id	sid	tid	token	lemma	pos
ijassa-1172	1	1	adv	adv	PROPN
ijassa-1172	1	2	syst	syst	PROPN
ijassa-1172	1	3	sci	sci	PROPN
ijassa-1172	1	4	appl	appl	PROPN
ijassa-1172	1	5	2022	2022	NUM
ijassa-1172	1	6	;	;	PUNCT
ijassa-1172	1	7	03:36–48	03:36–48	PROPN
ijassa-1172	1	8	published	publish	VERB
ijassa-1172	1	9	online	online	ADV
ijassa-1172	1	10	at	at	ADP
ijassa-1172	1	11	https://ijassa.ipu.ru	https://ijassa.ipu.ru	ADV
ijassa-1172	1	12	.	.	PUNCT
ijassa-1172	2	1	dynamics	dynamic	NOUN
ijassa-1172	2	2	of	of	ADP
ijassa-1172	2	3	a	a	DET
ijassa-1172	2	4	generalized	generalize	VERB
ijassa-1172	2	5	fractional	fractional	ADJ
ijassa-1172	2	6	epidemic	epidemic	NOUN
ijassa-1172	2	7	model	model	NOUN
ijassa-1172	2	8	of	of	ADP
ijassa-1172	2	9	covid-19	covid-19	PROPN
ijassa-1172	2	10	with	with	ADP
ijassa-1172	2	11	carrier	carrier	NOUN
ijassa-1172	2	12	effect	effect	PROPN
ijassa-1172	2	13	khalid	khalid	PROPN
ijassa-1172	2	14	hattaf1,2	hattaf1,2	PROPN
ijassa-1172	2	15	,	,	PUNCT
ijassa-1172	2	16	ahmed	ahmed	PROPN
ijassa-1172	2	17	a.	a.	PROPN
ijassa-1172	2	18	mohsen3,4	mohsen3,4	PROPN
ijassa-1172	2	19	1centre	1centre	PROPN
ijassa-1172	2	20	régional	régional	PROPN
ijassa-1172	2	21	des	des	X
ijassa-1172	2	22	métiers	métiers	PROPN
ijassa-1172	2	23	de	de	X
ijassa-1172	2	24	l’education	l’education	NOUN
ijassa-1172	2	25	et	et	NOUN
ijassa-1172	2	26	de	de	X
ijassa-1172	2	27	la	la	X
ijassa-1172	2	28	formation	formation	NOUN
ijassa-1172	2	29	(	(	PUNCT
ijassa-1172	2	30	crmef	crmef	PROPN
ijassa-1172	2	31	)	)	PUNCT
ijassa-1172	2	32	,	,	PUNCT
ijassa-1172	2	33	casablanca	casablanca	PROPN
ijassa-1172	2	34	,	,	PUNCT
ijassa-1172	2	35	morocco	morocco	PROPN
ijassa-1172	2	36	2	2	NUM
ijassa-1172	2	37	laboratory	laboratory	NOUN
ijassa-1172	2	38	of	of	ADP
ijassa-1172	2	39	analysis	analysis	NOUN
ijassa-1172	2	40	,	,	PUNCT
ijassa-1172	2	41	modeling	modeling	NOUN
ijassa-1172	2	42	and	and	CCONJ
ijassa-1172	2	43	simulation	simulation	NOUN
ijassa-1172	2	44	(	(	PUNCT
ijassa-1172	2	45	lams	lams	PROPN
ijassa-1172	2	46	)	)	PUNCT
ijassa-1172	2	47	,	,	PUNCT
ijassa-1172	2	48	hassan	hassan	PROPN
ijassa-1172	2	49	ii	ii	PROPN
ijassa-1172	2	50	university	university	PROPN
ijassa-1172	2	51	of	of	ADP
ijassa-1172	2	52	casablanca	casablanca	PROPN
ijassa-1172	2	53	,	,	PUNCT
ijassa-1172	2	54	casablanca	casablanca	PROPN
ijassa-1172	2	55	,	,	PUNCT
ijassa-1172	2	56	morocco	morocco	PROPN
ijassa-1172	2	57	3	3	NUM
ijassa-1172	2	58	department	department	NOUN
ijassa-1172	2	59	of	of	ADP
ijassa-1172	2	60	mathematics	mathematics	PROPN
ijassa-1172	2	61	,	,	PUNCT
ijassa-1172	2	62	college	college	NOUN
ijassa-1172	2	63	of	of	ADP
ijassa-1172	2	64	education	education	NOUN
ijassa-1172	2	65	for	for	ADP
ijassa-1172	2	66	pure	pure	ADJ
ijassa-1172	2	67	science	science	NOUN
ijassa-1172	2	68	(	(	PUNCT
ijassa-1172	2	69	ibn	ibn	PROPN
ijassa-1172	2	70	al	al	PROPN
ijassa-1172	2	71	-	-	PUNCT
ijassa-1172	2	72	haitham	haitham	PROPN
ijassa-1172	2	73	)	)	PUNCT
ijassa-1172	2	74	,	,	PUNCT
ijassa-1172	2	75	university	university	NOUN
ijassa-1172	2	76	of	of	ADP
ijassa-1172	2	77	baghdad	baghdad	PROPN
ijassa-1172	2	78	,	,	PUNCT
ijassa-1172	2	79	iraq	iraq	PROPN
ijassa-1172	2	80	2	2	NUM
ijassa-1172	2	81	ministry	ministry	PROPN
ijassa-1172	2	82	of	of	ADP
ijassa-1172	2	83	education	education	PROPN
ijassa-1172	2	84	,	,	PUNCT
ijassa-1172	2	85	baghdad	baghdad	PROPN
ijassa-1172	2	86	,	,	PUNCT
ijassa-1172	2	87	iraq	iraq	PROPN
ijassa-1172	2	88	abstract	abstract	NOUN
ijassa-1172	2	89	:	:	PUNCT
ijassa-1172	2	90	currently	currently	ADV
ijassa-1172	2	91	,	,	PUNCT
ijassa-1172	2	92	coronavirus	coronavirus	NOUN
ijassa-1172	2	93	disease	disease	NOUN
ijassa-1172	2	94	2019	2019	NUM
ijassa-1172	2	95	(	(	PUNCT
ijassa-1172	2	96	covid-19	covid-19	PROPN
ijassa-1172	2	97	)	)	PUNCT
ijassa-1172	2	98	continues	continue	VERB
ijassa-1172	2	99	to	to	PART
ijassa-1172	2	100	cause	cause	VERB
ijassa-1172	2	101	several	several	ADJ
ijassa-1172	2	102	new	new	ADJ
ijassa-1172	2	103	cases	case	NOUN
ijassa-1172	2	104	and	and	CCONJ
ijassa-1172	2	105	deaths	death	NOUN
ijassa-1172	2	106	,	,	PUNCT
ijassa-1172	2	107	and	and	CCONJ
ijassa-1172	2	108	further	further	ADJ
ijassa-1172	2	109	new	new	ADJ
ijassa-1172	2	110	variants	variant	NOUN
ijassa-1172	2	111	of	of	ADP
ijassa-1172	2	112	the	the	DET
ijassa-1172	2	113	virus	virus	NOUN
ijassa-1172	2	114	responsible	responsible	ADJ
ijassa-1172	2	115	this	this	DET
ijassa-1172	2	116	disease	disease	NOUN
ijassa-1172	2	117	appear	appear	VERB
ijassa-1172	2	118	.	.	PUNCT
ijassa-1172	3	1	the	the	DET
ijassa-1172	3	2	present	present	ADJ
ijassa-1172	3	3	paper	paper	NOUN
ijassa-1172	3	4	proposes	propose	VERB
ijassa-1172	3	5	a	a	DET
ijassa-1172	3	6	new	new	ADJ
ijassa-1172	3	7	mathematical	mathematical	ADJ
ijassa-1172	3	8	model	model	NOUN
ijassa-1172	3	9	to	to	PART
ijassa-1172	3	10	understand	understand	VERB
ijassa-1172	3	11	the	the	DET
ijassa-1172	3	12	mechanisms	mechanism	NOUN
ijassa-1172	3	13	of	of	ADP
ijassa-1172	3	14	the	the	DET
ijassa-1172	3	15	spread	spread	NOUN
ijassa-1172	3	16	of	of	ADP
ijassa-1172	3	17	covid-19	covid-19	PROPN
ijassa-1172	3	18	and	and	CCONJ
ijassa-1172	3	19	better	well	ADV
ijassa-1172	3	20	describe	describe	VERB
ijassa-1172	3	21	its	its	PRON
ijassa-1172	3	22	dynamics	dynamic	NOUN
ijassa-1172	3	23	.	.	PUNCT
ijassa-1172	4	1	the	the	DET
ijassa-1172	4	2	modes	mode	NOUN
ijassa-1172	4	3	of	of	ADP
ijassa-1172	4	4	infection	infection	NOUN
ijassa-1172	4	5	spread	spread	NOUN
ijassa-1172	4	6	of	of	ADP
ijassa-1172	4	7	covid-19	covid-19	PROPN
ijassa-1172	4	8	via	via	ADP
ijassa-1172	4	9	asymptomatic	asymptomatic	ADJ
ijassa-1172	4	10	and	and	CCONJ
ijassa-1172	4	11	symptomatic	symptomatic	ADJ
ijassa-1172	4	12	individuals	individual	NOUN
ijassa-1172	4	13	are	be	AUX
ijassa-1172	4	14	modeled	model	VERB
ijassa-1172	4	15	by	by	ADP
ijassa-1172	4	16	two	two	NUM
ijassa-1172	4	17	general	general	ADJ
ijassa-1172	4	18	nonlinear	nonlinear	ADJ
ijassa-1172	4	19	incidence	incidence	NOUN
ijassa-1172	4	20	functions	function	NOUN
ijassa-1172	4	21	in	in	ADP
ijassa-1172	4	22	order	order	NOUN
ijassa-1172	4	23	to	to	PART
ijassa-1172	4	24	include	include	VERB
ijassa-1172	4	25	several	several	ADJ
ijassa-1172	4	26	types	type	NOUN
ijassa-1172	4	27	of	of	ADP
ijassa-1172	4	28	incidence	incidence	NOUN
ijassa-1172	4	29	rates	rate	NOUN
ijassa-1172	4	30	existing	exist	VERB
ijassa-1172	4	31	in	in	ADP
ijassa-1172	4	32	the	the	DET
ijassa-1172	4	33	literature	literature	NOUN
ijassa-1172	4	34	.	.	PUNCT
ijassa-1172	5	1	when	when	SCONJ
ijassa-1172	5	2	a	a	DET
ijassa-1172	5	3	disease	disease	NOUN
ijassa-1172	5	4	outbreak	outbreak	NOUN
ijassa-1172	5	5	within	within	ADP
ijassa-1172	5	6	a	a	DET
ijassa-1172	5	7	community	community	NOUN
ijassa-1172	5	8	,	,	PUNCT
ijassa-1172	5	9	individuals	individual	NOUN
ijassa-1172	5	10	acquire	acquire	VERB
ijassa-1172	5	11	information	information	NOUN
ijassa-1172	5	12	about	about	ADP
ijassa-1172	5	13	this	this	DET
ijassa-1172	5	14	disease	disease	NOUN
ijassa-1172	5	15	.	.	PUNCT
ijassa-1172	6	1	therefore	therefore	ADV
ijassa-1172	6	2	,	,	PUNCT
ijassa-1172	6	3	the	the	DET
ijassa-1172	6	4	proposed	propose	VERB
ijassa-1172	6	5	model	model	NOUN
ijassa-1172	6	6	take	take	VERB
ijassa-1172	6	7	into	into	ADP
ijassa-1172	6	8	account	account	NOUN
ijassa-1172	6	9	the	the	DET
ijassa-1172	6	10	memory	memory	NOUN
ijassa-1172	6	11	effect	effect	NOUN
ijassa-1172	6	12	on	on	ADP
ijassa-1172	6	13	the	the	DET
ijassa-1172	6	14	outbreaks	outbreak	NOUN
ijassa-1172	6	15	of	of	ADP
ijassa-1172	6	16	covid-19	covid-19	PROPN
ijassa-1172	6	17	.	.	PUNCT
ijassa-1172	7	1	this	this	DET
ijassa-1172	7	2	effect	effect	NOUN
ijassa-1172	7	3	is	be	AUX
ijassa-1172	7	4	modeled	model	VERB
ijassa-1172	7	5	by	by	ADP
ijassa-1172	7	6	a	a	DET
ijassa-1172	7	7	fractional	fractional	ADJ
ijassa-1172	7	8	order	order	NOUN
ijassa-1172	7	9	derivative	derivative	NOUN
ijassa-1172	7	10	in	in	ADP
ijassa-1172	7	11	caputo	caputo	PROPN
ijassa-1172	7	12	sense	sense	NOUN
ijassa-1172	7	13	.	.	PUNCT
ijassa-1172	8	1	the	the	DET
ijassa-1172	8	2	mathematical	mathematical	ADJ
ijassa-1172	8	3	analysis	analysis	NOUN
ijassa-1172	8	4	of	of	ADP
ijassa-1172	8	5	the	the	DET
ijassa-1172	8	6	proposed	propose	VERB
ijassa-1172	8	7	model	model	NOUN
ijassa-1172	8	8	is	be	AUX
ijassa-1172	8	9	rigorously	rigorously	ADV
ijassa-1172	8	10	investigated	investigate	VERB
ijassa-1172	8	11	,	,	PUNCT
ijassa-1172	8	12	including	include	VERB
ijassa-1172	8	13	the	the	DET
ijassa-1172	8	14	computation	computation	NOUN
ijassa-1172	8	15	of	of	ADP
ijassa-1172	8	16	the	the	DET
ijassa-1172	8	17	basic	basic	ADJ
ijassa-1172	8	18	reproduction	reproduction	NOUN
ijassa-1172	8	19	number	number	NOUN
ijassa-1172	8	20	r0	r0	NOUN
ijassa-1172	8	21	and	and	CCONJ
ijassa-1172	8	22	the	the	DET
ijassa-1172	8	23	stability	stability	NOUN
ijassa-1172	8	24	of	of	ADP
ijassa-1172	8	25	equilibria	equilibrium	NOUN
ijassa-1172	8	26	.	.	PUNCT
ijassa-1172	9	1	keywords	keyword	NOUN
ijassa-1172	9	2	:	:	PUNCT
ijassa-1172	9	3	covid-19	covid-19	PROPN
ijassa-1172	9	4	,	,	PUNCT
ijassa-1172	9	5	sars	sar	NOUN
ijassa-1172	9	6	-	-	PUNCT
ijassa-1172	9	7	cov-2	cov-2	NOUN
ijassa-1172	9	8	,	,	PUNCT
ijassa-1172	9	9	incidence	incidence	NOUN
ijassa-1172	9	10	rate	rate	NOUN
ijassa-1172	9	11	,	,	PUNCT
ijassa-1172	9	12	basic	basic	ADJ
ijassa-1172	9	13	reproduction	reproduction	NOUN
ijassa-1172	9	14	number	number	NOUN
ijassa-1172	9	15	,	,	PUNCT
ijassa-1172	9	16	fractional	fractional	ADJ
ijassa-1172	9	17	differential	differential	ADJ
ijassa-1172	9	18	equations	equation	NOUN
ijassa-1172	9	19	,	,	PUNCT
ijassa-1172	9	20	stability	stability	NOUN
ijassa-1172	9	21	analysis	analysis	NOUN
ijassa-1172	9	22	.	.	PUNCT
ijassa-1172	10	1	1	1	X
ijassa-1172	10	2	.	.	X
ijassa-1172	10	3	introduction	introduction	NOUN
ijassa-1172	10	4	coronavirus	coronavirus	NOUN
ijassa-1172	10	5	disease	disease	NOUN
ijassa-1172	10	6	2019	2019	NUM
ijassa-1172	10	7	(	(	PUNCT
ijassa-1172	10	8	covid-19	covid-19	PROPN
ijassa-1172	10	9	)	)	PUNCT
ijassa-1172	10	10	is	be	AUX
ijassa-1172	10	11	a	a	DET
ijassa-1172	10	12	infectious	infectious	ADJ
ijassa-1172	10	13	disease	disease	NOUN
ijassa-1172	10	14	caused	cause	VERB
ijassa-1172	10	15	by	by	ADP
ijassa-1172	10	16	a	a	DET
ijassa-1172	10	17	new	new	ADJ
ijassa-1172	10	18	coronavirus	coronavirus	NOUN
ijassa-1172	10	19	called	call	VERB
ijassa-1172	10	20	severe	severe	ADJ
ijassa-1172	10	21	acute	acute	ADJ
ijassa-1172	10	22	respiratory	respiratory	ADJ
ijassa-1172	10	23	syndrome	syndrome	NOUN
ijassa-1172	10	24	coronavirus	coronavirus	NOUN
ijassa-1172	10	25	2	2	NUM
ijassa-1172	10	26	(	(	PUNCT
ijassa-1172	10	27	sars	sar	NOUN
ijassa-1172	10	28	-	-	PUNCT
ijassa-1172	10	29	cov-2	cov-2	NOUN
ijassa-1172	10	30	)	)	PUNCT
ijassa-1172	10	31	.	.	PUNCT
ijassa-1172	11	1	world	world	PROPN
ijassa-1172	11	2	health	health	PROPN
ijassa-1172	11	3	organization	organization	NOUN
ijassa-1172	11	4	(	(	PUNCT
ijassa-1172	11	5	who	who	PRON
ijassa-1172	11	6	)	)	PUNCT
ijassa-1172	11	7	first	first	ADV
ijassa-1172	11	8	learned	learn	VERB
ijassa-1172	11	9	of	of	ADP
ijassa-1172	11	10	this	this	DET
ijassa-1172	11	11	new	new	ADJ
ijassa-1172	11	12	virus	virus	NOUN
ijassa-1172	11	13	on	on	ADP
ijassa-1172	11	14	31	31	NUM
ijassa-1172	11	15	december	december	PROPN
ijassa-1172	11	16	2019	2019	NUM
ijassa-1172	11	17	,	,	PUNCT
ijassa-1172	11	18	following	follow	VERB
ijassa-1172	11	19	a	a	DET
ijassa-1172	11	20	report	report	NOUN
ijassa-1172	11	21	of	of	ADP
ijassa-1172	11	22	a	a	DET
ijassa-1172	11	23	cluster	cluster	NOUN
ijassa-1172	11	24	of	of	ADP
ijassa-1172	11	25	cases	case	NOUN
ijassa-1172	11	26	of	of	ADP
ijassa-1172	11	27	viral	viral	ADJ
ijassa-1172	11	28	pneumonia	pneumonia	NOUN
ijassa-1172	11	29	detected	detect	VERB
ijassa-1172	11	30	in	in	ADP
ijassa-1172	11	31	wuhan	wuhan	PROPN
ijassa-1172	11	32	city	city	PROPN
ijassa-1172	11	33	,	,	PUNCT
ijassa-1172	11	34	hubei	hubei	PROPN
ijassa-1172	11	35	province	province	PROPN
ijassa-1172	11	36	of	of	ADP
ijassa-1172	11	37	china	china	PROPN
ijassa-1172	11	38	.	.	PUNCT
ijassa-1172	12	1	due	due	ADP
ijassa-1172	12	2	to	to	ADP
ijassa-1172	12	3	rapid	rapid	ADJ
ijassa-1172	12	4	worldwide	worldwide	ADJ
ijassa-1172	12	5	outbreak	outbreak	NOUN
ijassa-1172	12	6	of	of	ADP
ijassa-1172	12	7	covid-19	covid-19	PROPN
ijassa-1172	12	8	,	,	PUNCT
ijassa-1172	12	9	the	the	PRON
ijassa-1172	12	10	who	who	PRON
ijassa-1172	12	11	declared	declare	VERB
ijassa-1172	12	12	it	it	PRON
ijassa-1172	12	13	a	a	DET
ijassa-1172	12	14	pandemic	pandemic	NOUN
ijassa-1172	12	15	on	on	ADP
ijassa-1172	12	16	11	11	NUM
ijassa-1172	12	17	march	march	NOUN
ijassa-1172	12	18	2020	2020	NUM
ijassa-1172	12	19	.	.	PUNCT
ijassa-1172	13	1	over	over	ADP
ijassa-1172	13	2	2.7	2.7	NUM
ijassa-1172	13	3	million	million	NUM
ijassa-1172	13	4	new	new	ADJ
ijassa-1172	13	5	cases	case	NOUN
ijassa-1172	13	6	were	be	AUX
ijassa-1172	13	7	reported	report	VERB
ijassa-1172	13	8	on	on	ADP
ijassa-1172	13	9	7	7	NUM
ijassa-1172	13	10	march	march	NOUN
ijassa-1172	13	11	2021	2021	NUM
ijassa-1172	13	12	with	with	ADP
ijassa-1172	13	13	116	116	NUM
ijassa-1172	13	14	166	166	NUM
ijassa-1172	13	15	652	652	NUM
ijassa-1172	13	16	cumulative	cumulative	ADJ
ijassa-1172	13	17	cases	case	NOUN
ijassa-1172	13	18	,	,	PUNCT
ijassa-1172	13	19	60	60	NUM
ijassa-1172	13	20	323	323	NUM
ijassa-1172	13	21	new	new	ADJ
ijassa-1172	13	22	deaths	death	NOUN
ijassa-1172	13	23	and	and	CCONJ
ijassa-1172	13	24	2	2	NUM
ijassa-1172	13	25	582	582	NUM
ijassa-1172	13	26	528	528	NUM
ijassa-1172	13	27	cumulative	cumulative	ADJ
ijassa-1172	13	28	deaths	death	NOUN
ijassa-1172	13	29	[	[	X
ijassa-1172	13	30	18	18	NUM
ijassa-1172	13	31	]	]	PUNCT
ijassa-1172	13	32	.	.	PUNCT
ijassa-1172	14	1	since	since	SCONJ
ijassa-1172	14	2	the	the	DET
ijassa-1172	14	3	appearance	appearance	NOUN
ijassa-1172	14	4	of	of	ADP
ijassa-1172	14	5	the	the	DET
ijassa-1172	14	6	first	first	ADJ
ijassa-1172	14	7	case	case	NOUN
ijassa-1172	14	8	in	in	ADP
ijassa-1172	14	9	morocco	morocco	PROPN
ijassa-1172	14	10	on	on	ADP
ijassa-1172	14	11	2	2	NUM
ijassa-1172	14	12	march	march	NOUN
ijassa-1172	14	13	2020	2020	NUM
ijassa-1172	14	14	in	in	ADP
ijassa-1172	14	15	the	the	DET
ijassa-1172	14	16	city	city	NOUN
ijassa-1172	14	17	of	of	ADP
ijassa-1172	14	18	casablanca	casablanca	PROPN
ijassa-1172	14	19	,	,	PUNCT
ijassa-1172	14	20	the	the	DET
ijassa-1172	14	21	disease	disease	NOUN
ijassa-1172	14	22	has	have	AUX
ijassa-1172	14	23	caused	cause	VERB
ijassa-1172	14	24	8	8	NUM
ijassa-1172	14	25	716	716	NUM
ijassa-1172	14	26	deaths	death	NOUN
ijassa-1172	14	27	on	on	ADP
ijassa-1172	14	28	12	12	NUM
ijassa-1172	14	29	march	march	NOUN
ijassa-1172	14	30	2021	2021	NUM
ijassa-1172	14	31	according	accord	VERB
ijassa-1172	14	32	to	to	ADP
ijassa-1172	14	33	new	new	ADJ
ijassa-1172	14	34	statistics	statistic	NOUN
ijassa-1172	14	35	given	give	VERB
ijassa-1172	14	36	by	by	ADP
ijassa-1172	14	37	moroccan	moroccan	PROPN
ijassa-1172	14	38	ministry	ministry	PROPN
ijassa-1172	14	39	of	of	ADP
ijassa-1172	14	40	health	health	PROPN
ijassa-1172	14	41	(	(	PUNCT
ijassa-1172	14	42	mmh	mmh	X
ijassa-1172	14	43	)	)	PUNCT
ijassa-1172	15	1	[	[	X
ijassa-1172	15	2	13	13	NUM
ijassa-1172	15	3	]	]	PUNCT
ijassa-1172	15	4	.	.	PUNCT
ijassa-1172	16	1	however	however	ADV
ijassa-1172	16	2	,	,	PUNCT
ijassa-1172	16	3	a	a	DET
ijassa-1172	16	4	total	total	NOUN
ijassa-1172	16	5	of	of	ADP
ijassa-1172	16	6	13	13	NUM
ijassa-1172	16	7	671	671	NUM
ijassa-1172	16	8	deaths	death	NOUN
ijassa-1172	16	9	was	be	AUX
ijassa-1172	16	10	confirmed	confirm	VERB
ijassa-1172	16	11	in	in	ADP
ijassa-1172	16	12	iraq	iraq	PROPN
ijassa-1172	16	13	on	on	ADP
ijassa-1172	16	14	the	the	DET
ijassa-1172	16	15	same	same	ADJ
ijassa-1172	16	16	day	day	NOUN
ijassa-1172	16	17	[	[	X
ijassa-1172	16	18	19	19	NUM
ijassa-1172	16	19	]	]	PUNCT
ijassa-1172	16	20	.	.	PUNCT
ijassa-1172	17	1	the	the	DET
ijassa-1172	17	2	outbreak	outbreak	NOUN
ijassa-1172	17	3	covid-19	covid-19	PROPN
ijassa-1172	17	4	impacted	impact	VERB
ijassa-1172	17	5	nearly	nearly	ADV
ijassa-1172	17	6	every	every	PRON
ijassa-1172	17	7	side	side	NOUN
ijassa-1172	17	8	of	of	ADP
ijassa-1172	17	9	society	society	NOUN
ijassa-1172	17	10	worldwide	worldwide	ADV
ijassa-1172	17	11	.	.	PUNCT
ijassa-1172	18	1	therefore	therefore	ADV
ijassa-1172	18	2	,	,	PUNCT
ijassa-1172	18	3	the	the	DET
ijassa-1172	18	4	pandemic	pandemic	NOUN
ijassa-1172	18	5	has	have	AUX
ijassa-1172	18	6	required	require	VERB
ijassa-1172	18	7	a	a	DET
ijassa-1172	18	8	substantial	substantial	ADJ
ijassa-1172	18	9	response	response	NOUN
ijassa-1172	18	10	by	by	ADP
ijassa-1172	18	11	public	public	ADJ
ijassa-1172	18	12	health	health	NOUN
ijassa-1172	18	13	authorities	authority	NOUN
ijassa-1172	18	14	at	at	ADP
ijassa-1172	18	15	every	every	DET
ijassa-1172	18	16	level	level	NOUN
ijassa-1172	18	17	and	and	CCONJ
ijassa-1172	18	18	considered	consider	VERB
ijassa-1172	18	19	aformidable	aformidable	ADJ
ijassa-1172	18	20	global	global	ADJ
ijassa-1172	18	21	public	public	ADJ
ijassa-1172	18	22	health	health	NOUN
ijassa-1172	18	23	challenge	challenge	NOUN
ijassa-1172	18	24	.	.	PUNCT
ijassa-1172	19	1	∗corresponding	∗corresponde	VERB
ijassa-1172	19	2	author	author	NOUN
ijassa-1172	19	3	:	:	PUNCT
ijassa-1172	19	4	corresponding	correspond	VERB
ijassa-1172	19	5	author	author	NOUN
ijassa-1172	19	6	:	:	PUNCT
ijassa-1172	19	7	aamuhseen@gmail.com	aamuhseen@gmail.com	X
ijassa-1172	19	8	dynamics	dynamic	NOUN
ijassa-1172	19	9	of	of	ADP
ijassa-1172	19	10	a	a	DET
ijassa-1172	19	11	generalized	generalize	VERB
ijassa-1172	19	12	fractional	fractional	ADJ
ijassa-1172	19	13	epidemic	epidemic	NOUN
ijassa-1172	19	14	model	model	NOUN
ijassa-1172	19	15	...	...	PUNCT
ijassa-1172	19	16	37	37	NUM
ijassa-1172	19	17	despite	despite	SCONJ
ijassa-1172	19	18	the	the	DET
ijassa-1172	19	19	prevention	prevention	NOUN
ijassa-1172	19	20	measures	measure	NOUN
ijassa-1172	19	21	,	,	PUNCT
ijassa-1172	19	22	such	such	ADJ
ijassa-1172	19	23	as	as	ADP
ijassa-1172	19	24	hand	hand	NOUN
ijassa-1172	19	25	washing	washing	NOUN
ijassa-1172	19	26	,	,	PUNCT
ijassa-1172	19	27	social	social	ADJ
ijassa-1172	19	28	distancing	distancing	NOUN
ijassa-1172	19	29	,	,	PUNCT
ijassa-1172	19	30	quarantine	quarantine	NOUN
ijassa-1172	19	31	of	of	ADP
ijassa-1172	19	32	suspected	suspect	VERB
ijassa-1172	19	33	cases	case	NOUN
ijassa-1172	19	34	,	,	PUNCT
ijassa-1172	19	35	isolation	isolation	NOUN
ijassa-1172	19	36	of	of	ADP
ijassa-1172	19	37	confirmed	confirm	VERB
ijassa-1172	19	38	cases	case	NOUN
ijassa-1172	19	39	,	,	PUNCT
ijassa-1172	19	40	the	the	DET
ijassa-1172	19	41	closure	closure	NOUN
ijassa-1172	19	42	of	of	ADP
ijassa-1172	19	43	schools	school	NOUN
ijassa-1172	19	44	and	and	CCONJ
ijassa-1172	19	45	non	non	ADJ
ijassa-1172	19	46	-	-	ADJ
ijassa-1172	19	47	essential	essential	ADJ
ijassa-1172	19	48	businesses	business	NOUN
ijassa-1172	19	49	and	and	CCONJ
ijassa-1172	19	50	services	service	NOUN
ijassa-1172	19	51	,	,	PUNCT
ijassa-1172	19	52	and	and	CCONJ
ijassa-1172	19	53	also	also	ADV
ijassa-1172	19	54	the	the	DET
ijassa-1172	19	55	start	start	NOUN
ijassa-1172	19	56	of	of	ADP
ijassa-1172	19	57	vaccination	vaccination	NOUN
ijassa-1172	19	58	in	in	ADP
ijassa-1172	19	59	some	some	DET
ijassa-1172	19	60	countries	country	NOUN
ijassa-1172	19	61	,	,	PUNCT
ijassa-1172	19	62	covid-19	covid-19	PROPN
ijassa-1172	19	63	continues	continue	VERB
ijassa-1172	19	64	to	to	PART
ijassa-1172	19	65	cause	cause	VERB
ijassa-1172	19	66	new	new	ADJ
ijassa-1172	19	67	cases	case	NOUN
ijassa-1172	19	68	and	and	CCONJ
ijassa-1172	19	69	deaths	death	NOUN
ijassa-1172	19	70	,	,	PUNCT
ijassa-1172	19	71	and	and	CCONJ
ijassa-1172	19	72	new	new	ADJ
ijassa-1172	19	73	variants	variant	NOUN
ijassa-1172	19	74	of	of	ADP
ijassa-1172	19	75	sars	sar	NOUN
ijassa-1172	19	76	-	-	PUNCT
ijassa-1172	19	77	cov-2	cov-2	PART
ijassa-1172	19	78	appear	appear	VERB
ijassa-1172	19	79	.	.	PUNCT
ijassa-1172	20	1	in	in	ADP
ijassa-1172	20	2	fact	fact	NOUN
ijassa-1172	20	3	,	,	PUNCT
ijassa-1172	20	4	the	the	DET
ijassa-1172	20	5	transmission	transmission	NOUN
ijassa-1172	20	6	or	or	CCONJ
ijassa-1172	20	7	outbreaks	outbreak	NOUN
ijassa-1172	20	8	of	of	ADP
ijassa-1172	20	9	the	the	DET
ijassa-1172	20	10	covid-19	covid-19	PROPN
ijassa-1172	20	11	consists	consist	VERB
ijassa-1172	20	12	of	of	ADP
ijassa-1172	20	13	direct	direct	ADJ
ijassa-1172	20	14	and	and	CCONJ
ijassa-1172	20	15	indirect	indirect	ADJ
ijassa-1172	20	16	contacts	contact	NOUN
ijassa-1172	20	17	between	between	ADP
ijassa-1172	20	18	infective	infective	ADJ
ijassa-1172	20	19	and	and	CCONJ
ijassa-1172	20	20	susceptible	susceptible	ADJ
ijassa-1172	20	21	individuals	individual	NOUN
ijassa-1172	20	22	,	,	PUNCT
ijassa-1172	20	23	it	it	PRON
ijassa-1172	20	24	seems	seem	VERB
ijassa-1172	20	25	that	that	SCONJ
ijassa-1172	20	26	the	the	DET
ijassa-1172	20	27	several	several	ADJ
ijassa-1172	20	28	mathematical	mathematical	ADJ
ijassa-1172	20	29	models	model	NOUN
ijassa-1172	20	30	well	well	ADV
ijassa-1172	20	31	described	describe	VERB
ijassa-1172	20	32	the	the	DET
ijassa-1172	20	33	covid-19	covid-19	PROPN
ijassa-1172	20	34	.	.	PUNCT
ijassa-1172	21	1	however	however	ADV
ijassa-1172	21	2	,	,	PUNCT
ijassa-1172	21	3	attention	attention	NOUN
ijassa-1172	21	4	must	must	AUX
ijassa-1172	21	5	be	be	AUX
ijassa-1172	21	6	paid	pay	VERB
ijassa-1172	21	7	to	to	ADP
ijassa-1172	21	8	the	the	DET
ijassa-1172	21	9	unique	unique	ADJ
ijassa-1172	21	10	traits	trait	NOUN
ijassa-1172	21	11	of	of	ADP
ijassa-1172	21	12	covid-19	covid-19	PROPN
ijassa-1172	21	13	.	.	PUNCT
ijassa-1172	22	1	these	these	DET
ijassa-1172	22	2	traits	trait	NOUN
ijassa-1172	22	3	are	be	AUX
ijassa-1172	22	4	(	(	PUNCT
ijassa-1172	22	5	a	a	X
ijassa-1172	22	6	)	)	PUNCT
ijassa-1172	22	7	the	the	DET
ijassa-1172	22	8	presence	presence	NOUN
ijassa-1172	22	9	of	of	ADP
ijassa-1172	22	10	individuals	individual	NOUN
ijassa-1172	22	11	who	who	PRON
ijassa-1172	22	12	exhibit	exhibit	VERB
ijassa-1172	22	13	infectious	infectious	ADJ
ijassa-1172	22	14	symptoms	symptom	NOUN
ijassa-1172	22	15	even	even	ADV
ijassa-1172	22	16	during	during	ADP
ijassa-1172	22	17	the	the	DET
ijassa-1172	22	18	incubation	incubation	NOUN
ijassa-1172	22	19	period	period	NOUN
ijassa-1172	22	20	,	,	PUNCT
ijassa-1172	22	21	such	such	ADJ
ijassa-1172	22	22	as	as	ADP
ijassa-1172	22	23	such	such	ADJ
ijassa-1172	22	24	as	as	ADP
ijassa-1172	22	25	high	high	ADJ
ijassa-1172	22	26	body	body	NOUN
ijassa-1172	22	27	temperature	temperature	NOUN
ijassa-1172	22	28	,	,	PUNCT
ijassa-1172	22	29	body	body	NOUN
ijassa-1172	22	30	aches	ache	NOUN
ijassa-1172	22	31	and	and	CCONJ
ijassa-1172	22	32	pains	pain	NOUN
ijassa-1172	22	33	,	,	PUNCT
ijassa-1172	22	34	queasiness	queasiness	NOUN
ijassa-1172	22	35	,	,	PUNCT
ijassa-1172	22	36	or	or	CCONJ
ijassa-1172	22	37	diarrhea	diarrhea	NOUN
ijassa-1172	22	38	typically	typically	ADV
ijassa-1172	22	39	(	(	PUNCT
ijassa-1172	22	40	2	2	NUM
ijassa-1172	22	41	14	14	NUM
ijassa-1172	22	42	)	)	PUNCT
ijassa-1172	22	43	days	day	NOUN
ijassa-1172	22	44	after	after	SCONJ
ijassa-1172	22	45	infective	infective	ADJ
ijassa-1172	22	46	to	to	ADP
ijassa-1172	22	47	the	the	DET
ijassa-1172	22	48	virus	virus	NOUN
ijassa-1172	22	49	,	,	PUNCT
ijassa-1172	22	50	(	(	PUNCT
ijassa-1172	22	51	b	b	X
ijassa-1172	22	52	)	)	PUNCT
ijassa-1172	22	53	there	there	PRON
ijassa-1172	22	54	are	be	VERB
ijassa-1172	22	55	asymptomatic	asymptomatic	ADJ
ijassa-1172	22	56	individuals	individual	NOUN
ijassa-1172	22	57	called	call	VERB
ijassa-1172	22	58	a	a	DET
ijassa-1172	22	59	carrier	carrier	NOUN
ijassa-1172	22	60	,	,	PUNCT
ijassa-1172	22	61	who	who	PRON
ijassa-1172	22	62	can	can	AUX
ijassa-1172	22	63	move	move	VERB
ijassa-1172	22	64	freely	freely	ADV
ijassa-1172	22	65	and	and	CCONJ
ijassa-1172	22	66	transmit	transmit	VERB
ijassa-1172	22	67	infection	infection	NOUN
ijassa-1172	22	68	,	,	PUNCT
ijassa-1172	22	69	which	which	PRON
ijassa-1172	22	70	makes	make	VERB
ijassa-1172	22	71	controlling	control	VERB
ijassa-1172	22	72	the	the	DET
ijassa-1172	22	73	situation	situation	NOUN
ijassa-1172	22	74	complicated	complicate	VERB
ijassa-1172	22	75	such	such	ADJ
ijassa-1172	22	76	that	that	SCONJ
ijassa-1172	22	77	,	,	PUNCT
ijassa-1172	22	78	approximately	approximately	ADV
ijassa-1172	22	79	half	half	NOUN
ijassa-1172	22	80	of	of	ADP
ijassa-1172	22	81	the	the	DET
ijassa-1172	22	82	covid-19	covid-19	PROPN
ijassa-1172	22	83	patients	patient	NOUN
ijassa-1172	22	84	(	(	PUNCT
ijassa-1172	22	85	40−	40−	NOUN
ijassa-1172	22	86	50	50	NUM
ijassa-1172	22	87	%	%	NOUN
ijassa-1172	22	88	)	)	PUNCT
ijassa-1172	22	89	in	in	ADP
ijassa-1172	22	90	one	one	NUM
ijassa-1172	22	91	study	study	NOUN
ijassa-1172	22	92	did	do	AUX
ijassa-1172	22	93	not	not	PART
ijassa-1172	22	94	show	show	VERB
ijassa-1172	22	95	any	any	DET
ijassa-1172	22	96	symptoms	symptom	NOUN
ijassa-1172	22	97	from	from	ADP
ijassa-1172	22	98	above	above	ADV
ijassa-1172	22	99	in	in	ADP
ijassa-1172	22	100	(	(	PUNCT
ijassa-1172	22	101	a	a	NOUN
ijassa-1172	22	102	)	)	PUNCT
ijassa-1172	22	103	.	.	PUNCT
ijassa-1172	23	1	for	for	ADP
ijassa-1172	23	2	example	example	NOUN
ijassa-1172	23	3	,	,	PUNCT
ijassa-1172	23	4	there	there	PRON
ijassa-1172	23	5	are	be	VERB
ijassa-1172	23	6	several	several	ADJ
ijassa-1172	23	7	reports	report	NOUN
ijassa-1172	23	8	that	that	PRON
ijassa-1172	23	9	have	have	AUX
ijassa-1172	23	10	investigated	investigate	VERB
ijassa-1172	23	11	the	the	DET
ijassa-1172	23	12	possibility	possibility	NOUN
ijassa-1172	23	13	of	of	ADP
ijassa-1172	23	14	asymptomatic	asymptomatic	ADJ
ijassa-1172	23	15	carriers	carrier	NOUN
ijassa-1172	23	16	of	of	ADP
ijassa-1172	23	17	covid-19	covid-19	PROPN
ijassa-1172	23	18	.	.	PUNCT
ijassa-1172	24	1	a	a	DET
ijassa-1172	24	2	report	report	NOUN
ijassa-1172	24	3	of	of	ADP
ijassa-1172	24	4	japanese	japanese	ADJ
ijassa-1172	24	5	travelers	traveler	NOUN
ijassa-1172	24	6	from	from	ADP
ijassa-1172	24	7	wuhan	wuhan	PROPN
ijassa-1172	24	8	,	,	PUNCT
ijassa-1172	24	9	china	china	PROPN
ijassa-1172	24	10	approximates	approximate	VERB
ijassa-1172	24	11	the	the	DET
ijassa-1172	24	12	proportion	proportion	NOUN
ijassa-1172	24	13	of	of	ADP
ijassa-1172	24	14	carriers	carrier	NOUN
ijassa-1172	24	15	of	of	ADP
ijassa-1172	24	16	disease	disease	NOUN
ijassa-1172	24	17	but	but	CCONJ
ijassa-1172	24	18	asymptomatic	asymptomatic	ADJ
ijassa-1172	24	19	at	at	ADP
ijassa-1172	24	20	30	30	NUM
ijassa-1172	24	21	%	%	NOUN
ijassa-1172	24	22	.	.	PUNCT
ijassa-1172	25	1	in	in	ADP
ijassa-1172	25	2	addition	addition	NOUN
ijassa-1172	25	3	,	,	PUNCT
ijassa-1172	25	4	studies	study	NOUN
ijassa-1172	25	5	of	of	ADP
ijassa-1172	25	6	every	every	DET
ijassa-1172	25	7	person	person	NOUN
ijassa-1172	25	8	in	in	ADP
ijassa-1172	25	9	an	an	DET
ijassa-1172	25	10	isolated	isolated	ADJ
ijassa-1172	25	11	italian	italian	ADJ
ijassa-1172	25	12	village	village	NOUN
ijassa-1172	25	13	revealed	reveal	VERB
ijassa-1172	25	14	that	that	SCONJ
ijassa-1172	25	15	50	50	NUM
ijassa-1172	25	16	-	-	SYM
ijassa-1172	25	17	75	75	NUM
ijassa-1172	25	18	%	%	NOUN
ijassa-1172	25	19	of	of	ADP
ijassa-1172	25	20	people	people	NOUN
ijassa-1172	25	21	were	be	AUX
ijassa-1172	25	22	asymptomatic	asymptomatic	ADJ
ijassa-1172	25	23	.	.	PUNCT
ijassa-1172	26	1	a	a	DET
ijassa-1172	26	2	report	report	NOUN
ijassa-1172	26	3	of	of	ADP
ijassa-1172	26	4	a	a	DET
ijassa-1172	26	5	nursing	nursing	NOUN
ijassa-1172	26	6	home	home	NOUN
ijassa-1172	26	7	in	in	ADP
ijassa-1172	26	8	the	the	DET
ijassa-1172	26	9	usa	usa	PROPN
ijassa-1172	26	10	found	find	VERB
ijassa-1172	26	11	30	30	NUM
ijassa-1172	26	12	%	%	NOUN
ijassa-1172	26	13	of	of	ADP
ijassa-1172	26	14	patients	patient	NOUN
ijassa-1172	26	15	were	be	AUX
ijassa-1172	26	16	asymptomatic	asymptomatic	ADJ
ijassa-1172	26	17	on	on	ADP
ijassa-1172	26	18	the	the	DET
ijassa-1172	26	19	day	day	NOUN
ijassa-1172	26	20	of	of	ADP
ijassa-1172	26	21	testing	testing	NOUN
ijassa-1172	26	22	for	for	ADP
ijassa-1172	26	23	covid-19	covid-19	PROPN
ijassa-1172	26	24	,	,	PUNCT
ijassa-1172	26	25	with	with	ADP
ijassa-1172	26	26	4	4	NUM
ijassa-1172	26	27	%	%	NOUN
ijassa-1172	26	28	remaining	remain	VERB
ijassa-1172	26	29	asymptomatic	asymptomatic	ADJ
ijassa-1172	26	30	upon	upon	SCONJ
ijassa-1172	26	31	follow	follow	NOUN
ijassa-1172	26	32	-	-	PUNCT
ijassa-1172	26	33	up	up	NOUN
ijassa-1172	26	34	a	a	DET
ijassa-1172	26	35	week	week	NOUN
ijassa-1172	26	36	later	later	ADV
ijassa-1172	26	37	.	.	PUNCT
ijassa-1172	27	1	a	a	DET
ijassa-1172	27	2	report	report	NOUN
ijassa-1172	27	3	in	in	ADP
ijassa-1172	27	4	hospitalized	hospitalize	VERB
ijassa-1172	27	5	patients	patient	NOUN
ijassa-1172	27	6	in	in	ADP
ijassa-1172	27	7	beijing	beijing	PROPN
ijassa-1172	27	8	found	find	VERB
ijassa-1172	27	9	that	that	SCONJ
ijassa-1172	27	10	5	5	NUM
ijassa-1172	27	11	%	%	NOUN
ijassa-1172	27	12	of	of	ADP
ijassa-1172	27	13	patients	patient	NOUN
ijassa-1172	27	14	testing	test	VERB
ijassa-1172	27	15	positive	positive	ADJ
ijassa-1172	27	16	for	for	ADP
ijassa-1172	27	17	covid-19	covid-19	PROPN
ijassa-1172	27	18	had	have	VERB
ijassa-1172	27	19	asymptomatic	asymptomatic	ADJ
ijassa-1172	27	20	infections	infection	NOUN
ijassa-1172	27	21	see	see	VERB
ijassa-1172	27	22	[	[	X
ijassa-1172	27	23	3–5,10,15	3–5,10,15	NUM
ijassa-1172	27	24	]	]	X
ijassa-1172	27	25	.	.	PUNCT
ijassa-1172	28	1	recently	recently	ADV
ijassa-1172	28	2	,	,	PUNCT
ijassa-1172	28	3	many	many	ADJ
ijassa-1172	28	4	mathematical	mathematical	ADJ
ijassa-1172	28	5	models	model	NOUN
ijassa-1172	28	6	have	have	AUX
ijassa-1172	28	7	been	be	AUX
ijassa-1172	28	8	developed	develop	VERB
ijassa-1172	28	9	to	to	PART
ijassa-1172	28	10	describe	describe	VERB
ijassa-1172	28	11	the	the	DET
ijassa-1172	28	12	spread	spread	NOUN
ijassa-1172	28	13	of	of	ADP
ijassa-1172	28	14	infectious	infectious	ADJ
ijassa-1172	28	15	diseases	disease	NOUN
ijassa-1172	28	16	,	,	PUNCT
ijassa-1172	28	17	such	such	ADJ
ijassa-1172	28	18	as	as	ADP
ijassa-1172	28	19	hepatitis	hepatitis	PROPN
ijassa-1172	28	20	b	b	PROPN
ijassa-1172	28	21	,	,	PUNCT
ijassa-1172	28	22	hiv	hiv	PROPN
ijassa-1172	28	23	transmission	transmission	NOUN
ijassa-1172	28	24	and	and	CCONJ
ijassa-1172	28	25	ebola	ebola	PROPN
ijassa-1172	28	26	virus	virus	NOUN
ijassa-1172	28	27	disease	disease	NOUN
ijassa-1172	28	28	.	.	PUNCT
ijassa-1172	29	1	for	for	ADP
ijassa-1172	29	2	more	more	ADJ
ijassa-1172	29	3	information	information	NOUN
ijassa-1172	29	4	,	,	PUNCT
ijassa-1172	29	5	we	we	PRON
ijassa-1172	29	6	refer	refer	VERB
ijassa-1172	29	7	the	the	DET
ijassa-1172	29	8	reader	reader	NOUN
ijassa-1172	29	9	to	to	ADP
ijassa-1172	29	10	the	the	DET
ijassa-1172	29	11	recent	recent	ADJ
ijassa-1172	29	12	book	book	NOUN
ijassa-1172	29	13	published	publish	VERB
ijassa-1172	29	14	by	by	ADP
ijassa-1172	29	15	hattaf	hattaf	NOUN
ijassa-1172	29	16	and	and	CCONJ
ijassa-1172	29	17	dutta	dutta	NOUN
ijassa-1172	29	18	about	about	ADP
ijassa-1172	29	19	this	this	DET
ijassa-1172	29	20	subject	subject	NOUN
ijassa-1172	29	21	in	in	ADP
ijassa-1172	29	22	[	[	X
ijassa-1172	29	23	7	7	NUM
ijassa-1172	29	24	]	]	PUNCT
ijassa-1172	29	25	.	.	PUNCT
ijassa-1172	30	1	for	for	ADP
ijassa-1172	30	2	covid-19	covid-19	PROPN
ijassa-1172	30	3	,	,	PUNCT
ijassa-1172	30	4	moussaoui	moussaoui	ADJ
ijassa-1172	30	5	and	and	CCONJ
ijassa-1172	30	6	zerga	zerga	NOUN
ijassa-1172	30	7	[	[	X
ijassa-1172	30	8	14	14	NUM
ijassa-1172	30	9	]	]	PUNCT
ijassa-1172	30	10	proposed	propose	VERB
ijassa-1172	30	11	a	a	DET
ijassa-1172	30	12	mathematical	mathematical	ADJ
ijassa-1172	30	13	model	model	NOUN
ijassa-1172	30	14	that	that	PRON
ijassa-1172	30	15	takes	take	VERB
ijassa-1172	30	16	into	into	ADP
ijassa-1172	30	17	account	account	NOUN
ijassa-1172	30	18	strategies	strategy	NOUN
ijassa-1172	30	19	against	against	ADP
ijassa-1172	30	20	covid-19	covid-19	PROPN
ijassa-1172	30	21	such	such	ADJ
ijassa-1172	30	22	as	as	ADP
ijassa-1172	30	23	wearing	wear	VERB
ijassa-1172	30	24	masks	mask	NOUN
ijassa-1172	30	25	and	and	CCONJ
ijassa-1172	30	26	respecting	respect	VERB
ijassa-1172	30	27	safety	safety	NOUN
ijassa-1172	30	28	distances	distance	NOUN
ijassa-1172	30	29	.	.	PUNCT
ijassa-1172	31	1	a	a	DET
ijassa-1172	31	2	model	model	NOUN
ijassa-1172	31	3	for	for	ADP
ijassa-1172	31	4	the	the	DET
ijassa-1172	31	5	dynamics	dynamic	NOUN
ijassa-1172	31	6	of	of	ADP
ijassa-1172	31	7	covid19	covid19	NOUN
ijassa-1172	31	8	with	with	ADP
ijassa-1172	31	9	quarantine	quarantine	NOUN
ijassa-1172	31	10	strategy	strategy	NOUN
ijassa-1172	31	11	and	and	CCONJ
ijassa-1172	31	12	media	medium	NOUN
ijassa-1172	31	13	coverage	coverage	NOUN
ijassa-1172	31	14	effect	effect	NOUN
ijassa-1172	31	15	was	be	AUX
ijassa-1172	31	16	presented	present	VERB
ijassa-1172	31	17	by	by	ADP
ijassa-1172	31	18	mohsen	mohsen	PROPN
ijassa-1172	31	19	et	et	PROPN
ijassa-1172	31	20	al	al	PROPN
ijassa-1172	31	21	.	.	PUNCT
ijassa-1172	32	1	in	in	ADP
ijassa-1172	32	2	[	[	X
ijassa-1172	32	3	12	12	NUM
ijassa-1172	32	4	]	]	PUNCT
ijassa-1172	32	5	,	,	PUNCT
ijassa-1172	32	6	while	while	SCONJ
ijassa-1172	32	7	a	a	DET
ijassa-1172	32	8	mathematical	mathematical	ADJ
ijassa-1172	32	9	model	model	NOUN
ijassa-1172	32	10	of	of	ADP
ijassa-1172	32	11	covid-19	covid-19	PROPN
ijassa-1172	32	12	pandemic	pandemic	NOUN
ijassa-1172	32	13	involving	involve	VERB
ijassa-1172	32	14	the	the	DET
ijassa-1172	32	15	infective	infective	ADJ
ijassa-1172	32	16	immigrants	immigrant	NOUN
ijassa-1172	32	17	was	be	AUX
ijassa-1172	32	18	also	also	ADV
ijassa-1172	32	19	studied	study	VERB
ijassa-1172	32	20	in	in	ADP
ijassa-1172	32	21	[	[	X
ijassa-1172	32	22	11	11	NUM
ijassa-1172	32	23	]	]	PUNCT
ijassa-1172	32	24	.	.	PUNCT
ijassa-1172	33	1	a	a	DET
ijassa-1172	33	2	fractional	fractional	ADJ
ijassa-1172	33	3	order	order	NOUN
ijassa-1172	33	4	model	model	NOUN
ijassa-1172	33	5	with	with	ADP
ijassa-1172	33	6	bilinear	bilinear	ADJ
ijassa-1172	33	7	incidence	incidence	NOUN
ijassa-1172	33	8	rate	rate	NOUN
ijassa-1172	33	9	was	be	AUX
ijassa-1172	33	10	introduced	introduce	VERB
ijassa-1172	33	11	in	in	ADP
ijassa-1172	33	12	[	[	X
ijassa-1172	33	13	2	2	NUM
ijassa-1172	33	14	]	]	PUNCT
ijassa-1172	33	15	to	to	PART
ijassa-1172	33	16	study	study	VERB
ijassa-1172	33	17	transmission	transmission	NOUN
ijassa-1172	33	18	dynamics	dynamic	NOUN
ijassa-1172	33	19	of	of	ADP
ijassa-1172	33	20	covid-19	covid-19	PROPN
ijassa-1172	33	21	in	in	ADP
ijassa-1172	33	22	japan	japan	PROPN
ijassa-1172	33	23	2020	2020	NUM
ijassa-1172	33	24	.	.	PUNCT
ijassa-1172	34	1	modeling	modeling	NOUN
ijassa-1172	34	2	of	of	ADP
ijassa-1172	34	3	the	the	DET
ijassa-1172	34	4	dynamics	dynamic	NOUN
ijassa-1172	34	5	of	of	ADP
ijassa-1172	34	6	covid-19	covid-19	PROPN
ijassa-1172	34	7	inside	inside	ADP
ijassa-1172	34	8	the	the	DET
ijassa-1172	34	9	human	human	ADJ
ijassa-1172	34	10	body	body	NOUN
ijassa-1172	34	11	was	be	AUX
ijassa-1172	34	12	presented	present	VERB
ijassa-1172	34	13	in	in	ADP
ijassa-1172	34	14	[	[	X
ijassa-1172	34	15	9	9	NUM
ijassa-1172	34	16	]	]	PUNCT
ijassa-1172	34	17	.	.	PUNCT
ijassa-1172	35	1	the	the	DET
ijassa-1172	35	2	main	main	ADJ
ijassa-1172	35	3	objective	objective	NOUN
ijassa-1172	35	4	of	of	ADP
ijassa-1172	35	5	this	this	DET
ijassa-1172	35	6	study	study	NOUN
ijassa-1172	35	7	is	be	AUX
ijassa-1172	35	8	to	to	PART
ijassa-1172	35	9	develop	develop	VERB
ijassa-1172	35	10	a	a	DET
ijassa-1172	35	11	new	new	ADJ
ijassa-1172	35	12	mathematical	mathematical	ADJ
ijassa-1172	35	13	model	model	NOUN
ijassa-1172	35	14	in	in	ADP
ijassa-1172	35	15	order	order	NOUN
ijassa-1172	35	16	to	to	PART
ijassa-1172	35	17	better	well	ADV
ijassa-1172	35	18	describe	describe	VERB
ijassa-1172	35	19	the	the	DET
ijassa-1172	35	20	dynamics	dynamic	NOUN
ijassa-1172	35	21	of	of	ADP
ijassa-1172	35	22	the	the	DET
ijassa-1172	35	23	covid-19	covid-19	PROPN
ijassa-1172	35	24	in	in	ADP
ijassa-1172	35	25	human	human	ADJ
ijassa-1172	35	26	population	population	NOUN
ijassa-1172	35	27	by	by	ADP
ijassa-1172	35	28	taking	take	VERB
ijassa-1172	35	29	into	into	ADP
ijassa-1172	35	30	account	account	NOUN
ijassa-1172	35	31	the	the	DET
ijassa-1172	35	32	effects	effect	NOUN
ijassa-1172	35	33	of	of	ADP
ijassa-1172	35	34	memory	memory	NOUN
ijassa-1172	35	35	and	and	CCONJ
ijassa-1172	35	36	carrier	carrier	NOUN
ijassa-1172	35	37	,	,	PUNCT
ijassa-1172	35	38	and	and	CCONJ
ijassa-1172	35	39	also	also	ADV
ijassa-1172	35	40	others	other	NOUN
ijassa-1172	35	41	aspects	aspect	NOUN
ijassa-1172	35	42	such	such	ADJ
ijassa-1172	35	43	as	as	ADP
ijassa-1172	35	44	the	the	DET
ijassa-1172	35	45	nonlinearity	nonlinearity	NOUN
ijassa-1172	35	46	of	of	ADP
ijassa-1172	35	47	the	the	DET
ijassa-1172	35	48	incidence	incidence	NOUN
ijassa-1172	35	49	function	function	NOUN
ijassa-1172	35	50	,	,	PUNCT
ijassa-1172	35	51	the	the	DET
ijassa-1172	35	52	death	death	NOUN
ijassa-1172	35	53	rate	rate	NOUN
ijassa-1172	35	54	due	due	ADP
ijassa-1172	35	55	to	to	ADP
ijassa-1172	35	56	covid-19	covid-19	PROPN
ijassa-1172	35	57	and	and	CCONJ
ijassa-1172	35	58	the	the	DET
ijassa-1172	35	59	recovery	recovery	NOUN
ijassa-1172	35	60	rates	rate	NOUN
ijassa-1172	35	61	of	of	ADP
ijassa-1172	35	62	the	the	DET
ijassa-1172	35	63	asymptomatic	asymptomatic	ADJ
ijassa-1172	35	64	and	and	CCONJ
ijassa-1172	35	65	symptomatic	symptomatic	ADJ
ijassa-1172	35	66	individuals	individual	NOUN
ijassa-1172	35	67	.	.	PUNCT
ijassa-1172	36	1	to	to	PART
ijassa-1172	36	2	do	do	VERB
ijassa-1172	36	3	this	this	PRON
ijassa-1172	36	4	,	,	PUNCT
ijassa-1172	36	5	the	the	DET
ijassa-1172	36	6	structure	structure	NOUN
ijassa-1172	36	7	of	of	ADP
ijassa-1172	36	8	this	this	DET
ijassa-1172	36	9	work	work	NOUN
ijassa-1172	36	10	is	be	AUX
ijassa-1172	36	11	outlined	outline	VERB
ijassa-1172	36	12	as	as	SCONJ
ijassa-1172	36	13	follows	follow	VERB
ijassa-1172	36	14	.	.	PUNCT
ijassa-1172	37	1	the	the	DET
ijassa-1172	37	2	next	next	ADJ
ijassa-1172	37	3	section	section	NOUN
ijassa-1172	37	4	is	be	AUX
ijassa-1172	37	5	devoted	devote	VERB
ijassa-1172	37	6	to	to	ADP
ijassa-1172	37	7	the	the	DET
ijassa-1172	37	8	formulation	formulation	NOUN
ijassa-1172	37	9	of	of	ADP
ijassa-1172	37	10	model	model	NOUN
ijassa-1172	37	11	,	,	PUNCT
ijassa-1172	37	12	the	the	DET
ijassa-1172	37	13	existence	existence	NOUN
ijassa-1172	37	14	of	of	ADP
ijassa-1172	37	15	equilibria	equilibrium	NOUN
ijassa-1172	37	16	and	and	CCONJ
ijassa-1172	37	17	the	the	DET
ijassa-1172	37	18	computation	computation	NOUN
ijassa-1172	37	19	of	of	ADP
ijassa-1172	37	20	the	the	DET
ijassa-1172	37	21	basic	basic	ADJ
ijassa-1172	37	22	reproduction	reproduction	NOUN
ijassa-1172	37	23	number	number	NOUN
ijassa-1172	37	24	r0	r0	NOUN
ijassa-1172	37	25	.	.	PUNCT
ijassa-1172	38	1	in	in	ADP
ijassa-1172	38	2	section	section	NOUN
ijassa-1172	38	3	3	3	NUM
ijassa-1172	38	4	,	,	PUNCT
ijassa-1172	38	5	we	we	PRON
ijassa-1172	38	6	use	use	VERB
ijassa-1172	38	7	mathematical	mathematical	ADJ
ijassa-1172	38	8	analysis	analysis	NOUN
ijassa-1172	38	9	to	to	PART
ijassa-1172	38	10	establish	establish	VERB
ijassa-1172	38	11	global	global	ADJ
ijassa-1172	38	12	stability	stability	NOUN
ijassa-1172	38	13	results	result	NOUN
ijassa-1172	38	14	for	for	ADP
ijassa-1172	38	15	the	the	DET
ijassa-1172	38	16	proposed	propose	VERB
ijassa-1172	38	17	model	model	NOUN
ijassa-1172	38	18	.	.	PUNCT
ijassa-1172	39	1	some	some	DET
ijassa-1172	39	2	applications	application	NOUN
ijassa-1172	39	3	of	of	ADP
ijassa-1172	39	4	our	our	PRON
ijassa-1172	39	5	main	main	ADJ
ijassa-1172	39	6	results	result	NOUN
ijassa-1172	39	7	are	be	AUX
ijassa-1172	39	8	presented	present	VERB
ijassa-1172	39	9	in	in	ADP
ijassa-1172	39	10	section	section	NOUN
ijassa-1172	39	11	4	4	NUM
ijassa-1172	39	12	.	.	PUNCT
ijassa-1172	40	1	eventually	eventually	ADV
ijassa-1172	40	2	,	,	PUNCT
ijassa-1172	40	3	the	the	DET
ijassa-1172	40	4	conclusion	conclusion	NOUN
ijassa-1172	40	5	is	be	AUX
ijassa-1172	40	6	given	give	VERB
ijassa-1172	40	7	in	in	ADP
ijassa-1172	40	8	section	section	NOUN
ijassa-1172	40	9	5	5	NUM
ijassa-1172	40	10	.	.	SYM
ijassa-1172	40	11	2	2	NUM
ijassa-1172	40	12	.	.	NOUN
ijassa-1172	40	13	model	model	NOUN
ijassa-1172	40	14	formulation	formulation	NOUN
ijassa-1172	40	15	and	and	CCONJ
ijassa-1172	40	16	equilibria	equilibrium	NOUN
ijassa-1172	40	17	in	in	ADP
ijassa-1172	40	18	this	this	DET
ijassa-1172	40	19	section	section	NOUN
ijassa-1172	40	20	,	,	PUNCT
ijassa-1172	40	21	we	we	PRON
ijassa-1172	40	22	first	first	ADV
ijassa-1172	40	23	construct	construct	VERB
ijassa-1172	40	24	a	a	DET
ijassa-1172	40	25	mathematical	mathematical	ADJ
ijassa-1172	40	26	model	model	NOUN
ijassa-1172	40	27	that	that	PRON
ijassa-1172	40	28	takes	take	VERB
ijassa-1172	40	29	into	into	ADP
ijassa-1172	40	30	account	account	NOUN
ijassa-1172	40	31	the	the	DET
ijassa-1172	40	32	effects	effect	NOUN
ijassa-1172	40	33	of	of	ADP
ijassa-1172	40	34	memory	memory	NOUN
ijassa-1172	40	35	and	and	CCONJ
ijassa-1172	40	36	carrier	carrier	NOUN
ijassa-1172	40	37	.	.	PUNCT
ijassa-1172	41	1	then	then	ADV
ijassa-1172	41	2	we	we	PRON
ijassa-1172	41	3	divide	divide	VERB
ijassa-1172	41	4	the	the	DET
ijassa-1172	41	5	total	total	ADJ
ijassa-1172	41	6	population	population	NOUN
ijassa-1172	41	7	into	into	ADP
ijassa-1172	41	8	four	four	NUM
ijassa-1172	41	9	classes	class	NOUN
ijassa-1172	41	10	s(t	s(t	PROPN
ijassa-1172	41	11	)	)	PUNCT
ijassa-1172	41	12	,	,	PUNCT
ijassa-1172	41	13	c(t	c(t	PROPN
ijassa-1172	41	14	)	)	PUNCT
ijassa-1172	41	15	,	,	PUNCT
ijassa-1172	41	16	i(t	i(t	PROPN
ijassa-1172	41	17	)	)	PUNCT
ijassa-1172	41	18	and	and	CCONJ
ijassa-1172	41	19	r(t	r(t	NOUN
ijassa-1172	41	20	)	)	PUNCT
ijassa-1172	41	21	that	that	PRON
ijassa-1172	41	22	represent	represent	VERB
ijassa-1172	41	23	susceptible	susceptible	ADJ
ijassa-1172	41	24	,	,	PUNCT
ijassa-1172	41	25	carrier	carrier	NOUN
ijassa-1172	41	26	(	(	PUNCT
ijassa-1172	41	27	asymptomatic	asymptomatic	ADJ
ijassa-1172	41	28	infected	infect	VERB
ijassa-1172	41	29	individuals	individual	NOUN
ijassa-1172	41	30	)	)	PUNCT
ijassa-1172	41	31	,	,	PUNCT
ijassa-1172	41	32	infected	infect	VERB
ijassa-1172	41	33	and	and	CCONJ
ijassa-1172	41	34	recovered	recover	VERB
ijassa-1172	41	35	individuals	individual	NOUN
ijassa-1172	41	36	at	at	ADP
ijassa-1172	41	37	time	time	NOUN
ijassa-1172	41	38	t	t	PROPN
ijassa-1172	41	39	,	,	PUNCT
ijassa-1172	41	40	respectively	respectively	ADV
ijassa-1172	41	41	.	.	PUNCT
ijassa-1172	42	1	the	the	DET
ijassa-1172	42	2	dynamics	dynamic	NOUN
ijassa-1172	42	3	copyright	copyright	NOUN
ijassa-1172	42	4	©	©	PROPN
ijassa-1172	42	5	2022	2022	NUM
ijassa-1172	42	6	assa	assa	NOUN
ijassa-1172	42	7	.	.	PUNCT
ijassa-1172	43	1	adv	adv	PROPN
ijassa-1172	43	2	syst	syst	PROPN
ijassa-1172	43	3	sci	sci	PROPN
ijassa-1172	43	4	appl	appl	PROPN
ijassa-1172	43	5	(	(	PUNCT
ijassa-1172	43	6	2022	2022	NUM
ijassa-1172	43	7	)	)	PUNCT
ijassa-1172	43	8	38	38	NUM
ijassa-1172	43	9	khalid	khalid	PROPN
ijassa-1172	43	10	hattaf	hattaf	NOUN
ijassa-1172	43	11	,	,	PUNCT
ijassa-1172	43	12	ahmed	ahmed	PROPN
ijassa-1172	43	13	a.	a.	PROPN
ijassa-1172	43	14	mohsen	mohsen	PROPN
ijassa-1172	43	15	of	of	ADP
ijassa-1172	43	16	the	the	DET
ijassa-1172	43	17	four	four	NUM
ijassa-1172	43	18	classes	class	NOUN
ijassa-1172	43	19	is	be	AUX
ijassa-1172	43	20	governed	govern	VERB
ijassa-1172	43	21	by	by	ADP
ijassa-1172	43	22	the	the	DET
ijassa-1172	43	23	following	following	ADJ
ijassa-1172	43	24	nonlinear	nonlinear	ADJ
ijassa-1172	43	25	system	system	NOUN
ijassa-1172	43	26	of	of	ADP
ijassa-1172	43	27	fdes:	fdes:	PROPN
ijassa-1172	43	28	dαs(t	dαs(t	PROPN
ijassa-1172	43	29	)	)	PUNCT
ijassa-1172	43	30	=	=	SYM
ijassa-1172	43	31	a−	a−	NOUN
ijassa-1172	43	32	νs	νs	NOUN
ijassa-1172	43	33	−	−	ADP
ijassa-1172	43	34	φ(s	φ(s	NOUN
ijassa-1172	43	35	,	,	PUNCT
ijassa-1172	43	36	c)c	c)c	X
ijassa-1172	43	37	−ψ(s	−ψ(s	NOUN
ijassa-1172	43	38	,	,	PUNCT
ijassa-1172	43	39	i)i	i)i	NOUN
ijassa-1172	43	40	,	,	PUNCT
ijassa-1172	43	41	dαc(t	dαc(t	NOUN
ijassa-1172	43	42	)	)	PUNCT
ijassa-1172	43	43	=	=	PUNCT
ijassa-1172	43	44	φ(s	φ(s	NOUN
ijassa-1172	43	45	,	,	PUNCT
ijassa-1172	43	46	c)c	c)c	PUNCT
ijassa-1172	44	1	+	+	ADJ
ijassa-1172	44	2	ψ(s	ψ(s	ADJ
ijassa-1172	44	3	,	,	PUNCT
ijassa-1172	44	4	i)i	i)i	NOUN
ijassa-1172	44	5	−	−	PROPN
ijassa-1172	44	6	(	(	PUNCT
ijassa-1172	44	7	ν	ν	X
ijassa-1172	44	8	+	+	CCONJ
ijassa-1172	44	9	d+	d+	NOUN
ijassa-1172	44	10	γ	γ	PROPN
ijassa-1172	44	11	+	+	SYM
ijassa-1172	44	12	r1)c	r1)c	NOUN
ijassa-1172	44	13	,	,	PUNCT
ijassa-1172	44	14	dαi(t	dαi(t	PROPN
ijassa-1172	44	15	)	)	PUNCT
ijassa-1172	45	1	=	=	SYM
ijassa-1172	45	2	γc	γc	PROPN
ijassa-1172	45	3	−	−	NOUN
ijassa-1172	45	4	(	(	PUNCT
ijassa-1172	45	5	ν	ν	X
ijassa-1172	45	6	+	+	CCONJ
ijassa-1172	45	7	d+	d+	X
ijassa-1172	45	8	r2)i	r2)i	PROPN
ijassa-1172	45	9	,	,	PUNCT
ijassa-1172	45	10	dαr(t	dαr(t	PROPN
ijassa-1172	45	11	)	)	PUNCT
ijassa-1172	45	12	=	=	PUNCT
ijassa-1172	46	1	r1c	r1c	NOUN
ijassa-1172	46	2	+	+	CCONJ
ijassa-1172	46	3	r2i	r2i	PROPN
ijassa-1172	46	4	−	−	PROPN
ijassa-1172	46	5	νr	νr	NOUN
ijassa-1172	46	6	,	,	PUNCT
ijassa-1172	46	7	(	(	PUNCT
ijassa-1172	46	8	2.1	2.1	NUM
ijassa-1172	46	9	)	)	PUNCT
ijassa-1172	46	10	where	where	SCONJ
ijassa-1172	46	11	the	the	DET
ijassa-1172	46	12	susceptible	susceptible	ADJ
ijassa-1172	46	13	individuals	individual	NOUN
ijassa-1172	46	14	are	be	AUX
ijassa-1172	46	15	recruited	recruit	VERB
ijassa-1172	46	16	at	at	ADP
ijassa-1172	46	17	a	a	DET
ijassa-1172	46	18	rate	rate	NOUN
ijassa-1172	46	19	a	a	PRON
ijassa-1172	46	20	and	and	CCONJ
ijassa-1172	46	21	become	become	VERB
ijassa-1172	46	22	infected	infect	VERB
ijassa-1172	46	23	either	either	CCONJ
ijassa-1172	46	24	by	by	ADP
ijassa-1172	46	25	effective	effective	ADJ
ijassa-1172	46	26	contact	contact	NOUN
ijassa-1172	46	27	with	with	ADP
ijassa-1172	46	28	carrier	carrier	NOUN
ijassa-1172	46	29	at	at	ADP
ijassa-1172	46	30	rate	rate	NOUN
ijassa-1172	46	31	φ(s	φ(s	NOUN
ijassa-1172	46	32	,	,	PUNCT
ijassa-1172	46	33	c)c	c)c	NOUN
ijassa-1172	46	34	or	or	CCONJ
ijassa-1172	46	35	by	by	ADP
ijassa-1172	46	36	effective	effective	ADJ
ijassa-1172	46	37	contact	contact	NOUN
ijassa-1172	46	38	with	with	ADP
ijassa-1172	46	39	infected	infected	ADJ
ijassa-1172	46	40	individuals	individual	NOUN
ijassa-1172	46	41	at	at	ADP
ijassa-1172	46	42	rate	rate	NOUN
ijassa-1172	46	43	ψ(s	ψ(s	PROPN
ijassa-1172	46	44	,	,	PUNCT
ijassa-1172	46	45	i)i	i)i	NOUN
ijassa-1172	46	46	.	.	PUNCT
ijassa-1172	47	1	so	so	ADV
ijassa-1172	47	2	,	,	PUNCT
ijassa-1172	47	3	the	the	DET
ijassa-1172	47	4	term	term	NOUN
ijassa-1172	47	5	φ(s	φ(s	NOUN
ijassa-1172	47	6	,	,	PUNCT
ijassa-1172	47	7	c)c	c)c	X
ijassa-1172	47	8	+	+	CCONJ
ijassa-1172	47	9	ψ(s	ψ(s	ADJ
ijassa-1172	47	10	,	,	PUNCT
ijassa-1172	47	11	i)i	i)i	NOUN
ijassa-1172	47	12	denotes	denote	VERB
ijassa-1172	47	13	the	the	DET
ijassa-1172	47	14	total	total	ADJ
ijassa-1172	47	15	asymptomatic	asymptomatic	ADJ
ijassa-1172	47	16	infection	infection	NOUN
ijassa-1172	47	17	rate	rate	NOUN
ijassa-1172	47	18	of	of	ADP
ijassa-1172	47	19	susceptible	susceptible	ADJ
ijassa-1172	47	20	individuals	individual	NOUN
ijassa-1172	47	21	.	.	PUNCT
ijassa-1172	48	1	the	the	DET
ijassa-1172	48	2	natural	natural	ADJ
ijassa-1172	48	3	death	death	NOUN
ijassa-1172	48	4	rate	rate	NOUN
ijassa-1172	48	5	in	in	ADP
ijassa-1172	48	6	all	all	DET
ijassa-1172	48	7	classes	class	NOUN
ijassa-1172	48	8	is	be	AUX
ijassa-1172	48	9	denoted	denote	VERB
ijassa-1172	48	10	by	by	ADP
ijassa-1172	48	11	ν	ν	NOUN
ijassa-1172	48	12	,	,	PUNCT
ijassa-1172	48	13	while	while	SCONJ
ijassa-1172	48	14	d	d	NOUN
ijassa-1172	48	15	is	be	AUX
ijassa-1172	48	16	the	the	DET
ijassa-1172	48	17	death	death	NOUN
ijassa-1172	48	18	rate	rate	NOUN
ijassa-1172	48	19	due	due	ADP
ijassa-1172	48	20	to	to	ADP
ijassa-1172	48	21	covid-19	covid-19	PROPN
ijassa-1172	48	22	.	.	PUNCT
ijassa-1172	49	1	the	the	DET
ijassa-1172	49	2	rate	rate	NOUN
ijassa-1172	49	3	of	of	ADP
ijassa-1172	49	4	transfer	transfer	NOUN
ijassa-1172	49	5	from	from	ADP
ijassa-1172	49	6	the	the	DET
ijassa-1172	49	7	asymptomatic	asymptomatic	NOUN
ijassa-1172	49	8	to	to	ADP
ijassa-1172	49	9	symptomatic	symptomatic	ADJ
ijassa-1172	49	10	is	be	AUX
ijassa-1172	49	11	denoted	denote	VERB
ijassa-1172	49	12	by	by	ADP
ijassa-1172	49	13	γ	γ	PROPN
ijassa-1172	49	14	.	.	PUNCT
ijassa-1172	50	1	the	the	DET
ijassa-1172	50	2	parameters	parameter	NOUN
ijassa-1172	50	3	r1	r1	PROPN
ijassa-1172	50	4	and	and	CCONJ
ijassa-1172	50	5	r2	r2	PROPN
ijassa-1172	50	6	are	be	AUX
ijassa-1172	50	7	recovery	recovery	NOUN
ijassa-1172	50	8	rates	rate	NOUN
ijassa-1172	50	9	of	of	ADP
ijassa-1172	50	10	the	the	DET
ijassa-1172	50	11	asymptomatic	asymptomatic	ADJ
ijassa-1172	50	12	and	and	CCONJ
ijassa-1172	50	13	symptomatic	symptomatic	ADJ
ijassa-1172	50	14	individuals	individual	NOUN
ijassa-1172	50	15	,	,	PUNCT
ijassa-1172	50	16	respectively	respectively	ADV
ijassa-1172	50	17	.	.	PUNCT
ijassa-1172	51	1	finally	finally	ADV
ijassa-1172	51	2	,	,	PUNCT
ijassa-1172	51	3	dα	dα	PROPN
ijassa-1172	51	4	denotes	denote	VERB
ijassa-1172	51	5	the	the	DET
ijassa-1172	51	6	caputo	caputo	PROPN
ijassa-1172	51	7	fractional	fractional	PROPN
ijassa-1172	51	8	derivative	derivative	NOUN
ijassa-1172	51	9	with	with	ADP
ijassa-1172	51	10	α	α	PROPN
ijassa-1172	51	11	∈	∈	PROPN
ijassa-1172	51	12	(	(	PUNCT
ijassa-1172	51	13	0	0	NUM
ijassa-1172	51	14	,	,	PUNCT
ijassa-1172	51	15	1	1	NUM
ijassa-1172	51	16	]	]	PUNCT
ijassa-1172	51	17	that	that	PRON
ijassa-1172	51	18	describes	describe	VERB
ijassa-1172	51	19	the	the	DET
ijassa-1172	51	20	memory	memory	NOUN
ijassa-1172	51	21	effect	effect	NOUN
ijassa-1172	51	22	.	.	PUNCT
ijassa-1172	52	1	a	a	DET
ijassa-1172	52	2	schematic	schematic	ADJ
ijassa-1172	52	3	diagram	diagram	NOUN
ijassa-1172	52	4	of	of	ADP
ijassa-1172	52	5	model	model	NOUN
ijassa-1172	52	6	(	(	PUNCT
ijassa-1172	52	7	2.1	2.1	NUM
ijassa-1172	52	8	)	)	PUNCT
ijassa-1172	52	9	is	be	AUX
ijassa-1172	52	10	shown	show	VERB
ijassa-1172	52	11	in	in	ADP
ijassa-1172	52	12	figure	figure	NOUN
ijassa-1172	52	13	1	1	NUM
ijassa-1172	52	14	.	.	PUNCT
ijassa-1172	52	15	fig	fig	NOUN
ijassa-1172	52	16	.	.	PUNCT
ijassa-1172	53	1	2.1	2.1	NUM
ijassa-1172	53	2	.	.	PUNCT
ijassa-1172	54	1	the	the	DET
ijassa-1172	54	2	diagram	diagram	NOUN
ijassa-1172	54	3	of	of	ADP
ijassa-1172	54	4	model	model	NOUN
ijassa-1172	54	5	(	(	PUNCT
ijassa-1172	54	6	2.1	2.1	NUM
ijassa-1172	54	7	)	)	PUNCT
ijassa-1172	54	8	.	.	PUNCT
ijassa-1172	55	1	obviously	obviously	ADV
ijassa-1172	55	2	,	,	PUNCT
ijassa-1172	55	3	the	the	DET
ijassa-1172	55	4	first	first	ADJ
ijassa-1172	55	5	three	three	NUM
ijassa-1172	55	6	equations	equation	NOUN
ijassa-1172	55	7	of	of	ADP
ijassa-1172	55	8	(	(	PUNCT
ijassa-1172	55	9	2.1	2.1	NUM
ijassa-1172	55	10	)	)	PUNCT
ijassa-1172	55	11	do	do	AUX
ijassa-1172	55	12	not	not	PART
ijassa-1172	55	13	depend	depend	VERB
ijassa-1172	55	14	on	on	ADP
ijassa-1172	55	15	the	the	DET
ijassa-1172	55	16	variable	variable	ADJ
ijassa-1172	55	17	r	r	NOUN
ijassa-1172	55	18	,	,	PUNCT
ijassa-1172	55	19	model	model	NOUN
ijassa-1172	55	20	(	(	PUNCT
ijassa-1172	55	21	2.1	2.1	NUM
ijassa-1172	55	22	)	)	PUNCT
ijassa-1172	55	23	can	can	AUX
ijassa-1172	55	24	be	be	AUX
ijassa-1172	55	25	rewrite	rewrite	VERB
ijassa-1172	55	26	by	by	ADP
ijassa-1172	55	27	following	follow	VERB
ijassa-1172	55	28	format	format	NOUN
ijassa-1172	55	29	{	{	PUNCT
ijassa-1172	55	30	dαs(t	dαs(t	NOUN
ijassa-1172	55	31	)	)	PUNCT
ijassa-1172	55	32	=	=	SYM
ijassa-1172	55	33	a−	a−	NOUN
ijassa-1172	55	34	νs	νs	NOUN
ijassa-1172	55	35	−	−	ADP
ijassa-1172	55	36	φ(s	φ(s	NOUN
ijassa-1172	55	37	,	,	PUNCT
ijassa-1172	55	38	c)c	c)c	X
ijassa-1172	55	39	−ψ(s	−ψ(s	NOUN
ijassa-1172	55	40	,	,	PUNCT
ijassa-1172	55	41	i)i	i)i	NOUN
ijassa-1172	55	42	,	,	PUNCT
ijassa-1172	55	43	dαc(t	dαc(t	NOUN
ijassa-1172	55	44	)	)	PUNCT
ijassa-1172	55	45	=	=	PUNCT
ijassa-1172	55	46	φ(s	φ(s	NOUN
ijassa-1172	55	47	,	,	PUNCT
ijassa-1172	55	48	c)c	c)c	PUNCT
ijassa-1172	56	1	+	+	ADJ
ijassa-1172	56	2	ψ(s	ψ(s	ADJ
ijassa-1172	56	3	,	,	PUNCT
ijassa-1172	56	4	i)i	i)i	NOUN
ijassa-1172	56	5	−	−	PROPN
ijassa-1172	56	6	d1c	d1c	PROPN
ijassa-1172	56	7	,	,	PUNCT
ijassa-1172	56	8	dαi(t	dαi(t	PROPN
ijassa-1172	56	9	)	)	PUNCT
ijassa-1172	56	10	=	=	SYM
ijassa-1172	56	11	γc	γc	PROPN
ijassa-1172	56	12	−	−	NUM
ijassa-1172	56	13	d2i	d2i	NOUN
ijassa-1172	56	14	,	,	PUNCT
ijassa-1172	56	15	(	(	PUNCT
ijassa-1172	56	16	2.2	2.2	NUM
ijassa-1172	56	17	)	)	PUNCT
ijassa-1172	56	18	where	where	SCONJ
ijassa-1172	56	19	d1	d1	NOUN
ijassa-1172	56	20	=	=	PUNCT
ijassa-1172	57	1	ν	ν	X
ijassa-1172	57	2	+	+	CCONJ
ijassa-1172	57	3	d+	d+	NOUN
ijassa-1172	57	4	γ	γ	X
ijassa-1172	57	5	+	+	CCONJ
ijassa-1172	57	6	r1	r1	PROPN
ijassa-1172	57	7	and	and	CCONJ
ijassa-1172	57	8	d2	d2	PROPN
ijassa-1172	57	9	=	=	SYM
ijassa-1172	57	10	ν	ν	NOUN
ijassa-1172	57	11	+	+	CCONJ
ijassa-1172	57	12	d+	d+	NOUN
ijassa-1172	57	13	r2	r2	NOUN
ijassa-1172	57	14	.	.	PUNCT
ijassa-1172	58	1	furthermore	furthermore	ADV
ijassa-1172	58	2	and	and	CCONJ
ijassa-1172	58	3	according	accord	VERB
ijassa-1172	58	4	to	to	ADP
ijassa-1172	58	5	[	[	X
ijassa-1172	58	6	8	8	NUM
ijassa-1172	58	7	]	]	PUNCT
ijassa-1172	58	8	,	,	PUNCT
ijassa-1172	58	9	we	we	PRON
ijassa-1172	58	10	assume	assume	VERB
ijassa-1172	58	11	that	that	SCONJ
ijassa-1172	58	12	the	the	DET
ijassa-1172	58	13	infection	infection	NOUN
ijassa-1172	58	14	rate	rate	NOUN
ijassa-1172	58	15	is	be	AUX
ijassa-1172	58	16	the	the	DET
ijassa-1172	58	17	general	general	ADJ
ijassa-1172	58	18	incidences	incidence	NOUN
ijassa-1172	58	19	φ	φ	PROPN
ijassa-1172	58	20	and	and	CCONJ
ijassa-1172	58	21	ψ	ψ	NOUN
ijassa-1172	58	22	are	be	AUX
ijassa-1172	58	23	continuously	continuously	ADV
ijassa-1172	58	24	differentiable	differentiable	ADJ
ijassa-1172	58	25	in	in	ADP
ijassa-1172	58	26	the	the	DET
ijassa-1172	58	27	interior	interior	NOUN
ijassa-1172	58	28	of	of	ADP
ijassa-1172	58	29	ir2	ir2	PROPN
ijassa-1172	58	30	+	+	CCONJ
ijassa-1172	58	31	and	and	CCONJ
ijassa-1172	58	32	satisfy	satisfy	VERB
ijassa-1172	58	33	the	the	DET
ijassa-1172	58	34	two	two	NUM
ijassa-1172	58	35	following	follow	VERB
ijassa-1172	58	36	conditions	condition	NOUN
ijassa-1172	58	37	:	:	PUNCT
ijassa-1172	58	38	(	(	PUNCT
ijassa-1172	58	39	h1	h1	NOUN
ijassa-1172	58	40	)	)	PUNCT
ijassa-1172	58	41	φ(0	φ(0	ADJ
ijassa-1172	58	42	,	,	PUNCT
ijassa-1172	58	43	c	c	NOUN
ijassa-1172	58	44	)	)	PUNCT
ijassa-1172	58	45	=	=	SYM
ijassa-1172	59	1	0	0	NUM
ijassa-1172	59	2	,	,	PUNCT
ijassa-1172	59	3	∂φ	∂φ	PROPN
ijassa-1172	60	1	∂s	∂s	PROPN
ijassa-1172	60	2	(	(	PUNCT
ijassa-1172	60	3	s	s	X
ijassa-1172	60	4	,	,	PUNCT
ijassa-1172	60	5	c	c	NOUN
ijassa-1172	60	6	)	)	PUNCT
ijassa-1172	60	7	>	>	X
ijassa-1172	60	8	0	0	NUM
ijassa-1172	60	9	,	,	PUNCT
ijassa-1172	60	10	∂φ	∂φ	PROPN
ijassa-1172	60	11	∂c	∂c	PROPN
ijassa-1172	61	1	(	(	PUNCT
ijassa-1172	61	2	s	s	X
ijassa-1172	61	3	,	,	PUNCT
ijassa-1172	61	4	c	c	NOUN
ijassa-1172	61	5	)	)	PUNCT
ijassa-1172	61	6	≤	≤	NOUN
ijassa-1172	61	7	0	0	NUM
ijassa-1172	62	1	for	for	ADP
ijassa-1172	62	2	all	all	DET
ijassa-1172	62	3	s	s	NOUN
ijassa-1172	62	4	,	,	PUNCT
ijassa-1172	62	5	c	c	X
ijassa-1172	62	6	≥	≥	PROPN
ijassa-1172	62	7	0	0	NUM
ijassa-1172	62	8	.	.	PUNCT
ijassa-1172	63	1	copyright	copyright	NOUN
ijassa-1172	63	2	©	©	PROPN
ijassa-1172	63	3	2022	2022	NUM
ijassa-1172	63	4	assa	assa	NOUN
ijassa-1172	63	5	.	.	PUNCT
ijassa-1172	64	1	adv	adv	PROPN
ijassa-1172	64	2	syst	syst	PROPN
ijassa-1172	64	3	sci	sci	PROPN
ijassa-1172	64	4	appl	appl	PROPN
ijassa-1172	64	5	(	(	PUNCT
ijassa-1172	64	6	2022	2022	NUM
ijassa-1172	64	7	)	)	PUNCT
ijassa-1172	64	8	dynamics	dynamic	NOUN
ijassa-1172	64	9	of	of	ADP
ijassa-1172	64	10	a	a	DET
ijassa-1172	64	11	generalized	generalize	VERB
ijassa-1172	64	12	fractional	fractional	ADJ
ijassa-1172	64	13	epidemic	epidemic	NOUN
ijassa-1172	64	14	model	model	NOUN
ijassa-1172	64	15	...	...	PUNCT
ijassa-1172	64	16	39	39	NUM
ijassa-1172	64	17	(	(	PUNCT
ijassa-1172	64	18	h2	h2	NOUN
ijassa-1172	64	19	)	)	PUNCT
ijassa-1172	64	20	ψ(0	ψ(0	PROPN
ijassa-1172	64	21	,	,	PUNCT
ijassa-1172	64	22	i	i	NOUN
ijassa-1172	64	23	)	)	PUNCT
ijassa-1172	64	24	=	=	SYM
ijassa-1172	64	25	0	0	NUM
ijassa-1172	64	26	,	,	PUNCT
ijassa-1172	64	27	∂ψ	∂ψ	VERB
ijassa-1172	65	1	∂s	∂s	PROPN
ijassa-1172	65	2	(	(	PUNCT
ijassa-1172	65	3	s	s	X
ijassa-1172	65	4	,	,	PUNCT
ijassa-1172	65	5	i	i	PROPN
ijassa-1172	65	6	)	)	PUNCT
ijassa-1172	65	7	>	>	X
ijassa-1172	65	8	0	0	NUM
ijassa-1172	65	9	,	,	PUNCT
ijassa-1172	65	10	∂ψ	∂ψ	PROPN
ijassa-1172	65	11	∂i	∂i	PROPN
ijassa-1172	65	12	(	(	PUNCT
ijassa-1172	65	13	s	s	PROPN
ijassa-1172	65	14	,	,	PUNCT
ijassa-1172	65	15	i	i	NOUN
ijassa-1172	65	16	)	)	PUNCT
ijassa-1172	65	17	≤	≤	NOUN
ijassa-1172	65	18	0	0	NUM
ijassa-1172	66	1	for	for	ADP
ijassa-1172	66	2	all	all	DET
ijassa-1172	66	3	s	s	NOUN
ijassa-1172	66	4	,	,	PUNCT
ijassa-1172	66	5	i	i	PRON
ijassa-1172	66	6	≥	≥	VERB
ijassa-1172	66	7	0	0	NUM
ijassa-1172	66	8	.	.	PUNCT
ijassa-1172	67	1	next	next	ADV
ijassa-1172	67	2	,	,	PUNCT
ijassa-1172	67	3	we	we	PRON
ijassa-1172	67	4	study	study	VERB
ijassa-1172	67	5	the	the	DET
ijassa-1172	67	6	existence	existence	NOUN
ijassa-1172	67	7	of	of	ADP
ijassa-1172	67	8	equilibria	equilibria	PROPN
ijassa-1172	67	9	.	.	PUNCT
ijassa-1172	68	1	clearly	clearly	ADV
ijassa-1172	68	2	,	,	PUNCT
ijassa-1172	68	3	ef	ef	PROPN
ijassa-1172	68	4	(	(	PUNCT
ijassa-1172	68	5	s0	s0	PROPN
ijassa-1172	68	6	,	,	PUNCT
ijassa-1172	68	7	0	0	NUM
ijassa-1172	68	8	,	,	PUNCT
ijassa-1172	68	9	0	0	NUM
ijassa-1172	68	10	)	)	PUNCT
ijassa-1172	68	11	is	be	AUX
ijassa-1172	68	12	a	a	DET
ijassa-1172	68	13	disease	disease	NOUN
ijassa-1172	68	14	-	-	PUNCT
ijassa-1172	68	15	free	free	ADJ
ijassa-1172	68	16	equilibrium	equilibrium	NOUN
ijassa-1172	68	17	of	of	ADP
ijassa-1172	68	18	(	(	PUNCT
ijassa-1172	68	19	2.2	2.2	NUM
ijassa-1172	68	20	)	)	PUNCT
ijassa-1172	68	21	,	,	PUNCT
ijassa-1172	68	22	where	where	SCONJ
ijassa-1172	68	23	s0	s0	PROPN
ijassa-1172	68	24	=	=	PUNCT
ijassa-1172	68	25	a	a	DET
ijassa-1172	68	26	ν	ν	NOUN
ijassa-1172	68	27	.	.	PUNCT
ijassa-1172	69	1	hence	hence	ADV
ijassa-1172	69	2	,	,	PUNCT
ijassa-1172	69	3	the	the	DET
ijassa-1172	69	4	basic	basic	ADJ
ijassa-1172	69	5	reproduction	reproduction	NOUN
ijassa-1172	69	6	number	number	NOUN
ijassa-1172	69	7	of	of	ADP
ijassa-1172	69	8	(	(	PUNCT
ijassa-1172	69	9	2.2	2.2	NUM
ijassa-1172	69	10	)	)	PUNCT
ijassa-1172	69	11	can	can	AUX
ijassa-1172	69	12	be	be	AUX
ijassa-1172	69	13	calculated	calculate	VERB
ijassa-1172	69	14	as	as	SCONJ
ijassa-1172	69	15	follows	follow	VERB
ijassa-1172	69	16	r0	r0	NOUN
ijassa-1172	69	17	=	=	SYM
ijassa-1172	69	18	d2φ(s0	d2φ(s0	PROPN
ijassa-1172	69	19	,	,	PUNCT
ijassa-1172	69	20	0	0	NUM
ijassa-1172	69	21	)	)	PUNCT
ijassa-1172	70	1	+	+	CCONJ
ijassa-1172	70	2	γg(s0	γg(s0	PROPN
ijassa-1172	70	3	,	,	PUNCT
ijassa-1172	70	4	0	0	NUM
ijassa-1172	70	5	)	)	PUNCT
ijassa-1172	70	6	d1d2	d1d2	VERB
ijassa-1172	70	7	,	,	PUNCT
ijassa-1172	70	8	(	(	PUNCT
ijassa-1172	70	9	2.3	2.3	NUM
ijassa-1172	70	10	)	)	PUNCT
ijassa-1172	70	11	which	which	PRON
ijassa-1172	70	12	can	can	AUX
ijassa-1172	70	13	be	be	AUX
ijassa-1172	70	14	rewritten	rewrite	VERB
ijassa-1172	70	15	as	as	ADP
ijassa-1172	70	16	r01+r02	r01+r02	NOUN
ijassa-1172	70	17	,	,	PUNCT
ijassa-1172	70	18	where	where	SCONJ
ijassa-1172	70	19	�	�	PROPN
ijassa-1172	70	20	r01	r01	PROPN
ijassa-1172	70	21	=	=	PUNCT
ijassa-1172	70	22	φ(s0	φ(s0	PROPN
ijassa-1172	70	23	,	,	PUNCT
ijassa-1172	70	24	0	0	X
ijassa-1172	70	25	)	)	PUNCT
ijassa-1172	70	26	d1	d1	NOUN
ijassa-1172	70	27	is	be	AUX
ijassa-1172	70	28	the	the	DET
ijassa-1172	70	29	basic	basic	ADJ
ijassa-1172	70	30	reproduction	reproduction	NOUN
ijassa-1172	70	31	number	number	NOUN
ijassa-1172	70	32	for	for	ADP
ijassa-1172	70	33	asymptomatic	asymptomatic	ADJ
ijassa-1172	70	34	mode	mode	NOUN
ijassa-1172	70	35	of	of	ADP
ijassa-1172	70	36	transmission	transmission	NOUN
ijassa-1172	70	37	.	.	PUNCT
ijassa-1172	71	1	�	�	PROPN
ijassa-1172	71	2	r02	r02	PROPN
ijassa-1172	71	3	=	=	SYM
ijassa-1172	71	4	γψ(s0	γψ(s0	ADJ
ijassa-1172	71	5	,	,	PUNCT
ijassa-1172	71	6	0	0	NUM
ijassa-1172	71	7	)	)	PUNCT
ijassa-1172	71	8	d1d2	d1d2	NOUN
ijassa-1172	71	9	is	be	AUX
ijassa-1172	71	10	the	the	DET
ijassa-1172	71	11	basic	basic	ADJ
ijassa-1172	71	12	reproduction	reproduction	NOUN
ijassa-1172	71	13	number	number	NOUN
ijassa-1172	71	14	for	for	ADP
ijassa-1172	71	15	symptomatic	symptomatic	ADJ
ijassa-1172	71	16	mode	mode	NOUN
ijassa-1172	71	17	of	of	ADP
ijassa-1172	71	18	transmission	transmission	NOUN
ijassa-1172	71	19	.	.	PUNCT
ijassa-1172	72	1	the	the	DET
ijassa-1172	72	2	second	second	ADJ
ijassa-1172	72	3	equilibrium	equilibrium	NOUN
ijassa-1172	72	4	point	point	NOUN
ijassa-1172	72	5	of	of	ADP
ijassa-1172	72	6	(	(	PUNCT
ijassa-1172	72	7	2.2	2.2	NUM
ijassa-1172	72	8	)	)	PUNCT
ijassa-1172	72	9	satisfies	satisfy	VERB
ijassa-1172	72	10	the	the	DET
ijassa-1172	72	11	following	follow	VERB
ijassa-1172	72	12	system	system	NOUN
ijassa-1172	72	13	of	of	ADP
ijassa-1172	72	14	equations	equation	NOUN
ijassa-1172	72	15	a−	a−	PROPN
ijassa-1172	72	16	νs	νs	NOUN
ijassa-1172	72	17	−	−	ADP
ijassa-1172	72	18	φ(s	φ(s	NOUN
ijassa-1172	72	19	,	,	PUNCT
ijassa-1172	72	20	c)c	c)c	X
ijassa-1172	72	21	−ψ(s	−ψ(s	NOUN
ijassa-1172	72	22	,	,	PUNCT
ijassa-1172	72	23	i)i	i)i	NOUN
ijassa-1172	72	24	=	=	NOUN
ijassa-1172	72	25	0	0	NUM
ijassa-1172	72	26	,	,	PUNCT
ijassa-1172	72	27	(	(	PUNCT
ijassa-1172	72	28	2.4	2.4	NUM
ijassa-1172	72	29	)	)	PUNCT
ijassa-1172	72	30	φ(s	φ(s	NOUN
ijassa-1172	72	31	,	,	PUNCT
ijassa-1172	72	32	c)c	c)c	PUNCT
ijassa-1172	73	1	+	+	ADJ
ijassa-1172	73	2	ψ(s	ψ(s	ADJ
ijassa-1172	73	3	,	,	PUNCT
ijassa-1172	73	4	i)i	i)i	NOUN
ijassa-1172	73	5	−	−	PROPN
ijassa-1172	73	6	d1c	d1c	ADJ
ijassa-1172	74	1	=	=	SYM
ijassa-1172	74	2	0	0	PROPN
ijassa-1172	74	3	,	,	PUNCT
ijassa-1172	74	4	(	(	PUNCT
ijassa-1172	74	5	2.5	2.5	NUM
ijassa-1172	74	6	)	)	PUNCT
ijassa-1172	74	7	γc	γc	PROPN
ijassa-1172	75	1	−	−	NOUN
ijassa-1172	75	2	d2i	d2i	NOUN
ijassa-1172	76	1	=	=	NOUN
ijassa-1172	76	2	0	0	X
ijassa-1172	76	3	.	.	PUNCT
ijassa-1172	76	4	(	(	PUNCT
ijassa-1172	76	5	2.6	2.6	NUM
ijassa-1172	76	6	)	)	PUNCT
ijassa-1172	76	7	hence	hence	ADV
ijassa-1172	76	8	,	,	PUNCT
ijassa-1172	76	9	c	c	NOUN
ijassa-1172	76	10	=	=	SYM
ijassa-1172	76	11	a−	a−	PROPN
ijassa-1172	76	12	νs	νs	NOUN
ijassa-1172	76	13	d1	d1	PROPN
ijassa-1172	76	14	,	,	PUNCT
ijassa-1172	76	15	i	i	PRON
ijassa-1172	76	16	=	=	SYM
ijassa-1172	76	17	γc	γc	PROPN
ijassa-1172	76	18	d2	d2	PROPN
ijassa-1172	76	19	=	=	SYM
ijassa-1172	76	20	γ(a−	γ(a−	PROPN
ijassa-1172	76	21	νs	νs	NOUN
ijassa-1172	76	22	)	)	PUNCT
ijassa-1172	76	23	d1d2	d1d2	PROPN
ijassa-1172	76	24	and	and	CCONJ
ijassa-1172	76	25	d2φ(s	d2φ(s	PROPN
ijassa-1172	76	26	,	,	PUNCT
ijassa-1172	76	27	a−	a−	PROPN
ijassa-1172	76	28	νs	νs	NOUN
ijassa-1172	76	29	d1	d1	PROPN
ijassa-1172	76	30	)	)	PUNCT
ijassa-1172	77	1	+	+	CCONJ
ijassa-1172	77	2	γψ(s	γψ(s	PUNCT
ijassa-1172	77	3	,	,	PUNCT
ijassa-1172	77	4	γ(a−	γ(a−	PROPN
ijassa-1172	77	5	νs	νs	NOUN
ijassa-1172	77	6	)	)	PUNCT
ijassa-1172	77	7	d1d2	d1d2	X
ijassa-1172	77	8	)	)	PUNCT
ijassa-1172	77	9	=	=	SYM
ijassa-1172	78	1	d1d2	d1d2	NOUN
ijassa-1172	78	2	.	.	NOUN
ijassa-1172	78	3	since	since	SCONJ
ijassa-1172	78	4	c	c	NOUN
ijassa-1172	78	5	=	=	SYM
ijassa-1172	78	6	a−	a−	PROPN
ijassa-1172	78	7	νs	νs	NOUN
ijassa-1172	78	8	d1	d1	PROPN
ijassa-1172	78	9	≥	≥	NUM
ijassa-1172	78	10	0	0	NUM
ijassa-1172	78	11	,	,	PUNCT
ijassa-1172	78	12	we	we	PRON
ijassa-1172	78	13	have	have	VERB
ijassa-1172	78	14	s	s	VERB
ijassa-1172	78	15	≤	≤	NOUN
ijassa-1172	78	16	a	a	DET
ijassa-1172	78	17	ν	ν	NOUN
ijassa-1172	78	18	.	.	PUNCT
ijassa-1172	79	1	then	then	ADV
ijassa-1172	79	2	there	there	PRON
ijassa-1172	79	3	is	be	VERB
ijassa-1172	79	4	no	no	DET
ijassa-1172	79	5	epidemiological	epidemiological	ADJ
ijassa-1172	79	6	equilibrium	equilibrium	NOUN
ijassa-1172	79	7	when	when	SCONJ
ijassa-1172	79	8	s	s	VERB
ijassa-1172	79	9	>	>	X
ijassa-1172	79	10	a	a	DET
ijassa-1172	79	11	ν	ν	NOUN
ijassa-1172	79	12	.	.	PUNCT
ijassa-1172	80	1	the	the	DET
ijassa-1172	80	2	function	function	NOUN
ijassa-1172	80	3	h	h	NOUN
ijassa-1172	80	4	can	can	AUX
ijassa-1172	80	5	be	be	AUX
ijassa-1172	80	6	define	define	VERB
ijassa-1172	80	7	on	on	ADP
ijassa-1172	80	8	the	the	DET
ijassa-1172	80	9	closed	closed	ADJ
ijassa-1172	80	10	interval	interval	NOUN
ijassa-1172	80	11	[	[	X
ijassa-1172	80	12	0	0	NUM
ijassa-1172	80	13	,	,	PUNCT
ijassa-1172	80	14	a	a	DET
ijassa-1172	80	15	ν	ν	NOUN
ijassa-1172	80	16	]	]	PUNCT
ijassa-1172	80	17	by	by	ADP
ijassa-1172	80	18	h(s	h(	NOUN
ijassa-1172	80	19	)	)	PUNCT
ijassa-1172	81	1	=	=	SYM
ijassa-1172	81	2	d2φ(s	d2φ(s	PROPN
ijassa-1172	81	3	,	,	PUNCT
ijassa-1172	81	4	a−	a−	PROPN
ijassa-1172	81	5	νs	νs	NOUN
ijassa-1172	81	6	d1	d1	PROPN
ijassa-1172	81	7	)	)	PUNCT
ijassa-1172	82	1	+	+	CCONJ
ijassa-1172	82	2	γψ(s	γψ(s	PUNCT
ijassa-1172	82	3	,	,	PUNCT
ijassa-1172	82	4	γ(a−	γ(a−	PROPN
ijassa-1172	82	5	νs	νs	NOUN
ijassa-1172	82	6	)	)	PUNCT
ijassa-1172	82	7	d1d2	d1d2	NOUN
ijassa-1172	82	8	)	)	PUNCT
ijassa-1172	82	9	−	−	PROPN
ijassa-1172	83	1	d1d2	d1d2	NOUN
ijassa-1172	83	2	.	.	PUNCT
ijassa-1172	84	1	we	we	PRON
ijassa-1172	84	2	have	have	VERB
ijassa-1172	84	3	h(0	h(0	PROPN
ijassa-1172	84	4	)	)	PUNCT
ijassa-1172	84	5	=	=	PUNCT
ijassa-1172	85	1	−d1d2	−d1d2	PROPN
ijassa-1172	85	2	<	<	X
ijassa-1172	85	3	0	0	NUM
ijassa-1172	85	4	,	,	PUNCT
ijassa-1172	85	5	h	h	NOUN
ijassa-1172	85	6	(	(	PUNCT
ijassa-1172	85	7	a	a	DET
ijassa-1172	85	8	ν	ν	NOUN
ijassa-1172	85	9	)	)	PUNCT
ijassa-1172	86	1	=	=	SYM
ijassa-1172	86	2	d1d2(r0	d1d2(r0	NOUN
ijassa-1172	86	3	−	−	NOUN
ijassa-1172	86	4	1	1	NUM
ijassa-1172	86	5	)	)	PUNCT
ijassa-1172	86	6	and	and	CCONJ
ijassa-1172	86	7	h′(s	h′(s	ADJ
ijassa-1172	86	8	)	)	PUNCT
ijassa-1172	86	9	=	=	SYM
ijassa-1172	86	10	d2	d2	PROPN
ijassa-1172	86	11	(	(	PUNCT
ijassa-1172	86	12	∂φ	∂φ	PROPN
ijassa-1172	86	13	∂s	∂s	PROPN
ijassa-1172	87	1	−	−	PROPN
ijassa-1172	87	2	ν	ν	NOUN
ijassa-1172	87	3	d1	d1	PROPN
ijassa-1172	87	4	∂φ	∂φ	PROPN
ijassa-1172	87	5	∂c	∂c	PRON
ijassa-1172	87	6	)	)	PUNCT
ijassa-1172	88	1	+	+	CCONJ
ijassa-1172	88	2	γ	γ	X
ijassa-1172	88	3	(	(	PUNCT
ijassa-1172	88	4	∂ψ	∂ψ	PROPN
ijassa-1172	88	5	∂s	∂s	PROPN
ijassa-1172	88	6	−	−	NOUN
ijassa-1172	88	7	γν	γν	PRON
ijassa-1172	88	8	d1d2	d1d2	PUNCT
ijassa-1172	88	9	∂ψ	∂ψ	PROPN
ijassa-1172	88	10	∂i	∂i	PROPN
ijassa-1172	88	11	)	)	PUNCT
ijassa-1172	88	12	>	>	X
ijassa-1172	88	13	0	0	X
ijassa-1172	88	14	.	.	PUNCT
ijassa-1172	89	1	therefore	therefore	ADV
ijassa-1172	89	2	,	,	PUNCT
ijassa-1172	89	3	the	the	DET
ijassa-1172	89	4	equation	equation	NOUN
ijassa-1172	89	5	h(s	h(s	PROPN
ijassa-1172	89	6	)	)	PUNCT
ijassa-1172	90	1	=	=	SYM
ijassa-1172	90	2	0	0	PUNCT
ijassa-1172	90	3	has	have	VERB
ijassa-1172	90	4	a	a	DET
ijassa-1172	90	5	unique	unique	ADJ
ijassa-1172	90	6	root	root	NOUN
ijassa-1172	90	7	s∗	s∗	PROPN
ijassa-1172	90	8	∈	∈	PROPN
ijassa-1172	90	9	(	(	PUNCT
ijassa-1172	90	10	0	0	NUM
ijassa-1172	90	11	,	,	PUNCT
ijassa-1172	90	12	a	a	DET
ijassa-1172	90	13	ν	ν	NOUN
ijassa-1172	90	14	)	)	PUNCT
ijassa-1172	90	15	when	when	SCONJ
ijassa-1172	90	16	r0	r0	NOUN
ijassa-1172	90	17	>	>	X
ijassa-1172	90	18	1	1	X
ijassa-1172	90	19	.	.	PUNCT
ijassa-1172	91	1	this	this	PRON
ijassa-1172	91	2	implies	imply	VERB
ijassa-1172	91	3	that	that	SCONJ
ijassa-1172	91	4	our	our	PRON
ijassa-1172	91	5	model	model	NOUN
ijassa-1172	91	6	has	have	VERB
ijassa-1172	91	7	a	a	DET
ijassa-1172	91	8	unique	unique	ADJ
ijassa-1172	91	9	endemic	endemic	ADJ
ijassa-1172	91	10	equilibrium	equilibrium	NOUN
ijassa-1172	91	11	e∗(s∗	e∗(s∗	NOUN
ijassa-1172	91	12	,	,	PUNCT
ijassa-1172	91	13	c∗	c∗	NOUN
ijassa-1172	91	14	,	,	PUNCT
ijassa-1172	91	15	i∗	i∗	NOUN
ijassa-1172	91	16	)	)	PUNCT
ijassa-1172	91	17	under	under	ADP
ijassa-1172	91	18	the	the	DET
ijassa-1172	91	19	condition	condition	NOUN
ijassa-1172	91	20	r0	r0	NOUN
ijassa-1172	91	21	>	>	X
ijassa-1172	91	22	1	1	X
ijassa-1172	91	23	.	.	X
ijassa-1172	91	24	summarizing	summarize	VERB
ijassa-1172	91	25	all	all	DET
ijassa-1172	91	26	the	the	DET
ijassa-1172	91	27	above	above	ADJ
ijassa-1172	91	28	cases	case	NOUN
ijassa-1172	91	29	in	in	ADP
ijassa-1172	91	30	the	the	DET
ijassa-1172	91	31	following	follow	VERB
ijassa-1172	91	32	result	result	NOUN
ijassa-1172	91	33	.	.	PUNCT
ijassa-1172	92	1	theorem	theorem	VERB
ijassa-1172	92	2	2.1	2.1	NUM
ijassa-1172	92	3	:	:	PUNCT
ijassa-1172	92	4	let	let	VERB
ijassa-1172	92	5	r0	r0	NOUN
ijassa-1172	92	6	be	be	AUX
ijassa-1172	92	7	defined	define	VERB
ijassa-1172	92	8	by	by	ADP
ijassa-1172	92	9	(	(	PUNCT
ijassa-1172	92	10	2.3	2.3	NUM
ijassa-1172	92	11	)	)	PUNCT
ijassa-1172	92	12	.	.	PUNCT
ijassa-1172	93	1	copyright	copyright	NOUN
ijassa-1172	93	2	©	©	PROPN
ijassa-1172	93	3	2022	2022	NUM
ijassa-1172	93	4	assa	assa	NOUN
ijassa-1172	93	5	.	.	PUNCT
ijassa-1172	94	1	adv	adv	PROPN
ijassa-1172	94	2	syst	syst	PROPN
ijassa-1172	94	3	sci	sci	PROPN
ijassa-1172	94	4	appl	appl	PROPN
ijassa-1172	94	5	(	(	PUNCT
ijassa-1172	94	6	2022	2022	NUM
ijassa-1172	94	7	)	)	PUNCT
ijassa-1172	94	8	40	40	NUM
ijassa-1172	94	9	khalid	khalid	PROPN
ijassa-1172	94	10	hattaf	hattaf	NOUN
ijassa-1172	94	11	,	,	PUNCT
ijassa-1172	94	12	ahmed	ahmed	PROPN
ijassa-1172	94	13	a.	a.	PROPN
ijassa-1172	94	14	mohsen	mohsen	PROPN
ijassa-1172	94	15	(	(	PUNCT
ijassa-1172	94	16	i	i	NOUN
ijassa-1172	94	17	)	)	PUNCT
ijassa-1172	94	18	if	if	SCONJ
ijassa-1172	94	19	r0	r0	NOUN
ijassa-1172	94	20	≤	≤	NOUN
ijassa-1172	94	21	1	1	NUM
ijassa-1172	94	22	,	,	PUNCT
ijassa-1172	94	23	then	then	ADV
ijassa-1172	94	24	system	system	NOUN
ijassa-1172	94	25	(	(	PUNCT
ijassa-1172	94	26	2.2	2.2	NUM
ijassa-1172	94	27	)	)	PUNCT
ijassa-1172	94	28	has	have	VERB
ijassa-1172	94	29	only	only	ADV
ijassa-1172	94	30	disease	disease	NOUN
ijassa-1172	94	31	-	-	PUNCT
ijassa-1172	94	32	free	free	ADJ
ijassa-1172	94	33	equilibrium	equilibrium	NOUN
ijassa-1172	94	34	of	of	ADP
ijassa-1172	94	35	the	the	DET
ijassa-1172	94	36	form	form	NOUN
ijassa-1172	94	37	ef	ef	X
ijassa-1172	94	38	(	(	PUNCT
ijassa-1172	94	39	s0	s0	PROPN
ijassa-1172	94	40	,	,	PUNCT
ijassa-1172	94	41	0	0	NUM
ijassa-1172	94	42	,	,	PUNCT
ijassa-1172	94	43	0	0	NUM
ijassa-1172	94	44	)	)	PUNCT
ijassa-1172	94	45	,	,	PUNCT
ijassa-1172	94	46	where	where	SCONJ
ijassa-1172	94	47	s0	s0	PROPN
ijassa-1172	94	48	=	=	PUNCT
ijassa-1172	94	49	a	a	DET
ijassa-1172	94	50	ν	ν	NOUN
ijassa-1172	94	51	.	.	PUNCT
ijassa-1172	95	1	(	(	PUNCT
ijassa-1172	95	2	ii	ii	NOUN
ijassa-1172	95	3	)	)	PUNCT
ijassa-1172	95	4	if	if	SCONJ
ijassa-1172	95	5	r0	r0	NOUN
ijassa-1172	95	6	>	>	X
ijassa-1172	95	7	1	1	NUM
ijassa-1172	95	8	,	,	PUNCT
ijassa-1172	95	9	then	then	ADV
ijassa-1172	95	10	system	system	NOUN
ijassa-1172	95	11	(	(	PUNCT
ijassa-1172	95	12	2.2	2.2	NUM
ijassa-1172	95	13	)	)	PUNCT
ijassa-1172	95	14	has	have	VERB
ijassa-1172	95	15	a	a	DET
ijassa-1172	95	16	unique	unique	ADJ
ijassa-1172	95	17	endemic	endemic	ADJ
ijassa-1172	95	18	equilibrium	equilibrium	NOUN
ijassa-1172	95	19	e∗(s∗	e∗(s∗	NOUN
ijassa-1172	95	20	,	,	PUNCT
ijassa-1172	95	21	c∗	c∗	NOUN
ijassa-1172	95	22	,	,	PUNCT
ijassa-1172	95	23	i∗	i∗	NOUN
ijassa-1172	95	24	)	)	PUNCT
ijassa-1172	95	25	,	,	PUNCT
ijassa-1172	95	26	where	where	SCONJ
ijassa-1172	95	27	s∗	s∗	PROPN
ijassa-1172	95	28	∈	∈	PROPN
ijassa-1172	95	29	(	(	PUNCT
ijassa-1172	95	30	0	0	NUM
ijassa-1172	95	31	,	,	PUNCT
ijassa-1172	95	32	a	a	DET
ijassa-1172	95	33	ν	ν	NOUN
ijassa-1172	95	34	)	)	PUNCT
ijassa-1172	95	35	,	,	PUNCT
ijassa-1172	95	36	c∗	c∗	NOUN
ijassa-1172	95	37	=	=	SYM
ijassa-1172	95	38	a−	a−	PROPN
ijassa-1172	95	39	νs∗	νs∗	NOUN
ijassa-1172	95	40	d1	d1	PROPN
ijassa-1172	95	41	and	and	CCONJ
ijassa-1172	95	42	i∗	i∗	NOUN
ijassa-1172	95	43	=	=	SYM
ijassa-1172	95	44	γ(a−	γ(a−	NOUN
ijassa-1172	95	45	νs∗	νs∗	NOUN
ijassa-1172	95	46	)	)	PUNCT
ijassa-1172	96	1	d1d2	d1d2	VERB
ijassa-1172	96	2	.	.	PUNCT
ijassa-1172	97	1	also	also	ADV
ijassa-1172	97	2	,	,	PUNCT
ijassa-1172	97	3	the	the	DET
ijassa-1172	97	4	disease	disease	NOUN
ijassa-1172	97	5	-	-	PUNCT
ijassa-1172	97	6	free	free	ADJ
ijassa-1172	97	7	equilibrium	equilibrium	NOUN
ijassa-1172	97	8	exists	exist	VERB
ijassa-1172	97	9	.	.	PUNCT
ijassa-1172	98	1	3	3	X
ijassa-1172	98	2	.	.	X
ijassa-1172	98	3	global	global	ADJ
ijassa-1172	98	4	dynamics	dynamic	NOUN
ijassa-1172	98	5	in	in	ADP
ijassa-1172	98	6	this	this	DET
ijassa-1172	98	7	section	section	NOUN
ijassa-1172	98	8	,	,	PUNCT
ijassa-1172	98	9	we	we	PRON
ijassa-1172	98	10	analyze	analyze	VERB
ijassa-1172	98	11	the	the	DET
ijassa-1172	98	12	stability	stability	NOUN
ijassa-1172	98	13	of	of	ADP
ijassa-1172	98	14	the	the	DET
ijassa-1172	98	15	disease	disease	NOUN
ijassa-1172	98	16	-	-	PUNCT
ijassa-1172	98	17	free	free	ADJ
ijassa-1172	98	18	equilibrium	equilibrium	NOUN
ijassa-1172	98	19	ef	ef	X
ijassa-1172	98	20	and	and	CCONJ
ijassa-1172	98	21	the	the	DET
ijassa-1172	98	22	endemic	endemic	ADJ
ijassa-1172	98	23	equilibrium	equilibrium	NOUN
ijassa-1172	98	24	e∗.	e∗.	NOUN
ijassa-1172	98	25	theorem	theorem	VERB
ijassa-1172	98	26	3.1	3.1	NUM
ijassa-1172	98	27	:	:	PUNCT
ijassa-1172	98	28	the	the	DET
ijassa-1172	98	29	disease	disease	NOUN
ijassa-1172	98	30	-	-	PUNCT
ijassa-1172	98	31	free	free	ADJ
ijassa-1172	98	32	equilibrium	equilibrium	NOUN
ijassa-1172	98	33	ef	ef	PROPN
ijassa-1172	98	34	is	be	AUX
ijassa-1172	98	35	globally	globally	ADV
ijassa-1172	98	36	asymptotically	asymptotically	ADV
ijassa-1172	98	37	stable	stable	ADJ
ijassa-1172	98	38	if	if	SCONJ
ijassa-1172	98	39	r0	r0	NOUN
ijassa-1172	98	40	≤	≤	NOUN
ijassa-1172	98	41	1	1	NUM
ijassa-1172	98	42	and	and	CCONJ
ijassa-1172	98	43	unstable	unstable	ADJ
ijassa-1172	98	44	if	if	SCONJ
ijassa-1172	98	45	r0	r0	NOUN
ijassa-1172	98	46	>	>	X
ijassa-1172	98	47	1	1	X
ijassa-1172	98	48	.	.	PUNCT
ijassa-1172	99	1	proof	proof	NOUN
ijassa-1172	99	2	we	we	PRON
ijassa-1172	99	3	define	define	VERB
ijassa-1172	99	4	the	the	DET
ijassa-1172	99	5	following	follow	VERB
ijassa-1172	99	6	lyapunov	lyapunov	ADJ
ijassa-1172	99	7	functional	functional	ADJ
ijassa-1172	99	8	f(t	f(t	NOUN
ijassa-1172	99	9	)	)	PUNCT
ijassa-1172	99	10	=	=	SYM
ijassa-1172	100	1	c	c	PROPN
ijassa-1172	100	2	+	+	NUM
ijassa-1172	100	3	ψ(s0	ψ(s0	PROPN
ijassa-1172	100	4	,	,	PUNCT
ijassa-1172	100	5	0	0	NUM
ijassa-1172	100	6	)	)	PUNCT
ijassa-1172	100	7	d2	d2	PROPN
ijassa-1172	100	8	i.	i.	NOUN
ijassa-1172	100	9	then	then	ADV
ijassa-1172	100	10	dαf(t	dαf(t	PROPN
ijassa-1172	100	11	)	)	PUNCT
ijassa-1172	100	12	=	=	SYM
ijassa-1172	100	13	dαc	dαc	NOUN
ijassa-1172	101	1	+	+	X
ijassa-1172	101	2	ψ(s0	ψ(s0	NOUN
ijassa-1172	101	3	,	,	PUNCT
ijassa-1172	101	4	0	0	NUM
ijassa-1172	101	5	)	)	PUNCT
ijassa-1172	101	6	d2	d2	NOUN
ijassa-1172	101	7	dαi	dαi	NOUN
ijassa-1172	101	8	=	=	PUNCT
ijassa-1172	101	9	φ(s	φ(s	NOUN
ijassa-1172	101	10	,	,	PUNCT
ijassa-1172	101	11	c)c	c)c	PUNCT
ijassa-1172	102	1	+	+	ADJ
ijassa-1172	102	2	ψ(s	ψ(s	ADJ
ijassa-1172	102	3	,	,	PUNCT
ijassa-1172	102	4	i)i	i)i	NOUN
ijassa-1172	102	5	−	−	PROPN
ijassa-1172	102	6	d1c	d1c	X
ijassa-1172	103	1	+	+	NUM
ijassa-1172	103	2	ψ(s0	ψ(s0	PROPN
ijassa-1172	103	3	,	,	PUNCT
ijassa-1172	103	4	0	0	NUM
ijassa-1172	103	5	)	)	PUNCT
ijassa-1172	103	6	d2	d2	NOUN
ijassa-1172	103	7	(	(	PUNCT
ijassa-1172	103	8	γc	γc	PROPN
ijassa-1172	103	9	−	−	PROPN
ijassa-1172	103	10	d2i	d2i	NOUN
ijassa-1172	103	11	)	)	PUNCT
ijassa-1172	103	12	.	.	PUNCT
ijassa-1172	104	1	using	use	VERB
ijassa-1172	104	2	a	a	DET
ijassa-1172	104	3	=	=	SYM
ijassa-1172	104	4	νs0	νs0	PROPN
ijassa-1172	104	5	,	,	PUNCT
ijassa-1172	104	6	we	we	PRON
ijassa-1172	104	7	get	get	VERB
ijassa-1172	104	8	dαf(t	dαf(t	NOUN
ijassa-1172	104	9	)	)	PUNCT
ijassa-1172	104	10	=	=	SYM
ijassa-1172	104	11	d1	d1	PROPN
ijassa-1172	104	12	(	(	PUNCT
ijassa-1172	104	13	d2φ(s	d2φ(s	PROPN
ijassa-1172	104	14	,	,	PUNCT
ijassa-1172	104	15	i	i	PROPN
ijassa-1172	104	16	)	)	PUNCT
ijassa-1172	104	17	+	+	CCONJ
ijassa-1172	104	18	γψ(s0	γψ(s0	ADJ
ijassa-1172	104	19	,	,	PUNCT
ijassa-1172	104	20	0	0	NUM
ijassa-1172	104	21	)	)	PUNCT
ijassa-1172	104	22	d1d2	d1d2	VERB
ijassa-1172	104	23	−	−	NOUN
ijassa-1172	104	24	1	1	NUM
ijassa-1172	104	25	)	)	PUNCT
ijassa-1172	104	26	c	c	NOUN
ijassa-1172	105	1	+	+	CCONJ
ijassa-1172	105	2	(	(	PUNCT
ijassa-1172	105	3	ψ(s	ψ(s	PROPN
ijassa-1172	105	4	,	,	PUNCT
ijassa-1172	105	5	i)−ψ(s0	i)−ψ(s0	PROPN
ijassa-1172	105	6	,	,	PUNCT
ijassa-1172	105	7	0	0	NUM
ijassa-1172	105	8	)	)	PUNCT
ijassa-1172	105	9	)	)	PUNCT
ijassa-1172	105	10	i.	i.	NOUN
ijassa-1172	105	11	from	from	ADP
ijassa-1172	105	12	the	the	DET
ijassa-1172	105	13	first	first	ADJ
ijassa-1172	105	14	equation	equation	NOUN
ijassa-1172	105	15	of	of	ADP
ijassa-1172	105	16	(	(	PUNCT
ijassa-1172	105	17	2.2	2.2	NUM
ijassa-1172	105	18	)	)	PUNCT
ijassa-1172	105	19	,	,	PUNCT
ijassa-1172	105	20	we	we	PRON
ijassa-1172	105	21	have	have	AUX
ijassa-1172	105	22	dαs	dα	VERB
ijassa-1172	105	23	≤	≤	NUM
ijassa-1172	105	24	a−	a−	PROPN
ijassa-1172	105	25	νs	νs	NOUN
ijassa-1172	105	26	,	,	PUNCT
ijassa-1172	105	27	which	which	PRON
ijassa-1172	105	28	leads	lead	VERB
ijassa-1172	105	29	to	to	ADP
ijassa-1172	105	30	s(t	s(t	PROPN
ijassa-1172	105	31	)	)	PUNCT
ijassa-1172	105	32	≤	≤	NUM
ijassa-1172	105	33	s(0)eα(−νtα	s(0)eα(−νtα	NOUN
ijassa-1172	105	34	)	)	PUNCT
ijassa-1172	106	1	+	+	CCONJ
ijassa-1172	106	2	a	a	DET
ijassa-1172	106	3	ν	ν	NOUN
ijassa-1172	106	4	[	[	X
ijassa-1172	106	5	1−	1−	NUM
ijassa-1172	106	6	eα(−νtα	eα(−νtα	NOUN
ijassa-1172	106	7	)	)	PUNCT
ijassa-1172	106	8	]	]	PUNCT
ijassa-1172	106	9	,	,	PUNCT
ijassa-1172	106	10	where	where	SCONJ
ijassa-1172	106	11	eα(z	eα(z	VERB
ijassa-1172	106	12	)	)	PUNCT
ijassa-1172	106	13	=	=	PUNCT
ijassa-1172	107	1	∞∑	∞∑	NUM
ijassa-1172	107	2	k=0	k=0	PUNCT
ijassa-1172	107	3	zα	zα	PROPN
ijassa-1172	107	4	γ(αk	γ(αk	PROPN
ijassa-1172	107	5	+	+	CCONJ
ijassa-1172	107	6	1	1	X
ijassa-1172	107	7	)	)	PUNCT
ijassa-1172	107	8	denotes	denote	VERB
ijassa-1172	107	9	the	the	DET
ijassa-1172	107	10	mittag	mittag	ADJ
ijassa-1172	107	11	-	-	PUNCT
ijassa-1172	107	12	leffler	leffler	NOUN
ijassa-1172	107	13	function	function	NOUN
ijassa-1172	107	14	of	of	ADP
ijassa-1172	107	15	one	one	NUM
ijassa-1172	107	16	parameter	parameter	NOUN
ijassa-1172	107	17	α	α	NOUN
ijassa-1172	107	18	.	.	PUNCT
ijassa-1172	108	1	then	then	ADV
ijassa-1172	108	2	lim	lim	PROPN
ijassa-1172	108	3	sup	sup	PROPN
ijassa-1172	108	4	t→∞	t→∞	NUM
ijassa-1172	108	5	s(t	s(t	PROPN
ijassa-1172	108	6	)	)	PUNCT
ijassa-1172	108	7	≤	≤	NOUN
ijassa-1172	108	8	a	a	DET
ijassa-1172	108	9	ν	ν	NOUN
ijassa-1172	108	10	,	,	PUNCT
ijassa-1172	108	11	which	which	PRON
ijassa-1172	108	12	involves	involve	VERB
ijassa-1172	108	13	that	that	SCONJ
ijassa-1172	108	14	all	all	DET
ijassa-1172	108	15	omega	omega	NOUN
ijassa-1172	108	16	limit	limit	NOUN
ijassa-1172	108	17	points	point	NOUN
ijassa-1172	108	18	verify	verify	VERB
ijassa-1172	108	19	s(t	s(t	NUM
ijassa-1172	108	20	)	)	PUNCT
ijassa-1172	108	21	≤	≤	NUM
ijassa-1172	108	22	s0	s0	NOUN
ijassa-1172	108	23	.	.	PUNCT
ijassa-1172	109	1	thus	thus	ADV
ijassa-1172	109	2	,	,	PUNCT
ijassa-1172	109	3	it	it	PRON
ijassa-1172	109	4	suffices	suffice	VERB
ijassa-1172	109	5	to	to	PART
ijassa-1172	109	6	consider	consider	VERB
ijassa-1172	109	7	solutions	solution	NOUN
ijassa-1172	109	8	for	for	ADP
ijassa-1172	109	9	which	which	PRON
ijassa-1172	109	10	s(t	s(t	PROPN
ijassa-1172	109	11	)	)	PUNCT
ijassa-1172	109	12	≤	≤	NUM
ijassa-1172	109	13	s0	s0	NOUN
ijassa-1172	109	14	.	.	PUNCT
ijassa-1172	110	1	according	accord	VERB
ijassa-1172	110	2	to	to	ADP
ijassa-1172	110	3	the	the	DET
ijassa-1172	110	4	explicit	explicit	ADJ
ijassa-1172	110	5	expression	expression	NOUN
ijassa-1172	110	6	of	of	ADP
ijassa-1172	110	7	r0	r0	NOUN
ijassa-1172	110	8	given	give	VERB
ijassa-1172	110	9	in	in	ADP
ijassa-1172	110	10	(	(	PUNCT
ijassa-1172	110	11	2.3	2.3	NUM
ijassa-1172	110	12	)	)	PUNCT
ijassa-1172	110	13	and	and	CCONJ
ijassa-1172	110	14	(	(	PUNCT
ijassa-1172	110	15	h1)-(h2	h1)-(h2	NOUN
ijassa-1172	110	16	)	)	PUNCT
ijassa-1172	110	17	,	,	PUNCT
ijassa-1172	110	18	we	we	PRON
ijassa-1172	110	19	obtain	obtain	VERB
ijassa-1172	110	20	dαf(t	dαf(t	NOUN
ijassa-1172	110	21	)	)	PUNCT
ijassa-1172	110	22	≤	≤	NOUN
ijassa-1172	110	23	d1	d1	NOUN
ijassa-1172	110	24	(	(	PUNCT
ijassa-1172	110	25	r0	r0	NOUN
ijassa-1172	110	26	−	−	PROPN
ijassa-1172	110	27	1	1	NUM
ijassa-1172	110	28	)	)	PUNCT
ijassa-1172	110	29	c.	c.	NOUN
ijassa-1172	110	30	then	then	ADV
ijassa-1172	110	31	dαf(t	dαf(t	PROPN
ijassa-1172	110	32	)	)	PUNCT
ijassa-1172	110	33	≤	≤	NOUN
ijassa-1172	110	34	0	0	PUNCT
ijassa-1172	110	35	when	when	SCONJ
ijassa-1172	110	36	r0	r0	NOUN
ijassa-1172	110	37	≤	≤	NOUN
ijassa-1172	110	38	1	1	NUM
ijassa-1172	110	39	.	.	PUNCT
ijassa-1172	111	1	furthermore	furthermore	ADV
ijassa-1172	111	2	,	,	PUNCT
ijassa-1172	111	3	dαf(t	dαf(t	PROPN
ijassa-1172	111	4	)	)	PUNCT
ijassa-1172	111	5	=	=	SYM
ijassa-1172	111	6	0	0	PUNCT
ijassa-1172	112	1	if	if	SCONJ
ijassa-1172	112	2	and	and	CCONJ
ijassa-1172	112	3	only	only	ADV
ijassa-1172	112	4	if	if	SCONJ
ijassa-1172	112	5	s	s	NOUN
ijassa-1172	112	6	=	=	SYM
ijassa-1172	112	7	s0	s0	PROPN
ijassa-1172	112	8	,	,	PUNCT
ijassa-1172	112	9	c	c	NOUN
ijassa-1172	112	10	=	=	SYM
ijassa-1172	112	11	0	0	PUNCT
ijassa-1172	113	1	and	and	CCONJ
ijassa-1172	113	2	i	i	PRON
ijassa-1172	113	3	=	=	NOUN
ijassa-1172	113	4	0	0	X
ijassa-1172	113	5	.	.	PUNCT
ijassa-1172	114	1	therefore	therefore	ADV
ijassa-1172	114	2	,	,	PUNCT
ijassa-1172	114	3	{	{	PUNCT
ijassa-1172	114	4	ef	ef	PART
ijassa-1172	114	5	}	}	PUNCT
ijassa-1172	114	6	is	be	AUX
ijassa-1172	114	7	the	the	DET
ijassa-1172	114	8	largest	large	ADJ
ijassa-1172	114	9	invariant	invariant	ADJ
ijassa-1172	114	10	set	set	NOUN
ijassa-1172	114	11	in	in	ADP
ijassa-1172	114	12	{	{	PUNCT
ijassa-1172	114	13	(	(	PUNCT
ijassa-1172	114	14	s	s	PROPN
ijassa-1172	114	15	,	,	PUNCT
ijassa-1172	114	16	c	c	X
ijassa-1172	114	17	,	,	PUNCT
ijassa-1172	114	18	i	i	NOUN
ijassa-1172	114	19	)	)	PUNCT
ijassa-1172	114	20	|	|	ADV
ijassa-1172	114	21	dαf(t	dαf(t	VERB
ijassa-1172	114	22	)	)	PUNCT
ijassa-1172	114	23	=	=	SYM
ijassa-1172	114	24	0	0	NUM
ijassa-1172	114	25	}	}	PUNCT
ijassa-1172	114	26	.	.	PUNCT
ijassa-1172	115	1	it	it	PRON
ijassa-1172	115	2	copyright	copyright	NOUN
ijassa-1172	115	3	©	©	PROPN
ijassa-1172	115	4	2022	2022	NUM
ijassa-1172	115	5	assa	assa	NOUN
ijassa-1172	115	6	.	.	PUNCT
ijassa-1172	116	1	adv	adv	PROPN
ijassa-1172	116	2	syst	syst	PROPN
ijassa-1172	116	3	sci	sci	PROPN
ijassa-1172	116	4	appl	appl	PROPN
ijassa-1172	116	5	(	(	PUNCT
ijassa-1172	116	6	2022	2022	NUM
ijassa-1172	116	7	)	)	PUNCT
ijassa-1172	116	8	dynamics	dynamic	NOUN
ijassa-1172	116	9	of	of	ADP
ijassa-1172	116	10	a	a	DET
ijassa-1172	116	11	generalized	generalize	VERB
ijassa-1172	116	12	fractional	fractional	ADJ
ijassa-1172	116	13	epidemic	epidemic	NOUN
ijassa-1172	116	14	model	model	NOUN
ijassa-1172	116	15	...	...	PUNCT
ijassa-1172	116	16	41	41	NUM
ijassa-1172	116	17	follows	follow	VERB
ijassa-1172	116	18	from	from	ADP
ijassa-1172	116	19	the	the	DET
ijassa-1172	116	20	lasalle	lasalle	NOUN
ijassa-1172	116	21	’s	’s	PART
ijassa-1172	116	22	invariance	invariance	NOUN
ijassa-1172	116	23	principale	principale	NOUN
ijassa-1172	116	24	[	[	X
ijassa-1172	116	25	16	16	NUM
ijassa-1172	116	26	]	]	PUNCT
ijassa-1172	116	27	that	that	SCONJ
ijassa-1172	116	28	ef	ef	PROPN
ijassa-1172	116	29	is	be	AUX
ijassa-1172	116	30	globally	globally	ADV
ijassa-1172	116	31	asymptotically	asymptotically	ADV
ijassa-1172	116	32	stable	stable	ADJ
ijassa-1172	116	33	when	when	SCONJ
ijassa-1172	116	34	r0	r0	NOUN
ijassa-1172	116	35	≤	≤	NOUN
ijassa-1172	116	36	1	1	NUM
ijassa-1172	116	37	.	.	PUNCT
ijassa-1172	117	1	on	on	ADP
ijassa-1172	117	2	the	the	DET
ijassa-1172	117	3	other	other	ADJ
ijassa-1172	117	4	hand	hand	NOUN
ijassa-1172	117	5	,	,	PUNCT
ijassa-1172	117	6	the	the	DET
ijassa-1172	117	7	characteristic	characteristic	ADJ
ijassa-1172	117	8	equation	equation	NOUN
ijassa-1172	117	9	of	of	ADP
ijassa-1172	117	10	model	model	NOUN
ijassa-1172	117	11	(	(	PUNCT
ijassa-1172	117	12	2.2	2.2	NUM
ijassa-1172	117	13	)	)	PUNCT
ijassa-1172	117	14	at	at	ADP
ijassa-1172	117	15	ef	ef	PROPN
ijassa-1172	117	16	is	be	AUX
ijassa-1172	117	17	given	give	VERB
ijassa-1172	117	18	by	by	ADP
ijassa-1172	117	19	(	(	PUNCT
ijassa-1172	117	20	ν	ν	X
ijassa-1172	117	21	+	+	CCONJ
ijassa-1172	117	22	λ)(α	λ)(α	PROPN
ijassa-1172	117	23	+	+	NUM
ijassa-1172	117	24	ν	ν	X
ijassa-1172	117	25	+	+	X
ijassa-1172	117	26	λ)θ(λ	λ)θ(λ	NOUN
ijassa-1172	117	27	)	)	PUNCT
ijassa-1172	117	28	=	=	SYM
ijassa-1172	117	29	0	0	NUM
ijassa-1172	117	30	,	,	PUNCT
ijassa-1172	117	31	(	(	PUNCT
ijassa-1172	117	32	3.7	3.7	NUM
ijassa-1172	117	33	)	)	PUNCT
ijassa-1172	117	34	where	where	SCONJ
ijassa-1172	117	35	θ(λ	θ(λ	NOUN
ijassa-1172	118	1	)	)	PUNCT
ijassa-1172	118	2	=	=	SYM
ijassa-1172	118	3	λ2	λ2	NOUN
ijassa-1172	118	4	+	+	CCONJ
ijassa-1172	118	5	(	(	PUNCT
ijassa-1172	118	6	d1	d1	NOUN
ijassa-1172	118	7	+	+	CCONJ
ijassa-1172	118	8	d1	d1	PROPN
ijassa-1172	118	9	−	−	PROPN
ijassa-1172	118	10	φ(s0	φ(s0	NOUN
ijassa-1172	118	11	,	,	PUNCT
ijassa-1172	118	12	0	0	NUM
ijassa-1172	118	13	)	)	PUNCT
ijassa-1172	118	14	)	)	PUNCT
ijassa-1172	119	1	λ+	λ+	PUNCT
ijassa-1172	119	2	d1d2(1−r0	d1d2(1−r0	NOUN
ijassa-1172	119	3	)	)	PUNCT
ijassa-1172	119	4	.	.	PUNCT
ijassa-1172	120	1	we	we	PRON
ijassa-1172	120	2	have	have	VERB
ijassa-1172	120	3	lim	lim	PROPN
ijassa-1172	120	4	λ→+∞	λ→+∞	PROPN
ijassa-1172	120	5	θ(λ	θ(λ	PROPN
ijassa-1172	120	6	)	)	PUNCT
ijassa-1172	120	7	=	=	PUNCT
ijassa-1172	121	1	+	+	PUNCT
ijassa-1172	121	2	∞	∞	NUM
ijassa-1172	121	3	and	and	CCONJ
ijassa-1172	121	4	θ(0	θ(0	PROPN
ijassa-1172	121	5	)	)	PUNCT
ijassa-1172	121	6	=	=	SYM
ijassa-1172	121	7	d1d2(1−r0	d1d2(1−r0	NOUN
ijassa-1172	121	8	)	)	PUNCT
ijassa-1172	121	9	<	<	X
ijassa-1172	121	10	0	0	PUNCT
ijassa-1172	121	11	when	when	SCONJ
ijassa-1172	121	12	r0	r0	NOUN
ijassa-1172	121	13	>	>	X
ijassa-1172	121	14	1	1	X
ijassa-1172	121	15	.	.	PUNCT
ijassa-1172	122	1	then	then	ADV
ijassa-1172	122	2	we	we	PRON
ijassa-1172	122	3	have	have	VERB
ijassa-1172	122	4	the	the	DET
ijassa-1172	122	5	eigenvalue	eigenvalue	PROPN
ijassa-1172	122	6	λ0	λ0	NOUN
ijassa-1172	122	7	∈	∈	NOUN
ijassa-1172	122	8	(	(	PUNCT
ijassa-1172	122	9	0,+∞	0,+∞	NUM
ijassa-1172	122	10	)	)	PUNCT
ijassa-1172	122	11	such	such	ADJ
ijassa-1172	122	12	that	that	DET
ijassa-1172	122	13	θ(λ0	θ(λ0	NOUN
ijassa-1172	122	14	)	)	PUNCT
ijassa-1172	122	15	=	=	SYM
ijassa-1172	122	16	0	0	NUM
ijassa-1172	122	17	,	,	PUNCT
ijassa-1172	122	18	which	which	PRON
ijassa-1172	122	19	implies	imply	VERB
ijassa-1172	122	20	that	that	SCONJ
ijassa-1172	122	21	the	the	DET
ijassa-1172	122	22	equation	equation	NOUN
ijassa-1172	122	23	(	(	PUNCT
ijassa-1172	122	24	3.7	3.7	NUM
ijassa-1172	122	25	)	)	PUNCT
ijassa-1172	122	26	at	at	ADP
ijassa-1172	122	27	ef	ef	PROPN
ijassa-1172	122	28	has	have	VERB
ijassa-1172	122	29	at	at	ADV
ijassa-1172	122	30	least	least	ADV
ijassa-1172	122	31	one	one	NUM
ijassa-1172	122	32	positive	positive	ADJ
ijassa-1172	122	33	root	root	NOUN
ijassa-1172	122	34	if	if	SCONJ
ijassa-1172	122	35	r0	r0	NOUN
ijassa-1172	122	36	>	>	X
ijassa-1172	122	37	1	1	X
ijassa-1172	122	38	.	.	PUNCT
ijassa-1172	123	1	therefore	therefore	ADV
ijassa-1172	123	2	,	,	PUNCT
ijassa-1172	123	3	ef	ef	PROPN
ijassa-1172	123	4	is	be	AUX
ijassa-1172	123	5	saddle	saddle	ADJ
ijassa-1172	123	6	(	(	PUNCT
ijassa-1172	123	7	unstable	unstable	ADJ
ijassa-1172	123	8	)	)	PUNCT
ijassa-1172	123	9	whenever	whenever	SCONJ
ijassa-1172	123	10	r0	r0	NOUN
ijassa-1172	123	11	>	>	X
ijassa-1172	123	12	1	1	X
ijassa-1172	123	13	.	.	PUNCT
ijassa-1172	124	1	in	in	ADP
ijassa-1172	124	2	the	the	DET
ijassa-1172	124	3	following	following	NOUN
ijassa-1172	124	4	,	,	PUNCT
ijassa-1172	124	5	we	we	PRON
ijassa-1172	124	6	assume	assume	VERB
ijassa-1172	124	7	that	that	SCONJ
ijassa-1172	124	8	r0	r0	NOUN
ijassa-1172	124	9	>	>	X
ijassa-1172	124	10	1	1	NUM
ijassa-1172	124	11	and	and	CCONJ
ijassa-1172	124	12	the	the	DET
ijassa-1172	124	13	incidence	incidence	NOUN
ijassa-1172	124	14	functions	function	NOUN
ijassa-1172	124	15	φ	φ	NOUN
ijassa-1172	124	16	and	and	CCONJ
ijassa-1172	124	17	ψ	ψ	X
ijassa-1172	124	18	satisfy	satisfy	NOUN
ijassa-1172	124	19	,	,	PUNCT
ijassa-1172	124	20	for	for	ADP
ijassa-1172	124	21	all	all	DET
ijassa-1172	124	22	s	s	PROPN
ijassa-1172	124	23	,	,	PUNCT
ijassa-1172	124	24	c	c	AUX
ijassa-1172	124	25	,	,	PUNCT
ijassa-1172	124	26	i	i	PRON
ijassa-1172	124	27	>	>	X
ijassa-1172	124	28	0	0	PROPN
ijassa-1172	124	29	,	,	PUNCT
ijassa-1172	124	30	the	the	DET
ijassa-1172	124	31	following	follow	VERB
ijassa-1172	124	32	assumption	assumption	NOUN
ijassa-1172	124	33	(	(	PUNCT
ijassa-1172	124	34	1−	1−	NUM
ijassa-1172	124	35	φ(s	φ(s	NOUN
ijassa-1172	124	36	,	,	PUNCT
ijassa-1172	124	37	c	c	NOUN
ijassa-1172	124	38	)	)	PUNCT
ijassa-1172	124	39	φ(s	φ(s	NOUN
ijassa-1172	124	40	,	,	PUNCT
ijassa-1172	124	41	c∗	c∗	PROPN
ijassa-1172	124	42	)	)	PUNCT
ijassa-1172	124	43	)	)	PUNCT
ijassa-1172	124	44	(	(	PUNCT
ijassa-1172	124	45	φ(s	φ(s	NOUN
ijassa-1172	124	46	,	,	PUNCT
ijassa-1172	124	47	c∗	c∗	PROPN
ijassa-1172	124	48	)	)	PUNCT
ijassa-1172	124	49	φ(s	φ(s	NOUN
ijassa-1172	124	50	,	,	PUNCT
ijassa-1172	124	51	c	c	NOUN
ijassa-1172	124	52	)	)	PUNCT
ijassa-1172	125	1	−	−	PROPN
ijassa-1172	125	2	c	c	PROPN
ijassa-1172	125	3	c∗	c∗	PROPN
ijassa-1172	125	4	)	)	PUNCT
ijassa-1172	125	5	≤	≤	NOUN
ijassa-1172	125	6	0	0	NUM
ijassa-1172	125	7	,	,	PUNCT
ijassa-1172	125	8	(	(	PUNCT
ijassa-1172	125	9	1−	1−	NUM
ijassa-1172	125	10	φ(s∗	φ(s∗	X
ijassa-1172	125	11	,	,	PUNCT
ijassa-1172	125	12	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	125	13	,	,	PUNCT
ijassa-1172	125	14	i	i	PROPN
ijassa-1172	125	15	)	)	PUNCT
ijassa-1172	125	16	φ(s	φ(s	PROPN
ijassa-1172	125	17	,	,	PUNCT
ijassa-1172	125	18	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	125	19	,	,	PUNCT
ijassa-1172	125	20	i∗	i∗	NOUN
ijassa-1172	125	21	)	)	PUNCT
ijassa-1172	125	22	)	)	PUNCT
ijassa-1172	125	23	(	(	PUNCT
ijassa-1172	125	24	φ(s	φ(s	NOUN
ijassa-1172	125	25	,	,	PUNCT
ijassa-1172	125	26	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	125	27	,	,	PUNCT
ijassa-1172	125	28	i∗	i∗	NOUN
ijassa-1172	125	29	)	)	PUNCT
ijassa-1172	125	30	φ(s∗	φ(s∗	NOUN
ijassa-1172	125	31	,	,	PUNCT
ijassa-1172	125	32	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	125	33	,	,	PUNCT
ijassa-1172	125	34	i	i	PROPN
ijassa-1172	125	35	)	)	PUNCT
ijassa-1172	125	36	−	−	PROPN
ijassa-1172	126	1	i	i	PRON
ijassa-1172	126	2	i∗	i∗	VERB
ijassa-1172	126	3	)	)	PUNCT
ijassa-1172	126	4	≤	≤	NOUN
ijassa-1172	126	5	0	0	NUM
ijassa-1172	126	6	.	.	PUNCT
ijassa-1172	127	1	(	(	PUNCT
ijassa-1172	127	2	h4	h4	PROPN
ijassa-1172	127	3	)	)	PUNCT
ijassa-1172	127	4	to	to	PART
ijassa-1172	127	5	establish	establish	VERB
ijassa-1172	127	6	the	the	DET
ijassa-1172	127	7	global	global	ADJ
ijassa-1172	127	8	dynamics	dynamic	NOUN
ijassa-1172	127	9	of	of	ADP
ijassa-1172	127	10	system	system	NOUN
ijassa-1172	127	11	(	(	PUNCT
ijassa-1172	127	12	2.2	2.2	NUM
ijassa-1172	127	13	)	)	PUNCT
ijassa-1172	127	14	when	when	SCONJ
ijassa-1172	127	15	r0	r0	NOUN
ijassa-1172	127	16	>	>	X
ijassa-1172	127	17	1	1	NUM
ijassa-1172	127	18	,	,	PUNCT
ijassa-1172	127	19	we	we	PRON
ijassa-1172	127	20	need	need	VERB
ijassa-1172	127	21	the	the	DET
ijassa-1172	127	22	following	follow	VERB
ijassa-1172	127	23	lemma	lemma	PROPN
ijassa-1172	127	24	.	.	PUNCT
ijassa-1172	128	1	lemma	lemma	PROPN
ijassa-1172	128	2	3.1	3.1	NUM
ijassa-1172	128	3	:	:	PUNCT
ijassa-1172	128	4	let	let	VERB
ijassa-1172	128	5	k	k	PRON
ijassa-1172	128	6	be	be	AUX
ijassa-1172	128	7	a	a	DET
ijassa-1172	128	8	continuous	continuous	ADJ
ijassa-1172	128	9	function	function	NOUN
ijassa-1172	128	10	and	and	CCONJ
ijassa-1172	128	11	u	u	NOUN
ijassa-1172	128	12	be	be	VERB
ijassa-1172	128	13	a	a	DET
ijassa-1172	128	14	continuously	continuously	ADV
ijassa-1172	128	15	differentiable	differentiable	ADJ
ijassa-1172	128	16	function	function	NOUN
ijassa-1172	128	17	.	.	PUNCT
ijassa-1172	129	1	for	for	ADP
ijassa-1172	129	2	any	any	DET
ijassa-1172	129	3	constant	constant	ADJ
ijassa-1172	129	4	c	c	NOUN
ijassa-1172	129	5	and	and	CCONJ
ijassa-1172	129	6	α	α	PRON
ijassa-1172	129	7	∈	∈	PROPN
ijassa-1172	129	8	(	(	PUNCT
ijassa-1172	129	9	0	0	NUM
ijassa-1172	129	10	,	,	PUNCT
ijassa-1172	129	11	1	1	NUM
ijassa-1172	129	12	]	]	PUNCT
ijassa-1172	129	13	,	,	PUNCT
ijassa-1172	129	14	the	the	DET
ijassa-1172	129	15	caputo	caputo	PROPN
ijassa-1172	129	16	fractional	fractional	PROPN
ijassa-1172	129	17	derivative	derivative	NOUN
ijassa-1172	129	18	of	of	ADP
ijassa-1172	129	19	the	the	DET
ijassa-1172	129	20	function	function	NOUN
ijassa-1172	130	1	k	k	PROPN
ijassa-1172	130	2	defined	define	VERB
ijassa-1172	130	3	by	by	ADP
ijassa-1172	130	4	k(t	k(t	NOUN
ijassa-1172	130	5	)	)	PUNCT
ijassa-1172	130	6	=	=	SYM
ijassa-1172	130	7	∫	∫	PROPN
ijassa-1172	130	8	u(t	u(t	PROPN
ijassa-1172	130	9	)	)	PUNCT
ijassa-1172	130	10	c	c	PROPN
ijassa-1172	130	11	k(x)dx	k(x)dx	PROPN
ijassa-1172	130	12	,	,	PUNCT
ijassa-1172	130	13	(	(	PUNCT
ijassa-1172	130	14	3.8	3.8	NUM
ijassa-1172	130	15	)	)	PUNCT
ijassa-1172	130	16	satisfies	satisfy	VERB
ijassa-1172	130	17	the	the	DET
ijassa-1172	130	18	following	follow	VERB
ijassa-1172	130	19	property	property	NOUN
ijassa-1172	130	20	dαk(t	dαk(t	PROPN
ijassa-1172	130	21	)	)	PUNCT
ijassa-1172	131	1	=	=	PUNCT
ijassa-1172	131	2	k(u(t))dαu(t)−	k(u(t))dαu(t)−	PROPN
ijassa-1172	131	3	t−αv(0	t−αv(0	PROPN
ijassa-1172	131	4	)	)	PUNCT
ijassa-1172	131	5	γ(1−	γ(1−	NOUN
ijassa-1172	132	1	α	α	X
ijassa-1172	132	2	)	)	PUNCT
ijassa-1172	132	3	−	−	PROPN
ijassa-1172	132	4	α	α	NUM
ijassa-1172	132	5	γ(1−	γ(1−	NOUN
ijassa-1172	132	6	α	α	X
ijassa-1172	132	7	)	)	PUNCT
ijassa-1172	132	8	∫	∫	PROPN
ijassa-1172	132	9	t	t	PROPN
ijassa-1172	132	10	0	0	NUM
ijassa-1172	132	11	(	(	PUNCT
ijassa-1172	132	12	t−	t−	PROPN
ijassa-1172	132	13	τ)−α−1v(τ)dτ	τ)−α−1v(τ)dτ	PROPN
ijassa-1172	132	14	,	,	PUNCT
ijassa-1172	132	15	(	(	PUNCT
ijassa-1172	132	16	3.9	3.9	NUM
ijassa-1172	132	17	)	)	PUNCT
ijassa-1172	132	18	where	where	SCONJ
ijassa-1172	132	19	v(τ	v(τ	PROPN
ijassa-1172	132	20	)	)	PUNCT
ijassa-1172	133	1	=	=	SYM
ijassa-1172	133	2	k(u(t	k(u(t	PROPN
ijassa-1172	133	3	)	)	PUNCT
ijassa-1172	133	4	)	)	PUNCT
ijassa-1172	134	1	(	(	PUNCT
ijassa-1172	134	2	u(t)−	u(t)−	PROPN
ijassa-1172	134	3	u(τ	u(τ	PROPN
ijassa-1172	134	4	)	)	PUNCT
ijassa-1172	134	5	)	)	PUNCT
ijassa-1172	135	1	+	+	CCONJ
ijassa-1172	135	2	∫	∫	PROPN
ijassa-1172	135	3	u(τ	u(τ	ADJ
ijassa-1172	135	4	)	)	PUNCT
ijassa-1172	135	5	u(t	u(t	NOUN
ijassa-1172	135	6	)	)	PUNCT
ijassa-1172	135	7	k(x)dx	k(x)dx	NOUN
ijassa-1172	135	8	.	.	PUNCT
ijassa-1172	136	1	in	in	ADP
ijassa-1172	136	2	particular	particular	ADJ
ijassa-1172	136	3	,	,	PUNCT
ijassa-1172	136	4	we	we	PRON
ijassa-1172	136	5	have	have	VERB
ijassa-1172	136	6	dαk(t	dαk(t	PROPN
ijassa-1172	136	7	)	)	PUNCT
ijassa-1172	137	1	=	=	PRON
ijassa-1172	137	2	k(u(t))dαu(t	k(u(t))dαu(t	NOUN
ijassa-1172	137	3	)	)	PUNCT
ijassa-1172	137	4	when	when	SCONJ
ijassa-1172	137	5	α	α	PROPN
ijassa-1172	137	6	=	=	NOUN
ijassa-1172	137	7	1	1	X
ijassa-1172	137	8	.	.	PUNCT
ijassa-1172	138	1	moreover	moreover	ADV
ijassa-1172	138	2	,	,	PUNCT
ijassa-1172	138	3	(	(	PUNCT
ijassa-1172	138	4	i	i	NOUN
ijassa-1172	138	5	)	)	PUNCT
ijassa-1172	138	6	if	if	SCONJ
ijassa-1172	138	7	k	k	PROPN
ijassa-1172	138	8	is	be	AUX
ijassa-1172	138	9	a	a	DET
ijassa-1172	138	10	increasing	increase	VERB
ijassa-1172	138	11	function	function	NOUN
ijassa-1172	138	12	,	,	PUNCT
ijassa-1172	138	13	then	then	ADV
ijassa-1172	138	14	dαk(u(t	dαk(u(t	PROPN
ijassa-1172	138	15	)	)	PUNCT
ijassa-1172	138	16	)	)	PUNCT
ijassa-1172	138	17	≤	≤	NUM
ijassa-1172	138	18	k(u(t))dαu(t	k(u(t))dαu(t	VERB
ijassa-1172	138	19	)	)	PUNCT
ijassa-1172	138	20	.	.	PUNCT
ijassa-1172	139	1	(	(	PUNCT
ijassa-1172	139	2	3.10	3.10	NUM
ijassa-1172	139	3	)	)	PUNCT
ijassa-1172	139	4	(	(	PUNCT
ijassa-1172	139	5	ii	ii	NOUN
ijassa-1172	139	6	)	)	PUNCT
ijassa-1172	139	7	if	if	SCONJ
ijassa-1172	139	8	k	k	PROPN
ijassa-1172	139	9	is	be	AUX
ijassa-1172	139	10	a	a	DET
ijassa-1172	139	11	decreasing	decrease	VERB
ijassa-1172	139	12	function	function	NOUN
ijassa-1172	139	13	,	,	PUNCT
ijassa-1172	139	14	then	then	ADV
ijassa-1172	139	15	dαk(u(t	dαk(u(t	PROPN
ijassa-1172	139	16	)	)	PUNCT
ijassa-1172	139	17	)	)	PUNCT
ijassa-1172	139	18	≥	≥	NOUN
ijassa-1172	139	19	k(u(t))dαu(t	k(u(t))dαu(t	VERB
ijassa-1172	139	20	)	)	PUNCT
ijassa-1172	139	21	.	.	PUNCT
ijassa-1172	140	1	(	(	PUNCT
ijassa-1172	140	2	3.11	3.11	NUM
ijassa-1172	140	3	)	)	PUNCT
ijassa-1172	140	4	proof	proof	NOUN
ijassa-1172	140	5	by	by	ADP
ijassa-1172	140	6	using	use	VERB
ijassa-1172	140	7	the	the	DET
ijassa-1172	140	8	definition	definition	NOUN
ijassa-1172	140	9	of	of	ADP
ijassa-1172	140	10	the	the	DET
ijassa-1172	140	11	caputo	caputo	PROPN
ijassa-1172	140	12	fractional	fractional	PROPN
ijassa-1172	140	13	derivative	derivative	NOUN
ijassa-1172	140	14	,	,	PUNCT
ijassa-1172	140	15	we	we	PRON
ijassa-1172	140	16	have	have	VERB
ijassa-1172	140	17	dαk(t)−	dαk(t)−	PROPN
ijassa-1172	140	18	k(u(t))dαu(t	k(u(t))dαu(t	X
ijassa-1172	140	19	)	)	PUNCT
ijassa-1172	140	20	=	=	SYM
ijassa-1172	141	1	1	1	NUM
ijassa-1172	141	2	γ(1−	γ(1−	NOUN
ijassa-1172	141	3	α	α	NUM
ijassa-1172	141	4	)	)	PUNCT
ijassa-1172	141	5	∫	∫	PROPN
ijassa-1172	141	6	t	t	PROPN
ijassa-1172	141	7	0	0	NUM
ijassa-1172	141	8	(	(	PUNCT
ijassa-1172	141	9	t−	t−	PROPN
ijassa-1172	141	10	τ)−αu′(τ	τ)−αu′(τ	NOUN
ijassa-1172	141	11	)	)	PUNCT
ijassa-1172	141	12	(	(	PUNCT
ijassa-1172	141	13	k(u(τ))−	k(u(τ))−	NOUN
ijassa-1172	141	14	k(u(t	k(u(t	PROPN
ijassa-1172	141	15	)	)	PUNCT
ijassa-1172	141	16	)	)	PUNCT
ijassa-1172	141	17	)	)	PUNCT
ijassa-1172	142	1	dτ	dτ	PROPN
ijassa-1172	142	2	.	.	PUNCT
ijassa-1172	142	3	copyright	copyright	NOUN
ijassa-1172	142	4	©	©	PROPN
ijassa-1172	142	5	2022	2022	NUM
ijassa-1172	142	6	assa	assa	NOUN
ijassa-1172	142	7	.	.	PUNCT
ijassa-1172	143	1	adv	adv	PROPN
ijassa-1172	143	2	syst	syst	PROPN
ijassa-1172	143	3	sci	sci	PROPN
ijassa-1172	143	4	appl	appl	PROPN
ijassa-1172	143	5	(	(	PUNCT
ijassa-1172	143	6	2022	2022	NUM
ijassa-1172	143	7	)	)	PUNCT
ijassa-1172	143	8	42	42	NUM
ijassa-1172	143	9	khalid	khalid	PROPN
ijassa-1172	143	10	hattaf	hattaf	NOUN
ijassa-1172	143	11	,	,	PUNCT
ijassa-1172	143	12	ahmed	ahmed	PROPN
ijassa-1172	143	13	a.	a.	PROPN
ijassa-1172	143	14	mohsen	mohsen	PROPN
ijassa-1172	143	15	clearly	clearly	ADV
ijassa-1172	143	16	,	,	PUNCT
ijassa-1172	143	17	v′(τ	v′(τ	ADJ
ijassa-1172	143	18	)	)	PUNCT
ijassa-1172	143	19	=	=	SYM
ijassa-1172	143	20	u′(τ	u′(τ	X
ijassa-1172	143	21	)	)	PUNCT
ijassa-1172	143	22	(	(	PUNCT
ijassa-1172	143	23	k(u(τ))−	k(u(τ))−	NOUN
ijassa-1172	143	24	k(u(t	k(u(t	PROPN
ijassa-1172	143	25	)	)	PUNCT
ijassa-1172	143	26	)	)	PUNCT
ijassa-1172	143	27	)	)	PUNCT
ijassa-1172	143	28	and	and	CCONJ
ijassa-1172	143	29	v(t	v(t	NOUN
ijassa-1172	143	30	)	)	PUNCT
ijassa-1172	143	31	=	=	SYM
ijassa-1172	143	32	0	0	X
ijassa-1172	143	33	.	.	X
ijassa-1172	143	34	integrating	integrate	VERB
ijassa-1172	143	35	by	by	ADP
ijassa-1172	143	36	parts	part	NOUN
ijassa-1172	143	37	the	the	DET
ijassa-1172	143	38	last	last	ADJ
ijassa-1172	143	39	integral	integral	NOUN
ijassa-1172	143	40	,	,	PUNCT
ijassa-1172	143	41	we	we	PRON
ijassa-1172	143	42	get	get	VERB
ijassa-1172	143	43	dαk(t)−	dαk(t)−	PROPN
ijassa-1172	143	44	k(u(t))dαu(t	k(u(t))dαu(t	X
ijassa-1172	143	45	)	)	PUNCT
ijassa-1172	143	46	=	=	SYM
ijassa-1172	143	47	1	1	NUM
ijassa-1172	143	48	γ(1−	γ(1−	NOUN
ijassa-1172	143	49	α	α	NUM
ijassa-1172	143	50	)	)	PUNCT
ijassa-1172	143	51	[	[	PUNCT
ijassa-1172	143	52	(	(	PUNCT
ijassa-1172	143	53	t−	t−	PROPN
ijassa-1172	143	54	τ)−αv(τ	τ)−αv(τ	PROPN
ijassa-1172	143	55	)	)	PUNCT
ijassa-1172	143	56	]	]	PUNCT
ijassa-1172	144	1	τ	τ	X
ijassa-1172	144	2	=	=	NOUN
ijassa-1172	144	3	t	t	X
ijassa-1172	144	4	τ=0	τ=0	PUNCT
ijassa-1172	144	5	−	−	PROPN
ijassa-1172	144	6	α	α	NUM
ijassa-1172	144	7	γ(1−	γ(1−	NOUN
ijassa-1172	144	8	α	α	X
ijassa-1172	144	9	)	)	PUNCT
ijassa-1172	144	10	∫	∫	PROPN
ijassa-1172	144	11	t	t	PROPN
ijassa-1172	144	12	0	0	NUM
ijassa-1172	144	13	(	(	PUNCT
ijassa-1172	144	14	t−	t−	PROPN
ijassa-1172	144	15	τ)−α−1v(τ)dτ	τ)−α−1v(τ)dτ	PROPN
ijassa-1172	144	16	.	.	PUNCT
ijassa-1172	145	1	from	from	ADP
ijassa-1172	145	2	l’hopital	l’hopital	PROPN
ijassa-1172	145	3	’s	’s	PART
ijassa-1172	145	4	rule	rule	NOUN
ijassa-1172	145	5	,	,	PUNCT
ijassa-1172	145	6	we	we	PRON
ijassa-1172	145	7	have	have	VERB
ijassa-1172	145	8	lim	lim	PROPN
ijassa-1172	145	9	τ→t	τ→t	NUM
ijassa-1172	145	10	(	(	PUNCT
ijassa-1172	145	11	t−	t−	PROPN
ijassa-1172	145	12	τ)−αv(τ	τ)−αv(τ	PROPN
ijassa-1172	145	13	)	)	PUNCT
ijassa-1172	146	1	=	=	PROPN
ijassa-1172	146	2	lim	lim	PROPN
ijassa-1172	146	3	τ→t	τ→t	PUNCT
ijassa-1172	146	4	v(τ	v(τ	PROPN
ijassa-1172	146	5	)	)	PUNCT
ijassa-1172	146	6	(	(	PUNCT
ijassa-1172	146	7	t−	t−	NOUN
ijassa-1172	146	8	τ)α	τ)α	PUNCT
ijassa-1172	146	9	=	=	SYM
ijassa-1172	146	10	lim	lim	PROPN
ijassa-1172	146	11	τ→t	τ→t	PUNCT
ijassa-1172	146	12	u′(τ	u′(τ	PROPN
ijassa-1172	146	13	)	)	PUNCT
ijassa-1172	146	14	(	(	PUNCT
ijassa-1172	146	15	k(u(τ))−	k(u(τ))−	NOUN
ijassa-1172	146	16	k(u(t	k(u(t	PROPN
ijassa-1172	146	17	)	)	PUNCT
ijassa-1172	146	18	)	)	PUNCT
ijassa-1172	146	19	)	)	PUNCT
ijassa-1172	147	1	−α(t−	−α(t−	PRON
ijassa-1172	147	2	τ)α−1	τ)α−1	NOUN
ijassa-1172	148	1	=	=	SYM
ijassa-1172	148	2	lim	lim	PROPN
ijassa-1172	148	3	τ→t	τ→t	PUNCT
ijassa-1172	148	4	−	−	PROPN
ijassa-1172	148	5	1	1	NUM
ijassa-1172	148	6	α	α	NOUN
ijassa-1172	148	7	(	(	PUNCT
ijassa-1172	148	8	t−	t−	PROPN
ijassa-1172	148	9	τ)1−αu′(τ	τ)1−αu′(τ	NOUN
ijassa-1172	148	10	)	)	PUNCT
ijassa-1172	148	11	(	(	PUNCT
ijassa-1172	148	12	k(u(τ))−	k(u(τ))−	NOUN
ijassa-1172	148	13	k(u(t	k(u(t	PROPN
ijassa-1172	148	14	)	)	PUNCT
ijassa-1172	148	15	)	)	PUNCT
ijassa-1172	148	16	)	)	PUNCT
ijassa-1172	149	1	=	=	PUNCT
ijassa-1172	149	2	0	0	X
ijassa-1172	149	3	.	.	PUNCT
ijassa-1172	150	1	hence	hence	ADV
ijassa-1172	150	2	,	,	PUNCT
ijassa-1172	150	3	dαk(t	dαk(t	PROPN
ijassa-1172	150	4	)	)	PUNCT
ijassa-1172	150	5	=	=	PUNCT
ijassa-1172	150	6	k(u(t))dαu(t)−	k(u(t))dαu(t)−	PROPN
ijassa-1172	150	7	t−αv(0	t−αv(0	PROPN
ijassa-1172	150	8	)	)	PUNCT
ijassa-1172	150	9	γ(1−	γ(1−	NOUN
ijassa-1172	151	1	α	α	X
ijassa-1172	151	2	)	)	PUNCT
ijassa-1172	151	3	−	−	PROPN
ijassa-1172	151	4	α	α	NUM
ijassa-1172	151	5	γ(1−	γ(1−	NOUN
ijassa-1172	151	6	α	α	X
ijassa-1172	151	7	)	)	PUNCT
ijassa-1172	151	8	∫	∫	PROPN
ijassa-1172	151	9	t	t	PROPN
ijassa-1172	151	10	0	0	NUM
ijassa-1172	151	11	(	(	PUNCT
ijassa-1172	151	12	t−	t−	PROPN
ijassa-1172	151	13	τ)−α−1v(τ)dτ	τ)−α−1v(τ)dτ	PROPN
ijassa-1172	151	14	.	.	PUNCT
ijassa-1172	152	1	now	now	ADV
ijassa-1172	152	2	let	let	VERB
ijassa-1172	152	3	us	we	PRON
ijassa-1172	152	4	consider	consider	VERB
ijassa-1172	152	5	the	the	DET
ijassa-1172	152	6	following	follow	VERB
ijassa-1172	152	7	function	function	NOUN
ijassa-1172	152	8	ψc	ψc	PROPN
ijassa-1172	152	9	,	,	PUNCT
ijassa-1172	152	10	k(τ	k(τ	PROPN
ijassa-1172	152	11	)	)	PUNCT
ijassa-1172	153	1	=	=	SYM
ijassa-1172	153	2	k(c	k(c	PROPN
ijassa-1172	153	3	)	)	PUNCT
ijassa-1172	153	4	(	(	PUNCT
ijassa-1172	153	5	c−	c−	X
ijassa-1172	153	6	τ	τ	X
ijassa-1172	153	7	)	)	PUNCT
ijassa-1172	153	8	+	+	CCONJ
ijassa-1172	153	9	∫	∫	PROPN
ijassa-1172	153	10	τ	τ	PROPN
ijassa-1172	153	11	c	c	PROPN
ijassa-1172	153	12	k(x)dx	k(x)dx	PROPN
ijassa-1172	153	13	.	.	PUNCT
ijassa-1172	154	1	it	it	PRON
ijassa-1172	154	2	is	be	AUX
ijassa-1172	154	3	obvious	obvious	ADJ
ijassa-1172	154	4	that	that	SCONJ
ijassa-1172	154	5	ψ′	ψ′	PUNCT
ijassa-1172	154	6	c	c	NOUN
ijassa-1172	154	7	,	,	PUNCT
ijassa-1172	154	8	k(τ	k(τ	PROPN
ijassa-1172	154	9	)	)	PUNCT
ijassa-1172	155	1	=	=	PUNCT
ijassa-1172	155	2	k(τ)−	k(τ)−	PROPN
ijassa-1172	155	3	k(c	k(c	PROPN
ijassa-1172	155	4	)	)	PUNCT
ijassa-1172	155	5	.	.	PUNCT
ijassa-1172	156	1	assume	assume	VERB
ijassa-1172	156	2	that	that	SCONJ
ijassa-1172	156	3	k	k	PROPN
ijassa-1172	156	4	is	be	AUX
ijassa-1172	156	5	a	a	DET
ijassa-1172	156	6	increasing	increase	VERB
ijassa-1172	156	7	function	function	NOUN
ijassa-1172	156	8	.	.	PUNCT
ijassa-1172	157	1	in	in	ADP
ijassa-1172	157	2	this	this	DET
ijassa-1172	157	3	case	case	NOUN
ijassa-1172	157	4	,	,	PUNCT
ijassa-1172	157	5	the	the	DET
ijassa-1172	157	6	function	function	NOUN
ijassa-1172	157	7	ψc	ψc	VERB
ijassa-1172	157	8	,	,	PUNCT
ijassa-1172	157	9	k(τ	k(τ	PROPN
ijassa-1172	157	10	)	)	PUNCT
ijassa-1172	157	11	is	be	AUX
ijassa-1172	157	12	decreasing	decrease	VERB
ijassa-1172	157	13	on	on	ADP
ijassa-1172	157	14	the	the	DET
ijassa-1172	157	15	interval	interval	NOUN
ijassa-1172	157	16	(	(	PUNCT
ijassa-1172	157	17	−∞	−∞	NOUN
ijassa-1172	157	18	,	,	PUNCT
ijassa-1172	157	19	c	c	X
ijassa-1172	157	20	]	]	PUNCT
ijassa-1172	157	21	and	and	CCONJ
ijassa-1172	157	22	increasing	increase	VERB
ijassa-1172	157	23	on	on	ADP
ijassa-1172	157	24	[	[	X
ijassa-1172	157	25	c,+∞	c,+∞	PROPN
ijassa-1172	157	26	)	)	PUNCT
ijassa-1172	157	27	with	with	ADP
ijassa-1172	157	28	ψc	ψc	ADP
ijassa-1172	157	29	,	,	PUNCT
ijassa-1172	157	30	k(c	k(c	PROPN
ijassa-1172	157	31	)	)	PUNCT
ijassa-1172	157	32	=	=	PUNCT
ijassa-1172	158	1	0	0	X
ijassa-1172	158	2	.	.	PUNCT
ijassa-1172	158	3	then	then	ADV
ijassa-1172	158	4	ψc	ψc	VERB
ijassa-1172	158	5	,	,	PUNCT
ijassa-1172	158	6	k(τ	k(τ	PROPN
ijassa-1172	158	7	)	)	PUNCT
ijassa-1172	158	8	has	have	VERB
ijassa-1172	158	9	the	the	DET
ijassa-1172	158	10	global	global	ADJ
ijassa-1172	158	11	minimum	minimum	NOUN
ijassa-1172	158	12	at	at	ADP
ijassa-1172	158	13	τ	τ	PROPN
ijassa-1172	158	14	=	=	PROPN
ijassa-1172	158	15	c.	c.	PROPN
ijassa-1172	158	16	thus	thus	ADV
ijassa-1172	158	17	,	,	PUNCT
ijassa-1172	158	18	ψc	ψc	INTJ
ijassa-1172	158	19	,	,	PUNCT
ijassa-1172	158	20	k(τ	k(τ	PROPN
ijassa-1172	158	21	)	)	PUNCT
ijassa-1172	158	22	≥	≥	NOUN
ijassa-1172	158	23	0	0	NUM
ijassa-1172	158	24	,	,	PUNCT
ijassa-1172	158	25	for	for	ADP
ijassa-1172	158	26	all	all	DET
ijassa-1172	158	27	(	(	PUNCT
ijassa-1172	158	28	c	c	PROPN
ijassa-1172	158	29	,	,	PUNCT
ijassa-1172	158	30	τ	τ	NOUN
ijassa-1172	158	31	)	)	PUNCT
ijassa-1172	158	32	∈	∈	PROPN
ijassa-1172	158	33	ir2	ir2	NOUN
ijassa-1172	158	34	.	.	PUNCT
ijassa-1172	159	1	since	since	SCONJ
ijassa-1172	159	2	v(τ	v(τ	PROPN
ijassa-1172	159	3	)	)	PUNCT
ijassa-1172	159	4	=	=	SYM
ijassa-1172	159	5	ψu(t),k(u(τ	ψu(t),k(u(τ	VERB
ijassa-1172	159	6	)	)	PUNCT
ijassa-1172	159	7	)	)	PUNCT
ijassa-1172	159	8	,	,	PUNCT
ijassa-1172	159	9	we	we	PRON
ijassa-1172	159	10	deduce	deduce	VERB
ijassa-1172	159	11	that	that	DET
ijassa-1172	159	12	v(τ	v(τ	PROPN
ijassa-1172	159	13	)	)	PUNCT
ijassa-1172	159	14	≥	≥	NOUN
ijassa-1172	159	15	0	0	NUM
ijassa-1172	160	1	and	and	CCONJ
ijassa-1172	160	2	then	then	ADV
ijassa-1172	160	3	(	(	PUNCT
ijassa-1172	160	4	i	i	NOUN
ijassa-1172	160	5	)	)	PUNCT
ijassa-1172	160	6	.	.	PUNCT
ijassa-1172	161	1	by	by	ADP
ijassa-1172	161	2	using	use	VERB
ijassa-1172	161	3	the	the	DET
ijassa-1172	161	4	same	same	ADJ
ijassa-1172	161	5	technique	technique	NOUN
ijassa-1172	161	6	,	,	PUNCT
ijassa-1172	161	7	we	we	PRON
ijassa-1172	161	8	can	can	AUX
ijassa-1172	161	9	easily	easily	ADV
ijassa-1172	161	10	prove	prove	VERB
ijassa-1172	161	11	(	(	PUNCT
ijassa-1172	161	12	ii	ii	NOUN
ijassa-1172	161	13	)	)	PUNCT
ijassa-1172	161	14	.	.	PUNCT
ijassa-1172	162	1	by	by	ADP
ijassa-1172	162	2	applying	apply	VERB
ijassa-1172	162	3	of	of	ADP
ijassa-1172	162	4	the	the	DET
ijassa-1172	162	5	fundamental	fundamental	ADJ
ijassa-1172	162	6	theorem	theorem	NOUN
ijassa-1172	162	7	of	of	ADP
ijassa-1172	162	8	analysis	analysis	NOUN
ijassa-1172	162	9	,	,	PUNCT
ijassa-1172	162	10	we	we	PRON
ijassa-1172	162	11	obtain	obtain	VERB
ijassa-1172	162	12	dαk(t	dαk(t	PROPN
ijassa-1172	162	13	)	)	PUNCT
ijassa-1172	163	1	=	=	PRON
ijassa-1172	163	2	k(u(t))dαu(t	k(u(t))dαu(t	X
ijassa-1172	163	3	)	)	PUNCT
ijassa-1172	163	4	,	,	PUNCT
ijassa-1172	163	5	when	when	SCONJ
ijassa-1172	163	6	α	α	PROPN
ijassa-1172	163	7	=	=	SYM
ijassa-1172	163	8	1	1	X
ijassa-1172	163	9	.	.	PUNCT
ijassa-1172	163	10	(	(	PUNCT
ijassa-1172	163	11	3.12	3.12	NUM
ijassa-1172	163	12	)	)	PUNCT
ijassa-1172	163	13	this	this	PRON
ijassa-1172	163	14	completes	complete	VERB
ijassa-1172	163	15	the	the	DET
ijassa-1172	163	16	proof	proof	NOUN
ijassa-1172	163	17	.	.	PUNCT
ijassa-1172	164	1	remark	remark	VERB
ijassa-1172	164	2	3.1	3.1	NUM
ijassa-1172	164	3	:	:	PUNCT
ijassa-1172	164	4	lemma	lemma	PROPN
ijassa-1172	164	5	3.1	3.1	NUM
ijassa-1172	164	6	extends	extend	VERB
ijassa-1172	164	7	the	the	DET
ijassa-1172	164	8	lemma	lemma	PROPN
ijassa-1172	164	9	presented	present	VERB
ijassa-1172	164	10	in	in	ADP
ijassa-1172	164	11	[	[	X
ijassa-1172	164	12	1	1	X
ijassa-1172	164	13	]	]	PUNCT
ijassa-1172	164	14	that	that	PRON
ijassa-1172	164	15	allows	allow	VERB
ijassa-1172	164	16	to	to	PART
ijassa-1172	164	17	find	find	VERB
ijassa-1172	164	18	lyapunov	lyapunov	ADJ
ijassa-1172	164	19	candidate	candidate	NOUN
ijassa-1172	164	20	functions	function	NOUN
ijassa-1172	164	21	for	for	ADP
ijassa-1172	164	22	demonstrating	demonstrate	VERB
ijassa-1172	164	23	the	the	DET
ijassa-1172	164	24	stability	stability	NOUN
ijassa-1172	164	25	of	of	ADP
ijassa-1172	164	26	many	many	ADJ
ijassa-1172	164	27	fractional	fractional	ADJ
ijassa-1172	164	28	order	order	NOUN
ijassa-1172	164	29	systems	system	NOUN
ijassa-1172	164	30	,	,	PUNCT
ijassa-1172	164	31	and	and	CCONJ
ijassa-1172	164	32	also	also	ADV
ijassa-1172	164	33	the	the	DET
ijassa-1172	164	34	elementary	elementary	ADJ
ijassa-1172	164	35	lemma	lemma	PROPN
ijassa-1172	164	36	given	give	VERB
ijassa-1172	164	37	in	in	ADP
ijassa-1172	164	38	[	[	X
ijassa-1172	164	39	17	17	NUM
ijassa-1172	164	40	]	]	PUNCT
ijassa-1172	164	41	that	that	PRON
ijassa-1172	164	42	estimates	estimate	VERB
ijassa-1172	164	43	fractional	fractional	ADJ
ijassa-1172	164	44	derivatives	derivative	NOUN
ijassa-1172	164	45	of	of	ADP
ijassa-1172	164	46	volterra	volterra	NOUN
ijassa-1172	164	47	-	-	PUNCT
ijassa-1172	164	48	type	type	NOUN
ijassa-1172	164	49	lyapunov	lyapunov	NOUN
ijassa-1172	164	50	functions	function	NOUN
ijassa-1172	164	51	in	in	ADP
ijassa-1172	164	52	the	the	DET
ijassa-1172	164	53	sense	sense	NOUN
ijassa-1172	164	54	caputo	caputo	NOUN
ijassa-1172	164	55	for	for	ADP
ijassa-1172	164	56	any	any	DET
ijassa-1172	164	57	monotone	monotone	ADJ
ijassa-1172	164	58	function	function	NOUN
ijassa-1172	164	59	.	.	PUNCT
ijassa-1172	165	1	in	in	ADP
ijassa-1172	165	2	fact	fact	NOUN
ijassa-1172	165	3	,	,	PUNCT
ijassa-1172	165	4	we	we	PRON
ijassa-1172	165	5	have	have	VERB
ijassa-1172	165	6	1	1	NUM
ijassa-1172	165	7	2	2	NUM
ijassa-1172	165	8	dαu2(t	dαu2(t	NUM
ijassa-1172	165	9	)	)	PUNCT
ijassa-1172	165	10	=	=	PUNCT
ijassa-1172	166	1	dα	dα	PRON
ijassa-1172	166	2	∫	∫	PROPN
ijassa-1172	166	3	u(t	u(t	PROPN
ijassa-1172	166	4	)	)	PUNCT
ijassa-1172	166	5	c	c	PROPN
ijassa-1172	166	6	xdx	xdx	PROPN
ijassa-1172	166	7	≤	≤	PROPN
ijassa-1172	166	8	u(t)dαu(t	u(t)dαu(t	PROPN
ijassa-1172	166	9	)	)	PUNCT
ijassa-1172	166	10	,	,	PUNCT
ijassa-1172	166	11	(	(	PUNCT
ijassa-1172	166	12	3.13	3.13	NUM
ijassa-1172	166	13	)	)	PUNCT
ijassa-1172	166	14	and	and	CCONJ
ijassa-1172	166	15	dα	dα	X
ijassa-1172	166	16	[	[	PUNCT
ijassa-1172	166	17	u(t)−	u(t)−	PROPN
ijassa-1172	166	18	u∗	u∗	PROPN
ijassa-1172	166	19	−	−	PROPN
ijassa-1172	166	20	u∗	u∗	ADJ
ijassa-1172	166	21	ln	ln	ADJ
ijassa-1172	166	22	u(t	u(t	PROPN
ijassa-1172	166	23	)	)	PUNCT
ijassa-1172	166	24	u∗	u∗	ADV
ijassa-1172	166	25	]	]	PUNCT
ijassa-1172	166	26	=	=	PUNCT
ijassa-1172	167	1	dαu(t)−	dαu(t)−	PROPN
ijassa-1172	167	2	u∗dα	u∗dα	PROPN
ijassa-1172	167	3	∫	∫	PROPN
ijassa-1172	167	4	u(t	u(t	PROPN
ijassa-1172	167	5	)	)	PUNCT
ijassa-1172	167	6	u∗	u∗	NOUN
ijassa-1172	167	7	1	1	NUM
ijassa-1172	167	8	x	x	SYM
ijassa-1172	167	9	dx	dx	PROPN
ijassa-1172	167	10	≤	≤	NUM
ijassa-1172	167	11	(	(	PUNCT
ijassa-1172	167	12	1−	1−	NUM
ijassa-1172	167	13	u∗	u∗	ADJ
ijassa-1172	167	14	u(t	u(t	NOUN
ijassa-1172	167	15	)	)	PUNCT
ijassa-1172	167	16	)	)	PUNCT
ijassa-1172	168	1	dαu(t	dαu(t	PROPN
ijassa-1172	168	2	)	)	PUNCT
ijassa-1172	168	3	.	.	PUNCT
ijassa-1172	169	1	(	(	PUNCT
ijassa-1172	169	2	3.14	3.14	NUM
ijassa-1172	169	3	)	)	PUNCT
ijassa-1172	169	4	copyright	copyright	NOUN
ijassa-1172	169	5	©	©	PROPN
ijassa-1172	169	6	2022	2022	NUM
ijassa-1172	169	7	assa	assa	NOUN
ijassa-1172	169	8	.	.	PUNCT
ijassa-1172	170	1	adv	adv	PROPN
ijassa-1172	170	2	syst	syst	PROPN
ijassa-1172	170	3	sci	sci	PROPN
ijassa-1172	170	4	appl	appl	PROPN
ijassa-1172	170	5	(	(	PUNCT
ijassa-1172	170	6	2022	2022	NUM
ijassa-1172	170	7	)	)	PUNCT
ijassa-1172	170	8	dynamics	dynamic	NOUN
ijassa-1172	170	9	of	of	ADP
ijassa-1172	170	10	a	a	DET
ijassa-1172	170	11	generalized	generalize	VERB
ijassa-1172	170	12	fractional	fractional	ADJ
ijassa-1172	170	13	epidemic	epidemic	NOUN
ijassa-1172	170	14	model	model	NOUN
ijassa-1172	170	15	...	...	PUNCT
ijassa-1172	170	16	43	43	NUM
ijassa-1172	170	17	theorem	theorem	NOUN
ijassa-1172	170	18	3.2	3.2	NUM
ijassa-1172	170	19	:	:	PUNCT
ijassa-1172	170	20	assume	assume	VERB
ijassa-1172	170	21	that	that	SCONJ
ijassa-1172	170	22	r0	r0	NOUN
ijassa-1172	170	23	>	>	X
ijassa-1172	170	24	1	1	NUM
ijassa-1172	170	25	and	and	CCONJ
ijassa-1172	170	26	(	(	PUNCT
ijassa-1172	170	27	h4	h4	PROPN
ijassa-1172	170	28	)	)	PUNCT
ijassa-1172	170	29	holds	hold	VERB
ijassa-1172	170	30	.	.	PUNCT
ijassa-1172	171	1	then	then	ADV
ijassa-1172	171	2	the	the	DET
ijassa-1172	171	3	endemic	endemic	ADJ
ijassa-1172	171	4	equilibrium	equilibrium	NOUN
ijassa-1172	171	5	e∗	e∗	NOUN
ijassa-1172	171	6	is	be	AUX
ijassa-1172	171	7	globally	globally	ADV
ijassa-1172	171	8	asymptotically	asymptotically	ADV
ijassa-1172	171	9	stable	stable	ADJ
ijassa-1172	171	10	.	.	PUNCT
ijassa-1172	172	1	proof	proof	NOUN
ijassa-1172	172	2	we	we	PRON
ijassa-1172	172	3	define	define	VERB
ijassa-1172	172	4	a	a	DET
ijassa-1172	172	5	lyapunov	lyapunov	NOUN
ijassa-1172	172	6	functional	functional	NOUN
ijassa-1172	172	7	as	as	SCONJ
ijassa-1172	172	8	follows	follow	VERB
ijassa-1172	172	9	g(t	g(t	PROPN
ijassa-1172	172	10	)	)	PUNCT
ijassa-1172	173	1	=	=	SYM
ijassa-1172	173	2	s	s	PART
ijassa-1172	173	3	−	−	NOUN
ijassa-1172	173	4	s∗	s∗	PROPN
ijassa-1172	173	5	−	−	PROPN
ijassa-1172	173	6	∫	∫	PROPN
ijassa-1172	173	7	s	s	PART
ijassa-1172	173	8	s∗	s∗	PROPN
ijassa-1172	173	9	φ(s∗	φ(s∗	SYM
ijassa-1172	173	10	,	,	PUNCT
ijassa-1172	173	11	c∗	c∗	NOUN
ijassa-1172	173	12	)	)	PUNCT
ijassa-1172	173	13	φ(x	φ(x	PROPN
ijassa-1172	173	14	,	,	PUNCT
ijassa-1172	173	15	c∗	c∗	PROPN
ijassa-1172	173	16	)	)	PUNCT
ijassa-1172	173	17	dx	dx	PROPN
ijassa-1172	174	1	+	+	CCONJ
ijassa-1172	174	2	c∗χ	c∗χ	X
ijassa-1172	174	3	(	(	PUNCT
ijassa-1172	174	4	c	c	NOUN
ijassa-1172	174	5	c∗	c∗	PROPN
ijassa-1172	174	6	)	)	PUNCT
ijassa-1172	175	1	+	+	CCONJ
ijassa-1172	175	2	ψ(s∗	ψ(s∗	X
ijassa-1172	175	3	,	,	PUNCT
ijassa-1172	175	4	i∗	i∗	NOUN
ijassa-1172	175	5	)	)	PUNCT
ijassa-1172	175	6	d2	d2	PROPN
ijassa-1172	175	7	i∗χ	i∗χ	NUM
ijassa-1172	175	8	(	(	PUNCT
ijassa-1172	175	9	i	i	PRON
ijassa-1172	175	10	i∗	i∗	PROPN
ijassa-1172	175	11	)	)	PUNCT
ijassa-1172	175	12	,	,	PUNCT
ijassa-1172	175	13	(	(	PUNCT
ijassa-1172	175	14	3.15	3.15	NUM
ijassa-1172	175	15	)	)	PUNCT
ijassa-1172	176	1	where	where	SCONJ
ijassa-1172	176	2	χ(x̂	χ(x̂	NOUN
ijassa-1172	176	3	)	)	PUNCT
ijassa-1172	176	4	=	=	PUNCT
ijassa-1172	177	1	x̂−	x̂−	PROPN
ijassa-1172	177	2	1−	1−	NUM
ijassa-1172	177	3	ln	ln	ADJ
ijassa-1172	177	4	x̂	x̂	NOUN
ijassa-1172	177	5	for	for	ADP
ijassa-1172	177	6	x̂	x̂	X
ijassa-1172	177	7	>	>	X
ijassa-1172	177	8	0	0	X
ijassa-1172	177	9	.	.	PUNCT
ijassa-1172	178	1	since	since	SCONJ
ijassa-1172	178	2	s	s	PROPN
ijassa-1172	178	3	7→	7→	NUM
ijassa-1172	178	4	1	1	NUM
ijassa-1172	178	5	φ(x	φ(x	NOUN
ijassa-1172	178	6	,	,	PUNCT
ijassa-1172	178	7	c∗	c∗	NOUN
ijassa-1172	178	8	)	)	PUNCT
ijassa-1172	178	9	is	be	AUX
ijassa-1172	178	10	a	a	DET
ijassa-1172	178	11	decreasing	decrease	VERB
ijassa-1172	178	12	function	function	NOUN
ijassa-1172	178	13	,	,	PUNCT
ijassa-1172	178	14	it	it	PRON
ijassa-1172	178	15	follows	follow	VERB
ijassa-1172	178	16	from	from	ADP
ijassa-1172	178	17	(	(	PUNCT
ijassa-1172	178	18	ii	ii	NOUN
ijassa-1172	178	19	)	)	PUNCT
ijassa-1172	178	20	of	of	ADP
ijassa-1172	178	21	lemma	lemma	PROPN
ijassa-1172	178	22	3.1	3.1	NUM
ijassa-1172	178	23	and	and	CCONJ
ijassa-1172	178	24	(	(	PUNCT
ijassa-1172	178	25	3.14	3.14	NUM
ijassa-1172	178	26	)	)	PUNCT
ijassa-1172	178	27	that	that	DET
ijassa-1172	178	28	dαg(t	dαg(t	NOUN
ijassa-1172	178	29	)	)	PUNCT
ijassa-1172	178	30	≤	≤	NOUN
ijassa-1172	178	31	(	(	PUNCT
ijassa-1172	178	32	1−	1−	NUM
ijassa-1172	178	33	φ(s∗	φ(s∗	NOUN
ijassa-1172	178	34	,	,	PUNCT
ijassa-1172	178	35	c∗	c∗	NOUN
ijassa-1172	178	36	)	)	PUNCT
ijassa-1172	178	37	φ(s	φ(s	NOUN
ijassa-1172	178	38	,	,	PUNCT
ijassa-1172	178	39	c∗	c∗	NOUN
ijassa-1172	178	40	)	)	PUNCT
ijassa-1172	178	41	)	)	PUNCT
ijassa-1172	179	1	dαs(t	dαs(t	PROPN
ijassa-1172	179	2	)	)	PUNCT
ijassa-1172	179	3	+	+	CCONJ
ijassa-1172	179	4	(	(	PUNCT
ijassa-1172	179	5	1−	1−	NUM
ijassa-1172	179	6	c∗	c∗	PROPN
ijassa-1172	179	7	c	c	PROPN
ijassa-1172	179	8	)	)	PUNCT
ijassa-1172	179	9	dαc(t	dαc(t	NOUN
ijassa-1172	179	10	)	)	PUNCT
ijassa-1172	179	11	+	+	CCONJ
ijassa-1172	179	12	ψ(s∗	ψ(s∗	X
ijassa-1172	179	13	,	,	PUNCT
ijassa-1172	179	14	i∗	i∗	NOUN
ijassa-1172	179	15	)	)	PUNCT
ijassa-1172	179	16	d2	d2	PROPN
ijassa-1172	179	17	(	(	PUNCT
ijassa-1172	179	18	1−	1−	NUM
ijassa-1172	179	19	i∗	i∗	NOUN
ijassa-1172	179	20	i	i	NOUN
ijassa-1172	179	21	)	)	PUNCT
ijassa-1172	180	1	dαi(t	dαi(t	PROPN
ijassa-1172	180	2	)	)	PUNCT
ijassa-1172	180	3	=	=	SYM
ijassa-1172	180	4	(	(	PUNCT
ijassa-1172	180	5	1−	1−	NUM
ijassa-1172	180	6	φ(s∗	φ(s∗	NOUN
ijassa-1172	180	7	,	,	PUNCT
ijassa-1172	180	8	c∗	c∗	NOUN
ijassa-1172	180	9	)	)	PUNCT
ijassa-1172	180	10	φ(s	φ(s	NOUN
ijassa-1172	180	11	,	,	PUNCT
ijassa-1172	180	12	c∗	c∗	NOUN
ijassa-1172	180	13	)	)	PUNCT
ijassa-1172	180	14	)	)	PUNCT
ijassa-1172	181	1	(	(	PUNCT
ijassa-1172	181	2	a−	a−	PROPN
ijassa-1172	181	3	νs	νs	NOUN
ijassa-1172	181	4	−	−	PROPN
ijassa-1172	181	5	φ(s	φ(s	NOUN
ijassa-1172	181	6	,	,	PUNCT
ijassa-1172	181	7	c)c	c)c	X
ijassa-1172	181	8	−ψ(s	−ψ(s	NOUN
ijassa-1172	181	9	,	,	PUNCT
ijassa-1172	181	10	i)i	i)i	NOUN
ijassa-1172	181	11	)	)	PUNCT
ijassa-1172	182	1	+	+	CCONJ
ijassa-1172	182	2	(	(	PUNCT
ijassa-1172	182	3	1−	1−	NUM
ijassa-1172	182	4	c∗	c∗	PROPN
ijassa-1172	182	5	c	c	PROPN
ijassa-1172	182	6	)	)	PUNCT
ijassa-1172	182	7	(	(	PUNCT
ijassa-1172	182	8	φ(s	φ(s	NOUN
ijassa-1172	182	9	,	,	PUNCT
ijassa-1172	182	10	c)c	c)c	PUNCT
ijassa-1172	183	1	+	+	ADJ
ijassa-1172	183	2	ψ(s	ψ(s	ADJ
ijassa-1172	183	3	,	,	PUNCT
ijassa-1172	183	4	i)i	i)i	NOUN
ijassa-1172	183	5	−	−	PROPN
ijassa-1172	183	6	d1c	d1c	PROPN
ijassa-1172	183	7	)	)	PUNCT
ijassa-1172	184	1	+	+	CCONJ
ijassa-1172	184	2	ψ(s∗	ψ(s∗	X
ijassa-1172	184	3	,	,	PUNCT
ijassa-1172	184	4	i∗	i∗	NOUN
ijassa-1172	184	5	)	)	PUNCT
ijassa-1172	184	6	d2	d2	PROPN
ijassa-1172	184	7	(	(	PUNCT
ijassa-1172	184	8	1−	1−	NUM
ijassa-1172	184	9	i∗	i∗	NOUN
ijassa-1172	184	10	i	i	NOUN
ijassa-1172	184	11	)	)	PUNCT
ijassa-1172	184	12	(	(	PUNCT
ijassa-1172	184	13	γc	γc	PROPN
ijassa-1172	184	14	−	−	PROPN
ijassa-1172	184	15	d2i	d2i	PRON
ijassa-1172	184	16	)	)	PUNCT
ijassa-1172	184	17	.	.	PUNCT
ijassa-1172	185	1	since	since	SCONJ
ijassa-1172	185	2	a	a	DET
ijassa-1172	185	3	=	=	X
ijassa-1172	185	4	νs∗	νs∗	NOUN
ijassa-1172	185	5	+	+	CCONJ
ijassa-1172	185	6	φ(s∗	φ(s∗	X
ijassa-1172	185	7	,	,	PUNCT
ijassa-1172	185	8	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	185	9	+	+	NOUN
ijassa-1172	185	10	ψ(s∗	ψ(s∗	ADJ
ijassa-1172	185	11	,	,	PUNCT
ijassa-1172	185	12	i∗)i∗	i∗)i∗	X
ijassa-1172	185	13	,	,	PUNCT
ijassa-1172	185	14	φ(s∗	φ(s∗	X
ijassa-1172	185	15	,	,	PUNCT
ijassa-1172	185	16	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	185	17	+	+	NOUN
ijassa-1172	185	18	ψ(s∗	ψ(s∗	ADJ
ijassa-1172	185	19	,	,	PUNCT
ijassa-1172	185	20	i∗)i∗	i∗)i∗	X
ijassa-1172	185	21	=	=	PUNCT
ijassa-1172	185	22	d1c∗	d1c∗	PROPN
ijassa-1172	185	23	and	and	CCONJ
ijassa-1172	185	24	γc∗	γc∗	ADJ
ijassa-1172	185	25	=	=	SYM
ijassa-1172	185	26	d2i∗	d2i∗	NOUN
ijassa-1172	185	27	,	,	PUNCT
ijassa-1172	185	28	we	we	PRON
ijassa-1172	185	29	have	have	VERB
ijassa-1172	185	30	dαg(t	dαg(t	PROPN
ijassa-1172	185	31	)	)	PUNCT
ijassa-1172	185	32	≤	≤	NUM
ijassa-1172	185	33	νs∗	νs∗	NOUN
ijassa-1172	185	34	(	(	PUNCT
ijassa-1172	185	35	1−	1−	NUM
ijassa-1172	185	36	s	s	PART
ijassa-1172	185	37	s∗	s∗	PROPN
ijassa-1172	185	38	)	)	PUNCT
ijassa-1172	185	39	(	(	PUNCT
ijassa-1172	185	40	1−	1−	NUM
ijassa-1172	185	41	φ(s∗	φ(s∗	NOUN
ijassa-1172	185	42	,	,	PUNCT
ijassa-1172	185	43	c∗	c∗	NOUN
ijassa-1172	185	44	)	)	PUNCT
ijassa-1172	185	45	φ(s	φ(s	NOUN
ijassa-1172	185	46	,	,	PUNCT
ijassa-1172	185	47	c∗	c∗	NOUN
ijassa-1172	185	48	)	)	PUNCT
ijassa-1172	185	49	)	)	PUNCT
ijassa-1172	186	1	+	+	ADV
ijassa-1172	186	2	φ(s∗	φ(s∗	X
ijassa-1172	186	3	,	,	PUNCT
ijassa-1172	186	4	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	186	5	(	(	PUNCT
ijassa-1172	186	6	2−	2−	NUM
ijassa-1172	186	7	φ(s∗	φ(s∗	NOUN
ijassa-1172	186	8	,	,	PUNCT
ijassa-1172	186	9	c∗	c∗	NOUN
ijassa-1172	186	10	)	)	PUNCT
ijassa-1172	186	11	φ(s	φ(s	NOUN
ijassa-1172	186	12	,	,	PUNCT
ijassa-1172	186	13	c∗	c∗	PROPN
ijassa-1172	186	14	)	)	PUNCT
ijassa-1172	186	15	+	+	CCONJ
ijassa-1172	186	16	φ(s	φ(s	NOUN
ijassa-1172	186	17	,	,	PUNCT
ijassa-1172	186	18	c)c	c)c	PUNCT
ijassa-1172	186	19	φ(s	φ(s	NOUN
ijassa-1172	186	20	,	,	PUNCT
ijassa-1172	186	21	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	186	22	−	−	PROPN
ijassa-1172	186	23	c	c	PROPN
ijassa-1172	186	24	c∗	c∗	PROPN
ijassa-1172	186	25	)	)	PUNCT
ijassa-1172	187	1	+	+	NOUN
ijassa-1172	187	2	ψ(s∗	ψ(s∗	X
ijassa-1172	187	3	,	,	PUNCT
ijassa-1172	187	4	i∗)i∗	i∗)i∗	PUNCT
ijassa-1172	187	5	(	(	PUNCT
ijassa-1172	187	6	3−	3−	NUM
ijassa-1172	187	7	φ(s∗	φ(s∗	NOUN
ijassa-1172	187	8	,	,	PUNCT
ijassa-1172	187	9	c∗	c∗	NOUN
ijassa-1172	187	10	)	)	PUNCT
ijassa-1172	187	11	φ(s	φ(s	NOUN
ijassa-1172	187	12	,	,	PUNCT
ijassa-1172	187	13	c∗	c∗	NOUN
ijassa-1172	187	14	)	)	PUNCT
ijassa-1172	187	15	+	+	CCONJ
ijassa-1172	187	16	φ(s∗	φ(s∗	NUM
ijassa-1172	187	17	,	,	PUNCT
ijassa-1172	187	18	c∗)ψ(s	c∗)ψ(s	ADJ
ijassa-1172	187	19	,	,	PUNCT
ijassa-1172	187	20	i)i	i)i	NOUN
ijassa-1172	187	21	φ(s	φ(s	NOUN
ijassa-1172	187	22	,	,	PUNCT
ijassa-1172	187	23	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	187	24	,	,	PUNCT
ijassa-1172	187	25	i∗)i∗	i∗)i∗	X
ijassa-1172	187	26	−	−	PROPN
ijassa-1172	188	1	i	i	PRON
ijassa-1172	188	2	i∗	i∗	VERB
ijassa-1172	188	3	−	−	PROPN
ijassa-1172	189	1	ψ(s	ψ(s	ADV
ijassa-1172	189	2	,	,	PUNCT
ijassa-1172	189	3	i)ic∗	i)ic∗	X
ijassa-1172	189	4	ψ(s∗	ψ(s∗	NOUN
ijassa-1172	189	5	,	,	PUNCT
ijassa-1172	189	6	i∗)i∗c	i∗)i∗c	VERB
ijassa-1172	189	7	−	−	NOUN
ijassa-1172	189	8	i∗c	i∗c	PUNCT
ijassa-1172	189	9	ic∗	ic∗	NOUN
ijassa-1172	189	10	)	)	PUNCT
ijassa-1172	189	11	.	.	PUNCT
ijassa-1172	190	1	copyright	copyright	NOUN
ijassa-1172	190	2	©	©	PROPN
ijassa-1172	190	3	2022	2022	NUM
ijassa-1172	190	4	assa	assa	NOUN
ijassa-1172	190	5	.	.	PUNCT
ijassa-1172	191	1	adv	adv	PROPN
ijassa-1172	191	2	syst	syst	PROPN
ijassa-1172	191	3	sci	sci	PROPN
ijassa-1172	191	4	appl	appl	PROPN
ijassa-1172	191	5	(	(	PUNCT
ijassa-1172	191	6	2022	2022	NUM
ijassa-1172	191	7	)	)	PUNCT
ijassa-1172	191	8	44	44	NUM
ijassa-1172	191	9	khalid	khalid	PROPN
ijassa-1172	191	10	hattaf	hattaf	NOUN
ijassa-1172	191	11	,	,	PUNCT
ijassa-1172	191	12	ahmed	ahmed	PROPN
ijassa-1172	191	13	a.	a.	PROPN
ijassa-1172	191	14	mohsen	mohsen	PROPN
ijassa-1172	191	15	hence	hence	ADV
ijassa-1172	191	16	,	,	PUNCT
ijassa-1172	191	17	dαg(t	dαg(t	PROPN
ijassa-1172	191	18	)	)	PUNCT
ijassa-1172	191	19	≤	≤	NUM
ijassa-1172	191	20	νs∗	νs∗	NOUN
ijassa-1172	191	21	(	(	PUNCT
ijassa-1172	191	22	1−	1−	NUM
ijassa-1172	191	23	s	s	PART
ijassa-1172	191	24	s∗	s∗	PROPN
ijassa-1172	191	25	)	)	PUNCT
ijassa-1172	191	26	(	(	PUNCT
ijassa-1172	191	27	1−	1−	NUM
ijassa-1172	191	28	φ(s∗	φ(s∗	NOUN
ijassa-1172	191	29	,	,	PUNCT
ijassa-1172	191	30	c∗	c∗	NOUN
ijassa-1172	191	31	)	)	PUNCT
ijassa-1172	191	32	φ(s	φ(s	NOUN
ijassa-1172	191	33	,	,	PUNCT
ijassa-1172	191	34	c∗	c∗	NOUN
ijassa-1172	191	35	)	)	PUNCT
ijassa-1172	191	36	)	)	PUNCT
ijassa-1172	192	1	+	+	ADV
ijassa-1172	192	2	φ(s∗	φ(s∗	X
ijassa-1172	192	3	,	,	PUNCT
ijassa-1172	192	4	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	192	5	(	(	PUNCT
ijassa-1172	192	6	−	−	PROPN
ijassa-1172	192	7	1−	1−	NUM
ijassa-1172	192	8	c	c	PROPN
ijassa-1172	192	9	c∗	c∗	NOUN
ijassa-1172	192	10	+	+	CCONJ
ijassa-1172	192	11	φ(s	φ(s	NOUN
ijassa-1172	192	12	,	,	PUNCT
ijassa-1172	192	13	c∗	c∗	NOUN
ijassa-1172	192	14	)	)	PUNCT
ijassa-1172	192	15	φ(s	φ(s	NOUN
ijassa-1172	192	16	,	,	PUNCT
ijassa-1172	192	17	c	c	NOUN
ijassa-1172	192	18	)	)	PUNCT
ijassa-1172	192	19	+	+	CCONJ
ijassa-1172	192	20	φ(s	φ(s	NOUN
ijassa-1172	192	21	,	,	PUNCT
ijassa-1172	192	22	c)c	c)c	PUNCT
ijassa-1172	192	23	φ(s	φ(s	NOUN
ijassa-1172	192	24	,	,	PUNCT
ijassa-1172	192	25	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	192	26	)	)	PUNCT
ijassa-1172	193	1	+	+	ADV
ijassa-1172	193	2	ψ(s∗	ψ(s∗	X
ijassa-1172	193	3	,	,	PUNCT
ijassa-1172	193	4	i∗)i∗	i∗)i∗	PUNCT
ijassa-1172	193	5	(	(	PUNCT
ijassa-1172	193	6	−	−	PROPN
ijassa-1172	193	7	1−	1−	NUM
ijassa-1172	193	8	i	i	PRON
ijassa-1172	193	9	i∗	i∗	VERB
ijassa-1172	193	10	+	+	CCONJ
ijassa-1172	193	11	φ(s	φ(s	NOUN
ijassa-1172	193	12	,	,	PUNCT
ijassa-1172	193	13	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	193	14	,	,	PUNCT
ijassa-1172	193	15	i∗	i∗	NOUN
ijassa-1172	193	16	)	)	PUNCT
ijassa-1172	193	17	φ(s∗	φ(s∗	NOUN
ijassa-1172	193	18	,	,	PUNCT
ijassa-1172	193	19	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	193	20	,	,	PUNCT
ijassa-1172	193	21	i	i	PROPN
ijassa-1172	193	22	)	)	PUNCT
ijassa-1172	194	1	+	+	CCONJ
ijassa-1172	194	2	φ(s∗	φ(s∗	NUM
ijassa-1172	194	3	,	,	PUNCT
ijassa-1172	194	4	c∗)ψ(s	c∗)ψ(s	ADJ
ijassa-1172	194	5	,	,	PUNCT
ijassa-1172	194	6	i)i	i)i	NOUN
ijassa-1172	194	7	φ(s	φ(s	NOUN
ijassa-1172	194	8	,	,	PUNCT
ijassa-1172	194	9	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	194	10	,	,	PUNCT
ijassa-1172	194	11	i∗)i∗	i∗)i∗	PUNCT
ijassa-1172	194	12	)	)	PUNCT
ijassa-1172	195	1	+	+	ADJ
ijassa-1172	195	2	φ(s∗	φ(s∗	X
ijassa-1172	195	3	,	,	PUNCT
ijassa-1172	195	4	c∗)c∗	c∗)c∗	PUNCT
ijassa-1172	195	5	(	(	PUNCT
ijassa-1172	195	6	3−	3−	NUM
ijassa-1172	195	7	φ(s	φ(s	NOUN
ijassa-1172	195	8	,	,	PUNCT
ijassa-1172	195	9	c∗	c∗	NOUN
ijassa-1172	195	10	)	)	PUNCT
ijassa-1172	195	11	φ(s	φ(s	NOUN
ijassa-1172	195	12	,	,	PUNCT
ijassa-1172	195	13	c	c	NOUN
ijassa-1172	195	14	)	)	PUNCT
ijassa-1172	195	15	−	−	PRON
ijassa-1172	195	16	φ(s∗	φ(s∗	NOUN
ijassa-1172	195	17	,	,	PUNCT
ijassa-1172	195	18	c∗	c∗	NOUN
ijassa-1172	195	19	)	)	PUNCT
ijassa-1172	195	20	φ(s	φ(s	NOUN
ijassa-1172	195	21	,	,	PUNCT
ijassa-1172	195	22	c∗	c∗	PROPN
ijassa-1172	195	23	)	)	PUNCT
ijassa-1172	195	24	−	−	ADP
ijassa-1172	195	25	φ(s	φ(s	NOUN
ijassa-1172	195	26	,	,	PUNCT
ijassa-1172	195	27	c	c	NOUN
ijassa-1172	195	28	)	)	PUNCT
ijassa-1172	195	29	φ(s∗	φ(s∗	NOUN
ijassa-1172	195	30	,	,	PUNCT
ijassa-1172	195	31	c∗	c∗	NOUN
ijassa-1172	195	32	)	)	PUNCT
ijassa-1172	195	33	)	)	PUNCT
ijassa-1172	196	1	+	+	ADV
ijassa-1172	196	2	ψ(s∗	ψ(s∗	X
ijassa-1172	196	3	,	,	PUNCT
ijassa-1172	196	4	i∗)i∗	i∗)i∗	PUNCT
ijassa-1172	196	5	(	(	PUNCT
ijassa-1172	196	6	4−	4−	NOUN
ijassa-1172	196	7	φ(s∗	φ(s∗	NOUN
ijassa-1172	196	8	,	,	PUNCT
ijassa-1172	196	9	c∗	c∗	NOUN
ijassa-1172	196	10	)	)	PUNCT
ijassa-1172	196	11	φ(s	φ(s	NOUN
ijassa-1172	196	12	,	,	PUNCT
ijassa-1172	196	13	c∗	c∗	PROPN
ijassa-1172	196	14	)	)	PUNCT
ijassa-1172	196	15	−	−	ADP
ijassa-1172	196	16	φ(s	φ(s	NOUN
ijassa-1172	196	17	,	,	PUNCT
ijassa-1172	196	18	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	196	19	,	,	PUNCT
ijassa-1172	196	20	i∗	i∗	NOUN
ijassa-1172	196	21	)	)	PUNCT
ijassa-1172	196	22	φ(s∗	φ(s∗	NOUN
ijassa-1172	196	23	,	,	PUNCT
ijassa-1172	196	24	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	196	25	,	,	PUNCT
ijassa-1172	196	26	i	i	PROPN
ijassa-1172	196	27	)	)	PUNCT
ijassa-1172	196	28	−	−	PROPN
ijassa-1172	197	1	ψ(s	ψ(s	PROPN
ijassa-1172	197	2	,	,	PUNCT
ijassa-1172	197	3	i)ic∗	i)ic∗	X
ijassa-1172	197	4	ψ(s∗	ψ(s∗	NOUN
ijassa-1172	197	5	,	,	PUNCT
ijassa-1172	197	6	i∗)i∗c	i∗)i∗c	ADP
ijassa-1172	197	7	−	−	PROPN
ijassa-1172	197	8	ci∗	ci∗	NOUN
ijassa-1172	197	9	c∗i	c∗i	PUNCT
ijassa-1172	197	10	)	)	PUNCT
ijassa-1172	197	11	.	.	PUNCT
ijassa-1172	198	1	from	from	ADP
ijassa-1172	198	2	above	above	ADP
ijassa-1172	198	3	inequality	inequality	NOUN
ijassa-1172	198	4	,	,	PUNCT
ijassa-1172	198	5	we	we	PRON
ijassa-1172	198	6	get	get	VERB
ijassa-1172	198	7	that	that	SCONJ
ijassa-1172	198	8	3−	3−	NUM
ijassa-1172	198	9	φ(s	φ(s	NOUN
ijassa-1172	198	10	,	,	PUNCT
ijassa-1172	198	11	c∗	c∗	NOUN
ijassa-1172	198	12	)	)	PUNCT
ijassa-1172	198	13	φ(s	φ(s	NOUN
ijassa-1172	198	14	,	,	PUNCT
ijassa-1172	198	15	c	c	NOUN
ijassa-1172	198	16	)	)	PUNCT
ijassa-1172	198	17	−	−	PRON
ijassa-1172	198	18	φ(s∗	φ(s∗	NOUN
ijassa-1172	198	19	,	,	PUNCT
ijassa-1172	198	20	c∗	c∗	NOUN
ijassa-1172	198	21	)	)	PUNCT
ijassa-1172	198	22	φ(s	φ(s	NOUN
ijassa-1172	198	23	,	,	PUNCT
ijassa-1172	198	24	c∗	c∗	PROPN
ijassa-1172	198	25	)	)	PUNCT
ijassa-1172	198	26	−	−	ADP
ijassa-1172	198	27	φ(s	φ(s	NOUN
ijassa-1172	198	28	,	,	PUNCT
ijassa-1172	198	29	c	c	NOUN
ijassa-1172	198	30	)	)	PUNCT
ijassa-1172	198	31	φ(s∗	φ(s∗	NOUN
ijassa-1172	198	32	,	,	PUNCT
ijassa-1172	198	33	c∗	c∗	NOUN
ijassa-1172	198	34	)	)	PUNCT
ijassa-1172	198	35	≤	≤	NOUN
ijassa-1172	198	36	0	0	NUM
ijassa-1172	198	37	,	,	PUNCT
ijassa-1172	198	38	and	and	CCONJ
ijassa-1172	198	39	4−	4−	NUM
ijassa-1172	198	40	φ(s∗	φ(s∗	NOUN
ijassa-1172	198	41	,	,	PUNCT
ijassa-1172	198	42	c∗	c∗	NOUN
ijassa-1172	198	43	)	)	PUNCT
ijassa-1172	198	44	φ(s	φ(s	NOUN
ijassa-1172	198	45	,	,	PUNCT
ijassa-1172	198	46	c∗	c∗	PROPN
ijassa-1172	198	47	)	)	PUNCT
ijassa-1172	198	48	−	−	ADP
ijassa-1172	198	49	φ(s	φ(s	NOUN
ijassa-1172	198	50	,	,	PUNCT
ijassa-1172	198	51	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	198	52	,	,	PUNCT
ijassa-1172	198	53	i∗	i∗	NOUN
ijassa-1172	198	54	)	)	PUNCT
ijassa-1172	198	55	φ(s∗	φ(s∗	NOUN
ijassa-1172	198	56	,	,	PUNCT
ijassa-1172	198	57	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	198	58	,	,	PUNCT
ijassa-1172	198	59	i	i	PROPN
ijassa-1172	198	60	)	)	PUNCT
ijassa-1172	198	61	−	−	PROPN
ijassa-1172	199	1	ψ(s	ψ(s	PROPN
ijassa-1172	199	2	,	,	PUNCT
ijassa-1172	199	3	i)ic∗	i)ic∗	X
ijassa-1172	199	4	ψ(s∗	ψ(s∗	NOUN
ijassa-1172	199	5	,	,	PUNCT
ijassa-1172	199	6	i∗)i∗c	i∗)i∗c	ADP
ijassa-1172	199	7	−	−	PROPN
ijassa-1172	199	8	ci∗	ci∗	NOUN
ijassa-1172	199	9	c∗i	c∗i	VERB
ijassa-1172	199	10	≤	≤	NUM
ijassa-1172	199	11	0	0	NUM
ijassa-1172	199	12	.	.	PUNCT
ijassa-1172	200	1	the	the	DET
ijassa-1172	200	2	hypothesis	hypothesis	NOUN
ijassa-1172	200	3	(	(	PUNCT
ijassa-1172	200	4	h1	h1	PROPN
ijassa-1172	200	5	)	)	PUNCT
ijassa-1172	200	6	leads	lead	VERB
ijassa-1172	200	7	to	to	ADP
ijassa-1172	200	8	(	(	PUNCT
ijassa-1172	200	9	1−	1−	NUM
ijassa-1172	200	10	s	s	NOUN
ijassa-1172	200	11	s∗	s∗	PROPN
ijassa-1172	200	12	)	)	PUNCT
ijassa-1172	200	13	(	(	PUNCT
ijassa-1172	200	14	1−	1−	NUM
ijassa-1172	200	15	φ(s∗	φ(s∗	NOUN
ijassa-1172	200	16	,	,	PUNCT
ijassa-1172	200	17	c∗	c∗	NOUN
ijassa-1172	200	18	)	)	PUNCT
ijassa-1172	200	19	φ(s	φ(s	NOUN
ijassa-1172	200	20	,	,	PUNCT
ijassa-1172	200	21	c∗	c∗	NOUN
ijassa-1172	200	22	)	)	PUNCT
ijassa-1172	200	23	)	)	PUNCT
ijassa-1172	201	1	≤	≤	ADV
ijassa-1172	201	2	0	0	X
ijassa-1172	201	3	.	.	PUNCT
ijassa-1172	202	1	by	by	ADP
ijassa-1172	202	2	(	(	PUNCT
ijassa-1172	202	3	h4	h4	PROPN
ijassa-1172	202	4	)	)	PUNCT
ijassa-1172	202	5	,	,	PUNCT
ijassa-1172	202	6	we	we	PRON
ijassa-1172	202	7	get	get	VERB
ijassa-1172	202	8	−1−	−1−	PROPN
ijassa-1172	202	9	c	c	PROPN
ijassa-1172	202	10	c∗	c∗	PROPN
ijassa-1172	202	11	+	+	CCONJ
ijassa-1172	202	12	φ(s	φ(s	NOUN
ijassa-1172	202	13	,	,	PUNCT
ijassa-1172	202	14	c∗	c∗	NOUN
ijassa-1172	202	15	)	)	PUNCT
ijassa-1172	202	16	φ(s	φ(s	NOUN
ijassa-1172	202	17	,	,	PUNCT
ijassa-1172	202	18	c	c	NOUN
ijassa-1172	202	19	)	)	PUNCT
ijassa-1172	203	1	+	+	CCONJ
ijassa-1172	203	2	φ(s	φ(s	NOUN
ijassa-1172	203	3	,	,	PUNCT
ijassa-1172	203	4	c)c	c)c	PUNCT
ijassa-1172	203	5	φ(s	φ(s	NOUN
ijassa-1172	203	6	,	,	PUNCT
ijassa-1172	203	7	c∗)c∗	c∗)c∗	X
ijassa-1172	203	8	=	=	PUNCT
ijassa-1172	203	9	(	(	PUNCT
ijassa-1172	203	10	1−	1−	NUM
ijassa-1172	203	11	φ(s	φ(s	NOUN
ijassa-1172	203	12	,	,	PUNCT
ijassa-1172	203	13	c	c	NOUN
ijassa-1172	203	14	)	)	PUNCT
ijassa-1172	203	15	φ(s	φ(s	NOUN
ijassa-1172	203	16	,	,	PUNCT
ijassa-1172	203	17	c∗	c∗	PROPN
ijassa-1172	203	18	)	)	PUNCT
ijassa-1172	203	19	)	)	PUNCT
ijassa-1172	203	20	(	(	PUNCT
ijassa-1172	203	21	φ(s	φ(s	NOUN
ijassa-1172	203	22	,	,	PUNCT
ijassa-1172	203	23	c∗	c∗	PROPN
ijassa-1172	203	24	)	)	PUNCT
ijassa-1172	203	25	φ(s	φ(s	NOUN
ijassa-1172	203	26	,	,	PUNCT
ijassa-1172	203	27	c	c	NOUN
ijassa-1172	203	28	)	)	PUNCT
ijassa-1172	203	29	−	−	PROPN
ijassa-1172	203	30	c	c	PROPN
ijassa-1172	203	31	c∗	c∗	PROPN
ijassa-1172	203	32	)	)	PUNCT
ijassa-1172	203	33	≤	≤	NOUN
ijassa-1172	203	34	0	0	NUM
ijassa-1172	204	1	and	and	CCONJ
ijassa-1172	205	1	−1−	−1−	NOUN
ijassa-1172	205	2	i	i	PRON
ijassa-1172	205	3	i∗	i∗	VERB
ijassa-1172	205	4	+	+	CCONJ
ijassa-1172	205	5	φ(s	φ(s	NOUN
ijassa-1172	205	6	,	,	PUNCT
ijassa-1172	205	7	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	205	8	,	,	PUNCT
ijassa-1172	205	9	i∗	i∗	NOUN
ijassa-1172	205	10	)	)	PUNCT
ijassa-1172	205	11	φ(s∗	φ(s∗	NOUN
ijassa-1172	205	12	,	,	PUNCT
ijassa-1172	205	13	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	205	14	,	,	PUNCT
ijassa-1172	205	15	i	i	PROPN
ijassa-1172	205	16	)	)	PUNCT
ijassa-1172	206	1	+	+	CCONJ
ijassa-1172	206	2	φ(s∗	φ(s∗	NUM
ijassa-1172	206	3	,	,	PUNCT
ijassa-1172	206	4	c∗)ψ(s	c∗)ψ(s	ADJ
ijassa-1172	206	5	,	,	PUNCT
ijassa-1172	206	6	i)i	i)i	NOUN
ijassa-1172	206	7	φ(s	φ(s	NOUN
ijassa-1172	206	8	,	,	PUNCT
ijassa-1172	206	9	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	206	10	,	,	PUNCT
ijassa-1172	206	11	i∗)i∗	i∗)i∗	X
ijassa-1172	206	12	=	=	PUNCT
ijassa-1172	206	13	(	(	PUNCT
ijassa-1172	206	14	1−	1−	NUM
ijassa-1172	206	15	φ(s∗	φ(s∗	NOUN
ijassa-1172	206	16	,	,	PUNCT
ijassa-1172	206	17	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	206	18	,	,	PUNCT
ijassa-1172	206	19	i	i	PROPN
ijassa-1172	206	20	)	)	PUNCT
ijassa-1172	206	21	φ(s	φ(s	PROPN
ijassa-1172	206	22	,	,	PUNCT
ijassa-1172	206	23	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	206	24	,	,	PUNCT
ijassa-1172	206	25	i∗	i∗	NOUN
ijassa-1172	206	26	)	)	PUNCT
ijassa-1172	206	27	)	)	PUNCT
ijassa-1172	206	28	(	(	PUNCT
ijassa-1172	206	29	φ(s	φ(s	NOUN
ijassa-1172	206	30	,	,	PUNCT
ijassa-1172	206	31	c∗)ψ(s∗	c∗)ψ(s∗	PROPN
ijassa-1172	206	32	,	,	PUNCT
ijassa-1172	206	33	i∗	i∗	NOUN
ijassa-1172	206	34	)	)	PUNCT
ijassa-1172	206	35	φ(s∗	φ(s∗	NOUN
ijassa-1172	206	36	,	,	PUNCT
ijassa-1172	206	37	c∗)ψ(s	c∗)ψ(s	PROPN
ijassa-1172	206	38	,	,	PUNCT
ijassa-1172	206	39	i	i	PROPN
ijassa-1172	206	40	)	)	PUNCT
ijassa-1172	207	1	−	−	PROPN
ijassa-1172	208	1	i	i	PRON
ijassa-1172	208	2	i∗	i∗	VERB
ijassa-1172	208	3	)	)	PUNCT
ijassa-1172	208	4	≤	≤	NUM
ijassa-1172	208	5	0	0	NUM
ijassa-1172	208	6	.	.	PUNCT
ijassa-1172	209	1	therefore	therefore	ADV
ijassa-1172	209	2	,	,	PUNCT
ijassa-1172	209	3	dαg(t	dαg(t	PROPN
ijassa-1172	209	4	)	)	PUNCT
ijassa-1172	209	5	≤	≤	NOUN
ijassa-1172	209	6	0	0	NUM
ijassa-1172	210	1	with	with	ADP
ijassa-1172	210	2	equality	equality	NOUN
ijassa-1172	210	3	holds	hold	VERB
ijassa-1172	210	4	if	if	SCONJ
ijassa-1172	210	5	and	and	CCONJ
ijassa-1172	210	6	only	only	ADV
ijassa-1172	210	7	if	if	SCONJ
ijassa-1172	210	8	s	s	NOUN
ijassa-1172	210	9	=	=	PUNCT
ijassa-1172	210	10	s∗	s∗	PROPN
ijassa-1172	210	11	,	,	PUNCT
ijassa-1172	210	12	c	c	NOUN
ijassa-1172	210	13	=	=	SYM
ijassa-1172	210	14	c∗	c∗	PROPN
ijassa-1172	211	1	and	and	CCONJ
ijassa-1172	211	2	i	i	PRON
ijassa-1172	211	3	=	=	PUNCT
ijassa-1172	211	4	i∗.	i∗.	PROPN
ijassa-1172	211	5	from	from	ADP
ijassa-1172	211	6	lasalle	lasalle	PROPN
ijassa-1172	211	7	’s	’s	PART
ijassa-1172	211	8	invariance	invariance	NOUN
ijassa-1172	211	9	principle	principle	NOUN
ijassa-1172	211	10	,	,	PUNCT
ijassa-1172	211	11	we	we	PRON
ijassa-1172	211	12	conclude	conclude	VERB
ijassa-1172	211	13	that	that	PRON
ijassa-1172	211	14	e∗	e∗	PROPN
ijassa-1172	211	15	is	be	AUX
ijassa-1172	211	16	globally	globally	ADV
ijassa-1172	211	17	asymptotically	asymptotically	ADV
ijassa-1172	211	18	stable	stable	ADJ
ijassa-1172	211	19	.	.	PUNCT
ijassa-1172	212	1	4	4	X
ijassa-1172	212	2	.	.	X
ijassa-1172	212	3	applications	application	NOUN
ijassa-1172	212	4	and	and	CCONJ
ijassa-1172	212	5	numerical	numerical	ADJ
ijassa-1172	212	6	simulations	simulation	NOUN
ijassa-1172	212	7	in	in	ADP
ijassa-1172	212	8	this	this	DET
ijassa-1172	212	9	section	section	NOUN
ijassa-1172	212	10	,	,	PUNCT
ijassa-1172	212	11	we	we	PRON
ijassa-1172	212	12	first	first	ADV
ijassa-1172	212	13	apply	apply	VERB
ijassa-1172	212	14	our	our	PRON
ijassa-1172	212	15	above	above	ADJ
ijassa-1172	212	16	analytical	analytical	ADJ
ijassa-1172	212	17	results	result	NOUN
ijassa-1172	212	18	to	to	ADP
ijassa-1172	212	19	the	the	DET
ijassa-1172	212	20	following	follow	VERB
ijassa-1172	212	21	fractional	fractional	ADJ
ijassa-1172	212	22	covid-19	covid-19	PROPN
ijassa-1172	212	23	model	model	NOUN
ijassa-1172	212	24			PROPN
ijassa-1172	212	25	dαs(t	dαs(t	PROPN
ijassa-1172	212	26	)	)	PUNCT
ijassa-1172	212	27	=	=	SYM
ijassa-1172	212	28	a−	a−	NOUN
ijassa-1172	212	29	νs	νs	NOUN
ijassa-1172	212	30	−	−	NOUN
ijassa-1172	212	31	β1sc	β1sc	PUNCT
ijassa-1172	213	1	1	1	NUM
ijassa-1172	213	2	+	+	CCONJ
ijassa-1172	213	3	ϵ1c	ϵ1c	SYM
ijassa-1172	213	4	−	−	NOUN
ijassa-1172	213	5	β2si	β2si	PUNCT
ijassa-1172	214	1	1	1	NUM
ijassa-1172	214	2	+	+	CCONJ
ijassa-1172	214	3	ϵ2i	ϵ2i	ADJ
ijassa-1172	214	4	,	,	PUNCT
ijassa-1172	214	5	dαc(t	dαc(t	NOUN
ijassa-1172	214	6	)	)	PUNCT
ijassa-1172	214	7	=	=	SYM
ijassa-1172	214	8	β1sc	β1sc	PUNCT
ijassa-1172	215	1	1	1	NUM
ijassa-1172	215	2	+	+	CCONJ
ijassa-1172	215	3	ϵ1c	ϵ1c	PUNCT
ijassa-1172	215	4	+	+	CCONJ
ijassa-1172	215	5	β2si	β2si	X
ijassa-1172	215	6	1	1	NUM
ijassa-1172	215	7	+	+	NUM
ijassa-1172	215	8	ϵ2i	ϵ2i	PROPN
ijassa-1172	215	9	−	−	PROPN
ijassa-1172	215	10	d1c	d1c	PROPN
ijassa-1172	215	11	,	,	PUNCT
ijassa-1172	215	12	dαi(t	dαi(t	PROPN
ijassa-1172	215	13	)	)	PUNCT
ijassa-1172	215	14	=	=	SYM
ijassa-1172	215	15	γc	γc	PROPN
ijassa-1172	215	16	−	−	NOUN
ijassa-1172	215	17	d2i	d2i	NOUN
ijassa-1172	215	18	.	.	PUNCT
ijassa-1172	216	1	(	(	PUNCT
ijassa-1172	216	2	4.16	4.16	NUM
ijassa-1172	216	3	)	)	PUNCT
ijassa-1172	216	4	copyright	copyright	NOUN
ijassa-1172	216	5	©	©	PROPN
ijassa-1172	216	6	2022	2022	NUM
ijassa-1172	216	7	assa	assa	NOUN
ijassa-1172	216	8	.	.	PUNCT
ijassa-1172	217	1	adv	adv	PROPN
ijassa-1172	217	2	syst	syst	PROPN
ijassa-1172	217	3	sci	sci	PROPN
ijassa-1172	217	4	appl	appl	PROPN
ijassa-1172	217	5	(	(	PUNCT
ijassa-1172	217	6	2022	2022	NUM
ijassa-1172	217	7	)	)	PUNCT
ijassa-1172	217	8	dynamics	dynamic	NOUN
ijassa-1172	217	9	of	of	ADP
ijassa-1172	217	10	a	a	DET
ijassa-1172	217	11	generalized	generalize	VERB
ijassa-1172	217	12	fractional	fractional	ADJ
ijassa-1172	217	13	epidemic	epidemic	NOUN
ijassa-1172	217	14	model	model	NOUN
ijassa-1172	217	15	...	...	PUNCT
ijassa-1172	217	16	45	45	NUM
ijassa-1172	217	17	where	where	SCONJ
ijassa-1172	217	18	β1	β1	PROPN
ijassa-1172	217	19	and	and	CCONJ
ijassa-1172	217	20	β2	β2	NOUN
ijassa-1172	217	21	are	be	AUX
ijassa-1172	217	22	the	the	DET
ijassa-1172	217	23	infection	infection	NOUN
ijassa-1172	217	24	rates	rate	NOUN
ijassa-1172	217	25	caused	cause	VERB
ijassa-1172	217	26	by	by	ADP
ijassa-1172	217	27	carrier	carrier	NOUN
ijassa-1172	217	28	and	and	CCONJ
ijassa-1172	217	29	infected	infected	ADJ
ijassa-1172	217	30	individuals	individual	NOUN
ijassa-1172	217	31	,	,	PUNCT
ijassa-1172	217	32	respectively	respectively	ADV
ijassa-1172	217	33	.	.	PUNCT
ijassa-1172	218	1	the	the	DET
ijassa-1172	218	2	nonnegative	nonnegative	ADJ
ijassa-1172	218	3	constants	constant	NOUN
ijassa-1172	218	4	ϵ1	ϵ1	VERB
ijassa-1172	218	5	and	and	CCONJ
ijassa-1172	218	6	ϵ2	ϵ2	PROPN
ijassa-1172	218	7	are	be	AUX
ijassa-1172	218	8	the	the	DET
ijassa-1172	218	9	saturation	saturation	NOUN
ijassa-1172	218	10	rates	rate	NOUN
ijassa-1172	218	11	.	.	PUNCT
ijassa-1172	219	1	model	model	NOUN
ijassa-1172	219	2	(	(	PUNCT
ijassa-1172	219	3	4.16	4.16	NUM
ijassa-1172	219	4	)	)	PUNCT
ijassa-1172	219	5	is	be	AUX
ijassa-1172	219	6	a	a	DET
ijassa-1172	219	7	special	special	ADJ
ijassa-1172	219	8	case	case	NOUN
ijassa-1172	219	9	of	of	ADP
ijassa-1172	219	10	system	system	NOUN
ijassa-1172	219	11	(	(	PUNCT
ijassa-1172	219	12	2.2	2.2	NUM
ijassa-1172	219	13	)	)	PUNCT
ijassa-1172	219	14	with	with	ADP
ijassa-1172	219	15	φ(s	φ(s	NOUN
ijassa-1172	219	16	,	,	PUNCT
ijassa-1172	219	17	c	c	NOUN
ijassa-1172	219	18	)	)	PUNCT
ijassa-1172	219	19	=	=	SYM
ijassa-1172	220	1	β1s	β1s	PUNCT
ijassa-1172	220	2	1	1	NUM
ijassa-1172	220	3	+	+	CCONJ
ijassa-1172	220	4	ϵ1c	ϵ1c	PUNCT
ijassa-1172	220	5	and	and	CCONJ
ijassa-1172	220	6	ψ(s	ψ(s	PROPN
ijassa-1172	220	7	,	,	PUNCT
ijassa-1172	220	8	i	i	NOUN
ijassa-1172	220	9	)	)	PUNCT
ijassa-1172	220	10	=	=	PUNCT
ijassa-1172	221	1	β2s	β2	NOUN
ijassa-1172	221	2	1	1	X
ijassa-1172	222	1	+	+	CCONJ
ijassa-1172	222	2	ϵ2i	ϵ2i	PROPN
ijassa-1172	222	3	.	.	PUNCT
ijassa-1172	223	1	it	it	PRON
ijassa-1172	223	2	easy	easy	ADJ
ijassa-1172	223	3	that	that	PRON
ijassa-1172	223	4	see	see	VERB
ijassa-1172	223	5	the	the	DET
ijassa-1172	223	6	basic	basic	ADJ
ijassa-1172	223	7	reproduction	reproduction	NOUN
ijassa-1172	223	8	number	number	NOUN
ijassa-1172	223	9	r0	r0	NOUN
ijassa-1172	223	10	is	be	AUX
ijassa-1172	223	11	given	give	VERB
ijassa-1172	223	12	by	by	ADP
ijassa-1172	223	13	r0	r0	NOUN
ijassa-1172	223	14	=	=	PUNCT
ijassa-1172	224	1	a(d2β1	a(d2β1	PROPN
ijassa-1172	225	1	+	+	CCONJ
ijassa-1172	225	2	γβ2	γβ2	X
ijassa-1172	225	3	)	)	PUNCT
ijassa-1172	225	4	νd1d2	νd1d2	PROPN
ijassa-1172	225	5	.	.	PUNCT
ijassa-1172	226	1	(	(	PUNCT
ijassa-1172	226	2	4.17	4.17	NUM
ijassa-1172	226	3	)	)	PUNCT
ijassa-1172	226	4	evidently	evidently	ADV
ijassa-1172	226	5	,	,	PUNCT
ijassa-1172	226	6	the	the	DET
ijassa-1172	226	7	infected	infect	VERB
ijassa-1172	226	8	rates	rate	NOUN
ijassa-1172	226	9	φ	φ	PROPN
ijassa-1172	226	10	and	and	CCONJ
ijassa-1172	226	11	ψ	ψ	PART
ijassa-1172	226	12	satisfy	satisfy	VERB
ijassa-1172	226	13	the	the	DET
ijassa-1172	226	14	conditions	condition	NOUN
ijassa-1172	226	15	(	(	PUNCT
ijassa-1172	226	16	h1)−	h1)−	PROPN
ijassa-1172	226	17	(	(	PUNCT
ijassa-1172	226	18	h3	h3	NOUN
ijassa-1172	226	19	)	)	PUNCT
ijassa-1172	226	20	.	.	PUNCT
ijassa-1172	227	1	according	accord	VERB
ijassa-1172	227	2	to	to	ADP
ijassa-1172	227	3	theorems	theorems	PROPN
ijassa-1172	227	4	3.1	3.1	NUM
ijassa-1172	227	5	and	and	CCONJ
ijassa-1172	227	6	3.2	3.2	NUM
ijassa-1172	227	7	,	,	PUNCT
ijassa-1172	227	8	we	we	PRON
ijassa-1172	227	9	get	get	VERB
ijassa-1172	227	10	the	the	DET
ijassa-1172	227	11	following	follow	VERB
ijassa-1172	227	12	result	result	NOUN
ijassa-1172	227	13	.	.	PUNCT
ijassa-1172	228	1	corollary	corollary	ADJ
ijassa-1172	228	2	4.1	4.1	NUM
ijassa-1172	228	3	:	:	PUNCT
ijassa-1172	228	4	(	(	PUNCT
ijassa-1172	228	5	i	i	NOUN
ijassa-1172	228	6	)	)	PUNCT
ijassa-1172	228	7	when	when	SCONJ
ijassa-1172	228	8	r0	r0	NOUN
ijassa-1172	228	9	≤	≤	NOUN
ijassa-1172	228	10	1	1	NUM
ijassa-1172	228	11	,	,	PUNCT
ijassa-1172	228	12	the	the	DET
ijassa-1172	228	13	disease	disease	NOUN
ijassa-1172	228	14	-	-	PUNCT
ijassa-1172	228	15	free	free	ADJ
ijassa-1172	228	16	equilibrium	equilibrium	NOUN
ijassa-1172	228	17	ef	ef	PROPN
ijassa-1172	228	18	of	of	ADP
ijassa-1172	228	19	model	model	NOUN
ijassa-1172	228	20	(	(	PUNCT
ijassa-1172	228	21	4.16	4.16	NUM
ijassa-1172	228	22	)	)	PUNCT
ijassa-1172	228	23	is	be	AUX
ijassa-1172	228	24	globally	globally	ADV
ijassa-1172	228	25	asymptotically	asymptotically	ADV
ijassa-1172	228	26	stable	stable	ADJ
ijassa-1172	228	27	.	.	PUNCT
ijassa-1172	229	1	(	(	PUNCT
ijassa-1172	229	2	ii	ii	NOUN
ijassa-1172	229	3	)	)	PUNCT
ijassa-1172	229	4	when	when	SCONJ
ijassa-1172	229	5	r0	r0	NOUN
ijassa-1172	229	6	>	>	X
ijassa-1172	229	7	1	1	NUM
ijassa-1172	229	8	,	,	PUNCT
ijassa-1172	229	9	the	the	DET
ijassa-1172	229	10	equilibrium	equilibrium	NOUN
ijassa-1172	229	11	ef	ef	NOUN
ijassa-1172	229	12	gets	get	VERB
ijassa-1172	229	13	unstable	unstable	ADJ
ijassa-1172	229	14	and	and	CCONJ
ijassa-1172	229	15	the	the	DET
ijassa-1172	229	16	endemic	endemic	ADJ
ijassa-1172	229	17	equilibrium	equilibrium	NOUN
ijassa-1172	229	18	e∗	e∗	PROPN
ijassa-1172	229	19	of	of	ADP
ijassa-1172	229	20	model	model	NOUN
ijassa-1172	229	21	(	(	PUNCT
ijassa-1172	229	22	4.16	4.16	NUM
ijassa-1172	229	23	)	)	PUNCT
ijassa-1172	229	24	is	be	AUX
ijassa-1172	229	25	globally	globally	ADV
ijassa-1172	229	26	asymptotically	asymptotically	ADV
ijassa-1172	229	27	stable	stable	ADJ
ijassa-1172	229	28	.	.	PUNCT
ijassa-1172	230	1	the	the	DET
ijassa-1172	230	2	second	second	ADJ
ijassa-1172	230	3	application	application	NOUN
ijassa-1172	230	4	of	of	ADP
ijassa-1172	230	5	our	our	PRON
ijassa-1172	230	6	results	result	NOUN
ijassa-1172	230	7	concerns	concern	VERB
ijassa-1172	230	8	a	a	DET
ijassa-1172	230	9	fractional	fractional	ADJ
ijassa-1172	230	10	model	model	NOUN
ijassa-1172	230	11	proposed	propose	VERB
ijassa-1172	230	12	in	in	ADP
ijassa-1172	230	13	2020	2020	NUM
ijassa-1172	230	14	by	by	ADP
ijassa-1172	230	15	das	das	PROPN
ijassa-1172	230	16	and	and	CCONJ
ijassa-1172	230	17	samanta	samanta	PROPN
ijassa-1172	231	1	[	[	X
ijassa-1172	231	2	2	2	NUM
ijassa-1172	231	3	]	]	PUNCT
ijassa-1172	231	4	to	to	PART
ijassa-1172	231	5	study	study	VERB
ijassa-1172	231	6	the	the	DET
ijassa-1172	231	7	transmission	transmission	NOUN
ijassa-1172	231	8	dynamics	dynamic	NOUN
ijassa-1172	231	9	of	of	ADP
ijassa-1172	231	10	covid-19	covid-19	PROPN
ijassa-1172	231	11	in	in	ADP
ijassa-1172	231	12	japan	japan	PROPN
ijassa-1172	231	13	.	.	PUNCT
ijassa-1172	232	1	this	this	DET
ijassa-1172	232	2	recent	recent	ADJ
ijassa-1172	232	3	fractional	fractional	ADJ
ijassa-1172	232	4	model	model	NOUN
ijassa-1172	232	5	is	be	AUX
ijassa-1172	232	6	a	a	DET
ijassa-1172	232	7	particular	particular	ADJ
ijassa-1172	232	8	case	case	NOUN
ijassa-1172	232	9	of	of	ADP
ijassa-1172	232	10	system	system	NOUN
ijassa-1172	232	11	(	(	PUNCT
ijassa-1172	232	12	4.16	4.16	NUM
ijassa-1172	232	13	)	)	PUNCT
ijassa-1172	232	14	,	,	PUNCT
ijassa-1172	232	15	it	it	PRON
ijassa-1172	232	16	suffices	suffice	VERB
ijassa-1172	232	17	to	to	PART
ijassa-1172	232	18	take	take	VERB
ijassa-1172	232	19	ϵ1	ϵ1	NOUN
ijassa-1172	232	20	=	=	PUNCT
ijassa-1172	232	21	ϵ2	ϵ2	NOUN
ijassa-1172	232	22	=	=	SYM
ijassa-1172	232	23	0	0	NUM
ijassa-1172	232	24	and	and	CCONJ
ijassa-1172	232	25	r1	r1	PROPN
ijassa-1172	232	26	=	=	SYM
ijassa-1172	232	27	r2	r2	PROPN
ijassa-1172	232	28	=	=	PUNCT
ijassa-1172	233	1	d	d	NOUN
ijassa-1172	233	2	=	=	SYM
ijassa-1172	233	3	0	0	X
ijassa-1172	233	4	.	.	PUNCT
ijassa-1172	234	1	in	in	ADP
ijassa-1172	234	2	[	[	X
ijassa-1172	234	3	2	2	NUM
ijassa-1172	234	4	]	]	PUNCT
ijassa-1172	234	5	,	,	PUNCT
ijassa-1172	234	6	the	the	DET
ijassa-1172	234	7	authors	author	NOUN
ijassa-1172	234	8	just	just	ADV
ijassa-1172	234	9	discussed	discuss	VERB
ijassa-1172	234	10	the	the	DET
ijassa-1172	234	11	local	local	ADJ
ijassa-1172	234	12	stability	stability	NOUN
ijassa-1172	234	13	of	of	ADP
ijassa-1172	234	14	the	the	DET
ijassa-1172	234	15	equilibria	equilibrium	NOUN
ijassa-1172	234	16	and	and	CCONJ
ijassa-1172	234	17	calculated	calculate	VERB
ijassa-1172	234	18	epidemic	epidemic	NOUN
ijassa-1172	234	19	peak	peak	NOUN
ijassa-1172	234	20	in	in	ADP
ijassa-1172	234	21	japan	japan	PROPN
ijassa-1172	234	22	scenario	scenario	PROPN
ijassa-1172	234	23	.	.	PUNCT
ijassa-1172	235	1	they	they	PRON
ijassa-1172	235	2	proved	prove	VERB
ijassa-1172	235	3	that	that	SCONJ
ijassa-1172	235	4	the	the	DET
ijassa-1172	235	5	disease	disease	NOUN
ijassa-1172	235	6	-	-	PUNCT
ijassa-1172	235	7	free	free	ADJ
ijassa-1172	235	8	equilibrium	equilibrium	NOUN
ijassa-1172	235	9	is	be	AUX
ijassa-1172	235	10	locally	locally	ADV
ijassa-1172	235	11	asymptotically	asymptotically	ADV
ijassa-1172	235	12	stable	stable	ADJ
ijassa-1172	235	13	if	if	SCONJ
ijassa-1172	235	14	r0	r0	NOUN
ijassa-1172	235	15	<	<	X
ijassa-1172	235	16	1	1	NUM
ijassa-1172	235	17	and	and	CCONJ
ijassa-1172	235	18	the	the	DET
ijassa-1172	235	19	endemic	endemic	ADJ
ijassa-1172	235	20	equilibrium	equilibrium	NOUN
ijassa-1172	235	21	is	be	AUX
ijassa-1172	235	22	locally	locally	ADV
ijassa-1172	235	23	asymptotically	asymptotically	ADV
ijassa-1172	235	24	stable	stable	ADJ
ijassa-1172	235	25	under	under	ADP
ijassa-1172	235	26	r0	r0	NOUN
ijassa-1172	235	27	>	>	X
ijassa-1172	235	28	1	1	NUM
ijassa-1172	235	29	and	and	CCONJ
ijassa-1172	235	30	others	other	NOUN
ijassa-1172	235	31	three	three	NUM
ijassa-1172	235	32	conditions	condition	NOUN
ijassa-1172	235	33	(	(	PUNCT
ijassa-1172	235	34	see	see	VERB
ijassa-1172	235	35	,	,	PUNCT
ijassa-1172	235	36	theorem	theorem	VERB
ijassa-1172	235	37	2	2	NUM
ijassa-1172	235	38	[	[	X
ijassa-1172	235	39	2	2	NUM
ijassa-1172	235	40	]	]	PUNCT
ijassa-1172	235	41	)	)	PUNCT
ijassa-1172	235	42	.	.	PUNCT
ijassa-1172	236	1	by	by	ADP
ijassa-1172	236	2	applying	apply	VERB
ijassa-1172	236	3	corollary	corollary	NOUN
ijassa-1172	236	4	4.1	4.1	NUM
ijassa-1172	236	5	,	,	PUNCT
ijassa-1172	236	6	we	we	PRON
ijassa-1172	236	7	deduce	deduce	VERB
ijassa-1172	236	8	that	that	SCONJ
ijassa-1172	236	9	when	when	SCONJ
ijassa-1172	236	10	r0	r0	NOUN
ijassa-1172	236	11	≤	≤	NOUN
ijassa-1172	236	12	1	1	NUM
ijassa-1172	236	13	,	,	PUNCT
ijassa-1172	236	14	the	the	DET
ijassa-1172	236	15	disease	disease	NOUN
ijassa-1172	236	16	-	-	PUNCT
ijassa-1172	236	17	free	free	ADJ
ijassa-1172	236	18	equilibrium	equilibrium	NOUN
ijassa-1172	236	19	of	of	ADP
ijassa-1172	236	20	model	model	NOUN
ijassa-1172	236	21	[	[	X
ijassa-1172	236	22	2	2	NUM
ijassa-1172	236	23	]	]	PUNCT
ijassa-1172	236	24	is	be	AUX
ijassa-1172	236	25	globally	globally	ADV
ijassa-1172	236	26	asymptotically	asymptotically	ADV
ijassa-1172	236	27	stable	stable	ADJ
ijassa-1172	236	28	.	.	PUNCT
ijassa-1172	237	1	however	however	ADV
ijassa-1172	237	2	,	,	PUNCT
ijassa-1172	237	3	this	this	DET
ijassa-1172	237	4	equilibrium	equilibrium	NOUN
ijassa-1172	237	5	becomes	become	VERB
ijassa-1172	237	6	unstable	unstable	ADJ
ijassa-1172	237	7	and	and	CCONJ
ijassa-1172	237	8	the	the	DET
ijassa-1172	237	9	endemic	endemic	ADJ
ijassa-1172	237	10	equilibrium	equilibrium	NOUN
ijassa-1172	237	11	is	be	AUX
ijassa-1172	237	12	globally	globally	ADV
ijassa-1172	237	13	asymptotically	asymptotically	ADV
ijassa-1172	237	14	stable	stable	ADJ
ijassa-1172	237	15	under	under	ADP
ijassa-1172	237	16	only	only	ADV
ijassa-1172	237	17	r0	r0	NOUN
ijassa-1172	237	18	>	>	X
ijassa-1172	237	19	1	1	X
ijassa-1172	237	20	.	.	PUNCT
ijassa-1172	238	1	this	this	PRON
ijassa-1172	238	2	improve	improve	VERB
ijassa-1172	238	3	the	the	DET
ijassa-1172	238	4	asymptotic	asymptotic	ADJ
ijassa-1172	238	5	stability	stability	NOUN
ijassa-1172	238	6	results	result	NOUN
ijassa-1172	238	7	presented	present	VERB
ijassa-1172	238	8	recently	recently	ADV
ijassa-1172	238	9	in	in	ADP
ijassa-1172	238	10	[	[	X
ijassa-1172	238	11	2	2	NUM
ijassa-1172	238	12	]	]	PUNCT
ijassa-1172	238	13	.	.	PUNCT
ijassa-1172	239	1	for	for	ADP
ijassa-1172	239	2	numerical	numerical	ADJ
ijassa-1172	239	3	simulations	simulation	NOUN
ijassa-1172	239	4	,	,	PUNCT
ijassa-1172	239	5	we	we	PRON
ijassa-1172	239	6	take	take	VERB
ijassa-1172	239	7	the	the	DET
ijassa-1172	239	8	some	some	DET
ijassa-1172	239	9	parameters	parameter	NOUN
ijassa-1172	239	10	of	of	ADP
ijassa-1172	239	11	model	model	NOUN
ijassa-1172	239	12	(	(	PUNCT
ijassa-1172	239	13	4.16	4.16	NUM
ijassa-1172	239	14	)	)	PUNCT
ijassa-1172	239	15	accordingly	accordingly	ADV
ijassa-1172	239	16	to	to	ADP
ijassa-1172	239	17	the	the	DET
ijassa-1172	239	18	world	world	PROPN
ijassa-1172	239	19	health	health	PROPN
ijassa-1172	239	20	organization	organization	NOUN
ijassa-1172	239	21	reports	report	VERB
ijassa-1172	239	22	as	as	ADV
ijassa-1172	239	23	well	well	ADV
ijassa-1172	239	24	as	as	ADP
ijassa-1172	239	25	adopted	adopt	VERB
ijassa-1172	239	26	by	by	ADP
ijassa-1172	239	27	many	many	ADJ
ijassa-1172	239	28	references	reference	NOUN
ijassa-1172	239	29	as	as	SCONJ
ijassa-1172	239	30	stated	state	VERB
ijassa-1172	239	31	in	in	ADP
ijassa-1172	239	32	table	table	NOUN
ijassa-1172	239	33	4.1	4.1	NUM
ijassa-1172	239	34	below	below	ADV
ijassa-1172	239	35	.	.	PUNCT
ijassa-1172	240	1	parameter	parameter	NOUN
ijassa-1172	240	2	definition	definition	NOUN
ijassa-1172	240	3	value	value	VERB
ijassa-1172	240	4	a	a	DET
ijassa-1172	240	5	recruitment	recruitment	NOUN
ijassa-1172	240	6	rate	rate	NOUN
ijassa-1172	240	7	50	50	NUM
ijassa-1172	240	8	β1	β1	NOUN
ijassa-1172	240	9	transmission	transmission	NOUN
ijassa-1172	240	10	contact	contact	NOUN
ijassa-1172	240	11	rate	rate	NOUN
ijassa-1172	240	12	between	between	ADP
ijassa-1172	240	13	s	s	PRON
ijassa-1172	240	14	and	and	CCONJ
ijassa-1172	240	15	c	c	PROPN
ijassa-1172	240	16	1.2×	1.2×	NUM
ijassa-1172	240	17	10−5	10−5	NUM
ijassa-1172	240	18	β2	β2	NOUN
ijassa-1172	240	19	transmission	transmission	NOUN
ijassa-1172	240	20	contact	contact	NOUN
ijassa-1172	240	21	rate	rate	NOUN
ijassa-1172	240	22	between	between	ADP
ijassa-1172	240	23	s	s	PRON
ijassa-1172	241	1	and	and	CCONJ
ijassa-1172	241	2	i	i	PRON
ijassa-1172	241	3	varied	vary	VERB
ijassa-1172	241	4	γ	γ	PROPN
ijassa-1172	241	5	symptoms	symptom	NOUN
ijassa-1172	241	6	period	period	VERB
ijassa-1172	241	7	1/7	1/7	NUM
ijassa-1172	242	1	day−1	day−1	PROPN
ijassa-1172	242	2	ϵ1	ϵ1	VERB
ijassa-1172	242	3	the	the	DET
ijassa-1172	242	4	measure	measure	NOUN
ijassa-1172	242	5	of	of	ADP
ijassa-1172	242	6	inhibition	inhibition	NOUN
ijassa-1172	242	7	effect	effect	NOUN
ijassa-1172	242	8	for	for	ADP
ijassa-1172	242	9	carrier	carrier	NOUN
ijassa-1172	242	10	0.04	0.04	NUM
ijassa-1172	242	11	ϵ2	ϵ2	NOUN
ijassa-1172	242	12	the	the	DET
ijassa-1172	242	13	measure	measure	NOUN
ijassa-1172	242	14	of	of	ADP
ijassa-1172	242	15	inhibition	inhibition	NOUN
ijassa-1172	242	16	effect	effect	NOUN
ijassa-1172	242	17	for	for	ADP
ijassa-1172	242	18	infected	infected	ADJ
ijassa-1172	242	19	0.01	0.01	NUM
ijassa-1172	242	20	ν	ν	NOUN
ijassa-1172	242	21	natural	natural	ADJ
ijassa-1172	242	22	death	death	NOUN
ijassa-1172	242	23	rate	rate	NOUN
ijassa-1172	242	24	0.01	0.01	NUM
ijassa-1172	242	25	day−1	day−1	PROPN
ijassa-1172	242	26	r1	r1	PROPN
ijassa-1172	242	27	recovery	recovery	NOUN
ijassa-1172	242	28	rate	rate	NOUN
ijassa-1172	242	29	from	from	ADP
ijassa-1172	242	30	carrier	carrier	NOUN
ijassa-1172	242	31	1/21	1/21	NUM
ijassa-1172	242	32	day−1	day−1	PROPN
ijassa-1172	242	33	r2	r2	PROPN
ijassa-1172	242	34	recovery	recovery	NOUN
ijassa-1172	242	35	rate	rate	NOUN
ijassa-1172	242	36	from	from	ADP
ijassa-1172	242	37	infected	infected	ADJ
ijassa-1172	242	38	1/15	1/15	NUM
ijassa-1172	242	39	day−1	day−1	PROPN
ijassa-1172	243	1	d	d	PRON
ijassa-1172	243	2	death	death	NOUN
ijassa-1172	243	3	due	due	ADP
ijassa-1172	243	4	to	to	ADP
ijassa-1172	243	5	disease	disease	NOUN
ijassa-1172	243	6	rate	rate	NOUN
ijassa-1172	243	7	0.1	0.1	NUM
ijassa-1172	243	8	day−1	day−1	PROPN
ijassa-1172	243	9	table	table	NOUN
ijassa-1172	243	10	4.1	4.1	NUM
ijassa-1172	243	11	.	.	PUNCT
ijassa-1172	244	1	parameter	parameter	NOUN
ijassa-1172	244	2	values	value	NOUN
ijassa-1172	244	3	of	of	ADP
ijassa-1172	244	4	model	model	NOUN
ijassa-1172	244	5	(	(	PUNCT
ijassa-1172	244	6	4.16	4.16	NUM
ijassa-1172	244	7	)	)	PUNCT
ijassa-1172	244	8	.	.	PUNCT
ijassa-1172	245	1	for	for	ADP
ijassa-1172	245	2	β2	β2	NOUN
ijassa-1172	245	3	=	=	PROPN
ijassa-1172	245	4	5.5×	5.5×	NUM
ijassa-1172	245	5	10−5	10−5	NUM
ijassa-1172	245	6	,	,	PUNCT
ijassa-1172	245	7	we	we	PRON
ijassa-1172	245	8	have	have	VERB
ijassa-1172	245	9	r0	r0	NOUN
ijassa-1172	245	10	=	=	NOUN
ijassa-1172	245	11	0.9532	0.9532	NUM
ijassa-1172	245	12	≤	≤	NUM
ijassa-1172	245	13	1	1	NUM
ijassa-1172	245	14	.	.	PUNCT
ijassa-1172	246	1	then	then	ADV
ijassa-1172	246	2	model	model	NOUN
ijassa-1172	246	3	(	(	PUNCT
ijassa-1172	246	4	4.16	4.16	NUM
ijassa-1172	246	5	)	)	PUNCT
ijassa-1172	246	6	has	have	VERB
ijassa-1172	246	7	a	a	DET
ijassa-1172	246	8	diseasefree	diseasefree	ADJ
ijassa-1172	246	9	equilibrium	equilibrium	NOUN
ijassa-1172	246	10	ef	ef	X
ijassa-1172	246	11	(	(	PUNCT
ijassa-1172	246	12	5000	5000	NUM
ijassa-1172	246	13	,	,	PUNCT
ijassa-1172	246	14	0	0	NUM
ijassa-1172	246	15	,	,	PUNCT
ijassa-1172	246	16	0	0	NUM
ijassa-1172	246	17	,	,	PUNCT
ijassa-1172	246	18	0	0	NUM
ijassa-1172	246	19	)	)	PUNCT
ijassa-1172	246	20	.	.	PUNCT
ijassa-1172	247	1	by	by	ADP
ijassa-1172	247	2	corollary	corollary	ADJ
ijassa-1172	247	3	4.1	4.1	NUM
ijassa-1172	247	4	(	(	PUNCT
ijassa-1172	247	5	i	i	NOUN
ijassa-1172	247	6	)	)	PUNCT
ijassa-1172	247	7	,	,	PUNCT
ijassa-1172	247	8	we	we	PRON
ijassa-1172	247	9	know	know	VERB
ijassa-1172	247	10	that	that	SCONJ
ijassa-1172	247	11	ef	ef	PROPN
ijassa-1172	247	12	is	be	AUX
ijassa-1172	247	13	globally	globally	ADV
ijassa-1172	247	14	asymptotically	asymptotically	ADV
ijassa-1172	247	15	stable	stable	ADJ
ijassa-1172	247	16	.	.	PUNCT
ijassa-1172	248	1	figure	figure	NOUN
ijassa-1172	248	2	2	2	NUM
ijassa-1172	248	3	demonstrates	demonstrate	VERB
ijassa-1172	248	4	this	this	DET
ijassa-1172	248	5	result	result	NOUN
ijassa-1172	248	6	.	.	PUNCT
ijassa-1172	249	1	copyright	copyright	NOUN
ijassa-1172	249	2	©	©	PROPN
ijassa-1172	249	3	2022	2022	NUM
ijassa-1172	249	4	assa	assa	NOUN
ijassa-1172	249	5	.	.	PUNCT
ijassa-1172	250	1	adv	adv	PROPN
ijassa-1172	250	2	syst	syst	PROPN
ijassa-1172	250	3	sci	sci	PROPN
ijassa-1172	250	4	appl	appl	PROPN
ijassa-1172	250	5	(	(	PUNCT
ijassa-1172	250	6	2022	2022	NUM
ijassa-1172	250	7	)	)	PUNCT
ijassa-1172	250	8	46	46	NUM
ijassa-1172	250	9	khalid	khalid	PROPN
ijassa-1172	250	10	hattaf	hattaf	NOUN
ijassa-1172	250	11	,	,	PUNCT
ijassa-1172	250	12	ahmed	ahmed	PROPN
ijassa-1172	250	13	a.	a.	PROPN
ijassa-1172	250	14	mohsen	mohsen	PROPN
ijassa-1172	250	15	0	0	NUM
ijassa-1172	250	16	100	100	NUM
ijassa-1172	250	17	200	200	NUM
ijassa-1172	250	18	300	300	NUM
ijassa-1172	250	19	400	400	NUM
ijassa-1172	250	20	500	500	NUM
ijassa-1172	250	21	3500	3500	NUM
ijassa-1172	250	22	4000	4000	NUM
ijassa-1172	250	23	4500	4500	NUM
ijassa-1172	250	24	5000	5000	NUM
ijassa-1172	250	25	days	day	NOUN
ijassa-1172	250	26	s	s	AUX
ijassa-1172	250	27	u	u	NOUN
ijassa-1172	250	28	s	s	PRON
ijassa-1172	250	29	c	c	NOUN
ijassa-1172	250	30	e	e	NOUN
ijassa-1172	250	31	p	p	X
ijassa-1172	250	32	ti	ti	PROPN
ijassa-1172	250	33	b	b	X
ijassa-1172	250	34	le	le	X
ijassa-1172	250	35	(	(	PUNCT
ijassa-1172	250	36	s	s	NOUN
ijassa-1172	250	37	)	)	PUNCT
ijassa-1172	250	38	α=0.3	α=0.3	PUNCT
ijassa-1172	250	39	α=0.5	α=0.5	VERB
ijassa-1172	250	40	α=0.8	α=0.8	NOUN
ijassa-1172	250	41	α=1	α=1	ADP
ijassa-1172	250	42	0	0	NUM
ijassa-1172	250	43	100	100	NUM
ijassa-1172	250	44	200	200	NUM
ijassa-1172	250	45	300	300	NUM
ijassa-1172	250	46	400	400	NUM
ijassa-1172	250	47	500	500	NUM
ijassa-1172	250	48	0	0	NUM
ijassa-1172	250	49	200	200	NUM
ijassa-1172	250	50	400	400	NUM
ijassa-1172	250	51	600	600	NUM
ijassa-1172	250	52	days	day	NOUN
ijassa-1172	250	53	c	c	NOUN
ijassa-1172	250	54	a	a	PRON
ijassa-1172	250	55	rr	rr	X
ijassa-1172	250	56	ie	ie	X
ijassa-1172	250	57	r	r	NOUN
ijassa-1172	250	58	(	(	PUNCT
ijassa-1172	250	59	c	c	NOUN
ijassa-1172	250	60	)	)	PUNCT
ijassa-1172	250	61	α=0.3	α=0.3	PUNCT
ijassa-1172	250	62	α=0.5	α=0.5	VERB
ijassa-1172	250	63	α=0.8	α=0.8	NOUN
ijassa-1172	250	64	α=1	α=1	ADP
ijassa-1172	250	65	0	0	NUM
ijassa-1172	250	66	100	100	NUM
ijassa-1172	250	67	200	200	NUM
ijassa-1172	250	68	300	300	NUM
ijassa-1172	250	69	400	400	NUM
ijassa-1172	251	1	500	500	NUM
ijassa-1172	251	2	0	0	NUM
ijassa-1172	251	3	100	100	NUM
ijassa-1172	251	4	200	200	NUM
ijassa-1172	251	5	300	300	NUM
ijassa-1172	251	6	days	day	NOUN
ijassa-1172	251	7	in	in	ADP
ijassa-1172	251	8	fe	fe	X
ijassa-1172	251	9	c	c	X
ijassa-1172	251	10	te	te	PROPN
ijassa-1172	251	11	d	d	X
ijassa-1172	251	12	(	(	PUNCT
ijassa-1172	251	13	i	i	NOUN
ijassa-1172	251	14	)	)	PUNCT
ijassa-1172	251	15	α=0.3	α=0.3	PUNCT
ijassa-1172	251	16	α=0.5	α=0.5	VERB
ijassa-1172	251	17	α=0.8	α=0.8	NOUN
ijassa-1172	251	18	α=1	α=1	SYM
ijassa-1172	251	19	3500	3500	NUM
ijassa-1172	251	20	4000	4000	NUM
ijassa-1172	251	21	4500	4500	NUM
ijassa-1172	251	22	5000	5000	NUM
ijassa-1172	251	23	0	0	NUM
ijassa-1172	251	24	500	500	NUM
ijassa-1172	251	25	0	0	NUM
ijassa-1172	251	26	200	200	NUM
ijassa-1172	251	27	400	400	NUM
ijassa-1172	251	28	sc	sc	NOUN
ijassa-1172	251	29	i	i	PRON
ijassa-1172	251	30	fig	fig	NOUN
ijassa-1172	251	31	.	.	PUNCT
ijassa-1172	252	1	4.2	4.2	NUM
ijassa-1172	252	2	.	.	PUNCT
ijassa-1172	252	3	dynamics	dynamic	NOUN
ijassa-1172	252	4	of	of	ADP
ijassa-1172	252	5	the	the	DET
ijassa-1172	252	6	model	model	NOUN
ijassa-1172	252	7	(	(	PUNCT
ijassa-1172	252	8	4.16	4.16	NUM
ijassa-1172	252	9	)	)	PUNCT
ijassa-1172	252	10	when	when	SCONJ
ijassa-1172	252	11	r0	r0	NOUN
ijassa-1172	252	12	=	=	NOUN
ijassa-1172	252	13	0.9532	0.9532	NUM
ijassa-1172	252	14	≤	≤	NUM
ijassa-1172	252	15	1	1	NUM
ijassa-1172	252	16	.	.	NOUN
ijassa-1172	252	17	0	0	NUM
ijassa-1172	253	1	50	50	NUM
ijassa-1172	253	2	100	100	NUM
ijassa-1172	253	3	150	150	NUM
ijassa-1172	253	4	200	200	NUM
ijassa-1172	253	5	3400	3400	NUM
ijassa-1172	253	6	3600	3600	NUM
ijassa-1172	253	7	3800	3800	NUM
ijassa-1172	253	8	4000	4000	NUM
ijassa-1172	253	9	days	day	NOUN
ijassa-1172	253	10	s	s	PART
ijassa-1172	253	11	u	u	NOUN
ijassa-1172	253	12	s	s	PRON
ijassa-1172	253	13	c	c	NOUN
ijassa-1172	253	14	e	e	NOUN
ijassa-1172	253	15	p	p	X
ijassa-1172	253	16	ti	ti	PROPN
ijassa-1172	253	17	b	b	X
ijassa-1172	253	18	le	le	X
ijassa-1172	253	19	(	(	PUNCT
ijassa-1172	253	20	s	s	NOUN
ijassa-1172	253	21	)	)	PUNCT
ijassa-1172	253	22	α=0.3	α=0.3	PUNCT
ijassa-1172	253	23	α=0.5	α=0.5	VERB
ijassa-1172	253	24	α=0.8	α=0.8	NOUN
ijassa-1172	253	25	α=1	α=1	SYM
ijassa-1172	253	26	0	0	NUM
ijassa-1172	253	27	50	50	NUM
ijassa-1172	253	28	100	100	NUM
ijassa-1172	253	29	150	150	NUM
ijassa-1172	253	30	200	200	NUM
ijassa-1172	253	31	0	0	NUM
ijassa-1172	253	32	200	200	NUM
ijassa-1172	253	33	400	400	NUM
ijassa-1172	253	34	600	600	NUM
ijassa-1172	253	35	days	day	NOUN
ijassa-1172	253	36	c	c	NOUN
ijassa-1172	253	37	a	a	PRON
ijassa-1172	253	38	rr	rr	X
ijassa-1172	253	39	ie	ie	X
ijassa-1172	253	40	r	r	NOUN
ijassa-1172	253	41	(	(	PUNCT
ijassa-1172	253	42	c	c	NOUN
ijassa-1172	253	43	)	)	PUNCT
ijassa-1172	253	44	α=0.3	α=0.3	PUNCT
ijassa-1172	253	45	α=0.5	α=0.5	VERB
ijassa-1172	253	46	α=0.8	α=0.8	ADV
ijassa-1172	253	47	α=1	α=1	SYM
ijassa-1172	253	48	0	0	NUM
ijassa-1172	253	49	50	50	NUM
ijassa-1172	253	50	100	100	NUM
ijassa-1172	253	51	150	150	NUM
ijassa-1172	253	52	200	200	NUM
ijassa-1172	253	53	0	0	NUM
ijassa-1172	253	54	100	100	NUM
ijassa-1172	253	55	200	200	NUM
ijassa-1172	253	56	300	300	NUM
ijassa-1172	253	57	days	day	NOUN
ijassa-1172	253	58	in	in	ADP
ijassa-1172	253	59	fe	fe	X
ijassa-1172	253	60	c	c	X
ijassa-1172	253	61	te	te	PROPN
ijassa-1172	253	62	d	d	X
ijassa-1172	253	63	(	(	PUNCT
ijassa-1172	253	64	i	i	NOUN
ijassa-1172	253	65	)	)	PUNCT
ijassa-1172	253	66	α=0.3	α=0.3	PUNCT
ijassa-1172	253	67	α=0.5	α=0.5	VERB
ijassa-1172	253	68	α=0.8	α=0.8	NOUN
ijassa-1172	253	69	α=1	α=1	ADP
ijassa-1172	253	70	3400	3400	NUM
ijassa-1172	253	71	3600	3600	NUM
ijassa-1172	253	72	3800	3800	NUM
ijassa-1172	253	73	4000	4000	NUM
ijassa-1172	253	74	0	0	NUM
ijassa-1172	253	75	500	500	NUM
ijassa-1172	253	76	0	0	NUM
ijassa-1172	253	77	200	200	NUM
ijassa-1172	253	78	400	400	NUM
ijassa-1172	253	79	sc	sc	NOUN
ijassa-1172	253	80	i	i	PRON
ijassa-1172	253	81	fig	fig	NOUN
ijassa-1172	253	82	.	.	PUNCT
ijassa-1172	254	1	4.3	4.3	NUM
ijassa-1172	254	2	.	.	PUNCT
ijassa-1172	254	3	dynamics	dynamic	NOUN
ijassa-1172	254	4	of	of	ADP
ijassa-1172	254	5	the	the	DET
ijassa-1172	254	6	model	model	NOUN
ijassa-1172	254	7	(	(	PUNCT
ijassa-1172	254	8	4.16	4.16	NUM
ijassa-1172	254	9	)	)	PUNCT
ijassa-1172	254	10	when	when	SCONJ
ijassa-1172	254	11	r0	r0	NOUN
ijassa-1172	254	12	=	=	PROPN
ijassa-1172	254	13	1.9489	1.9489	NUM
ijassa-1172	254	14	>	>	SYM
ijassa-1172	254	15	1	1	NUM
ijassa-1172	254	16	.	.	X
ijassa-1172	254	17	for	for	ADP
ijassa-1172	254	18	β2	β2	NOUN
ijassa-1172	254	19	=	=	PROPN
ijassa-1172	254	20	1.3×	1.3×	NUM
ijassa-1172	254	21	10−4	10−4	NUM
ijassa-1172	254	22	,	,	PUNCT
ijassa-1172	254	23	we	we	PRON
ijassa-1172	254	24	have	have	VERB
ijassa-1172	254	25	r0	r0	NOUN
ijassa-1172	254	26	=	=	SYM
ijassa-1172	254	27	1.9489	1.9489	NUM
ijassa-1172	254	28	>	>	SYM
ijassa-1172	254	29	1	1	X
ijassa-1172	254	30	.	.	PUNCT
ijassa-1172	254	31	figure	figure	NOUN
ijassa-1172	254	32	3	3	NUM
ijassa-1172	254	33	shows	show	VERB
ijassa-1172	254	34	that	that	SCONJ
ijassa-1172	254	35	the	the	DET
ijassa-1172	254	36	trajectories	trajectory	NOUN
ijassa-1172	254	37	of	of	ADP
ijassa-1172	254	38	model	model	NOUN
ijassa-1172	254	39	(	(	PUNCT
ijassa-1172	254	40	4.16	4.16	NUM
ijassa-1172	254	41	)	)	PUNCT
ijassa-1172	254	42	converge	converge	NOUN
ijassa-1172	254	43	to	to	ADP
ijassa-1172	254	44	e∗(3670.8515	e∗(3670.8515	NOUN
ijassa-1172	254	45	,	,	PUNCT
ijassa-1172	254	46	35.1197	35.1197	NUM
ijassa-1172	254	47	,	,	PUNCT
ijassa-1172	254	48	43.4314	43.4314	NUM
ijassa-1172	254	49	)	)	PUNCT
ijassa-1172	254	50	from	from	ADP
ijassa-1172	254	51	different	different	ADJ
ijassa-1172	254	52	values	value	NOUN
ijassa-1172	254	53	of	of	ADP
ijassa-1172	254	54	order	order	NOUN
ijassa-1172	254	55	fractional	fractional	ADJ
ijassa-1172	254	56	derivative	derivative	ADJ
ijassa-1172	254	57	α	α	NOUN
ijassa-1172	254	58	.	.	PUNCT
ijassa-1172	255	1	this	this	PRON
ijassa-1172	255	2	confirms	confirm	VERB
ijassa-1172	255	3	the	the	DET
ijassa-1172	255	4	global	global	ADJ
ijassa-1172	255	5	stability	stability	NOUN
ijassa-1172	255	6	result	result	NOUN
ijassa-1172	255	7	given	give	VERB
ijassa-1172	255	8	by	by	ADP
ijassa-1172	255	9	corollary	corollary	ADJ
ijassa-1172	255	10	4.1	4.1	NUM
ijassa-1172	255	11	(	(	PUNCT
ijassa-1172	255	12	ii	ii	NOUN
ijassa-1172	255	13	)	)	PUNCT
ijassa-1172	255	14	.	.	PUNCT
ijassa-1172	256	1	5	5	X
ijassa-1172	256	2	.	.	X
ijassa-1172	256	3	conclusion	conclusion	NOUN
ijassa-1172	256	4	in	in	ADP
ijassa-1172	256	5	this	this	DET
ijassa-1172	256	6	work	work	NOUN
ijassa-1172	256	7	,	,	PUNCT
ijassa-1172	256	8	we	we	PRON
ijassa-1172	256	9	have	have	AUX
ijassa-1172	256	10	presented	present	VERB
ijassa-1172	256	11	a	a	DET
ijassa-1172	256	12	new	new	ADJ
ijassa-1172	256	13	fractional	fractional	ADJ
ijassa-1172	256	14	-	-	PUNCT
ijassa-1172	256	15	order	order	NOUN
ijassa-1172	256	16	model	model	NOUN
ijassa-1172	256	17	of	of	ADP
ijassa-1172	256	18	covid-19	covid-19	PROPN
ijassa-1172	256	19	that	that	PRON
ijassa-1172	256	20	takes	take	VERB
ijassa-1172	256	21	into	into	ADP
ijassa-1172	256	22	modes	mode	NOUN
ijassa-1172	256	23	of	of	ADP
ijassa-1172	256	24	infection	infection	NOUN
ijassa-1172	256	25	spread	spread	NOUN
ijassa-1172	256	26	of	of	ADP
ijassa-1172	256	27	covid-19	covid-19	PROPN
ijassa-1172	256	28	due	due	ADP
ijassa-1172	256	29	to	to	ADP
ijassa-1172	256	30	the	the	DET
ijassa-1172	256	31	direct	direct	ADJ
ijassa-1172	256	32	contact	contact	NOUN
ijassa-1172	256	33	between	between	ADP
ijassa-1172	256	34	the	the	DET
ijassa-1172	256	35	susceptible	susceptible	ADJ
ijassa-1172	256	36	individuals	individual	NOUN
ijassa-1172	256	37	,	,	PUNCT
ijassa-1172	256	38	with	with	ADP
ijassa-1172	256	39	symptomatic	symptomatic	ADJ
ijassa-1172	256	40	and	and	CCONJ
ijassa-1172	256	41	asymptomatic	asymptomatic	ADJ
ijassa-1172	256	42	individuals	individual	NOUN
ijassa-1172	256	43	.	.	PUNCT
ijassa-1172	257	1	the	the	DET
ijassa-1172	257	2	incidence	incidence	NOUN
ijassa-1172	257	3	of	of	ADP
ijassa-1172	257	4	infection	infection	NOUN
ijassa-1172	257	5	is	be	AUX
ijassa-1172	257	6	described	describe	VERB
ijassa-1172	257	7	by	by	ADP
ijassa-1172	257	8	general	general	ADJ
ijassa-1172	257	9	nonlinear	nonlinear	ADJ
ijassa-1172	257	10	functions	function	NOUN
ijassa-1172	257	11	.	.	PUNCT
ijassa-1172	258	1	it	it	PRON
ijassa-1172	258	2	is	be	AUX
ijassa-1172	258	3	proved	prove	VERB
ijassa-1172	258	4	that	that	SCONJ
ijassa-1172	258	5	the	the	DET
ijassa-1172	258	6	proposed	propose	VERB
ijassa-1172	258	7	model	model	NOUN
ijassa-1172	258	8	has	have	VERB
ijassa-1172	258	9	two	two	NUM
ijassa-1172	258	10	steady	steady	ADJ
ijassa-1172	258	11	states	state	NOUN
ijassa-1172	258	12	namely	namely	ADV
ijassa-1172	258	13	disease	disease	NOUN
ijassa-1172	258	14	-	-	PUNCT
ijassa-1172	258	15	free	free	ADJ
ijassa-1172	258	16	equilibrium	equilibrium	NOUN
ijassa-1172	258	17	and	and	CCONJ
ijassa-1172	258	18	endemic	endemic	ADJ
ijassa-1172	258	19	equilibrium	equilibrium	NOUN
ijassa-1172	258	20	.	.	PUNCT
ijassa-1172	259	1	the	the	DET
ijassa-1172	259	2	stability	stability	NOUN
ijassa-1172	259	3	analysis	analysis	NOUN
ijassa-1172	259	4	of	of	ADP
ijassa-1172	259	5	the	the	DET
ijassa-1172	259	6	model	model	NOUN
ijassa-1172	259	7	shows	show	VERB
ijassa-1172	259	8	that	that	SCONJ
ijassa-1172	259	9	the	the	DET
ijassa-1172	259	10	disease	disease	NOUN
ijassa-1172	259	11	-	-	PUNCT
ijassa-1172	259	12	free	free	ADJ
ijassa-1172	259	13	equilibrium	equilibrium	NOUN
ijassa-1172	259	14	is	be	AUX
ijassa-1172	259	15	globally	globally	ADV
ijassa-1172	259	16	asymptotically	asymptotically	ADV
ijassa-1172	259	17	stable	stable	ADJ
ijassa-1172	259	18	if	if	SCONJ
ijassa-1172	259	19	the	the	DET
ijassa-1172	259	20	basic	basic	ADJ
ijassa-1172	259	21	reproduction	reproduction	NOUN
ijassa-1172	259	22	numberr0	numberr0	NOUN
ijassa-1172	259	23	≤	≤	NOUN
ijassa-1172	259	24	1	1	NUM
ijassa-1172	259	25	,	,	PUNCT
ijassa-1172	259	26	which	which	PRON
ijassa-1172	259	27	biologically	biologically	ADV
ijassa-1172	259	28	means	mean	VERB
ijassa-1172	259	29	that	that	SCONJ
ijassa-1172	259	30	the	the	DET
ijassa-1172	259	31	covid-19	covid-19	PROPN
ijassa-1172	259	32	is	be	AUX
ijassa-1172	259	33	cleared	clear	VERB
ijassa-1172	259	34	.	.	PUNCT
ijassa-1172	260	1	while	while	SCONJ
ijassa-1172	260	2	the	the	DET
ijassa-1172	260	3	endemic	endemic	ADJ
ijassa-1172	260	4	equilibrium	equilibrium	NOUN
ijassa-1172	260	5	is	be	AUX
ijassa-1172	260	6	globally	globally	ADV
ijassa-1172	260	7	asymptotically	asymptotically	ADV
ijassa-1172	260	8	stable	stable	ADJ
ijassa-1172	260	9	when	when	SCONJ
ijassa-1172	260	10	r0	r0	NOUN
ijassa-1172	260	11	>	>	X
ijassa-1172	260	12	1	1	X
ijassa-1172	260	13	.	.	PUNCT
ijassa-1172	261	1	in	in	ADP
ijassa-1172	261	2	this	this	DET
ijassa-1172	261	3	case	case	NOUN
ijassa-1172	261	4	,	,	PUNCT
ijassa-1172	261	5	the	the	DET
ijassa-1172	261	6	covid-19	covid-19	PROPN
ijassa-1172	261	7	persists	persist	VERB
ijassa-1172	261	8	in	in	ADP
ijassa-1172	261	9	the	the	DET
ijassa-1172	261	10	world	world	NOUN
ijassa-1172	261	11	and	and	CCONJ
ijassa-1172	261	12	the	the	DET
ijassa-1172	261	13	infection	infection	NOUN
ijassa-1172	261	14	copyright	copyright	NOUN
ijassa-1172	261	15	©	©	PROPN
ijassa-1172	261	16	2022	2022	NUM
ijassa-1172	261	17	assa	assa	NOUN
ijassa-1172	261	18	.	.	PUNCT
ijassa-1172	262	1	adv	adv	PROPN
ijassa-1172	262	2	syst	syst	PROPN
ijassa-1172	262	3	sci	sci	PROPN
ijassa-1172	262	4	appl	appl	PROPN
ijassa-1172	262	5	(	(	PUNCT
ijassa-1172	262	6	2022	2022	NUM
ijassa-1172	262	7	)	)	PUNCT
ijassa-1172	262	8	dynamics	dynamic	NOUN
ijassa-1172	262	9	of	of	ADP
ijassa-1172	262	10	a	a	DET
ijassa-1172	262	11	generalized	generalize	VERB
ijassa-1172	262	12	fractional	fractional	ADJ
ijassa-1172	262	13	epidemic	epidemic	NOUN
ijassa-1172	262	14	model	model	NOUN
ijassa-1172	262	15	...	...	PUNCT
ijassa-1172	262	16	47	47	NUM
ijassa-1172	262	17	becomes	become	VERB
ijassa-1172	262	18	similar	similar	ADJ
ijassa-1172	262	19	to	to	ADP
ijassa-1172	262	20	many	many	ADJ
ijassa-1172	262	21	endemic	endemic	ADJ
ijassa-1172	262	22	viral	viral	ADJ
ijassa-1172	262	23	diseases	disease	NOUN
ijassa-1172	262	24	to	to	ADP
ijassa-1172	262	25	date	date	NOUN
ijassa-1172	262	26	,	,	PUNCT
ijassa-1172	262	27	such	such	ADJ
ijassa-1172	262	28	as	as	ADP
ijassa-1172	262	29	influenza	influenza	NOUN
ijassa-1172	262	30	,	,	PUNCT
ijassa-1172	262	31	aids	aid	NOUN
ijassa-1172	262	32	and	and	CCONJ
ijassa-1172	262	33	viral	viral	ADJ
ijassa-1172	262	34	hepatitis	hepatitis	NOUN
ijassa-1172	262	35	.	.	PUNCT
ijassa-1172	263	1	on	on	ADP
ijassa-1172	263	2	the	the	DET
ijassa-1172	263	3	other	other	ADJ
ijassa-1172	263	4	hand	hand	NOUN
ijassa-1172	263	5	,	,	PUNCT
ijassa-1172	263	6	we	we	PRON
ijassa-1172	263	7	have	have	AUX
ijassa-1172	263	8	established	establish	VERB
ijassa-1172	263	9	a	a	DET
ijassa-1172	263	10	new	new	ADJ
ijassa-1172	263	11	generalized	generalized	ADJ
ijassa-1172	263	12	lemma	lemma	PROPN
ijassa-1172	263	13	to	to	PART
ijassa-1172	263	14	analyze	analyze	VERB
ijassa-1172	263	15	the	the	DET
ijassa-1172	263	16	global	global	ADJ
ijassa-1172	263	17	stability	stability	NOUN
ijassa-1172	263	18	of	of	ADP
ijassa-1172	263	19	the	the	DET
ijassa-1172	263	20	endemic	endemic	ADJ
ijassa-1172	263	21	equilibrium	equilibrium	NOUN
ijassa-1172	263	22	.	.	PUNCT
ijassa-1172	264	1	this	this	DET
ijassa-1172	264	2	lemma	lemma	PROPN
ijassa-1172	264	3	presents	present	VERB
ijassa-1172	264	4	a	a	DET
ijassa-1172	264	5	new	new	ADJ
ijassa-1172	264	6	property	property	NOUN
ijassa-1172	264	7	for	for	ADP
ijassa-1172	264	8	caputo	caputo	PROPN
ijassa-1172	264	9	fractional	fractional	PROPN
ijassa-1172	264	10	derivative	derivative	NOUN
ijassa-1172	264	11	when	when	SCONJ
ijassa-1172	264	12	α	α	X
ijassa-1172	264	13	∈	∈	PROPN
ijassa-1172	264	14	(	(	PUNCT
ijassa-1172	264	15	0	0	NUM
ijassa-1172	264	16	,	,	PUNCT
ijassa-1172	264	17	1	1	NUM
ijassa-1172	264	18	]	]	PUNCT
ijassa-1172	264	19	and	and	CCONJ
ijassa-1172	264	20	extends	extend	VERB
ijassa-1172	264	21	the	the	DET
ijassa-1172	264	22	results	result	NOUN
ijassa-1172	264	23	presented	present	VERB
ijassa-1172	264	24	in	in	ADP
ijassa-1172	264	25	[	[	X
ijassa-1172	264	26	1	1	NUM
ijassa-1172	264	27	,	,	PUNCT
ijassa-1172	264	28	17	17	NUM
ijassa-1172	264	29	]	]	PUNCT
ijassa-1172	264	30	for	for	ADP
ijassa-1172	264	31	any	any	DET
ijassa-1172	264	32	monotone	monotone	ADJ
ijassa-1172	264	33	function	function	NOUN
ijassa-1172	264	34	.	.	PUNCT
ijassa-1172	265	1	also	also	ADV
ijassa-1172	265	2	,	,	PUNCT
ijassa-1172	265	3	the	the	DET
ijassa-1172	265	4	new	new	ADJ
ijassa-1172	265	5	generalized	generalize	VERB
ijassa-1172	265	6	lemma	lemma	PROPN
ijassa-1172	265	7	can	can	AUX
ijassa-1172	265	8	be	be	AUX
ijassa-1172	265	9	an	an	DET
ijassa-1172	265	10	important	important	ADJ
ijassa-1172	265	11	tool	tool	NOUN
ijassa-1172	265	12	to	to	PART
ijassa-1172	265	13	study	study	VERB
ijassa-1172	265	14	the	the	DET
ijassa-1172	265	15	global	global	ADJ
ijassa-1172	265	16	dynamics	dynamic	NOUN
ijassa-1172	265	17	of	of	ADP
ijassa-1172	265	18	many	many	ADJ
ijassa-1172	265	19	fractional	fractional	ADJ
ijassa-1172	265	20	order	order	NOUN
ijassa-1172	265	21	systems	system	NOUN
ijassa-1172	265	22	modeling	model	VERB
ijassa-1172	265	23	infectious	infectious	ADJ
ijassa-1172	265	24	diseases	disease	NOUN
ijassa-1172	265	25	such	such	ADJ
ijassa-1172	265	26	as	as	ADP
ijassa-1172	265	27	covid-19	covid-19	PROPN
ijassa-1172	265	28	.	.	PUNCT
ijassa-1172	266	1	furthermore	furthermore	ADV
ijassa-1172	266	2	,	,	PUNCT
ijassa-1172	266	3	the	the	DET
ijassa-1172	266	4	asymptotic	asymptotic	ADJ
ijassa-1172	266	5	stability	stability	NOUN
ijassa-1172	266	6	results	result	NOUN
ijassa-1172	266	7	of	of	ADP
ijassa-1172	266	8	the	the	DET
ijassa-1172	266	9	recent	recent	ADJ
ijassa-1172	266	10	model	model	NOUN
ijassa-1172	266	11	of	of	ADP
ijassa-1172	266	12	covid-19	covid-19	PROPN
ijassa-1172	266	13	in	in	ADP
ijassa-1172	266	14	japan	japan	PROPN
ijassa-1172	266	15	[	[	X
ijassa-1172	266	16	2	2	NUM
ijassa-1172	266	17	]	]	PUNCT
ijassa-1172	266	18	are	be	AUX
ijassa-1172	266	19	improved	improve	VERB
ijassa-1172	266	20	.	.	PUNCT
ijassa-1172	267	1	several	several	ADJ
ijassa-1172	267	2	countries	country	NOUN
ijassa-1172	267	3	around	around	ADP
ijassa-1172	267	4	the	the	DET
ijassa-1172	267	5	world	world	NOUN
ijassa-1172	267	6	have	have	AUX
ijassa-1172	267	7	started	start	VERB
ijassa-1172	267	8	vaccination	vaccination	NOUN
ijassa-1172	267	9	against	against	ADP
ijassa-1172	267	10	covid-19	covid-19	PROPN
ijassa-1172	267	11	in	in	ADP
ijassa-1172	267	12	order	order	NOUN
ijassa-1172	267	13	to	to	PART
ijassa-1172	267	14	reduce	reduce	VERB
ijassa-1172	267	15	the	the	DET
ijassa-1172	267	16	prevalence	prevalence	NOUN
ijassa-1172	267	17	of	of	ADP
ijassa-1172	267	18	this	this	DET
ijassa-1172	267	19	dangerous	dangerous	ADJ
ijassa-1172	267	20	disease	disease	NOUN
ijassa-1172	267	21	.	.	PUNCT
ijassa-1172	268	1	we	we	PRON
ijassa-1172	268	2	will	will	AUX
ijassa-1172	268	3	extend	extend	VERB
ijassa-1172	268	4	our	our	PRON
ijassa-1172	268	5	model	model	NOUN
ijassa-1172	268	6	to	to	PART
ijassa-1172	268	7	study	study	VERB
ijassa-1172	268	8	the	the	DET
ijassa-1172	268	9	impact	impact	NOUN
ijassa-1172	268	10	of	of	ADP
ijassa-1172	268	11	vaccination	vaccination	NOUN
ijassa-1172	268	12	on	on	ADP
ijassa-1172	268	13	the	the	DET
ijassa-1172	268	14	propagation	propagation	NOUN
ijassa-1172	268	15	of	of	ADP
ijassa-1172	268	16	covid-19	covid-19	PROPN
ijassa-1172	268	17	.	.	PUNCT
ijassa-1172	269	1	also	also	ADV
ijassa-1172	269	2	,	,	PUNCT
ijassa-1172	269	3	our	our	PRON
ijassa-1172	269	4	model	model	NOUN
ijassa-1172	269	5	used	use	VERB
ijassa-1172	269	6	a	a	DET
ijassa-1172	269	7	fractional	fractional	ADJ
ijassa-1172	269	8	derivative	derivative	NOUN
ijassa-1172	269	9	with	with	ADP
ijassa-1172	269	10	singular	singular	ADJ
ijassa-1172	269	11	kernel	kernel	NOUN
ijassa-1172	269	12	.	.	PUNCT
ijassa-1172	270	1	it	it	PRON
ijassa-1172	270	2	will	will	AUX
ijassa-1172	270	3	be	be	AUX
ijassa-1172	270	4	more	more	ADV
ijassa-1172	270	5	interesting	interesting	ADJ
ijassa-1172	270	6	to	to	PART
ijassa-1172	270	7	extend	extend	VERB
ijassa-1172	270	8	the	the	DET
ijassa-1172	270	9	model	model	NOUN
ijassa-1172	270	10	by	by	ADP
ijassa-1172	270	11	using	use	VERB
ijassa-1172	270	12	a	a	DET
ijassa-1172	270	13	fractional	fractional	ADJ
ijassa-1172	270	14	derivative	derivative	NOUN
ijassa-1172	270	15	with	with	ADP
ijassa-1172	270	16	non	non	ADJ
ijassa-1172	270	17	-	-	ADJ
ijassa-1172	270	18	singular	singular	ADJ
ijassa-1172	270	19	kernel	kernel	NOUN
ijassa-1172	270	20	as	as	ADP
ijassa-1172	270	21	in	in	ADP
ijassa-1172	270	22	[	[	X
ijassa-1172	270	23	6	6	NUM
ijassa-1172	270	24	]	]	PUNCT
ijassa-1172	270	25	.	.	PUNCT
ijassa-1172	271	1	data	datum	NOUN
ijassa-1172	271	2	availability	availability	NOUN
ijassa-1172	271	3	the	the	DET
ijassa-1172	271	4	data	datum	NOUN
ijassa-1172	271	5	used	use	VERB
ijassa-1172	271	6	to	to	PART
ijassa-1172	271	7	support	support	VERB
ijassa-1172	271	8	the	the	DET
ijassa-1172	271	9	findings	finding	NOUN
ijassa-1172	271	10	of	of	ADP
ijassa-1172	271	11	this	this	DET
ijassa-1172	271	12	study	study	NOUN
ijassa-1172	271	13	are	be	AUX
ijassa-1172	271	14	available	available	ADJ
ijassa-1172	271	15	from	from	ADP
ijassa-1172	271	16	the	the	DET
ijassa-1172	271	17	corresponding	corresponding	ADJ
ijassa-1172	271	18	author	author	NOUN
ijassa-1172	271	19	upon	upon	SCONJ
ijassa-1172	271	20	request	request	NOUN
ijassa-1172	271	21	.	.	PUNCT
ijassa-1172	272	1	the	the	DET
ijassa-1172	272	2	articles	article	NOUN
ijassa-1172	272	3	used	use	VERB
ijassa-1172	272	4	to	to	PART
ijassa-1172	272	5	support	support	VERB
ijassa-1172	272	6	the	the	DET
ijassa-1172	272	7	findings	finding	NOUN
ijassa-1172	272	8	of	of	ADP
ijassa-1172	272	9	this	this	DET
ijassa-1172	272	10	study	study	NOUN
ijassa-1172	272	11	are	be	AUX
ijassa-1172	272	12	included	include	VERB
ijassa-1172	272	13	within	within	ADP
ijassa-1172	272	14	the	the	DET
ijassa-1172	272	15	article	article	NOUN
ijassa-1172	272	16	and	and	CCONJ
ijassa-1172	272	17	are	be	AUX
ijassa-1172	272	18	cited	cite	VERB
ijassa-1172	272	19	at	at	ADP
ijassa-1172	272	20	relevant	relevant	ADJ
ijassa-1172	272	21	places	place	NOUN
ijassa-1172	272	22	within	within	ADP
ijassa-1172	272	23	the	the	DET
ijassa-1172	272	24	text	text	NOUN
ijassa-1172	272	25	as	as	ADP
ijassa-1172	272	26	references	reference	NOUN
ijassa-1172	272	27	.	.	PUNCT
ijassa-1172	273	1	conflicts	conflict	NOUN
ijassa-1172	273	2	of	of	ADP
ijassa-1172	273	3	interest	interest	NOUN
ijassa-1172	273	4	the	the	DET
ijassa-1172	273	5	authors	author	NOUN
ijassa-1172	273	6	declare	declare	VERB
ijassa-1172	273	7	that	that	SCONJ
ijassa-1172	273	8	they	they	PRON
ijassa-1172	273	9	have	have	VERB
ijassa-1172	273	10	no	no	DET
ijassa-1172	273	11	conflicts	conflict	NOUN
ijassa-1172	273	12	of	of	ADP
ijassa-1172	273	13	interest	interest	NOUN
ijassa-1172	273	14	.	.	PUNCT
ijassa-1172	274	1	acknowledgements	acknowledgement	NOUN
ijassa-1172	274	2	the	the	DET
ijassa-1172	274	3	authors	author	NOUN
ijassa-1172	274	4	would	would	AUX
ijassa-1172	274	5	like	like	VERB
ijassa-1172	274	6	to	to	PART
ijassa-1172	274	7	thank	thank	VERB
ijassa-1172	274	8	the	the	DET
ijassa-1172	274	9	editor	editor	NOUN
ijassa-1172	274	10	and	and	CCONJ
ijassa-1172	274	11	anonymous	anonymous	ADJ
ijassa-1172	274	12	reviewers	reviewer	NOUN
ijassa-1172	274	13	for	for	ADP
ijassa-1172	274	14	their	their	PRON
ijassa-1172	274	15	constructive	constructive	ADJ
ijassa-1172	274	16	comments	comment	NOUN
ijassa-1172	274	17	and	and	CCONJ
ijassa-1172	274	18	suggestions	suggestion	NOUN
ijassa-1172	274	19	,	,	PUNCT
ijassa-1172	274	20	which	which	PRON
ijassa-1172	274	21	have	have	AUX
ijassa-1172	274	22	improved	improve	VERB
ijassa-1172	274	23	the	the	DET
ijassa-1172	274	24	quality	quality	NOUN
ijassa-1172	274	25	of	of	ADP
ijassa-1172	274	26	the	the	DET
ijassa-1172	274	27	manuscript	manuscript	NOUN
ijassa-1172	274	28	.	.	PUNCT
ijassa-1172	275	1	references	reference	NOUN
ijassa-1172	275	2	1	1	NUM
ijassa-1172	275	3	.	.	PUNCT
ijassa-1172	275	4	aguila	aguila	PROPN
ijassa-1172	275	5	-	-	PUNCT
ijassa-1172	275	6	camacho	camacho	PROPN
ijassa-1172	275	7	,	,	PUNCT
ijassa-1172	275	8	n.	n.	NOUN
ijassa-1172	275	9	,	,	PUNCT
ijassa-1172	275	10	duarte	duarte	PROPN
ijassa-1172	275	11	-	-	PUNCT
ijassa-1172	275	12	mermoud	mermoud	PROPN
ijassa-1172	275	13	,	,	PUNCT
ijassa-1172	275	14	m.	m.	NOUN
ijassa-1172	275	15	a.	a.	PROPN
ijassa-1172	275	16	,	,	PUNCT
ijassa-1172	275	17	&	&	CCONJ
ijassa-1172	275	18	gallegos	gallegos	PROPN
ijassa-1172	275	19	,	,	PUNCT
ijassa-1172	275	20	j.	j.	PROPN
ijassa-1172	275	21	a.	a.	PROPN
ijassa-1172	275	22	(	(	PUNCT
ijassa-1172	275	23	2014	2014	NUM
ijassa-1172	275	24	)	)	PUNCT
ijassa-1172	275	25	a	a	DET
ijassa-1172	275	26	lyapunov	lyapunov	NOUN
ijassa-1172	275	27	functions	function	NOUN
ijassa-1172	275	28	for	for	ADP
ijassa-1172	275	29	fractional	fractional	ADJ
ijassa-1172	275	30	order	order	NOUN
ijassa-1172	275	31	systems	system	NOUN
ijassa-1172	275	32	.	.	PUNCT
ijassa-1172	276	1	commun	commun	PROPN
ijassa-1172	276	2	.	.	PUNCT
ijassa-1172	277	1	nonlinear	nonlinear	PROPN
ijassa-1172	277	2	.	.	PUNCT
ijassa-1172	278	1	sci	sci	PROPN
ijassa-1172	278	2	.	.	PUNCT
ijassa-1172	278	3	numer	numer	PROPN
ijassa-1172	278	4	.	.	PUNCT
ijassa-1172	279	1	simulat	simulat	NOUN
ijassa-1172	279	2	.	.	PUNCT
ijassa-1172	280	1	19	19	NUM
ijassa-1172	280	2	,	,	PUNCT
ijassa-1172	280	3	2951–2957	2951–2957	NOUN
ijassa-1172	280	4	.	.	PUNCT
ijassa-1172	281	1	2	2	X
ijassa-1172	281	2	.	.	X
ijassa-1172	281	3	das	das	PROPN
ijassa-1172	281	4	,	,	PUNCT
ijassa-1172	281	5	m.	m.	NOUN
ijassa-1172	281	6	,	,	PUNCT
ijassa-1172	281	7	samanta	samanta	PROPN
ijassa-1172	281	8	,	,	PUNCT
ijassa-1172	281	9	g.	g.	PROPN
ijassa-1172	281	10	p.	p.	PROPN
ijassa-1172	281	11	(	(	PUNCT
ijassa-1172	281	12	2020	2020	NUM
ijassa-1172	281	13	)	)	PUNCT
ijassa-1172	281	14	a	a	DET
ijassa-1172	281	15	fractional	fractional	ADJ
ijassa-1172	281	16	order	order	NOUN
ijassa-1172	281	17	covid-19	covid-19	PROPN
ijassa-1172	281	18	epidemic	epidemic	NOUN
ijassa-1172	281	19	transmission	transmission	NOUN
ijassa-1172	281	20	model	model	NOUN
ijassa-1172	281	21	:	:	PUNCT
ijassa-1172	281	22	stability	stability	NOUN
ijassa-1172	281	23	analysis	analysis	NOUN
ijassa-1172	281	24	and	and	CCONJ
ijassa-1172	281	25	optimal	optimal	ADJ
ijassa-1172	281	26	control	control	NOUN
ijassa-1172	281	27	.	.	PUNCT
ijassa-1172	282	1	[	[	X
ijassa-1172	282	2	online	online	X
ijassa-1172	282	3	]	]	X
ijassa-1172	282	4	.	.	PUNCT
ijassa-1172	283	1	available	available	ADJ
ijassa-1172	283	2	https://ssrn.com/abstract=3635938	https://ssrn.com/abstract=3635938	ADV
ijassa-1172	283	3	.	.	PUNCT
ijassa-1172	284	1	3	3	X
ijassa-1172	284	2	.	.	X
ijassa-1172	284	3	dehingia	dehingia	NOUN
ijassa-1172	284	4	,	,	PUNCT
ijassa-1172	284	5	k.	k.	PROPN
ijassa-1172	284	6	,	,	PUNCT
ijassa-1172	284	7	mohsen	mohsen	PROPN
ijassa-1172	284	8	,	,	PUNCT
ijassa-1172	284	9	a.a	a.a	PROPN
ijassa-1172	284	10	.	.	PROPN
ijassa-1172	284	11	,	,	PUNCT
ijassa-1172	284	12	alharbi	alharbi	PROPN
ijassa-1172	284	13	,	,	PUNCT
ijassa-1172	284	14	s.a	s.a	PROPN
ijassa-1172	284	15	.	.	PROPN
ijassa-1172	284	16	,	,	PUNCT
ijassa-1172	284	17	alsemiry	alsemiry	NOUN
ijassa-1172	284	18	,	,	PUNCT
ijassa-1172	284	19	r.d	r.d	PROPN
ijassa-1172	284	20	.	.	PROPN
ijassa-1172	284	21	,	,	PUNCT
ijassa-1172	284	22	&	&	CCONJ
ijassa-1172	284	23	rezapour	rezapour	PROPN
ijassa-1172	284	24	,	,	PUNCT
ijassa-1172	284	25	s.	s.	PROPN
ijassa-1172	284	26	(	(	PUNCT
ijassa-1172	284	27	2022	2022	NUM
ijassa-1172	284	28	)	)	PUNCT
ijassa-1172	284	29	dynamical	dynamical	ADJ
ijassa-1172	284	30	behavior	behavior	NOUN
ijassa-1172	284	31	of	of	ADP
ijassa-1172	284	32	a	a	DET
ijassa-1172	284	33	fractional	fractional	ADJ
ijassa-1172	284	34	order	order	NOUN
ijassa-1172	284	35	model	model	NOUN
ijassa-1172	284	36	for	for	ADP
ijassa-1172	284	37	within	within	ADP
ijassa-1172	284	38	-	-	PUNCT
ijassa-1172	284	39	host	host	NOUN
ijassa-1172	284	40	sars	sar	NOUN
ijassa-1172	284	41	-	-	PUNCT
ijassa-1172	284	42	cov-2	cov-2	NOUN
ijassa-1172	284	43	.	.	PUNCT
ijassa-1172	285	1	mathematics	mathematic	NOUN
ijassa-1172	285	2	,	,	PUNCT
ijassa-1172	285	3	10	10	NUM
ijassa-1172	285	4	,	,	PUNCT
ijassa-1172	285	5	2344	2344	NUM
ijassa-1172	285	6	.	.	PUNCT
ijassa-1172	286	1	4	4	X
ijassa-1172	286	2	.	.	X
ijassa-1172	286	3	dobrovolny	dobrovolny	NOUN
ijassa-1172	286	4	,	,	PUNCT
ijassa-1172	286	5	h.	h.	PROPN
ijassa-1172	286	6	m.	m.	NOUN
ijassa-1172	286	7	(	(	PUNCT
ijassa-1172	286	8	2020	2020	NUM
ijassa-1172	286	9	)	)	PUNCT
ijassa-1172	286	10	modeling	model	VERB
ijassa-1172	286	11	the	the	DET
ijassa-1172	286	12	role	role	NOUN
ijassa-1172	286	13	of	of	ADP
ijassa-1172	286	14	asymptomatics	asymptomatic	NOUN
ijassa-1172	286	15	in	in	ADP
ijassa-1172	286	16	infection	infection	NOUN
ijassa-1172	286	17	spread	spread	VERB
ijassa-1172	286	18	with	with	ADP
ijassa-1172	286	19	application	application	NOUN
ijassa-1172	286	20	to	to	ADP
ijassa-1172	286	21	sars	sar	NOUN
ijassa-1172	286	22	-	-	PUNCT
ijassa-1172	286	23	cov-2	cov-2	NOUN
ijassa-1172	286	24	,	,	PUNCT
ijassa-1172	286	25	plose	plose	ADJ
ijassa-1172	286	26	one	one	NUM
ijassa-1172	286	27	journal	journal	NOUN
ijassa-1172	286	28	,	,	PUNCT
ijassa-1172	286	29	15(8	15(8	NOUN
ijassa-1172	286	30	)	)	PUNCT
ijassa-1172	286	31	,	,	PUNCT
ijassa-1172	286	32	1–14	1–14	PROPN
ijassa-1172	286	33	.	.	PROPN
ijassa-1172	286	34	5	5	NUM
ijassa-1172	286	35	.	.	PUNCT
ijassa-1172	287	1	feng	feng	PROPN
ijassa-1172	287	2	,	,	PUNCT
ijassa-1172	287	3	y	y	PROPN
ijassa-1172	287	4	,	,	PUNCT
ijassa-1172	287	5	xu	xu	PROPN
ijassa-1172	287	6	,	,	PUNCT
ijassa-1172	287	7	s.	s.	PROPN
ijassa-1172	287	8	,	,	PUNCT
ijassa-1172	287	9	rong	rong	PROPN
ijassa-1172	287	10	,	,	PUNCT
ijassa-1172	287	11	z.	z.	PROPN
ijassa-1172	287	12	,	,	PUNCT
ijassa-1172	287	13	xu	xu	PROPN
ijassa-1172	287	14	,	,	PUNCT
ijassa-1172	287	15	r.	r.	PROPN
ijassa-1172	287	16	,	,	PUNCT
ijassa-1172	287	17	liu	liu	PROPN
ijassa-1172	287	18	,	,	PUNCT
ijassa-1172	287	19	x.	x.	PROPN
ijassa-1172	287	20	et	et	PROPN
ijassa-1172	287	21	al	al	PROPN
ijassa-1172	287	22	.	.	PROPN
ijassa-1172	287	23	(	(	PUNCT
ijassa-1172	287	24	2020	2020	NUM
ijassa-1172	287	25	)	)	PUNCT
ijassa-1172	287	26	delivery	delivery	NOUN
ijassa-1172	287	27	of	of	ADP
ijassa-1172	287	28	infection	infection	NOUN
ijassa-1172	287	29	from	from	ADP
ijassa-1172	287	30	asymptomatic	asymptomatic	ADJ
ijassa-1172	287	31	carriers	carrier	NOUN
ijassa-1172	287	32	of	of	ADP
ijassa-1172	287	33	covid-19	covid-19	PROPN
ijassa-1172	287	34	in	in	ADP
ijassa-1172	287	35	a	a	DET
ijassa-1172	287	36	familial	familial	ADJ
ijassa-1172	287	37	cluster	cluster	NOUN
ijassa-1172	287	38	.	.	PUNCT
ijassa-1172	288	1	international	international	ADJ
ijassa-1172	288	2	journal	journal	NOUN
ijassa-1172	288	3	of	of	ADP
ijassa-1172	288	4	infectious	infectious	ADJ
ijassa-1172	288	5	diseases	disease	NOUN
ijassa-1172	288	6	,	,	PUNCT
ijassa-1172	288	7	94	94	NUM
ijassa-1172	288	8	,	,	PUNCT
ijassa-1172	288	9	133–138	133–138	NUM
ijassa-1172	288	10	.	.	PUNCT
ijassa-1172	289	1	6	6	NUM
ijassa-1172	289	2	.	.	PUNCT
ijassa-1172	289	3	hattaf	hattaf	NOUN
ijassa-1172	289	4	,	,	PUNCT
ijassa-1172	289	5	k.	k.	PROPN
ijassa-1172	289	6	(	(	PUNCT
ijassa-1172	289	7	2020	2020	NUM
ijassa-1172	289	8	)	)	PUNCT
ijassa-1172	289	9	a	a	DET
ijassa-1172	289	10	new	new	ADJ
ijassa-1172	289	11	generalized	generalized	ADJ
ijassa-1172	289	12	definition	definition	NOUN
ijassa-1172	289	13	of	of	ADP
ijassa-1172	289	14	fractional	fractional	ADJ
ijassa-1172	289	15	derivative	derivative	NOUN
ijassa-1172	289	16	with	with	ADP
ijassa-1172	289	17	nonsingular	nonsingular	ADJ
ijassa-1172	289	18	kernel	kernel	PROPN
ijassa-1172	289	19	.	.	PUNCT
ijassa-1172	290	1	computation	computation	NOUN
ijassa-1172	290	2	,	,	PUNCT
ijassa-1172	290	3	8(2	8(2	NUM
ijassa-1172	290	4	)	)	PUNCT
ijassa-1172	290	5	,	,	PUNCT
ijassa-1172	290	6	1–9	1–9	NOUN
ijassa-1172	290	7	.	.	NOUN
ijassa-1172	290	8	7	7	NUM
ijassa-1172	290	9	.	.	PUNCT
ijassa-1172	291	1	hattaf	hattaf	NOUN
ijassa-1172	291	2	,	,	PUNCT
ijassa-1172	291	3	k.	k.	PROPN
ijassa-1172	291	4	,	,	PUNCT
ijassa-1172	291	5	&	&	CCONJ
ijassa-1172	291	6	dutta	dutta	PROPN
ijassa-1172	291	7	,	,	PUNCT
ijassa-1172	291	8	h.	h.	PROPN
ijassa-1172	291	9	(	(	PUNCT
ijassa-1172	291	10	2020	2020	NUM
ijassa-1172	291	11	)	)	PUNCT
ijassa-1172	291	12	mathematical	mathematical	ADJ
ijassa-1172	291	13	modelling	modelling	NOUN
ijassa-1172	291	14	and	and	CCONJ
ijassa-1172	291	15	analysis	analysis	NOUN
ijassa-1172	291	16	of	of	ADP
ijassa-1172	291	17	infectious	infectious	ADJ
ijassa-1172	291	18	diseases	disease	NOUN
ijassa-1172	291	19	,	,	PUNCT
ijassa-1172	291	20	studies	study	NOUN
ijassa-1172	291	21	in	in	ADP
ijassa-1172	291	22	systems	system	NOUN
ijassa-1172	291	23	,	,	PUNCT
ijassa-1172	291	24	decision	decision	NOUN
ijassa-1172	291	25	and	and	CCONJ
ijassa-1172	291	26	control	control	NOUN
ijassa-1172	291	27	,	,	PUNCT
ijassa-1172	291	28	springer	springer	NOUN
ijassa-1172	291	29	.	.	PUNCT
ijassa-1172	292	1	copyright	copyright	NOUN
ijassa-1172	292	2	©	©	PROPN
ijassa-1172	292	3	2022	2022	NUM
ijassa-1172	292	4	assa	assa	NOUN
ijassa-1172	292	5	.	.	PUNCT
ijassa-1172	293	1	adv	adv	PROPN
ijassa-1172	293	2	syst	syst	PROPN
ijassa-1172	293	3	sci	sci	PROPN
ijassa-1172	293	4	appl	appl	PROPN
ijassa-1172	293	5	(	(	PUNCT
ijassa-1172	293	6	2022	2022	NUM
ijassa-1172	293	7	)	)	PUNCT
ijassa-1172	293	8	48	48	NUM
ijassa-1172	293	9	khalid	khalid	PROPN
ijassa-1172	293	10	hattaf	hattaf	NOUN
ijassa-1172	293	11	,	,	PUNCT
ijassa-1172	293	12	ahmed	ahmed	PROPN
ijassa-1172	293	13	a.	a.	PROPN
ijassa-1172	293	14	mohsen	mohsen	PROPN
ijassa-1172	293	15	8	8	PROPN
ijassa-1172	293	16	.	.	PUNCT
ijassa-1172	294	1	hattaf	hattaf	NOUN
ijassa-1172	294	2	,	,	PUNCT
ijassa-1172	294	3	k.	k.	PROPN
ijassa-1172	294	4	,	,	PUNCT
ijassa-1172	294	5	lashari	lashari	PROPN
ijassa-1172	294	6	,	,	PUNCT
ijassa-1172	294	7	a.	a.	NOUN
ijassa-1172	294	8	a.	a.	PROPN
ijassa-1172	294	9	,	,	PUNCT
ijassa-1172	294	10	louartassi	louartassi	NOUN
ijassa-1172	294	11	,	,	PUNCT
ijassa-1172	294	12	y.	y.	PROPN
ijassa-1172	294	13	,	,	PUNCT
ijassa-1172	294	14	&	&	CCONJ
ijassa-1172	294	15	yousfi	yousfi	ADJ
ijassa-1172	294	16	,	,	PUNCT
ijassa-1172	294	17	n.	n.	PROPN
ijassa-1172	294	18	(	(	PUNCT
ijassa-1172	294	19	2013	2013	NUM
ijassa-1172	294	20	)	)	PUNCT
ijassa-1172	294	21	a	a	DET
ijassa-1172	294	22	delayed	delay	VERB
ijassa-1172	294	23	sir	sir	NOUN
ijassa-1172	294	24	epidemic	epidemic	NOUN
ijassa-1172	294	25	model	model	NOUN
ijassa-1172	294	26	with	with	ADP
ijassa-1172	294	27	general	general	ADJ
ijassa-1172	294	28	incidence	incidence	NOUN
ijassa-1172	294	29	rate	rate	NOUN
ijassa-1172	294	30	.	.	PUNCT
ijassa-1172	295	1	electronic	electronic	ADJ
ijassa-1172	295	2	journal	journal	NOUN
ijassa-1172	295	3	of	of	ADP
ijassa-1172	295	4	qualitative	qualitative	ADJ
ijassa-1172	295	5	theory	theory	NOUN
ijassa-1172	295	6	of	of	ADP
ijassa-1172	295	7	differential	differential	ADJ
ijassa-1172	295	8	equations	equation	NOUN
ijassa-1172	295	9	,	,	PUNCT
ijassa-1172	295	10	3	3	NUM
ijassa-1172	295	11	,	,	PUNCT
ijassa-1172	295	12	1–9	1–9	NOUN
ijassa-1172	295	13	.	.	NOUN
ijassa-1172	295	14	9	9	NUM
ijassa-1172	295	15	.	.	PUNCT
ijassa-1172	296	1	hattaf	hattaf	NOUN
ijassa-1172	296	2	,	,	PUNCT
ijassa-1172	296	3	k.	k.	PROPN
ijassa-1172	296	4	,	,	PUNCT
ijassa-1172	296	5	&	&	CCONJ
ijassa-1172	296	6	yousfi	yousfi	PROPN
ijassa-1172	296	7	,	,	PUNCT
ijassa-1172	296	8	n.	n.	PROPN
ijassa-1172	296	9	(	(	PUNCT
ijassa-1172	296	10	2020	2020	NUM
ijassa-1172	296	11	)	)	PUNCT
ijassa-1172	296	12	dynamics	dynamic	NOUN
ijassa-1172	296	13	of	of	ADP
ijassa-1172	296	14	sars	sar	NOUN
ijassa-1172	296	15	-	-	PUNCT
ijassa-1172	296	16	cov-2	cov-2	PART
ijassa-1172	296	17	infection	infection	NOUN
ijassa-1172	296	18	model	model	NOUN
ijassa-1172	296	19	with	with	ADP
ijassa-1172	296	20	two	two	NUM
ijassa-1172	296	21	modes	mode	NOUN
ijassa-1172	296	22	of	of	ADP
ijassa-1172	296	23	transmission	transmission	NOUN
ijassa-1172	296	24	and	and	CCONJ
ijassa-1172	296	25	immune	immune	ADJ
ijassa-1172	296	26	response	response	NOUN
ijassa-1172	296	27	.	.	PUNCT
ijassa-1172	297	1	mathematical	mathematical	ADJ
ijassa-1172	297	2	biosciences	bioscience	NOUN
ijassa-1172	297	3	and	and	CCONJ
ijassa-1172	297	4	engineering	engineering	NOUN
ijassa-1172	297	5	,	,	PUNCT
ijassa-1172	297	6	17	17	NUM
ijassa-1172	297	7	(	(	PUNCT
ijassa-1172	297	8	5	5	NUM
ijassa-1172	297	9	)	)	PUNCT
ijassa-1172	297	10	,	,	PUNCT
ijassa-1172	297	11	5326–5340	5326–5340	NUM
ijassa-1172	297	12	.	.	PUNCT
ijassa-1172	298	1	10	10	NUM
ijassa-1172	298	2	.	.	PUNCT
ijassa-1172	299	1	kim	kim	PROPN
ijassa-1172	299	2	,	,	PUNCT
ijassa-1172	299	3	b.	b.	PROPN
ijassa-1172	299	4	n.	n.	PROPN
ijassa-1172	299	5	,	,	PUNCT
ijassa-1172	299	6	kim	kim	PROPN
ijassa-1172	299	7	,	,	PUNCT
ijassa-1172	299	8	e.	e.	PROPN
ijassa-1172	299	9	,	,	PUNCT
ijassa-1172	299	10	lee	lee	PROPN
ijassa-1172	299	11	,	,	PUNCT
ijassa-1172	299	12	s.	s.	PROPN
ijassa-1172	299	13	,	,	PUNCT
ijassa-1172	299	14	&	&	CCONJ
ijassa-1172	299	15	oh	oh	INTJ
ijassa-1172	299	16	,	,	PUNCT
ijassa-1172	299	17	c.	c.	PROPN
ijassa-1172	299	18	(	(	PUNCT
ijassa-1172	299	19	2020	2020	NUM
ijassa-1172	299	20	)	)	PUNCT
ijassa-1172	299	21	mathematical	mathematical	ADJ
ijassa-1172	299	22	model	model	NOUN
ijassa-1172	299	23	of	of	ADP
ijassa-1172	299	24	covid-19	covid-19	PROPN
ijassa-1172	299	25	transmission	transmission	NOUN
ijassa-1172	299	26	dynamics	dynamic	NOUN
ijassa-1172	299	27	in	in	ADP
ijassa-1172	299	28	south	south	PROPN
ijassa-1172	299	29	korea	korea	PROPN
ijassa-1172	299	30	:	:	PUNCT
ijassa-1172	299	31	the	the	DET
ijassa-1172	299	32	impacts	impact	NOUN
ijassa-1172	299	33	of	of	ADP
ijassa-1172	299	34	travel	travel	NOUN
ijassa-1172	299	35	restrictions	restriction	NOUN
ijassa-1172	299	36	,	,	PUNCT
ijassa-1172	299	37	social	social	ADJ
ijassa-1172	299	38	distancing	distancing	NOUN
ijassa-1172	299	39	,	,	PUNCT
ijassa-1172	299	40	and	and	CCONJ
ijassa-1172	299	41	early	early	ADJ
ijassa-1172	299	42	detection	detection	NOUN
ijassa-1172	299	43	.	.	PUNCT
ijassa-1172	300	1	mdpi	mdpi	ADJ
ijassa-1172	300	2	processes	process	NOUN
ijassa-1172	300	3	journal	journal	NOUN
ijassa-1172	300	4	,	,	PUNCT
ijassa-1172	300	5	8	8	NUM
ijassa-1172	300	6	.	.	NOUN
ijassa-1172	300	7	11	11	NUM
ijassa-1172	300	8	.	.	PUNCT
ijassa-1172	301	1	mohsen	mohsen	PROPN
ijassa-1172	301	2	,	,	PUNCT
ijassa-1172	301	3	a.	a.	PROPN
ijassa-1172	301	4	a.	a.	PROPN
ijassa-1172	301	5	,	,	PUNCT
ijassa-1172	301	6	al	al	PROPN
ijassa-1172	301	7	-	-	PUNCT
ijassa-1172	301	8	husseiny	husseiny	PROPN
ijassa-1172	301	9	,	,	PUNCT
ijassa-1172	301	10	h.	h.	PROPN
ijassa-1172	301	11	f.	f.	PROPN
ijassa-1172	301	12	,	,	PUNCT
ijassa-1172	301	13	hattaf	hattaf	NOUN
ijassa-1172	301	14	,	,	PUNCT
ijassa-1172	301	15	k.	k.	PROPN
ijassa-1172	301	16	,	,	PUNCT
ijassa-1172	301	17	&	&	CCONJ
ijassa-1172	301	18	boulfoul	boulfoul	PROPN
ijassa-1172	301	19	,	,	PUNCT
ijassa-1172	301	20	b.	b.	PROPN
ijassa-1172	301	21	(	(	PUNCT
ijassa-1172	301	22	2021	2021	NUM
ijassa-1172	301	23	)	)	PUNCT
ijassa-1172	301	24	a	a	DET
ijassa-1172	301	25	mathematical	mathematical	ADJ
ijassa-1172	301	26	model	model	NOUN
ijassa-1172	301	27	for	for	ADP
ijassa-1172	301	28	the	the	DET
ijassa-1172	301	29	dynamics	dynamic	NOUN
ijassa-1172	301	30	of	of	ADP
ijassa-1172	301	31	covid-19	covid-19	PROPN
ijassa-1172	301	32	pandemic	pandemic	NOUN
ijassa-1172	301	33	involving	involve	VERB
ijassa-1172	301	34	the	the	DET
ijassa-1172	301	35	infective	infective	ADJ
ijassa-1172	301	36	immigrants	immigrant	NOUN
ijassa-1172	301	37	.	.	PUNCT
ijassa-1172	302	1	iraqi	iraqi	ADJ
ijassa-1172	302	2	journals	journal	NOUN
ijassa-1172	302	3	of	of	ADP
ijassa-1172	302	4	science	science	NOUN
ijassa-1172	302	5	62(1	62(1	NUM
ijassa-1172	302	6	)	)	PUNCT
ijassa-1172	302	7	,	,	PUNCT
ijassa-1172	302	8	295–307	295–307	NUM
ijassa-1172	302	9	.	.	PUNCT
ijassa-1172	303	1	12	12	NUM
ijassa-1172	303	2	.	.	PUNCT
ijassa-1172	304	1	mohsen	mohsen	PROPN
ijassa-1172	304	2	,	,	PUNCT
ijassa-1172	304	3	a.	a.	PROPN
ijassa-1172	304	4	a.	a.	PROPN
ijassa-1172	304	5	,	,	PUNCT
ijassa-1172	304	6	al	al	PROPN
ijassa-1172	304	7	-	-	PUNCT
ijassa-1172	304	8	husseiny	husseiny	PROPN
ijassa-1172	304	9	,	,	PUNCT
ijassa-1172	304	10	h.	h.	PROPN
ijassa-1172	304	11	f.	f.	PROPN
ijassa-1172	304	12	,	,	PUNCT
ijassa-1172	304	13	zhou	zhou	PROPN
ijassa-1172	304	14	,	,	PUNCT
ijassa-1172	304	15	x.	x.	NOUN
ijassa-1172	304	16	,	,	PUNCT
ijassa-1172	304	17	&	&	CCONJ
ijassa-1172	304	18	hattaf	hattaf	PROPN
ijassa-1172	304	19	,	,	PUNCT
ijassa-1172	304	20	k.	k.	PROPN
ijassa-1172	304	21	(	(	PUNCT
ijassa-1172	304	22	2020	2020	NUM
ijassa-1172	304	23	)	)	PUNCT
ijassa-1172	304	24	global	global	ADJ
ijassa-1172	304	25	stability	stability	NOUN
ijassa-1172	304	26	of	of	ADP
ijassa-1172	304	27	covid-19	covid-19	PROPN
ijassa-1172	304	28	model	model	NOUN
ijassa-1172	304	29	involving	involve	VERB
ijassa-1172	304	30	the	the	DET
ijassa-1172	304	31	quarantine	quarantine	NOUN
ijassa-1172	304	32	strategy	strategy	NOUN
ijassa-1172	304	33	and	and	CCONJ
ijassa-1172	304	34	media	medium	NOUN
ijassa-1172	304	35	coverage	coverage	NOUN
ijassa-1172	304	36	effects	effect	NOUN
ijassa-1172	304	37	.	.	PUNCT
ijassa-1172	305	1	aims	aim	VERB
ijassa-1172	305	2	public	public	ADJ
ijassa-1172	305	3	health	health	NOUN
ijassa-1172	305	4	journal	journal	NOUN
ijassa-1172	305	5	7(3	7(3	NUM
ijassa-1172	305	6	)	)	PUNCT
ijassa-1172	305	7	,	,	PUNCT
ijassa-1172	305	8	587–605	587–605	NUM
ijassa-1172	305	9	.	.	NOUN
ijassa-1172	305	10	13	13	NUM
ijassa-1172	305	11	.	.	PUNCT
ijassa-1172	306	1	moroccan	moroccan	PROPN
ijassa-1172	306	2	ministry	ministry	PROPN
ijassa-1172	306	3	of	of	ADP
ijassa-1172	306	4	health	health	PROPN
ijassa-1172	306	5	(	(	PUNCT
ijassa-1172	306	6	mmh	mmh	PROPN
ijassa-1172	306	7	)	)	PUNCT
ijassa-1172	306	8	,	,	PUNCT
ijassa-1172	306	9	the	the	DET
ijassa-1172	306	10	official	official	ADJ
ijassa-1172	306	11	coronavirus	coronavirus	NOUN
ijassa-1172	306	12	portal	portal	NOUN
ijassa-1172	306	13	in	in	ADP
ijassa-1172	306	14	morocco	morocco	PROPN
ijassa-1172	306	15	.	.	PUNCT
ijassa-1172	307	1	[	[	X
ijassa-1172	307	2	online	online	X
ijassa-1172	307	3	]	]	X
ijassa-1172	307	4	.	.	PUNCT
ijassa-1172	308	1	available	available	ADJ
ijassa-1172	308	2	http://www.covidmaroc.ma/pages/accueilfr.aspx	http://www.covidmaroc.ma/pages/accueilfr.aspx	NOUN
ijassa-1172	308	3	.	.	PUNCT
ijassa-1172	309	1	14	14	NUM
ijassa-1172	309	2	.	.	PUNCT
ijassa-1172	310	1	moussaoui	moussaoui	ADJ
ijassa-1172	310	2	,	,	PUNCT
ijassa-1172	310	3	a.	a.	NOUN
ijassa-1172	310	4	,	,	PUNCT
ijassa-1172	310	5	&	&	CCONJ
ijassa-1172	310	6	zerga	zerga	PROPN
ijassa-1172	310	7	,	,	PUNCT
ijassa-1172	310	8	e.	e.	PROPN
ijassa-1172	310	9	h.	h.	PROPN
ijassa-1172	310	10	(	(	PUNCT
ijassa-1172	310	11	2020	2020	NUM
ijassa-1172	310	12	)	)	PUNCT
ijassa-1172	310	13	transmission	transmission	NOUN
ijassa-1172	310	14	dynamics	dynamic	NOUN
ijassa-1172	310	15	of	of	ADP
ijassa-1172	310	16	covid-19	covid-19	PROPN
ijassa-1172	310	17	in	in	ADP
ijassa-1172	310	18	algeria	algeria	PROPN
ijassa-1172	310	19	:	:	PUNCT
ijassa-1172	310	20	the	the	DET
ijassa-1172	310	21	impact	impact	NOUN
ijassa-1172	310	22	of	of	ADP
ijassa-1172	310	23	physical	physical	ADJ
ijassa-1172	310	24	distancing	distancing	NOUN
ijassa-1172	310	25	and	and	CCONJ
ijassa-1172	310	26	face	face	VERB
ijassa-1172	310	27	masks	mask	NOUN
ijassa-1172	310	28	.	.	PUNCT
ijassa-1172	311	1	aims	aim	VERB
ijassa-1172	311	2	public	public	ADJ
ijassa-1172	311	3	health	health	NOUN
ijassa-1172	311	4	,	,	PUNCT
ijassa-1172	311	5	7(4	7(4	NUM
ijassa-1172	311	6	)	)	PUNCT
ijassa-1172	311	7	,	,	PUNCT
ijassa-1172	311	8	816–827	816–827	NUM
ijassa-1172	311	9	.	.	PUNCT
ijassa-1172	312	1	15	15	NUM
ijassa-1172	312	2	.	.	X
ijassa-1172	312	3	peirlink	peirlink	VERB
ijassa-1172	312	4	,	,	PUNCT
ijassa-1172	312	5	m.	m.	NOUN
ijassa-1172	312	6	,	,	PUNCT
ijassa-1172	312	7	linka	linka	PROPN
ijassa-1172	312	8	,	,	PUNCT
ijassa-1172	312	9	k.	k.	PROPN
ijassa-1172	312	10	,	,	PUNCT
ijassa-1172	312	11	costable	costable	PROPN
ijassa-1172	312	12	,	,	PUNCT
ijassa-1172	312	13	f.	f.	PROPN
ijassa-1172	312	14	s.	s.	PROPN
ijassa-1172	312	15	,	,	PUNCT
ijassa-1172	312	16	bhattacharya	bhattacharya	PROPN
ijassa-1172	312	17	,	,	PUNCT
ijassa-1172	312	18	j.	j.	PROPN
ijassa-1172	312	19	,	,	PUNCT
ijassa-1172	312	20	bendavid	bendavid	PROPN
ijassa-1172	312	21	,	,	PUNCT
ijassa-1172	312	22	e.	e.	PROPN
ijassa-1172	312	23	et	et	PROPN
ijassa-1172	312	24	al	al	PROPN
ijassa-1172	312	25	.	.	PROPN
ijassa-1172	313	1	(	(	PUNCT
ijassa-1172	313	2	2020	2020	NUM
ijassa-1172	313	3	)	)	PUNCT
ijassa-1172	313	4	visualizing	visualize	VERB
ijassa-1172	313	5	the	the	DET
ijassa-1172	313	6	invisible	invisible	ADJ
ijassa-1172	313	7	:	:	PUNCT
ijassa-1172	313	8	the	the	DET
ijassa-1172	313	9	effect	effect	NOUN
ijassa-1172	313	10	of	of	ADP
ijassa-1172	313	11	asymptomatic	asymptomatic	ADJ
ijassa-1172	313	12	transmission	transmission	NOUN
ijassa-1172	313	13	on	on	ADP
ijassa-1172	313	14	the	the	DET
ijassa-1172	313	15	outbreak	outbreak	NOUN
ijassa-1172	313	16	dynamics	dynamic	NOUN
ijassa-1172	313	17	of	of	ADP
ijassa-1172	313	18	covid-19	covid-19	PROPN
ijassa-1172	313	19	.	.	PUNCT
ijassa-1172	313	20	comput	comput	PROPN
ijassa-1172	313	21	.	.	PUNCT
ijassa-1172	314	1	methods	method	NOUN
ijassa-1172	314	2	appl	appl	PROPN
ijassa-1172	314	3	.	.	PROPN
ijassa-1172	314	4	mech	mech	PROPN
ijassa-1172	314	5	.	.	PUNCT
ijassa-1172	315	1	engrg	engrg	PROPN
ijassa-1172	315	2	.	.	PUNCT
ijassa-1172	316	1	j.	j.	PROPN
ijassa-1172	316	2	,	,	PUNCT
ijassa-1172	316	3	372	372	NUM
ijassa-1172	316	4	.	.	NOUN
ijassa-1172	316	5	16	16	NUM
ijassa-1172	316	6	.	.	PUNCT
ijassa-1172	317	1	huo	huo	PROPN
ijassa-1172	317	2	,	,	PUNCT
ijassa-1172	317	3	j.	j.	PROPN
ijassa-1172	317	4	,	,	PUNCT
ijassa-1172	317	5	zhao	zhao	PROPN
ijassa-1172	317	6	,	,	PUNCT
ijassa-1172	317	7	h.	h.	PROPN
ijassa-1172	317	8	,	,	PUNCT
ijassa-1172	317	9	&	&	CCONJ
ijassa-1172	317	10	zhu	zhu	PROPN
ijassa-1172	317	11	,	,	PUNCT
ijassa-1172	317	12	l.	l.	PROPN
ijassa-1172	317	13	(	(	PUNCT
ijassa-1172	317	14	2015	2015	NUM
ijassa-1172	317	15	)	)	PUNCT
ijassa-1172	317	16	the	the	DET
ijassa-1172	317	17	effect	effect	NOUN
ijassa-1172	317	18	of	of	ADP
ijassa-1172	317	19	vaccines	vaccine	NOUN
ijassa-1172	317	20	on	on	ADP
ijassa-1172	317	21	backward	backward	ADJ
ijassa-1172	317	22	bifurcation	bifurcation	NOUN
ijassa-1172	317	23	in	in	ADP
ijassa-1172	317	24	a	a	DET
ijassa-1172	317	25	fractional	fractional	ADJ
ijassa-1172	317	26	order	order	NOUN
ijassa-1172	317	27	hiv	hiv	PROPN
ijassa-1172	317	28	model	model	PROPN
ijassa-1172	317	29	.	.	PUNCT
ijassa-1172	318	1	nonlinear	nonlinear	ADJ
ijassa-1172	318	2	analysis	analysis	NOUN
ijassa-1172	318	3	:	:	PUNCT
ijassa-1172	318	4	real	real	ADJ
ijassa-1172	318	5	world	world	NOUN
ijassa-1172	318	6	applications	application	NOUN
ijassa-1172	318	7	,	,	PUNCT
ijassa-1172	318	8	26	26	NUM
ijassa-1172	318	9	,	,	PUNCT
ijassa-1172	318	10	289–305	289–305	NUM
ijassa-1172	318	11	.	.	PUNCT
ijassa-1172	319	1	17	17	NUM
ijassa-1172	319	2	.	.	X
ijassa-1172	319	3	vargas	vargas	NOUN
ijassa-1172	319	4	-	-	PUNCT
ijassa-1172	319	5	de	de	NOUN
ijassa-1172	319	6	-	-	NOUN
ijassa-1172	319	7	león	león	PROPN
ijassa-1172	319	8	,	,	PUNCT
ijassa-1172	319	9	c.	c.	NOUN
ijassa-1172	319	10	(	(	PUNCT
ijassa-1172	319	11	2015	2015	NUM
ijassa-1172	319	12	)	)	PUNCT
ijassa-1172	319	13	volterra	volterra	NOUN
ijassa-1172	319	14	-	-	PUNCT
ijassa-1172	319	15	type	type	NOUN
ijassa-1172	319	16	lyapunov	lyapunov	NOUN
ijassa-1172	319	17	functions	function	NOUN
ijassa-1172	319	18	for	for	ADP
ijassa-1172	319	19	fractional	fractional	ADJ
ijassa-1172	319	20	-	-	PUNCT
ijassa-1172	319	21	order	order	NOUN
ijassa-1172	319	22	epidemic	epidemic	NOUN
ijassa-1172	319	23	systems	system	NOUN
ijassa-1172	319	24	.	.	PUNCT
ijassa-1172	320	1	commun	commun	PROPN
ijassa-1172	320	2	.	.	PUNCT
ijassa-1172	321	1	nonlinear	nonlinear	PROPN
ijassa-1172	321	2	.	.	PUNCT
ijassa-1172	322	1	sci	sci	PROPN
ijassa-1172	322	2	.	.	PUNCT
ijassa-1172	322	3	numer	numer	PROPN
ijassa-1172	322	4	.	.	PUNCT
ijassa-1172	323	1	simulat	simulat	PROPN
ijassa-1172	323	2	.	.	PUNCT
ijassa-1172	324	1	,	,	PUNCT
ijassa-1172	324	2	24	24	NUM
ijassa-1172	324	3	,	,	PUNCT
ijassa-1172	324	4	75–85	75–85	NUM
ijassa-1172	324	5	.	.	PUNCT
ijassa-1172	325	1	18	18	NUM
ijassa-1172	325	2	.	.	PUNCT
ijassa-1172	325	3	world	world	PROPN
ijassa-1172	325	4	health	health	PROPN
ijassa-1172	325	5	organization	organization	NOUN
ijassa-1172	325	6	(	(	PUNCT
ijassa-1172	325	7	who	who	PRON
ijassa-1172	325	8	)	)	PUNCT
ijassa-1172	325	9	(	(	PUNCT
ijassa-1172	325	10	2021	2021	NUM
ijassa-1172	325	11	,	,	PUNCT
ijassa-1172	325	12	march	march	PROPN
ijassa-1172	325	13	)	)	PUNCT
ijassa-1172	325	14	.	.	PUNCT
ijassa-1172	326	1	weekly	weekly	ADJ
ijassa-1172	326	2	epidemiological	epidemiological	ADJ
ijassa-1172	326	3	update	update	NOUN
ijassa-1172	326	4	.	.	PUNCT
ijassa-1172	327	1	[	[	X
ijassa-1172	327	2	online	online	X
ijassa-1172	327	3	]	]	X
ijassa-1172	327	4	.	.	PUNCT
ijassa-1172	328	1	available	available	ADJ
ijassa-1172	328	2	https://www.who.int/docs/defaultsource/coronaviruse/situation-reports/20210309	https://www.who.int/docs/defaultsource/coronaviruse/situation-reports/20210309	PROPN
ijassa-1172	328	3	.	.	PUNCT
ijassa-1172	329	1	19	19	NUM
ijassa-1172	329	2	.	.	X
ijassa-1172	329	3	worldometer	worldometer	PROPN
ijassa-1172	329	4	,	,	PUNCT
ijassa-1172	329	5	covid-19	covid-19	PROPN
ijassa-1172	329	6	coronavirus	coronavirus	NOUN
ijassa-1172	329	7	pandemic	pandemic	NOUN
ijassa-1172	329	8	.	.	PUNCT
ijassa-1172	330	1	[	[	X
ijassa-1172	330	2	online	online	X
ijassa-1172	330	3	]	]	X
ijassa-1172	330	4	.	.	PUNCT
ijassa-1172	331	1	available	available	ADJ
ijassa-1172	331	2	https://www.worldometers.info/coronavirus/country/iraq/.	https://www.worldometers.info/coronavirus/country/iraq/.	PROPN
ijassa-1172	331	3	copyright	copyright	NOUN
ijassa-1172	331	4	©	©	PROPN
ijassa-1172	331	5	2022	2022	NUM
ijassa-1172	331	6	assa	assa	NOUN
ijassa-1172	331	7	.	.	PUNCT
ijassa-1172	332	1	adv	adv	PROPN
ijassa-1172	332	2	syst	syst	PROPN
ijassa-1172	332	3	sci	sci	PROPN
ijassa-1172	332	4	appl	appl	PROPN
ijassa-1172	332	5	(	(	PUNCT
ijassa-1172	332	6	2022	2022	NUM
ijassa-1172	332	7	)	)	PUNCT
ijassa-1172	332	8	introduction	introduction	NOUN
ijassa-1172	332	9	model	model	NOUN
ijassa-1172	332	10	formulation	formulation	NOUN
ijassa-1172	332	11	and	and	CCONJ
ijassa-1172	332	12	equilibria	equilibrium	NOUN
ijassa-1172	332	13	global	global	ADJ
ijassa-1172	332	14	dynamics	dynamic	NOUN
ijassa-1172	332	15	applications	application	NOUN
ijassa-1172	332	16	and	and	CCONJ
ijassa-1172	332	17	numerical	numerical	ADJ
ijassa-1172	332	18	simulations	simulation	NOUN
ijassa-1172	332	19	conclusion	conclusion	NOUN