id	sid	tid	token	lemma	pos
cana-4839	1	1	communications	communication	NOUN
cana-4839	1	2	on	on	ADP
cana-4839	1	3	applied	apply	VERB
cana-4839	1	4	nonlinear	nonlinear	ADJ
cana-4839	1	5	analysis	analysis	NOUN
cana-4839	1	6	issn	issn	NOUN
cana-4839	1	7	:	:	PUNCT
cana-4839	1	8	1074	1074	NUM
cana-4839	1	9	-	-	PUNCT
cana-4839	1	10	133x	133x	NUM
cana-4839	1	11	vol	vol	VERB
cana-4839	1	12	32	32	NUM
cana-4839	1	13	no	no	NOUN
cana-4839	1	14	.	.	PUNCT
cana-4839	2	1	10s	10	NOUN
cana-4839	2	2	(	(	PUNCT
cana-4839	2	3	2025	2025	NUM
cana-4839	2	4	)	)	PUNCT
cana-4839	2	5	437	437	NUM
cana-4839	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	2	7	exploring	explore	VERB
cana-4839	2	8	bifurcation	bifurcation	NOUN
cana-4839	2	9	theory	theory	NOUN
cana-4839	2	10	in	in	ADP
cana-4839	2	11	epidemiological	epidemiological	ADJ
cana-4839	2	12	models	model	NOUN
cana-4839	2	13	:	:	PUNCT
cana-4839	2	14	a	a	DET
cana-4839	2	15	case	case	NOUN
cana-4839	2	16	study	study	NOUN
cana-4839	2	17	on	on	ADP
cana-4839	2	18	covid-19	covid-19	PROPN
cana-4839	2	19	spread	spread	VERB
cana-4839	2	20	1anant	1anant	NUM
cana-4839	2	21	manish	manish	PROPN
cana-4839	2	22	singh	singh	PROPN
cana-4839	2	23	(	(	PUNCT
cana-4839	2	24	corresponding	corresponding	ADJ
cana-4839	2	25	author	author	NOUN
cana-4839	2	26	)	)	PUNCT
cana-4839	2	27	,	,	PUNCT
cana-4839	2	28	2krishna	2krishna	NUM
cana-4839	2	29	jitendra	jitendra	PROPN
cana-4839	2	30	jaiswal	jaiswal	PROPN
cana-4839	2	31	,	,	PUNCT
cana-4839	2	32	3arya	3arya	NUM
cana-4839	2	33	brijesh	brijesh	PROPN
cana-4839	2	34	tiwari	tiwari	NOUN
cana-4839	2	35	,	,	PUNCT
cana-4839	2	36	4shifa	4shifa	NUM
cana-4839	2	37	siraj	siraj	PROPN
cana-4839	2	38	khan	khan	PROPN
cana-4839	2	39	,	,	PUNCT
cana-4839	2	40	5sanika	5sanika	NUM
cana-4839	2	41	satish	satish	ADJ
cana-4839	2	42	lad	lad	NOUN
cana-4839	2	43	,	,	PUNCT
cana-4839	2	44	6darshit	6darshit	NUM
cana-4839	2	45	sandeep	sandeep	PROPN
cana-4839	2	46	raut	raut	PROPN
cana-4839	2	47	,	,	PUNCT
cana-4839	2	48	7sanika	7sanika	NUM
cana-4839	2	49	rajan	rajan	PROPN
cana-4839	2	50	shete	shete	PROPN
cana-4839	2	51	,	,	PUNCT
cana-4839	2	52	8disha	8disha	NUM
cana-4839	2	53	satyan	satyan	PROPN
cana-4839	2	54	dahanukar	dahanukar	PROPN
cana-4839	2	55	1department	1department	NUM
cana-4839	2	56	of	of	ADP
cana-4839	2	57	computer	computer	NOUN
cana-4839	2	58	engineering	engineering	NOUN
cana-4839	2	59	,	,	PUNCT
cana-4839	2	60	thakur	thakur	PROPN
cana-4839	2	61	college	college	PROPN
cana-4839	2	62	of	of	ADP
cana-4839	2	63	engineering	engineering	NOUN
cana-4839	2	64	and	and	CCONJ
cana-4839	2	65	technology	technology	NOUN
cana-4839	2	66	(	(	PUNCT
cana-4839	2	67	tcet	tcet	PROPN
cana-4839	2	68	)	)	PUNCT
cana-4839	2	69	,	,	PUNCT
cana-4839	2	70	mumbai	mumbai	PROPN
cana-4839	2	71	,	,	PUNCT
cana-4839	2	72	maharashtra	maharashtra	PROPN
cana-4839	2	73	,	,	PUNCT
cana-4839	2	74	india	india	PROPN
cana-4839	2	75	anantsingh1302@gmail.com	anantsingh1302@gmail.com	PUNCT
cana-4839	3	1	2department	2department	NUM
cana-4839	3	2	of	of	ADP
cana-4839	3	3	computer	computer	NOUN
cana-4839	3	4	engineering	engineering	NOUN
cana-4839	3	5	,	,	PUNCT
cana-4839	3	6	thakur	thakur	PROPN
cana-4839	3	7	college	college	PROPN
cana-4839	3	8	of	of	ADP
cana-4839	3	9	engineering	engineering	NOUN
cana-4839	3	10	and	and	CCONJ
cana-4839	3	11	technology	technology	NOUN
cana-4839	3	12	(	(	PUNCT
cana-4839	3	13	tcet	tcet	PROPN
cana-4839	3	14	)	)	PUNCT
cana-4839	3	15	,	,	PUNCT
cana-4839	3	16	mumbai	mumbai	PROPN
cana-4839	3	17	,	,	PUNCT
cana-4839	3	18	maharashtra	maharashtra	PROPN
cana-4839	3	19	,	,	PUNCT
cana-4839	3	20	india	india	PROPN
cana-4839	3	21	krishnajaiswal2512@gmail.com	krishnajaiswal2512@gmail.com	PROPN
cana-4839	4	1	3department	3department	NUM
cana-4839	4	2	of	of	ADP
cana-4839	4	3	computer	computer	NOUN
cana-4839	4	4	engineering	engineering	NOUN
cana-4839	4	5	,	,	PUNCT
cana-4839	4	6	thakur	thakur	PROPN
cana-4839	4	7	college	college	PROPN
cana-4839	4	8	of	of	ADP
cana-4839	4	9	engineering	engineering	NOUN
cana-4839	4	10	and	and	CCONJ
cana-4839	4	11	technology	technology	NOUN
cana-4839	4	12	(	(	PUNCT
cana-4839	4	13	tcet	tcet	PROPN
cana-4839	4	14	)	)	PUNCT
cana-4839	4	15	,	,	PUNCT
cana-4839	4	16	mumbai	mumbai	PROPN
cana-4839	4	17	,	,	PUNCT
cana-4839	4	18	maharashtra	maharashtra	PROPN
cana-4839	4	19	,	,	PUNCT
cana-4839	4	20	india	india	PROPN
cana-4839	4	21	aryabbrijeshtiwarii@gmail.com	aryabbrijeshtiwarii@gmail.com	PROPN
cana-4839	5	1	4department	4department	NUM
cana-4839	5	2	of	of	ADP
cana-4839	5	3	information	information	NOUN
cana-4839	5	4	technology	technology	NOUN
cana-4839	5	5	,	,	PUNCT
cana-4839	5	6	thakur	thakur	PROPN
cana-4839	5	7	college	college	PROPN
cana-4839	5	8	of	of	ADP
cana-4839	5	9	engineering	engineering	NOUN
cana-4839	5	10	and	and	CCONJ
cana-4839	5	11	technology	technology	NOUN
cana-4839	5	12	(	(	PUNCT
cana-4839	5	13	tcet	tcet	PROPN
cana-4839	5	14	)	)	PUNCT
cana-4839	5	15	,	,	PUNCT
cana-4839	5	16	mumbai	mumbai	PROPN
cana-4839	5	17	,	,	PUNCT
cana-4839	5	18	maharashtra	maharashtra	PROPN
cana-4839	5	19	,	,	PUNCT
cana-4839	5	20	india	india	PROPN
cana-4839	5	21	shifakhan.work@gmail.com	shifakhan.work@gmail.com	X
cana-4839	5	22	5department	5department	NUM
cana-4839	5	23	of	of	ADP
cana-4839	5	24	computer	computer	NOUN
cana-4839	5	25	engineering	engineering	NOUN
cana-4839	5	26	,	,	PUNCT
cana-4839	5	27	thakur	thakur	PROPN
cana-4839	5	28	college	college	PROPN
cana-4839	5	29	of	of	ADP
cana-4839	5	30	engineering	engineering	NOUN
cana-4839	5	31	and	and	CCONJ
cana-4839	5	32	technology	technology	NOUN
cana-4839	5	33	(	(	PUNCT
cana-4839	5	34	tcet	tcet	PROPN
cana-4839	5	35	)	)	PUNCT
cana-4839	5	36	,	,	PUNCT
cana-4839	5	37	mumbai	mumbai	PROPN
cana-4839	5	38	,	,	PUNCT
cana-4839	5	39	maharashtra	maharashtra	PROPN
cana-4839	5	40	,	,	PUNCT
cana-4839	5	41	india	india	PROPN
cana-4839	5	42	ladsanika01@gmail.com	ladsanika01@gmail.com	X
cana-4839	6	1	6department	6department	NUM
cana-4839	6	2	of	of	ADP
cana-4839	6	3	electronics	electronic	NOUN
cana-4839	6	4	&	&	CCONJ
cana-4839	6	5	telecommunication	telecommunication	NOUN
cana-4839	6	6	engineering(extc	engineering(extc	PROPN
cana-4839	6	7	)	)	PUNCT
cana-4839	6	8	,	,	PUNCT
cana-4839	6	9	thakur	thakur	PROPN
cana-4839	6	10	college	college	PROPN
cana-4839	6	11	of	of	ADP
cana-4839	6	12	engineering	engineering	NOUN
cana-4839	6	13	and	and	CCONJ
cana-4839	6	14	technology	technology	NOUN
cana-4839	6	15	(	(	PUNCT
cana-4839	6	16	tcet	tcet	PROPN
cana-4839	6	17	)	)	PUNCT
cana-4839	6	18	,	,	PUNCT
cana-4839	6	19	mumbai	mumbai	PROPN
cana-4839	6	20	,	,	PUNCT
cana-4839	6	21	maharashtra	maharashtra	PROPN
cana-4839	6	22	,	,	PUNCT
cana-4839	6	23	india	india	PROPN
cana-4839	6	24	darshitraut28@gmail.com	darshitraut28@gmail.com	X
cana-4839	7	1	7department	7department	NUM
cana-4839	7	2	of	of	ADP
cana-4839	7	3	computer	computer	NOUN
cana-4839	7	4	engineering	engineering	NOUN
cana-4839	7	5	,	,	PUNCT
cana-4839	7	6	thakur	thakur	PROPN
cana-4839	7	7	college	college	PROPN
cana-4839	7	8	of	of	ADP
cana-4839	7	9	engineering	engineering	NOUN
cana-4839	7	10	and	and	CCONJ
cana-4839	7	11	technology	technology	NOUN
cana-4839	7	12	(	(	PUNCT
cana-4839	7	13	tcet	tcet	PROPN
cana-4839	7	14	)	)	PUNCT
cana-4839	7	15	,	,	PUNCT
cana-4839	7	16	mumbai	mumbai	PROPN
cana-4839	7	17	,	,	PUNCT
cana-4839	7	18	maharashtra	maharashtra	PROPN
cana-4839	7	19	,	,	PUNCT
cana-4839	7	20	india	india	PROPN
cana-4839	7	21	sanika.shetee@gmail.com	sanika.shetee@gmail.com	X
cana-4839	7	22	8department	8department	NUM
cana-4839	7	23	of	of	ADP
cana-4839	7	24	computer	computer	NOUN
cana-4839	7	25	engineering	engineering	NOUN
cana-4839	7	26	,	,	PUNCT
cana-4839	7	27	thakur	thakur	PROPN
cana-4839	7	28	college	college	PROPN
cana-4839	7	29	of	of	ADP
cana-4839	7	30	engineering	engineering	NOUN
cana-4839	7	31	and	and	CCONJ
cana-4839	7	32	technology	technology	NOUN
cana-4839	7	33	(	(	PUNCT
cana-4839	7	34	tcet	tcet	PROPN
cana-4839	7	35	)	)	PUNCT
cana-4839	7	36	,	,	PUNCT
cana-4839	7	37	mumbai	mumbai	PROPN
cana-4839	7	38	,	,	PUNCT
cana-4839	7	39	maharashtra	maharashtra	PROPN
cana-4839	7	40	,	,	PUNCT
cana-4839	7	41	india	india	PROPN
cana-4839	7	42	dishadahanukar@gmail.com	dishadahanukar@gmail.com	PROPN
cana-4839	7	43	article	article	NOUN
cana-4839	7	44	history	history	NOUN
cana-4839	7	45	:	:	PUNCT
cana-4839	7	46	received	receive	VERB
cana-4839	7	47	:	:	PUNCT
cana-4839	7	48	12	12	NUM
cana-4839	7	49	-	-	SYM
cana-4839	7	50	01	01	NUM
cana-4839	7	51	-	-	PUNCT
cana-4839	7	52	2025	2025	NUM
cana-4839	7	53	revised	revise	VERB
cana-4839	7	54	:	:	PUNCT
cana-4839	7	55	15	15	NUM
cana-4839	7	56	-	-	NUM
cana-4839	7	57	02	02	NUM
cana-4839	7	58	-	-	PUNCT
cana-4839	7	59	2025	2025	NUM
cana-4839	7	60	accepted	accept	VERB
cana-4839	7	61	:	:	PUNCT
cana-4839	7	62	01	01	NUM
cana-4839	7	63	-	-	SYM
cana-4839	7	64	03	03	NUM
cana-4839	7	65	-	-	PUNCT
cana-4839	7	66	2025	2025	NUM
cana-4839	7	67	abstract	abstract	NOUN
cana-4839	7	68	:	:	PUNCT
cana-4839	7	69	bifurcation	bifurcation	NOUN
cana-4839	7	70	theory	theory	NOUN
cana-4839	7	71	plays	play	VERB
cana-4839	7	72	a	a	DET
cana-4839	7	73	crucial	crucial	ADJ
cana-4839	7	74	role	role	NOUN
cana-4839	7	75	in	in	ADP
cana-4839	7	76	understanding	understand	VERB
cana-4839	7	77	the	the	DET
cana-4839	7	78	nonlinear	nonlinear	ADJ
cana-4839	7	79	dynamics	dynamic	NOUN
cana-4839	7	80	of	of	ADP
cana-4839	7	81	infectious	infectious	ADJ
cana-4839	7	82	disease	disease	NOUN
cana-4839	7	83	transmission	transmission	NOUN
cana-4839	7	84	particularly	particularly	ADV
cana-4839	7	85	in	in	ADP
cana-4839	7	86	the	the	DET
cana-4839	7	87	context	context	NOUN
cana-4839	7	88	of	of	ADP
cana-4839	7	89	global	global	ADJ
cana-4839	7	90	pandemics	pandemic	NOUN
cana-4839	7	91	such	such	ADJ
cana-4839	7	92	as	as	ADP
cana-4839	7	93	covid-19	covid-19	PROPN
cana-4839	7	94	.	.	PUNCT
cana-4839	8	1	this	this	DET
cana-4839	8	2	study	study	NOUN
cana-4839	8	3	explores	explore	VERB
cana-4839	8	4	the	the	DET
cana-4839	8	5	application	application	NOUN
cana-4839	8	6	of	of	ADP
cana-4839	8	7	bifurcation	bifurcation	NOUN
cana-4839	8	8	analysis	analysis	NOUN
cana-4839	8	9	to	to	ADP
cana-4839	8	10	epidemiological	epidemiological	ADJ
cana-4839	8	11	models	model	NOUN
cana-4839	8	12	specifically	specifically	ADV
cana-4839	8	13	the	the	DET
cana-4839	8	14	susceptible	susceptible	ADJ
cana-4839	8	15	-	-	PUNCT
cana-4839	8	16	infected	infect	VERB
cana-4839	8	17	-	-	PUNCT
cana-4839	8	18	recovered	recover	VERB
cana-4839	8	19	-	-	PUNCT
cana-4839	8	20	susceptible	susceptible	ADJ
cana-4839	8	21	(	(	PUNCT
cana-4839	8	22	sirs	sir	NOUN
cana-4839	8	23	)	)	PUNCT
cana-4839	8	24	framework	framework	NOUN
cana-4839	8	25	to	to	PART
cana-4839	8	26	identify	identify	VERB
cana-4839	8	27	critical	critical	ADJ
cana-4839	8	28	threshold	threshold	NOUN
cana-4839	8	29	conditions	condition	NOUN
cana-4839	8	30	that	that	PRON
cana-4839	8	31	govern	govern	VERB
cana-4839	8	32	disease	disease	NOUN
cana-4839	8	33	spread	spread	VERB
cana-4839	8	34	.	.	PUNCT
cana-4839	9	1	existing	exist	VERB
cana-4839	9	2	epidemiological	epidemiological	ADJ
cana-4839	9	3	models	model	NOUN
cana-4839	9	4	often	often	ADV
cana-4839	9	5	rely	rely	VERB
cana-4839	9	6	on	on	ADP
cana-4839	9	7	deterministic	deterministic	ADJ
cana-4839	9	8	or	or	CCONJ
cana-4839	9	9	stochastic	stochastic	ADJ
cana-4839	9	10	approaches	approach	NOUN
cana-4839	9	11	but	but	CCONJ
cana-4839	9	12	lack	lack	VERB
cana-4839	9	13	a	a	DET
cana-4839	9	14	mailto:anantsingh1302@gmail.com	mailto:anantsingh1302@gmail.com	NUM
cana-4839	9	15	mailto:krishnajaiswal2512@gmail.com	mailto:krishnajaiswal2512@gmail.com	NOUN
cana-4839	9	16	mailto:aryabbrijeshtiwarii@gmail.com	mailto:aryabbrijeshtiwarii@gmail.com	PROPN
cana-4839	9	17	mailto:shifakhan.work@gmail.com	mailto:shifakhan.work@gmail.com	X
cana-4839	9	18	mailto:ladsanika01@gmail.com	mailto:ladsanika01@gmail.com	PROPN
cana-4839	9	19	mailto:darshitraut28@gmail.com	mailto:darshitraut28@gmail.com	PROPN
cana-4839	9	20	mailto:sanika.shetee@gmail.com	mailto:sanika.shetee@gmail.com	PROPN
cana-4839	10	1	mailto:dishadahanukar@gmail.com	mailto:dishadahanukar@gmail.com	PROPN
cana-4839	10	2	communications	communication	NOUN
cana-4839	10	3	on	on	ADP
cana-4839	10	4	applied	apply	VERB
cana-4839	10	5	nonlinear	nonlinear	ADJ
cana-4839	10	6	analysis	analysis	NOUN
cana-4839	10	7	issn	issn	NOUN
cana-4839	10	8	:	:	PUNCT
cana-4839	10	9	1074	1074	NUM
cana-4839	10	10	-	-	PUNCT
cana-4839	10	11	133x	133x	NUM
cana-4839	10	12	vol	vol	VERB
cana-4839	10	13	32	32	NUM
cana-4839	10	14	no	no	NOUN
cana-4839	10	15	.	.	PUNCT
cana-4839	11	1	10s	10	NOUN
cana-4839	11	2	(	(	PUNCT
cana-4839	11	3	2025	2025	NUM
cana-4839	11	4	)	)	PUNCT
cana-4839	11	5	438	438	NUM
cana-4839	11	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	11	7	systematic	systematic	ADJ
cana-4839	11	8	bifurcation	bifurcation	NOUN
cana-4839	11	9	analysis	analysis	NOUN
cana-4839	11	10	that	that	SCONJ
cana-4839	11	11	explicitly	explicitly	ADV
cana-4839	11	12	captures	capture	VERB
cana-4839	11	13	transitions	transition	NOUN
cana-4839	11	14	between	between	ADP
cana-4839	11	15	controlled	control	VERB
cana-4839	11	16	outbreaks	outbreak	NOUN
cana-4839	11	17	and	and	CCONJ
cana-4839	11	18	widespread	widespread	ADJ
cana-4839	11	19	pandemics	pandemic	NOUN
cana-4839	11	20	.	.	PUNCT
cana-4839	12	1	this	this	DET
cana-4839	12	2	research	research	NOUN
cana-4839	12	3	bridges	bridge	NOUN
cana-4839	12	4	this	this	DET
cana-4839	12	5	gap	gap	NOUN
cana-4839	12	6	by	by	ADP
cana-4839	12	7	conducting	conduct	VERB
cana-4839	12	8	a	a	DET
cana-4839	12	9	comprehensive	comprehensive	ADJ
cana-4839	12	10	stability	stability	NOUN
cana-4839	12	11	analysis	analysis	NOUN
cana-4839	12	12	of	of	ADP
cana-4839	12	13	equilibrium	equilibrium	NOUN
cana-4839	12	14	states	state	NOUN
cana-4839	12	15	,	,	PUNCT
cana-4839	12	16	examining	examine	VERB
cana-4839	12	17	the	the	DET
cana-4839	12	18	impact	impact	NOUN
cana-4839	12	19	of	of	ADP
cana-4839	12	20	key	key	ADJ
cana-4839	12	21	parameters	parameter	NOUN
cana-4839	12	22	such	such	ADJ
cana-4839	12	23	as	as	ADP
cana-4839	12	24	the	the	DET
cana-4839	12	25	recruitment	recruitment	NOUN
cana-4839	12	26	rate	rate	NOUN
cana-4839	12	27	and	and	CCONJ
cana-4839	12	28	the	the	DET
cana-4839	12	29	basic	basic	ADJ
cana-4839	12	30	reproduction	reproduction	NOUN
cana-4839	12	31	number	number	NOUN
cana-4839	12	32	(	(	PUNCT
cana-4839	12	33	r₀	r₀	NOUN
cana-4839	12	34	)	)	PUNCT
cana-4839	12	35	.	.	PUNCT
cana-4839	13	1	by	by	ADP
cana-4839	13	2	integrating	integrate	VERB
cana-4839	13	3	bifurcation	bifurcation	NOUN
cana-4839	13	4	theory	theory	NOUN
cana-4839	13	5	with	with	ADP
cana-4839	13	6	real	real	ADJ
cana-4839	13	7	-	-	PUNCT
cana-4839	13	8	world	world	NOUN
cana-4839	13	9	data	datum	NOUN
cana-4839	13	10	,	,	PUNCT
cana-4839	13	11	we	we	PRON
cana-4839	13	12	demonstrate	demonstrate	VERB
cana-4839	13	13	how	how	SCONJ
cana-4839	13	14	small	small	ADJ
cana-4839	13	15	changes	change	NOUN
cana-4839	13	16	in	in	ADP
cana-4839	13	17	public	public	ADJ
cana-4839	13	18	health	health	NOUN
cana-4839	13	19	policies	policy	NOUN
cana-4839	13	20	such	such	ADJ
cana-4839	13	21	as	as	ADP
cana-4839	13	22	vaccination	vaccination	NOUN
cana-4839	13	23	strategies	strategy	NOUN
cana-4839	13	24	and	and	CCONJ
cana-4839	13	25	non	non	ADJ
cana-4839	13	26	-	-	ADJ
cana-4839	13	27	pharmaceutical	pharmaceutical	ADJ
cana-4839	13	28	interventions	intervention	NOUN
cana-4839	13	29	,	,	PUNCT
cana-4839	13	30	can	can	AUX
cana-4839	13	31	induce	induce	VERB
cana-4839	13	32	significant	significant	ADJ
cana-4839	13	33	shifts	shift	NOUN
cana-4839	13	34	in	in	ADP
cana-4839	13	35	epidemic	epidemic	NOUN
cana-4839	13	36	trajectories	trajectory	NOUN
cana-4839	13	37	.	.	PUNCT
cana-4839	14	1	the	the	DET
cana-4839	14	2	findings	finding	NOUN
cana-4839	14	3	reveal	reveal	VERB
cana-4839	14	4	that	that	SCONJ
cana-4839	14	5	timely	timely	ADJ
cana-4839	14	6	interventions	intervention	NOUN
cana-4839	14	7	,	,	PUNCT
cana-4839	14	8	guided	guide	VERB
cana-4839	14	9	by	by	ADP
cana-4839	14	10	bifurcation	bifurcation	NOUN
cana-4839	14	11	thresholds	threshold	NOUN
cana-4839	14	12	could	could	AUX
cana-4839	14	13	have	have	AUX
cana-4839	14	14	substantially	substantially	ADV
cana-4839	14	15	altered	alter	VERB
cana-4839	14	16	the	the	DET
cana-4839	14	17	course	course	NOUN
cana-4839	14	18	of	of	ADP
cana-4839	14	19	the	the	DET
cana-4839	14	20	covid-19	covid-19	PROPN
cana-4839	14	21	pandemic	pandemic	NOUN
cana-4839	14	22	,	,	PUNCT
cana-4839	14	23	reinforcing	reinforce	VERB
cana-4839	14	24	the	the	DET
cana-4839	14	25	necessity	necessity	NOUN
cana-4839	14	26	of	of	ADP
cana-4839	14	27	mathematical	mathematical	ADJ
cana-4839	14	28	modeling	modeling	NOUN
cana-4839	14	29	in	in	ADP
cana-4839	14	30	public	public	ADJ
cana-4839	14	31	health	health	NOUN
cana-4839	14	32	decision	decision	NOUN
cana-4839	14	33	-	-	PUNCT
cana-4839	14	34	making	making	NOUN
cana-4839	14	35	.	.	PUNCT
cana-4839	15	1	this	this	DET
cana-4839	15	2	study	study	NOUN
cana-4839	15	3	also	also	ADV
cana-4839	15	4	highlights	highlight	VERB
cana-4839	15	5	the	the	DET
cana-4839	15	6	challenges	challenge	NOUN
cana-4839	15	7	in	in	ADP
cana-4839	15	8	existing	exist	VERB
cana-4839	15	9	methodologies	methodology	NOUN
cana-4839	15	10	including	include	VERB
cana-4839	15	11	the	the	DET
cana-4839	15	12	limitations	limitation	NOUN
cana-4839	15	13	of	of	ADP
cana-4839	15	14	homogeneous	homogeneous	ADJ
cana-4839	15	15	modeling	modeling	NOUN
cana-4839	15	16	approaches	approach	NOUN
cana-4839	15	17	that	that	PRON
cana-4839	15	18	fail	fail	VERB
cana-4839	15	19	to	to	PART
cana-4839	15	20	account	account	VERB
cana-4839	15	21	for	for	ADP
cana-4839	15	22	population	population	NOUN
cana-4839	15	23	heterogeneity	heterogeneity	NOUN
cana-4839	15	24	and	and	CCONJ
cana-4839	15	25	policy	policy	NOUN
cana-4839	15	26	compliance	compliance	NOUN
cana-4839	15	27	variations	variation	NOUN
cana-4839	15	28	.	.	PUNCT
cana-4839	16	1	our	our	PRON
cana-4839	16	2	methodology	methodology	NOUN
cana-4839	16	3	employs	employ	VERB
cana-4839	16	4	numerical	numerical	ADJ
cana-4839	16	5	bifurcation	bifurcation	NOUN
cana-4839	16	6	analysis	analysis	NOUN
cana-4839	16	7	,	,	PUNCT
cana-4839	16	8	phase	phase	NOUN
cana-4839	16	9	plane	plane	NOUN
cana-4839	16	10	analysis	analysis	NOUN
cana-4839	16	11	and	and	CCONJ
cana-4839	16	12	eigenvalue	eigenvalue	NOUN
cana-4839	16	13	computations	computation	NOUN
cana-4839	16	14	to	to	PART
cana-4839	16	15	systematically	systematically	ADV
cana-4839	16	16	explore	explore	VERB
cana-4839	16	17	stability	stability	NOUN
cana-4839	16	18	transitions	transition	NOUN
cana-4839	16	19	within	within	ADP
cana-4839	16	20	the	the	DET
cana-4839	16	21	sirs	sir	NOUN
cana-4839	16	22	framework	framework	NOUN
cana-4839	16	23	.	.	PUNCT
cana-4839	17	1	the	the	DET
cana-4839	17	2	practical	practical	ADJ
cana-4839	17	3	implications	implication	NOUN
cana-4839	17	4	of	of	ADP
cana-4839	17	5	this	this	DET
cana-4839	17	6	research	research	NOUN
cana-4839	17	7	extend	extend	NOUN
cana-4839	17	8	to	to	ADP
cana-4839	17	9	optimizing	optimize	VERB
cana-4839	17	10	control	control	NOUN
cana-4839	17	11	measures	measure	NOUN
cana-4839	17	12	,	,	PUNCT
cana-4839	17	13	refining	refine	VERB
cana-4839	17	14	epidemic	epidemic	NOUN
cana-4839	17	15	forecasting	forecasting	NOUN
cana-4839	17	16	and	and	CCONJ
cana-4839	17	17	informing	inform	VERB
cana-4839	17	18	global	global	ADJ
cana-4839	17	19	health	health	NOUN
cana-4839	17	20	strategies	strategy	NOUN
cana-4839	17	21	for	for	ADP
cana-4839	17	22	future	future	ADJ
cana-4839	17	23	outbreaks	outbreak	NOUN
cana-4839	17	24	.	.	PUNCT
cana-4839	18	1	by	by	ADP
cana-4839	18	2	advancing	advance	VERB
cana-4839	18	3	the	the	DET
cana-4839	18	4	theoretical	theoretical	ADJ
cana-4839	18	5	understanding	understanding	NOUN
cana-4839	18	6	of	of	ADP
cana-4839	18	7	bifurcations	bifurcation	NOUN
cana-4839	18	8	in	in	ADP
cana-4839	18	9	epidemiological	epidemiological	ADJ
cana-4839	18	10	systems	system	NOUN
cana-4839	18	11	,	,	PUNCT
cana-4839	18	12	this	this	DET
cana-4839	18	13	study	study	NOUN
cana-4839	18	14	contributes	contribute	VERB
cana-4839	18	15	to	to	ADP
cana-4839	18	16	the	the	DET
cana-4839	18	17	broader	broad	ADJ
cana-4839	18	18	field	field	NOUN
cana-4839	18	19	of	of	ADP
cana-4839	18	20	applied	apply	VERB
cana-4839	18	21	nonlinear	nonlinear	ADJ
cana-4839	18	22	analysis	analysis	NOUN
cana-4839	18	23	and	and	CCONJ
cana-4839	18	24	underscores	underscore	VERB
cana-4839	18	25	the	the	DET
cana-4839	18	26	need	need	NOUN
cana-4839	18	27	for	for	ADP
cana-4839	18	28	interdisciplinary	interdisciplinary	ADJ
cana-4839	18	29	approaches	approach	NOUN
cana-4839	18	30	in	in	ADP
cana-4839	18	31	mathematical	mathematical	ADJ
cana-4839	18	32	epidemiology	epidemiology	NOUN
cana-4839	18	33	.	.	PUNCT
cana-4839	19	1	keywords	keyword	NOUN
cana-4839	19	2	:	:	PUNCT
cana-4839	19	3	communication	communication	NOUN
cana-4839	19	4	analysis	analysis	NOUN
cana-4839	19	5	,	,	PUNCT
cana-4839	19	6	political	political	ADJ
cana-4839	19	7	communication	communication	NOUN
cana-4839	19	8	,	,	PUNCT
cana-4839	19	9	community	community	NOUN
cana-4839	19	10	aspiration	aspiration	NOUN
cana-4839	19	11	.	.	PUNCT
cana-4839	20	1	background	background	NOUN
cana-4839	20	2	bifurcation	bifurcation	NOUN
cana-4839	20	3	theory	theory	NOUN
cana-4839	20	4	plays	play	VERB
cana-4839	20	5	a	a	DET
cana-4839	20	6	crucial	crucial	ADJ
cana-4839	20	7	role	role	NOUN
cana-4839	20	8	in	in	ADP
cana-4839	20	9	understanding	understand	VERB
cana-4839	20	10	the	the	DET
cana-4839	20	11	dynamics	dynamic	NOUN
cana-4839	20	12	of	of	ADP
cana-4839	20	13	infectious	infectious	ADJ
cana-4839	20	14	disease	disease	NOUN
cana-4839	20	15	spread	spread	VERB
cana-4839	20	16	particularly	particularly	ADV
cana-4839	20	17	during	during	ADP
cana-4839	20	18	significant	significant	ADJ
cana-4839	20	19	public	public	ADJ
cana-4839	20	20	health	health	NOUN
cana-4839	20	21	events	event	NOUN
cana-4839	20	22	such	such	ADJ
cana-4839	20	23	as	as	ADP
cana-4839	20	24	the	the	DET
cana-4839	20	25	covid-19	covid-19	PROPN
cana-4839	20	26	pandemic	pandemic	NOUN
cana-4839	20	27	.	.	PUNCT
cana-4839	21	1	a	a	DET
cana-4839	21	2	bifurcation	bifurcation	NOUN
cana-4839	21	3	is	be	AUX
cana-4839	21	4	defined	define	VERB
cana-4839	21	5	as	as	ADP
cana-4839	21	6	a	a	DET
cana-4839	21	7	qualitative	qualitative	ADJ
cana-4839	21	8	or	or	CCONJ
cana-4839	21	9	quantitative	quantitative	ADJ
cana-4839	21	10	change	change	NOUN
cana-4839	21	11	in	in	ADP
cana-4839	21	12	the	the	DET
cana-4839	21	13	dynamics	dynamic	NOUN
cana-4839	21	14	of	of	ADP
cana-4839	21	15	a	a	DET
cana-4839	21	16	system	system	NOUN
cana-4839	21	17	that	that	PRON
cana-4839	21	18	occurs	occur	VERB
cana-4839	21	19	when	when	SCONJ
cana-4839	21	20	a	a	DET
cana-4839	21	21	parameter	parameter	NOUN
cana-4839	21	22	crosses	cross	VERB
cana-4839	21	23	a	a	DET
cana-4839	21	24	critical	critical	ADJ
cana-4839	21	25	threshold	threshold	NOUN
cana-4839	21	26	.	.	PUNCT
cana-4839	22	1	this	this	DET
cana-4839	22	2	phenomenon	phenomenon	NOUN
cana-4839	22	3	can	can	AUX
cana-4839	22	4	result	result	VERB
cana-4839	22	5	in	in	ADP
cana-4839	22	6	substantial	substantial	ADJ
cana-4839	22	7	shifts	shift	NOUN
cana-4839	22	8	in	in	ADP
cana-4839	22	9	the	the	DET
cana-4839	22	10	course	course	NOUN
cana-4839	22	11	of	of	ADP
cana-4839	22	12	an	an	DET
cana-4839	22	13	epidemic	epidemic	NOUN
cana-4839	22	14	,	,	PUNCT
cana-4839	22	15	transitioning	transition	VERB
cana-4839	22	16	from	from	ADP
cana-4839	22	17	a	a	DET
cana-4839	22	18	controlled	control	VERB
cana-4839	22	19	outbreak	outbreak	NOUN
cana-4839	22	20	to	to	ADP
cana-4839	22	21	a	a	DET
cana-4839	22	22	major	major	ADJ
cana-4839	22	23	pandemic	pandemic	ADJ
cana-4839	23	1	[	[	X
cana-4839	23	2	1	1	X
cana-4839	23	3	]	]	X
cana-4839	23	4	[	[	X
cana-4839	23	5	2	2	NUM
cana-4839	23	6	]	]	PUNCT
cana-4839	23	7	in	in	ADP
cana-4839	23	8	the	the	DET
cana-4839	23	9	context	context	NOUN
cana-4839	23	10	of	of	ADP
cana-4839	23	11	epidemiological	epidemiological	ADJ
cana-4839	23	12	modeling	modeling	NOUN
cana-4839	23	13	,	,	PUNCT
cana-4839	23	14	bifurcations	bifurcation	NOUN
cana-4839	23	15	are	be	AUX
cana-4839	23	16	essential	essential	ADJ
cana-4839	23	17	for	for	ADP
cana-4839	23	18	identifying	identify	VERB
cana-4839	23	19	the	the	DET
cana-4839	23	20	conditions	condition	NOUN
cana-4839	23	21	under	under	ADP
cana-4839	23	22	which	which	PRON
cana-4839	23	23	an	an	DET
cana-4839	23	24	outbreak	outbreak	NOUN
cana-4839	23	25	may	may	AUX
cana-4839	23	26	escalate	escalate	VERB
cana-4839	23	27	or	or	CCONJ
cana-4839	23	28	diminish	diminish	VERB
cana-4839	23	29	.	.	PUNCT
cana-4839	24	1	for	for	ADP
cana-4839	24	2	example	example	NOUN
cana-4839	24	3	,	,	PUNCT
cana-4839	24	4	by	by	ADP
cana-4839	24	5	studying	study	VERB
cana-4839	24	6	the	the	DET
cana-4839	24	7	bifurcation	bifurcation	NOUN
cana-4839	24	8	structures	structure	NOUN
cana-4839	24	9	of	of	ADP
cana-4839	24	10	models	model	NOUN
cana-4839	24	11	like	like	ADP
cana-4839	24	12	the	the	DET
cana-4839	24	13	sirs	sir	NOUN
cana-4839	24	14	(	(	PUNCT
cana-4839	24	15	susceptible	susceptible	ADJ
cana-4839	24	16	-	-	PUNCT
cana-4839	24	17	infected	infect	VERB
cana-4839	24	18	-	-	PUNCT
cana-4839	24	19	recovered	recover	VERB
cana-4839	24	20	-	-	PUNCT
cana-4839	24	21	susceptible	susceptible	ADJ
cana-4839	24	22	)	)	PUNCT
cana-4839	24	23	system	system	NOUN
cana-4839	24	24	,	,	PUNCT
cana-4839	24	25	researchers	researcher	NOUN
cana-4839	24	26	can	can	AUX
cana-4839	24	27	analyze	analyze	VERB
cana-4839	24	28	how	how	SCONJ
cana-4839	24	29	variations	variation	NOUN
cana-4839	24	30	in	in	ADP
cana-4839	24	31	parameters	parameter	NOUN
cana-4839	24	32	impact	impact	VERB
cana-4839	24	33	the	the	DET
cana-4839	24	34	dynamics	dynamic	NOUN
cana-4839	24	35	of	of	ADP
cana-4839	24	36	disease	disease	NOUN
cana-4839	24	37	transmission	transmission	NOUN
cana-4839	25	1	[	[	X
cana-4839	25	2	3][4	3][4	X
cana-4839	25	3	]	]	X
cana-4839	25	4	the	the	DET
cana-4839	25	5	two	two	NUM
cana-4839	25	6	-	-	PUNCT
cana-4839	25	7	dimensional	dimensional	ADJ
cana-4839	25	8	parameter	parameter	NOUN
cana-4839	25	9	space	space	NOUN
cana-4839	25	10	,	,	PUNCT
cana-4839	25	11	specifically	specifically	ADV
cana-4839	25	12	,	,	PUNCT
cana-4839	25	13	allows	allow	VERB
cana-4839	25	14	for	for	ADP
cana-4839	25	15	a	a	DET
cana-4839	25	16	detailed	detailed	ADJ
cana-4839	25	17	examination	examination	NOUN
cana-4839	25	18	of	of	ADP
cana-4839	25	19	key	key	ADJ
cana-4839	25	20	variables	variable	NOUN
cana-4839	25	21	that	that	PRON
cana-4839	25	22	govern	govern	VERB
cana-4839	25	23	epidemic	epidemic	ADJ
cana-4839	25	24	behavior	behavior	NOUN
cana-4839	25	25	.	.	PUNCT
cana-4839	26	1	one	one	NUM
cana-4839	26	2	such	such	ADJ
cana-4839	26	3	critical	critical	ADJ
cana-4839	26	4	parameter	parameter	NOUN
cana-4839	26	5	is	be	AUX
cana-4839	26	6	the	the	DET
cana-4839	26	7	recruitment	recruitment	NOUN
cana-4839	26	8	rate	rate	NOUN
cana-4839	26	9	of	of	ADP
cana-4839	26	10	individuals	individual	NOUN
cana-4839	26	11	into	into	ADP
cana-4839	26	12	the	the	DET
cana-4839	26	13	population	population	NOUN
cana-4839	26	14	which	which	PRON
cana-4839	26	15	can	can	AUX
cana-4839	26	16	be	be	AUX
cana-4839	26	17	influenced	influence	VERB
cana-4839	26	18	by	by	ADP
cana-4839	26	19	factors	factor	NOUN
cana-4839	26	20	such	such	ADJ
cana-4839	26	21	as	as	ADP
cana-4839	26	22	immigration	immigration	NOUN
cana-4839	26	23	policies	policy	NOUN
cana-4839	26	24	[	[	X
cana-4839	26	25	5][6	5][6	NUM
cana-4839	26	26	]	]	PUNCT
cana-4839	26	27	.	.	PUNCT
cana-4839	27	1	understanding	understand	VERB
cana-4839	27	2	how	how	SCONJ
cana-4839	27	3	these	these	DET
cana-4839	27	4	parameters	parameter	NOUN
cana-4839	27	5	interact	interact	VERB
cana-4839	27	6	and	and	CCONJ
cana-4839	27	7	lead	lead	VERB
cana-4839	27	8	to	to	ADP
cana-4839	27	9	bifurcations	bifurcation	NOUN
cana-4839	27	10	can	can	AUX
cana-4839	27	11	inform	inform	VERB
cana-4839	27	12	public	public	ADJ
cana-4839	27	13	health	health	NOUN
cana-4839	27	14	strategies	strategy	NOUN
cana-4839	27	15	aimed	aim	VERB
cana-4839	27	16	at	at	ADP
cana-4839	27	17	controlling	control	VERB
cana-4839	27	18	the	the	DET
cana-4839	27	19	spread	spread	NOUN
cana-4839	27	20	of	of	ADP
cana-4839	27	21	diseases	disease	NOUN
cana-4839	27	22	.	.	PUNCT
cana-4839	28	1	the	the	DET
cana-4839	28	2	implications	implication	NOUN
cana-4839	28	3	of	of	ADP
cana-4839	28	4	these	these	DET
cana-4839	28	5	changes	change	NOUN
cana-4839	28	6	are	be	AUX
cana-4839	28	7	significant	significant	ADJ
cana-4839	28	8	as	as	SCONJ
cana-4839	28	9	they	they	PRON
cana-4839	28	10	can	can	AUX
cana-4839	28	11	guide	guide	VERB
cana-4839	28	12	the	the	DET
cana-4839	28	13	development	development	NOUN
cana-4839	28	14	of	of	ADP
cana-4839	28	15	more	more	ADV
cana-4839	28	16	effective	effective	ADJ
cana-4839	28	17	control	control	NOUN
cana-4839	28	18	measures	measure	NOUN
cana-4839	28	19	during	during	ADP
cana-4839	28	20	outbreaks	outbreak	NOUN
cana-4839	28	21	[	[	X
cana-4839	28	22	7	7	NUM
cana-4839	28	23	]	]	PUNCT
cana-4839	28	24	.	.	PUNCT
cana-4839	29	1	during	during	ADP
cana-4839	29	2	the	the	DET
cana-4839	29	3	covid-19	covid-19	PROPN
cana-4839	29	4	pandemic	pandemic	NOUN
cana-4839	29	5	,	,	PUNCT
cana-4839	29	6	for	for	ADP
cana-4839	29	7	instance	instance	NOUN
cana-4839	29	8	,	,	PUNCT
cana-4839	29	9	analyses	analysis	NOUN
cana-4839	29	10	revealed	reveal	VERB
cana-4839	29	11	that	that	SCONJ
cana-4839	29	12	effective	effective	ADJ
cana-4839	29	13	control	control	NOUN
cana-4839	29	14	measures	measure	NOUN
cana-4839	29	15	could	could	AUX
cana-4839	29	16	have	have	AUX
cana-4839	29	17	led	lead	VERB
cana-4839	29	18	to	to	ADP
cana-4839	29	19	better	well	ADJ
cana-4839	29	20	management	management	NOUN
cana-4839	29	21	of	of	ADP
cana-4839	29	22	the	the	DET
cana-4839	29	23	disease	disease	NOUN
cana-4839	29	24	spread	spread	VERB
cana-4839	29	25	.	.	PUNCT
cana-4839	30	1	graphical	graphical	ADJ
cana-4839	30	2	representations	representation	NOUN
cana-4839	30	3	of	of	ADP
cana-4839	30	4	deterministic	deterministic	ADJ
cana-4839	30	5	models	model	NOUN
cana-4839	30	6	with	with	ADP
cana-4839	30	7	and	and	CCONJ
cana-4839	30	8	without	without	ADP
cana-4839	30	9	control	control	NOUN
cana-4839	30	10	measures	measure	NOUN
cana-4839	30	11	illustrated	illustrate	VERB
cana-4839	30	12	the	the	DET
cana-4839	30	13	potential	potential	ADJ
cana-4839	30	14	outcomes	outcome	NOUN
cana-4839	30	15	based	base	VERB
cana-4839	30	16	on	on	ADP
cana-4839	30	17	various	various	ADJ
cana-4839	30	18	scenarios	scenario	NOUN
cana-4839	30	19	.	.	PUNCT
cana-4839	31	1	the	the	DET
cana-4839	31	2	basic	basic	ADJ
cana-4839	31	3	reproduction	reproduction	NOUN
cana-4839	31	4	number	number	NOUN
cana-4839	31	5	,	,	PUNCT
cana-4839	31	6	an	an	DET
cana-4839	31	7	important	important	ADJ
cana-4839	31	8	indicator	indicator	NOUN
cana-4839	31	9	in	in	ADP
cana-4839	31	10	epidemiology	epidemiology	NOUN
cana-4839	31	11	,	,	PUNCT
cana-4839	31	12	being	be	AUX
cana-4839	31	13	less	less	ADJ
cana-4839	31	14	than	than	ADP
cana-4839	31	15	1	1	NUM
cana-4839	31	16	suggests	suggest	VERB
cana-4839	31	17	that	that	SCONJ
cana-4839	31	18	the	the	DET
cana-4839	31	19	disease	disease	NOUN
cana-4839	31	20	could	could	AUX
cana-4839	31	21	decline	decline	VERB
cana-4839	31	22	under	under	ADP
cana-4839	31	23	certain	certain	ADJ
cana-4839	31	24	controlled	control	VERB
cana-4839	31	25	conditions	condition	NOUN
cana-4839	31	26	[	[	X
cana-4839	31	27	8][9][10	8][9][10	NUM
cana-4839	31	28	]	]	PUNCT
cana-4839	31	29	communications	communication	NOUN
cana-4839	31	30	on	on	ADP
cana-4839	31	31	applied	apply	VERB
cana-4839	31	32	nonlinear	nonlinear	ADJ
cana-4839	31	33	analysis	analysis	NOUN
cana-4839	31	34	issn	issn	NOUN
cana-4839	31	35	:	:	PUNCT
cana-4839	31	36	1074	1074	NUM
cana-4839	31	37	-	-	PUNCT
cana-4839	31	38	133x	133x	NUM
cana-4839	31	39	vol	vol	VERB
cana-4839	31	40	32	32	NUM
cana-4839	31	41	no	no	NOUN
cana-4839	31	42	.	.	PUNCT
cana-4839	32	1	10s	10	NOUN
cana-4839	32	2	(	(	PUNCT
cana-4839	32	3	2025	2025	NUM
cana-4839	32	4	)	)	PUNCT
cana-4839	32	5	439	439	NUM
cana-4839	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	32	7	bifurcation	bifurcation	NOUN
cana-4839	32	8	theory	theory	NOUN
cana-4839	32	9	in	in	ADP
cana-4839	32	10	epidemiology	epidemiology	NOUN
cana-4839	32	11	bifurcation	bifurcation	NOUN
cana-4839	32	12	theory	theory	NOUN
cana-4839	32	13	is	be	AUX
cana-4839	32	14	crucial	crucial	ADJ
cana-4839	32	15	in	in	ADP
cana-4839	32	16	understanding	understand	VERB
cana-4839	32	17	the	the	DET
cana-4839	32	18	dynamics	dynamic	NOUN
cana-4839	32	19	of	of	ADP
cana-4839	32	20	infectious	infectious	ADJ
cana-4839	32	21	diseases	disease	NOUN
cana-4839	32	22	as	as	SCONJ
cana-4839	32	23	it	it	PRON
cana-4839	32	24	describes	describe	VERB
cana-4839	32	25	qualitative	qualitative	ADJ
cana-4839	32	26	or	or	CCONJ
cana-4839	32	27	quantitative	quantitative	ADJ
cana-4839	32	28	changes	change	NOUN
cana-4839	32	29	that	that	PRON
cana-4839	32	30	occur	occur	VERB
cana-4839	32	31	in	in	ADP
cana-4839	32	32	a	a	DET
cana-4839	32	33	system	system	NOUN
cana-4839	32	34	when	when	SCONJ
cana-4839	32	35	a	a	DET
cana-4839	32	36	parameter	parameter	NOUN
cana-4839	32	37	crosses	cross	VERB
cana-4839	32	38	a	a	DET
cana-4839	32	39	critical	critical	ADJ
cana-4839	32	40	threshold	threshold	NOUN
cana-4839	32	41	.	.	PUNCT
cana-4839	33	1	this	this	DET
cana-4839	33	2	phenomenon	phenomenon	NOUN
cana-4839	33	3	can	can	AUX
cana-4839	33	4	significantly	significantly	ADV
cana-4839	33	5	alter	alter	VERB
cana-4839	33	6	the	the	DET
cana-4839	33	7	course	course	NOUN
cana-4839	33	8	of	of	ADP
cana-4839	33	9	an	an	DET
cana-4839	33	10	epidemic	epidemic	NOUN
cana-4839	33	11	,	,	PUNCT
cana-4839	33	12	potentially	potentially	ADV
cana-4839	33	13	transforming	transform	VERB
cana-4839	33	14	a	a	DET
cana-4839	33	15	controlled	control	VERB
cana-4839	33	16	outbreak	outbreak	NOUN
cana-4839	33	17	into	into	ADP
cana-4839	33	18	a	a	DET
cana-4839	33	19	widespread	widespread	ADJ
cana-4839	33	20	pandemic[11][12	pandemic[11][12	PROPN
cana-4839	33	21	]	]	PUNCT
cana-4839	33	22	by	by	ADP
cana-4839	33	23	analyzing	analyze	VERB
cana-4839	33	24	these	these	DET
cana-4839	33	25	bifurcations	bifurcation	NOUN
cana-4839	33	26	,	,	PUNCT
cana-4839	33	27	researchers	researcher	NOUN
cana-4839	33	28	can	can	AUX
cana-4839	33	29	develop	develop	VERB
cana-4839	33	30	more	more	ADV
cana-4839	33	31	effective	effective	ADJ
cana-4839	33	32	strategies	strategy	NOUN
cana-4839	33	33	for	for	ADP
cana-4839	33	34	managing	manage	VERB
cana-4839	33	35	infectious	infectious	ADJ
cana-4839	33	36	diseases	disease	NOUN
cana-4839	33	37	.	.	PUNCT
cana-4839	34	1	sirs	sir	NOUN
cana-4839	34	2	model	model	VERB
cana-4839	34	3	and	and	CCONJ
cana-4839	34	4	parameter	parameter	NOUN
cana-4839	34	5	analysis	analysis	NOUN
cana-4839	34	6	in	in	ADP
cana-4839	34	7	the	the	DET
cana-4839	34	8	context	context	NOUN
cana-4839	34	9	of	of	ADP
cana-4839	34	10	epidemiological	epidemiological	ADJ
cana-4839	34	11	modeling	modeling	NOUN
cana-4839	34	12	,	,	PUNCT
cana-4839	34	13	the	the	DET
cana-4839	34	14	sirs	sir	NOUN
cana-4839	34	15	(	(	PUNCT
cana-4839	34	16	susceptible	susceptible	ADJ
cana-4839	34	17	-	-	PUNCT
cana-4839	34	18	infectious	infectious	ADJ
cana-4839	34	19	-	-	PUNCT
cana-4839	34	20	recovered	recover	VERB
cana-4839	34	21	-	-	PUNCT
cana-4839	34	22	susceptible	susceptible	ADJ
cana-4839	34	23	)	)	PUNCT
cana-4839	34	24	model	model	NOUN
cana-4839	34	25	serves	serve	VERB
cana-4839	34	26	as	as	ADP
cana-4839	34	27	a	a	DET
cana-4839	34	28	foundational	foundational	ADJ
cana-4839	34	29	framework	framework	NOUN
cana-4839	34	30	.	.	PUNCT
cana-4839	35	1	it	it	PRON
cana-4839	35	2	allows	allow	VERB
cana-4839	35	3	for	for	ADP
cana-4839	35	4	the	the	DET
cana-4839	35	5	examination	examination	NOUN
cana-4839	35	6	of	of	ADP
cana-4839	35	7	bifurcation	bifurcation	NOUN
cana-4839	35	8	structures	structure	NOUN
cana-4839	35	9	in	in	ADP
cana-4839	35	10	a	a	DET
cana-4839	35	11	two	two	NUM
cana-4839	35	12	-	-	PUNCT
cana-4839	35	13	dimensional	dimensional	ADJ
cana-4839	35	14	parameter	parameter	NOUN
cana-4839	35	15	space	space	NOUN
cana-4839	35	16	specifically	specifically	ADV
cana-4839	35	17	focusing	focus	VERB
cana-4839	35	18	on	on	ADP
cana-4839	35	19	the	the	DET
cana-4839	35	20	parameters	parameter	NOUN
cana-4839	35	21	that	that	PRON
cana-4839	35	22	influence	influence	VERB
cana-4839	35	23	epidemic	epidemic	NOUN
cana-4839	35	24	dynamics	dynamic	NOUN
cana-4839	35	25	.	.	PUNCT
cana-4839	36	1	notably	notably	ADV
cana-4839	36	2	,	,	PUNCT
cana-4839	36	3	parameter	parameter	NOUN
cana-4839	36	4	a	a	PRON
cana-4839	36	5	is	be	AUX
cana-4839	36	6	instrumental	instrumental	ADJ
cana-4839	36	7	as	as	SCONJ
cana-4839	36	8	it	it	PRON
cana-4839	36	9	governs	govern	VERB
cana-4839	36	10	the	the	DET
cana-4839	36	11	recruitment	recruitment	NOUN
cana-4839	36	12	rate	rate	NOUN
cana-4839	36	13	of	of	ADP
cana-4839	36	14	individuals	individual	NOUN
cana-4839	36	15	into	into	ADP
cana-4839	36	16	the	the	DET
cana-4839	36	17	population	population	NOUN
cana-4839	36	18	,	,	PUNCT
cana-4839	36	19	a	a	DET
cana-4839	36	20	factor	factor	NOUN
cana-4839	36	21	that	that	PRON
cana-4839	36	22	can	can	AUX
cana-4839	36	23	be	be	AUX
cana-4839	36	24	influenced	influence	VERB
cana-4839	36	25	by	by	ADP
cana-4839	36	26	immigration	immigration	NOUN
cana-4839	36	27	policies	policy	NOUN
cana-4839	36	28	.	.	PUNCT
cana-4839	37	1	the	the	DET
cana-4839	37	2	selection	selection	NOUN
cana-4839	37	3	of	of	ADP
cana-4839	37	4	the	the	DET
cana-4839	37	5	appropriate	appropriate	ADJ
cana-4839	37	6	parameter	parameter	NOUN
cana-4839	37	7	space	space	NOUN
cana-4839	37	8	for	for	ADP
cana-4839	37	9	bifurcation	bifurcation	NOUN
cana-4839	37	10	analysis	analysis	NOUN
cana-4839	37	11	is	be	AUX
cana-4839	37	12	essential	essential	ADJ
cana-4839	37	13	as	as	SCONJ
cana-4839	37	14	it	it	PRON
cana-4839	37	15	enables	enable	VERB
cana-4839	37	16	researchers	researcher	NOUN
cana-4839	37	17	to	to	PART
cana-4839	37	18	identify	identify	VERB
cana-4839	37	19	conditions	condition	NOUN
cana-4839	37	20	leading	lead	VERB
cana-4839	37	21	to	to	ADP
cana-4839	37	22	significant	significant	ADJ
cana-4839	37	23	shifts	shift	NOUN
cana-4839	37	24	in	in	ADP
cana-4839	37	25	disease	disease	NOUN
cana-4839	37	26	spread	spread	VERB
cana-4839	37	27	.	.	PUNCT
cana-4839	38	1	integration	integration	NOUN
cana-4839	38	2	of	of	ADP
cana-4839	38	3	qualitative	qualitative	NOUN
cana-4839	38	4	and	and	CCONJ
cana-4839	38	5	bifurcation	bifurcation	NOUN
cana-4839	38	6	analysis	analysis	NOUN
cana-4839	38	7	a	a	DET
cana-4839	38	8	comprehensive	comprehensive	ADJ
cana-4839	38	9	approach	approach	NOUN
cana-4839	38	10	that	that	PRON
cana-4839	38	11	integrates	integrate	VERB
cana-4839	38	12	qualitative	qualitative	ADJ
cana-4839	38	13	analysis	analysis	NOUN
cana-4839	38	14	with	with	ADP
cana-4839	38	15	bifurcation	bifurcation	NOUN
cana-4839	38	16	analysis	analysis	NOUN
cana-4839	38	17	enhances	enhance	VERB
cana-4839	38	18	the	the	DET
cana-4839	38	19	understanding	understanding	NOUN
cana-4839	38	20	of	of	ADP
cana-4839	38	21	both	both	DET
cana-4839	38	22	social	social	ADJ
cana-4839	38	23	dynamics	dynamic	NOUN
cana-4839	38	24	and	and	CCONJ
cana-4839	38	25	mathematical	mathematical	ADJ
cana-4839	38	26	modeling	modeling	NOUN
cana-4839	38	27	of	of	ADP
cana-4839	38	28	disease	disease	NOUN
cana-4839	38	29	transmission	transmission	NOUN
cana-4839	38	30	.	.	PUNCT
cana-4839	39	1	this	this	DET
cana-4839	39	2	interdisciplinary	interdisciplinary	ADJ
cana-4839	39	3	methodology	methodology	NOUN
cana-4839	39	4	draws	draw	VERB
cana-4839	39	5	insights	insight	NOUN
cana-4839	39	6	from	from	ADP
cana-4839	39	7	social	social	ADJ
cana-4839	39	8	sciences	science	NOUN
cana-4839	39	9	alongside	alongside	ADP
cana-4839	39	10	epidemiological	epidemiological	ADJ
cana-4839	39	11	modeling	modeling	NOUN
cana-4839	39	12	,	,	PUNCT
cana-4839	39	13	thereby	thereby	ADV
cana-4839	39	14	improving	improve	VERB
cana-4839	39	15	the	the	DET
cana-4839	39	16	reliability	reliability	NOUN
cana-4839	39	17	of	of	ADP
cana-4839	39	18	predictions	prediction	NOUN
cana-4839	39	19	.	.	PUNCT
cana-4839	40	1	by	by	ADP
cana-4839	40	2	justifying	justify	VERB
cana-4839	40	3	the	the	DET
cana-4839	40	4	use	use	NOUN
cana-4839	40	5	of	of	ADP
cana-4839	40	6	both	both	PRON
cana-4839	40	7	qualitative	qualitative	ADJ
cana-4839	40	8	and	and	CCONJ
cana-4839	40	9	bifurcation	bifurcation	NOUN
cana-4839	40	10	analyses	analysis	NOUN
cana-4839	40	11	,	,	PUNCT
cana-4839	40	12	researchers	researcher	NOUN
cana-4839	40	13	can	can	AUX
cana-4839	40	14	effectively	effectively	ADV
cana-4839	40	15	bridge	bridge	VERB
cana-4839	40	16	theoretical	theoretical	ADJ
cana-4839	40	17	modeling	modeling	NOUN
cana-4839	40	18	with	with	ADP
cana-4839	40	19	practical	practical	ADJ
cana-4839	40	20	public	public	ADJ
cana-4839	40	21	health	health	NOUN
cana-4839	40	22	strategies	strategy	NOUN
cana-4839	40	23	particularly	particularly	ADV
cana-4839	40	24	in	in	ADP
cana-4839	40	25	regions	region	NOUN
cana-4839	40	26	like	like	ADP
cana-4839	40	27	bangladesh	bangladesh	NOUN
cana-4839	40	28	where	where	SCONJ
cana-4839	40	29	such	such	ADJ
cana-4839	40	30	dynamics	dynamic	NOUN
cana-4839	40	31	are	be	AUX
cana-4839	40	32	increasingly	increasingly	ADV
cana-4839	40	33	relevant[12][13	relevant[12][13	PROPN
cana-4839	40	34	]	]	PUNCT
cana-4839	40	35	stability	stability	NOUN
cana-4839	40	36	analysis	analysis	NOUN
cana-4839	40	37	and	and	CCONJ
cana-4839	40	38	control	control	NOUN
cana-4839	40	39	strategies	strategy	NOUN
cana-4839	40	40	the	the	DET
cana-4839	40	41	stability	stability	NOUN
cana-4839	40	42	of	of	ADP
cana-4839	40	43	equilibrium	equilibrium	NOUN
cana-4839	40	44	points	point	NOUN
cana-4839	40	45	within	within	ADP
cana-4839	40	46	the	the	DET
cana-4839	40	47	sirs	sir	NOUN
cana-4839	40	48	model	model	NOUN
cana-4839	40	49	is	be	AUX
cana-4839	40	50	determined	determine	VERB
cana-4839	40	51	by	by	ADP
cana-4839	40	52	analyzing	analyze	VERB
cana-4839	40	53	the	the	DET
cana-4839	40	54	eigenvalues	eigenvalue	NOUN
cana-4839	40	55	associated	associate	VERB
cana-4839	40	56	with	with	ADP
cana-4839	40	57	the	the	DET
cana-4839	40	58	jacobian	jacobian	ADJ
cana-4839	40	59	matrix	matrix	NOUN
cana-4839	40	60	at	at	ADP
cana-4839	40	61	fixed	fix	VERB
cana-4839	40	62	points	point	NOUN
cana-4839	40	63	.	.	PUNCT
cana-4839	41	1	if	if	SCONJ
cana-4839	41	2	the	the	DET
cana-4839	41	3	absolute	absolute	ADJ
cana-4839	41	4	values	value	NOUN
cana-4839	41	5	of	of	ADP
cana-4839	41	6	the	the	DET
cana-4839	41	7	eigenvalues	eigenvalue	NOUN
cana-4839	41	8	are	be	AUX
cana-4839	41	9	less	less	ADJ
cana-4839	41	10	than	than	ADP
cana-4839	41	11	one	one	NUM
cana-4839	41	12	,	,	PUNCT
cana-4839	41	13	the	the	DET
cana-4839	41	14	equilibrium	equilibrium	NOUN
cana-4839	41	15	point	point	NOUN
cana-4839	41	16	is	be	AUX
cana-4839	41	17	considered	consider	VERB
cana-4839	41	18	asymptotically	asymptotically	ADV
cana-4839	41	19	stable	stable	ADJ
cana-4839	41	20	.	.	PUNCT
cana-4839	42	1	conversely	conversely	ADV
cana-4839	42	2	,	,	PUNCT
cana-4839	42	3	if	if	SCONJ
cana-4839	42	4	at	at	ADV
cana-4839	42	5	least	least	ADV
cana-4839	42	6	one	one	NUM
cana-4839	42	7	eigenvalue	eigenvalue	NOUN
cana-4839	42	8	exceeds	exceed	VERB
cana-4839	42	9	one	one	NUM
cana-4839	42	10	in	in	ADP
cana-4839	42	11	absolute	absolute	ADJ
cana-4839	42	12	value	value	NOUN
cana-4839	42	13	,	,	PUNCT
cana-4839	42	14	the	the	DET
cana-4839	42	15	equilibrium	equilibrium	NOUN
cana-4839	42	16	point	point	NOUN
cana-4839	42	17	becomes	become	VERB
cana-4839	42	18	unstable[14][15	unstable[14][15	PROPN
cana-4839	42	19	]	]	PUNCT
cana-4839	42	20	.	.	PUNCT
cana-4839	43	1	this	this	DET
cana-4839	43	2	analysis	analysis	NOUN
cana-4839	43	3	is	be	AUX
cana-4839	43	4	typically	typically	ADV
cana-4839	43	5	numerical	numerical	ADJ
cana-4839	43	6	due	due	ADP
cana-4839	43	7	to	to	ADP
cana-4839	43	8	the	the	DET
cana-4839	43	9	complexities	complexity	NOUN
cana-4839	43	10	of	of	ADP
cana-4839	43	11	nonlinear	nonlinear	ADJ
cana-4839	43	12	systems	system	NOUN
cana-4839	43	13	,	,	PUNCT
cana-4839	43	14	though	though	SCONJ
cana-4839	43	15	some	some	DET
cana-4839	43	16	analytical	analytical	ADJ
cana-4839	43	17	results	result	NOUN
cana-4839	43	18	can	can	AUX
cana-4839	43	19	still	still	ADV
cana-4839	43	20	be	be	AUX
cana-4839	43	21	established	establish	VERB
cana-4839	43	22	.	.	PUNCT
cana-4839	44	1	future	future	ADJ
cana-4839	44	2	research	research	NOUN
cana-4839	44	3	directions	direction	NOUN
cana-4839	44	4	involve	involve	VERB
cana-4839	44	5	a	a	DET
cana-4839	44	6	deeper	deep	ADJ
cana-4839	44	7	exploration	exploration	NOUN
cana-4839	44	8	of	of	ADP
cana-4839	44	9	bifurcation	bifurcation	NOUN
cana-4839	44	10	curves	curve	NOUN
cana-4839	44	11	and	and	CCONJ
cana-4839	44	12	their	their	PRON
cana-4839	44	13	implications	implication	NOUN
cana-4839	44	14	for	for	ADP
cana-4839	44	15	control	control	NOUN
cana-4839	44	16	strategies	strategy	NOUN
cana-4839	44	17	.	.	PUNCT
cana-4839	45	1	investigating	investigate	VERB
cana-4839	45	2	the	the	DET
cana-4839	45	3	stability	stability	NOUN
cana-4839	45	4	domain	domain	NOUN
cana-4839	45	5	of	of	ADP
cana-4839	45	6	periodic	periodic	ADJ
cana-4839	45	7	points	point	NOUN
cana-4839	45	8	such	such	ADJ
cana-4839	45	9	as	as	ADP
cana-4839	45	10	period	period	NOUN
cana-4839	45	11	-	-	PUNCT
cana-4839	45	12	doubling	double	VERB
cana-4839	45	13	bifurcations	bifurcation	NOUN
cana-4839	45	14	,	,	PUNCT
cana-4839	45	15	poses	pose	VERB
cana-4839	45	16	analytical	analytical	ADJ
cana-4839	45	17	challenges	challenge	NOUN
cana-4839	45	18	but	but	CCONJ
cana-4839	45	19	may	may	AUX
cana-4839	45	20	yield	yield	VERB
cana-4839	45	21	critical	critical	ADJ
cana-4839	45	22	insights	insight	NOUN
cana-4839	45	23	for	for	ADP
cana-4839	45	24	optimizing	optimize	VERB
cana-4839	45	25	control	control	NOUN
cana-4839	45	26	measures	measure	NOUN
cana-4839	45	27	during	during	ADP
cana-4839	45	28	epidemics[16	epidemics[16	PROPN
cana-4839	45	29	]	]	PUNCT
cana-4839	45	30	.	.	PUNCT
cana-4839	46	1	understanding	understand	VERB
cana-4839	46	2	these	these	DET
cana-4839	46	3	bifurcations	bifurcation	NOUN
cana-4839	46	4	and	and	CCONJ
cana-4839	46	5	their	their	PRON
cana-4839	46	6	consequences	consequence	NOUN
cana-4839	46	7	is	be	AUX
cana-4839	46	8	pivotal	pivotal	ADJ
cana-4839	46	9	for	for	ADP
cana-4839	46	10	translating	translate	VERB
cana-4839	46	11	theoretical	theoretical	ADJ
cana-4839	46	12	findings	finding	NOUN
cana-4839	46	13	into	into	ADP
cana-4839	46	14	actionable	actionable	ADJ
cana-4839	46	15	public	public	ADJ
cana-4839	46	16	health	health	NOUN
cana-4839	46	17	interventions	intervention	NOUN
cana-4839	46	18	aimed	aim	VERB
cana-4839	46	19	at	at	ADP
cana-4839	46	20	controlling	control	VERB
cana-4839	46	21	infectious	infectious	ADJ
cana-4839	46	22	disease	disease	NOUN
cana-4839	46	23	spread	spread	VERB
cana-4839	46	24	.	.	PUNCT
cana-4839	47	1	covid-19	covid-19	PROPN
cana-4839	47	2	case	case	NOUN
cana-4839	47	3	study	study	VERB
cana-4839	47	4	the	the	DET
cana-4839	47	5	spread	spread	NOUN
cana-4839	47	6	of	of	ADP
cana-4839	47	7	covid-19	covid-19	PROPN
cana-4839	47	8	has	have	AUX
cana-4839	47	9	been	be	AUX
cana-4839	47	10	extensively	extensively	ADV
cana-4839	47	11	analyzed	analyze	VERB
cana-4839	47	12	using	use	VERB
cana-4839	47	13	various	various	ADJ
cana-4839	47	14	mathematical	mathematical	ADJ
cana-4839	47	15	models	model	NOUN
cana-4839	47	16	to	to	PART
cana-4839	47	17	understand	understand	VERB
cana-4839	47	18	its	its	PRON
cana-4839	47	19	dynamics	dynamic	NOUN
cana-4839	47	20	and	and	CCONJ
cana-4839	47	21	inform	inform	VERB
cana-4839	47	22	public	public	ADJ
cana-4839	47	23	health	health	NOUN
cana-4839	47	24	interventions	intervention	NOUN
cana-4839	47	25	.	.	PUNCT
cana-4839	48	1	researchers	researcher	NOUN
cana-4839	48	2	have	have	AUX
cana-4839	48	3	utilized	utilize	VERB
cana-4839	48	4	modified	modify	VERB
cana-4839	48	5	sir	sir	NOUN
cana-4839	48	6	(	(	PUNCT
cana-4839	48	7	susceptible	susceptible	ADJ
cana-4839	48	8	,	,	PUNCT
cana-4839	48	9	infected	infect	VERB
cana-4839	48	10	,	,	PUNCT
cana-4839	48	11	recovered	recover	VERB
cana-4839	48	12	)	)	PUNCT
cana-4839	48	13	and	and	CCONJ
cana-4839	48	14	sirs	sir	NOUN
cana-4839	48	15	(	(	PUNCT
cana-4839	48	16	susceptible	susceptible	ADJ
cana-4839	48	17	,	,	PUNCT
cana-4839	48	18	infected	infect	VERB
cana-4839	48	19	,	,	PUNCT
cana-4839	48	20	recovered	recovered	ADJ
cana-4839	48	21	and	and	CCONJ
cana-4839	48	22	susceptible	susceptible	ADJ
cana-4839	48	23	)	)	PUNCT
cana-4839	48	24	models	model	NOUN
cana-4839	48	25	to	to	PART
cana-4839	48	26	estimate	estimate	VERB
cana-4839	48	27	infection	infection	NOUN
cana-4839	48	28	rates	rate	NOUN
cana-4839	48	29	and	and	CCONJ
cana-4839	48	30	evaluate	evaluate	VERB
cana-4839	48	31	strategies	strategy	NOUN
cana-4839	48	32	for	for	ADP
cana-4839	48	33	controlling	control	VERB
cana-4839	48	34	the	the	DET
cana-4839	48	35	rapid	rapid	ADJ
cana-4839	48	36	spread	spread	NOUN
cana-4839	48	37	of	of	ADP
cana-4839	48	38	the	the	DET
cana-4839	48	39	communications	communication	NOUN
cana-4839	48	40	on	on	ADP
cana-4839	48	41	applied	apply	VERB
cana-4839	48	42	nonlinear	nonlinear	ADJ
cana-4839	48	43	analysis	analysis	NOUN
cana-4839	48	44	issn	issn	NOUN
cana-4839	48	45	:	:	PUNCT
cana-4839	48	46	1074	1074	NUM
cana-4839	48	47	-	-	PUNCT
cana-4839	48	48	133x	133x	NUM
cana-4839	48	49	vol	vol	VERB
cana-4839	48	50	32	32	NUM
cana-4839	48	51	no	no	NOUN
cana-4839	48	52	.	.	PUNCT
cana-4839	49	1	10s	10	NOUN
cana-4839	49	2	(	(	PUNCT
cana-4839	49	3	2025	2025	NUM
cana-4839	49	4	)	)	PUNCT
cana-4839	49	5	440	440	NUM
cana-4839	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	49	7	virus[17	virus[17	PROPN
cana-4839	49	8	]	]	PUNCT
cana-4839	49	9	.	.	PUNCT
cana-4839	50	1	these	these	DET
cana-4839	50	2	models	model	NOUN
cana-4839	50	3	enable	enable	VERB
cana-4839	50	4	a	a	DET
cana-4839	50	5	detailed	detailed	ADJ
cana-4839	50	6	exploration	exploration	NOUN
cana-4839	50	7	of	of	ADP
cana-4839	50	8	how	how	SCONJ
cana-4839	50	9	changes	change	NOUN
cana-4839	50	10	in	in	ADP
cana-4839	50	11	parameters	parameter	NOUN
cana-4839	50	12	can	can	AUX
cana-4839	50	13	affect	affect	VERB
cana-4839	50	14	the	the	DET
cana-4839	50	15	epidemic	epidemic	NOUN
cana-4839	50	16	's	's	PART
cana-4839	50	17	progression	progression	NOUN
cana-4839	50	18	.	.	PUNCT
cana-4839	51	1	mathematical	mathematical	ADJ
cana-4839	51	2	modeling	modeling	NOUN
cana-4839	51	3	approaches	approach	VERB
cana-4839	51	4	many	many	ADJ
cana-4839	51	5	studies	study	NOUN
cana-4839	51	6	have	have	AUX
cana-4839	51	7	adopted	adopt	VERB
cana-4839	51	8	different	different	ADJ
cana-4839	51	9	modeling	modeling	NOUN
cana-4839	51	10	approaches	approach	NOUN
cana-4839	51	11	,	,	PUNCT
cana-4839	51	12	including	include	VERB
cana-4839	51	13	continuous	continuous	ADJ
cana-4839	51	14	and	and	CCONJ
cana-4839	51	15	discrete	discrete	ADJ
cana-4839	51	16	-	-	PUNCT
cana-4839	51	17	time	time	NOUN
cana-4839	51	18	models	model	NOUN
cana-4839	51	19	as	as	ADV
cana-4839	51	20	well	well	ADV
cana-4839	51	21	as	as	ADP
cana-4839	51	22	stochastic	stochastic	ADJ
cana-4839	51	23	modeling	modeling	NOUN
cana-4839	51	24	to	to	PART
cana-4839	51	25	simulate	simulate	VERB
cana-4839	51	26	the	the	DET
cana-4839	51	27	dynamics	dynamic	NOUN
cana-4839	51	28	of	of	ADP
cana-4839	51	29	covid-19	covid-19	PROPN
cana-4839	51	30	transmission[18	transmission[18	PROPN
cana-4839	51	31	]	]	PUNCT
cana-4839	51	32	.	.	PUNCT
cana-4839	52	1	the	the	DET
cana-4839	52	2	sirs	sirs	PROPN
cana-4839	52	3	epidemic	epidemic	NOUN
cana-4839	52	4	model	model	NOUN
cana-4839	52	5	,	,	PUNCT
cana-4839	52	6	in	in	ADP
cana-4839	52	7	particular	particular	ADJ
cana-4839	52	8	,	,	PUNCT
cana-4839	52	9	has	have	AUX
cana-4839	52	10	been	be	AUX
cana-4839	52	11	highlighted	highlight	VERB
cana-4839	52	12	for	for	ADP
cana-4839	52	13	its	its	PRON
cana-4839	52	14	ability	ability	NOUN
cana-4839	52	15	to	to	PART
cana-4839	52	16	capture	capture	VERB
cana-4839	52	17	the	the	DET
cana-4839	52	18	cyclic	cyclic	ADJ
cana-4839	52	19	nature	nature	NOUN
cana-4839	52	20	of	of	ADP
cana-4839	52	21	infection	infection	NOUN
cana-4839	52	22	dynamics	dynamic	NOUN
cana-4839	52	23	,	,	PUNCT
cana-4839	52	24	where	where	SCONJ
cana-4839	52	25	individuals	individual	NOUN
cana-4839	52	26	may	may	AUX
cana-4839	52	27	transition	transition	VERB
cana-4839	52	28	back	back	ADV
cana-4839	52	29	to	to	ADP
cana-4839	52	30	a	a	DET
cana-4839	52	31	susceptible	susceptible	ADJ
cana-4839	52	32	state	state	NOUN
cana-4839	52	33	after	after	SCONJ
cana-4839	52	34	recovery	recovery	NOUN
cana-4839	52	35	bifurcation	bifurcation	NOUN
cana-4839	52	36	analysis	analysis	NOUN
cana-4839	52	37	bifurcation	bifurcation	NOUN
cana-4839	52	38	analysis	analysis	NOUN
cana-4839	52	39	has	have	AUX
cana-4839	52	40	been	be	AUX
cana-4839	52	41	a	a	DET
cana-4839	52	42	crucial	crucial	ADJ
cana-4839	52	43	tool	tool	NOUN
cana-4839	52	44	in	in	ADP
cana-4839	52	45	examining	examine	VERB
cana-4839	52	46	the	the	DET
cana-4839	52	47	stability	stability	NOUN
cana-4839	52	48	and	and	CCONJ
cana-4839	52	49	behavior	behavior	NOUN
cana-4839	52	50	of	of	ADP
cana-4839	52	51	the	the	DET
cana-4839	52	52	sirs	sirs	PROPN
cana-4839	52	53	model	model	NOUN
cana-4839	52	54	under	under	ADP
cana-4839	52	55	varying	vary	VERB
cana-4839	52	56	parameters	parameter	NOUN
cana-4839	52	57	.	.	PUNCT
cana-4839	53	1	researchers	researcher	NOUN
cana-4839	53	2	have	have	AUX
cana-4839	53	3	identified	identify	VERB
cana-4839	53	4	critical	critical	ADJ
cana-4839	53	5	points	point	NOUN
cana-4839	53	6	at	at	ADP
cana-4839	53	7	which	which	PRON
cana-4839	53	8	small	small	ADJ
cana-4839	53	9	changes	change	NOUN
cana-4839	53	10	can	can	AUX
cana-4839	53	11	lead	lead	VERB
cana-4839	53	12	to	to	ADP
cana-4839	53	13	significant	significant	ADJ
cana-4839	53	14	shifts	shift	NOUN
cana-4839	53	15	in	in	ADP
cana-4839	53	16	the	the	DET
cana-4839	53	17	system	system	NOUN
cana-4839	53	18	's	's	PART
cana-4839	53	19	dynamics	dynamic	NOUN
cana-4839	53	20	thereby	thereby	ADV
cana-4839	53	21	providing	provide	VERB
cana-4839	53	22	insights	insight	NOUN
cana-4839	53	23	into	into	ADP
cana-4839	53	24	the	the	DET
cana-4839	53	25	conditions	condition	NOUN
cana-4839	53	26	that	that	PRON
cana-4839	53	27	could	could	AUX
cana-4839	53	28	lead	lead	VERB
cana-4839	53	29	to	to	ADP
cana-4839	53	30	either	either	DET
cana-4839	53	31	outbreaks	outbreak	NOUN
cana-4839	53	32	or	or	CCONJ
cana-4839	53	33	declines	decline	NOUN
cana-4839	53	34	in	in	ADP
cana-4839	53	35	infection	infection	NOUN
cana-4839	53	36	rates[19	rates[19	NOUN
cana-4839	53	37	]	]	PUNCT
cana-4839	53	38	.	.	PUNCT
cana-4839	54	1	this	this	DET
cana-4839	54	2	analysis	analysis	NOUN
cana-4839	54	3	not	not	PART
cana-4839	54	4	only	only	ADV
cana-4839	54	5	aids	aid	NOUN
cana-4839	54	6	in	in	ADP
cana-4839	54	7	understanding	understand	VERB
cana-4839	54	8	the	the	DET
cana-4839	54	9	existing	exist	VERB
cana-4839	54	10	data	datum	NOUN
cana-4839	54	11	but	but	CCONJ
cana-4839	54	12	also	also	ADV
cana-4839	54	13	helps	help	VERB
cana-4839	54	14	in	in	ADP
cana-4839	54	15	predicting	predict	VERB
cana-4839	54	16	future	future	ADJ
cana-4839	54	17	trends	trend	NOUN
cana-4839	54	18	in	in	ADP
cana-4839	54	19	covid-19	covid-19	PROPN
cana-4839	54	20	spread	spread	NOUN
cana-4839	54	21	.	.	PUNCT
cana-4839	55	1	application	application	NOUN
cana-4839	55	2	to	to	ADP
cana-4839	55	3	real	real	ADJ
cana-4839	55	4	-	-	PUNCT
cana-4839	55	5	world	world	NOUN
cana-4839	55	6	data	data	PROPN
cana-4839	55	7	numerical	numerical	ADJ
cana-4839	55	8	simulations	simulation	NOUN
cana-4839	55	9	have	have	AUX
cana-4839	55	10	been	be	AUX
cana-4839	55	11	conducted	conduct	VERB
cana-4839	55	12	to	to	PART
cana-4839	55	13	enhance	enhance	VERB
cana-4839	55	14	the	the	DET
cana-4839	55	15	understanding	understanding	NOUN
cana-4839	55	16	of	of	ADP
cana-4839	55	17	stability	stability	NOUN
cana-4839	55	18	regions	region	NOUN
cana-4839	55	19	within	within	ADP
cana-4839	55	20	the	the	DET
cana-4839	55	21	parameter	parameter	NOUN
cana-4839	55	22	space	space	NOUN
cana-4839	55	23	.	.	PUNCT
cana-4839	56	1	these	these	DET
cana-4839	56	2	simulations	simulation	NOUN
cana-4839	56	3	have	have	AUX
cana-4839	56	4	utilized	utilize	VERB
cana-4839	56	5	real	real	ADJ
cana-4839	56	6	-	-	PUNCT
cana-4839	56	7	world	world	NOUN
cana-4839	56	8	data	datum	NOUN
cana-4839	56	9	from	from	ADP
cana-4839	56	10	countries	country	NOUN
cana-4839	56	11	such	such	ADJ
cana-4839	56	12	as	as	ADP
cana-4839	56	13	algeria	algeria	PROPN
cana-4839	56	14	and	and	CCONJ
cana-4839	56	15	the	the	DET
cana-4839	56	16	usa	usa	PROPN
cana-4839	56	17	to	to	PART
cana-4839	56	18	compare	compare	VERB
cana-4839	56	19	observed	observe	VERB
cana-4839	56	20	infection	infection	NOUN
cana-4839	56	21	rates	rate	NOUN
cana-4839	56	22	against	against	ADP
cana-4839	56	23	model	model	NOUN
cana-4839	56	24	predictions[20][21	predictions[20][21	PROPN
cana-4839	56	25	]	]	PUNCT
cana-4839	56	26	.	.	PUNCT
cana-4839	57	1	by	by	ADP
cana-4839	57	2	employing	employ	VERB
cana-4839	57	3	next	next	ADJ
cana-4839	57	4	-	-	PUNCT
cana-4839	57	5	generation	generation	NOUN
cana-4839	57	6	matrix	matrix	NOUN
cana-4839	57	7	approaches	approach	NOUN
cana-4839	57	8	,	,	PUNCT
cana-4839	57	9	researchers	researcher	NOUN
cana-4839	57	10	have	have	AUX
cana-4839	57	11	calculated	calculate	VERB
cana-4839	57	12	the	the	DET
cana-4839	57	13	basic	basic	ADJ
cana-4839	57	14	reproduction	reproduction	NOUN
cana-4839	57	15	number	number	NOUN
cana-4839	57	16	(	(	PUNCT
cana-4839	57	17	r0	r0	NOUN
cana-4839	57	18	)	)	PUNCT
cana-4839	57	19	which	which	PRON
cana-4839	57	20	serves	serve	VERB
cana-4839	57	21	as	as	ADP
cana-4839	57	22	a	a	DET
cana-4839	57	23	key	key	ADJ
cana-4839	57	24	indicator	indicator	NOUN
cana-4839	57	25	of	of	ADP
cana-4839	57	26	the	the	DET
cana-4839	57	27	transmission	transmission	NOUN
cana-4839	57	28	potential	potential	NOUN
cana-4839	57	29	of	of	ADP
cana-4839	57	30	the	the	DET
cana-4839	57	31	virus[22	virus[22	NOUN
cana-4839	57	32	]	]	PUNCT
cana-4839	57	33	.	.	PUNCT
cana-4839	58	1	this	this	DET
cana-4839	58	2	number	number	NOUN
cana-4839	58	3	has	have	AUX
cana-4839	58	4	been	be	AUX
cana-4839	58	5	essential	essential	ADJ
cana-4839	58	6	in	in	ADP
cana-4839	58	7	guiding	guide	VERB
cana-4839	58	8	policymakers	policymaker	NOUN
cana-4839	58	9	to	to	PART
cana-4839	58	10	implement	implement	VERB
cana-4839	58	11	effective	effective	ADJ
cana-4839	58	12	public	public	ADJ
cana-4839	58	13	health	health	NOUN
cana-4839	58	14	measures	measure	NOUN
cana-4839	58	15	.	.	PUNCT
cana-4839	59	1	vaccination	vaccination	NOUN
cana-4839	59	2	strategies	strategy	NOUN
cana-4839	59	3	and	and	CCONJ
cana-4839	59	4	their	their	PRON
cana-4839	59	5	impact	impact	NOUN
cana-4839	59	6	vaccination	vaccination	NOUN
cana-4839	59	7	plays	play	VERB
cana-4839	59	8	a	a	DET
cana-4839	59	9	crucial	crucial	ADJ
cana-4839	59	10	role	role	NOUN
cana-4839	59	11	in	in	ADP
cana-4839	59	12	controlling	control	VERB
cana-4839	59	13	the	the	DET
cana-4839	59	14	spread	spread	NOUN
cana-4839	59	15	of	of	ADP
cana-4839	59	16	infectious	infectious	ADJ
cana-4839	59	17	diseases	disease	NOUN
cana-4839	59	18	including	include	VERB
cana-4839	59	19	covid-19	covid-19	PROPN
cana-4839	59	20	.	.	PUNCT
cana-4839	60	1	vaccination	vaccination	NOUN
cana-4839	60	2	campaigns	campaign	NOUN
cana-4839	60	3	exert	exert	VERB
cana-4839	60	4	both	both	CCONJ
cana-4839	60	5	direct	direct	ADJ
cana-4839	60	6	and	and	CCONJ
cana-4839	60	7	indirect	indirect	ADJ
cana-4839	60	8	effects	effect	NOUN
cana-4839	60	9	,	,	PUNCT
cana-4839	60	10	significantly	significantly	ADV
cana-4839	60	11	influencing	influence	VERB
cana-4839	60	12	disease	disease	NOUN
cana-4839	60	13	dynamics	dynamic	NOUN
cana-4839	60	14	within	within	ADP
cana-4839	60	15	populations[23	populations[23	NOUN
cana-4839	60	16	]	]	X
cana-4839	60	17	.	.	PUNCT
cana-4839	61	1	the	the	DET
cana-4839	61	2	development	development	NOUN
cana-4839	61	3	and	and	CCONJ
cana-4839	61	4	deployment	deployment	NOUN
cana-4839	61	5	of	of	ADP
cana-4839	61	6	covid-19	covid-19	PROPN
cana-4839	61	7	vaccines	vaccine	NOUN
cana-4839	61	8	such	such	ADJ
cana-4839	61	9	as	as	ADP
cana-4839	61	10	pfizer	pfizer	NOUN
cana-4839	61	11	,	,	PUNCT
cana-4839	61	12	moderna	moderna	PROPN
cana-4839	61	13	astrazeneca	astrazeneca	PROPN
cana-4839	61	14	(	(	PUNCT
cana-4839	61	15	azd1222	azd1222	NOUN
cana-4839	61	16	)	)	PUNCT
cana-4839	61	17	and	and	CCONJ
cana-4839	61	18	johnson	johnson	PROPN
cana-4839	61	19	&	&	CCONJ
cana-4839	61	20	johnson	johnson	PROPN
cana-4839	61	21	(	(	PUNCT
cana-4839	61	22	j&j	j&j	PROPN
cana-4839	61	23	ad26.cov2.s	ad26.cov2.s	NUM
cana-4839	61	24	)	)	PUNCT
cana-4839	61	25	marked	mark	VERB
cana-4839	61	26	a	a	DET
cana-4839	61	27	significant	significant	ADJ
cana-4839	61	28	milestone	milestone	NOUN
cana-4839	61	29	in	in	ADP
cana-4839	61	30	the	the	DET
cana-4839	61	31	fight	fight	NOUN
cana-4839	61	32	against	against	ADP
cana-4839	61	33	the	the	DET
cana-4839	61	34	virus	virus	NOUN
cana-4839	61	35	taking	take	VERB
cana-4839	61	36	over	over	ADP
cana-4839	61	37	one	one	NUM
cana-4839	61	38	and	and	CCONJ
cana-4839	61	39	a	a	DET
cana-4839	61	40	half	half	NOUN
cana-4839	61	41	years	year	NOUN
cana-4839	61	42	from	from	ADP
cana-4839	61	43	the	the	DET
cana-4839	61	44	inception	inception	NOUN
cana-4839	61	45	of	of	ADP
cana-4839	61	46	research	research	NOUN
cana-4839	61	47	to	to	ADP
cana-4839	61	48	the	the	DET
cana-4839	61	49	first	first	ADJ
cana-4839	61	50	vaccines	vaccine	NOUN
cana-4839	61	51	being	be	AUX
cana-4839	61	52	approved	approve	VERB
cana-4839	61	53	for	for	ADP
cana-4839	61	54	public	public	ADJ
cana-4839	61	55	use	use	NOUN
cana-4839	61	56	in	in	ADP
cana-4839	61	57	march	march	PROPN
cana-4839	61	58	2021[18	2021[18	NUM
cana-4839	61	59	]	]	PUNCT
cana-4839	61	60	.	.	PUNCT
cana-4839	62	1	the	the	DET
cana-4839	62	2	efficacy	efficacy	NOUN
cana-4839	62	3	of	of	ADP
cana-4839	62	4	these	these	DET
cana-4839	62	5	vaccines	vaccine	NOUN
cana-4839	62	6	has	have	AUX
cana-4839	62	7	been	be	AUX
cana-4839	62	8	reported	report	VERB
cana-4839	62	9	to	to	PART
cana-4839	62	10	be	be	AUX
cana-4839	62	11	high	high	ADJ
cana-4839	62	12	,	,	PUNCT
cana-4839	62	13	with	with	ADP
cana-4839	62	14	figures	figure	NOUN
cana-4839	62	15	around	around	ADP
cana-4839	62	16	94	94	NUM
cana-4839	62	17	%	%	NOUN
cana-4839	62	18	for	for	ADP
cana-4839	62	19	moderna	moderna	NOUN
cana-4839	62	20	,	,	PUNCT
cana-4839	62	21	95	95	NUM
cana-4839	62	22	%	%	NOUN
cana-4839	62	23	for	for	ADP
cana-4839	62	24	pfizer	pfizer	NOUN
cana-4839	62	25	,	,	PUNCT
cana-4839	62	26	81.3	81.3	NUM
cana-4839	62	27	%	%	NOUN
cana-4839	62	28	for	for	ADP
cana-4839	62	29	astrazeneca	astrazeneca	NOUN
cana-4839	62	30	and	and	CCONJ
cana-4839	62	31	85	85	NUM
cana-4839	62	32	%	%	NOUN
cana-4839	62	33	for	for	ADP
cana-4839	62	34	j&j	j&j	PROPN
cana-4839	62	35	.	.	PUNCT
cana-4839	63	1	these	these	DET
cana-4839	63	2	vaccines	vaccine	NOUN
cana-4839	63	3	not	not	PART
cana-4839	63	4	only	only	ADV
cana-4839	63	5	help	help	AUX
cana-4839	63	6	prevent	prevent	VERB
cana-4839	63	7	severe	severe	ADJ
cana-4839	63	8	disease	disease	NOUN
cana-4839	63	9	and	and	CCONJ
cana-4839	63	10	mortality	mortality	NOUN
cana-4839	63	11	but	but	CCONJ
cana-4839	63	12	also	also	ADV
cana-4839	63	13	enhance	enhance	VERB
cana-4839	63	14	the	the	DET
cana-4839	63	15	immunity	immunity	NOUN
cana-4839	63	16	of	of	ADP
cana-4839	63	17	the	the	DET
cana-4839	63	18	vaccinated	vaccinated	ADJ
cana-4839	63	19	population	population	NOUN
cana-4839	63	20	against	against	ADP
cana-4839	63	21	the	the	DET
cana-4839	63	22	virus	virus	NOUN
cana-4839	63	23	,	,	PUNCT
cana-4839	63	24	contributing	contribute	VERB
cana-4839	63	25	to	to	ADP
cana-4839	63	26	overall	overall	ADJ
cana-4839	63	27	public	public	ADJ
cana-4839	63	28	health	health	NOUN
cana-4839	63	29	safety	safety	NOUN
cana-4839	63	30	.	.	PUNCT
cana-4839	64	1	in	in	ADP
cana-4839	64	2	conjunction	conjunction	NOUN
cana-4839	64	3	with	with	ADP
cana-4839	64	4	vaccination	vaccination	NOUN
cana-4839	64	5	,	,	PUNCT
cana-4839	64	6	public	public	ADJ
cana-4839	64	7	health	health	NOUN
cana-4839	64	8	measures	measure	NOUN
cana-4839	64	9	such	such	ADJ
cana-4839	64	10	as	as	ADP
cana-4839	64	11	lockdowns	lockdown	NOUN
cana-4839	64	12	,	,	PUNCT
cana-4839	64	13	social	social	ADJ
cana-4839	64	14	distancing	distancing	NOUN
cana-4839	64	15	and	and	CCONJ
cana-4839	64	16	mask	mask	NOUN
cana-4839	64	17	-	-	PUNCT
cana-4839	64	18	wearing	wearing	NOUN
cana-4839	64	19	have	have	AUX
cana-4839	64	20	also	also	ADV
cana-4839	64	21	been	be	AUX
cana-4839	64	22	shown	show	VERB
cana-4839	64	23	to	to	PART
cana-4839	64	24	reduce	reduce	VERB
cana-4839	64	25	the	the	DET
cana-4839	64	26	growth	growth	NOUN
cana-4839	64	27	rate	rate	NOUN
cana-4839	64	28	of	of	ADP
cana-4839	64	29	confirmed	confirm	VERB
cana-4839	64	30	covid19	covid19	NOUN
cana-4839	64	31	cases	case	NOUN
cana-4839	64	32	.	.	PUNCT
cana-4839	65	1	studies	study	NOUN
cana-4839	65	2	indicate	indicate	VERB
cana-4839	65	3	that	that	SCONJ
cana-4839	65	4	these	these	DET
cana-4839	65	5	non	non	ADJ
cana-4839	65	6	-	-	ADJ
cana-4839	65	7	pharmaceutical	pharmaceutical	ADJ
cana-4839	65	8	interventions	intervention	NOUN
cana-4839	65	9	can	can	AUX
cana-4839	65	10	be	be	AUX
cana-4839	65	11	more	more	ADV
cana-4839	65	12	effective	effective	ADJ
cana-4839	65	13	when	when	SCONJ
cana-4839	65	14	combined	combine	VERB
cana-4839	65	15	with	with	ADP
cana-4839	65	16	vaccination	vaccination	NOUN
cana-4839	65	17	strategies[27	strategies[27	NOUN
cana-4839	65	18	]	]	PUNCT
cana-4839	65	19	.	.	PUNCT
cana-4839	66	1	for	for	ADP
cana-4839	66	2	instance	instance	NOUN
cana-4839	66	3	,	,	PUNCT
cana-4839	66	4	effective	effective	ADJ
cana-4839	66	5	lockdowns	lockdown	NOUN
cana-4839	66	6	were	be	AUX
cana-4839	66	7	found	find	VERB
cana-4839	66	8	to	to	PART
cana-4839	66	9	be	be	AUX
cana-4839	66	10	associated	associate	VERB
cana-4839	66	11	with	with	ADP
cana-4839	66	12	varying	vary	VERB
cana-4839	66	13	levels	level	NOUN
cana-4839	66	14	of	of	ADP
cana-4839	66	15	compliance	compliance	NOUN
cana-4839	66	16	and	and	CCONJ
cana-4839	66	17	the	the	DET
cana-4839	66	18	implementation	implementation	NOUN
cana-4839	66	19	of	of	ADP
cana-4839	66	20	other	other	ADJ
cana-4839	66	21	measures	measure	NOUN
cana-4839	66	22	highlighting	highlight	VERB
cana-4839	66	23	the	the	DET
cana-4839	66	24	need	need	NOUN
cana-4839	66	25	for	for	ADP
cana-4839	66	26	a	a	DET
cana-4839	66	27	multifaceted	multifaceted	ADJ
cana-4839	66	28	approach	approach	NOUN
cana-4839	66	29	to	to	ADP
cana-4839	66	30	epidemic	epidemic	NOUN
cana-4839	66	31	control	control	NOUN
cana-4839	66	32	.	.	PUNCT
cana-4839	67	1	moreover	moreover	ADV
cana-4839	67	2	,	,	PUNCT
cana-4839	67	3	understanding	understand	VERB
cana-4839	67	4	the	the	DET
cana-4839	67	5	dynamics	dynamic	NOUN
cana-4839	67	6	of	of	ADP
cana-4839	67	7	vaccination	vaccination	NOUN
cana-4839	67	8	is	be	AUX
cana-4839	67	9	essential	essential	ADJ
cana-4839	67	10	as	as	SCONJ
cana-4839	67	11	it	it	PRON
cana-4839	67	12	relates	relate	VERB
cana-4839	67	13	to	to	ADP
cana-4839	67	14	the	the	DET
cana-4839	67	15	concepts	concept	NOUN
cana-4839	67	16	of	of	ADP
cana-4839	67	17	bifurcation	bifurcation	NOUN
cana-4839	67	18	within	within	ADP
cana-4839	67	19	epidemiological	epidemiological	ADJ
cana-4839	67	20	models	model	NOUN
cana-4839	67	21	.	.	PUNCT
cana-4839	68	1	bifurcation	bifurcation	NOUN
cana-4839	68	2	theory	theory	NOUN
cana-4839	68	3	helps	help	VERB
cana-4839	68	4	in	in	ADP
cana-4839	68	5	analyzing	analyze	VERB
cana-4839	68	6	how	how	SCONJ
cana-4839	68	7	changes	change	NOUN
cana-4839	68	8	in	in	ADP
cana-4839	68	9	vaccination	vaccination	NOUN
cana-4839	68	10	coverage	coverage	NOUN
cana-4839	68	11	can	can	AUX
cana-4839	68	12	lead	lead	VERB
cana-4839	68	13	to	to	ADP
cana-4839	68	14	significant	significant	ADJ
cana-4839	68	15	shifts	shift	NOUN
cana-4839	68	16	in	in	ADP
cana-4839	68	17	epidemic	epidemic	ADJ
cana-4839	68	18	outcomes	outcome	NOUN
cana-4839	68	19	such	such	ADJ
cana-4839	68	20	as	as	ADP
cana-4839	68	21	transitioning	transition	VERB
cana-4839	68	22	from	from	ADP
cana-4839	68	23	a	a	DET
cana-4839	68	24	controlled	control	VERB
cana-4839	68	25	outbreak	outbreak	NOUN
cana-4839	68	26	to	to	ADP
cana-4839	68	27	a	a	DET
cana-4839	68	28	pandemic	pandemic	ADJ
cana-4839	68	29	state	state	NOUN
cana-4839	68	30	.	.	PUNCT
cana-4839	69	1	by	by	ADP
cana-4839	69	2	communications	communication	NOUN
cana-4839	69	3	on	on	ADP
cana-4839	69	4	applied	apply	VERB
cana-4839	69	5	nonlinear	nonlinear	ADJ
cana-4839	69	6	analysis	analysis	NOUN
cana-4839	69	7	issn	issn	NOUN
cana-4839	69	8	:	:	PUNCT
cana-4839	69	9	1074	1074	NUM
cana-4839	69	10	-	-	PUNCT
cana-4839	69	11	133x	133x	NUM
cana-4839	69	12	vol	vol	VERB
cana-4839	69	13	32	32	NUM
cana-4839	69	14	no	no	NOUN
cana-4839	69	15	.	.	PUNCT
cana-4839	70	1	10s	10	NOUN
cana-4839	70	2	(	(	PUNCT
cana-4839	70	3	2025	2025	NUM
cana-4839	70	4	)	)	PUNCT
cana-4839	70	5	441	441	NUM
cana-4839	70	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	70	7	investigating	investigate	VERB
cana-4839	70	8	these	these	DET
cana-4839	70	9	dynamics	dynamic	NOUN
cana-4839	70	10	,	,	PUNCT
cana-4839	70	11	researchers	researcher	NOUN
cana-4839	70	12	can	can	AUX
cana-4839	70	13	better	well	ADV
cana-4839	70	14	develop	develop	VERB
cana-4839	70	15	strategies	strategy	NOUN
cana-4839	70	16	to	to	PART
cana-4839	70	17	optimize	optimize	VERB
cana-4839	70	18	vaccination	vaccination	NOUN
cana-4839	70	19	campaigns	campaign	NOUN
cana-4839	70	20	and	and	CCONJ
cana-4839	70	21	improve	improve	VERB
cana-4839	70	22	public	public	ADJ
cana-4839	70	23	health	health	NOUN
cana-4839	70	24	responses	response	NOUN
cana-4839	70	25	conclusion	conclusion	NOUN
cana-4839	70	26	and	and	CCONJ
cana-4839	70	27	future	future	ADJ
cana-4839	70	28	direction	direction	NOUN
cana-4839	70	29	bifurcation	bifurcation	NOUN
cana-4839	70	30	theory	theory	NOUN
cana-4839	70	31	provides	provide	VERB
cana-4839	70	32	significant	significant	ADJ
cana-4839	70	33	insights	insight	NOUN
cana-4839	70	34	into	into	ADP
cana-4839	70	35	the	the	DET
cana-4839	70	36	dynamics	dynamic	NOUN
cana-4839	70	37	of	of	ADP
cana-4839	70	38	infectious	infectious	ADJ
cana-4839	70	39	disease	disease	NOUN
cana-4839	70	40	spread	spread	VERB
cana-4839	70	41	particularly	particularly	ADV
cana-4839	70	42	in	in	ADP
cana-4839	70	43	the	the	DET
cana-4839	70	44	context	context	NOUN
cana-4839	70	45	of	of	ADP
cana-4839	70	46	the	the	DET
cana-4839	70	47	covid-19	covid-19	PROPN
cana-4839	70	48	pandemic	pandemic	NOUN
cana-4839	70	49	.	.	PUNCT
cana-4839	71	1	it	it	PRON
cana-4839	71	2	highlights	highlight	VERB
cana-4839	71	3	how	how	SCONJ
cana-4839	71	4	changes	change	NOUN
cana-4839	71	5	in	in	ADP
cana-4839	71	6	model	model	NOUN
cana-4839	71	7	parameters	parameter	NOUN
cana-4839	71	8	can	can	AUX
cana-4839	71	9	lead	lead	VERB
cana-4839	71	10	to	to	ADP
cana-4839	71	11	qualitative	qualitative	ADJ
cana-4839	71	12	shifts	shift	NOUN
cana-4839	71	13	in	in	ADP
cana-4839	71	14	epidemic	epidemic	ADJ
cana-4839	71	15	behavior	behavior	NOUN
cana-4839	71	16	such	such	ADJ
cana-4839	71	17	as	as	ADP
cana-4839	71	18	transitions	transition	NOUN
cana-4839	71	19	from	from	ADP
cana-4839	71	20	controlled	control	VERB
cana-4839	71	21	outbreaks	outbreak	NOUN
cana-4839	71	22	to	to	ADP
cana-4839	71	23	widespread	widespread	ADJ
cana-4839	71	24	pandemics	pandemic	NOUN
cana-4839	71	25	.	.	PUNCT
cana-4839	72	1	understanding	understand	VERB
cana-4839	72	2	these	these	DET
cana-4839	72	3	bifurcations	bifurcation	NOUN
cana-4839	72	4	is	be	AUX
cana-4839	72	5	crucial	crucial	ADJ
cana-4839	72	6	for	for	ADP
cana-4839	72	7	developing	develop	VERB
cana-4839	72	8	effective	effective	ADJ
cana-4839	72	9	public	public	ADJ
cana-4839	72	10	health	health	NOUN
cana-4839	72	11	interventions	intervention	NOUN
cana-4839	72	12	and	and	CCONJ
cana-4839	72	13	control	control	NOUN
cana-4839	72	14	strategies	strategy	NOUN
cana-4839	72	15	.	.	PUNCT
cana-4839	73	1	numerical	numerical	ADJ
cana-4839	73	2	simulations	simulation	NOUN
cana-4839	73	3	have	have	AUX
cana-4839	73	4	been	be	AUX
cana-4839	73	5	conducted	conduct	VERB
cana-4839	73	6	to	to	PART
cana-4839	73	7	explore	explore	VERB
cana-4839	73	8	the	the	DET
cana-4839	73	9	stability	stability	NOUN
cana-4839	73	10	regions	region	NOUN
cana-4839	73	11	of	of	ADP
cana-4839	73	12	various	various	ADJ
cana-4839	73	13	parameters	parameter	NOUN
cana-4839	73	14	particularly	particularly	ADV
cana-4839	73	15	focusing	focus	VERB
cana-4839	73	16	on	on	ADP
cana-4839	73	17	the	the	DET
cana-4839	73	18	influence	influence	NOUN
cana-4839	73	19	of	of	ADP
cana-4839	73	20	the	the	DET
cana-4839	73	21	recruitment	recruitment	NOUN
cana-4839	73	22	rate	rate	NOUN
cana-4839	73	23	of	of	ADP
cana-4839	73	24	individuals	individual	NOUN
cana-4839	73	25	into	into	ADP
cana-4839	73	26	the	the	DET
cana-4839	73	27	population	population	NOUN
cana-4839	73	28	which	which	PRON
cana-4839	73	29	can	can	AUX
cana-4839	73	30	be	be	AUX
cana-4839	73	31	modulated	modulate	VERB
cana-4839	73	32	through	through	ADP
cana-4839	73	33	immigration	immigration	NOUN
cana-4839	73	34	policies	policy	NOUN
cana-4839	73	35	.	.	PUNCT
cana-4839	74	1	such	such	ADJ
cana-4839	74	2	parameters	parameter	NOUN
cana-4839	74	3	are	be	AUX
cana-4839	74	4	pivotal	pivotal	ADJ
cana-4839	74	5	in	in	ADP
cana-4839	74	6	determining	determine	VERB
cana-4839	74	7	the	the	DET
cana-4839	74	8	basic	basic	ADJ
cana-4839	74	9	reproduction	reproduction	NOUN
cana-4839	74	10	number	number	NOUN
cana-4839	74	11	(	(	PUNCT
cana-4839	74	12	r0	r0	NOUN
cana-4839	74	13	)	)	PUNCT
cana-4839	74	14	which	which	PRON
cana-4839	74	15	is	be	AUX
cana-4839	74	16	a	a	DET
cana-4839	74	17	critical	critical	ADJ
cana-4839	74	18	metric	metric	NOUN
cana-4839	74	19	for	for	ADP
cana-4839	74	20	assessing	assess	VERB
cana-4839	74	21	the	the	DET
cana-4839	74	22	potential	potential	NOUN
cana-4839	74	23	for	for	SCONJ
cana-4839	74	24	an	an	DET
cana-4839	74	25	epidemic	epidemic	NOUN
cana-4839	74	26	to	to	PART
cana-4839	74	27	grow	grow	VERB
cana-4839	74	28	or	or	CCONJ
cana-4839	74	29	decline	decline	VERB
cana-4839	74	30	.	.	PUNCT
cana-4839	75	1	by	by	ADP
cana-4839	75	2	analyzing	analyze	VERB
cana-4839	75	3	these	these	DET
cana-4839	75	4	stability	stability	NOUN
cana-4839	75	5	regions	region	NOUN
cana-4839	75	6	,	,	PUNCT
cana-4839	75	7	researchers	researcher	NOUN
cana-4839	75	8	can	can	AUX
cana-4839	75	9	predict	predict	VERB
cana-4839	75	10	potential	potential	ADJ
cana-4839	75	11	outbreaks	outbreak	NOUN
cana-4839	75	12	and	and	CCONJ
cana-4839	75	13	formulate	formulate	VERB
cana-4839	75	14	timely	timely	ADJ
cana-4839	75	15	responses	response	NOUN
cana-4839	75	16	.	.	PUNCT
cana-4839	76	1	this	this	DET
cana-4839	76	2	systematic	systematic	ADJ
cana-4839	76	3	review	review	NOUN
cana-4839	76	4	identified	identify	VERB
cana-4839	76	5	several	several	ADJ
cana-4839	76	6	limitations	limitation	NOUN
cana-4839	76	7	in	in	ADP
cana-4839	76	8	the	the	DET
cana-4839	76	9	existing	exist	VERB
cana-4839	76	10	studies	study	NOUN
cana-4839	76	11	on	on	ADP
cana-4839	76	12	public	public	ADJ
cana-4839	76	13	health	health	NOUN
cana-4839	76	14	interventions	intervention	NOUN
cana-4839	76	15	.	.	PUNCT
cana-4839	77	1	the	the	DET
cana-4839	77	2	heterogeneous	heterogeneous	ADJ
cana-4839	77	3	methodologies	methodology	NOUN
cana-4839	77	4	employed	employ	VERB
cana-4839	77	5	,	,	PUNCT
cana-4839	77	6	along	along	ADP
cana-4839	77	7	with	with	ADP
cana-4839	77	8	a	a	DET
cana-4839	77	9	lack	lack	NOUN
cana-4839	77	10	of	of	ADP
cana-4839	77	11	control	control	NOUN
cana-4839	77	12	groups	group	NOUN
cana-4839	77	13	in	in	ADP
cana-4839	77	14	many	many	ADJ
cana-4839	77	15	studies	study	NOUN
cana-4839	77	16	,	,	PUNCT
cana-4839	77	17	hinder	hinder	VERB
cana-4839	77	18	the	the	DET
cana-4839	77	19	accurate	accurate	ADJ
cana-4839	77	20	evaluation	evaluation	NOUN
cana-4839	77	21	of	of	ADP
cana-4839	77	22	the	the	DET
cana-4839	77	23	individual	individual	ADJ
cana-4839	77	24	impact	impact	NOUN
cana-4839	77	25	of	of	ADP
cana-4839	77	26	interventions	intervention	NOUN
cana-4839	77	27	.	.	PUNCT
cana-4839	78	1	most	most	ADJ
cana-4839	78	2	studies	study	NOUN
cana-4839	78	3	were	be	AUX
cana-4839	78	4	categorized	categorize	VERB
cana-4839	78	5	as	as	ADP
cana-4839	78	6	having	have	VERB
cana-4839	78	7	moderate	moderate	ADJ
cana-4839	78	8	quality	quality	NOUN
cana-4839	78	9	,	,	PUNCT
cana-4839	78	10	with	with	ADP
cana-4839	78	11	some	some	PRON
cana-4839	78	12	showing	show	VERB
cana-4839	78	13	low	low	ADJ
cana-4839	78	14	levels	level	NOUN
cana-4839	78	15	of	of	ADP
cana-4839	78	16	evidence	evidence	NOUN
cana-4839	78	17	.	.	PUNCT
cana-4839	79	1	furthermore	furthermore	ADV
cana-4839	79	2	,	,	PUNCT
cana-4839	79	3	the	the	DET
cana-4839	79	4	simultaneous	simultaneous	ADJ
cana-4839	79	5	implementation	implementation	NOUN
cana-4839	79	6	of	of	ADP
cana-4839	79	7	multiple	multiple	ADJ
cana-4839	79	8	interventions	intervention	NOUN
cana-4839	79	9	complicates	complicate	VERB
cana-4839	79	10	the	the	DET
cana-4839	79	11	assessment	assessment	NOUN
cana-4839	79	12	of	of	ADP
cana-4839	79	13	their	their	PRON
cana-4839	79	14	individual	individual	ADJ
cana-4839	79	15	effectiveness	effectiveness	NOUN
cana-4839	79	16	,	,	PUNCT
cana-4839	79	17	potentially	potentially	ADV
cana-4839	79	18	leading	lead	VERB
cana-4839	79	19	to	to	ADP
cana-4839	79	20	misestimations	misestimation	NOUN
cana-4839	79	21	of	of	ADP
cana-4839	79	22	their	their	PRON
cana-4839	79	23	overall	overall	ADJ
cana-4839	79	24	impact	impact	NOUN
cana-4839	79	25	on	on	ADP
cana-4839	79	26	covid-19	covid-19	PROPN
cana-4839	79	27	transmission	transmission	NOUN
cana-4839	79	28	dynamics	dynamic	NOUN
cana-4839	79	29	.	.	PUNCT
cana-4839	80	1	future	future	ADJ
cana-4839	80	2	research	research	NOUN
cana-4839	80	3	should	should	AUX
cana-4839	80	4	focus	focus	VERB
cana-4839	80	5	on	on	ADP
cana-4839	80	6	the	the	DET
cana-4839	80	7	need	need	NOUN
cana-4839	80	8	for	for	ADP
cana-4839	80	9	rigorous	rigorous	ADJ
cana-4839	80	10	methodologies	methodology	NOUN
cana-4839	80	11	particularly	particularly	ADV
cana-4839	80	12	experimental	experimental	ADJ
cana-4839	80	13	and	and	CCONJ
cana-4839	80	14	quasi	quasi	ADJ
cana-4839	80	15	-	-	ADJ
cana-4839	80	16	experimental	experimental	ADJ
cana-4839	80	17	designs	design	NOUN
cana-4839	80	18	to	to	PART
cana-4839	80	19	accurately	accurately	ADV
cana-4839	80	20	assess	assess	VERB
cana-4839	80	21	the	the	DET
cana-4839	80	22	outcomes	outcome	NOUN
cana-4839	80	23	of	of	ADP
cana-4839	80	24	non	non	ADJ
cana-4839	80	25	-	-	ADJ
cana-4839	80	26	pharmaceutical	pharmaceutical	ADJ
cana-4839	80	27	interventions	intervention	NOUN
cana-4839	80	28	and	and	CCONJ
cana-4839	80	29	public	public	ADJ
cana-4839	80	30	health	health	NOUN
cana-4839	80	31	measures	measure	NOUN
cana-4839	80	32	.	.	PUNCT
cana-4839	81	1	such	such	ADJ
cana-4839	81	2	studies	study	NOUN
cana-4839	81	3	will	will	AUX
cana-4839	81	4	be	be	AUX
cana-4839	81	5	vital	vital	ADJ
cana-4839	81	6	for	for	ADP
cana-4839	81	7	refining	refine	VERB
cana-4839	81	8	epidemic	epidemic	NOUN
cana-4839	81	9	models	model	NOUN
cana-4839	81	10	and	and	CCONJ
cana-4839	81	11	improving	improve	VERB
cana-4839	81	12	the	the	DET
cana-4839	81	13	efficacy	efficacy	NOUN
cana-4839	81	14	of	of	ADP
cana-4839	81	15	strategies	strategy	NOUN
cana-4839	81	16	aimed	aim	VERB
cana-4839	81	17	at	at	ADP
cana-4839	81	18	controlling	control	VERB
cana-4839	81	19	the	the	DET
cana-4839	81	20	spread	spread	NOUN
cana-4839	81	21	of	of	ADP
cana-4839	81	22	infectious	infectious	ADJ
cana-4839	81	23	diseases	disease	NOUN
cana-4839	81	24	including	include	VERB
cana-4839	81	25	covid-19	covid-19	PROPN
cana-4839	81	26	.	.	PUNCT
cana-4839	82	1	understanding	understand	VERB
cana-4839	82	2	the	the	DET
cana-4839	82	3	underlying	underlie	VERB
cana-4839	82	4	bifurcation	bifurcation	NOUN
cana-4839	82	5	structures	structure	NOUN
cana-4839	82	6	in	in	ADP
cana-4839	82	7	these	these	DET
cana-4839	82	8	models	model	NOUN
cana-4839	82	9	will	will	AUX
cana-4839	82	10	not	not	PART
cana-4839	82	11	only	only	ADV
cana-4839	82	12	enhance	enhance	VERB
cana-4839	82	13	predictive	predictive	ADJ
cana-4839	82	14	capabilities	capability	NOUN
cana-4839	82	15	but	but	CCONJ
cana-4839	82	16	also	also	ADV
cana-4839	82	17	assist	assist	VERB
cana-4839	82	18	policymakers	policymaker	NOUN
cana-4839	82	19	in	in	ADP
cana-4839	82	20	making	make	VERB
cana-4839	82	21	informed	informed	ADJ
cana-4839	82	22	decisions	decision	NOUN
cana-4839	82	23	regarding	regard	VERB
cana-4839	82	24	public	public	ADJ
cana-4839	82	25	health	health	NOUN
cana-4839	82	26	interventions	intervention	NOUN
cana-4839	82	27	.	.	PUNCT
cana-4839	83	1	references	reference	NOUN
cana-4839	83	2	:	:	PUNCT
cana-4839	84	1	[	[	X
cana-4839	84	2	1	1	NUM
cana-4839	84	3	]	]	PUNCT
cana-4839	84	4	.	.	PUNCT
cana-4839	84	5	akhtar	akhtar	PROPN
cana-4839	84	6	,	,	PUNCT
cana-4839	84	7	m.	m.	PROPN
cana-4839	84	8	w.	w.	PROPN
cana-4839	84	9	,	,	PUNCT
cana-4839	84	10	bashir	bashir	PROPN
cana-4839	84	11	,	,	PUNCT
cana-4839	84	12	z.	z.	PROPN
cana-4839	84	13	,	,	PUNCT
cana-4839	84	14	&	&	CCONJ
cana-4839	84	15	malik	malik	PROPN
cana-4839	84	16	,	,	PUNCT
cana-4839	84	17	m.	m.	NOUN
cana-4839	84	18	g.	g.	PROPN
cana-4839	84	19	(	(	PUNCT
cana-4839	84	20	2025	2025	NUM
cana-4839	84	21	)	)	PUNCT
cana-4839	84	22	.	.	PUNCT
cana-4839	85	1	exploring	explore	VERB
cana-4839	85	2	bifurcations	bifurcation	NOUN
cana-4839	85	3	and	and	CCONJ
cana-4839	85	4	chaos	chaos	NOUN
cana-4839	85	5	in	in	ADP
cana-4839	85	6	an	an	DET
cana-4839	85	7	ecoepidemiological	ecoepidemiological	ADJ
cana-4839	85	8	prey	prey	NOUN
cana-4839	85	9	-	-	PUNCT
cana-4839	85	10	predator	predator	NOUN
cana-4839	85	11	model	model	NOUN
cana-4839	85	12	with	with	ADP
cana-4839	85	13	infected	infected	ADJ
cana-4839	85	14	prey	prey	NOUN
cana-4839	85	15	:	:	PUNCT
cana-4839	85	16	optimization	optimization	NOUN
cana-4839	85	17	with	with	ADP
cana-4839	85	18	rbfnn	rbfnn	NOUN
cana-4839	85	19	.	.	PUNCT
cana-4839	86	1	journal	journal	NOUN
cana-4839	86	2	of	of	ADP
cana-4839	86	3	dynamical	dynamical	ADJ
cana-4839	86	4	and	and	CCONJ
cana-4839	86	5	control	control	NOUN
cana-4839	86	6	systems	system	NOUN
cana-4839	86	7	,	,	PUNCT
cana-4839	86	8	31(1	31(1	NUM
cana-4839	86	9	)	)	PUNCT
cana-4839	86	10	,	,	PUNCT
cana-4839	86	11	1	1	NUM
cana-4839	86	12	-	-	SYM
cana-4839	86	13	35	35	NUM
cana-4839	86	14	.	.	PUNCT
cana-4839	87	1	[	[	X
cana-4839	87	2	2	2	NUM
cana-4839	87	3	]	]	PUNCT
cana-4839	87	4	.	.	PUNCT
cana-4839	88	1	nadim	nadim	PROPN
cana-4839	88	2	,	,	PUNCT
cana-4839	88	3	s.	s.	PROPN
cana-4839	88	4	s.	s.	PROPN
cana-4839	88	5	,	,	PUNCT
cana-4839	88	6	&	&	CCONJ
cana-4839	88	7	chattopadhyay	chattopadhyay	PROPN
cana-4839	88	8	,	,	PUNCT
cana-4839	88	9	j.	j.	PROPN
cana-4839	88	10	(	(	PUNCT
cana-4839	88	11	2020	2020	NUM
cana-4839	88	12	)	)	PUNCT
cana-4839	88	13	.	.	PUNCT
cana-4839	89	1	occurrence	occurrence	NOUN
cana-4839	89	2	of	of	ADP
cana-4839	89	3	backward	backward	ADJ
cana-4839	89	4	bifurcation	bifurcation	NOUN
cana-4839	89	5	and	and	CCONJ
cana-4839	89	6	prediction	prediction	NOUN
cana-4839	89	7	of	of	ADP
cana-4839	89	8	disease	disease	NOUN
cana-4839	89	9	transmission	transmission	NOUN
cana-4839	89	10	with	with	ADP
cana-4839	89	11	imperfect	imperfect	ADJ
cana-4839	89	12	lockdown	lockdown	NOUN
cana-4839	89	13	:	:	PUNCT
cana-4839	89	14	a	a	DET
cana-4839	89	15	case	case	NOUN
cana-4839	89	16	study	study	NOUN
cana-4839	89	17	on	on	ADP
cana-4839	89	18	covid-19	covid-19	PROPN
cana-4839	89	19	.	.	PUNCT
cana-4839	89	20	chaos	chaos	PROPN
cana-4839	89	21	,	,	PUNCT
cana-4839	89	22	solitons	soliton	NOUN
cana-4839	89	23	&	&	CCONJ
cana-4839	89	24	fractals	fractal	NOUN
cana-4839	89	25	,	,	PUNCT
cana-4839	89	26	140	140	NUM
cana-4839	89	27	,	,	PUNCT
cana-4839	89	28	110163	110163	NUM
cana-4839	89	29	.	.	PUNCT
cana-4839	90	1	[	[	X
cana-4839	90	2	3	3	NUM
cana-4839	90	3	]	]	PUNCT
cana-4839	90	4	.	.	PUNCT
cana-4839	91	1	kiseleva	kiseleva	PROPN
cana-4839	91	2	,	,	PUNCT
cana-4839	91	3	o.	o.	PROPN
cana-4839	91	4	,	,	PUNCT
cana-4839	91	5	yakovlev	yakovlev	PROPN
cana-4839	91	6	,	,	PUNCT
cana-4839	91	7	s.	s.	PROPN
cana-4839	91	8	,	,	PUNCT
cana-4839	91	9	chumachenko	chumachenko	PROPN
cana-4839	91	10	,	,	PUNCT
cana-4839	91	11	d.	d.	PROPN
cana-4839	91	12	,	,	PUNCT
cana-4839	91	13	&	&	CCONJ
cana-4839	91	14	kuzenkov	kuzenkov	PROPN
cana-4839	91	15	,	,	PUNCT
cana-4839	91	16	o.	o.	PROPN
cana-4839	91	17	(	(	PUNCT
cana-4839	91	18	2024	2024	NUM
cana-4839	91	19	)	)	PUNCT
cana-4839	91	20	.	.	PUNCT
cana-4839	92	1	exploring	explore	VERB
cana-4839	92	2	bifurcation	bifurcation	NOUN
cana-4839	92	3	in	in	ADP
cana-4839	92	4	the	the	DET
cana-4839	92	5	compartmental	compartmental	ADJ
cana-4839	92	6	mathematical	mathematical	ADJ
cana-4839	92	7	model	model	NOUN
cana-4839	92	8	of	of	ADP
cana-4839	92	9	covid-19	covid-19	PROPN
cana-4839	92	10	transmission	transmission	NOUN
cana-4839	92	11	.	.	PUNCT
cana-4839	93	1	computation	computation	NOUN
cana-4839	93	2	,	,	PUNCT
cana-4839	93	3	12(9	12(9	NUM
cana-4839	93	4	)	)	PUNCT
cana-4839	93	5	,	,	PUNCT
cana-4839	93	6	186	186	NUM
cana-4839	93	7	.	.	PUNCT
cana-4839	94	1	[	[	X
cana-4839	94	2	4	4	NUM
cana-4839	94	3	]	]	PUNCT
cana-4839	94	4	.	.	PUNCT
cana-4839	95	1	ahmed	ahmed	PROPN
cana-4839	95	2	,	,	PUNCT
cana-4839	95	3	m.	m.	NOUN
cana-4839	95	4	,	,	PUNCT
cana-4839	95	5	masud	masud	PROPN
cana-4839	95	6	,	,	PUNCT
cana-4839	95	7	m.	m.	NOUN
cana-4839	95	8	a.	a.	PROPN
cana-4839	95	9	b.	b.	PROPN
cana-4839	95	10	,	,	PUNCT
cana-4839	95	11	&	&	CCONJ
cana-4839	95	12	sarker	sarker	PROPN
cana-4839	95	13	,	,	PUNCT
cana-4839	95	14	m.	m.	NOUN
cana-4839	95	15	m.	m.	NOUN
cana-4839	95	16	a.	a.	PROPN
cana-4839	95	17	(	(	PUNCT
cana-4839	95	18	2023	2023	NUM
cana-4839	95	19	)	)	PUNCT
cana-4839	95	20	.	.	PUNCT
cana-4839	96	1	bifurcation	bifurcation	NOUN
cana-4839	96	2	analysis	analysis	NOUN
cana-4839	96	3	and	and	CCONJ
cana-4839	96	4	optimal	optimal	ADJ
cana-4839	96	5	control	control	NOUN
cana-4839	96	6	of	of	ADP
cana-4839	96	7	discrete	discrete	ADJ
cana-4839	96	8	sir	sir	NOUN
cana-4839	96	9	model	model	NOUN
cana-4839	96	10	for	for	ADP
cana-4839	96	11	covid-19	covid-19	PROPN
cana-4839	96	12	.	.	PUNCT
cana-4839	96	13	chaos	chaos	PROPN
cana-4839	96	14	,	,	PUNCT
cana-4839	96	15	solitons	soliton	NOUN
cana-4839	96	16	&	&	CCONJ
cana-4839	96	17	fractals	fractal	NOUN
cana-4839	96	18	,	,	PUNCT
cana-4839	96	19	174	174	NUM
cana-4839	96	20	,	,	PUNCT
cana-4839	96	21	113899	113899	NUM
cana-4839	96	22	.	.	PUNCT
cana-4839	97	1	[	[	X
cana-4839	97	2	5	5	NUM
cana-4839	97	3	]	]	PUNCT
cana-4839	97	4	.	.	PUNCT
cana-4839	98	1	idisi	idisi	PROPN
cana-4839	98	2	,	,	PUNCT
cana-4839	98	3	o.	o.	PROPN
cana-4839	98	4	i.	i.	PROPN
cana-4839	98	5	,	,	PUNCT
cana-4839	98	6	yusuf	yusuf	PROPN
cana-4839	98	7	,	,	PUNCT
cana-4839	98	8	t.	t.	PROPN
cana-4839	98	9	t.	t.	PROPN
cana-4839	98	10	,	,	PUNCT
cana-4839	98	11	owolabi	owolabi	NOUN
cana-4839	98	12	,	,	PUNCT
cana-4839	98	13	k.	k.	PROPN
cana-4839	98	14	m.	m.	PROPN
cana-4839	98	15	,	,	PUNCT
cana-4839	98	16	&	&	CCONJ
cana-4839	98	17	ojokoh	ojokoh	PROPN
cana-4839	98	18	,	,	PUNCT
cana-4839	98	19	b.	b.	PROPN
cana-4839	98	20	a.	a.	PROPN
cana-4839	98	21	(	(	PUNCT
cana-4839	98	22	2023	2023	NUM
cana-4839	98	23	)	)	PUNCT
cana-4839	98	24	.	.	PUNCT
cana-4839	99	1	a	a	DET
cana-4839	99	2	bifurcation	bifurcation	NOUN
cana-4839	99	3	analysis	analysis	NOUN
cana-4839	99	4	and	and	CCONJ
cana-4839	99	5	model	model	NOUN
cana-4839	99	6	of	of	ADP
cana-4839	99	7	covid-19	covid-19	PROPN
cana-4839	99	8	transmission	transmission	NOUN
cana-4839	99	9	dynamics	dynamic	NOUN
cana-4839	99	10	with	with	ADP
cana-4839	99	11	post	post	ADJ
cana-4839	99	12	-	-	ADJ
cana-4839	99	13	vaccination	vaccination	ADJ
cana-4839	99	14	infection	infection	NOUN
cana-4839	99	15	impact	impact	NOUN
cana-4839	99	16	.	.	PUNCT
cana-4839	100	1	healthcare	healthcare	NOUN
cana-4839	100	2	analytics	analytic	NOUN
cana-4839	100	3	,	,	PUNCT
cana-4839	100	4	3	3	NUM
cana-4839	100	5	,	,	PUNCT
cana-4839	100	6	100157	100157	NUM
cana-4839	100	7	.	.	PUNCT
cana-4839	101	1	communications	communication	NOUN
cana-4839	101	2	on	on	ADP
cana-4839	101	3	applied	apply	VERB
cana-4839	101	4	nonlinear	nonlinear	ADJ
cana-4839	101	5	analysis	analysis	NOUN
cana-4839	101	6	issn	issn	NOUN
cana-4839	101	7	:	:	PUNCT
cana-4839	101	8	1074	1074	NUM
cana-4839	101	9	-	-	PUNCT
cana-4839	101	10	133x	133x	NUM
cana-4839	101	11	vol	vol	VERB
cana-4839	101	12	32	32	NUM
cana-4839	101	13	no	no	NOUN
cana-4839	101	14	.	.	PUNCT
cana-4839	102	1	10s	10	NOUN
cana-4839	102	2	(	(	PUNCT
cana-4839	102	3	2025	2025	NUM
cana-4839	102	4	)	)	PUNCT
cana-4839	102	5	442	442	NUM
cana-4839	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	103	1	[	[	X
cana-4839	103	2	6	6	NUM
cana-4839	103	3	]	]	PUNCT
cana-4839	103	4	.	.	PUNCT
cana-4839	104	1	kuddus	kuddus	NOUN
cana-4839	104	2	,	,	PUNCT
cana-4839	104	3	m.	m.	NOUN
cana-4839	104	4	a.	a.	PROPN
cana-4839	104	5	,	,	PUNCT
cana-4839	104	6	mohiuddin	mohiuddin	NOUN
cana-4839	104	7	,	,	PUNCT
cana-4839	104	8	m.	m.	NOUN
cana-4839	104	9	,	,	PUNCT
cana-4839	104	10	paul	paul	PROPN
cana-4839	104	11	,	,	PUNCT
cana-4839	104	12	a.	a.	PROPN
cana-4839	104	13	k.	k.	PROPN
cana-4839	104	14	,	,	PUNCT
cana-4839	104	15	&	&	CCONJ
cana-4839	104	16	rahman	rahman	PROPN
cana-4839	104	17	,	,	PUNCT
cana-4839	104	18	a.	a.	NOUN
cana-4839	104	19	(	(	PUNCT
cana-4839	104	20	2024	2024	NUM
cana-4839	104	21	)	)	PUNCT
cana-4839	104	22	.	.	PUNCT
cana-4839	105	1	insights	insight	NOUN
cana-4839	105	2	from	from	ADP
cana-4839	105	3	qualitative	qualitative	NOUN
cana-4839	105	4	and	and	CCONJ
cana-4839	105	5	bifurcation	bifurcation	NOUN
cana-4839	105	6	analysis	analysis	NOUN
cana-4839	105	7	of	of	ADP
cana-4839	105	8	covid-19	covid-19	PROPN
cana-4839	105	9	vaccination	vaccination	NOUN
cana-4839	105	10	model	model	NOUN
cana-4839	105	11	in	in	ADP
cana-4839	105	12	bangladesh	bangladesh	PROPN
cana-4839	105	13	.	.	PUNCT
cana-4839	106	1	plos	plos	PROPN
cana-4839	106	2	one	one	NUM
cana-4839	106	3	,	,	PUNCT
cana-4839	106	4	19(11	19(11	NUM
cana-4839	106	5	)	)	PUNCT
cana-4839	106	6	,	,	PUNCT
cana-4839	106	7	e0312780	e0312780	NOUN
cana-4839	106	8	.	.	PUNCT
cana-4839	107	1	[	[	X
cana-4839	107	2	7	7	NUM
cana-4839	107	3	]	]	PUNCT
cana-4839	107	4	.	.	PUNCT
cana-4839	107	5	wanjala	wanjala	PROPN
cana-4839	107	6	,	,	PUNCT
cana-4839	107	7	h.	h.	PROPN
cana-4839	107	8	,	,	PUNCT
cana-4839	107	9	okongo	okongo	PROPN
cana-4839	107	10	,	,	PUNCT
cana-4839	107	11	m.	m.	NOUN
cana-4839	107	12	,	,	PUNCT
cana-4839	107	13	&	&	CCONJ
cana-4839	107	14	ochwach	ochwach	PROPN
cana-4839	107	15	,	,	PUNCT
cana-4839	107	16	j.	j.	PROPN
cana-4839	107	17	(	(	PUNCT
cana-4839	107	18	2024	2024	NUM
cana-4839	107	19	)	)	PUNCT
cana-4839	107	20	.	.	PUNCT
cana-4839	108	1	bifurcation	bifurcation	NOUN
cana-4839	108	2	analysis	analysis	NOUN
cana-4839	108	3	of	of	ADP
cana-4839	108	4	an	an	DET
cana-4839	108	5	infectious	infectious	ADJ
cana-4839	108	6	respiratory	respiratory	ADJ
cana-4839	108	7	disease	disease	NOUN
cana-4839	108	8	with	with	ADP
cana-4839	108	9	lockdown	lockdown	NOUN
cana-4839	108	10	and	and	CCONJ
cana-4839	108	11	social	social	ADJ
cana-4839	108	12	distancing	distancing	NOUN
cana-4839	108	13	.	.	PUNCT
cana-4839	109	1	arrus	arrus	NOUN
cana-4839	109	2	journal	journal	NOUN
cana-4839	109	3	of	of	ADP
cana-4839	109	4	mathematics	mathematic	NOUN
cana-4839	109	5	and	and	CCONJ
cana-4839	109	6	applied	apply	VERB
cana-4839	109	7	science	science	NOUN
cana-4839	109	8	,	,	PUNCT
cana-4839	109	9	4(2	4(2	NUM
cana-4839	109	10	)	)	PUNCT
cana-4839	109	11	,	,	PUNCT
cana-4839	109	12	118	118	NUM
cana-4839	109	13	-	-	SYM
cana-4839	109	14	134	134	NUM
cana-4839	109	15	.	.	PUNCT
cana-4839	110	1	[	[	X
cana-4839	110	2	8	8	NUM
cana-4839	110	3	]	]	PUNCT
cana-4839	110	4	.	.	PUNCT
cana-4839	111	1	sharbayta	sharbayta	NOUN
cana-4839	111	2	,	,	PUNCT
cana-4839	111	3	s.	s.	PROPN
cana-4839	111	4	s.	s.	PROPN
cana-4839	111	5	,	,	PUNCT
cana-4839	111	6	desta	desta	PROPN
cana-4839	111	7	,	,	PUNCT
cana-4839	111	8	h.	h.	PROPN
cana-4839	111	9	d.	d.	PROPN
cana-4839	111	10	,	,	PUNCT
cana-4839	111	11	&	&	CCONJ
cana-4839	111	12	abdi	abdi	PROPN
cana-4839	111	13	,	,	PUNCT
cana-4839	111	14	t.	t.	PROPN
cana-4839	111	15	(	(	PUNCT
cana-4839	111	16	2023	2023	NUM
cana-4839	111	17	)	)	PUNCT
cana-4839	111	18	.	.	PUNCT
cana-4839	112	1	mathematical	mathematical	ADJ
cana-4839	112	2	modelling	modelling	NOUN
cana-4839	112	3	of	of	ADP
cana-4839	112	4	covid‐19	covid‐19	PROPN
cana-4839	112	5	transmission	transmission	NOUN
cana-4839	112	6	dynamics	dynamic	NOUN
cana-4839	112	7	with	with	ADP
cana-4839	112	8	vaccination	vaccination	NOUN
cana-4839	112	9	:	:	PUNCT
cana-4839	112	10	a	a	DET
cana-4839	112	11	case	case	NOUN
cana-4839	112	12	study	study	NOUN
cana-4839	112	13	in	in	ADP
cana-4839	112	14	ethiopia	ethiopia	PROPN
cana-4839	112	15	.	.	PUNCT
cana-4839	113	1	discrete	discrete	ADJ
cana-4839	113	2	dynamics	dynamic	NOUN
cana-4839	113	3	in	in	ADP
cana-4839	113	4	nature	nature	NOUN
cana-4839	113	5	and	and	CCONJ
cana-4839	113	6	society	society	NOUN
cana-4839	113	7	,	,	PUNCT
cana-4839	113	8	2023(1	2023(1	PROPN
cana-4839	113	9	)	)	PUNCT
cana-4839	113	10	,	,	PUNCT
cana-4839	113	11	2972164	2972164	NUM
cana-4839	113	12	.	.	PUNCT
cana-4839	114	1	[	[	X
cana-4839	114	2	9	9	NUM
cana-4839	114	3	]	]	PUNCT
cana-4839	114	4	.	.	PUNCT
cana-4839	115	1	radha	radha	PROPN
cana-4839	115	2	,	,	PUNCT
cana-4839	115	3	m.	m.	NOUN
cana-4839	115	4	,	,	PUNCT
cana-4839	115	5	&	&	CCONJ
cana-4839	115	6	balamuralitharan	balamuralitharan	PROPN
cana-4839	115	7	,	,	PUNCT
cana-4839	115	8	s.	s.	PROPN
cana-4839	115	9	(	(	PUNCT
cana-4839	115	10	2020	2020	NUM
cana-4839	115	11	)	)	PUNCT
cana-4839	115	12	.	.	PUNCT
cana-4839	116	1	a	a	DET
cana-4839	116	2	study	study	NOUN
cana-4839	116	3	on	on	ADP
cana-4839	116	4	covid-19	covid-19	PROPN
cana-4839	116	5	transmission	transmission	NOUN
cana-4839	116	6	dynamics	dynamic	NOUN
cana-4839	116	7	:	:	PUNCT
cana-4839	116	8	stability	stability	NOUN
cana-4839	116	9	analysis	analysis	NOUN
cana-4839	116	10	of	of	ADP
cana-4839	116	11	seir	seir	ADJ
cana-4839	116	12	model	model	NOUN
cana-4839	116	13	with	with	ADP
cana-4839	116	14	hopf	hopf	ADJ
cana-4839	116	15	bifurcation	bifurcation	NOUN
cana-4839	116	16	for	for	ADP
cana-4839	116	17	effect	effect	NOUN
cana-4839	116	18	of	of	ADP
cana-4839	116	19	time	time	NOUN
cana-4839	116	20	delay	delay	NOUN
cana-4839	116	21	.	.	PUNCT
cana-4839	117	1	advances	advance	NOUN
cana-4839	117	2	in	in	ADP
cana-4839	117	3	difference	difference	NOUN
cana-4839	117	4	equations	equation	NOUN
cana-4839	117	5	,	,	PUNCT
cana-4839	117	6	2020	2020	NUM
cana-4839	117	7	,	,	PUNCT
cana-4839	117	8	1	1	NUM
cana-4839	117	9	-	-	SYM
cana-4839	117	10	20	20	NUM
cana-4839	117	11	.	.	PUNCT
cana-4839	118	1	[	[	X
cana-4839	118	2	10	10	NUM
cana-4839	118	3	]	]	PUNCT
cana-4839	118	4	.	.	PUNCT
cana-4839	119	1	wang	wang	PROPN
cana-4839	119	2	,	,	PUNCT
cana-4839	119	3	x.	x.	PROPN
cana-4839	119	4	,	,	PUNCT
cana-4839	119	5	wang	wang	PROPN
cana-4839	119	6	,	,	PUNCT
cana-4839	119	7	s.	s.	PROPN
cana-4839	119	8	,	,	PUNCT
cana-4839	119	9	wang	wang	PROPN
cana-4839	119	10	,	,	PUNCT
cana-4839	119	11	j.	j.	PROPN
cana-4839	119	12	,	,	PUNCT
cana-4839	119	13	&	&	CCONJ
cana-4839	119	14	rong	rong	PROPN
cana-4839	119	15	,	,	PUNCT
cana-4839	119	16	l.	l.	PROPN
cana-4839	119	17	(	(	PUNCT
cana-4839	119	18	2022	2022	NUM
cana-4839	119	19	)	)	PUNCT
cana-4839	119	20	.	.	PUNCT
cana-4839	120	1	a	a	DET
cana-4839	120	2	multiscale	multiscale	ADJ
cana-4839	120	3	model	model	NOUN
cana-4839	120	4	of	of	ADP
cana-4839	120	5	covid-19	covid-19	PROPN
cana-4839	120	6	dynamics	dynamic	NOUN
cana-4839	120	7	.	.	PUNCT
cana-4839	121	1	bulletin	bulletin	NOUN
cana-4839	121	2	of	of	ADP
cana-4839	121	3	mathematical	mathematical	ADJ
cana-4839	121	4	biology	biology	NOUN
cana-4839	121	5	,	,	PUNCT
cana-4839	121	6	84(9	84(9	NUM
cana-4839	121	7	)	)	PUNCT
cana-4839	121	8	,	,	PUNCT
cana-4839	121	9	99	99	NUM
cana-4839	121	10	.	.	PUNCT
cana-4839	122	1	[	[	X
cana-4839	122	2	11	11	NUM
cana-4839	122	3	]	]	PUNCT
cana-4839	122	4	.	.	PUNCT
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cana-4839	123	2	,	,	PUNCT
cana-4839	123	3	b.	b.	PROPN
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cana-4839	123	7	,	,	PUNCT
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cana-4839	123	9	w.	w.	PROPN
cana-4839	123	10	,	,	PUNCT
cana-4839	123	11	&	&	CCONJ
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cana-4839	123	13	,	,	PUNCT
cana-4839	123	14	y.	y.	PROPN
cana-4839	123	15	f.	f.	PROPN
cana-4839	123	16	(	(	PUNCT
cana-4839	123	17	2023	2023	NUM
cana-4839	123	18	)	)	PUNCT
cana-4839	123	19	.	.	PUNCT
cana-4839	124	1	bifurcation	bifurcation	NOUN
cana-4839	124	2	and	and	CCONJ
cana-4839	124	3	optimal	optimal	ADJ
cana-4839	124	4	control	control	NOUN
cana-4839	124	5	analysis	analysis	NOUN
cana-4839	124	6	of	of	ADP
cana-4839	124	7	hiv	hiv	PROPN
cana-4839	124	8	/	/	SYM
cana-4839	124	9	aids	aids	PROPN
cana-4839	124	10	and	and	CCONJ
cana-4839	124	11	covid-19	covid-19	PROPN
cana-4839	124	12	co	co	ADJ
cana-4839	124	13	-	-	ADJ
cana-4839	124	14	infection	infection	NOUN
cana-4839	124	15	model	model	NOUN
cana-4839	124	16	with	with	ADP
cana-4839	124	17	numerical	numerical	PROPN
cana-4839	124	18	simulation	simulation	PROPN
cana-4839	124	19	.	.	PUNCT
cana-4839	125	1	plos	plos	PROPN
cana-4839	125	2	one	one	NUM
cana-4839	125	3	,	,	PUNCT
cana-4839	125	4	18(5	18(5	NUM
cana-4839	125	5	)	)	PUNCT
cana-4839	125	6	,	,	PUNCT
cana-4839	125	7	e0284759	e0284759	PROPN
cana-4839	125	8	.	.	PUNCT
cana-4839	126	1	[	[	X
cana-4839	126	2	12	12	NUM
cana-4839	126	3	]	]	PUNCT
cana-4839	126	4	.	.	PUNCT
cana-4839	127	1	small	small	ADJ
cana-4839	127	2	,	,	PUNCT
cana-4839	127	3	m.	m.	NOUN
cana-4839	127	4	,	,	PUNCT
cana-4839	127	5	&	&	CCONJ
cana-4839	127	6	cavanagh	cavanagh	PROPN
cana-4839	127	7	,	,	PUNCT
cana-4839	127	8	d.	d.	PROPN
cana-4839	127	9	(	(	PUNCT
cana-4839	127	10	2020	2020	NUM
cana-4839	127	11	)	)	PUNCT
cana-4839	127	12	.	.	PUNCT
cana-4839	128	1	modelling	model	VERB
cana-4839	128	2	strong	strong	ADJ
cana-4839	128	3	control	control	NOUN
cana-4839	128	4	measures	measure	NOUN
cana-4839	128	5	for	for	ADP
cana-4839	128	6	epidemic	epidemic	ADJ
cana-4839	128	7	propagation	propagation	NOUN
cana-4839	128	8	with	with	ADP
cana-4839	128	9	networks	network	NOUN
cana-4839	128	10	—	—	PUNCT
cana-4839	128	11	a	a	DET
cana-4839	128	12	covid-19	covid-19	PROPN
cana-4839	128	13	case	case	NOUN
cana-4839	128	14	study	study	NOUN
cana-4839	128	15	.	.	PUNCT
cana-4839	129	1	ieee	ieee	NOUN
cana-4839	129	2	access	access	NOUN
cana-4839	129	3	,	,	PUNCT
cana-4839	129	4	8	8	NUM
cana-4839	129	5	,	,	PUNCT
cana-4839	129	6	109719	109719	NUM
cana-4839	129	7	-	-	SYM
cana-4839	129	8	109731	109731	NUM
cana-4839	129	9	.	.	PUNCT
cana-4839	130	1	[	[	X
cana-4839	130	2	13	13	NUM
cana-4839	130	3	]	]	PUNCT
cana-4839	130	4	.	.	PUNCT
cana-4839	131	1	samma	samma	PROPN
cana-4839	131	2	,	,	PUNCT
cana-4839	131	3	f.	f.	PROPN
cana-4839	131	4	m.	m.	PROPN
cana-4839	131	5	a.	a.	PROPN
cana-4839	131	6	,	,	PUNCT
cana-4839	131	7	avinash	avinash	NOUN
cana-4839	131	8	,	,	PUNCT
cana-4839	131	9	n.	n.	NOUN
cana-4839	131	10	,	,	PUNCT
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cana-4839	131	12	,	,	PUNCT
cana-4839	131	13	p.	p.	NOUN
cana-4839	131	14	,	,	PUNCT
cana-4839	131	15	albasheir	albasheir	PROPN
cana-4839	131	16	,	,	PUNCT
cana-4839	131	17	n.	n.	PROPN
cana-4839	131	18	a.	a.	PROPN
cana-4839	131	19	,	,	PUNCT
cana-4839	131	20	gargouri	gargouri	PROPN
cana-4839	131	21	,	,	PUNCT
cana-4839	131	22	a.	a.	NOUN
cana-4839	131	23	,	,	PUNCT
cana-4839	131	24	britto	britto	PROPN
cana-4839	131	25	antony	antony	PROPN
cana-4839	131	26	xavier	xavier	PROPN
cana-4839	131	27	,	,	PUNCT
cana-4839	131	28	g.	g.	PROPN
cana-4839	131	29	,	,	PUNCT
cana-4839	131	30	&	&	CCONJ
cana-4839	131	31	almazah	almazah	PROPN
cana-4839	131	32	,	,	PUNCT
cana-4839	131	33	m.	m.	NOUN
cana-4839	131	34	m.	m.	NOUN
cana-4839	131	35	(	(	PUNCT
cana-4839	131	36	2024	2024	NUM
cana-4839	131	37	)	)	PUNCT
cana-4839	131	38	.	.	PUNCT
cana-4839	132	1	exploring	explore	VERB
cana-4839	132	2	symmetry	symmetry	NOUN
cana-4839	132	3	in	in	ADP
cana-4839	132	4	an	an	DET
cana-4839	132	5	epidemiological	epidemiological	ADJ
cana-4839	132	6	model	model	NOUN
cana-4839	132	7	:	:	PUNCT
cana-4839	132	8	numerical	numerical	ADJ
cana-4839	132	9	analysis	analysis	NOUN
cana-4839	132	10	of	of	ADP
cana-4839	132	11	backward	backward	ADJ
cana-4839	132	12	bifurcation	bifurcation	NOUN
cana-4839	132	13	and	and	CCONJ
cana-4839	132	14	sensitivity	sensitivity	NOUN
cana-4839	132	15	indices	index	NOUN
cana-4839	132	16	.	.	PUNCT
cana-4839	133	1	symmetry	symmetry	NOUN
cana-4839	133	2	,	,	PUNCT
cana-4839	133	3	16(12	16(12	NUM
cana-4839	133	4	)	)	PUNCT
cana-4839	133	5	,	,	PUNCT
cana-4839	133	6	1579	1579	NUM
cana-4839	133	7	.	.	PUNCT
cana-4839	134	1	[	[	X
cana-4839	134	2	14	14	NUM
cana-4839	134	3	]	]	PUNCT
cana-4839	134	4	.	.	PUNCT
cana-4839	135	1	ullah	ullah	PROPN
cana-4839	135	2	,	,	PUNCT
cana-4839	135	3	i.	i.	PROPN
cana-4839	135	4	,	,	PUNCT
cana-4839	135	5	ali	ali	PROPN
cana-4839	135	6	,	,	PUNCT
cana-4839	135	7	n.	n.	PROPN
cana-4839	135	8	,	,	PUNCT
cana-4839	135	9	haq	haq	PROPN
cana-4839	135	10	,	,	PUNCT
cana-4839	135	11	i.	i.	PROPN
cana-4839	135	12	u.	u.	PROPN
cana-4839	135	13	,	,	PUNCT
cana-4839	135	14	ahmad	ahmad	PROPN
cana-4839	135	15	,	,	PUNCT
cana-4839	135	16	i.	i.	PROPN
cana-4839	135	17	,	,	PUNCT
cana-4839	135	18	albalwi	albalwi	NOUN
cana-4839	135	19	,	,	PUNCT
cana-4839	135	20	m.	m.	NOUN
cana-4839	135	21	d.	d.	PROPN
cana-4839	135	22	,	,	PUNCT
cana-4839	135	23	&	&	CCONJ
cana-4839	135	24	ali	ali	PROPN
cana-4839	135	25	biswas	biswas	PROPN
cana-4839	135	26	,	,	PUNCT
cana-4839	135	27	m.	m.	PROPN
cana-4839	135	28	h.	h.	PROPN
cana-4839	135	29	(	(	PUNCT
cana-4839	135	30	2024	2024	NUM
cana-4839	135	31	)	)	PUNCT
cana-4839	135	32	.	.	PUNCT
cana-4839	136	1	analysis	analysis	NOUN
cana-4839	136	2	of	of	ADP
cana-4839	136	3	covid‐19	covid‐19	PROPN
cana-4839	136	4	disease	disease	NOUN
cana-4839	136	5	model	model	NOUN
cana-4839	136	6	:	:	PUNCT
cana-4839	136	7	backward	backward	ADJ
cana-4839	136	8	bifurcation	bifurcation	NOUN
cana-4839	136	9	and	and	CCONJ
cana-4839	136	10	impact	impact	NOUN
cana-4839	136	11	of	of	ADP
cana-4839	136	12	pharmaceutical	pharmaceutical	ADJ
cana-4839	136	13	and	and	CCONJ
cana-4839	136	14	nonpharmaceutical	nonpharmaceutical	ADJ
cana-4839	136	15	interventions	intervention	NOUN
cana-4839	136	16	.	.	PUNCT
cana-4839	137	1	international	international	ADJ
cana-4839	137	2	journal	journal	NOUN
cana-4839	137	3	of	of	ADP
cana-4839	137	4	mathematics	mathematics	PROPN
cana-4839	137	5	and	and	CCONJ
cana-4839	137	6	mathematical	mathematical	ADJ
cana-4839	137	7	sciences	science	NOUN
cana-4839	137	8	,	,	PUNCT
cana-4839	137	9	2024(1	2024(1	NUM
cana-4839	137	10	)	)	PUNCT
cana-4839	137	11	,	,	PUNCT
cana-4839	137	12	6069996	6069996	NUM
cana-4839	137	13	.	.	PUNCT
cana-4839	138	1	[	[	X
cana-4839	138	2	15	15	NUM
cana-4839	138	3	]	]	PUNCT
cana-4839	138	4	.	.	PUNCT
cana-4839	139	1	alqahtani	alqahtani	PROPN
cana-4839	139	2	,	,	PUNCT
cana-4839	139	3	h.	h.	PROPN
cana-4839	139	4	,	,	PUNCT
cana-4839	139	5	badshah	badshah	PROPN
cana-4839	139	6	,	,	PUNCT
cana-4839	139	7	q.	q.	PROPN
cana-4839	139	8	,	,	PUNCT
cana-4839	139	9	rahman	rahman	PROPN
cana-4839	139	10	,	,	PUNCT
cana-4839	139	11	g.	g.	PROPN
cana-4839	139	12	u.	u.	PROPN
cana-4839	139	13	,	,	PUNCT
cana-4839	139	14	baleanu	baleanu	PROPN
cana-4839	139	15	,	,	PUNCT
cana-4839	139	16	d.	d.	PROPN
cana-4839	139	17	,	,	PUNCT
cana-4839	139	18	&	&	CCONJ
cana-4839	139	19	sakhi	sakhi	PROPN
cana-4839	139	20	,	,	PUNCT
cana-4839	139	21	s.	s.	PROPN
cana-4839	139	22	(	(	PUNCT
cana-4839	139	23	2023	2023	NUM
cana-4839	139	24	)	)	PUNCT
cana-4839	139	25	.	.	PUNCT
cana-4839	140	1	threshold	threshold	NOUN
cana-4839	140	2	dynamics	dynamic	NOUN
cana-4839	140	3	and	and	CCONJ
cana-4839	140	4	bifurcation	bifurcation	NOUN
cana-4839	140	5	analysis	analysis	NOUN
cana-4839	140	6	of	of	ADP
cana-4839	140	7	the	the	DET
cana-4839	140	8	epidemic	epidemic	NOUN
cana-4839	140	9	model	model	NOUN
cana-4839	140	10	of	of	ADP
cana-4839	140	11	mers	mers	PROPN
cana-4839	140	12	-	-	PUNCT
cana-4839	140	13	cov	cov	NOUN
cana-4839	140	14	.	.	PUNCT
cana-4839	141	1	fractals	fractal	NOUN
cana-4839	141	2	,	,	PUNCT
cana-4839	141	3	31(10	31(10	NUM
cana-4839	141	4	)	)	PUNCT
cana-4839	141	5	,	,	PUNCT
cana-4839	141	6	2340167	2340167	NUM
cana-4839	141	7	.	.	PUNCT
cana-4839	142	1	[	[	X
cana-4839	142	2	16	16	NUM
cana-4839	142	3	]	]	PUNCT
cana-4839	142	4	.	.	PUNCT
cana-4839	143	1	zhang	zhang	PROPN
cana-4839	143	2	,	,	PUNCT
cana-4839	143	3	j.	j.	PROPN
cana-4839	143	4	,	,	PUNCT
cana-4839	143	5	&	&	CCONJ
cana-4839	143	6	qiao	qiao	PROPN
cana-4839	143	7	,	,	PUNCT
cana-4839	143	8	y.	y.	PROPN
cana-4839	143	9	(	(	PUNCT
cana-4839	143	10	2023	2023	NUM
cana-4839	143	11	)	)	PUNCT
cana-4839	143	12	.	.	PUNCT
cana-4839	144	1	bifurcation	bifurcation	NOUN
cana-4839	144	2	analysis	analysis	NOUN
cana-4839	144	3	of	of	ADP
cana-4839	144	4	an	an	DET
cana-4839	144	5	sir	sir	NOUN
cana-4839	144	6	model	model	NOUN
cana-4839	144	7	considering	consider	VERB
cana-4839	144	8	hospital	hospital	NOUN
cana-4839	144	9	resources	resource	NOUN
cana-4839	144	10	and	and	CCONJ
cana-4839	144	11	vaccination	vaccination	NOUN
cana-4839	144	12	.	.	PUNCT
cana-4839	145	1	mathematics	mathematic	NOUN
cana-4839	145	2	and	and	CCONJ
cana-4839	145	3	computers	computer	NOUN
cana-4839	145	4	in	in	ADP
cana-4839	145	5	simulation	simulation	NOUN
cana-4839	145	6	,	,	PUNCT
cana-4839	145	7	208	208	NUM
cana-4839	145	8	,	,	PUNCT
cana-4839	145	9	157	157	NUM
cana-4839	145	10	-	-	SYM
cana-4839	145	11	185	185	NUM
cana-4839	145	12	.	.	PUNCT
cana-4839	146	1	[	[	X
cana-4839	146	2	17	17	NUM
cana-4839	146	3	]	]	PUNCT
cana-4839	146	4	.	.	PUNCT
cana-4839	147	1	khan	khan	PROPN
cana-4839	147	2	,	,	PUNCT
cana-4839	147	3	a.	a.	PROPN
cana-4839	147	4	q.	q.	PROPN
cana-4839	147	5	,	,	PUNCT
cana-4839	147	6	tasneem	tasneem	PROPN
cana-4839	147	7	,	,	PUNCT
cana-4839	147	8	m.	m.	NOUN
cana-4839	147	9	,	,	PUNCT
cana-4839	147	10	&	&	CCONJ
cana-4839	147	11	almatrafi	almatrafi	NUM
cana-4839	147	12	,	,	PUNCT
cana-4839	147	13	m.	m.	PROPN
cana-4839	147	14	b.	b.	PROPN
cana-4839	147	15	(	(	PUNCT
cana-4839	147	16	2021	2021	NUM
cana-4839	147	17	)	)	PUNCT
cana-4839	147	18	.	.	PUNCT
cana-4839	148	1	discrete	discrete	ADJ
cana-4839	148	2	-	-	PUNCT
cana-4839	148	3	time	time	NOUN
cana-4839	148	4	covid-19	covid-19	PROPN
cana-4839	148	5	epidemic	epidemic	NOUN
cana-4839	148	6	model	model	NOUN
cana-4839	148	7	with	with	ADP
cana-4839	148	8	bifurcation	bifurcation	NOUN
cana-4839	148	9	and	and	CCONJ
cana-4839	148	10	control	control	NOUN
cana-4839	148	11	.	.	PUNCT
cana-4839	149	1	math	math	PROPN
cana-4839	149	2	biosci	biosci	PROPN
cana-4839	149	3	eng	eng	PROPN
cana-4839	149	4	,	,	PUNCT
cana-4839	149	5	19	19	NUM
cana-4839	149	6	,	,	PUNCT
cana-4839	149	7	1944	1944	NUM
cana-4839	149	8	-	-	SYM
cana-4839	149	9	1969	1969	NUM
cana-4839	149	10	.	.	PUNCT
cana-4839	150	1	[	[	X
cana-4839	150	2	18	18	NUM
cana-4839	150	3	]	]	PUNCT
cana-4839	150	4	.	.	PUNCT
cana-4839	151	1	yaseen	yaseen	PROPN
cana-4839	151	2	,	,	PUNCT
cana-4839	151	3	r.	r.	PROPN
cana-4839	151	4	m.	m.	PROPN
cana-4839	151	5	,	,	PUNCT
cana-4839	151	6	mohsen	mohsen	PROPN
cana-4839	151	7	,	,	PUNCT
cana-4839	151	8	a.	a.	NOUN
cana-4839	151	9	a.	a.	PROPN
cana-4839	151	10	,	,	PUNCT
cana-4839	151	11	al	al	PROPN
cana-4839	151	12	-	-	PUNCT
cana-4839	151	13	husseiny	husseiny	PROPN
cana-4839	151	14	,	,	PUNCT
cana-4839	151	15	h.	h.	PROPN
cana-4839	151	16	f.	f.	PROPN
cana-4839	151	17	,	,	PUNCT
cana-4839	151	18	&	&	CCONJ
cana-4839	151	19	hattaf	hattaf	PROPN
cana-4839	151	20	,	,	PUNCT
cana-4839	151	21	k.	k.	PROPN
cana-4839	151	22	(	(	PUNCT
cana-4839	151	23	2023	2023	NUM
cana-4839	151	24	)	)	PUNCT
cana-4839	151	25	.	.	PUNCT
cana-4839	152	1	stability	stability	NOUN
cana-4839	152	2	and	and	CCONJ
cana-4839	152	3	hopf	hopf	ADJ
cana-4839	152	4	bifurcation	bifurcation	NOUN
cana-4839	152	5	of	of	ADP
cana-4839	152	6	an	an	DET
cana-4839	152	7	epidemiological	epidemiological	ADJ
cana-4839	152	8	model	model	NOUN
cana-4839	152	9	with	with	ADP
cana-4839	152	10	effect	effect	NOUN
cana-4839	152	11	of	of	ADP
cana-4839	152	12	delay	delay	NOUN
cana-4839	152	13	the	the	DET
cana-4839	152	14	awareness	awareness	NOUN
cana-4839	152	15	programs	program	NOUN
cana-4839	152	16	and	and	CCONJ
cana-4839	152	17	vaccination	vaccination	NOUN
cana-4839	152	18	:	:	PUNCT
cana-4839	152	19	analysis	analysis	NOUN
cana-4839	152	20	and	and	CCONJ
cana-4839	152	21	simulation	simulation	NOUN
cana-4839	152	22	.	.	PUNCT
cana-4839	153	1	commun	commun	PROPN
cana-4839	153	2	.	.	PUNCT
cana-4839	154	1	math	math	NOUN
cana-4839	154	2	.	.	PUNCT
cana-4839	155	1	biol	biol	PROPN
cana-4839	155	2	.	.	PUNCT
cana-4839	156	1	neurosci	neurosci	PROPN
cana-4839	156	2	.	.	PROPN
cana-4839	156	3	,	,	PUNCT
cana-4839	156	4	2023	2023	NUM
cana-4839	156	5	,	,	PUNCT
cana-4839	156	6	article	article	NOUN
cana-4839	156	7	-	-	PUNCT
cana-4839	156	8	id	id	NOUN
cana-4839	156	9	.	.	PUNCT
cana-4839	157	1	[	[	X
cana-4839	157	2	19	19	NUM
cana-4839	157	3	]	]	PUNCT
cana-4839	157	4	.	.	PUNCT
cana-4839	158	1	zerefe	zerefe	PROPN
cana-4839	158	2	,	,	PUNCT
cana-4839	158	3	s.	s.	PROPN
cana-4839	158	4	b.	b.	PROPN
cana-4839	158	5	,	,	PUNCT
cana-4839	158	6	ayele	ayele	PROPN
cana-4839	158	7	,	,	PUNCT
cana-4839	158	8	t.	t.	PROPN
cana-4839	158	9	k.	k.	PROPN
cana-4839	158	10	,	,	PUNCT
cana-4839	158	11	&	&	CCONJ
cana-4839	158	12	tilahun	tilahun	PROPN
cana-4839	158	13	,	,	PUNCT
cana-4839	158	14	s.	s.	PROPN
cana-4839	158	15	l.	l.	PROPN
cana-4839	158	16	(	(	PUNCT
cana-4839	158	17	2024	2024	NUM
cana-4839	158	18	)	)	PUNCT
cana-4839	158	19	.	.	PUNCT
cana-4839	159	1	mathematical	mathematical	ADJ
cana-4839	159	2	modeling	modeling	NOUN
cana-4839	159	3	of	of	ADP
cana-4839	159	4	covid‐19	covid‐19	PROPN
cana-4839	159	5	disease	disease	NOUN
cana-4839	159	6	dynamics	dynamic	NOUN
cana-4839	159	7	with	with	ADP
cana-4839	159	8	contact	contact	NOUN
cana-4839	159	9	tracing	tracing	NOUN
cana-4839	159	10	:	:	PUNCT
cana-4839	159	11	a	a	DET
cana-4839	159	12	case	case	NOUN
cana-4839	159	13	study	study	NOUN
cana-4839	159	14	in	in	ADP
cana-4839	159	15	ethiopia	ethiopia	PROPN
cana-4839	159	16	.	.	PUNCT
cana-4839	160	1	journal	journal	PROPN
cana-4839	160	2	of	of	ADP
cana-4839	160	3	applied	apply	VERB
cana-4839	160	4	mathematics	mathematic	NOUN
cana-4839	160	5	,	,	PUNCT
cana-4839	160	6	2024(1	2024(1	NOUN
cana-4839	160	7	)	)	PUNCT
cana-4839	160	8	,	,	PUNCT
cana-4839	160	9	5556734	5556734	NUM
cana-4839	160	10	.	.	PUNCT
cana-4839	161	1	[	[	X
cana-4839	161	2	20	20	NUM
cana-4839	161	3	]	]	PUNCT
cana-4839	161	4	.	.	PUNCT
cana-4839	161	5	wangari	wangari	PROPN
cana-4839	161	6	,	,	PUNCT
cana-4839	161	7	i.	i.	PROPN
cana-4839	161	8	m.	m.	PROPN
cana-4839	161	9	(	(	PUNCT
cana-4839	161	10	2024	2024	NUM
cana-4839	161	11	)	)	PUNCT
cana-4839	161	12	.	.	PUNCT
cana-4839	162	1	emergence	emergence	NOUN
cana-4839	162	2	of	of	ADP
cana-4839	162	3	a	a	DET
cana-4839	162	4	reversed	reverse	VERB
cana-4839	162	5	backward	backward	ADJ
cana-4839	162	6	bifurcation	bifurcation	NOUN
cana-4839	162	7	,	,	PUNCT
cana-4839	162	8	reversed	reverse	VERB
cana-4839	162	9	hysteresis	hysteresis	NOUN
cana-4839	162	10	effect	effect	NOUN
cana-4839	162	11	,	,	PUNCT
cana-4839	162	12	and	and	CCONJ
cana-4839	162	13	backward	backward	ADJ
cana-4839	162	14	bifurcation	bifurcation	NOUN
cana-4839	162	15	phenomenon	phenomenon	NOUN
cana-4839	162	16	in	in	ADP
cana-4839	162	17	a	a	DET
cana-4839	162	18	covid‐19	covid‐19	PROPN
cana-4839	162	19	mathematical	mathematical	ADJ
cana-4839	162	20	model	model	NOUN
cana-4839	162	21	.	.	PUNCT
cana-4839	163	1	mathematical	mathematical	ADJ
cana-4839	163	2	methods	method	NOUN
cana-4839	163	3	in	in	ADP
cana-4839	163	4	the	the	DET
cana-4839	163	5	applied	apply	VERB
cana-4839	163	6	sciences	science	NOUN
cana-4839	163	7	,	,	PUNCT
cana-4839	163	8	47(4	47(4	NUM
cana-4839	163	9	)	)	PUNCT
cana-4839	163	10	,	,	PUNCT
cana-4839	163	11	2250	2250	NUM
cana-4839	163	12	-	-	SYM
cana-4839	163	13	2272	2272	NUM
cana-4839	163	14	.	.	PUNCT
cana-4839	164	1	communications	communication	NOUN
cana-4839	164	2	on	on	ADP
cana-4839	164	3	applied	apply	VERB
cana-4839	164	4	nonlinear	nonlinear	ADJ
cana-4839	164	5	analysis	analysis	NOUN
cana-4839	164	6	issn	issn	NOUN
cana-4839	164	7	:	:	PUNCT
cana-4839	164	8	1074	1074	NUM
cana-4839	164	9	-	-	PUNCT
cana-4839	164	10	133x	133x	NUM
cana-4839	164	11	vol	vol	VERB
cana-4839	164	12	32	32	NUM
cana-4839	164	13	no	no	NOUN
cana-4839	164	14	.	.	PUNCT
cana-4839	165	1	10s	10	NOUN
cana-4839	165	2	(	(	PUNCT
cana-4839	165	3	2025	2025	NUM
cana-4839	165	4	)	)	PUNCT
cana-4839	165	5	443	443	NUM
cana-4839	165	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4839	166	1	[	[	X
cana-4839	166	2	21	21	NUM
cana-4839	166	3	]	]	PUNCT
cana-4839	166	4	.	.	PUNCT
cana-4839	167	1	ahmad	ahmad	PROPN
cana-4839	167	2	,	,	PUNCT
cana-4839	167	3	w.	w.	PROPN
cana-4839	167	4	,	,	PUNCT
cana-4839	167	5	rafiq	rafiq	PROPN
cana-4839	167	6	,	,	PUNCT
cana-4839	167	7	m.	m.	NOUN
cana-4839	167	8	,	,	PUNCT
cana-4839	167	9	butt	butt	PROPN
cana-4839	167	10	,	,	PUNCT
cana-4839	167	11	a.	a.	PROPN
cana-4839	167	12	i.	i.	PROPN
cana-4839	167	13	k.	k.	PROPN
cana-4839	167	14	,	,	PUNCT
cana-4839	167	15	ahmad	ahmad	PROPN
cana-4839	167	16	,	,	PUNCT
cana-4839	167	17	n.	n.	NOUN
cana-4839	167	18	,	,	PUNCT
cana-4839	167	19	ismaeel	ismaeel	NOUN
cana-4839	167	20	,	,	PUNCT
cana-4839	167	21	t.	t.	PROPN
cana-4839	167	22	,	,	PUNCT
cana-4839	167	23	malik	malik	PROPN
cana-4839	167	24	,	,	PUNCT
cana-4839	167	25	s.	s.	PROPN
cana-4839	167	26	,	,	PUNCT
cana-4839	167	27	...	...	PUNCT
cana-4839	167	28	&	&	CCONJ
cana-4839	167	29	asif	asif	PROPN
cana-4839	167	30	,	,	PUNCT
cana-4839	167	31	z.	z.	PROPN
cana-4839	167	32	(	(	PUNCT
cana-4839	167	33	2024	2024	NUM
cana-4839	167	34	)	)	PUNCT
cana-4839	167	35	.	.	PUNCT
cana-4839	168	1	analytical	analytical	ADJ
cana-4839	168	2	and	and	CCONJ
cana-4839	168	3	numerical	numerical	ADJ
cana-4839	168	4	explorations	exploration	NOUN
cana-4839	168	5	of	of	ADP
cana-4839	168	6	optimal	optimal	ADJ
cana-4839	168	7	control	control	NOUN
cana-4839	168	8	techniques	technique	NOUN
cana-4839	168	9	for	for	ADP
cana-4839	168	10	the	the	DET
cana-4839	168	11	bi	bi	ADJ
cana-4839	168	12	-	-	ADJ
cana-4839	168	13	modal	modal	ADJ
cana-4839	168	14	dynamics	dynamic	NOUN
cana-4839	168	15	of	of	ADP
cana-4839	168	16	covid-19	covid-19	PROPN
cana-4839	168	17	.	.	PUNCT
cana-4839	169	1	nonlinear	nonlinear	ADJ
cana-4839	169	2	dynamics	dynamic	NOUN
cana-4839	169	3	,	,	PUNCT
cana-4839	169	4	112(5	112(5	NUM
cana-4839	169	5	)	)	PUNCT
cana-4839	169	6	,	,	PUNCT
cana-4839	169	7	3977	3977	NUM
cana-4839	169	8	-	-	SYM
cana-4839	169	9	4006	4006	NUM
cana-4839	169	10	.	.	PUNCT
cana-4839	170	1	[	[	X
cana-4839	170	2	22	22	NUM
cana-4839	170	3	]	]	PUNCT
cana-4839	170	4	.	.	PUNCT
cana-4839	171	1	zhang	zhang	PROPN
cana-4839	171	2	,	,	PUNCT
cana-4839	171	3	j.	j.	PROPN
cana-4839	171	4	,	,	PUNCT
cana-4839	171	5	qiao	qiao	PROPN
cana-4839	171	6	,	,	PUNCT
cana-4839	171	7	y.	y.	PROPN
cana-4839	171	8	,	,	PUNCT
cana-4839	171	9	&	&	CCONJ
cana-4839	171	10	zhang	zhang	PROPN
cana-4839	171	11	,	,	PUNCT
cana-4839	171	12	y.	y.	PROPN
cana-4839	171	13	(	(	PUNCT
cana-4839	171	14	2023	2023	NUM
cana-4839	171	15	)	)	PUNCT
cana-4839	171	16	.	.	PUNCT
cana-4839	172	1	bifurcation	bifurcation	NOUN
cana-4839	172	2	analysis	analysis	NOUN
cana-4839	172	3	and	and	CCONJ
cana-4839	172	4	optimal	optimal	ADJ
cana-4839	172	5	control	control	NOUN
cana-4839	172	6	of	of	ADP
cana-4839	172	7	covid-19	covid-19	PROPN
cana-4839	172	8	with	with	ADP
cana-4839	172	9	exogenous	exogenous	ADJ
cana-4839	172	10	reinfection	reinfection	NOUN
cana-4839	172	11	and	and	CCONJ
cana-4839	172	12	media	media	NOUN
cana-4839	172	13	coverages	coverage	NOUN
cana-4839	172	14	.	.	PUNCT
cana-4839	173	1	international	international	ADJ
cana-4839	173	2	journal	journal	NOUN
cana-4839	173	3	of	of	ADP
cana-4839	173	4	biomathematics	biomathematic	NOUN
cana-4839	173	5	,	,	PUNCT
cana-4839	173	6	16(03	16(03	PROPN
cana-4839	173	7	)	)	PUNCT
cana-4839	173	8	,	,	PUNCT
cana-4839	173	9	2250086	2250086	NUM
cana-4839	173	10	.	.	PUNCT
cana-4839	174	1	[	[	X
cana-4839	174	2	23	23	NUM
cana-4839	174	3	]	]	PUNCT
cana-4839	174	4	.	.	PUNCT
cana-4839	175	1	nazarimehr	nazarimehr	NOUN
cana-4839	175	2	,	,	PUNCT
cana-4839	175	3	f.	f.	PROPN
cana-4839	175	4	,	,	PUNCT
cana-4839	175	5	pham	pham	PROPN
cana-4839	175	6	,	,	PUNCT
cana-4839	175	7	v.	v.	PROPN
cana-4839	175	8	t.	t.	PROPN
cana-4839	175	9	,	,	PUNCT
cana-4839	175	10	&	&	CCONJ
cana-4839	175	11	kapitaniak	kapitaniak	PROPN
cana-4839	175	12	,	,	PUNCT
cana-4839	175	13	t.	t.	PROPN
cana-4839	175	14	(	(	PUNCT
cana-4839	175	15	2020	2020	NUM
cana-4839	175	16	)	)	PUNCT
cana-4839	175	17	.	.	PUNCT
cana-4839	176	1	prediction	prediction	NOUN
cana-4839	176	2	of	of	ADP
cana-4839	176	3	bifurcations	bifurcation	NOUN
cana-4839	176	4	by	by	ADP
cana-4839	176	5	varying	vary	VERB
cana-4839	176	6	critical	critical	ADJ
cana-4839	176	7	parameters	parameter	NOUN
cana-4839	176	8	of	of	ADP
cana-4839	176	9	covid-19	covid-19	PROPN
cana-4839	176	10	.	.	PUNCT
cana-4839	177	1	nonlinear	nonlinear	ADJ
cana-4839	177	2	dynamics	dynamic	NOUN
cana-4839	177	3	,	,	PUNCT
cana-4839	177	4	101	101	NUM
cana-4839	177	5	,	,	PUNCT
cana-4839	177	6	1681	1681	NUM
cana-4839	177	7	-	-	SYM
cana-4839	177	8	1692	1692	NUM
cana-4839	177	9	.	.	PUNCT
cana-4839	178	1	[	[	X
cana-4839	178	2	24	24	NUM
cana-4839	178	3	]	]	PUNCT
cana-4839	178	4	.	.	PUNCT
cana-4839	179	1	kumar	kumar	PROPN
cana-4839	179	2	,	,	PUNCT
cana-4839	179	3	a.	a.	PROPN
cana-4839	179	4	,	,	PUNCT
cana-4839	179	5	gupta	gupta	PROPN
cana-4839	179	6	,	,	PUNCT
cana-4839	179	7	a.	a.	PROPN
cana-4839	179	8	,	,	PUNCT
cana-4839	179	9	dubey	dubey	PROPN
cana-4839	179	10	,	,	PUNCT
cana-4839	179	11	u.	u.	PROPN
cana-4839	179	12	s.	s.	PROPN
cana-4839	179	13	,	,	PUNCT
cana-4839	179	14	&	&	CCONJ
cana-4839	179	15	dubey	dubey	PROPN
cana-4839	179	16	,	,	PUNCT
cana-4839	179	17	b.	b.	PROPN
cana-4839	179	18	(	(	PUNCT
cana-4839	179	19	2023	2023	NUM
cana-4839	179	20	)	)	PUNCT
cana-4839	179	21	.	.	PUNCT
cana-4839	180	1	stability	stability	NOUN
cana-4839	180	2	and	and	CCONJ
cana-4839	180	3	bifurcation	bifurcation	NOUN
cana-4839	180	4	analysis	analysis	NOUN
cana-4839	180	5	of	of	ADP
cana-4839	180	6	an	an	DET
cana-4839	180	7	infectious	infectious	ADJ
cana-4839	180	8	disease	disease	NOUN
cana-4839	180	9	model	model	NOUN
cana-4839	180	10	with	with	ADP
cana-4839	180	11	different	different	ADJ
cana-4839	180	12	optimal	optimal	ADJ
cana-4839	180	13	control	control	NOUN
cana-4839	180	14	strategies	strategy	NOUN
cana-4839	180	15	.	.	PUNCT
cana-4839	181	1	mathematics	mathematic	NOUN
cana-4839	181	2	and	and	CCONJ
cana-4839	181	3	computers	computer	NOUN
cana-4839	181	4	in	in	ADP
cana-4839	181	5	simulation	simulation	NOUN
cana-4839	181	6	,	,	PUNCT
cana-4839	181	7	213	213	NUM
cana-4839	181	8	,	,	PUNCT
cana-4839	181	9	78	78	NUM
cana-4839	181	10	-	-	SYM
cana-4839	181	11	114	114	NUM
cana-4839	181	12	.	.	PUNCT
cana-4839	182	1	[	[	X
cana-4839	182	2	25	25	NUM
cana-4839	182	3	]	]	PUNCT
cana-4839	182	4	.	.	PUNCT
cana-4839	183	1	borah	borah	PROPN
cana-4839	183	2	,	,	PUNCT
cana-4839	183	3	m.	m.	NOUN
cana-4839	183	4	,	,	PUNCT
cana-4839	183	5	gayan	gayan	PROPN
cana-4839	183	6	,	,	PUNCT
cana-4839	183	7	a.	a.	PROPN
cana-4839	183	8	,	,	PUNCT
cana-4839	183	9	sharma	sharma	PROPN
cana-4839	183	10	,	,	PUNCT
cana-4839	183	11	j.	j.	PROPN
cana-4839	183	12	s.	s.	PROPN
cana-4839	183	13	,	,	PUNCT
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cana-4839	183	15	,	,	PUNCT
cana-4839	183	16	y.	y.	PROPN
cana-4839	183	17	,	,	PUNCT
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cana-4839	183	19	,	,	PUNCT
cana-4839	183	20	z.	z.	PROPN
cana-4839	183	21	,	,	PUNCT
cana-4839	183	22	&	&	CCONJ
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cana-4839	183	24	,	,	PUNCT
cana-4839	183	25	v.	v.	PROPN
cana-4839	183	26	t.	t.	PROPN
cana-4839	183	27	(	(	PUNCT
cana-4839	183	28	2022	2022	NUM
cana-4839	183	29	)	)	PUNCT
cana-4839	183	30	.	.	PUNCT
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cana-4839	183	32	fractional	fractional	ADJ
cana-4839	183	33	-	-	PUNCT
cana-4839	183	34	order	order	NOUN
cana-4839	183	35	chaos	chaos	NOUN
cana-4839	183	36	theory	theory	NOUN
cana-4839	183	37	the	the	DET
cana-4839	183	38	new	new	ADJ
cana-4839	183	39	tool	tool	NOUN
cana-4839	183	40	to	to	PART
cana-4839	183	41	model	model	VERB
cana-4839	183	42	chaotic	chaotic	ADJ
cana-4839	183	43	pandemics	pandemic	NOUN
cana-4839	183	44	as	as	ADP
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cana-4839	183	46	?	?	PROPN
cana-4839	183	47	.	.	PUNCT
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cana-4839	184	5	)	)	PUNCT
cana-4839	184	6	,	,	PUNCT
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cana-4839	184	8	-	-	SYM
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cana-4839	184	10	.	.	PUNCT
cana-4839	185	1	[	[	X
cana-4839	185	2	26	26	NUM
cana-4839	185	3	]	]	PUNCT
cana-4839	185	4	.	.	PUNCT
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cana-4839	186	2	,	,	PUNCT
cana-4839	186	3	m.	m.	PROPN
cana-4839	186	4	c.	c.	PROPN
cana-4839	186	5	,	,	PUNCT
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cana-4839	186	8	m.	m.	PROPN
cana-4839	186	9	l.	l.	PROPN
cana-4839	186	10	,	,	PUNCT
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cana-4839	186	12	,	,	PUNCT
cana-4839	186	13	o.	o.	PROPN
cana-4839	186	14	,	,	PUNCT
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cana-4839	186	19	(	(	PUNCT
cana-4839	186	20	2024	2024	NUM
cana-4839	186	21	)	)	PUNCT
cana-4839	186	22	.	.	PUNCT
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cana-4839	187	2	bifurcations	bifurcation	NOUN
cana-4839	187	3	and	and	CCONJ
cana-4839	187	4	optimal	optimal	ADJ
cana-4839	187	5	control	control	NOUN
cana-4839	187	6	strategies	strategy	NOUN
cana-4839	187	7	in	in	ADP
cana-4839	187	8	sirs	sir	NOUN
cana-4839	187	9	epidemic	epidemic	NOUN
cana-4839	187	10	models	model	NOUN
cana-4839	187	11	:	:	PUNCT
cana-4839	187	12	insights	insight	NOUN
cana-4839	187	13	from	from	ADP
cana-4839	187	14	theory	theory	NOUN
cana-4839	187	15	and	and	CCONJ
cana-4839	187	16	covid-19	covid-19	PROPN
cana-4839	187	17	data	datum	NOUN
cana-4839	187	18	.	.	PUNCT
cana-4839	188	1	mathematical	mathematical	ADJ
cana-4839	188	2	and	and	CCONJ
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cana-4839	188	5	,	,	PUNCT
cana-4839	188	6	29(4	29(4	NOUN
cana-4839	188	7	)	)	PUNCT
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cana-4839	188	10	.	.	PUNCT
cana-4839	189	1	[	[	X
cana-4839	189	2	27	27	NUM
cana-4839	189	3	]	]	PUNCT
cana-4839	189	4	.	.	PUNCT
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cana-4839	190	2	,	,	PUNCT
cana-4839	190	3	t.	t.	PROPN
cana-4839	190	4	c.	c.	PROPN
cana-4839	190	5	,	,	PUNCT
cana-4839	190	6	darassi	darassi	ADJ
cana-4839	190	7	,	,	PUNCT
cana-4839	190	8	m.	m.	PROPN
cana-4839	190	9	h.	h.	PROPN
cana-4839	190	10	,	,	PUNCT
cana-4839	190	11	alfwzan	alfwzan	PROPN
cana-4839	190	12	,	,	PUNCT
cana-4839	190	13	w.	w.	PROPN
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cana-4839	190	22	,	,	PUNCT
cana-4839	190	23	a.	a.	PROPN
cana-4839	190	24	s.	s.	PROPN
cana-4839	190	25	,	,	PUNCT
cana-4839	190	26	alshahrani	alshahrani	PROPN
cana-4839	190	27	,	,	PUNCT
cana-4839	190	28	s.	s.	PROPN
cana-4839	190	29	s.	s.	PROPN
cana-4839	190	30	,	,	PUNCT
cana-4839	190	31	&	&	CCONJ
cana-4839	190	32	muhammad	muhammad	PROPN
cana-4839	190	33	,	,	PUNCT
cana-4839	190	34	t.	t.	PROPN
cana-4839	190	35	(	(	PUNCT
cana-4839	190	36	2023	2023	NUM
cana-4839	190	37	)	)	PUNCT
cana-4839	190	38	.	.	PUNCT
cana-4839	191	1	mathematical	mathematical	ADJ
cana-4839	191	2	modeling	modeling	NOUN
cana-4839	191	3	of	of	ADP
cana-4839	191	4	covid-19	covid-19	PROPN
cana-4839	191	5	with	with	ADP
cana-4839	191	6	vaccination	vaccination	NOUN
cana-4839	191	7	using	use	VERB
cana-4839	191	8	fractional	fractional	ADJ
cana-4839	191	9	derivative	derivative	NOUN
cana-4839	191	10	:	:	PUNCT
cana-4839	191	11	a	a	DET
cana-4839	191	12	case	case	NOUN
cana-4839	191	13	study	study	NOUN
cana-4839	191	14	.	.	PUNCT
cana-4839	192	1	fractal	fractal	PROPN
cana-4839	192	2	and	and	CCONJ
cana-4839	192	3	fractional	fractional	ADJ
cana-4839	192	4	,	,	PUNCT
cana-4839	192	5	7(3	7(3	NUM
cana-4839	192	6	)	)	PUNCT
cana-4839	192	7	,	,	PUNCT
cana-4839	192	8	234	234	NUM
cana-4839	192	9	.	.	PUNCT
cana-4839	193	1	[	[	X
cana-4839	193	2	28	28	NUM
cana-4839	193	3	]	]	PUNCT
cana-4839	193	4	.	.	PUNCT
cana-4839	194	1	ajbar	ajbar	PROPN
cana-4839	194	2	,	,	PUNCT
cana-4839	194	3	a.	a.	NOUN
cana-4839	194	4	,	,	PUNCT
cana-4839	194	5	alqahtani	alqahtani	PROPN
cana-4839	194	6	,	,	PUNCT
cana-4839	194	7	r.	r.	PROPN
cana-4839	194	8	t.	t.	PROPN
cana-4839	194	9	,	,	PUNCT
cana-4839	194	10	&	&	CCONJ
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cana-4839	194	14	(	(	PUNCT
cana-4839	194	15	2021	2021	NUM
cana-4839	194	16	)	)	PUNCT
cana-4839	194	17	.	.	PUNCT
cana-4839	195	1	dynamics	dynamic	NOUN
cana-4839	195	2	of	of	ADP
cana-4839	195	3	an	an	DET
cana-4839	195	4	sir	sir	NOUN
cana-4839	195	5	-	-	PUNCT
cana-4839	195	6	based	base	VERB
cana-4839	195	7	covid-19	covid-19	PROPN
cana-4839	195	8	model	model	NOUN
cana-4839	195	9	with	with	ADP
cana-4839	195	10	linear	linear	ADJ
cana-4839	195	11	incidence	incidence	NOUN
cana-4839	195	12	rate	rate	NOUN
cana-4839	195	13	,	,	PUNCT
cana-4839	195	14	nonlinear	nonlinear	ADJ
cana-4839	195	15	removal	removal	NOUN
cana-4839	195	16	rate	rate	NOUN
cana-4839	195	17	,	,	PUNCT
cana-4839	195	18	and	and	CCONJ
cana-4839	195	19	public	public	ADJ
cana-4839	195	20	awareness	awareness	NOUN
cana-4839	195	21	.	.	PUNCT
cana-4839	196	1	frontiers	frontier	NOUN
cana-4839	196	2	in	in	ADP
cana-4839	196	3	physics	physics	PROPN
cana-4839	196	4	,	,	PUNCT
cana-4839	196	5	9	9	NUM
cana-4839	196	6	,	,	PUNCT
cana-4839	196	7	634251	634251	NUM
cana-4839	196	8	.	.	PUNCT
cana-4839	197	1	[	[	X
cana-4839	197	2	29	29	NUM
cana-4839	197	3	]	]	PUNCT
cana-4839	197	4	.	.	PUNCT
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cana-4839	197	6	,	,	PUNCT
cana-4839	197	7	s.	s.	PROPN
cana-4839	197	8	,	,	PUNCT
cana-4839	197	9	rashid	rashid	PROPN
cana-4839	197	10	,	,	PUNCT
cana-4839	197	11	s.	s.	PROPN
cana-4839	197	12	,	,	PUNCT
cana-4839	197	13	shah	shah	NOUN
cana-4839	197	14	,	,	PUNCT
cana-4839	197	15	m.	m.	NOUN
cana-4839	197	16	a.	a.	PROPN
cana-4839	197	17	,	,	PUNCT
cana-4839	197	18	&	&	CCONJ
cana-4839	197	19	elagan	elagan	PROPN
cana-4839	197	20	,	,	PUNCT
cana-4839	197	21	s.	s.	PROPN
cana-4839	197	22	k.	k.	PROPN
cana-4839	197	23	(	(	PUNCT
cana-4839	197	24	2024	2024	NUM
cana-4839	197	25	)	)	PUNCT
cana-4839	197	26	.	.	PUNCT
cana-4839	198	1	exploring	explore	VERB
cana-4839	198	2	the	the	DET
cana-4839	198	3	dynamical	dynamical	ADJ
cana-4839	198	4	bifurcation	bifurcation	NOUN
cana-4839	198	5	and	and	CCONJ
cana-4839	198	6	stability	stability	NOUN
cana-4839	198	7	analysis	analysis	NOUN
cana-4839	198	8	of	of	ADP
cana-4839	198	9	nipah	nipah	PROPN
cana-4839	198	10	virus	virus	NOUN
cana-4839	198	11	;	;	PUNCT
cana-4839	198	12	novel	novel	ADJ
cana-4839	198	13	perspectives	perspective	NOUN
cana-4839	198	14	utilizing	utilize	VERB
cana-4839	198	15	fractional	fractional	ADJ
cana-4839	198	16	calculus	calculus	NOUN
cana-4839	198	17	.	.	PUNCT
cana-4839	199	1	modeling	model	VERB
cana-4839	199	2	earth	earth	NOUN
cana-4839	199	3	systems	system	NOUN
cana-4839	199	4	and	and	CCONJ
cana-4839	199	5	environment	environment	NOUN
cana-4839	199	6	,	,	PUNCT
cana-4839	199	7	10(4	10(4	NUM
cana-4839	199	8	)	)	PUNCT
cana-4839	199	9	,	,	PUNCT
cana-4839	199	10	5427	5427	NUM
cana-4839	199	11	-	-	SYM
cana-4839	199	12	5448	5448	NUM
cana-4839	199	13	.	.	PUNCT
cana-4839	200	1	[	[	X
cana-4839	200	2	30	30	NUM
cana-4839	200	3	]	]	PUNCT
cana-4839	200	4	.	.	PUNCT
cana-4839	201	1	verma	verma	PROPN
cana-4839	201	2	,	,	PUNCT
cana-4839	201	3	t.	t.	PROPN
cana-4839	201	4	,	,	PUNCT
cana-4839	201	5	&	&	CCONJ
cana-4839	201	6	gupta	gupta	PROPN
cana-4839	201	7	,	,	PUNCT
cana-4839	201	8	a.	a.	PROPN
cana-4839	201	9	k.	k.	PROPN
cana-4839	201	10	(	(	PUNCT
cana-4839	201	11	2021	2021	NUM
cana-4839	201	12	)	)	PUNCT
cana-4839	201	13	.	.	PUNCT
cana-4839	202	1	network	network	NOUN
cana-4839	202	2	synchronization	synchronization	NOUN
cana-4839	202	3	,	,	PUNCT
cana-4839	202	4	stability	stability	NOUN
cana-4839	202	5	and	and	CCONJ
cana-4839	202	6	rhythmic	rhythmic	ADJ
cana-4839	202	7	processes	process	NOUN
cana-4839	202	8	in	in	ADP
cana-4839	202	9	a	a	DET
cana-4839	202	10	diffusive	diffusive	ADJ
cana-4839	202	11	mean	mean	ADJ
cana-4839	202	12	-	-	PUNCT
cana-4839	202	13	field	field	NOUN
cana-4839	202	14	coupled	couple	VERB
cana-4839	202	15	seir	seir	ADJ
cana-4839	202	16	model	model	NOUN
cana-4839	202	17	.	.	PUNCT
cana-4839	203	1	communications	communication	NOUN
cana-4839	203	2	in	in	ADP
cana-4839	203	3	nonlinear	nonlinear	ADJ
cana-4839	203	4	science	science	NOUN
cana-4839	203	5	and	and	CCONJ
cana-4839	203	6	numerical	numerical	PROPN
cana-4839	203	7	simulation	simulation	PROPN
cana-4839	203	8	,	,	PUNCT
cana-4839	203	9	102	102	NUM
cana-4839	203	10	,	,	PUNCT
cana-4839	203	11	105927	105927	NUM
cana-4839	203	12	.	.	PUNCT
cana-4839	203	13	.	.	PUNCT