative alternative JJ cord-308115-bjyr6ehq 43 5 definition definition NN cord-308115-bjyr6ehq 43 6 , , , cord-308115-bjyr6ehq 43 7 which which WDT cord-308115-bjyr6ehq 43 8 has have VBZ cord-308115-bjyr6ehq 43 9 the the DT cord-308115-bjyr6ehq 43 10 advantage advantage NN cord-308115-bjyr6ehq 43 11 of of IN cord-308115-bjyr6ehq 43 12 defining define VBG cord-308115-bjyr6ehq 43 13 integer integ JJR cord-308115-bjyr6ehq 43 14 order order NN cord-308115-bjyr6ehq 43 15 initial initial JJ cord-308115-bjyr6ehq 43 16 condition condition NN cord-308115-bjyr6ehq 43 17 for for IN cord-308115-bjyr6ehq 43 18 fractional fractional JJ cord-308115-bjyr6ehq 43 19 differential differential JJ cord-308115-bjyr6ehq 43 20 equations equation NNS cord-308115-bjyr6ehq 43 21 [ [ -LRB- cord-308115-bjyr6ehq 43 22 15 15 CD cord-308115-bjyr6ehq 43 23 ] ] -RRB- cord-308115-bjyr6ehq 43 24 . . . cord-308115-bjyr6ehq 44 1 In in IN cord-308115-bjyr6ehq 44 2 mostly mostly RB cord-308115-bjyr6ehq 44 3 poor poor JJ cord-308115-bjyr6ehq 44 4 and and CC cord-308115-bjyr6ehq 44 5 underdeveloped underdeveloped JJ cord-308115-bjyr6ehq 44 6 territories territory NNS cord-308115-bjyr6ehq 44 7 where where WRB cord-308115-bjyr6ehq 44 8 there there EX cord-308115-bjyr6ehq 44 9 is be VBZ cord-308115-bjyr6ehq 44 10 no no DT cord-308115-bjyr6ehq 44 11 capacity capacity NN cord-308115-bjyr6ehq 44 12 of of IN cord-308115-bjyr6ehq 44 13 rapid rapid JJ cord-308115-bjyr6ehq 44 14 diagnostic diagnostic JJ cord-308115-bjyr6ehq 44 15 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 44 16 test test NN cord-308115-bjyr6ehq 44 17 due due IN cord-308115-bjyr6ehq 44 18 to to IN cord-308115-bjyr6ehq 44 19 insufficiency insufficiency NN cord-308115-bjyr6ehq 44 20 of of IN cord-308115-bjyr6ehq 44 21 testing testing NN cord-308115-bjyr6ehq 44 22 kits kit NNS cord-308115-bjyr6ehq 44 23 , , , cord-308115-bjyr6ehq 44 24 the the DT cord-308115-bjyr6ehq 44 25 Mild mild JJ cord-308115-bjyr6ehq 44 26 cases case NNS cord-308115-bjyr6ehq 44 27 ( ( -LRB- cord-308115-bjyr6ehq 44 28 who who WP cord-308115-bjyr6ehq 44 29 usually usually RB cord-308115-bjyr6ehq 44 30 show show VBP cord-308115-bjyr6ehq 44 31 no no DT cord-308115-bjyr6ehq 44 32 symptoms symptom NNS cord-308115-bjyr6ehq 44 33 of of IN cord-308115-bjyr6ehq 44 34 the the DT cord-308115-bjyr6ehq 44 35 infection infection NN cord-308115-bjyr6ehq 44 36 due due IN cord-308115-bjyr6ehq 44 37 to to IN cord-308115-bjyr6ehq 44 38 their -PRON- PRP$ cord-308115-bjyr6ehq 44 39 strong strong JJ cord-308115-bjyr6ehq 44 40 and and CC cord-308115-bjyr6ehq 44 41 active active JJ cord-308115-bjyr6ehq 44 42 immune immune JJ cord-308115-bjyr6ehq 44 43 system system NN cord-308115-bjyr6ehq 44 44 up up IN cord-308115-bjyr6ehq 44 45 to to IN cord-308115-bjyr6ehq 44 46 their -PRON- PRP$ cord-308115-bjyr6ehq 44 47 recovery recovery NN cord-308115-bjyr6ehq 44 48 period period NN cord-308115-bjyr6ehq 44 49 ) ) -RRB- cord-308115-bjyr6ehq 44 50 play play VBP cord-308115-bjyr6ehq 44 51 a a DT cord-308115-bjyr6ehq 44 52 major major JJ cord-308115-bjyr6ehq 44 53 role role NN cord-308115-bjyr6ehq 44 54 as as IN cord-308115-bjyr6ehq 44 55 a a DT cord-308115-bjyr6ehq 44 56 route route NN cord-308115-bjyr6ehq 44 57 that that WDT cord-308115-bjyr6ehq 44 58 spread spread VBD cord-308115-bjyr6ehq 44 59 the the DT cord-308115-bjyr6ehq 44 60 disease disease NN cord-308115-bjyr6ehq 44 61 to to IN cord-308115-bjyr6ehq 44 62 healthy healthy JJ cord-308115-bjyr6ehq 44 63 individuals individual NNS cord-308115-bjyr6ehq 44 64 . . . cord-308115-bjyr6ehq 45 1 Here here RB cord-308115-bjyr6ehq 45 2 we -PRON- PRP cord-308115-bjyr6ehq 45 3 build build VBP cord-308115-bjyr6ehq 45 4 our -PRON- PRP$ cord-308115-bjyr6ehq 45 5 model model NN cord-308115-bjyr6ehq 45 6 by by IN cord-308115-bjyr6ehq 45 7 incorporating incorporate VBG cord-308115-bjyr6ehq 45 8 the the DT cord-308115-bjyr6ehq 45 9 population population NN cord-308115-bjyr6ehq 45 10 of of IN cord-308115-bjyr6ehq 45 11 mild mild JJ cord-308115-bjyr6ehq 45 12 individuals individual NNS cord-308115-bjyr6ehq 45 13 into into IN cord-308115-bjyr6ehq 45 14 the the DT cord-308115-bjyr6ehq 45 15 compartmental compartmental JJ cord-308115-bjyr6ehq 45 16 SIR SIR NNP cord-308115-bjyr6ehq 45 17 model model NN cord-308115-bjyr6ehq 45 18 to to TO cord-308115-bjyr6ehq 45 19 become become VB cord-308115-bjyr6ehq 45 20 SMIR SMIR NNP cord-308115-bjyr6ehq 45 21 model model NN cord-308115-bjyr6ehq 45 22 in in IN cord-308115-bjyr6ehq 45 23 form form NN cord-308115-bjyr6ehq 45 24 of of IN cord-308115-bjyr6ehq 45 25 system system NN cord-308115-bjyr6ehq 45 26 of of IN cord-308115-bjyr6ehq 45 27 fractional fractional JJ cord-308115-bjyr6ehq 45 28 order order NN cord-308115-bjyr6ehq 45 29 differential differential NN cord-308115-bjyr6ehq 45 30 equations equation NNS cord-308115-bjyr6ehq 45 31 ( ( -LRB- cord-308115-bjyr6ehq 45 32 FODE FODE NNP cord-308115-bjyr6ehq 45 33 ) ) -RRB- cord-308115-bjyr6ehq 45 34 in in IN cord-308115-bjyr6ehq 45 35 the the DT cord-308115-bjyr6ehq 45 36 Caputo Caputo NNP cord-308115-bjyr6ehq 45 37 sense sense NN cord-308115-bjyr6ehq 45 38 . . . cord-308115-bjyr6ehq 46 1 It -PRON- PRP cord-308115-bjyr6ehq 46 2 should should MD cord-308115-bjyr6ehq 46 3 be be VB cord-308115-bjyr6ehq 46 4 noted note VBN cord-308115-bjyr6ehq 46 5 that that IN cord-308115-bjyr6ehq 46 6 to to IN cord-308115-bjyr6ehq 46 7 our -PRON- PRP$ cord-308115-bjyr6ehq 46 8 knowledge knowledge NN cord-308115-bjyr6ehq 46 9 no no DT cord-308115-bjyr6ehq 46 10 model model NN cord-308115-bjyr6ehq 46 11 in in IN cord-308115-bjyr6ehq 46 12 literature literature NN cord-308115-bjyr6ehq 46 13 considered consider VBD cord-308115-bjyr6ehq 46 14 the the DT cord-308115-bjyr6ehq 46 15 contribution contribution NN cord-308115-bjyr6ehq 46 16 of of IN cord-308115-bjyr6ehq 46 17 the the DT cord-308115-bjyr6ehq 46 18 mild mild JJ cord-308115-bjyr6ehq 46 19 cases case NNS cord-308115-bjyr6ehq 46 20 of of IN cord-308115-bjyr6ehq 46 21 COVID COVID NNP cord-308115-bjyr6ehq 46 22 -19 -19 NN cord-308115-bjyr6ehq 46 23 in in IN cord-308115-bjyr6ehq 46 24 the the DT cord-308115-bjyr6ehq 46 25 proliferation proliferation NN cord-308115-bjyr6ehq 46 26 of of IN cord-308115-bjyr6ehq 46 27 the the DT cord-308115-bjyr6ehq 46 28 pandemic pandemic NN cord-308115-bjyr6ehq 46 29 . . . cord-308115-bjyr6ehq 47 1 The the DT cord-308115-bjyr6ehq 47 2 paper paper NN cord-308115-bjyr6ehq 47 3 is be VBZ cord-308115-bjyr6ehq 47 4 organized organize VBN cord-308115-bjyr6ehq 47 5 as as IN cord-308115-bjyr6ehq 47 6 follows follow VBZ cord-308115-bjyr6ehq 47 7 : : : cord-308115-bjyr6ehq 47 8 section section NN cord-308115-bjyr6ehq 47 9 1is 1is NN cord-308115-bjyr6ehq 47 10 the the DT cord-308115-bjyr6ehq 47 11 introduction introduction NN cord-308115-bjyr6ehq 47 12 , , , cord-308115-bjyr6ehq 47 13 section section NN cord-308115-bjyr6ehq 47 14 2 2 CD cord-308115-bjyr6ehq 47 15 is be VBZ cord-308115-bjyr6ehq 47 16 the the DT cord-308115-bjyr6ehq 47 17 preliminary preliminary JJ cord-308115-bjyr6ehq 47 18 definitions definition NNS cord-308115-bjyr6ehq 47 19 and and CC cord-308115-bjyr6ehq 47 20 theorems theorem NNS cord-308115-bjyr6ehq 47 21 , , , cord-308115-bjyr6ehq 47 22 the the DT cord-308115-bjyr6ehq 47 23 model model NN cord-308115-bjyr6ehq 47 24 formulation formulation NN cord-308115-bjyr6ehq 47 25 followed follow VBD cord-308115-bjyr6ehq 47 26 in in IN cord-308115-bjyr6ehq 47 27 section section NN cord-308115-bjyr6ehq 47 28 3 3 CD cord-308115-bjyr6ehq 47 29 , , , cord-308115-bjyr6ehq 47 30 section section NN cord-308115-bjyr6ehq 47 31 4 4 CD cord-308115-bjyr6ehq 47 32 is be VBZ cord-308115-bjyr6ehq 47 33 the the DT cord-308115-bjyr6ehq 47 34 stability stability NN cord-308115-bjyr6ehq 47 35 analysis analysis NN cord-308115-bjyr6ehq 47 36 , , , cord-308115-bjyr6ehq 47 37 section section NN cord-308115-bjyr6ehq 47 38 5 5 CD cord-308115-bjyr6ehq 47 39 gives give VBZ cord-308115-bjyr6ehq 47 40 numerical numerical JJ cord-308115-bjyr6ehq 47 41 simulations simulation NNS cord-308115-bjyr6ehq 47 42 and and CC cord-308115-bjyr6ehq 47 43 discussions discussion NNS cord-308115-bjyr6ehq 47 44 and and CC cord-308115-bjyr6ehq 47 45 section section NN cord-308115-bjyr6ehq 47 46 6 6 CD cord-308115-bjyr6ehq 47 47 gives give VBZ cord-308115-bjyr6ehq 47 48 the the DT cord-308115-bjyr6ehq 47 49 conclusion conclusion NN cord-308115-bjyr6ehq 47 50 . . . cord-308115-bjyr6ehq 48 1 The the DT cord-308115-bjyr6ehq 48 2 notion notion NN cord-308115-bjyr6ehq 48 3 of of IN cord-308115-bjyr6ehq 48 4 convergence convergence NN cord-308115-bjyr6ehq 48 5 of of IN cord-308115-bjyr6ehq 48 6 mittag mittag NNP cord-308115-bjyr6ehq 48 7 - - HYPH cord-308115-bjyr6ehq 48 8 Leffler Leffler NNP cord-308115-bjyr6ehq 48 9 function function NN cord-308115-bjyr6ehq 48 10 is be VBZ cord-308115-bjyr6ehq 48 11 fully fully RB cord-308115-bjyr6ehq 48 12 discussed discuss VBN cord-308115-bjyr6ehq 48 13 in in IN cord-308115-bjyr6ehq 48 14 [ [ -LRB- cord-308115-bjyr6ehq 48 15 15 15 CD cord-308115-bjyr6ehq 48 16 ] ] -RRB- cord-308115-bjyr6ehq 48 17 . . . cord-308115-bjyr6ehq 49 1 Theorem theorem NN cord-308115-bjyr6ehq 49 2 1 1 CD cord-308115-bjyr6ehq 49 3 [ [ -LRB- cord-308115-bjyr6ehq 49 4 19 19 CD cord-308115-bjyr6ehq 49 5 ] ] -RRB- cord-308115-bjyr6ehq 49 6 : : : cord-308115-bjyr6ehq 49 7 The the DT cord-308115-bjyr6ehq 49 8 equilibrium equilibrium NN cord-308115-bjyr6ehq 49 9 solutions solution NNS cord-308115-bjyr6ehq 49 10 of of IN cord-308115-bjyr6ehq 49 11 the the DT cord-308115-bjyr6ehq 49 12 system system NN cord-308115-bjyr6ehq 49 13 is be VBZ cord-308115-bjyr6ehq 49 14 locally locally RB cord-308115-bjyr6ehq 49 15 asymptotically asymptotically RB cord-308115-bjyr6ehq 49 16 stable stable JJ cord-308115-bjyr6ehq 49 17 if if IN cord-308115-bjyr6ehq 49 18 all all PDT cord-308115-bjyr6ehq 49 19 the the DT cord-308115-bjyr6ehq 49 20 eigenvalues eigenvalue NNS cord-308115-bjyr6ehq 49 21 of of IN cord-308115-bjyr6ehq 49 22 the the DT cord-308115-bjyr6ehq 49 23 Jacobian jacobian JJ cord-308115-bjyr6ehq 49 24 matrix matrix NN cord-308115-bjyr6ehq 49 25 evaluated evaluate VBN cord-308115-bjyr6ehq 49 26 at at IN cord-308115-bjyr6ehq 49 27 the the DT cord-308115-bjyr6ehq 49 28 equilibrium equilibrium NN cord-308115-bjyr6ehq 49 29 points point NNS cord-308115-bjyr6ehq 49 30 satisfy satisfy VBP cord-308115-bjyr6ehq 49 31 _SP cord-308115-bjyr6ehq 49 32 Despite despite IN cord-308115-bjyr6ehq 49 33 the the DT cord-308115-bjyr6ehq 49 34 fact fact NN cord-308115-bjyr6ehq 49 35 that that IN cord-308115-bjyr6ehq 49 36 almost almost RB cord-308115-bjyr6ehq 49 37 80 80 CD cord-308115-bjyr6ehq 49 38 % % NN cord-308115-bjyr6ehq 49 39 of of IN cord-308115-bjyr6ehq 49 40 the the DT cord-308115-bjyr6ehq 49 41 COVID COVID NNP cord-308115-bjyr6ehq 49 42 -19cases -19case NNS cord-308115-bjyr6ehq 49 43 are be VBP cord-308115-bjyr6ehq 49 44 mild mild JJ cord-308115-bjyr6ehq 49 45 who who WP cord-308115-bjyr6ehq 49 46 recover recover VBP cord-308115-bjyr6ehq 49 47 naturally naturally RB cord-308115-bjyr6ehq 49 48 ( ( -LRB- cord-308115-bjyr6ehq 49 49 due due IN cord-308115-bjyr6ehq 49 50 to to IN cord-308115-bjyr6ehq 49 51 stronger strong JJR cord-308115-bjyr6ehq 49 52 and and CC cord-308115-bjyr6ehq 49 53 active active JJ cord-308115-bjyr6ehq 49 54 immune immune JJ cord-308115-bjyr6ehq 49 55 system system NN cord-308115-bjyr6ehq 49 56 that that WDT cord-308115-bjyr6ehq 49 57 fight fight VBP cord-308115-bjyr6ehq 49 58 against against IN cord-308115-bjyr6ehq 49 59 the the DT cord-308115-bjyr6ehq 49 60 virus virus NN cord-308115-bjyr6ehq 49 61 ) ) -RRB- cord-308115-bjyr6ehq 49 62 , , , cord-308115-bjyr6ehq 49 63 they -PRON- PRP cord-308115-bjyr6ehq 49 64 still still RB cord-308115-bjyr6ehq 49 65 play play VBP cord-308115-bjyr6ehq 49 66 a a DT cord-308115-bjyr6ehq 49 67 role role NN cord-308115-bjyr6ehq 49 68 as as IN cord-308115-bjyr6ehq 49 69 a a DT cord-308115-bjyr6ehq 49 70 route route NN cord-308115-bjyr6ehq 49 71 of of IN cord-308115-bjyr6ehq 49 72 transmission transmission NN cord-308115-bjyr6ehq 49 73 of of IN cord-308115-bjyr6ehq 49 74 the the DT cord-308115-bjyr6ehq 49 75 infection infection NN cord-308115-bjyr6ehq 49 76 [ [ -LRB- cord-308115-bjyr6ehq 49 77 2 2 CD cord-308115-bjyr6ehq 49 78 ] ] -RRB- cord-308115-bjyr6ehq 49 79 . . . cord-308115-bjyr6ehq 50 1 The the DT cord-308115-bjyr6ehq 50 2 model model NN cord-308115-bjyr6ehq 50 3 was be VBD cord-308115-bjyr6ehq 50 4 formulated formulate VBN cord-308115-bjyr6ehq 50 5 based base VBN cord-308115-bjyr6ehq 50 6 on on IN cord-308115-bjyr6ehq 50 7 the the DT cord-308115-bjyr6ehq 50 8 assumption assumption NN cord-308115-bjyr6ehq 50 9 that that IN cord-308115-bjyr6ehq 50 10 new new JJ cord-308115-bjyr6ehq 50 11 born bear VBN cord-308115-bjyr6ehq 50 12 of of IN cord-308115-bjyr6ehq 50 13 human human NN cord-308115-bjyr6ehq 50 14 are be VBP cord-308115-bjyr6ehq 50 15 recruited recruit VBN cord-308115-bjyr6ehq 50 16 into into IN cord-308115-bjyr6ehq 50 17 susceptible susceptible JJ cord-308115-bjyr6ehq 50 18 class class NN cord-308115-bjyr6ehq 50 19 ( ( -LRB- cord-308115-bjyr6ehq 50 20 ) ) -RRB- cord-308115-bjyr6ehq 50 21 at at IN cord-308115-bjyr6ehq 50 22 the the DT cord-308115-bjyr6ehq 50 23 rate rate NN cord-308115-bjyr6ehq 50 24 . . . cord-308115-bjyr6ehq 51 1 The the DT cord-308115-bjyr6ehq 51 2 susceptible susceptible JJ cord-308115-bjyr6ehq 51 3 individual individual NN cord-308115-bjyr6ehq 51 4 who who WP cord-308115-bjyr6ehq 51 5 had have VBD cord-308115-bjyr6ehq 51 6 contact contact NN cord-308115-bjyr6ehq 51 7 with with IN cord-308115-bjyr6ehq 51 8 an an DT cord-308115-bjyr6ehq 51 9 infected infected JJ cord-308115-bjyr6ehq 51 10 at at IN cord-308115-bjyr6ehq 51 11 rate rate NN cord-308115-bjyr6ehq 51 12 can can MD cord-308115-bjyr6ehq 51 13 developed develop VBN cord-308115-bjyr6ehq 51 14 mild mild JJ cord-308115-bjyr6ehq 51 15 symptoms symptom NNS cord-308115-bjyr6ehq 51 16 and and CC cord-308115-bjyr6ehq 51 17 move move VBP cord-308115-bjyr6ehq 51 18 to to IN cord-308115-bjyr6ehq 51 19 mild mild JJ cord-308115-bjyr6ehq 51 20 class class NN cord-308115-bjyr6ehq 51 21 ( ( -LRB- cord-308115-bjyr6ehq 51 22 ) ) -RRB- cord-308115-bjyr6ehq 51 23 . . . cord-308115-bjyr6ehq 52 1 Based base VBN cord-308115-bjyr6ehq 52 2 on on IN cord-308115-bjyr6ehq 52 3 [ [ -LRB- cord-308115-bjyr6ehq 52 4 2 2 CD cord-308115-bjyr6ehq 52 5 ] ] -RRB- cord-308115-bjyr6ehq 52 6 , , , cord-308115-bjyr6ehq 52 7 the the DT cord-308115-bjyr6ehq 52 8 mild mild JJ cord-308115-bjyr6ehq 52 9 patients patient NNS cord-308115-bjyr6ehq 52 10 with with IN cord-308115-bjyr6ehq 52 11 infectivity infectivity NN cord-308115-bjyr6ehq 52 12 rate rate NN cord-308115-bjyr6ehq 52 13 play play VB cord-308115-bjyr6ehq 52 14 the the DT cord-308115-bjyr6ehq 52 15 role role NN cord-308115-bjyr6ehq 52 16 as as IN cord-308115-bjyr6ehq 52 17 a a DT cord-308115-bjyr6ehq 52 18 routine routine NN cord-308115-bjyr6ehq 52 19 that that WDT cord-308115-bjyr6ehq 52 20 spread spread VBD cord-308115-bjyr6ehq 52 21 the the DT cord-308115-bjyr6ehq 52 22 infection infection NN cord-308115-bjyr6ehq 52 23 , , , cord-308115-bjyr6ehq 52 24 those those DT cord-308115-bjyr6ehq 52 25 with with IN cord-308115-bjyr6ehq 52 26 strong strong JJ cord-308115-bjyr6ehq 52 27 immunity immunity NN cord-308115-bjyr6ehq 52 28 recovered recover VBD cord-308115-bjyr6ehq 52 29 naturally naturally RB cord-308115-bjyr6ehq 52 30 at at IN cord-308115-bjyr6ehq 52 31 the the DT cord-308115-bjyr6ehq 52 32 rate rate NN cord-308115-bjyr6ehq 52 33 , , , cord-308115-bjyr6ehq 52 34 While while IN cord-308115-bjyr6ehq 52 35 some some DT cord-308115-bjyr6ehq 52 36 with with IN cord-308115-bjyr6ehq 52 37 critical critical JJ cord-308115-bjyr6ehq 52 38 illness illness NN cord-308115-bjyr6ehq 52 39 became become VBD cord-308115-bjyr6ehq 52 40 infected infected JJ cord-308115-bjyr6ehq 52 41 and and CC cord-308115-bjyr6ehq 52 42 moved move VBN cord-308115-bjyr6ehq 52 43 to to IN cord-308115-bjyr6ehq 52 44 infectious infectious JJ cord-308115-bjyr6ehq 52 45 class class NN cord-308115-bjyr6ehq 52 46 ( ( -LRB- cord-308115-bjyr6ehq 52 47 ) ) -RRB- cord-308115-bjyr6ehq 52 48 after after IN cord-308115-bjyr6ehq 52 49 the the DT cord-308115-bjyr6ehq 52 50 incubation incubation NN cord-308115-bjyr6ehq 52 51 period period NN cord-308115-bjyr6ehq 52 52 . . . cord-308115-bjyr6ehq 53 1 The the DT cord-308115-bjyr6ehq 53 2 infectious infectious JJ cord-308115-bjyr6ehq 53 3 individual individual NN cord-308115-bjyr6ehq 53 4 may may MD cord-308115-bjyr6ehq 53 5 then then RB cord-308115-bjyr6ehq 53 6 recovered recover VBN cord-308115-bjyr6ehq 53 7 ( ( -LRB- cord-308115-bjyr6ehq 53 8 ) ) -RRB- cord-308115-bjyr6ehq 53 9 or or CC cord-308115-bjyr6ehq 53 10 died die VBD cord-308115-bjyr6ehq 53 11 at at IN cord-308115-bjyr6ehq 53 12 the the DT cord-308115-bjyr6ehq 53 13 rates rate NNS cord-308115-bjyr6ehq 53 14 and and CC cord-308115-bjyr6ehq 53 15 respectively respectively RB cord-308115-bjyr6ehq 53 16 . . . cord-308115-bjyr6ehq 54 1 Figure figure NN cord-308115-bjyr6ehq 54 2 1 1 CD cord-308115-bjyr6ehq 54 3 gives give VBZ cord-308115-bjyr6ehq 54 4 the the DT cord-308115-bjyr6ehq 54 5 schematic schematic JJ cord-308115-bjyr6ehq 54 6 diagram diagram NN cord-308115-bjyr6ehq 54 7 describing describe VBG cord-308115-bjyr6ehq 54 8 the the DT cord-308115-bjyr6ehq 54 9 transmission transmission NN cord-308115-bjyr6ehq 54 10 dynamics dynamic NNS cord-308115-bjyr6ehq 54 11 of of IN cord-308115-bjyr6ehq 54 12 the the DT cord-308115-bjyr6ehq 54 13 disease disease NN cord-308115-bjyr6ehq 54 14 . . . cord-308115-bjyr6ehq 54 15 _SP cord-308115-bjyr6ehq 55 1 With with IN cord-308115-bjyr6ehq 55 2 the the DT cord-308115-bjyr6ehq 55 3 total total JJ cord-308115-bjyr6ehq 55 4 population population NN cord-308115-bjyr6ehq 55 5 The the DT cord-308115-bjyr6ehq 55 6 linearity linearity NN cord-308115-bjyr6ehq 55 7 of of IN cord-308115-bjyr6ehq 55 8 the the DT cord-308115-bjyr6ehq 55 9 Caputo Caputo NNP cord-308115-bjyr6ehq 55 10 operator operator NN cord-308115-bjyr6ehq 55 11 yield yield NN cord-308115-bjyr6ehq 55 12 We -PRON- PRP cord-308115-bjyr6ehq 55 13 apply apply VBP cord-308115-bjyr6ehq 55 14 the the DT cord-308115-bjyr6ehq 55 15 Laplace Laplace NNP cord-308115-bjyr6ehq 56 1 transform transform VB cord-308115-bjyr6ehq 56 2 method method NN cord-308115-bjyr6ehq 56 3 to to TO cord-308115-bjyr6ehq 56 4 solve solve VB cord-308115-bjyr6ehq 56 5 the the DT cord-308115-bjyr6ehq 56 6 Gronwall Gronwall NNP cord-308115-bjyr6ehq 56 7 's 's POS cord-308115-bjyr6ehq 56 8 inequality inequality NN cord-308115-bjyr6ehq 56 9 ( ( -LRB- cord-308115-bjyr6ehq 56 10 6 6 CD cord-308115-bjyr6ehq 56 11 ) ) -RRB- cord-308115-bjyr6ehq 56 12 with with IN cord-308115-bjyr6ehq 56 13 initial initial JJ cord-308115-bjyr6ehq 56 14 condition condition NN cord-308115-bjyr6ehq 56 15 ( ( -LRB- cord-308115-bjyr6ehq 56 16 ) ) -RRB- cord-308115-bjyr6ehq 56 17 Linearity linearity NN cord-308115-bjyr6ehq 56 18 property property NN cord-308115-bjyr6ehq 56 19 of of IN cord-308115-bjyr6ehq 56 20 the the DT cord-308115-bjyr6ehq 56 21 Laplace Laplace NNP cord-308115-bjyr6ehq 56 22 transform transform VB cord-308115-bjyr6ehq 56 23 gives give VBZ cord-308115-bjyr6ehq 56 24 ( ( -LRB- cord-308115-bjyr6ehq 56 25 7 7 CD cord-308115-bjyr6ehq 56 26 ) ) -RRB- cord-308115-bjyr6ehq 56 27 to to IN cord-308115-bjyr6ehq 56 28 partial partial JJ cord-308115-bjyr6ehq 56 29 fraction fraction NN cord-308115-bjyr6ehq 56 30 gives give VBZ cord-308115-bjyr6ehq 56 31 Taking take VBG cord-308115-bjyr6ehq 56 32 the the DT cord-308115-bjyr6ehq 56 33 inverse inverse JJ cord-308115-bjyr6ehq 56 34 Laplace Laplace NNP cord-308115-bjyr6ehq 56 35 transform transform NN cord-308115-bjyr6ehq 56 36 of of IN cord-308115-bjyr6ehq 56 37 ( ( -LRB- cord-308115-bjyr6ehq 56 38 8) 8) NN cord-308115-bjyr6ehq 56 39 Where where WRB cord-308115-bjyr6ehq 56 40 ( ( -LRB- cord-308115-bjyr6ehq 56 41 ) ) -RRB- cord-308115-bjyr6ehq 56 42 , , , cord-308115-bjyr6ehq 56 43 ( ( -LRB- cord-308115-bjyr6ehq 56 44 ) ) -RRB- cord-308115-bjyr6ehq 56 45 are be VBP cord-308115-bjyr6ehq 56 46 the the DT cord-308115-bjyr6ehq 56 47 series series NN cord-308115-bjyr6ehq 56 48 of of IN cord-308115-bjyr6ehq 56 49 Mittag Mittag NNP cord-308115-bjyr6ehq 56 50 - - HYPH cord-308115-bjyr6ehq 56 51 Leffler Leffler NNP cord-308115-bjyr6ehq 56 52 function function NN cord-308115-bjyr6ehq 56 53 ( ( -LRB- cord-308115-bjyr6ehq 56 54 as as IN cord-308115-bjyr6ehq 56 55 in in IN cord-308115-bjyr6ehq 56 56 definition definition NN cord-308115-bjyr6ehq 56 57 4 4 CD cord-308115-bjyr6ehq 56 58 ) ) -RRB- cord-308115-bjyr6ehq 56 59 which which WDT cord-308115-bjyr6ehq 56 60 converges converge VBZ cord-308115-bjyr6ehq 56 61 for for IN cord-308115-bjyr6ehq 56 62 any any DT cord-308115-bjyr6ehq 56 63 argument argument NN cord-308115-bjyr6ehq 57 1 , , , cord-308115-bjyr6ehq 57 2 hence hence RB cord-308115-bjyr6ehq 57 3 we -PRON- PRP cord-308115-bjyr6ehq 57 4 say say VBP cord-308115-bjyr6ehq 57 5 that that IN cord-308115-bjyr6ehq 57 6 the the DT cord-308115-bjyr6ehq 57 7 solution solution NN cord-308115-bjyr6ehq 57 8 to to IN cord-308115-bjyr6ehq 57 9 the the DT cord-308115-bjyr6ehq 57 10 model model NN cord-308115-bjyr6ehq 57 11 is be VBZ cord-308115-bjyr6ehq 57 12 bounded bound VBN cord-308115-bjyr6ehq 57 13 . . . cord-308115-bjyr6ehq 58 1 Thus thus RB cord-308115-bjyr6ehq 58 2 , , , cord-308115-bjyr6ehq 58 3 Consider consider VB cord-308115-bjyr6ehq 58 4 the the DT cord-308115-bjyr6ehq 58 5 system system NN cord-308115-bjyr6ehq 58 6 ( ( -LRB- cord-308115-bjyr6ehq 58 7 1 1 CD cord-308115-bjyr6ehq 58 8 ) ) -RRB- cord-308115-bjyr6ehq 58 9 through through IN cord-308115-bjyr6ehq 58 10 ( ( -LRB- cord-308115-bjyr6ehq 58 11 4 4 CD cord-308115-bjyr6ehq 58 12 ) ) -RRB- cord-308115-bjyr6ehq 58 13 written write VBN cord-308115-bjyr6ehq 58 14 as as IN cord-308115-bjyr6ehq 58 15 ] ] -RRB- cord-308115-bjyr6ehq 59 1 Proof Proof NNP cord-308115-bjyr6ehq 59 2 Reference Reference NNP cord-308115-bjyr6ehq 59 3 to to IN cord-308115-bjyr6ehq 59 4 Picard Picard NNP cord-308115-bjyr6ehq 59 5 - - HYPH cord-308115-bjyr6ehq 59 6 Lindelof Lindelof NNP cord-308115-bjyr6ehq 59 7 theorem theorem NN cord-308115-bjyr6ehq 60 1 [ [ -LRB- cord-308115-bjyr6ehq 60 2 20 20 CD cord-308115-bjyr6ehq 60 3 ] ] -RRB- cord-308115-bjyr6ehq 60 4 we -PRON- PRP cord-308115-bjyr6ehq 60 5 establish establish VBP cord-308115-bjyr6ehq 60 6 the the DT cord-308115-bjyr6ehq 60 7 following following JJ cord-308115-bjyr6ehq 60 8 theorem theorem NN cord-308115-bjyr6ehq 60 9 . . . cord-308115-bjyr6ehq 60 10 _SP cord-308115-bjyr6ehq 61 1 Since since IN cord-308115-bjyr6ehq 61 2 and and CC cord-308115-bjyr6ehq 61 3 its -PRON- PRP$ cord-308115-bjyr6ehq 61 4 closed closed JJ cord-308115-bjyr6ehq 61 5 set set NN cord-308115-bjyr6ehq 61 6 , , , cord-308115-bjyr6ehq 61 7 then then RB cord-308115-bjyr6ehq 61 8 is be VBZ cord-308115-bjyr6ehq 61 9 complete complete JJ cord-308115-bjyr6ehq 61 10 metric metric JJ cord-308115-bjyr6ehq 61 11 space space NN cord-308115-bjyr6ehq 61 12 . . . cord-308115-bjyr6ehq 62 1 The the DT cord-308115-bjyr6ehq 62 2 continuous continuous JJ cord-308115-bjyr6ehq 62 3 system system NN cord-308115-bjyr6ehq 62 4 ( ( -LRB- cord-308115-bjyr6ehq 62 5 10 10 CD cord-308115-bjyr6ehq 62 6 ) ) -RRB- cord-308115-bjyr6ehq 62 7 can can MD cord-308115-bjyr6ehq 62 8 be be VB cord-308115-bjyr6ehq 62 9 transformed transform VBN cord-308115-bjyr6ehq 62 10 to to IN cord-308115-bjyr6ehq 62 11 equivalent equivalent JJ cord-308115-bjyr6ehq 62 12 integral integral JJ cord-308115-bjyr6ehq 62 13 equations equation NNS cord-308115-bjyr6ehq 62 14 as as IN cord-308115-bjyr6ehq 62 15 ; ; : cord-308115-bjyr6ehq 62 16 ( ( -LRB- cord-308115-bjyr6ehq 62 17 12 12 CD cord-308115-bjyr6ehq 62 18 ) ) -RRB- cord-308115-bjyr6ehq 62 19 is be VBZ cord-308115-bjyr6ehq 62 20 equivalent equivalent JJ cord-308115-bjyr6ehq 62 21 to to IN cord-308115-bjyr6ehq 62 22 Volterra Volterra NNP cord-308115-bjyr6ehq 62 23 integral integral JJ cord-308115-bjyr6ehq 62 24 equation equation NN cord-308115-bjyr6ehq 62 25 that that WDT cord-308115-bjyr6ehq 62 26 solves solve VBZ cord-308115-bjyr6ehq 62 27 ( ( -LRB- cord-308115-bjyr6ehq 62 28 10 10 CD cord-308115-bjyr6ehq 62 29 ) ) -RRB- cord-308115-bjyr6ehq 62 30 . . . cord-308115-bjyr6ehq 63 1 Define define VB cord-308115-bjyr6ehq 63 2 an an DT cord-308115-bjyr6ehq 63 3 operator operator NN cord-308115-bjyr6ehq 63 4 in in IN cord-308115-bjyr6ehq 63 5 Now now RB cord-308115-bjyr6ehq 63 6 , , , cord-308115-bjyr6ehq 63 7 we -PRON- PRP cord-308115-bjyr6ehq 63 8 need need VBP cord-308115-bjyr6ehq 63 9 to to TO cord-308115-bjyr6ehq 63 10 verify verify VB cord-308115-bjyr6ehq 63 11 that that DT cord-308115-bjyr6ehq 63 12 ( ( -LRB- cord-308115-bjyr6ehq 63 13 13 13 CD cord-308115-bjyr6ehq 63 14 ) ) -RRB- cord-308115-bjyr6ehq 63 15 satisfies satisfy VBZ cord-308115-bjyr6ehq 63 16 the the DT cord-308115-bjyr6ehq 63 17 hypothesis hypothesis NN cord-308115-bjyr6ehq 63 18 of of IN cord-308115-bjyr6ehq 63 19 contraction contraction NN cord-308115-bjyr6ehq 63 20 mapping mapping NN cord-308115-bjyr6ehq 63 21 principle principle NN cord-308115-bjyr6ehq 63 22 . . . cord-308115-bjyr6ehq 64 1 First first RB cord-308115-bjyr6ehq 64 2 to to TO cord-308115-bjyr6ehq 64 3 show show VB cord-308115-bjyr6ehq 64 4 Hence hence RB cord-308115-bjyr6ehq 64 5 the the DT cord-308115-bjyr6ehq 64 6 operator operator NN cord-308115-bjyr6ehq 64 7 maps map VBZ cord-308115-bjyr6ehq 64 8 onto onto IN cord-308115-bjyr6ehq 64 9 itself -PRON- PRP cord-308115-bjyr6ehq 64 10 . . . cord-308115-bjyr6ehq 65 1 Secondly secondly RB cord-308115-bjyr6ehq 65 2 , , , cord-308115-bjyr6ehq 65 3 to to TO cord-308115-bjyr6ehq 65 4 show show VB cord-308115-bjyr6ehq 65 5 that that DT cord-308115-bjyr6ehq 65 6 is be VBZ cord-308115-bjyr6ehq 65 7 a a DT cord-308115-bjyr6ehq 65 8 contraction contraction NN cord-308115-bjyr6ehq 65 9 , , , cord-308115-bjyr6ehq 65 10 we -PRON- PRP cord-308115-bjyr6ehq 65 11 have have VBP cord-308115-bjyr6ehq 65 12 Since since IN cord-308115-bjyr6ehq 65 13 by by IN cord-308115-bjyr6ehq 65 14 hypothesis hypothesis NN cord-308115-bjyr6ehq 65 15 , , , cord-308115-bjyr6ehq 65 16 then then RB cord-308115-bjyr6ehq 65 17 is be VBZ cord-308115-bjyr6ehq 65 18 a a DT cord-308115-bjyr6ehq 65 19 contraction contraction NN cord-308115-bjyr6ehq 65 20 and and CC cord-308115-bjyr6ehq 65 21 has have VBZ cord-308115-bjyr6ehq 65 22 a a DT cord-308115-bjyr6ehq 65 23 unique unique JJ cord-308115-bjyr6ehq 65 24 fixed fix VBN cord-308115-bjyr6ehq 65 25 point point NN cord-308115-bjyr6ehq 65 26 .. .. . cord-308115-bjyr6ehq 66 1 Thus thus RB cord-308115-bjyr6ehq 66 2 , , , cord-308115-bjyr6ehq 66 3 system system NN cord-308115-bjyr6ehq 66 4 ( ( -LRB- cord-308115-bjyr6ehq 66 5 10 10 CD cord-308115-bjyr6ehq 66 6 ) ) -RRB- cord-308115-bjyr6ehq 66 7 has have VBZ cord-308115-bjyr6ehq 66 8 unique unique JJ cord-308115-bjyr6ehq 66 9 solution solution NN cord-308115-bjyr6ehq 66 10 . . . cord-308115-bjyr6ehq 67 1 To to TO cord-308115-bjyr6ehq 67 2 obtain obtain VB cord-308115-bjyr6ehq 67 3 equilibrium equilibrium NN cord-308115-bjyr6ehq 67 4 solution solution NN cord-308115-bjyr6ehq 67 5 , , , cord-308115-bjyr6ehq 67 6 we -PRON- PRP cord-308115-bjyr6ehq 67 7 set set VBD cord-308115-bjyr6ehq 67 8 the the DT cord-308115-bjyr6ehq 67 9 system system NN cord-308115-bjyr6ehq 67 10 to to IN cord-308115-bjyr6ehq 67 11 zero zero CD cord-308115-bjyr6ehq 67 12 and and CC cord-308115-bjyr6ehq 67 13 solve solve VB cord-308115-bjyr6ehq 67 14 simultaneously simultaneously RB cord-308115-bjyr6ehq 67 15 as as IN cord-308115-bjyr6ehq 67 16 follows follow VBZ cord-308115-bjyr6ehq 67 17 ; ; : cord-308115-bjyr6ehq 68 1 Considering consider VBG cord-308115-bjyr6ehq 68 2 ( ( -LRB- cord-308115-bjyr6ehq 68 3 ) ) -RRB- cord-308115-bjyr6ehq 68 4 ( ( -LRB- cord-308115-bjyr6ehq 68 5 ) ) -RRB- cord-308115-bjyr6ehq 68 6 , , , cord-308115-bjyr6ehq 68 7 in in IN cord-308115-bjyr6ehq 68 8 ( ( -LRB- cord-308115-bjyr6ehq 68 9 11)- 11)- NNP cord-308115-bjyr6ehq 68 10 ( ( -LRB- cord-308115-bjyr6ehq 68 11 14 14 CD cord-308115-bjyr6ehq 68 12 ) ) -RRB- cord-308115-bjyr6ehq 68 13 we -PRON- PRP cord-308115-bjyr6ehq 68 14 find find VBP cord-308115-bjyr6ehq 68 15 the the DT cord-308115-bjyr6ehq 68 16 endemic endemic JJ cord-308115-bjyr6ehq 68 17 equilibrium equilibrium NN cord-308115-bjyr6ehq 68 18 , , , cord-308115-bjyr6ehq 68 19 Consider consider VB cord-308115-bjyr6ehq 68 20 system system NN cord-308115-bjyr6ehq 68 21 ( ( -LRB- cord-308115-bjyr6ehq 68 22 1 1 CD cord-308115-bjyr6ehq 68 23 ) ) -RRB- cord-308115-bjyr6ehq 69 1 then then RB cord-308115-bjyr6ehq 69 2 , , , cord-308115-bjyr6ehq 69 3 we -PRON- PRP cord-308115-bjyr6ehq 69 4 have have VBP cord-308115-bjyr6ehq 69 5 the the DT cord-308115-bjyr6ehq 69 6 following following JJ cord-308115-bjyr6ehq 69 7 Jacobian jacobian JJ cord-308115-bjyr6ehq 69 8 matrix matrix NN cord-308115-bjyr6ehq 70 1 Clearly clearly RB cord-308115-bjyr6ehq 70 2 , , , cord-308115-bjyr6ehq 70 3 allthe allthe DT cord-308115-bjyr6ehq 70 4 eigenvalues eigenvalue NNS cord-308115-bjyr6ehq 70 5 have have VBP cord-308115-bjyr6ehq 70 6 zero zero CD cord-308115-bjyr6ehq 70 7 imaginary imaginary JJ cord-308115-bjyr6ehq 70 8 part part NN cord-308115-bjyr6ehq 70 9 ( ( -LRB- cord-308115-bjyr6ehq 70 10 ) ) -RRB- cord-308115-bjyr6ehq 71 1 Therefore therefore RB cord-308115-bjyr6ehq 71 2 , , , cord-308115-bjyr6ehq 71 3 Hence hence RB cord-308115-bjyr6ehq 71 4 by by IN cord-308115-bjyr6ehq 71 5 theorem theorem NN cord-308115-bjyr6ehq 71 6 1 1 CD cord-308115-bjyr6ehq 71 7 above above RB cord-308115-bjyr6ehq 71 8 , , , cord-308115-bjyr6ehq 71 9 the the DT cord-308115-bjyr6ehq 71 10 disease disease NN cord-308115-bjyr6ehq 71 11 free free JJ cord-308115-bjyr6ehq 71 12 equilibrium equilibrium NN cord-308115-bjyr6ehq 71 13 is be VBZ cord-308115-bjyr6ehq 71 14 locally locally RB cord-308115-bjyr6ehq 71 15 asymptotically asymptotically RB cord-308115-bjyr6ehq 71 16 stable stable JJ cord-308115-bjyr6ehq 71 17 . . . cord-308115-bjyr6ehq 72 1 To to TO cord-308115-bjyr6ehq 72 2 obtain obtain VB cord-308115-bjyr6ehq 72 3 basic basic JJ cord-308115-bjyr6ehq 72 4 reproduction reproduction NN cord-308115-bjyr6ehq 72 5 number number NN cord-308115-bjyr6ehq 72 6 which which WDT cord-308115-bjyr6ehq 72 7 is be VBZ cord-308115-bjyr6ehq 72 8 a a DT cord-308115-bjyr6ehq 72 9 key key JJ cord-308115-bjyr6ehq 72 10 parameter parameter NN cord-308115-bjyr6ehq 72 11 describing describe VBG cord-308115-bjyr6ehq 72 12 the the DT cord-308115-bjyr6ehq 72 13 number number NN cord-308115-bjyr6ehq 72 14 of of IN cord-308115-bjyr6ehq 72 15 secondary secondary JJ cord-308115-bjyr6ehq 72 16 infections infection NNS cord-308115-bjyr6ehq 72 17 generated generate VBN cord-308115-bjyr6ehq 72 18 by by IN cord-308115-bjyr6ehq 72 19 a a DT cord-308115-bjyr6ehq 72 20 single single JJ cord-308115-bjyr6ehq 72 21 infectious infectious JJ cord-308115-bjyr6ehq 72 22 individual individual NN cord-308115-bjyr6ehq 72 23 , , , cord-308115-bjyr6ehq 72 24 we -PRON- PRP cord-308115-bjyr6ehq 72 25 consider consider VBP cord-308115-bjyr6ehq 72 26 the the DT cord-308115-bjyr6ehq 72 27 eigen eigen JJ cord-308115-bjyr6ehq 72 28 values value NNS cord-308115-bjyr6ehq 72 29 above above RB cord-308115-bjyr6ehq 72 30 . . . cord-308115-bjyr6ehq 73 1 For for IN cord-308115-bjyr6ehq 73 2 implies implie NNS cord-308115-bjyr6ehq 73 3 that that IN cord-308115-bjyr6ehq 73 4 This this DT cord-308115-bjyr6ehq 73 5 threshold threshold NN cord-308115-bjyr6ehq 73 6 quantity quantity NN cord-308115-bjyr6ehq 73 7 which which WDT cord-308115-bjyr6ehq 73 8 if if IN cord-308115-bjyr6ehq 73 9 less less JJR cord-308115-bjyr6ehq 73 10 than than IN cord-308115-bjyr6ehq 73 11 one one CD cord-308115-bjyr6ehq 73 12 disease disease NN cord-308115-bjyr6ehq 73 13 free free JJ cord-308115-bjyr6ehq 73 14 equilibrium equilibrium NN cord-308115-bjyr6ehq 73 15 will will MD cord-308115-bjyr6ehq 73 16 be be VB cord-308115-bjyr6ehq 73 17 stable stable JJ cord-308115-bjyr6ehq 73 18 and and CC cord-308115-bjyr6ehq 73 19 if if IN cord-308115-bjyr6ehq 73 20 greater great JJR cord-308115-bjyr6ehq 73 21 than than IN cord-308115-bjyr6ehq 73 22 one one NN cord-308115-bjyr6ehq 73 23 it -PRON- PRP cord-308115-bjyr6ehq 73 24 is be VBZ cord-308115-bjyr6ehq 73 25 unstable unstable JJ cord-308115-bjyr6ehq 73 26 is be VBZ cord-308115-bjyr6ehq 73 27 what what WP cord-308115-bjyr6ehq 73 28 we -PRON- PRP cord-308115-bjyr6ehq 73 29 termed term VBD cord-308115-bjyr6ehq 73 30 as as IN cord-308115-bjyr6ehq 73 31 basic basic JJ cord-308115-bjyr6ehq 73 32 reproduction reproduction NN cord-308115-bjyr6ehq 73 33 ratio ratio NN cord-308115-bjyr6ehq 73 34 . . . cord-308115-bjyr6ehq 74 1 Hence hence RB cord-308115-bjyr6ehq 74 2 we -PRON- PRP cord-308115-bjyr6ehq 74 3 define define VBP cord-308115-bjyr6ehq 74 4 The the DT cord-308115-bjyr6ehq 74 5 endemic endemic JJ cord-308115-bjyr6ehq 74 6 equilibrium equilibrium NN cord-308115-bjyr6ehq 74 7 points point NNS cord-308115-bjyr6ehq 74 8 can can MD cord-308115-bjyr6ehq 74 9 be be VB cord-308115-bjyr6ehq 74 10 rewritten rewrite VBN cord-308115-bjyr6ehq 74 11 in in IN cord-308115-bjyr6ehq 74 12 the the DT cord-308115-bjyr6ehq 74 13 form form NN cord-308115-bjyr6ehq 74 14 of of IN cord-308115-bjyr6ehq 74 15 _SP cord-308115-bjyr6ehq 74 16 To to TO cord-308115-bjyr6ehq 74 17 derive derive VB cord-308115-bjyr6ehq 74 18 the the DT cord-308115-bjyr6ehq 74 19 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 74 20 candidate candidate NN cord-308115-bjyr6ehq 74 21 function function NN cord-308115-bjyr6ehq 74 22 for for IN cord-308115-bjyr6ehq 74 23 fractional fractional JJ cord-308115-bjyr6ehq 74 24 order order NN cord-308115-bjyr6ehq 74 25 as as IN cord-308115-bjyr6ehq 74 26 in in IN cord-308115-bjyr6ehq 74 27 [ [ -LRB- cord-308115-bjyr6ehq 74 28 24 24 CD cord-308115-bjyr6ehq 74 29 ] ] -RRB- cord-308115-bjyr6ehq 74 30 , , , cord-308115-bjyr6ehq 74 31 consider consider VB cord-308115-bjyr6ehq 74 32 the the DT cord-308115-bjyr6ehq 74 33 family family NN cord-308115-bjyr6ehq 74 34 of of IN cord-308115-bjyr6ehq 74 35 quadratic quadratic JJ cord-308115-bjyr6ehq 74 36 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 74 37 function function NN cord-308115-bjyr6ehq 74 38 And and CC cord-308115-bjyr6ehq 74 39 define define VB cord-308115-bjyr6ehq 74 40 the the DT cord-308115-bjyr6ehq 74 41 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 74 42 candidate candidate NN cord-308115-bjyr6ehq 74 43 function function NN cord-308115-bjyr6ehq 74 44 as as IN cord-308115-bjyr6ehq 74 45 Applying apply VBG cord-308115-bjyr6ehq 74 46 lemma lemma NN cord-308115-bjyr6ehq 74 47 2 2 CD cord-308115-bjyr6ehq 74 48 [ [ -LRB- cord-308115-bjyr6ehq 74 49 23 23 CD cord-308115-bjyr6ehq 74 50 ] ] -RRB- cord-308115-bjyr6ehq 74 51 above above IN cord-308115-bjyr6ehq 74 52 _SP cord-308115-bjyr6ehq 74 53 Hence Hence NNP cord-308115-bjyr6ehq 74 54 by by IN cord-308115-bjyr6ehq 74 55 theorem theorem NN cord-308115-bjyr6ehq 74 56 3 3 CD cord-308115-bjyr6ehq 74 57 above above RB cord-308115-bjyr6ehq 74 58 , , , cord-308115-bjyr6ehq 74 59 the the DT cord-308115-bjyr6ehq 74 60 disease disease NN cord-308115-bjyr6ehq 74 61 free free JJ cord-308115-bjyr6ehq 74 62 equilibrium equilibrium NN cord-308115-bjyr6ehq 74 63 is be VBZ cord-308115-bjyr6ehq 74 64 globally globally RB cord-308115-bjyr6ehq 74 65 asymptotically asymptotically RB cord-308115-bjyr6ehq 74 66 stable stable JJ cord-308115-bjyr6ehq 74 67 . . . cord-308115-bjyr6ehq 75 1 Case case NN cord-308115-bjyr6ehq 75 2 2 2 CD cord-308115-bjyr6ehq 75 3 : : : cord-308115-bjyr6ehq 76 1 At at IN cord-308115-bjyr6ehq 76 2 the the DT cord-308115-bjyr6ehq 76 3 endemic endemic JJ cord-308115-bjyr6ehq 76 4 ( ( -LRB- cord-308115-bjyr6ehq 76 5 positive positive JJ cord-308115-bjyr6ehq 76 6 ) ) -RRB- cord-308115-bjyr6ehq 76 7 equilibrium equilibrium NN cord-308115-bjyr6ehq 76 8 , , , cord-308115-bjyr6ehq 76 9 ( ( -LRB- cord-308115-bjyr6ehq 76 10 28 28 CD cord-308115-bjyr6ehq 76 11 ) ) -RRB- cord-308115-bjyr6ehq 76 12 becomes become VBZ cord-308115-bjyr6ehq 76 13 We -PRON- PRP cord-308115-bjyr6ehq 76 14 had have VBD cord-308115-bjyr6ehq 76 15 earlier early RBR cord-308115-bjyr6ehq 76 16 established establish VBN cord-308115-bjyr6ehq 76 17 that that IN cord-308115-bjyr6ehq 76 18 the the DT cord-308115-bjyr6ehq 76 19 positive positive JJ cord-308115-bjyr6ehq 76 20 equilibrium equilibrium NN cord-308115-bjyr6ehq 76 21 is be VBZ cord-308115-bjyr6ehq 76 22 stable stable JJ cord-308115-bjyr6ehq 76 23 if if IN cord-308115-bjyr6ehq 76 24 . . . cord-308115-bjyr6ehq 77 1 Now now RB cord-308115-bjyr6ehq 77 2 consider consider VB cord-308115-bjyr6ehq 77 3 the the DT cord-308115-bjyr6ehq 77 4 relations relation NNS cord-308115-bjyr6ehq 77 5 Therefore therefore RB cord-308115-bjyr6ehq 77 6 back back RB cord-308115-bjyr6ehq 77 7 substitution substitution NN cord-308115-bjyr6ehq 77 8 the the DT cord-308115-bjyr6ehq 77 9 above above JJ cord-308115-bjyr6ehq 77 10 relations relation NNS cord-308115-bjyr6ehq 77 11 into into IN cord-308115-bjyr6ehq 77 12 ( ( -LRB- cord-308115-bjyr6ehq 77 13 30 30 CD cord-308115-bjyr6ehq 77 14 ) ) -RRB- cord-308115-bjyr6ehq 77 15 yields yield NNS cord-308115-bjyr6ehq 78 1 where where WRB cord-308115-bjyr6ehq 79 1 ( ( -LRB- cord-308115-bjyr6ehq 79 2 ( ( -LRB- cord-308115-bjyr6ehq 79 3 ) ) -RRB- cord-308115-bjyr6ehq 79 4 ) ) -RRB- cord-308115-bjyr6ehq 79 5 ( ( -LRB- cord-308115-bjyr6ehq 79 6 ) ) -RRB- cord-308115-bjyr6ehq 79 7 ( ( -LRB- cord-308115-bjyr6ehq 79 8 ) ) -RRB- cord-308115-bjyr6ehq 79 9 Hence hence RB cord-308115-bjyr6ehq 79 10 by by IN cord-308115-bjyr6ehq 79 11 theorem theorem NN cord-308115-bjyr6ehq 79 12 3 3 CD cord-308115-bjyr6ehq 79 13 above above RB cord-308115-bjyr6ehq 79 14 , , , cord-308115-bjyr6ehq 79 15 the the DT cord-308115-bjyr6ehq 79 16 endemic endemic JJ cord-308115-bjyr6ehq 79 17 equilibrium equilibrium NN cord-308115-bjyr6ehq 79 18 is be VBZ cord-308115-bjyr6ehq 79 19 globally globally RB cord-308115-bjyr6ehq 79 20 asymptotically asymptotically RB cord-308115-bjyr6ehq 79 21 stable stable JJ cord-308115-bjyr6ehq 79 22 . . . cord-308115-bjyr6ehq 80 1 In in IN cord-308115-bjyr6ehq 80 2 this this DT cord-308115-bjyr6ehq 80 3 chapter chapter NN cord-308115-bjyr6ehq 80 4 we -PRON- PRP cord-308115-bjyr6ehq 80 5 carry carry VBP cord-308115-bjyr6ehq 80 6 out out RP cord-308115-bjyr6ehq 80 7 numerical numerical JJ cord-308115-bjyr6ehq 80 8 examples example NNS cord-308115-bjyr6ehq 80 9 to to TO cord-308115-bjyr6ehq 80 10 support support VB cord-308115-bjyr6ehq 80 11 the the DT cord-308115-bjyr6ehq 80 12 analytic analytic JJ cord-308115-bjyr6ehq 80 13 results result NNS cord-308115-bjyr6ehq 80 14 using use VBG cord-308115-bjyr6ehq 80 15 parameter parameter NN cord-308115-bjyr6ehq 80 16 values value NNS cord-308115-bjyr6ehq 80 17 in in IN cord-308115-bjyr6ehq 80 18 Table Table NNP cord-308115-bjyr6ehq 80 19 1 1 CD cord-308115-bjyr6ehq 80 20 . . . cord-308115-bjyr6ehq 81 1 For for IN cord-308115-bjyr6ehq 81 2 the the DT cord-308115-bjyr6ehq 81 3 variables variable NNS cord-308115-bjyr6ehq 81 4 we -PRON- PRP cord-308115-bjyr6ehq 81 5 use use VBP cord-308115-bjyr6ehq 81 6 ( ( -LRB- cord-308115-bjyr6ehq 81 7 ) ) -RRB- cord-308115-bjyr6ehq 81 8 ( ( -LRB- cord-308115-bjyr6ehq 81 9 ) ) -RRB- cord-308115-bjyr6ehq 81 10 ( ( -LRB- cord-308115-bjyr6ehq 81 11 ) ) -RRB- cord-308115-bjyr6ehq 81 12 ( ( -LRB- cord-308115-bjyr6ehq 81 13 ) ) -RRB- cord-308115-bjyr6ehq 81 14 When when WRB cord-308115-bjyr6ehq 81 15 the the DT cord-308115-bjyr6ehq 81 16 rate rate NN cord-308115-bjyr6ehq 81 17 of of IN cord-308115-bjyr6ehq 81 18 infection infection NN cord-308115-bjyr6ehq 81 19 of of IN cord-308115-bjyr6ehq 81 20 the the DT cord-308115-bjyr6ehq 81 21 mild mild JJ cord-308115-bjyr6ehq 81 22 cases case NNS cord-308115-bjyr6ehq 81 23 increases increase NNS cord-308115-bjyr6ehq 81 24 , , , cord-308115-bjyr6ehq 81 25 there there EX cord-308115-bjyr6ehq 81 26 is be VBZ cord-308115-bjyr6ehq 81 27 equivalent equivalent JJ cord-308115-bjyr6ehq 81 28 increase increase NN cord-308115-bjyr6ehq 81 29 in in IN cord-308115-bjyr6ehq 81 30 the the DT cord-308115-bjyr6ehq 81 31 overall overall JJ cord-308115-bjyr6ehq 81 32 population population NN cord-308115-bjyr6ehq 81 33 of of IN cord-308115-bjyr6ehq 81 34 infected infected JJ cord-308115-bjyr6ehq 81 35 individuals individual NNS cord-308115-bjyr6ehq 81 36 . . . cord-308115-bjyr6ehq 82 1 Hence hence RB cord-308115-bjyr6ehq 82 2 to to TO cord-308115-bjyr6ehq 82 3 curtail curtail VB cord-308115-bjyr6ehq 82 4 the the DT cord-308115-bjyr6ehq 82 5 spread spread NN cord-308115-bjyr6ehq 82 6 of of IN cord-308115-bjyr6ehq 82 7 the the DT cord-308115-bjyr6ehq 82 8 disease disease NN cord-308115-bjyr6ehq 82 9 there there EX cord-308115-bjyr6ehq 82 10 is be VBZ cord-308115-bjyr6ehq 82 11 need need NN cord-308115-bjyr6ehq 82 12 to to TO cord-308115-bjyr6ehq 82 13 take take VB cord-308115-bjyr6ehq 82 14 care care NN cord-308115-bjyr6ehq 82 15 of of IN cord-308115-bjyr6ehq 82 16 the the DT cord-308115-bjyr6ehq 82 17 Mild mild JJ cord-308115-bjyr6ehq 82 18 cases case NNS cord-308115-bjyr6ehq 82 19 as as RB cord-308115-bjyr6ehq 82 20 well well RB cord-308115-bjyr6ehq 82 21 . . . cord-308115-bjyr6ehq 83 1 In in IN cord-308115-bjyr6ehq 83 2 mostly mostly RB cord-308115-bjyr6ehq 83 3 poor poor JJ cord-308115-bjyr6ehq 83 4 and and CC cord-308115-bjyr6ehq 83 5 underdeveloped underdeveloped JJ cord-308115-bjyr6ehq 83 6 territories territory NNS cord-308115-bjyr6ehq 83 7 where where WRB cord-308115-bjyr6ehq 83 8 there there EX cord-308115-bjyr6ehq 83 9 is be VBZ cord-308115-bjyr6ehq 83 10 no no DT cord-308115-bjyr6ehq 83 11 capacity capacity NN cord-308115-bjyr6ehq 83 12 of of IN cord-308115-bjyr6ehq 83 13 rapid rapid JJ cord-308115-bjyr6ehq 83 14 diagnostic diagnostic JJ cord-308115-bjyr6ehq 83 15 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 83 16 test test NN cord-308115-bjyr6ehq 83 17 due due IN cord-308115-bjyr6ehq 83 18 to to IN cord-308115-bjyr6ehq 83 19 insufficiency insufficiency NN cord-308115-bjyr6ehq 83 20 of of IN cord-308115-bjyr6ehq 83 21 testing testing NN cord-308115-bjyr6ehq 83 22 kits kit NNS cord-308115-bjyr6ehq 83 23 , , , cord-308115-bjyr6ehq 83 24 the the DT cord-308115-bjyr6ehq 83 25 Mild mild JJ cord-308115-bjyr6ehq 83 26 cases case NNS cord-308115-bjyr6ehq 83 27 ( ( -LRB- cord-308115-bjyr6ehq 83 28 who who WP cord-308115-bjyr6ehq 83 29 usually usually RB cord-308115-bjyr6ehq 83 30 show show VBP cord-308115-bjyr6ehq 83 31 no no DT cord-308115-bjyr6ehq 83 32 symptoms symptom NNS cord-308115-bjyr6ehq 83 33 of of IN cord-308115-bjyr6ehq 83 34 the the DT cord-308115-bjyr6ehq 83 35 infection infection NN cord-308115-bjyr6ehq 83 36 due due IN cord-308115-bjyr6ehq 83 37 to to IN cord-308115-bjyr6ehq 83 38 their -PRON- PRP$ cord-308115-bjyr6ehq 83 39 strong strong JJ cord-308115-bjyr6ehq 83 40 and and CC cord-308115-bjyr6ehq 83 41 active active JJ cord-308115-bjyr6ehq 83 42 immune immune JJ cord-308115-bjyr6ehq 83 43 system system NN cord-308115-bjyr6ehq 83 44 up up IN cord-308115-bjyr6ehq 83 45 to to IN cord-308115-bjyr6ehq 83 46 their -PRON- PRP$ cord-308115-bjyr6ehq 83 47 recovery recovery NN cord-308115-bjyr6ehq 83 48 period period NN cord-308115-bjyr6ehq 83 49 ) ) -RRB- cord-308115-bjyr6ehq 83 50 play play VBP cord-308115-bjyr6ehq 83 51 a a DT cord-308115-bjyr6ehq 83 52 major major JJ cord-308115-bjyr6ehq 83 53 role role NN cord-308115-bjyr6ehq 83 54 as as IN cord-308115-bjyr6ehq 83 55 a a DT cord-308115-bjyr6ehq 83 56 route route NN cord-308115-bjyr6ehq 83 57 that that WDT cord-308115-bjyr6ehq 83 58 spread spread VBD cord-308115-bjyr6ehq 83 59 the the DT cord-308115-bjyr6ehq 83 60 disease disease NN cord-308115-bjyr6ehq 83 61 to to IN cord-308115-bjyr6ehq 83 62 healthy healthy JJ cord-308115-bjyr6ehq 83 63 individuals individual NNS cord-308115-bjyr6ehq 83 64 . . . cord-308115-bjyr6ehq 84 1 Here here RB cord-308115-bjyr6ehq 84 2 we -PRON- PRP cord-308115-bjyr6ehq 84 3 build build VBP cord-308115-bjyr6ehq 84 4 our -PRON- PRP$ cord-308115-bjyr6ehq 84 5 model model NN cord-308115-bjyr6ehq 84 6 by by IN cord-308115-bjyr6ehq 84 7 incorporating incorporate VBG cord-308115-bjyr6ehq 84 8 the the DT cord-308115-bjyr6ehq 84 9 population population NN cord-308115-bjyr6ehq 84 10 of of IN cord-308115-bjyr6ehq 84 11 mild mild JJ cord-308115-bjyr6ehq 84 12 individuals individual NNS cord-308115-bjyr6ehq 84 13 into into IN cord-308115-bjyr6ehq 84 14 the the DT cord-308115-bjyr6ehq 84 15 compartmental compartmental JJ cord-308115-bjyr6ehq 84 16 SIR SIR NNP cord-308115-bjyr6ehq 84 17 model model NN cord-308115-bjyr6ehq 84 18 to to TO cord-308115-bjyr6ehq 84 19 become become VB cord-308115-bjyr6ehq 84 20 SMIR SMIR NNP cord-308115-bjyr6ehq 84 21 model model NN cord-308115-bjyr6ehq 84 22 in in IN cord-308115-bjyr6ehq 84 23 form form NN cord-308115-bjyr6ehq 84 24 of of IN cord-308115-bjyr6ehq 84 25 system system NN cord-308115-bjyr6ehq 84 26 of of IN cord-308115-bjyr6ehq 84 27 fractional fractional JJ cord-308115-bjyr6ehq 84 28 order order NN cord-308115-bjyr6ehq 84 29 differential differential NN cord-308115-bjyr6ehq 84 30 equations equation NNS cord-308115-bjyr6ehq 84 31 ( ( -LRB- cord-308115-bjyr6ehq 84 32 FODE FODE NNP cord-308115-bjyr6ehq 84 33 ) ) -RRB- cord-308115-bjyr6ehq 84 34 in in IN cord-308115-bjyr6ehq 84 35 the the DT cord-308115-bjyr6ehq 84 36 Caputo Caputo NNP cord-308115-bjyr6ehq 84 37 sense sense NN cord-308115-bjyr6ehq 84 38 . . . cord-308115-bjyr6ehq 85 1 The the DT cord-308115-bjyr6ehq 85 2 existence existence NN cord-308115-bjyr6ehq 85 3 of of IN cord-308115-bjyr6ehq 85 4 the the DT cord-308115-bjyr6ehq 85 5 solutions solution NNS cord-308115-bjyr6ehq 85 6 of of IN cord-308115-bjyr6ehq 85 7 the the DT cord-308115-bjyr6ehq 85 8 model model NN cord-308115-bjyr6ehq 85 9 was be VBD cord-308115-bjyr6ehq 85 10 shown show VBN cord-308115-bjyr6ehq 85 11 by by IN cord-308115-bjyr6ehq 85 12 solving solve VBG cord-308115-bjyr6ehq 85 13 the the DT cord-308115-bjyr6ehq 85 14 fractional fractional JJ cord-308115-bjyr6ehq 85 15 Gronwall Gronwall NNP cord-308115-bjyr6ehq 85 16 's 's POS cord-308115-bjyr6ehq 85 17 inequality inequality NN cord-308115-bjyr6ehq 85 18 using use VBG cord-308115-bjyr6ehq 85 19 the the DT cord-308115-bjyr6ehq 85 20 Laplace Laplace NNP cord-308115-bjyr6ehq 85 21 transform transform VB cord-308115-bjyr6ehq 85 22 approach approach NN cord-308115-bjyr6ehq 85 23 . . . cord-308115-bjyr6ehq 86 1 Two two CD cord-308115-bjyr6ehq 86 2 equilibrium equilibrium NN cord-308115-bjyr6ehq 86 3 solutions solution NNS cord-308115-bjyr6ehq 86 4 , , , cord-308115-bjyr6ehq 86 5 disease disease NN cord-308115-bjyr6ehq 86 6 free free JJ cord-308115-bjyr6ehq 86 7 and and CC cord-308115-bjyr6ehq 86 8 endemic endemic JJ cord-308115-bjyr6ehq 86 9 were be VBD cord-308115-bjyr6ehq 86 10 obtained obtain VBN cord-308115-bjyr6ehq 86 11 . . . cord-308115-bjyr6ehq 87 1 Both both DT cord-308115-bjyr6ehq 87 2 local local JJ cord-308115-bjyr6ehq 87 3 and and CC cord-308115-bjyr6ehq 87 4 global global JJ cord-308115-bjyr6ehq 87 5 stability stability NN cord-308115-bjyr6ehq 87 6 of of IN cord-308115-bjyr6ehq 87 7 the the DT cord-308115-bjyr6ehq 87 8 equilibria equilibrium NNS cord-308115-bjyr6ehq 87 9 were be VBD cord-308115-bjyr6ehq 87 10 shown show VBN cord-308115-bjyr6ehq 87 11 to to TO cord-308115-bjyr6ehq 87 12 depend depend VB cord-308115-bjyr6ehq 87 13 on on IN cord-308115-bjyr6ehq 87 14 the the DT cord-308115-bjyr6ehq 87 15 magnitude magnitude NN cord-308115-bjyr6ehq 87 16 of of IN cord-308115-bjyr6ehq 87 17 basic basic JJ cord-308115-bjyr6ehq 87 18 reproduction reproduction NN cord-308115-bjyr6ehq 87 19 ratio ratio NN cord-308115-bjyr6ehq 87 20 . . . cord-308115-bjyr6ehq 88 1 Numerical numerical JJ cord-308115-bjyr6ehq 88 2 simulations simulation NNS cord-308115-bjyr6ehq 88 3 were be VBD cord-308115-bjyr6ehq 88 4 carried carry VBN cord-308115-bjyr6ehq 88 5 out out RP cord-308115-bjyr6ehq 88 6 and and CC cord-308115-bjyr6ehq 88 7 dynamics dynamic NNS cord-308115-bjyr6ehq 88 8 of of IN cord-308115-bjyr6ehq 88 9 the the DT cord-308115-bjyr6ehq 88 10 populations population NNS cord-308115-bjyr6ehq 88 11 were be VBD cord-308115-bjyr6ehq 88 12 shown show VBN cord-308115-bjyr6ehq 88 13 to to TO cord-308115-bjyr6ehq 88 14 vary vary VB cord-308115-bjyr6ehq 88 15 for for IN cord-308115-bjyr6ehq 88 16 different different JJ cord-308115-bjyr6ehq 88 17 values value NNS cord-308115-bjyr6ehq 88 18 of of IN cord-308115-bjyr6ehq 88 19 It -PRON- PRP cord-308115-bjyr6ehq 88 20 was be VBD cord-308115-bjyr6ehq 88 21 also also RB cord-308115-bjyr6ehq 88 22 shown show VBN cord-308115-bjyr6ehq 88 23 that that IN cord-308115-bjyr6ehq 88 24 when when WRB cord-308115-bjyr6ehq 88 25 the the DT cord-308115-bjyr6ehq 88 26 rate rate NN cord-308115-bjyr6ehq 88 27 of of IN cord-308115-bjyr6ehq 88 28 infection infection NN cord-308115-bjyr6ehq 88 29 of of IN cord-308115-bjyr6ehq 88 30 the the DT cord-308115-bjyr6ehq 88 31 mild mild JJ cord-308115-bjyr6ehq 88 32 cases case NNS cord-308115-bjyr6ehq 88 33 increases increase NNS cord-308115-bjyr6ehq 88 34 , , , cord-308115-bjyr6ehq 88 35 there there EX cord-308115-bjyr6ehq 88 36 is be VBZ cord-308115-bjyr6ehq 88 37 equivalent equivalent JJ cord-308115-bjyr6ehq 88 38 increase increase NN cord-308115-bjyr6ehq 88 39 in in IN cord-308115-bjyr6ehq 88 40 the the DT cord-308115-bjyr6ehq 88 41 overall overall JJ cord-308115-bjyr6ehq 88 42 population population NN cord-308115-bjyr6ehq 88 43 of of IN cord-308115-bjyr6ehq 88 44 infected infected JJ cord-308115-bjyr6ehq 88 45 individuals individual NNS cord-308115-bjyr6ehq 88 46 . . . cord-308115-bjyr6ehq 89 1 Hence hence RB cord-308115-bjyr6ehq 89 2 to to TO cord-308115-bjyr6ehq 89 3 curtail curtail VB cord-308115-bjyr6ehq 89 4 the the DT cord-308115-bjyr6ehq 89 5 spread spread NN cord-308115-bjyr6ehq 89 6 of of IN cord-308115-bjyr6ehq 89 7 the the DT cord-308115-bjyr6ehq 89 8 disease disease NN cord-308115-bjyr6ehq 89 9 there there EX cord-308115-bjyr6ehq 89 10 is be VBZ cord-308115-bjyr6ehq 89 11 need need NN cord-308115-bjyr6ehq 89 12 to to TO cord-308115-bjyr6ehq 89 13 take take VB cord-308115-bjyr6ehq 89 14 care care NN cord-308115-bjyr6ehq 89 15 of of IN cord-308115-bjyr6ehq 89 16 the the DT cord-308115-bjyr6ehq 89 17 Mild mild JJ cord-308115-bjyr6ehq 89 18 cases case NNS cord-308115-bjyr6ehq 89 19 as as RB cord-308115-bjyr6ehq 89 20 well well RB cord-308115-bjyr6ehq 89 21 . . . cord-308115-bjyr6ehq 89 22 _SP cord-308115-bjyr6ehq 90 1 World World NNP cord-308115-bjyr6ehq 90 2 Health Health NNP cord-308115-bjyr6ehq 90 3 Organization Organization NNP cord-308115-bjyr6ehq 90 4 ( ( -LRB- cord-308115-bjyr6ehq 90 5 WHO who WP cord-308115-bjyr6ehq 90 6 ) ) -RRB- cord-308115-bjyr6ehq 90 7 . . . cord-308115-bjyr6ehq 91 1 Novel Novel NNP cord-308115-bjyr6ehq 91 2 Coronavirus Coronavirus NNP cord-308115-bjyr6ehq 91 3 - - HYPH cord-308115-bjyr6ehq 91 4 China China NNP cord-308115-bjyr6ehq 91 5 Modeling model VBG cord-308115-bjyr6ehq 91 6 the the DT cord-308115-bjyr6ehq 91 7 epidemic epidemic NN cord-308115-bjyr6ehq 91 8 dynamics dynamic NNS cord-308115-bjyr6ehq 91 9 and and CC cord-308115-bjyr6ehq 91 10 control control NN cord-308115-bjyr6ehq 91 11 of of IN cord-308115-bjyr6ehq 91 12 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 91 13 in in IN cord-308115-bjyr6ehq 91 14 China China NNP cord-308115-bjyr6ehq 92 1 The the DT cord-308115-bjyr6ehq 92 2 impact impact NN cord-308115-bjyr6ehq 92 3 of of IN cord-308115-bjyr6ehq 92 4 isolation isolation NN cord-308115-bjyr6ehq 92 5 on on IN cord-308115-bjyr6ehq 92 6 the the DT cord-308115-bjyr6ehq 92 7 transmission transmission NN cord-308115-bjyr6ehq 92 8 of of IN cord-308115-bjyr6ehq 92 9 COVID-19 covid-19 NN cord-308115-bjyr6ehq 92 10 and and CC cord-308115-bjyr6ehq 92 11 estimation estimation NN cord-308115-bjyr6ehq 92 12 of of IN cord-308115-bjyr6ehq 92 13 potential potential JJ cord-308115-bjyr6ehq 92 14 second second JJ cord-308115-bjyr6ehq 92 15 epidemic epidemic NN cord-308115-bjyr6ehq 92 16 in in IN cord-308115-bjyr6ehq 92 17 China China NNP cord-308115-bjyr6ehq 92 18 . . . cord-308115-bjyr6ehq 93 1 Preprints preprint NNS cord-308115-bjyr6ehq 93 2 ( ( -LRB- cord-308115-bjyr6ehq 93 3 www.preprints.org www.preprints.org NFP cord-308115-bjyr6ehq 93 4 ) ) -RRB- cord-308115-bjyr6ehq 93 5 Stability stability NN cord-308115-bjyr6ehq 93 6 behavior behavior NN cord-308115-bjyr6ehq 93 7 of of IN cord-308115-bjyr6ehq 93 8 mathematical mathematical JJ cord-308115-bjyr6ehq 93 9 model model NN cord-308115-bjyr6ehq 93 10 of of IN cord-308115-bjyr6ehq 93 11 MERS MERS NNP cord-308115-bjyr6ehq 93 12 Corona Corona NNP cord-308115-bjyr6ehq 93 13 virus virus NN cord-308115-bjyr6ehq 93 14 spread spread VBD cord-308115-bjyr6ehq 93 15 in in IN cord-308115-bjyr6ehq 93 16 population population NN cord-308115-bjyr6ehq 93 17 a a DT cord-308115-bjyr6ehq 93 18 mathematical mathematical JJ cord-308115-bjyr6ehq 93 19 model model NN cord-308115-bjyr6ehq 93 20 for for IN cord-308115-bjyr6ehq 93 21 simulating simulate VBG cord-308115-bjyr6ehq 93 22 the the DT cord-308115-bjyr6ehq 93 23 phased phase VBN cord-308115-bjyr6ehq 93 24 - - HYPH cord-308115-bjyr6ehq 93 25 based base VBN cord-308115-bjyr6ehq 93 26 transmissibility transmissibility NN cord-308115-bjyr6ehq 93 27 of of IN cord-308115-bjyr6ehq 93 28 a a DT cord-308115-bjyr6ehq 93 29 novel novel JJ cord-308115-bjyr6ehq 93 30 coronavirus coronavirus NN cord-308115-bjyr6ehq 93 31 Estimating estimate VBG cord-308115-bjyr6ehq 93 32 the the DT cord-308115-bjyr6ehq 93 33 reproductive reproductive JJ cord-308115-bjyr6ehq 93 34 number number NN cord-308115-bjyr6ehq 93 35 and and CC cord-308115-bjyr6ehq 93 36 the the DT cord-308115-bjyr6ehq 93 37 outbreak outbreak NN cord-308115-bjyr6ehq 93 38 size size NN cord-308115-bjyr6ehq 93 39 of of IN cord-308115-bjyr6ehq 93 40 novel novel JJ cord-308115-bjyr6ehq 93 41 coronavirus coronavirus NN cord-308115-bjyr6ehq 93 42 ( ( -LRB- cord-308115-bjyr6ehq 93 43 COVID-19 covid-19 VB cord-308115-bjyr6ehq 93 44 ) ) -RRB- cord-308115-bjyr6ehq 93 45 using use VBG cord-308115-bjyr6ehq 93 46 mathematical mathematical JJ cord-308115-bjyr6ehq 93 47 model model NN cord-308115-bjyr6ehq 93 48 republic republic NN cord-308115-bjyr6ehq 93 49 of of IN cord-308115-bjyr6ehq 93 50 Korea Korea NNP cord-308115-bjyr6ehq 93 51 A a DT cord-308115-bjyr6ehq 93 52 conceptual conceptual JJ cord-308115-bjyr6ehq 93 53 model model NN cord-308115-bjyr6ehq 93 54 for for IN cord-308115-bjyr6ehq 93 55 the the DT cord-308115-bjyr6ehq 93 56 Coronavirus Coronavirus NNP cord-308115-bjyr6ehq 93 57 disease disease NN cord-308115-bjyr6ehq 93 58 2019 2019 CD cord-308115-bjyr6ehq 93 59 ( ( -LRB- cord-308115-bjyr6ehq 93 60 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 93 61 ) ) -RRB- cord-308115-bjyr6ehq 93 62 outbreak outbreak NN cord-308115-bjyr6ehq 93 63 in in IN cord-308115-bjyr6ehq 93 64 Wuhan Wuhan NNP cord-308115-bjyr6ehq 93 65 , , , cord-308115-bjyr6ehq 93 66 China China NNP cord-308115-bjyr6ehq 93 67 with with IN cord-308115-bjyr6ehq 93 68 individual individual JJ cord-308115-bjyr6ehq 93 69 reaction reaction NN cord-308115-bjyr6ehq 93 70 and and CC cord-308115-bjyr6ehq 94 1 governmental governmental JJ cord-308115-bjyr6ehq 94 2 action action NN cord-308115-bjyr6ehq 94 3 A a DT cord-308115-bjyr6ehq 94 4 mathematical mathematical JJ cord-308115-bjyr6ehq 94 5 for for IN cord-308115-bjyr6ehq 94 6 the the DT cord-308115-bjyr6ehq 94 7 novel novel JJ cord-308115-bjyr6ehq 94 8 Coronavirus Coronavirus NNP cord-308115-bjyr6ehq 94 9 epidemic epidemic NN cord-308115-bjyr6ehq 94 10 in in IN cord-308115-bjyr6ehq 94 11 Wuhan Wuhan NNP cord-308115-bjyr6ehq 95 1 What what WP cord-308115-bjyr6ehq 95 2 is be VBZ cord-308115-bjyr6ehq 95 3 fractional fractional JJ cord-308115-bjyr6ehq 95 4 Calculus Calculus NNP cord-308115-bjyr6ehq 95 5 . . . cord-308115-bjyr6ehq 96 1 Cantor Cantor NNP cord-308115-bjyr6ehq 96 2 's 's POS cord-308115-bjyr6ehq 96 3 Paradise Paradise NNP cord-308115-bjyr6ehq 96 4 A A NNP cord-308115-bjyr6ehq 96 5 survey survey NN cord-308115-bjyr6ehq 96 6 on on IN cord-308115-bjyr6ehq 96 7 existence existence NN cord-308115-bjyr6ehq 96 8 result result NN cord-308115-bjyr6ehq 96 9 for for IN cord-308115-bjyr6ehq 96 10 boundary boundary JJ cord-308115-bjyr6ehq 96 11 value value NN cord-308115-bjyr6ehq 96 12 problem problem NN cord-308115-bjyr6ehq 96 13 of of IN cord-308115-bjyr6ehq 96 14 nonlinear nonlinear JJ cord-308115-bjyr6ehq 96 15 fractional fractional JJ cord-308115-bjyr6ehq 96 16 differential differential JJ cord-308115-bjyr6ehq 96 17 equations equation NNS cord-308115-bjyr6ehq 97 1 The the DT cord-308115-bjyr6ehq 97 2 Analysis Analysis NNP cord-308115-bjyr6ehq 97 3 of of IN cord-308115-bjyr6ehq 97 4 fractional fractional JJ cord-308115-bjyr6ehq 97 5 differential differential JJ cord-308115-bjyr6ehq 97 6 equations equation NNS cord-308115-bjyr6ehq 97 7 : : : cord-308115-bjyr6ehq 97 8 an an DT cord-308115-bjyr6ehq 97 9 application application NN cord-308115-bjyr6ehq 97 10 - - HYPH cord-308115-bjyr6ehq 97 11 oriented orient VBN cord-308115-bjyr6ehq 97 12 exposition exposition NN cord-308115-bjyr6ehq 97 13 using use VBG cord-308115-bjyr6ehq 97 14 differential differential JJ cord-308115-bjyr6ehq 97 15 operators operator NNS cord-308115-bjyr6ehq 97 16 of of IN cord-308115-bjyr6ehq 97 17 Caputo Caputo NNP cord-308115-bjyr6ehq 97 18 type type NN cord-308115-bjyr6ehq 97 19 stability stability NN cord-308115-bjyr6ehq 97 20 and and CC cord-308115-bjyr6ehq 97 21 dynamics dynamic NNS cord-308115-bjyr6ehq 97 22 of of IN cord-308115-bjyr6ehq 97 23 a a DT cord-308115-bjyr6ehq 97 24 fractional fractional JJ cord-308115-bjyr6ehq 97 25 order order NN cord-308115-bjyr6ehq 97 26 Leslie Leslie NNP cord-308115-bjyr6ehq 97 27 - - HYPH cord-308115-bjyr6ehq 97 28 Gower Gower NNP cord-308115-bjyr6ehq 97 29 model model NN cord-308115-bjyr6ehq 97 30 Fractional Fractional NNP cord-308115-bjyr6ehq 97 31 Differential Differential NNP cord-308115-bjyr6ehq 97 32 Equations Equations NNPS cord-308115-bjyr6ehq 97 33 Stability Stability NNP cord-308115-bjyr6ehq 97 34 analysis analysis NN cord-308115-bjyr6ehq 97 35 of of IN cord-308115-bjyr6ehq 97 36 fractional fractional JJ cord-308115-bjyr6ehq 97 37 differential differential JJ cord-308115-bjyr6ehq 97 38 system system NN cord-308115-bjyr6ehq 97 39 with with IN cord-308115-bjyr6ehq 97 40 Rieman Rieman NNP cord-308115-bjyr6ehq 97 41 - - HYPH cord-308115-bjyr6ehq 97 42 Liouville Liouville NNP cord-308115-bjyr6ehq 97 43 derivative derivative NN cord-308115-bjyr6ehq 97 44 Differential Differential NNP cord-308115-bjyr6ehq 97 45 Equations Equations NNPS cord-308115-bjyr6ehq 97 46 : : : cord-308115-bjyr6ehq 97 47 An an DT cord-308115-bjyr6ehq 97 48 Introduction introduction NN cord-308115-bjyr6ehq 97 49 to to IN cord-308115-bjyr6ehq 97 50 Differential Differential NNP cord-308115-bjyr6ehq 97 51 Equations Equations NNPS cord-308115-bjyr6ehq 97 52 , , , cord-308115-bjyr6ehq 97 53 creative creative JJ cord-308115-bjyr6ehq 97 54 commons common NNS cord-308115-bjyr6ehq 97 55 , , , cord-308115-bjyr6ehq 97 56 94105 94105 CD cord-308115-bjyr6ehq 97 57 Differential Differential NNP cord-308115-bjyr6ehq 97 58 Equation Equation NNP cord-308115-bjyr6ehq 97 59 of of IN cord-308115-bjyr6ehq 97 60 Fractional fractional JJ cord-308115-bjyr6ehq 97 61 Order order NN cord-308115-bjyr6ehq 97 62 : : : cord-308115-bjyr6ehq 97 63 Methods method NNS cord-308115-bjyr6ehq 97 64 , , , cord-308115-bjyr6ehq 97 65 Results Results NNPS cord-308115-bjyr6ehq 97 66 and and CC cord-308115-bjyr6ehq 97 67 Problems problem NNS cord-308115-bjyr6ehq 97 68 stability stability NN cord-308115-bjyr6ehq 97 69 result result NN cord-308115-bjyr6ehq 97 70 for for IN cord-308115-bjyr6ehq 97 71 fractional fractional JJ cord-308115-bjyr6ehq 97 72 differential differential JJ cord-308115-bjyr6ehq 97 73 equations equation NNS cord-308115-bjyr6ehq 97 74 with with IN cord-308115-bjyr6ehq 97 75 application application NN cord-308115-bjyr6ehq 97 76 to to TO cord-308115-bjyr6ehq 97 77 control control VB cord-308115-bjyr6ehq 97 78 processing processing NN cord-308115-bjyr6ehq 97 79 Stability Stability NNP cord-308115-bjyr6ehq 97 80 analysis analysis NN cord-308115-bjyr6ehq 97 81 of of IN cord-308115-bjyr6ehq 97 82 Caputo Caputo NNP cord-308115-bjyr6ehq 97 83 fractional fractional JJ cord-308115-bjyr6ehq 97 84 - - HYPH cord-308115-bjyr6ehq 97 85 order order NN cord-308115-bjyr6ehq 97 86 nonlinear nonlinear JJ cord-308115-bjyr6ehq 97 87 systems system NNS cord-308115-bjyr6ehq 97 88 revisited revisit VBN cord-308115-bjyr6ehq 98 1 Lyapunov lyapunov JJ cord-308115-bjyr6ehq 98 2 function function NN cord-308115-bjyr6ehq 98 3 for for IN cord-308115-bjyr6ehq 98 4 fractional fractional JJ cord-308115-bjyr6ehq 98 5 ; ; : cord-308115-bjyr6ehq 98 6 order order NN cord-308115-bjyr6ehq 98 7 systems system NNS cord-308115-bjyr6ehq 99 1 Volterra Volterra NNP cord-308115-bjyr6ehq 99 2 type type NN cord-308115-bjyr6ehq 99 3 Lyapunov Lyapunov NNP cord-308115-bjyr6ehq 99 4 function function NN cord-308115-bjyr6ehq 99 5 for for IN cord-308115-bjyr6ehq 99 6 fractional fractional JJ cord-308115-bjyr6ehq 99 7 order order NN cord-308115-bjyr6ehq 99 8 epidemic epidemic NN cord-308115-bjyr6ehq 99 9 model model NN cord-308115-bjyr6ehq 99 10 Analysis Analysis NNP cord-308115-bjyr6ehq 99 11 of of IN cord-308115-bjyr6ehq 99 12 Caputo Caputo NNP cord-308115-bjyr6ehq 99 13 fractional fractional JJ cord-308115-bjyr6ehq 99 14 - - HYPH cord-308115-bjyr6ehq 99 15 order order NN cord-308115-bjyr6ehq 99 16 model model NN cord-308115-bjyr6ehq 99 17 for for IN cord-308115-bjyr6ehq 99 18 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 99 19 with with IN cord-308115-bjyr6ehq 99 20 lockdown lockdown NN cord-308115-bjyr6ehq 99 21 Analysis Analysis NNP cord-308115-bjyr6ehq 99 22 of of IN cord-308115-bjyr6ehq 99 23 meningitis meningitis NN cord-308115-bjyr6ehq 99 24 model model NN cord-308115-bjyr6ehq 99 25 : : : cord-308115-bjyr6ehq 100 1 A a DT cord-308115-bjyr6ehq 100 2 case case NN cord-308115-bjyr6ehq 100 3 study study NN cord-308115-bjyr6ehq 100 4 of of IN cord-308115-bjyr6ehq 100 5 northern northern JJ cord-308115-bjyr6ehq 100 6 Nigeria Nigeria NNP cord-308115-bjyr6ehq 100 7 Mathematical Mathematical NNP cord-308115-bjyr6ehq 100 8 modeling modeling NN cord-308115-bjyr6ehq 100 9 for for IN cord-308115-bjyr6ehq 100 10 adsorption adsorption NN cord-308115-bjyr6ehq 100 11 process process NN cord-308115-bjyr6ehq 100 12 of of IN cord-308115-bjyr6ehq 100 13 dye dye NN cord-308115-bjyr6ehq 100 14 removal removal NN cord-308115-bjyr6ehq 100 15 nonlinear nonlinear JJ cord-308115-bjyr6ehq 100 16 equation equation NN cord-308115-bjyr6ehq 100 17 using use VBG cord-308115-bjyr6ehq 100 18 power power NN cord-308115-bjyr6ehq 100 19 law law NN cord-308115-bjyr6ehq 100 20 and and CC cord-308115-bjyr6ehq 100 21 exponentially exponentially RB cord-308115-bjyr6ehq 100 22 decaying decay VBG cord-308115-bjyr6ehq 100 23 kernels kernel NNS cord-308115-bjyr6ehq 100 24 Modeling model VBG cord-308115-bjyr6ehq 100 25 chickenpox chickenpox NN cord-308115-bjyr6ehq 100 26 disease disease NN cord-308115-bjyr6ehq 100 27 with with IN cord-308115-bjyr6ehq 100 28 fractional fractional JJ cord-308115-bjyr6ehq 100 29 derivatives derivative NNS cord-308115-bjyr6ehq 100 30 : : : cord-308115-bjyr6ehq 100 31 From from IN cord-308115-bjyr6ehq 100 32 Caputo Caputo NNP cord-308115-bjyr6ehq 100 33 to to IN cord-308115-bjyr6ehq 100 34 Atangana Atangana NNP cord-308115-bjyr6ehq 100 35 - - HYPH cord-308115-bjyr6ehq 100 36 Baleanu Baleanu NNP cord-308115-bjyr6ehq 101 1 A a DT cord-308115-bjyr6ehq 101 2 new new JJ cord-308115-bjyr6ehq 101 3 study study NN cord-308115-bjyr6ehq 101 4 on on IN cord-308115-bjyr6ehq 101 5 the the DT cord-308115-bjyr6ehq 101 6 mathematical mathematical JJ cord-308115-bjyr6ehq 101 7 modelling modelling NN cord-308115-bjyr6ehq 101 8 of of IN cord-308115-bjyr6ehq 101 9 human human JJ cord-308115-bjyr6ehq 101 10 liver liver NN cord-308115-bjyr6ehq 101 11 with with IN cord-308115-bjyr6ehq 101 12 Caputo Caputo NNP cord-308115-bjyr6ehq 101 13 - - HYPH cord-308115-bjyr6ehq 101 14 Fabrizio Fabrizio NNP cord-308115-bjyr6ehq 101 15 fractional fractional JJ cord-308115-bjyr6ehq 101 16 derivative derivative JJ cord-308115-bjyr6ehq 101 17 A a DT cord-308115-bjyr6ehq 101 18 mathematical mathematical JJ cord-308115-bjyr6ehq 101 19 model model NN cord-308115-bjyr6ehq 101 20 for for IN cord-308115-bjyr6ehq 101 21 COVID-19 covid-19 JJ cord-308115-bjyr6ehq 101 22 transmission transmission NN cord-308115-bjyr6ehq 101 23 by by IN cord-308115-bjyr6ehq 101 24 using use VBG cord-308115-bjyr6ehq 101 25 the the DT cord-308115-bjyr6ehq 101 26 Caputo Caputo NNP cord-308115-bjyr6ehq 101 27 fractional fractional JJ cord-308115-bjyr6ehq 101 28 derivative derivative JJ cord-308115-bjyr6ehq 101 29 Epidemiological epidemiological JJ cord-308115-bjyr6ehq 101 30 and and CC cord-308115-bjyr6ehq 101 31 clinical clinical JJ cord-308115-bjyr6ehq 101 32 characteristics characteristic NNS cord-308115-bjyr6ehq 101 33 of of IN cord-308115-bjyr6ehq 101 34 the the DT cord-308115-bjyr6ehq 101 35 COVID-19 covid-19 JJ cord-308115-bjyr6ehq 101 36 epidemic epidemic NN cord-308115-bjyr6ehq 101 37 in in IN cord-308115-bjyr6ehq 101 38 Brazil Brazil NNP cord-308115-bjyr6ehq 101 39 A a DT cord-308115-bjyr6ehq 101 40 fractional fractional JJ cord-308115-bjyr6ehq 101 41 differential differential JJ cord-308115-bjyr6ehq 101 42 equation equation NN cord-308115-bjyr6ehq 101 43 model model NN cord-308115-bjyr6ehq 101 44 for for IN cord-308115-bjyr6ehq 101 45 the the DT cord-308115-bjyr6ehq 101 46 COVID-19 covid-19 JJ cord-308115-bjyr6ehq 101 47 transmission transmission NN cord-308115-bjyr6ehq 101 48 by by IN cord-308115-bjyr6ehq 101 49 using use VBG cord-308115-bjyr6ehq 101 50 the the DT cord-308115-bjyr6ehq 101 51 Caputo Caputo NNP cord-308115-bjyr6ehq 101 52 - - HYPH cord-308115-bjyr6ehq 101 53 Fabrizio Fabrizio NNP cord-308115-bjyr6ehq 101 54 derivative derivative NN cord-308115-bjyr6ehq 101 55 On on IN cord-308115-bjyr6ehq 101 56 the the DT cord-308115-bjyr6ehq 101 57 mathematical mathematical JJ cord-308115-bjyr6ehq 101 58 model model NN cord-308115-bjyr6ehq 101 59 of of IN cord-308115-bjyr6ehq 101 60 Rabies Rabies NNP cord-308115-bjyr6ehq 101 61 by by IN cord-308115-bjyr6ehq 101 62 using use VBG cord-308115-bjyr6ehq 101 63 the the DT cord-308115-bjyr6ehq 101 64 fractional fractional JJ cord-308115-bjyr6ehq 101 65 Caputo Caputo NNP cord-308115-bjyr6ehq 101 66 - - HYPH cord-308115-bjyr6ehq 101 67 Fabrizio Fabrizio NNP cord-308115-bjyr6ehq 101 68 derivative derivative JJ cord-308115-bjyr6ehq 102 1 A a DT cord-308115-bjyr6ehq 102 2 mathematical mathematical JJ cord-308115-bjyr6ehq 102 3 theoretical theoretical JJ cord-308115-bjyr6ehq 102 4 study study NN cord-308115-bjyr6ehq 102 5 of of IN cord-308115-bjyr6ehq 102 6 a a DT cord-308115-bjyr6ehq 102 7 particular particular JJ cord-308115-bjyr6ehq 102 8 system system NN cord-308115-bjyr6ehq 102 9 of of IN cord-308115-bjyr6ehq 102 10 Caputo Caputo NNP cord-308115-bjyr6ehq 102 11 - - HYPH cord-308115-bjyr6ehq 102 12 Fabrizio Fabrizio NNP cord-308115-bjyr6ehq 102 13 fractional fractional JJ cord-308115-bjyr6ehq 102 14 differential differential JJ cord-308115-bjyr6ehq 102 15 equations equation NNS cord-308115-bjyr6ehq 102 16 for for IN cord-308115-bjyr6ehq 102 17 the the DT cord-308115-bjyr6ehq 102 18 Rubella Rubella NNP cord-308115-bjyr6ehq 102 19 disease disease NN cord-308115-bjyr6ehq 102 20 model model NN cord-308115-bjyr6ehq 102 21 Analysis Analysis NNP cord-308115-bjyr6ehq 102 22 of of IN cord-308115-bjyr6ehq 102 23 the the DT cord-308115-bjyr6ehq 102 24 model model NN cord-308115-bjyr6ehq 102 25 of of IN cord-308115-bjyr6ehq 102 26 HIV-1 HIV-1 NNP cord-308115-bjyr6ehq 102 27 infection infection NN cord-308115-bjyr6ehq 102 28 of of IN cord-308115-bjyr6ehq 102 29 CD4^{+ CD4^{+ NNP cord-308115-bjyr6ehq 102 30 } } -RRB- cord-308115-bjyr6ehq 102 31 T t NN cord-308115-bjyr6ehq 102 32 - - HYPH cord-308115-bjyr6ehq 102 33 cell cell NN cord-308115-bjyr6ehq 102 34 with with IN cord-308115-bjyr6ehq 102 35 a a DT cord-308115-bjyr6ehq 102 36 new new JJ cord-308115-bjyr6ehq 102 37 approach approach NN cord-308115-bjyr6ehq 102 38 of of IN cord-308115-bjyr6ehq 102 39 fractional fractional JJ cord-308115-bjyr6ehq 102 40 derivative derivative NN cord-308115-bjyr6ehq 102 41 We -PRON- PRP cord-308115-bjyr6ehq 102 42 write write VBP cord-308115-bjyr6ehq 102 43 to to TO cord-308115-bjyr6ehq 102 44 declare declare VB cord-308115-bjyr6ehq 102 45 our -PRON- PRP$ cord-308115-bjyr6ehq 102 46 interest interest NN cord-308115-bjyr6ehq 102 47 in in IN cord-308115-bjyr6ehq 102 48 publishing publish VBG cord-308115-bjyr6ehq 102 49 our -PRON- PRP$ cord-308115-bjyr6ehq 102 50 work work NN cord-308115-bjyr6ehq 102 51 titled title VBN cord-308115-bjyr6ehq 102 52 " " `` cord-308115-bjyr6ehq 102 53 Fractional Fractional NNP cord-308115-bjyr6ehq 102 54 Order Order NNP cord-308115-bjyr6ehq 102 55 Model Model NNP cord-308115-bjyr6ehq 102 56 for for IN cord-308115-bjyr6ehq 102 57 the the DT cord-308115-bjyr6ehq 102 58 Role role NN cord-308115-bjyr6ehq 102 59 of of IN cord-308115-bjyr6ehq 102 60 Mild mild JJ cord-308115-bjyr6ehq 102 61 Cases case NNS cord-308115-bjyr6ehq 102 62 in in IN cord-308115-bjyr6ehq 102 63 the the DT cord-308115-bjyr6ehq 102 64 Transmission Transmission NNP cord-308115-bjyr6ehq 102 65 of of IN cord-308115-bjyr6ehq 102 66 COVID-19 COVID-19 NNP cord-308115-bjyr6ehq 102 67 " " '' cord-308115-bjyr6ehq 102 68 with with IN cord-308115-bjyr6ehq 102 69 the the DT cord-308115-bjyr6ehq 102 70 journal journal NN cord-308115-bjyr6ehq 102 71 " " `` cord-308115-bjyr6ehq 102 72 Chaos Chaos NNP cord-308115-bjyr6ehq 102 73 , , , cord-308115-bjyr6ehq 102 74 Solitons Solitons NNPS cord-308115-bjyr6ehq 102 75 and and CC cord-308115-bjyr6ehq 102 76 Fractals Fractals NNPS cord-308115-bjyr6ehq 102 77 " " '' cord-308115-bjyr6ehq 102 78 . . .