id	sid	tid	token	lemma	pos
eajbcs-694	1	1	tinaw	tinaw	NOUN
eajbcs-694	1	2	tilahun	tilahun	PROPN
eajbcs-694	1	3	asmamaw1	asmamaw1	PROPN
eajbcs-694	1	4	,	,	PUNCT
eajbcs-694	1	5	kiros	kiros	PROPN
eajbcs-694	1	6	gebrearegawi	gebrearegawi	VERB
eajbcs-694	1	7	kebedow2,∗	kebedow2,∗	PROPN
eajbcs-694	1	8	1department	1department	NUM
eajbcs-694	1	9	of	of	ADP
eajbcs-694	1	10	mathematics	mathematic	NOUN
eajbcs-694	1	11	,	,	PUNCT
eajbcs-694	1	12	mezan	mezan	PROPN
eajbcs-694	1	13	tepi	tepi	PROPN
eajbcs-694	1	14	university	university	PROPN
eajbcs-694	1	15	,	,	PUNCT
eajbcs-694	1	16	ethiopia	ethiopia	PROPN
eajbcs-694	1	17	2department	2department	NUM
eajbcs-694	1	18	of	of	ADP
eajbcs-694	1	19	mathematics	mathematic	NOUN
eajbcs-694	1	20	,	,	PUNCT
eajbcs-694	1	21	hawassa	hawassa	PROPN
eajbcs-694	1	22	university	university	PROPN
eajbcs-694	1	23	,	,	PUNCT
eajbcs-694	1	24	ethiopia	ethiopia	PROPN
eajbcs-694	1	25	keywords	keyword	VERB
eajbcs-694	1	26	covid-19	covid-19	PROPN
eajbcs-694	1	27	;	;	PUNCT
eajbcs-694	1	28	protection	protection	NOUN
eajbcs-694	1	29	;	;	PUNCT
eajbcs-694	1	30	stability	stability	NOUN
eajbcs-694	1	31	analysis	analysis	NOUN
eajbcs-694	1	32	;	;	PUNCT
eajbcs-694	1	33	forward	forward	ADJ
eajbcs-694	1	34	bifurcation	bifurcation	NOUN
eajbcs-694	1	35	;	;	PUNCT
eajbcs-694	1	36	sensitivity	sensitivity	NOUN
eajbcs-694	1	37	analysis	analysis	NOUN
eajbcs-694	1	38	;	;	PUNCT
eajbcs-694	1	39	optimal	optimal	ADJ
eajbcs-694	1	40	control	control	NOUN
eajbcs-694	1	41	abstract	abstract	ADJ
eajbcs-694	1	42	∗	∗	PROPN
eajbcs-694	1	43	east	east	PROPN
eajbcs-694	1	44	afr	afr	PROPN
eajbcs-694	1	45	.	.	PUNCT
eajbcs-694	2	1	j.	j.	PROPN
eajbcs-694	2	2	biophys	biophys	PROPN
eajbcs-694	2	3	.	.	PUNCT
eajbcs-694	3	1	comput	comput	NOUN
eajbcs-694	3	2	.	.	PUNCT
eajbcs-694	4	1	sci	sci	PROPN
eajbcs-694	4	2	.	.	PUNCT
eajbcs-694	5	1	(	(	PUNCT
eajbcs-694	5	2	2024	2024	NUM
eajbcs-694	5	3	)	)	PUNCT
eajbcs-694	5	4	,	,	PUNCT
eajbcs-694	5	5	issue	issue	NOUN
eajbcs-694	5	6	5	5	NUM
eajbcs-694	5	7	,	,	PUNCT
eajbcs-694	5	8	no	no	INTJ
eajbcs-694	5	9	.	.	NOUN
eajbcs-694	5	10	2	2	NUM
eajbcs-694	5	11	,	,	PUNCT
eajbcs-694	5	12	58	58	NUM
eajbcs-694	5	13	-	-	SYM
eajbcs-694	5	14	86	86	NUM
eajbcs-694	5	15	58	58	NUM
eajbcs-694	5	16	in	in	ADP
eajbcs-694	5	17	this	this	DET
eajbcs-694	5	18	paper	paper	NOUN
eajbcs-694	5	19	,	,	PUNCT
eajbcs-694	5	20	we	we	PRON
eajbcs-694	5	21	propose	propose	VERB
eajbcs-694	5	22	a	a	DET
eajbcs-694	5	23	mathematical	mathematical	ADJ
eajbcs-694	5	24	model	model	NOUN
eajbcs-694	5	25	to	to	PART
eajbcs-694	5	26	investigate	investigate	VERB
eajbcs-694	5	27	coronavirus	coronavirus	NOUN
eajbcs-694	5	28	diseases	disease	NOUN
eajbcs-694	5	29	(	(	PUNCT
eajbcs-694	5	30	covid-19	covid-19	PROPN
eajbcs-694	5	31	)	)	PUNCT
eajbcs-694	5	32	transmission	transmission	NOUN
eajbcs-694	5	33	in	in	ADP
eajbcs-694	5	34	the	the	DET
eajbcs-694	5	35	presence	presence	NOUN
eajbcs-694	5	36	of	of	ADP
eajbcs-694	5	37	protected	protect	VERB
eajbcs-694	5	38	and	and	CCONJ
eajbcs-694	5	39	hospitalized	hospitalize	VERB
eajbcs-694	5	40	classes	class	NOUN
eajbcs-694	5	41	.	.	PUNCT
eajbcs-694	6	1	we	we	PRON
eajbcs-694	6	2	establish	establish	VERB
eajbcs-694	6	3	that	that	SCONJ
eajbcs-694	6	4	the	the	DET
eajbcs-694	6	5	solution	solution	NOUN
eajbcs-694	6	6	of	of	ADP
eajbcs-694	6	7	the	the	DET
eajbcs-694	6	8	dynamical	dynamical	ADJ
eajbcs-694	6	9	system	system	NOUN
eajbcs-694	6	10	remains	remain	VERB
eajbcs-694	6	11	positive	positive	ADJ
eajbcs-694	6	12	and	and	CCONJ
eajbcs-694	6	13	bounded	bound	VERB
eajbcs-694	6	14	.	.	PUNCT
eajbcs-694	7	1	we	we	PRON
eajbcs-694	7	2	compute	compute	VERB
eajbcs-694	7	3	the	the	DET
eajbcs-694	7	4	disease	disease	NOUN
eajbcs-694	7	5	free	free	ADJ
eajbcs-694	7	6	equilibrium	equilibrium	NOUN
eajbcs-694	7	7	point	point	NOUN
eajbcs-694	7	8	and	and	CCONJ
eajbcs-694	7	9	analyze	analyze	VERB
eajbcs-694	7	10	the	the	DET
eajbcs-694	7	11	stability	stability	NOUN
eajbcs-694	7	12	behavior	behavior	NOUN
eajbcs-694	7	13	of	of	ADP
eajbcs-694	7	14	the	the	DET
eajbcs-694	7	15	steady	steady	ADJ
eajbcs-694	7	16	state	state	NOUN
eajbcs-694	7	17	solutions.we	solutions.we	PRON
eajbcs-694	7	18	determine	determine	VERB
eajbcs-694	7	19	the	the	DET
eajbcs-694	7	20	basic	basic	ADJ
eajbcs-694	7	21	reproduction	reproduction	NOUN
eajbcs-694	7	22	number	number	NOUN
eajbcs-694	7	23	(	(	PUNCT
eajbcs-694	7	24	r0)and	r0)and	ADV
eajbcs-694	7	25	demonstrate	demonstrate	VERB
eajbcs-694	7	26	that	that	SCONJ
eajbcs-694	7	27	the	the	DET
eajbcs-694	7	28	disease	disease	NOUN
eajbcs-694	7	29	fades	fade	VERB
eajbcs-694	7	30	away	away	ADV
eajbcs-694	7	31	when	when	SCONJ
eajbcs-694	7	32	r0	r0	NOUN
eajbcs-694	7	33	<	<	X
eajbcs-694	7	34	1but	1but	NUM
eajbcs-694	7	35	persists	persist	VERB
eajbcs-694	7	36	in	in	ADP
eajbcs-694	7	37	the	the	DET
eajbcs-694	7	38	population	population	NOUN
eajbcs-694	7	39	when	when	SCONJ
eajbcs-694	7	40	r0	r0	VERB
eajbcs-694	7	41	>	>	X
eajbcs-694	7	42	1	1	X
eajbcs-694	7	43	.	.	PUNCT
eajbcs-694	8	1	the	the	DET
eajbcs-694	8	2	center	center	ADJ
eajbcs-694	8	3	manifold	manifold	ADJ
eajbcs-694	8	4	theory	theory	NOUN
eajbcs-694	8	5	is	be	AUX
eajbcs-694	8	6	used	use	VERB
eajbcs-694	8	7	to	to	PART
eajbcs-694	8	8	assess	assess	VERB
eajbcs-694	8	9	the	the	DET
eajbcs-694	8	10	local	local	ADJ
eajbcs-694	8	11	stability	stability	NOUN
eajbcs-694	8	12	of	of	ADP
eajbcs-694	8	13	the	the	DET
eajbcs-694	8	14	endemic	endemic	ADJ
eajbcs-694	8	15	equilibrium	equilibrium	NOUN
eajbcs-694	8	16	.	.	PUNCT
eajbcs-694	9	1	the	the	DET
eajbcs-694	9	2	model	model	NOUN
eajbcs-694	9	3	demonstrates	demonstrate	VERB
eajbcs-694	9	4	a	a	DET
eajbcs-694	9	5	forward	forward	ADJ
eajbcs-694	9	6	bifurcation	bifurcation	NOUN
eajbcs-694	9	7	,	,	PUNCT
eajbcs-694	9	8	and	and	CCONJ
eajbcs-694	9	9	a	a	DET
eajbcs-694	9	10	sensitivity	sensitivity	NOUN
eajbcs-694	9	11	analysis	analysis	NOUN
eajbcs-694	9	12	is	be	AUX
eajbcs-694	9	13	conducted	conduct	VERB
eajbcs-694	9	14	.	.	PUNCT
eajbcs-694	10	1	the	the	DET
eajbcs-694	10	2	sensitivity	sensitivity	NOUN
eajbcs-694	10	3	analysis	analysis	NOUN
eajbcs-694	10	4	reveals	reveal	VERB
eajbcs-694	10	5	that	that	SCONJ
eajbcs-694	10	6	r0	r0	NOUN
eajbcs-694	10	7	is	be	AUX
eajbcs-694	10	8	highly	highly	ADV
eajbcs-694	10	9	influenced	influence	VERB
eajbcs-694	10	10	by	by	ADP
eajbcs-694	10	11	the	the	DET
eajbcs-694	10	12	protection	protection	NOUN
eajbcs-694	10	13	rate	rate	NOUN
eajbcs-694	10	14	,	,	PUNCT
eajbcs-694	10	15	highlighting	highlight	VERB
eajbcs-694	10	16	the	the	DET
eajbcs-694	10	17	necessity	necessity	NOUN
eajbcs-694	10	18	of	of	ADP
eajbcs-694	10	19	maintaining	maintain	VERB
eajbcs-694	10	20	a	a	DET
eajbcs-694	10	21	high	high	ADJ
eajbcs-694	10	22	level	level	NOUN
eajbcs-694	10	23	of	of	ADP
eajbcs-694	10	24	protection	protection	NOUN
eajbcs-694	10	25	along	along	ADP
eajbcs-694	10	26	with	with	ADP
eajbcs-694	10	27	hospitalization	hospitalization	NOUN
eajbcs-694	10	28	to	to	PART
eajbcs-694	10	29	effectively	effectively	ADV
eajbcs-694	10	30	control	control	VERB
eajbcs-694	10	31	the	the	DET
eajbcs-694	10	32	disease	disease	NOUN
eajbcs-694	10	33	.	.	PUNCT
eajbcs-694	11	1	we	we	PRON
eajbcs-694	11	2	develop	develop	VERB
eajbcs-694	11	3	optimal	optimal	ADJ
eajbcs-694	11	4	strategies	strategy	NOUN
eajbcs-694	11	5	for	for	ADP
eajbcs-694	11	6	protection	protection	NOUN
eajbcs-694	11	7	and	and	CCONJ
eajbcs-694	11	8	hospitalization	hospitalization	NOUN
eajbcs-694	11	9	.	.	PUNCT
eajbcs-694	12	1	the	the	DET
eajbcs-694	12	2	characterization	characterization	NOUN
eajbcs-694	12	3	of	of	ADP
eajbcs-694	12	4	the	the	DET
eajbcs-694	12	5	optimal	optimal	ADJ
eajbcs-694	12	6	control	control	NOUN
eajbcs-694	12	7	is	be	AUX
eajbcs-694	12	8	derived	derive	VERB
eajbcs-694	12	9	using	use	VERB
eajbcs-694	12	10	pontryagin	pontryagin	NOUN
eajbcs-694	12	11	’s	’s	PART
eajbcs-694	12	12	maximum	maximum	ADJ
eajbcs-694	12	13	principle	principle	NOUN
eajbcs-694	12	14	.	.	PUNCT
eajbcs-694	13	1	numerical	numerical	ADJ
eajbcs-694	13	2	results	result	NOUN
eajbcs-694	13	3	for	for	ADP
eajbcs-694	13	4	the	the	DET
eajbcs-694	13	5	dynamics	dynamic	NOUN
eajbcs-694	13	6	of	of	ADP
eajbcs-694	13	7	the	the	DET
eajbcs-694	13	8	covid-19	covid-19	PROPN
eajbcs-694	13	9	outbreak	outbreak	NOUN
eajbcs-694	13	10	and	and	CCONJ
eajbcs-694	13	11	its	its	PRON
eajbcs-694	13	12	optimal	optimal	ADJ
eajbcs-694	13	13	control	control	NOUN
eajbcs-694	13	14	show	show	VERB
eajbcs-694	13	15	that	that	SCONJ
eajbcs-694	13	16	a	a	DET
eajbcs-694	13	17	combination	combination	NOUN
eajbcs-694	13	18	of	of	ADP
eajbcs-694	13	19	protection	protection	NOUN
eajbcs-694	13	20	and	and	CCONJ
eajbcs-694	13	21	hospitalization	hospitalization	NOUN
eajbcs-694	13	22	is	be	AUX
eajbcs-694	13	23	the	the	DET
eajbcs-694	13	24	most	most	ADV
eajbcs-694	13	25	effective	effective	ADJ
eajbcs-694	13	26	strategy	strategy	NOUN
eajbcs-694	13	27	for	for	ADP
eajbcs-694	13	28	reducing	reduce	VERB
eajbcs-694	13	29	the	the	DET
eajbcs-694	13	30	spread	spread	NOUN
eajbcs-694	13	31	of	of	ADP
eajbcs-694	13	32	covid-19	covid-19	PROPN
eajbcs-694	13	33	within	within	ADP
eajbcs-694	13	34	the	the	DET
eajbcs-694	13	35	population	population	NOUN
eajbcs-694	13	36	..	..	PUNCT
eajbcs-694	14	1	portation	portation	NOUN
eajbcs-694	14	2	lemecha	lemecha	ADV
eajbcs-694	14	3	obsu	obsu	ADV
eajbcs-694	14	4	and	and	CCONJ
eajbcs-694	14	5	feyissa	feyissa	ADJ
eajbcs-694	14	6	balcha	balcha	NOUN
eajbcs-694	14	7	(	(	PUNCT
eajbcs-694	14	8	2020	2020	NUM
eajbcs-694	14	9	)	)	PUNCT
eajbcs-694	14	10	.	.	PUNCT
eajbcs-694	15	1	the	the	DET
eajbcs-694	15	2	novel	novel	ADJ
eajbcs-694	15	3	coronavirusnow	coronavirusnow	NOUN
eajbcs-694	15	4	referred	refer	VERB
eajbcs-694	15	5	to	to	ADP
eajbcs-694	15	6	as	as	SCONJ
eajbcs-694	15	7	covid-19	covid-19	PROPN
eajbcs-694	15	8	is	be	AUX
eajbcs-694	15	9	caused	cause	VERB
eajbcs-694	15	10	by	by	ADP
eajbcs-694	15	11	severe	severe	ADJ
eajbcs-694	15	12	acute	acute	ADJ
eajbcs-694	15	13	respiratory	respiratory	ADJ
eajbcs-694	15	14	syndrome	syndrome	NOUN
eajbcs-694	15	15	(	(	PUNCT
eajbcs-694	15	16	sars	sar	NOUN
eajbcs-694	15	17	-	-	PUNCT
eajbcs-694	15	18	cov-2	cov-2	NOUN
eajbcs-694	15	19	)	)	PUNCT
eajbcs-694	15	20	and	and	CCONJ
eajbcs-694	15	21	consists	consist	VERB
eajbcs-694	15	22	of	of	ADP
eajbcs-694	15	23	single	single	ADJ
eajbcs-694	15	24	-	-	PUNCT
eajbcs-694	15	25	stranded	strand	VERB
eajbcs-694	15	26	ribonucleic	ribonucleic	ADJ
eajbcs-694	15	27	acid	acid	NOUN
eajbcs-694	15	28	(	(	PUNCT
eajbcs-694	15	29	rna	rna	NOUN
eajbcs-694	15	30	)	)	PUNCT
eajbcs-694	15	31	structure	structure	NOUN
eajbcs-694	15	32	sohrabi	sohrabi	NOUN
eajbcs-694	15	33	et	et	PROPN
eajbcs-694	15	34	al	al	PROPN
eajbcs-694	15	35	.	.	PROPN
eajbcs-694	16	1	(	(	PUNCT
eajbcs-694	16	2	2020	2020	NUM
eajbcs-694	16	3	)	)	PUNCT
eajbcs-694	16	4	.	.	PUNCT
eajbcs-694	17	1	the	the	DET
eajbcs-694	17	2	novel	novel	ADJ
eajbcs-694	17	3	coronavirus	coronavirus	NOUN
eajbcs-694	17	4	is	be	AUX
eajbcs-694	17	5	mainly	mainly	ADV
eajbcs-694	17	6	spread	spread	VERB
eajbcs-694	17	7	1	1	NUM
eajbcs-694	17	8	.	.	PUNCT
eajbcs-694	18	1	introduction	introduction	NOUN
eajbcs-694	18	2	coronavirus	coronavirus	NOUN
eajbcs-694	18	3	disease	disease	NOUN
eajbcs-694	18	4	2019	2019	NUM
eajbcs-694	18	5	(	(	PUNCT
eajbcs-694	18	6	covid-19	covid-19	PROPN
eajbcs-694	18	7	)	)	PUNCT
eajbcs-694	18	8	is	be	AUX
eajbcs-694	18	9	an	an	DET
eajbcs-694	18	10	infectious	infectious	ADJ
eajbcs-694	18	11	disease	disease	NOUN
eajbcs-694	18	12	caused	cause	VERB
eajbcs-694	18	13	by	by	ADP
eajbcs-694	18	14	a	a	DET
eajbcs-694	18	15	newly	newly	ADV
eajbcs-694	18	16	discovered	discover	VERB
eajbcs-694	18	17	coronavirus	coronavirus	NOUN
eajbcs-694	18	18	gurmu	gurmu	NOUN
eajbcs-694	18	19	et	et	PROPN
eajbcs-694	18	20	al	al	PROPN
eajbcs-694	18	21	.	.	PROPN
eajbcs-694	19	1	(	(	PUNCT
eajbcs-694	19	2	2020	2020	NUM
eajbcs-694	19	3	)	)	PUNCT
eajbcs-694	19	4	.	.	PUNCT
eajbcs-694	20	1	the	the	DET
eajbcs-694	20	2	new	new	ADJ
eajbcs-694	20	3	virus	virus	NOUN
eajbcs-694	20	4	was	be	AUX
eajbcs-694	20	5	first	first	ADV
eajbcs-694	20	6	appeared	appear	VERB
eajbcs-694	20	7	late	late	ADJ
eajbcs-694	20	8	december	december	PROPN
eajbcs-694	20	9	2019	2019	NUM
eajbcs-694	20	10	in	in	ADP
eajbcs-694	20	11	the	the	DET
eajbcs-694	20	12	chinese	chinese	ADJ
eajbcs-694	20	13	city	city	NOUN
eajbcs-694	20	14	of	of	ADP
eajbcs-694	20	15	wuhan	wuhan	PROPN
eajbcs-694	20	16	and	and	CCONJ
eajbcs-694	20	17	eventually	eventually	ADV
eajbcs-694	20	18	invaded	invade	VERB
eajbcs-694	20	19	the	the	DET
eajbcs-694	20	20	world	world	NOUN
eajbcs-694	20	21	due	due	ADP
eajbcs-694	20	22	to	to	ADP
eajbcs-694	20	23	fast	fast	VERB
eajbcs-694	20	24	modern	modern	ADJ
eajbcs-694	20	25	air	air	NOUN
eajbcs-694	20	26	transa	transa	VERB
eajbcs-694	20	27	mathematical	mathematical	ADJ
eajbcs-694	20	28	model	model	NOUN
eajbcs-694	20	29	for	for	ADP
eajbcs-694	20	30	the	the	DET
eajbcs-694	20	31	transmission	transmission	NOUN
eajbcs-694	20	32	dynamics	dynamic	NOUN
eajbcs-694	20	33	of	of	ADP
eajbcs-694	20	34	covid-19	covid-19	PROPN
eajbcs-694	20	35	pandemic	pandemic	ADJ
eajbcs-694	20	36	considering	considering	NOUN
eajbcs-694	20	37	protected	protect	VERB
eajbcs-694	20	38	and	and	CCONJ
eajbcs-694	20	39	hospitalized	hospitalize	VERB
eajbcs-694	20	40	with	with	ADP
eajbcs-694	20	41	optimal	optimal	ADJ
eajbcs-694	20	42	control	control	NOUN
eajbcs-694	20	43	corresponding	corresponding	ADJ
eajbcs-694	20	44	author	author	NOUN
eajbcs-694	20	45	email	email	NOUN
eajbcs-694	20	46	address	address	NOUN
eajbcs-694	20	47	:	:	PUNCT
eajbcs-694	21	1	kirosg@hu.edu.et	kirosg@hu.edu.et	ADJ
eajbcs-694	21	2	+251926528484	+251926528484	NOUN
eajbcs-694	21	3	https://dx.doi.org/10.4314/eajbcs.v5i2.5s	https://dx.doi.org/10.4314/eajbcs.v5i2.5s	NOUN
eajbcs-694	21	4	research	research	NOUN
eajbcs-694	21	5	article	article	NOUN
eajbcs-694	21	6	east	east	PROPN
eajbcs-694	21	7	afr	afr	PROPN
eajbcs-694	21	8	.	.	PUNCT
eajbcs-694	22	1	j.	j.	PROPN
eajbcs-694	22	2	biophys	biophys	PROPN
eajbcs-694	22	3	.	.	PUNCT
eajbcs-694	23	1	comput	comput	NOUN
eajbcs-694	23	2	.	.	PUNCT
eajbcs-694	24	1	sci	sci	PROPN
eajbcs-694	24	2	.	.	PUNCT
eajbcs-694	25	1	(	(	PUNCT
eajbcs-694	25	2	2024	2024	NUM
eajbcs-694	25	3	)	)	PUNCT
eajbcs-694	25	4	,	,	PUNCT
eajbcs-694	25	5	vol	vol	NOUN
eajbcs-694	25	6	.	.	PROPN
eajbcs-694	25	7	5	5	NUM
eajbcs-694	25	8	,	,	PUNCT
eajbcs-694	25	9	no	no	INTJ
eajbcs-694	25	10	.	.	NOUN
eajbcs-694	25	11	2	2	NUM
eajbcs-694	25	12	,	,	PUNCT
eajbcs-694	25	13	58	58	NUM
eajbcs-694	25	14	-	-	SYM
eajbcs-694	25	15	86	86	NUM
eajbcs-694	25	16	59	59	NUM
eajbcs-694	25	17	from	from	ADP
eajbcs-694	25	18	person	person	NOUN
eajbcs-694	25	19	to	to	ADP
eajbcs-694	25	20	person	person	NOUN
eajbcs-694	25	21	,	,	PUNCT
eajbcs-694	25	22	through	through	ADP
eajbcs-694	25	23	respiratory	respiratory	ADJ
eajbcs-694	25	24	droplets	droplet	NOUN
eajbcs-694	25	25	,	,	PUNCT
eajbcs-694	25	26	the	the	DET
eajbcs-694	25	27	spread	spread	NOUN
eajbcs-694	25	28	is	be	AUX
eajbcs-694	25	29	more	more	ADV
eajbcs-694	25	30	likely	likely	ADJ
eajbcs-694	25	31	when	when	SCONJ
eajbcs-694	25	32	people	people	NOUN
eajbcs-694	25	33	are	be	AUX
eajbcs-694	25	34	within	within	ADP
eajbcs-694	25	35	6	6	NUM
eajbcs-694	25	36	feet	foot	NOUN
eajbcs-694	25	37	of	of	ADP
eajbcs-694	25	38	each	each	DET
eajbcs-694	25	39	other	other	ADJ
eajbcs-694	25	40	toquero	toquero	NOUN
eajbcs-694	25	41	(	(	PUNCT
eajbcs-694	25	42	2020	2020	NUM
eajbcs-694	25	43	)	)	PUNCT
eajbcs-694	25	44	.	.	PUNCT
eajbcs-694	26	1	there	there	PRON
eajbcs-694	26	2	is	be	VERB
eajbcs-694	26	3	no	no	DET
eajbcs-694	26	4	known	know	VERB
eajbcs-694	26	5	curing	cure	VERB
eajbcs-694	26	6	medicine	medicine	NOUN
eajbcs-694	26	7	to	to	PART
eajbcs-694	26	8	combat	combat	VERB
eajbcs-694	26	9	the	the	DET
eajbcs-694	26	10	covid-19	covid-19	PROPN
eajbcs-694	26	11	pandemic	pandemic	NOUN
eajbcs-694	26	12	.	.	PUNCT
eajbcs-694	27	1	standard	standard	ADJ
eajbcs-694	27	2	recommendations	recommendation	NOUN
eajbcs-694	27	3	by	by	ADP
eajbcs-694	27	4	the	the	PRON
eajbcs-694	27	5	who	who	PRON
eajbcs-694	27	6	to	to	PART
eajbcs-694	27	7	prevent	prevent	VERB
eajbcs-694	27	8	the	the	DET
eajbcs-694	27	9	spread	spread	NOUN
eajbcs-694	27	10	of	of	ADP
eajbcs-694	27	11	covid-19	covid-19	PROPN
eajbcs-694	27	12	include	include	VERB
eajbcs-694	27	13	frequent	frequent	ADJ
eajbcs-694	27	14	cleaning	cleaning	NOUN
eajbcs-694	27	15	of	of	ADP
eajbcs-694	27	16	hands	hand	NOUN
eajbcs-694	27	17	using	use	VERB
eajbcs-694	27	18	soap	soap	NOUN
eajbcs-694	27	19	or	or	CCONJ
eajbcs-694	27	20	alcohol	alcohol	NOUN
eajbcs-694	27	21	-	-	PUNCT
eajbcs-694	27	22	based	base	VERB
eajbcs-694	27	23	sanitizer	sanitizer	NOUN
eajbcs-694	27	24	,	,	PUNCT
eajbcs-694	27	25	covering	cover	VERB
eajbcs-694	27	26	the	the	DET
eajbcs-694	27	27	nose	nose	NOUN
eajbcs-694	27	28	and	and	CCONJ
eajbcs-694	27	29	mouth	mouth	NOUN
eajbcs-694	27	30	with	with	ADP
eajbcs-694	27	31	a	a	DET
eajbcs-694	27	32	flexed	flexed	ADJ
eajbcs-694	27	33	elbow	elbow	NOUN
eajbcs-694	27	34	or	or	CCONJ
eajbcs-694	27	35	disposable	disposable	ADJ
eajbcs-694	27	36	tissue	tissue	NOUN
eajbcs-694	27	37	when	when	SCONJ
eajbcs-694	27	38	coughing	cough	VERB
eajbcs-694	27	39	and	and	CCONJ
eajbcs-694	27	40	sneezing	sneeze	VERB
eajbcs-694	27	41	and	and	CCONJ
eajbcs-694	27	42	avoiding	avoid	VERB
eajbcs-694	27	43	close	close	ADJ
eajbcs-694	27	44	contact	contact	NOUN
eajbcs-694	27	45	with	with	ADP
eajbcs-694	27	46	anyone	anyone	PRON
eajbcs-694	27	47	that	that	PRON
eajbcs-694	27	48	has	have	VERB
eajbcs-694	27	49	a	a	DET
eajbcs-694	27	50	fever	fever	NOUN
eajbcs-694	27	51	and	and	CCONJ
eajbcs-694	27	52	cough	cough	NOUN
eajbcs-694	27	53	toquero	toquero	NOUN
eajbcs-694	27	54	(	(	PUNCT
eajbcs-694	27	55	2020	2020	NUM
eajbcs-694	27	56	)	)	PUNCT
eajbcs-694	27	57	.	.	PUNCT
eajbcs-694	28	1	mathematical	mathematical	ADJ
eajbcs-694	28	2	modeling	modeling	NOUN
eajbcs-694	28	3	in	in	ADP
eajbcs-694	28	4	epidemiology	epidemiology	NOUN
eajbcs-694	28	5	helps	help	VERB
eajbcs-694	28	6	to	to	PART
eajbcs-694	28	7	understand	understand	VERB
eajbcs-694	28	8	the	the	DET
eajbcs-694	28	9	fundamental	fundamental	ADJ
eajbcs-694	28	10	mechanisms	mechanism	NOUN
eajbcs-694	28	11	that	that	PRON
eajbcs-694	28	12	drive	drive	VERB
eajbcs-694	28	13	the	the	DET
eajbcs-694	28	14	spread	spread	NOUN
eajbcs-694	28	15	of	of	ADP
eajbcs-694	28	16	disease	disease	NOUN
eajbcs-694	28	17	,	,	PUNCT
eajbcs-694	28	18	while	while	SCONJ
eajbcs-694	28	19	also	also	ADV
eajbcs-694	28	20	offering	offer	VERB
eajbcs-694	28	21	insights	insight	NOUN
eajbcs-694	28	22	into	into	ADP
eajbcs-694	28	23	potential	potential	ADJ
eajbcs-694	28	24	control	control	NOUN
eajbcs-694	28	25	strategies.the	strategies.the	DET
eajbcs-694	28	26	model	model	NOUN
eajbcs-694	28	27	formulation	formulation	NOUN
eajbcs-694	28	28	process	process	NOUN
eajbcs-694	28	29	clarifies	clarify	VERB
eajbcs-694	28	30	the	the	DET
eajbcs-694	28	31	assumptions	assumption	NOUN
eajbcs-694	28	32	,	,	PUNCT
eajbcs-694	28	33	variables	variable	NOUN
eajbcs-694	28	34	,	,	PUNCT
eajbcs-694	28	35	and	and	CCONJ
eajbcs-694	28	36	parameters	parameter	NOUN
eajbcs-694	28	37	involved	involve	VERB
eajbcs-694	28	38	.	.	PUNCT
eajbcs-694	29	1	furthermore	furthermore	ADV
eajbcs-694	29	2	,	,	PUNCT
eajbcs-694	29	3	models	model	NOUN
eajbcs-694	29	4	offer	offer	VERB
eajbcs-694	29	5	conceptual	conceptual	ADJ
eajbcs-694	29	6	insights	insight	NOUN
eajbcs-694	29	7	such	such	ADJ
eajbcs-694	29	8	as	as	ADP
eajbcs-694	29	9	thresholds	threshold	NOUN
eajbcs-694	29	10	,	,	PUNCT
eajbcs-694	29	11	basic	basic	ADJ
eajbcs-694	29	12	reproduction	reproduction	NOUN
eajbcs-694	29	13	numbers	number	NOUN
eajbcs-694	29	14	,	,	PUNCT
eajbcs-694	29	15	contact	contact	NOUN
eajbcs-694	29	16	numbers	number	NOUN
eajbcs-694	29	17	,	,	PUNCT
eajbcs-694	29	18	and	and	CCONJ
eajbcs-694	29	19	replacement	replacement	VERB
eajbcs-694	29	20	numbers.mathematical	numbers.mathematical	ADJ
eajbcs-694	29	21	models	model	NOUN
eajbcs-694	29	22	and	and	CCONJ
eajbcs-694	29	23	computer	computer	NOUN
eajbcs-694	29	24	simulations	simulation	NOUN
eajbcs-694	29	25	are	be	AUX
eajbcs-694	29	26	useful	useful	ADJ
eajbcs-694	29	27	experimental	experimental	ADJ
eajbcs-694	29	28	tools	tool	NOUN
eajbcs-694	29	29	for	for	ADP
eajbcs-694	29	30	building	building	NOUN
eajbcs-694	29	31	and	and	CCONJ
eajbcs-694	29	32	testing	testing	NOUN
eajbcs-694	29	33	theories	theory	NOUN
eajbcs-694	29	34	,	,	PUNCT
eajbcs-694	29	35	assessing	assess	VERB
eajbcs-694	29	36	quan	quan	PROPN
eajbcs-694	29	37	-	-	ADJ
eajbcs-694	29	38	titative	titative	ADJ
eajbcs-694	29	39	conjectures	conjecture	VERB
eajbcs-694	29	40	,	,	PUNCT
eajbcs-694	29	41	answering	answer	VERB
eajbcs-694	29	42	specific	specific	ADJ
eajbcs-694	29	43	ques	que	NOUN
eajbcs-694	29	44	-	-	PUNCT
eajbcs-694	29	45	tions	tion	NOUN
eajbcs-694	29	46	,	,	PUNCT
eajbcs-694	29	47	determining	determine	VERB
eajbcs-694	29	48	sensitivities	sensitivity	NOUN
eajbcs-694	29	49	to	to	ADP
eajbcs-694	29	50	changes	change	NOUN
eajbcs-694	29	51	in	in	ADP
eajbcs-694	29	52	parameter	parameter	NOUN
eajbcs-694	29	53	values	value	NOUN
eajbcs-694	29	54	,	,	PUNCT
eajbcs-694	29	55	and	and	CCONJ
eajbcs-694	29	56	estimating	estimate	VERB
eajbcs-694	29	57	parameters	parameter	NOUN
eajbcs-694	29	58	from	from	ADP
eajbcs-694	29	59	data	data	PROPN
eajbcs-694	29	60	brauer	brauer	PROPN
eajbcs-694	29	61	et	et	PROPN
eajbcs-694	29	62	al	al	PROPN
eajbcs-694	29	63	.	.	PROPN
eajbcs-694	30	1	(	(	PUNCT
eajbcs-694	30	2	2019	2019	NUM
eajbcs-694	30	3	)	)	PUNCT
eajbcs-694	30	4	.	.	PUNCT
eajbcs-694	31	1	the	the	DET
eajbcs-694	31	2	first	first	ADJ
eajbcs-694	31	3	m	m	VERB
eajbcs-694	31	4	athematical	athematical	ADJ
eajbcs-694	31	5	m	m	VERB
eajbcs-694	31	6	odel	odel	NOUN
eajbcs-694	31	7	w	w	PROPN
eajbcs-694	31	8	as	as	SCONJ
eajbcs-694	31	9	developed	develop	VERB
eajbcs-694	31	10	by	by	ADP
eajbcs-694	31	11	daniel	daniel	PROPN
eajbcs-694	31	12	bernoulli	bernoulli	PROPN
eajbcs-694	31	13	allen	allen	PROPN
eajbcs-694	31	14	et	et	PROPN
eajbcs-694	31	15	al	al	PROPN
eajbcs-694	31	16	.	.	PROPN
eajbcs-694	31	17	(	(	PUNCT
eajbcs-694	31	18	2008	2008	NUM
eajbcs-694	31	19	)	)	PUNCT
eajbcs-694	31	20	on	on	ADP
eajbcs-694	31	21	pandemic	pandemic	ADJ
eajbcs-694	31	22	of	of	ADP
eajbcs-694	31	23	smallpox	smallpox	NOUN
eajbcs-694	31	24	by	by	ADP
eajbcs-694	31	25	introducing	introduce	VERB
eajbcs-694	31	26	two	two	NUM
eajbcs-694	31	27	systems	system	NOUN
eajbcs-694	31	28	of	of	ADP
eajbcs-694	31	29	ordinary	ordinary	ADJ
eajbcs-694	31	30	differential	differential	ADJ
eajbcs-694	31	31	e	e	NOUN
eajbcs-694	31	32	quations	quation	NOUN
eajbcs-694	31	33	.	.	PUNCT
eajbcs-694	32	1	he	he	PRON
eajbcs-694	32	2	assumed	assume	VERB
eajbcs-694	32	3	that	that	SCONJ
eajbcs-694	32	4	recovery	recovery	NOUN
eajbcs-694	32	5	from	from	ADP
eajbcs-694	32	6	infection	infection	NOUN
eajbcs-694	32	7	confers	confer	NOUN
eajbcs-694	32	8	immunity	immunity	NOUN
eajbcs-694	32	9	(	(	PUNCT
eajbcs-694	32	10	no	no	DET
eajbcs-694	32	11	re	re	NOUN
eajbcs-694	32	12	-	-	NOUN
eajbcs-694	32	13	infection	infection	NOUN
eajbcs-694	32	14	)	)	PUNCT
eajbcs-694	32	15	.	.	PUNCT
eajbcs-694	33	1	he	he	PRON
eajbcs-694	33	2	also	also	ADV
eajbcs-694	33	3	assumed	assume	VERB
eajbcs-694	33	4	that	that	SCONJ
eajbcs-694	33	5	the	the	DET
eajbcs-694	33	6	probability	probability	NOUN
eajbcs-694	33	7	that	that	SCONJ
eajbcs-694	33	8	infected	infect	VERB
eajbcs-694	33	9	individuals	individual	NOUN
eajbcs-694	33	10	for	for	ADP
eajbcs-694	33	11	the	the	DET
eajbcs-694	33	12	first	first	ADJ
eajbcs-694	33	13	time	time	NOUN
eajbcs-694	33	14	die	die	VERB
eajbcs-694	33	15	does	do	AUX
eajbcs-694	33	16	not	not	PART
eajbcs-694	33	17	depend	depend	VERB
eajbcs-694	33	18	on	on	ADP
eajbcs-694	33	19	those	those	PRON
eajbcs-694	33	20	who	who	PRON
eajbcs-694	33	21	survive	survive	VERB
eajbcs-694	33	22	from	from	ADP
eajbcs-694	33	23	the	the	DET
eajbcs-694	33	24	infection	infection	NOUN
eajbcs-694	33	25	.	.	PUNCT
eajbcs-694	34	1	he	he	PRON
eajbcs-694	34	2	showed	show	VERB
eajbcs-694	34	3	that	that	SCONJ
eajbcs-694	34	4	inoculation	inoculation	NOUN
eajbcs-694	34	5	was	be	AUX
eajbcs-694	34	6	advantageous	advantageous	ADJ
eajbcs-694	34	7	if	if	SCONJ
eajbcs-694	34	8	the	the	DET
eajbcs-694	34	9	associated	associated	ADJ
eajbcs-694	34	10	risk	risk	NOUN
eajbcs-694	34	11	of	of	ADP
eajbcs-694	34	12	dying	die	VERB
eajbcs-694	34	13	was	be	AUX
eajbcs-694	34	14	less	less	ADJ
eajbcs-694	34	15	than	than	ADP
eajbcs-694	34	16	11	11	NUM
eajbcs-694	34	17	%	%	NOUN
eajbcs-694	34	18	.	.	PUNCT
eajbcs-694	35	1	a	a	DET
eajbcs-694	35	2	number	number	NOUN
eajbcs-694	35	3	of	of	ADP
eajbcs-694	35	4	compartmental	compartmental	ADJ
eajbcs-694	35	5	models	model	NOUN
eajbcs-694	35	6	have	have	AUX
eajbcs-694	35	7	been	be	AUX
eajbcs-694	35	8	proposed	propose	VERB
eajbcs-694	35	9	and	and	CCONJ
eajbcs-694	35	10	analyzed	analyze	VERB
eajbcs-694	35	11	for	for	ADP
eajbcs-694	35	12	the	the	DET
eajbcs-694	35	13	covid19	covid19	NOUN
eajbcs-694	35	14	outbreak	outbreak	NOUN
eajbcs-694	35	15	in	in	ADP
eajbcs-694	35	16	different	different	ADJ
eajbcs-694	35	17	c	c	NOUN
eajbcs-694	35	18	ountries	ountrie	NOUN
eajbcs-694	35	19	.	.	PUNCT
eajbcs-694	36	1	i	i	PRON
eajbcs-694	36	2	n	n	ADV
eajbcs-694	36	3	particular	particular	ADJ
eajbcs-694	36	4	,	,	PUNCT
eajbcs-694	36	5	yang	yang	PROPN
eajbcs-694	36	6	and	and	CCONJ
eajbcs-694	36	7	wang	wang	PROPN
eajbcs-694	36	8	yang	yang	PROPN
eajbcs-694	36	9	and	and	CCONJ
eajbcs-694	36	10	wang	wang	PROPN
eajbcs-694	36	11	(	(	PUNCT
eajbcs-694	36	12	2020	2020	NUM
eajbcs-694	36	13	)	)	PUNCT
eajbcs-694	36	14	proposed	propose	VERB
eajbcs-694	36	15	a	a	DET
eajbcs-694	36	16	mathematical	mathematical	ADJ
eajbcs-694	36	17	model	model	NOUN
eajbcs-694	36	18	for	for	ADP
eajbcs-694	36	19	covid-19	covid-19	PROPN
eajbcs-694	36	20	incorporating	incorporate	VERB
eajbcs-694	36	21	multiple	multiple	ADJ
eajbcs-694	36	22	transmission	transmission	NOUN
eajbcs-694	36	23	pathways	pathway	NOUN
eajbcs-694	36	24	,	,	PUNCT
eajbcs-694	36	25	including	include	VERB
eajbcs-694	36	26	both	both	CCONJ
eajbcs-694	36	27	human	human	ADJ
eajbcs-694	36	28	-	-	PUNCT
eajbcs-694	36	29	tohuman	tohuman	NOUN
eajbcs-694	36	30	and	and	CCONJ
eajbcs-694	36	31	environment	environment	NOUN
eajbcs-694	36	32	-	-	PUNCT
eajbcs-694	36	33	to	to	ADP
eajbcs-694	36	34	-	-	PUNCT
eajbcs-694	36	35	human	human	ADJ
eajbcs-694	36	36	transmission	transmission	NOUN
eajbcs-694	36	37	routes	route	NOUN
eajbcs-694	36	38	.	.	PUNCT
eajbcs-694	37	1	global	global	ADJ
eajbcs-694	37	2	stability	stability	NOUN
eajbcs-694	37	3	was	be	AUX
eajbcs-694	37	4	analysed	analyse	VERB
eajbcs-694	37	5	using	use	VERB
eajbcs-694	37	6	lyapunov	lyapunov	ADJ
eajbcs-694	37	7	function	function	NOUN
eajbcs-694	37	8	.	.	PUNCT
eajbcs-694	38	1	the	the	DET
eajbcs-694	38	2	authors	author	NOUN
eajbcs-694	38	3	employed	employ	VERB
eajbcs-694	38	4	a	a	DET
eajbcs-694	38	5	bilinear	bilinear	ADJ
eajbcs-694	38	6	incidence	incidence	NOUN
eajbcs-694	38	7	rate	rate	NOUN
eajbcs-694	38	8	based	base	VERB
eajbcs-694	38	9	on	on	ADP
eajbcs-694	38	10	the	the	DET
eajbcs-694	38	11	law	law	NOUN
eajbcs-694	38	12	of	of	ADP
eajbcs-694	38	13	mass	mass	ADJ
eajbcs-694	38	14	action	action	NOUN
eajbcs-694	38	15	and	and	CCONJ
eajbcs-694	38	16	fitted	fit	VERB
eajbcs-694	38	17	the	the	DET
eajbcs-694	38	18	model	model	NOUN
eajbcs-694	38	19	with	with	ADP
eajbcs-694	38	20	the	the	DET
eajbcs-694	38	21	data	datum	NOUN
eajbcs-694	38	22	of	of	ADP
eajbcs-694	38	23	wuhan	wuhan	PROPN
eajbcs-694	38	24	city	city	PROPN
eajbcs-694	38	25	of	of	ADP
eajbcs-694	38	26	china	china	PROPN
eajbcs-694	38	27	and	and	CCONJ
eajbcs-694	38	28	estimated	estimate	VERB
eajbcs-694	38	29	the	the	DET
eajbcs-694	38	30	reproduction	reproduction	NOUN
eajbcs-694	38	31	number	number	NOUN
eajbcs-694	38	32	.	.	PUNCT
eajbcs-694	39	1	in	in	ADP
eajbcs-694	39	2	2020	2020	NUM
eajbcs-694	39	3	,	,	PUNCT
eajbcs-694	39	4	haileyesus	haileyesus	NOUN
eajbcs-694	39	5	and	and	CCONJ
eajbcs-694	39	6	getachew	getachew	VERB
eajbcs-694	39	7	alemneh	alemneh	PROPN
eajbcs-694	39	8	and	and	CCONJ
eajbcs-694	39	9	telahun	telahun	PROPN
eajbcs-694	39	10	(	(	PUNCT
eajbcs-694	39	11	2020	2020	NUM
eajbcs-694	39	12	)	)	PUNCT
eajbcs-694	39	13	proposed	propose	VERB
eajbcs-694	39	14	a	a	DET
eajbcs-694	39	15	conceptual	conceptual	ADJ
eajbcs-694	39	16	seir	seir	NOUN
eajbcs-694	39	17	model	model	NOUN
eajbcs-694	39	18	to	to	PART
eajbcs-694	39	19	study	study	VERB
eajbcs-694	39	20	the	the	DET
eajbcs-694	39	21	pandemic	pandemic	ADJ
eajbcs-694	39	22	covid-19	covid-19	PROPN
eajbcs-694	39	23	transmission	transmission	NOUN
eajbcs-694	39	24	in	in	ADP
eajbcs-694	39	25	ethiopia	ethiopia	PROPN
eajbcs-694	39	26	.	.	PUNCT
eajbcs-694	40	1	global	global	ADJ
eajbcs-694	40	2	stability	stability	NOUN
eajbcs-694	40	3	was	be	AUX
eajbcs-694	40	4	analysed	analyse	VERB
eajbcs-694	40	5	using	use	VERB
eajbcs-694	40	6	lyapunov	lyapunov	ADJ
eajbcs-694	40	7	function	function	NOUN
eajbcs-694	40	8	.	.	PUNCT
eajbcs-694	41	1	additionally	additionally	ADV
eajbcs-694	41	2	,	,	PUNCT
eajbcs-694	41	3	they	they	PRON
eajbcs-694	41	4	incorporated	incorporate	VERB
eajbcs-694	41	5	timedependent	timedependent	NOUN
eajbcs-694	41	6	controls	control	NOUN
eajbcs-694	41	7	into	into	ADP
eajbcs-694	41	8	the	the	DET
eajbcs-694	41	9	basic	basic	ADJ
eajbcs-694	41	10	model	model	NOUN
eajbcs-694	41	11	and	and	CCONJ
eajbcs-694	41	12	extended	extend	VERB
eajbcs-694	41	13	it	it	PRON
eajbcs-694	41	14	to	to	ADP
eajbcs-694	41	15	an	an	DET
eajbcs-694	41	16	optimal	optimal	ADJ
eajbcs-694	41	17	control	control	NOUN
eajbcs-694	41	18	framework	framework	NOUN
eajbcs-694	41	19	for	for	ADP
eajbcs-694	41	20	the	the	DET
eajbcs-694	41	21	disease	disease	NOUN
eajbcs-694	41	22	.	.	PUNCT
eajbcs-694	42	1	an	an	DET
eajbcs-694	42	2	optimal	optimal	ADJ
eajbcs-694	42	3	control	control	NOUN
eajbcs-694	42	4	problem	problem	NOUN
eajbcs-694	42	5	was	be	AUX
eajbcs-694	42	6	formulated	formulate	VERB
eajbcs-694	42	7	and	and	CCONJ
eajbcs-694	42	8	analyzed	analyze	VERB
eajbcs-694	42	9	using	use	VERB
eajbcs-694	42	10	pontryagin	pontryagin	NOUN
eajbcs-694	42	11	’s	’s	PART
eajbcs-694	42	12	maximum	maximum	ADJ
eajbcs-694	42	13	principle	principle	NOUN
eajbcs-694	42	14	.	.	PUNCT
eajbcs-694	43	1	however	however	ADV
eajbcs-694	43	2	,	,	PUNCT
eajbcs-694	43	3	none	none	NOUN
eajbcs-694	43	4	of	of	ADP
eajbcs-694	43	5	the	the	DET
eajbcs-694	43	6	authors	author	NOUN
eajbcs-694	43	7	cited	cite	VERB
eajbcs-694	43	8	here	here	ADV
eajbcs-694	43	9	are	be	AUX
eajbcs-694	43	10	considering	consider	VERB
eajbcs-694	43	11	protected	protect	VERB
eajbcs-694	43	12	and	and	CCONJ
eajbcs-694	43	13	hospital	hospital	NOUN
eajbcs-694	43	14	-	-	PUNCT
eajbcs-694	43	15	ized	ize	VERB
eajbcs-694	43	16	classes	class	NOUN
eajbcs-694	43	17	with	with	ADP
eajbcs-694	43	18	optimal	optimal	ADJ
eajbcs-694	43	19	controls	control	NOUN
eajbcs-694	43	20	.	.	PUNCT
eajbcs-694	44	1	thus	thus	ADV
eajbcs-694	44	2	we	we	PRON
eajbcs-694	44	3	are	be	AUX
eajbcs-694	44	4	improved	improve	VERB
eajbcs-694	44	5	the	the	DET
eajbcs-694	44	6	existed	existed	ADJ
eajbcs-694	44	7	model	model	NOUN
eajbcs-694	44	8	,	,	PUNCT
eajbcs-694	44	9	by	by	ADP
eajbcs-694	44	10	including	include	VERB
eajbcs-694	44	11	the	the	DET
eajbcs-694	44	12	protected	protect	VERB
eajbcs-694	44	13	and	and	CCONJ
eajbcs-694	44	14	hospitalized	hospitalize	VERB
eajbcs-694	44	15	classes	class	NOUN
eajbcs-694	44	16	with	with	ADP
eajbcs-694	44	17	optimal	optimal	ADJ
eajbcs-694	44	18	controls	control	NOUN
eajbcs-694	44	19	.	.	PUNCT
eajbcs-694	45	1	the	the	DET
eajbcs-694	45	2	remainder	remainder	NOUN
eajbcs-694	45	3	of	of	ADP
eajbcs-694	45	4	the	the	DET
eajbcs-694	45	5	paper	paper	NOUN
eajbcs-694	45	6	is	be	AUX
eajbcs-694	45	7	organized	organize	VERB
eajbcs-694	45	8	as	as	SCONJ
eajbcs-694	45	9	follows	follow	VERB
eajbcs-694	45	10	.	.	PUNCT
eajbcs-694	46	1	in	in	ADP
eajbcs-694	46	2	section	section	NOUN
eajbcs-694	46	3	2	2	NUM
eajbcs-694	46	4	we	we	PRON
eajbcs-694	46	5	provided	provide	VERB
eajbcs-694	46	6	the	the	DET
eajbcs-694	46	7	description	description	NOUN
eajbcs-694	46	8	of	of	ADP
eajbcs-694	46	9	the	the	DET
eajbcs-694	46	10	problems	problem	NOUN
eajbcs-694	46	11	and	and	CCONJ
eajbcs-694	46	12	its	its	PRON
eajbcs-694	46	13	mathematical	mathematical	ADJ
eajbcs-694	46	14	model	model	NOUN
eajbcs-694	46	15	formulation	formulation	NOUN
eajbcs-694	46	16	.	.	PUNCT
eajbcs-694	47	1	section	section	NOUN
eajbcs-694	47	2	3	3	NUM
eajbcs-694	47	3	gives	give	VERB
eajbcs-694	47	4	details	detail	NOUN
eajbcs-694	47	5	the	the	DET
eajbcs-694	47	6	qualitative	qualitative	ADJ
eajbcs-694	47	7	analysis	analysis	NOUN
eajbcs-694	47	8	of	of	ADP
eajbcs-694	47	9	the	the	DET
eajbcs-694	47	10	model	model	NOUN
eajbcs-694	47	11	.	.	PUNCT
eajbcs-694	48	1	the	the	DET
eajbcs-694	48	2	numerical	numerical	ADJ
eajbcs-694	48	3	analysis	analysis	NOUN
eajbcs-694	48	4	of	of	ADP
eajbcs-694	48	5	our	our	PRON
eajbcs-694	48	6	model	model	NOUN
eajbcs-694	48	7	is	be	AUX
eajbcs-694	48	8	presented	present	VERB
eajbcs-694	48	9	in	in	ADP
eajbcs-694	48	10	section	section	NOUN
eajbcs-694	48	11	4	4	NUM
eajbcs-694	48	12	,	,	PUNCT
eajbcs-694	48	13	furthermore	furthermore	ADV
eajbcs-694	48	14	we	we	PRON
eajbcs-694	48	15	present	present	VERB
eajbcs-694	48	16	the	the	DET
eajbcs-694	48	17	extensions	extension	NOUN
eajbcs-694	48	18	of	of	ADP
eajbcs-694	48	19	the	the	DET
eajbcs-694	48	20	model	model	NOUN
eajbcs-694	48	21	to	to	ADP
eajbcs-694	48	22	optimal	optimal	ADJ
eajbcs-694	48	23	controls	control	NOUN
eajbcs-694	48	24	and	and	CCONJ
eajbcs-694	48	25	its	its	PRON
eajbcs-694	48	26	numerical	numerical	ADJ
eajbcs-694	48	27	simulations	simulation	NOUN
eajbcs-694	48	28	5	5	NUM
eajbcs-694	48	29	and	and	CCONJ
eajbcs-694	48	30	6	6	NUM
eajbcs-694	48	31	.	.	PUNCT
eajbcs-694	49	1	finally	finally	ADV
eajbcs-694	49	2	,	,	PUNCT
eajbcs-694	49	3	section	section	NOUN
eajbcs-694	49	4	7	7	NUM
eajbcs-694	49	5	concludes	conclude	VERB
eajbcs-694	49	6	the	the	DET
eajbcs-694	49	7	paper	paper	NOUN
eajbcs-694	49	8	.	.	PUNCT
eajbcs-694	50	1	2	2	X
eajbcs-694	50	2	.	.	X
eajbcs-694	50	3	mathematical	mathematical	ADJ
eajbcs-694	50	4	model	model	NOUN
eajbcs-694	50	5	formulation	formulation	NOUN
eajbcs-694	50	6	2.1	2.1	NUM
eajbcs-694	50	7	.	.	PUNCT
eajbcs-694	51	1	the	the	DET
eajbcs-694	51	2	modified	modify	VERB
eajbcs-694	51	3	mathematical	mathematical	ADJ
eajbcs-694	51	4	model	model	NOUN
eajbcs-694	51	5	in	in	ADP
eajbcs-694	51	6	this	this	DET
eajbcs-694	51	7	subsection	subsection	NOUN
eajbcs-694	51	8	,	,	PUNCT
eajbcs-694	51	9	we	we	PRON
eajbcs-694	51	10	modify	modify	VERB
eajbcs-694	51	11	the	the	DET
eajbcs-694	51	12	existing	exist	VERB
eajbcs-694	51	13	seir	seir	NOUN
eajbcs-694	51	14	model	model	NOUN
eajbcs-694	51	15	to	to	PART
eajbcs-694	51	16	study	study	VERB
eajbcs-694	51	17	the	the	DET
eajbcs-694	51	18	transmission	transmission	NOUN
eajbcs-694	51	19	dynamics	dynamic	NOUN
eajbcs-694	51	20	of	of	ADP
eajbcs-694	51	21	covid-19	covid-19	PROPN
eajbcs-694	51	22	infection	infection	NOUN
eajbcs-694	51	23	in	in	ADP
eajbcs-694	51	24	a	a	DET
eajbcs-694	51	25	population	population	NOUN
eajbcs-694	51	26	.	.	PUNCT
eajbcs-694	52	1	the	the	DET
eajbcs-694	52	2	model	model	NOUN
eajbcs-694	52	3	is	be	AUX
eajbcs-694	52	4	a	a	DET
eajbcs-694	52	5	modification	modification	NOUN
eajbcs-694	52	6	of	of	ADP
eajbcs-694	52	7	what	what	PRON
eajbcs-694	52	8	is	be	AUX
eajbcs-694	52	9	presented	present	VERB
eajbcs-694	52	10	in	in	ADP
eajbcs-694	52	11	alemneh	alemneh	PROPN
eajbcs-694	52	12	and	and	CCONJ
eajbcs-694	52	13	telahun	telahun	PROPN
eajbcs-694	52	14	(	(	PUNCT
eajbcs-694	52	15	2020	2020	NUM
eajbcs-694	52	16	)	)	PUNCT
eajbcs-694	52	17	.	.	PUNCT
eajbcs-694	53	1	in	in	ADP
eajbcs-694	53	2	the	the	DET
eajbcs-694	53	3	present	present	ADJ
eajbcs-694	53	4	model	model	NOUN
eajbcs-694	53	5	we	we	PRON
eajbcs-694	53	6	extended	extend	VERB
eajbcs-694	53	7	seir	seir	NOUN
eajbcs-694	53	8	model	model	NOUN
eajbcs-694	53	9	by	by	ADP
eajbcs-694	53	10	including	include	VERB
eajbcs-694	53	11	protected	protect	VERB
eajbcs-694	53	12	and	and	CCONJ
eajbcs-694	53	13	hospitalized	hospitalize	VERB
eajbcs-694	53	14	classes	class	NOUN
eajbcs-694	53	15	.	.	PUNCT
eajbcs-694	54	1	2.2	2.2	NUM
eajbcs-694	54	2	.	.	PUNCT
eajbcs-694	54	3	model	model	NOUN
eajbcs-694	54	4	formulation	formulation	NOUN
eajbcs-694	54	5	we	we	PRON
eajbcs-694	54	6	formulate	formulate	VERB
eajbcs-694	54	7	a	a	DET
eajbcs-694	54	8	mathematical	mathematical	ADJ
eajbcs-694	54	9	model	model	NOUN
eajbcs-694	54	10	to	to	PART
eajbcs-694	54	11	investigate	investigate	VERB
eajbcs-694	54	12	coronavirus	coronavirus	NOUN
eajbcs-694	54	13	diseases	disease	NOUN
eajbcs-694	54	14	(	(	PUNCT
eajbcs-694	54	15	covid-19	covid-19	PROPN
eajbcs-694	54	16	)	)	PUNCT
eajbcs-694	54	17	transmission	transmission	NOUN
eajbcs-694	54	18	in	in	ADP
eajbcs-694	54	19	the	the	DET
eajbcs-694	54	20	presence	presence	NOUN
eajbcs-694	54	21	of	of	ADP
eajbcs-694	54	22	protected	protect	VERB
eajbcs-694	54	23	and	and	CCONJ
eajbcs-694	54	24	hospitalized	hospitalize	VERB
eajbcs-694	54	25	classes	class	NOUN
eajbcs-694	54	26	.	.	PUNCT
eajbcs-694	55	1	the	the	DET
eajbcs-694	55	2	model	model	NOUN
eajbcs-694	55	3	divides	divide	VERB
eajbcs-694	55	4	the	the	DET
eajbcs-694	55	5	total	total	ADJ
eajbcs-694	55	6	population	population	NOUN
eajbcs-694	55	7	into	into	ADP
eajbcs-694	55	8	six	six	NUM
eajbcs-694	55	9	sub	sub	NOUN
eajbcs-694	55	10	-	-	NOUN
eajbcs-694	55	11	classes	class	NOUN
eajbcs-694	55	12	according	accord	VERB
eajbcs-694	55	13	to	to	ADP
eajbcs-694	55	14	their	their	PRON
eajbcs-694	55	15	disease	disease	NOUN
eajbcs-694	55	16	status	status	NOUN
eajbcs-694	55	17	.	.	PUNCT
eajbcs-694	56	1	susceptible	susceptible	ADJ
eajbcs-694	56	2	(	(	PUNCT
eajbcs-694	56	3	s	s	NOUN
eajbcs-694	56	4	)	)	PUNCT
eajbcs-694	56	5	,	,	PUNCT
eajbcs-694	56	6	protected	protect	VERB
eajbcs-694	56	7	(	(	PUNCT
eajbcs-694	56	8	p	p	NOUN
eajbcs-694	56	9	)	)	PUNCT
eajbcs-694	56	10	,	,	PUNCT
eajbcs-694	56	11	exposed	expose	VERB
eajbcs-694	56	12	(	(	PUNCT
eajbcs-694	56	13	e	e	NOUN
eajbcs-694	56	14	)	)	PUNCT
eajbcs-694	56	15	,	,	PUNCT
eajbcs-694	56	16	infected	infect	VERB
eajbcs-694	56	17	(	(	PUNCT
eajbcs-694	56	18	i	i	NOUN
eajbcs-694	56	19	)	)	PUNCT
eajbcs-694	56	20	,	,	PUNCT
eajbcs-694	56	21	hospitalized	hospitalize	VERB
eajbcs-694	56	22	(	(	PUNCT
eajbcs-694	56	23	h	h	NOUN
eajbcs-694	56	24	)	)	PUNCT
eajbcs-694	56	25	and	and	CCONJ
eajbcs-694	56	26	recovered	recover	VERB
eajbcs-694	56	27	(	(	PUNCT
eajbcs-694	56	28	r	r	NOUN
eajbcs-694	56	29	)	)	PUNCT
eajbcs-694	56	30	.	.	PUNCT
eajbcs-694	57	1	the	the	DET
eajbcs-694	57	2	following	follow	VERB
eajbcs-694	57	3	assumptions	assumption	NOUN
eajbcs-694	57	4	have	have	AUX
eajbcs-694	57	5	been	be	AUX
eajbcs-694	57	6	used	use	VERB
eajbcs-694	57	7	in	in	ADP
eajbcs-694	57	8	the	the	DET
eajbcs-694	57	9	formulation	formulation	NOUN
eajbcs-694	57	10	of	of	ADP
eajbcs-694	57	11	the	the	DET
eajbcs-694	57	12	model	model	NOUN
eajbcs-694	57	13	:	:	PUNCT
eajbcs-694	58	1	1	1	X
eajbcs-694	58	2	.	.	X
eajbcs-694	58	3	the	the	DET
eajbcs-694	58	4	population	population	NOUN
eajbcs-694	58	5	under	under	ADP
eajbcs-694	58	6	study	study	NOUN
eajbcs-694	58	7	is	be	AUX
eajbcs-694	58	8	heterogeneous	heterogeneous	ADJ
eajbcs-694	58	9	and	and	CCONJ
eajbcs-694	58	10	varying	vary	VERB
eajbcs-694	58	11	with	with	ADP
eajbcs-694	58	12	time	time	NOUN
eajbcs-694	58	13	.	.	PUNCT
eajbcs-694	59	1	2	2	X
eajbcs-694	59	2	.	.	X
eajbcs-694	59	3	all	all	PRON
eajbcs-694	59	4	recruited	recruit	VERB
eajbcs-694	59	5	human	human	ADJ
eajbcs-694	59	6	population	population	NOUN
eajbcs-694	59	7	is	be	AUX
eajbcs-694	59	8	susceptible	susceptible	ADJ
eajbcs-694	59	9	.	.	PUNCT
eajbcs-694	60	1	3	3	X
eajbcs-694	60	2	.	.	NOUN
eajbcs-694	60	3	susceptible	susceptible	ADJ
eajbcs-694	60	4	individuals	individual	NOUN
eajbcs-694	60	5	who	who	PRON
eajbcs-694	60	6	keeping	keep	VERB
eajbcs-694	60	7	social	social	ADJ
eajbcs-694	60	8	distancing	distancing	NOUN
eajbcs-694	60	9	,	,	PUNCT
eajbcs-694	60	10	using	use	VERB
eajbcs-694	60	11	an	an	DET
eajbcs-694	60	12	alcohol	alcohol	NOUN
eajbcs-694	60	13	-	-	PUNCT
eajbcs-694	60	14	based	base	VERB
eajbcs-694	60	15	hand	hand	NOUN
eajbcs-694	60	16	sanitizer	sanitizer	NOUN
eajbcs-694	60	17	and	and	CCONJ
eajbcs-694	60	18	wearing	wear	VERB
eajbcs-694	60	19	face	face	NOUN
eajbcs-694	60	20	masks	mask	NOUN
eajbcs-694	60	21	progress	progress	NOUN
eajbcs-694	60	22	into	into	ADP
eajbcs-694	60	23	protected	protect	VERB
eajbcs-694	60	24	class	class	NOUN
eajbcs-694	60	25	.	.	PUNCT
eajbcs-694	61	1	4	4	X
eajbcs-694	61	2	.	.	NUM
eajbcs-694	61	3	protected	protect	VERB
eajbcs-694	61	4	individuals	individual	NOUN
eajbcs-694	61	5	can	can	AUX
eajbcs-694	61	6	not	not	PART
eajbcs-694	61	7	acquire	acquire	VERB
eajbcs-694	61	8	infection	infection	NOUN
eajbcs-694	61	9	of	of	ADP
eajbcs-694	61	10	covid-19	covid-19	PROPN
eajbcs-694	61	11	disease	disease	NOUN
eajbcs-694	61	12	due	due	ADP
eajbcs-694	61	13	to	to	ADP
eajbcs-694	61	14	proper	proper	ADJ
eajbcs-694	61	15	use	use	NOUN
eajbcs-694	61	16	of	of	ADP
eajbcs-694	61	17	an	an	DET
eajbcs-694	61	18	alcohol	alcohol	NOUN
eajbcs-694	61	19	-	-	PUNCT
eajbcs-694	61	20	based	base	VERB
eajbcs-694	61	21	hand	hand	NOUN
eajbcs-694	61	22	sanitizer	sanitizer	NOUN
eajbcs-694	61	23	,	,	PUNCT
eajbcs-694	61	24	wearing	wear	VERB
eajbcs-694	61	25	face	face	NOUN
eajbcs-694	61	26	masks	mask	NOUN
eajbcs-694	61	27	and	and	CCONJ
eajbcs-694	61	28	keeping	keep	VERB
eajbcs-694	61	29	social	social	ADJ
eajbcs-694	61	30	distancing	distancing	NOUN
eajbcs-694	61	31	.	.	PUNCT
eajbcs-694	62	1	5	5	X
eajbcs-694	62	2	.	.	X
eajbcs-694	62	3	the	the	DET
eajbcs-694	62	4	latently	latently	ADV
eajbcs-694	62	5	infected	infect	VERB
eajbcs-694	62	6	individuals	individual	NOUN
eajbcs-694	62	7	(	(	PUNCT
eajbcs-694	62	8	exposed	expose	VERB
eajbcs-694	62	9	)	)	PUNCT
eajbcs-694	62	10	could	could	AUX
eajbcs-694	62	11	infect	infect	VERB
eajbcs-694	62	12	other	other	ADJ
eajbcs-694	62	13	people	people	NOUN
eajbcs-694	62	14	with	with	ADP
eajbcs-694	62	15	a	a	DET
eajbcs-694	62	16	higher	high	ADJ
eajbcs-694	62	17	probability	probability	NOUN
eajbcs-694	62	18	than	than	ADP
eajbcs-694	62	19	people	people	NOUN
eajbcs-694	62	20	in	in	ADP
eajbcs-694	62	21	the	the	DET
eajbcs-694	62	22	infected	infected	ADJ
eajbcs-694	62	23	class	class	NOUN
eajbcs-694	62	24	.	.	PUNCT
eajbcs-694	63	1	since	since	SCONJ
eajbcs-694	63	2	exposed	expose	VERB
eajbcs-694	63	3	individuals	individual	NOUN
eajbcs-694	63	4	show	show	VERB
eajbcs-694	63	5	no	no	DET
eajbcs-694	63	6	symptoms	symptom	NOUN
eajbcs-694	63	7	and	and	CCONJ
eajbcs-694	63	8	can	can	AUX
eajbcs-694	63	9	easily	easily	ADV
eajbcs-694	63	10	spread	spread	VERB
eajbcs-694	63	11	the	the	DET
eajbcs-694	63	12	infection	infection	NOUN
eajbcs-694	63	13	to	to	ADP
eajbcs-694	63	14	other	other	ADJ
eajbcs-694	63	15	people	people	NOUN
eajbcs-694	63	16	with	with	ADP
eajbcs-694	63	17	close	close	ADJ
eajbcs-694	63	18	contact	contact	NOUN
eajbcs-694	63	19	,	,	PUNCT
eajbcs-694	63	20	often	often	ADV
eajbcs-694	63	21	in	in	ADP
eajbcs-694	63	22	an	an	DET
eajbcs-694	63	23	unconscious	unconscious	ADJ
eajbcs-694	63	24	manner	manner	NOUN
eajbcs-694	63	25	.	.	PUNCT
eajbcs-694	64	1	6	6	NUM
eajbcs-694	64	2	.	.	X
eajbcs-694	64	3	transmission	transmission	NOUN
eajbcs-694	64	4	through	through	ADP
eajbcs-694	64	5	human	human	NOUN
eajbcs-694	64	6	-	-	PUNCT
eajbcs-694	64	7	to	to	ADP
eajbcs-694	64	8	-	-	PUNCT
eajbcs-694	64	9	human	human	ADJ
eajbcs-694	64	10	route	route	NOUN
eajbcs-694	64	11	is	be	AUX
eajbcs-694	64	12	alone	alone	ADV
eajbcs-694	64	13	considered	consider	VERB
eajbcs-694	64	14	.	.	PUNCT
eajbcs-694	65	1	other	other	ADJ
eajbcs-694	65	2	means	mean	NOUN
eajbcs-694	65	3	of	of	ADP
eajbcs-694	65	4	transmission	transmission	NOUN
eajbcs-694	65	5	are	be	AUX
eajbcs-694	65	6	ignored	ignore	VERB
eajbcs-694	65	7	.	.	PUNCT
eajbcs-694	66	1	7	7	X
eajbcs-694	66	2	.	.	X
eajbcs-694	66	3	we	we	PRON
eajbcs-694	66	4	assume	assume	VERB
eajbcs-694	66	5	that	that	SCONJ
eajbcs-694	66	6	individuals	individual	NOUN
eajbcs-694	66	7	have	have	VERB
eajbcs-694	66	8	no	no	DET
eajbcs-694	66	9	permanent	permanent	ADJ
eajbcs-694	66	10	immunity	immunity	NOUN
eajbcs-694	66	11	after	after	ADP
eajbcs-694	66	12	recovery	recovery	NOUN
eajbcs-694	66	13	from	from	ADP
eajbcs-694	66	14	the	the	DET
eajbcs-694	66	15	disease	disease	NOUN
eajbcs-694	66	16	,	,	PUNCT
eajbcs-694	66	17	that	that	PRON
eajbcs-694	66	18	is	be	AUX
eajbcs-694	66	19	the	the	DET
eajbcs-694	66	20	recovered	recover	VERB
eajbcs-694	66	21	individuals	individual	NOUN
eajbcs-694	66	22	have	have	VERB
eajbcs-694	66	23	a	a	DET
eajbcs-694	66	24	chance	chance	NOUN
eajbcs-694	66	25	to	to	PART
eajbcs-694	66	26	be	be	AUX
eajbcs-694	66	27	susceptible	susceptible	ADJ
eajbcs-694	66	28	again	again	ADV
eajbcs-694	66	29	.	.	PUNCT
eajbcs-694	67	1	8	8	NUM
eajbcs-694	67	2	the	the	DET
eajbcs-694	67	3	natural	natural	ADJ
eajbcs-694	67	4	mortality	mortality	NOUN
eajbcs-694	67	5	rates	rate	NOUN
eajbcs-694	67	6	are	be	AUX
eajbcs-694	67	7	assumed	assume	VERB
eajbcs-694	67	8	to	to	PART
eajbcs-694	67	9	be	be	AUX
eajbcs-694	67	10	the	the	DET
eajbcs-694	67	11	same	same	ADJ
eajbcs-694	67	12	for	for	ADP
eajbcs-694	67	13	all	all	DET
eajbcs-694	67	14	the	the	DET
eajbcs-694	67	15	compartments	compartment	NOUN
eajbcs-694	67	16	.	.	PUNCT
eajbcs-694	68	1	9	9	X
eajbcs-694	68	2	.	.	X
eajbcs-694	68	3	all	all	DET
eajbcs-694	68	4	parameters	parameter	NOUN
eajbcs-694	68	5	in	in	ADP
eajbcs-694	68	6	the	the	DET
eajbcs-694	68	7	model	model	NOUN
eajbcs-694	68	8	being	be	AUX
eajbcs-694	68	9	nonnegative	nonnegative	ADJ
eajbcs-694	68	10	.	.	PUNCT
eajbcs-694	69	1	we	we	PRON
eajbcs-694	69	2	consider	consider	VERB
eajbcs-694	69	3	the	the	DET
eajbcs-694	69	4	force	force	NOUN
eajbcs-694	69	5	of	of	ADP
eajbcs-694	69	6	infection	infection	NOUN
eajbcs-694	69	7	λ	λ	PROPN
eajbcs-694	69	8	which	which	PRON
eajbcs-694	69	9	is	be	AUX
eajbcs-694	69	10	given	give	VERB
eajbcs-694	69	11	by	by	ADP
eajbcs-694	69	12	λ	λ	PROPN
eajbcs-694	69	13	=	=	PUNCT
eajbcs-694	69	14	β(σ1i	β(σ1i	PUNCT
eajbcs-694	69	15	+	+	NUM
eajbcs-694	69	16	σ2e	σ2e	PROPN
eajbcs-694	69	17	+	+	CCONJ
eajbcs-694	69	18	σ3h	σ3h	NOUN
eajbcs-694	69	19	)	)	PUNCT
eajbcs-694	69	20	,	,	PUNCT
eajbcs-694	69	21	(	(	PUNCT
eajbcs-694	69	22	1	1	X
eajbcs-694	69	23	)	)	PUNCT
eajbcs-694	69	24	where	where	SCONJ
eajbcs-694	69	25	β	β	PROPN
eajbcs-694	69	26	is	be	AUX
eajbcs-694	69	27	the	the	DET
eajbcs-694	69	28	effective	effective	ADJ
eajbcs-694	69	29	contact	contact	NOUN
eajbcs-694	69	30	rate	rate	NOUN
eajbcs-694	69	31	,	,	PUNCT
eajbcs-694	69	32	while	while	SCONJ
eajbcs-694	69	33	σ1	σ1	PROPN
eajbcs-694	69	34	,	,	PUNCT
eajbcs-694	69	35	σ2	σ2	PROPN
eajbcs-694	69	36	and	and	CCONJ
eajbcs-694	69	37	σ3	σ3	PROPN
eajbcs-694	69	38	are	be	AUX
eajbcs-694	69	39	the	the	DET
eajbcs-694	69	40	relative	relative	ADJ
eajbcs-694	69	41	infectiousness	infectiousness	NOUN
eajbcs-694	69	42	parameters	parameter	NOUN
eajbcs-694	69	43	associated	associate	VERB
eajbcs-694	69	44	with	with	ADP
eajbcs-694	69	45	the	the	DET
eajbcs-694	69	46	infected	infect	VERB
eajbcs-694	69	47	,	,	PUNCT
eajbcs-694	69	48	exposed	expose	VERB
eajbcs-694	69	49	and	and	CCONJ
eajbcs-694	69	50	hospitalized	hospitalize	VERB
eajbcs-694	69	51	classes	class	NOUN
eajbcs-694	69	52	respectively	respectively	ADV
eajbcs-694	69	53	.	.	PUNCT
eajbcs-694	70	1	variables	variable	VERB
eajbcs-694	70	2	description	description	NOUN
eajbcs-694	70	3	of	of	ADP
eajbcs-694	70	4	the	the	DET
eajbcs-694	70	5	state	state	NOUN
eajbcs-694	70	6	variables	variable	NOUN
eajbcs-694	70	7	s	s	VERB
eajbcs-694	70	8	susceptible	susceptible	ADJ
eajbcs-694	70	9	individuals	individual	NOUN
eajbcs-694	70	10	p	p	PROPN
eajbcs-694	70	11	protected	protect	VERB
eajbcs-694	70	12	individuals	individual	NOUN
eajbcs-694	70	13	,	,	PUNCT
eajbcs-694	70	14	who	who	PRON
eajbcs-694	70	15	keeping	keep	VERB
eajbcs-694	70	16	social	social	ADJ
eajbcs-694	70	17	distancing	distancing	NOUN
eajbcs-694	70	18	,	,	PUNCT
eajbcs-694	70	19	using	use	VERB
eajbcs-694	70	20	an	an	DET
eajbcs-694	70	21	alcohol	alcohol	NOUN
eajbcs-694	70	22	-	-	PUNCT
eajbcs-694	70	23	based	base	VERB
eajbcs-694	70	24	hand	hand	NOUN
eajbcs-694	70	25	sanitizer	sanitizer	NOUN
eajbcs-694	70	26	and	and	CCONJ
eajbcs-694	70	27	wearing	wear	VERB
eajbcs-694	70	28	face	face	NOUN
eajbcs-694	70	29	masks	mask	NOUN
eajbcs-694	70	30	properly	properly	ADV
eajbcs-694	70	31	to	to	PART
eajbcs-694	70	32	protect	protect	VERB
eajbcs-694	70	33	themselves	themselves	PRON
eajbcs-694	70	34	from	from	ADP
eajbcs-694	70	35	the	the	DET
eajbcs-694	70	36	virus	virus	NOUN
eajbcs-694	70	37	e	e	NOUN
eajbcs-694	70	38	latently	latently	ADV
eajbcs-694	70	39	infected	infect	VERB
eajbcs-694	70	40	individuals	individual	NOUN
eajbcs-694	70	41	,	,	PUNCT
eajbcs-694	70	42	who	who	PRON
eajbcs-694	70	43	have	have	VERB
eajbcs-694	70	44	no	no	DET
eajbcs-694	70	45	symptoms	symptom	NOUN
eajbcs-694	70	46	of	of	ADP
eajbcs-694	70	47	covid-19	covid-19	PROPN
eajbcs-694	70	48	virus	virus	NOUN
eajbcs-694	70	49	disease	disease	NOUN
eajbcs-694	70	50	but	but	CCONJ
eajbcs-694	70	51	are	be	AUX
eajbcs-694	70	52	capable	capable	ADJ
eajbcs-694	70	53	of	of	ADP
eajbcs-694	70	54	infecting	infect	VERB
eajbcs-694	70	55	others	other	NOUN
eajbcs-694	70	56	i	i	PRON
eajbcs-694	70	57	infected	infect	VERB
eajbcs-694	70	58	individuals	individual	NOUN
eajbcs-694	70	59	,	,	PUNCT
eajbcs-694	70	60	who	who	PRON
eajbcs-694	70	61	have	have	VERB
eajbcs-694	70	62	active	active	ADJ
eajbcs-694	70	63	covid-19	covid-19	PROPN
eajbcs-694	70	64	virus	virus	NOUN
eajbcs-694	70	65	disease	disease	NOUN
eajbcs-694	70	66	and	and	CCONJ
eajbcs-694	70	67	can	can	AUX
eajbcs-694	70	68	infect	infect	VERB
eajbcs-694	70	69	other	other	ADJ
eajbcs-694	70	70	people	people	NOUN
eajbcs-694	70	71	h	h	PROPN
eajbcs-694	70	72	hospitalized	hospitalize	VERB
eajbcs-694	70	73	individuals	individual	NOUN
eajbcs-694	70	74	,	,	PUNCT
eajbcs-694	70	75	who	who	PRON
eajbcs-694	70	76	are	be	AUX
eajbcs-694	70	77	admitted	admit	VERB
eajbcs-694	70	78	to	to	ADP
eajbcs-694	70	79	health	health	NOUN
eajbcs-694	70	80	care	care	NOUN
eajbcs-694	70	81	facility	facility	NOUN
eajbcs-694	70	82	or	or	CCONJ
eajbcs-694	70	83	isolated	isolate	VERB
eajbcs-694	70	84	in	in	ADP
eajbcs-694	70	85	their	their	PRON
eajbcs-694	70	86	home	home	NOUN
eajbcs-694	70	87	due	due	ADP
eajbcs-694	70	88	to	to	ADP
eajbcs-694	70	89	virus	virus	NOUN
eajbcs-694	70	90	infection	infection	NOUN
eajbcs-694	70	91	active	active	ADJ
eajbcs-694	70	92	cases	case	NOUN
eajbcs-694	70	93	r	r	NOUN
eajbcs-694	70	94	recovered	recover	VERB
eajbcs-694	70	95	individuals	individual	NOUN
eajbcs-694	70	96	the	the	DET
eajbcs-694	70	97	flow	flow	NOUN
eajbcs-694	70	98	chart	chart	NOUN
eajbcs-694	70	99	of	of	ADP
eajbcs-694	70	100	the	the	DET
eajbcs-694	70	101	modified	modify	VERB
eajbcs-694	70	102	model	model	NOUN
eajbcs-694	70	103	is	be	AUX
eajbcs-694	70	104	illustrated	illustrate	VERB
eajbcs-694	70	105	in	in	ADP
eajbcs-694	70	106	figure	figure	NOUN
eajbcs-694	70	107	(	(	PUNCT
eajbcs-694	70	108	1	1	NUM
eajbcs-694	70	109	)	)	PUNCT
eajbcs-694	70	110	.	.	PUNCT
eajbcs-694	71	1	east	east	PROPN
eajbcs-694	71	2	afr	afr	PROPN
eajbcs-694	71	3	.	.	PUNCT
eajbcs-694	72	1	j.	j.	PROPN
eajbcs-694	72	2	biophys	biophys	PROPN
eajbcs-694	72	3	.	.	PUNCT
eajbcs-694	73	1	comput	comput	NOUN
eajbcs-694	73	2	.	.	PUNCT
eajbcs-694	74	1	sci	sci	PROPN
eajbcs-694	74	2	.	.	PUNCT
eajbcs-694	75	1	(	(	PUNCT
eajbcs-694	75	2	2024	2024	NUM
eajbcs-694	75	3	)	)	PUNCT
eajbcs-694	75	4	,	,	PUNCT
eajbcs-694	75	5	vol	vol	NOUN
eajbcs-694	75	6	.	.	PROPN
eajbcs-694	75	7	5	5	NUM
eajbcs-694	75	8	,	,	PUNCT
eajbcs-694	75	9	no	no	INTJ
eajbcs-694	75	10	.	.	NOUN
eajbcs-694	75	11	2	2	NUM
eajbcs-694	75	12	,	,	PUNCT
eajbcs-694	75	13	58	58	NUM
eajbcs-694	75	14	-	-	SYM
eajbcs-694	75	15	86	86	NUM
eajbcs-694	75	16	60	60	NUM
eajbcs-694	75	17	table	table	NOUN
eajbcs-694	75	18	1	1	NUM
eajbcs-694	75	19	:	:	PUNCT
eajbcs-694	75	20	the	the	DET
eajbcs-694	75	21	state	state	NOUN
eajbcs-694	75	22	variables	variable	NOUN
eajbcs-694	75	23	and	and	CCONJ
eajbcs-694	75	24	their	their	PRON
eajbcs-694	75	25	descriptions	description	NOUN
eajbcs-694	75	26	.	.	PUNCT
eajbcs-694	76	1	table	table	NOUN
eajbcs-694	76	2	2	2	NUM
eajbcs-694	76	3	:	:	PUNCT
eajbcs-694	76	4	parameters	parameter	NOUN
eajbcs-694	76	5	of	of	ADP
eajbcs-694	76	6	the	the	DET
eajbcs-694	76	7	modified	modify	VERB
eajbcs-694	76	8	model	model	NOUN
eajbcs-694	76	9	and	and	CCONJ
eajbcs-694	76	10	their	their	PRON
eajbcs-694	76	11	descriptions	description	NOUN
eajbcs-694	76	12	.	.	PUNCT
eajbcs-694	77	1	parameter	parameter	NOUN
eajbcs-694	77	2	description	description	NOUN
eajbcs-694	77	3	of	of	ADP
eajbcs-694	77	4	the	the	DET
eajbcs-694	77	5	parameter	parameter	NOUN
eajbcs-694	77	6	π	π	PROPN
eajbcs-694	77	7	recruitment	recruitment	NOUN
eajbcs-694	77	8	rate	rate	NOUN
eajbcs-694	77	9	of	of	ADP
eajbcs-694	77	10	susceptible	susceptible	ADJ
eajbcs-694	77	11	θ	θ	PROPN
eajbcs-694	77	12	protection	protection	NOUN
eajbcs-694	77	13	rate	rate	NOUN
eajbcs-694	77	14	of	of	ADP
eajbcs-694	77	15	susceptible	susceptible	ADJ
eajbcs-694	77	16	individuals	individual	NOUN
eajbcs-694	77	17	ν	ν	ADP
eajbcs-694	77	18	waning	wane	VERB
eajbcs-694	77	19	rate	rate	NOUN
eajbcs-694	77	20	of	of	ADP
eajbcs-694	77	21	protected	protect	VERB
eajbcs-694	77	22	individuals	individual	NOUN
eajbcs-694	77	23	to	to	PART
eajbcs-694	77	24	susceptible	susceptible	ADJ
eajbcs-694	77	25	class	class	NOUN
eajbcs-694	77	26	β	β	X
eajbcs-694	77	27	effective	effective	ADJ
eajbcs-694	77	28	contact	contact	NOUN
eajbcs-694	77	29	rate	rate	NOUN
eajbcs-694	77	30	σ1	σ1	PROPN
eajbcs-694	77	31	modification	modification	NOUN
eajbcs-694	77	32	parameter	parameter	NOUN
eajbcs-694	77	33	for	for	ADP
eajbcs-694	77	34	relative	relative	ADJ
eajbcs-694	77	35	infectiousness	infectiousness	NOUN
eajbcs-694	77	36	of	of	ADP
eajbcs-694	77	37	infected	infect	VERB
eajbcs-694	77	38	individuals	individual	NOUN
eajbcs-694	77	39	σ2	σ2	NOUN
eajbcs-694	77	40	modification	modification	NOUN
eajbcs-694	77	41	parameter	parameter	NOUN
eajbcs-694	77	42	for	for	ADP
eajbcs-694	77	43	relative	relative	ADJ
eajbcs-694	77	44	infectiousness	infectiousness	NOUN
eajbcs-694	77	45	of	of	ADP
eajbcs-694	77	46	exposed	expose	VERB
eajbcs-694	77	47	individuals	individual	NOUN
eajbcs-694	77	48	σ3	σ3	PROPN
eajbcs-694	77	49	modification	modification	NOUN
eajbcs-694	77	50	parameter	parameter	NOUN
eajbcs-694	77	51	for	for	ADP
eajbcs-694	77	52	relative	relative	ADJ
eajbcs-694	77	53	infectiousness	infectiousness	NOUN
eajbcs-694	77	54	of	of	ADP
eajbcs-694	77	55	hospitalized	hospitalize	VERB
eajbcs-694	77	56	individuals	individual	NOUN
eajbcs-694	77	57	δ	δ	PROPN
eajbcs-694	77	58	the	the	DET
eajbcs-694	77	59	exposed	expose	VERB
eajbcs-694	77	60	progression	progression	NOUN
eajbcs-694	77	61	rate	rate	NOUN
eajbcs-694	77	62	τ	τ	PROPN
eajbcs-694	77	63	proportion	proportion	NOUN
eajbcs-694	77	64	of	of	ADP
eajbcs-694	77	65	exposed	expose	VERB
eajbcs-694	77	66	individuals	individual	NOUN
eajbcs-694	77	67	who	who	PRON
eajbcs-694	77	68	join	join	VERB
eajbcs-694	77	69	infected	infected	ADJ
eajbcs-694	77	70	class	class	NOUN
eajbcs-694	77	71	1−	1−	NUM
eajbcs-694	77	72	τ	τ	PROPN
eajbcs-694	77	73	the	the	DET
eajbcs-694	77	74	progression	progression	NOUN
eajbcs-694	77	75	from	from	ADP
eajbcs-694	77	76	exposed	expose	VERB
eajbcs-694	77	77	individuals	individual	NOUN
eajbcs-694	77	78	to	to	PART
eajbcs-694	77	79	recovered	recover	VERB
eajbcs-694	77	80	class	class	NOUN
eajbcs-694	77	81	α	α	NOUN
eajbcs-694	77	82	hospitalization	hospitalization	NOUN
eajbcs-694	77	83	rate	rate	NOUN
eajbcs-694	77	84	of	of	ADP
eajbcs-694	77	85	infected	infected	ADJ
eajbcs-694	77	86	individuals	individual	NOUN
eajbcs-694	77	87	µ	µ	PRON
eajbcs-694	77	88	natural	natural	ADJ
eajbcs-694	77	89	death	death	NOUN
eajbcs-694	77	90	rate	rate	NOUN
eajbcs-694	77	91	ρ	ρ	PROPN
eajbcs-694	77	92	disease	disease	NOUN
eajbcs-694	77	93	-	-	PUNCT
eajbcs-694	77	94	induced	induce	VERB
eajbcs-694	77	95	death	death	NOUN
eajbcs-694	77	96	rate	rate	NOUN
eajbcs-694	77	97	of	of	ADP
eajbcs-694	77	98	infected	infected	ADJ
eajbcs-694	77	99	individuals	individual	NOUN
eajbcs-694	77	100	ξ	ξ	PRON
eajbcs-694	77	101	disease	disease	NOUN
eajbcs-694	77	102	-	-	PUNCT
eajbcs-694	77	103	induced	induce	VERB
eajbcs-694	77	104	death	death	NOUN
eajbcs-694	77	105	rate	rate	NOUN
eajbcs-694	77	106	of	of	ADP
eajbcs-694	77	107	hospitalized	hospitalize	VERB
eajbcs-694	77	108	individuals	individual	NOUN
eajbcs-694	77	109	ε	ε	PROPN
eajbcs-694	77	110	rate	rate	NOUN
eajbcs-694	77	111	of	of	ADP
eajbcs-694	77	112	recovery	recovery	NOUN
eajbcs-694	77	113	of	of	ADP
eajbcs-694	77	114	the	the	DET
eajbcs-694	77	115	individuals	individual	NOUN
eajbcs-694	77	116	from	from	ADP
eajbcs-694	77	117	infected	infected	ADJ
eajbcs-694	77	118	class	class	NOUN
eajbcs-694	77	119	γ	γ	PROPN
eajbcs-694	77	120	recovery	recovery	NOUN
eajbcs-694	77	121	rate	rate	NOUN
eajbcs-694	77	122	of	of	ADP
eajbcs-694	77	123	hospitalized	hospitalize	VERB
eajbcs-694	77	124	patients	patient	NOUN
eajbcs-694	77	125	ω	ω	NUM
eajbcs-694	77	126	waning	wane	VERB
eajbcs-694	77	127	immunity	immunity	NOUN
eajbcs-694	77	128	rate	rate	NOUN
eajbcs-694	77	129	η	η	PROPN
eajbcs-694	77	130	the	the	DET
eajbcs-694	77	131	proportion	proportion	NOUN
eajbcs-694	77	132	of	of	ADP
eajbcs-694	77	133	recovered	recover	VERB
eajbcs-694	77	134	individuals	individual	NOUN
eajbcs-694	77	135	that	that	PRON
eajbcs-694	77	136	become	become	VERB
eajbcs-694	77	137	susceptible	susceptible	ADJ
eajbcs-694	77	138	1−	1−	NUM
eajbcs-694	77	139	η	η	NOUN
eajbcs-694	77	140	the	the	DET
eajbcs-694	77	141	progression	progression	NOUN
eajbcs-694	77	142	from	from	ADP
eajbcs-694	77	143	recovered	recover	VERB
eajbcs-694	77	144	individuals	individual	NOUN
eajbcs-694	77	145	to	to	PART
eajbcs-694	77	146	protected	protect	VERB
eajbcs-694	77	147	class	class	NOUN
eajbcs-694	77	148	figure	figure	NOUN
eajbcs-694	77	149	1	1	NUM
eajbcs-694	77	150	:	:	PUNCT
eajbcs-694	77	151	the	the	DET
eajbcs-694	77	152	flow	flow	NOUN
eajbcs-694	77	153	chart	chart	NOUN
eajbcs-694	77	154	for	for	ADP
eajbcs-694	77	155	the	the	DET
eajbcs-694	77	156	modified	modify	VERB
eajbcs-694	77	157	model	model	NOUN
eajbcs-694	77	158	of	of	ADP
eajbcs-694	77	159	covid-19	covid-19	PROPN
eajbcs-694	77	160	pandemic	pandemic	NOUN
eajbcs-694	77	161	.	.	PUNCT
eajbcs-694	78	1	east	east	PROPN
eajbcs-694	78	2	afr	afr	PROPN
eajbcs-694	78	3	.	.	PUNCT
eajbcs-694	79	1	j.	j.	PROPN
eajbcs-694	79	2	biophys	biophys	PROPN
eajbcs-694	79	3	.	.	PUNCT
eajbcs-694	80	1	comput	comput	NOUN
eajbcs-694	80	2	.	.	PUNCT
eajbcs-694	81	1	sci	sci	PROPN
eajbcs-694	81	2	.	.	PUNCT
eajbcs-694	82	1	(	(	PUNCT
eajbcs-694	82	2	2024	2024	NUM
eajbcs-694	82	3	)	)	PUNCT
eajbcs-694	82	4	,	,	PUNCT
eajbcs-694	82	5	vol	vol	NOUN
eajbcs-694	82	6	.	.	PROPN
eajbcs-694	82	7	5	5	NUM
eajbcs-694	82	8	,	,	PUNCT
eajbcs-694	82	9	no	no	INTJ
eajbcs-694	82	10	.	.	NOUN
eajbcs-694	82	11	2	2	NUM
eajbcs-694	82	12	,	,	PUNCT
eajbcs-694	82	13	58	58	NUM
eajbcs-694	82	14	-	-	SYM
eajbcs-694	82	15	86	86	NUM
eajbcs-694	82	16	61	61	NUM
eajbcs-694	82	17	based	base	VERB
eajbcs-694	82	18	on	on	ADP
eajbcs-694	82	19	our	our	PRON
eajbcs-694	82	20	assumptions	assumption	NOUN
eajbcs-694	82	21	and	and	CCONJ
eajbcs-694	82	22	the	the	DET
eajbcs-694	82	23	flow	flow	NOUN
eajbcs-694	82	24	chart	chart	NOUN
eajbcs-694	82	25	(	(	PUNCT
eajbcs-694	82	26	1	1	NUM
eajbcs-694	82	27	)	)	PUNCT
eajbcs-694	82	28	,	,	PUNCT
eajbcs-694	82	29	the	the	DET
eajbcs-694	82	30	modified	modify	VERB
eajbcs-694	82	31	model	model	NOUN
eajbcs-694	82	32	for	for	ADP
eajbcs-694	82	33	the	the	DET
eajbcs-694	82	34	transmission	transmission	NOUN
eajbcs-694	82	35	dynamics	dynamic	NOUN
eajbcs-694	82	36	of	of	ADP
eajbcs-694	82	37	covid-19	covid-19	PROPN
eajbcs-694	82	38	is	be	AUX
eajbcs-694	82	39	given	give	VERB
eajbcs-694	82	40	by	by	ADP
eajbcs-694	82	41	the	the	DET
eajbcs-694	82	42	following	follow	VERB
eajbcs-694	82	43	deterministic	deterministic	ADJ
eajbcs-694	82	44	system	system	NOUN
eajbcs-694	82	45	of	of	ADP
eajbcs-694	82	46	non	non	ADJ
eajbcs-694	82	47	-	-	ADJ
eajbcs-694	82	48	linear	linear	ADJ
eajbcs-694	82	49	differential	differential	ADJ
eajbcs-694	82	50	equations	equation	NOUN
eajbcs-694	82	51	:	:	PUNCT
eajbcs-694	82	52	ds	ds	ADJ
eajbcs-694	82	53	dt	dt	X
eajbcs-694	82	54	=	=	PUNCT
eajbcs-694	82	55	π+	π+	PUNCT
eajbcs-694	82	56	ηωr	ηωr	PROPN
eajbcs-694	82	57	+	+	X
eajbcs-694	82	58	νp	νp	ADP
eajbcs-694	82	59	−	−	PROPN
eajbcs-694	82	60	β(σ1i	β(σ1i	NOUN
eajbcs-694	82	61	+	+	NUM
eajbcs-694	82	62	σ2e	σ2e	PROPN
eajbcs-694	82	63	+	+	CCONJ
eajbcs-694	83	1	σ3h)s	σ3h)s	X
eajbcs-694	83	2	−	−	PROPN
eajbcs-694	83	3	(	(	PUNCT
eajbcs-694	83	4	θ	θ	NOUN
eajbcs-694	83	5	+	+	NOUN
eajbcs-694	83	6	µ)s	µ)s	NOUN
eajbcs-694	83	7	,	,	PUNCT
eajbcs-694	83	8	(	(	PUNCT
eajbcs-694	83	9	2a	2a	NUM
eajbcs-694	83	10	)	)	PUNCT
eajbcs-694	83	11	dp	dp	NOUN
eajbcs-694	83	12	dt	dt	NOUN
eajbcs-694	83	13	=	=	PUNCT
eajbcs-694	83	14	θs	θs	PRON
eajbcs-694	83	15	+	+	X
eajbcs-694	83	16	(	(	PUNCT
eajbcs-694	83	17	1−	1−	NUM
eajbcs-694	83	18	η)ωr−	η)ωr−	PROPN
eajbcs-694	83	19	(	(	PUNCT
eajbcs-694	83	20	ν	ν	X
eajbcs-694	83	21	+	+	NOUN
eajbcs-694	83	22	µ)p	µ)p	NOUN
eajbcs-694	83	23	,	,	PUNCT
eajbcs-694	83	24	(	(	PUNCT
eajbcs-694	83	25	2b	2b	NOUN
eajbcs-694	83	26	)	)	PUNCT
eajbcs-694	83	27	de	de	NOUN
eajbcs-694	83	28	dt	dt	NOUN
eajbcs-694	83	29	=	=	PUNCT
eajbcs-694	83	30	β(σ1i	β(σ1i	PUNCT
eajbcs-694	83	31	+	+	NUM
eajbcs-694	83	32	σ2e	σ2e	PROPN
eajbcs-694	83	33	+	+	CCONJ
eajbcs-694	83	34	σ3h)s	σ3h)s	X
eajbcs-694	83	35	−	−	PROPN
eajbcs-694	83	36	(	(	PUNCT
eajbcs-694	83	37	δ	δ	NOUN
eajbcs-694	83	38	+	+	CCONJ
eajbcs-694	83	39	µ)e	µ)e	NOUN
eajbcs-694	83	40	,	,	PUNCT
eajbcs-694	83	41	(	(	PUNCT
eajbcs-694	83	42	2c	2c	NOUN
eajbcs-694	83	43	)	)	PUNCT
eajbcs-694	83	44	di	di	NOUN
eajbcs-694	83	45	dt	dt	NOUN
eajbcs-694	83	46	=	=	PUNCT
eajbcs-694	83	47	τδe	τδe	NOUN
eajbcs-694	83	48	−	−	PROPN
eajbcs-694	83	49	(	(	PUNCT
eajbcs-694	83	50	α+	α+	X
eajbcs-694	83	51	ε+	ε+	X
eajbcs-694	83	52	µ+	µ+	X
eajbcs-694	83	53	ρ)i	ρ)i	NOUN
eajbcs-694	83	54	,	,	PUNCT
eajbcs-694	83	55	(	(	PUNCT
eajbcs-694	83	56	2d	2d	NOUN
eajbcs-694	83	57	)	)	PUNCT
eajbcs-694	84	1	dh	dh	NOUN
eajbcs-694	84	2	dt	dt	NOUN
eajbcs-694	84	3	=	=	PUNCT
eajbcs-694	84	4	αi	αi	PROPN
eajbcs-694	84	5	−	−	PROPN
eajbcs-694	85	1	(	(	PUNCT
eajbcs-694	85	2	γ	γ	X
eajbcs-694	85	3	+	+	X
eajbcs-694	85	4	µ+	µ+	X
eajbcs-694	85	5	ξ)h	ξ)h	NUM
eajbcs-694	85	6	,	,	PUNCT
eajbcs-694	85	7	(	(	PUNCT
eajbcs-694	85	8	2e	2e	NOUN
eajbcs-694	85	9	)	)	PUNCT
eajbcs-694	85	10	dr	dr	PROPN
eajbcs-694	85	11	dt	dt	NOUN
eajbcs-694	85	12	=	=	PUNCT
eajbcs-694	85	13	(	(	PUNCT
eajbcs-694	85	14	1−	1−	NUM
eajbcs-694	85	15	τ)δe	τ)δe	PROPN
eajbcs-694	85	16	+	+	CCONJ
eajbcs-694	85	17	εi	εi	VERB
eajbcs-694	85	18	+	+	X
eajbcs-694	85	19	γh	γh	ADP
eajbcs-694	85	20	−	−	PROPN
eajbcs-694	85	21	(	(	PUNCT
eajbcs-694	85	22	ω	ω	NOUN
eajbcs-694	85	23	+	+	CCONJ
eajbcs-694	85	24	µ)r	µ)r	NOUN
eajbcs-694	85	25	,	,	PUNCT
eajbcs-694	85	26	(	(	PUNCT
eajbcs-694	85	27	2f	2f	NOUN
eajbcs-694	85	28	)	)	PUNCT
eajbcs-694	85	29	with	with	ADP
eajbcs-694	85	30	non	non	ADJ
eajbcs-694	85	31	-	-	ADJ
eajbcs-694	85	32	negative	negative	ADJ
eajbcs-694	85	33	initial	initial	ADJ
eajbcs-694	85	34	conditions	condition	NOUN
eajbcs-694	85	35	s(0	s(0	PROPN
eajbcs-694	85	36	)	)	PUNCT
eajbcs-694	85	37	=	=	SYM
eajbcs-694	85	38	s0	s0	PROPN
eajbcs-694	85	39	>	>	PUNCT
eajbcs-694	85	40	0	0	PROPN
eajbcs-694	85	41	,	,	PUNCT
eajbcs-694	85	42	p	p	X
eajbcs-694	85	43	(	(	PUNCT
eajbcs-694	85	44	0	0	NUM
eajbcs-694	85	45	)	)	PUNCT
eajbcs-694	85	46	=	=	NOUN
eajbcs-694	86	1	p0	p0	NOUN
eajbcs-694	86	2	≥	≥	NUM
eajbcs-694	86	3	0	0	NUM
eajbcs-694	86	4	,	,	PUNCT
eajbcs-694	86	5	e(0	e(0	NOUN
eajbcs-694	86	6	)	)	PUNCT
eajbcs-694	86	7	=	=	SYM
eajbcs-694	86	8	e0	e0	PROPN
eajbcs-694	86	9	≥	≥	NUM
eajbcs-694	86	10	0	0	NUM
eajbcs-694	86	11	,	,	PUNCT
eajbcs-694	86	12	i(0	i(0	PROPN
eajbcs-694	86	13	)	)	PUNCT
eajbcs-694	87	1	=	=	SYM
eajbcs-694	87	2	i0	i0	PROPN
eajbcs-694	87	3	≥	≥	NUM
eajbcs-694	87	4	0	0	NUM
eajbcs-694	87	5	,	,	PUNCT
eajbcs-694	87	6	h(0	h(0	PROPN
eajbcs-694	87	7	)	)	PUNCT
eajbcs-694	87	8	=	=	SYM
eajbcs-694	87	9	h0	h0	NOUN
eajbcs-694	87	10	≥	≥	NOUN
eajbcs-694	87	11	0	0	NUM
eajbcs-694	87	12	and	and	CCONJ
eajbcs-694	87	13	r(0	r(0	PROPN
eajbcs-694	87	14	)	)	PUNCT
eajbcs-694	88	1	=	=	PUNCT
eajbcs-694	88	2	r0	r0	NOUN
eajbcs-694	88	3	≥	≥	NOUN
eajbcs-694	88	4	0	0	NUM
eajbcs-694	88	5	.	.	PUNCT
eajbcs-694	89	1	3	3	X
eajbcs-694	89	2	.	.	X
eajbcs-694	89	3	qualitative	qualitative	ADJ
eajbcs-694	89	4	analysis	analysis	NOUN
eajbcs-694	89	5	of	of	ADP
eajbcs-694	89	6	the	the	DET
eajbcs-694	89	7	modified	modify	VERB
eajbcs-694	89	8	model	model	NOUN
eajbcs-694	89	9	in	in	ADP
eajbcs-694	89	10	this	this	DET
eajbcs-694	89	11	section	section	NOUN
eajbcs-694	90	1	,	,	PUNCT
eajbcs-694	90	2	we	we	PRON
eajbcs-694	90	3	present	present	VERB
eajbcs-694	90	4	some	some	DET
eajbcs-694	90	5	basic	basic	ADJ
eajbcs-694	90	6	qualitative	qualitative	ADJ
eajbcs-694	90	7	properties	property	NOUN
eajbcs-694	90	8	of	of	ADP
eajbcs-694	90	9	the	the	DET
eajbcs-694	90	10	modified	modify	VERB
eajbcs-694	90	11	model	model	NOUN
eajbcs-694	90	12	.	.	PUNCT
eajbcs-694	91	1	these	these	DET
eajbcs-694	91	2	analysis	analysis	NOUN
eajbcs-694	91	3	include	include	VERB
eajbcs-694	91	4	finding	find	VERB
eajbcs-694	91	5	t	t	X
eajbcs-694	91	6	he	he	PRON
eajbcs-694	91	7	s	s	VERB
eajbcs-694	91	8	et	et	NOUN
eajbcs-694	91	9	i	i	PRON
eajbcs-694	91	10	nside	nside	ADP
eajbcs-694	91	11	which	which	PRON
eajbcs-694	91	12	the	the	DET
eajbcs-694	91	13	model	model	NOUN
eajbcs-694	91	14	can	can	AUX
eajbcs-694	91	15	be	be	AUX
eajbcs-694	91	16	sufficiently	sufficiently	ADV
eajbcs-694	91	17	studied	study	VERB
eajbcs-694	91	18	(	(	PUNCT
eajbcs-694	91	19	i	i	NOUN
eajbcs-694	91	20	.e	.e	PROPN
eajbcs-694	91	21	.	.	PUNCT
eajbcs-694	92	1	,	,	PUNCT
eajbcs-694	92	2	the	the	DET
eajbcs-694	92	3	invariant	invariant	ADJ
eajbcs-694	92	4	region	region	NOUN
eajbcs-694	92	5	)	)	PUNCT
eajbcs-694	92	6	;	;	PUNCT
eajbcs-694	92	7	local	local	ADJ
eajbcs-694	92	8	and	and	CCONJ
eajbcs-694	92	9	global	global	ADJ
eajbcs-694	92	10	stability	stability	NOUN
eajbcs-694	92	11	of	of	ADP
eajbcs-694	92	12	equilibrium	equilibrium	NOUN
eajbcs-694	92	13	points	point	NOUN
eajbcs-694	92	14	of	of	ADP
eajbcs-694	92	15	the	the	DET
eajbcs-694	92	16	model	model	NOUN
eajbcs-694	92	17	(	(	PUNCT
eajbcs-694	92	18	2	2	NUM
eajbcs-694	92	19	)	)	PUNCT
eajbcs-694	92	20	.	.	PUNCT
eajbcs-694	93	1	3.1	3.1	NUM
eajbcs-694	93	2	.	.	PUNCT
eajbcs-694	94	1	well	well	ADJ
eajbcs-694	94	2	-	-	PUNCT
eajbcs-694	94	3	posedness	posedness	NOUN
eajbcs-694	94	4	since	since	SCONJ
eajbcs-694	94	5	all	all	DET
eajbcs-694	94	6	the	the	DET
eajbcs-694	94	7	functions	function	NOUN
eajbcs-694	94	8	on	on	ADP
eajbcs-694	94	9	the	the	DET
eajbcs-694	94	10	right	right	ADJ
eajbcs-694	94	11	hand	hand	NOUN
eajbcs-694	94	12	side	side	NOUN
eajbcs-694	94	13	of	of	ADP
eajbcs-694	94	14	the	the	DET
eajbcs-694	94	15	system	system	NOUN
eajbcs-694	94	16	(	(	PUNCT
eajbcs-694	94	17	2	2	X
eajbcs-694	94	18	)	)	PUNCT
eajbcs-694	94	19	are	be	AUX
eajbcs-694	94	20	continuously	continuously	ADV
eajbcs-694	94	21	differentiable	differentiable	ADJ
eajbcs-694	94	22	.	.	PUNCT
eajbcs-694	95	1	thus	thus	ADV
eajbcs-694	95	2	,	,	PUNCT
eajbcs-694	95	3	the	the	DET
eajbcs-694	95	4	existence	existence	NOUN
eajbcs-694	95	5	and	and	CCONJ
eajbcs-694	95	6	uniqueness	uniqueness	NOUN
eajbcs-694	95	7	of	of	ADP
eajbcs-694	95	8	the	the	DET
eajbcs-694	95	9	solutions	solution	NOUN
eajbcs-694	95	10	is	be	AUX
eajbcs-694	95	11	established	establish	VERB
eajbcs-694	95	12	by	by	ADP
eajbcs-694	95	13	the	the	DET
eajbcs-694	95	14	picard	picard	PROPN
eajbcs-694	95	15	’s	’s	PART
eajbcs-694	95	16	theorem	theorem	ADJ
eajbcs-694	95	17	valcher	valcher	NOUN
eajbcs-694	95	18	(	(	PUNCT
eajbcs-694	95	19	2002	2002	NUM
eajbcs-694	95	20	)	)	PUNCT
eajbcs-694	95	21	.	.	PUNCT
eajbcs-694	96	1	now	now	ADV
eajbcs-694	96	2	,	,	PUNCT
eajbcs-694	96	3	we	we	PRON
eajbcs-694	96	4	show	show	VERB
eajbcs-694	96	5	the	the	DET
eajbcs-694	96	6	positivity	positivity	NOUN
eajbcs-694	96	7	and	and	CCONJ
eajbcs-694	96	8	boundedness	boundedness	NOUN
eajbcs-694	96	9	of	of	ADP
eajbcs-694	96	10	solutions	solution	NOUN
eajbcs-694	96	11	.	.	PUNCT
eajbcs-694	97	1	theorem	theorem	VERB
eajbcs-694	97	2	3.1	3.1	NUM
eajbcs-694	97	3	.	.	PUNCT
eajbcs-694	98	1	(	(	PUNCT
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eajbcs-694	98	3	)	)	PUNCT
eajbcs-694	98	4	if	if	SCONJ
eajbcs-694	98	5	s(0	s(0	PROPN
eajbcs-694	98	6	)	)	PUNCT
eajbcs-694	98	7	>	>	X
eajbcs-694	98	8	0	0	NUM
eajbcs-694	98	9	,	,	PUNCT
eajbcs-694	98	10	p	p	X
eajbcs-694	98	11	(	(	PUNCT
eajbcs-694	98	12	0	0	NUM
eajbcs-694	98	13	)	)	PUNCT
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eajbcs-694	98	15	0	0	NUM
eajbcs-694	98	16	,	,	PUNCT
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eajbcs-694	98	18	)	)	PUNCT
eajbcs-694	98	19	≥	≥	NOUN
eajbcs-694	98	20	0	0	NUM
eajbcs-694	98	21	,	,	PUNCT
eajbcs-694	98	22	i(0	i(0	PROPN
eajbcs-694	98	23	)	)	PUNCT
eajbcs-694	98	24	≥	≥	NOUN
eajbcs-694	98	25	0	0	NUM
eajbcs-694	98	26	,	,	PUNCT
eajbcs-694	98	27	h(0	h(0	PROPN
eajbcs-694	98	28	)	)	PUNCT
eajbcs-694	98	29	≥	≥	NOUN
eajbcs-694	98	30	0	0	NUM
eajbcs-694	98	31	and	and	CCONJ
eajbcs-694	98	32	r(0	r(0	PROPN
eajbcs-694	98	33	)	)	PUNCT
eajbcs-694	98	34	≥	≥	NOUN
eajbcs-694	98	35	0	0	NUM
eajbcs-694	98	36	,	,	PUNCT
eajbcs-694	98	37	then	then	ADV
eajbcs-694	98	38	the	the	DET
eajbcs-694	98	39	solution	solution	NOUN
eajbcs-694	98	40	(	(	PUNCT
eajbcs-694	98	41	s(t	s(t	PROPN
eajbcs-694	98	42	)	)	PUNCT
eajbcs-694	98	43	,	,	PUNCT
eajbcs-694	98	44	p	p	X
eajbcs-694	98	45	(	(	PUNCT
eajbcs-694	98	46	t	t	PROPN
eajbcs-694	98	47	)	)	PUNCT
eajbcs-694	98	48	,	,	PUNCT
eajbcs-694	98	49	e(t	e(t	NOUN
eajbcs-694	98	50	)	)	PUNCT
eajbcs-694	98	51	,	,	PUNCT
eajbcs-694	98	52	i(t	i(t	PROPN
eajbcs-694	98	53	)	)	PUNCT
eajbcs-694	98	54	,	,	PUNCT
eajbcs-694	98	55	h(t	h(t	PROPN
eajbcs-694	98	56	)	)	PUNCT
eajbcs-694	98	57	,	,	PUNCT
eajbcs-694	98	58	r(t	r(t	NOUN
eajbcs-694	98	59	)	)	PUNCT
eajbcs-694	98	60	)	)	PUNCT
eajbcs-694	98	61	of	of	ADP
eajbcs-694	98	62	the	the	DET
eajbcs-694	98	63	dynamical	dynamical	ADJ
eajbcs-694	98	64	system	system	NOUN
eajbcs-694	98	65	(	(	PUNCT
eajbcs-694	98	66	2	2	X
eajbcs-694	98	67	)	)	PUNCT
eajbcs-694	98	68	is	be	AUX
eajbcs-694	98	69	non	non	ADJ
eajbcs-694	98	70	-	-	ADJ
eajbcs-694	98	71	negative	negative	ADJ
eajbcs-694	98	72	for	for	ADP
eajbcs-694	98	73	all	all	DET
eajbcs-694	98	74	time	time	NOUN
eajbcs-694	98	75	t	t	PROPN
eajbcs-694	98	76	≥	≥	NOUN
eajbcs-694	98	77	0	0	NUM
eajbcs-694	98	78	.	.	PUNCT
eajbcs-694	99	1	proof	proof	NOUN
eajbcs-694	99	2	.	.	PUNCT
eajbcs-694	100	1	to	to	PART
eajbcs-694	100	2	show	show	VERB
eajbcs-694	100	3	the	the	DET
eajbcs-694	100	4	positivity	positivity	NOUN
eajbcs-694	100	5	of	of	ADP
eajbcs-694	100	6	the	the	DET
eajbcs-694	100	7	solution	solution	NOUN
eajbcs-694	100	8	of	of	ADP
eajbcs-694	100	9	the	the	DET
eajbcs-694	100	10	dynamical	dynamical	ADJ
eajbcs-694	100	11	system	system	NOUN
eajbcs-694	100	12	(	(	PUNCT
eajbcs-694	100	13	2	2	NUM
eajbcs-694	100	14	)	)	PUNCT
eajbcs-694	100	15	,	,	PUNCT
eajbcs-694	100	16	we	we	PRON
eajbcs-694	100	17	will	will	AUX
eajbcs-694	100	18	perform	perform	VERB
eajbcs-694	100	19	the	the	DET
eajbcs-694	100	20	proof	proof	NOUN
eajbcs-694	100	21	by	by	ADP
eajbcs-694	100	22	using	use	VERB
eajbcs-694	100	23	contradiction	contradiction	NOUN
eajbcs-694	100	24	.	.	PUNCT
eajbcs-694	101	1	we	we	PRON
eajbcs-694	101	2	assume	assume	VERB
eajbcs-694	101	3	that	that	SCONJ
eajbcs-694	101	4	s(t	s(t	PROPN
eajbcs-694	101	5	)	)	PUNCT
eajbcs-694	101	6	≤	≤	NOUN
eajbcs-694	101	7	0	0	NUM
eajbcs-694	101	8	for	for	ADP
eajbcs-694	101	9	some	some	DET
eajbcs-694	101	10	t	t	NOUN
eajbcs-694	101	11	≥	≥	NOUN
eajbcs-694	101	12	0	0	NUM
eajbcs-694	101	13	,	,	PUNCT
eajbcs-694	101	14	that	that	PRON
eajbcs-694	101	15	is	be	AUX
eajbcs-694	101	16	there	there	PRON
eajbcs-694	101	17	exists	exist	VERB
eajbcs-694	101	18	small	small	ADJ
eajbcs-694	101	19	t0	t0	PROPN
eajbcs-694	101	20	>	>	X
eajbcs-694	101	21	0	0	PUNCT
eajbcs-694	101	22	such	such	ADJ
eajbcs-694	101	23	that	that	DET
eajbcs-694	101	24	s(t0	s(t0	NOUN
eajbcs-694	101	25	)	)	PUNCT
eajbcs-694	102	1	=	=	SYM
eajbcs-694	102	2	0	0	NUM
eajbcs-694	102	3	,	,	PUNCT
eajbcs-694	102	4	s	s	NOUN
eajbcs-694	102	5	′(t0	′(t0	NOUN
eajbcs-694	102	6	)	)	PUNCT
eajbcs-694	102	7	≤	≤	NOUN
eajbcs-694	102	8	0	0	NUM
eajbcs-694	102	9	and	and	CCONJ
eajbcs-694	102	10	s(t	s(t	NUM
eajbcs-694	102	11	)	)	PUNCT
eajbcs-694	102	12	>	>	X
eajbcs-694	102	13	0	0	PUNCT
eajbcs-694	103	1	for	for	ADP
eajbcs-694	103	2	t	t	PROPN
eajbcs-694	103	3	∈	∈	PROPN
eajbcs-694	104	1	[	[	X
eajbcs-694	104	2	0	0	NUM
eajbcs-694	104	3	,	,	PUNCT
eajbcs-694	104	4	t0	t0	PROPN
eajbcs-694	104	5	)	)	PUNCT
eajbcs-694	104	6	.	.	PUNCT
eajbcs-694	105	1	then	then	ADV
eajbcs-694	105	2	p	p	X
eajbcs-694	105	3	(	(	PUNCT
eajbcs-694	105	4	t	t	PROPN
eajbcs-694	105	5	)	)	PUNCT
eajbcs-694	105	6	≥	≥	NOUN
eajbcs-694	105	7	0	0	NUM
eajbcs-694	105	8	,	,	PUNCT
eajbcs-694	105	9	e(t	e(t	PROPN
eajbcs-694	105	10	)	)	PUNCT
eajbcs-694	105	11	≥	≥	NOUN
eajbcs-694	105	12	0	0	NUM
eajbcs-694	105	13	and	and	CCONJ
eajbcs-694	105	14	i(t	i(t	NOUN
eajbcs-694	105	15	)	)	PUNCT
eajbcs-694	105	16	≥	≥	NOUN
eajbcs-694	105	17	0	0	NUM
eajbcs-694	105	18	for	for	ADP
eajbcs-694	105	19	t	t	PROPN
eajbcs-694	105	20	∈	∈	PROPN
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eajbcs-694	106	2	0	0	NUM
eajbcs-694	106	3	,	,	PUNCT
eajbcs-694	106	4	t0	t0	PROPN
eajbcs-694	106	5	]	]	PUNCT
eajbcs-694	106	6	.	.	PUNCT
eajbcs-694	107	1	if	if	SCONJ
eajbcs-694	107	2	this	this	PRON
eajbcs-694	107	3	be	be	VERB
eajbcs-694	107	4	not	not	PART
eajbcs-694	107	5	the	the	DET
eajbcs-694	107	6	case	case	NOUN
eajbcs-694	107	7	,	,	PUNCT
eajbcs-694	107	8	there	there	PRON
eajbcs-694	107	9	exists	exist	VERB
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eajbcs-694	107	11	-	-	PUNCT
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eajbcs-694	108	6	such	such	ADJ
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eajbcs-694	108	11	)	)	PUNCT
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eajbcs-694	108	19	0	0	PUNCT
eajbcs-694	108	20	and	and	CCONJ
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eajbcs-694	108	23	t	t	PROPN
eajbcs-694	108	24	)	)	PUNCT
eajbcs-694	108	25	>	>	X
eajbcs-694	108	26	0	0	PUNCT
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eajbcs-694	128	3	(	(	PUNCT
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eajbcs-694	128	5	ε+	ε+	X
eajbcs-694	128	6	µ+	µ+	X
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eajbcs-694	129	5	′(t3	′(t3	VERB
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eajbcs-694	130	5	0	0	NUM
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eajbcs-694	131	1	this	this	PRON
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eajbcs-694	131	3	a	a	DET
eajbcs-694	131	4	contradiction	contradiction	NOUN
eajbcs-694	131	5	.	.	PUNCT
eajbcs-694	132	1	integration	integration	NOUN
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eajbcs-694	132	6	)	)	PUNCT
eajbcs-694	132	7	leads	lead	VERB
eajbcs-694	132	8	to	to	ADP
eajbcs-694	132	9	h(t	h(t	NUM
eajbcs-694	132	10	)	)	PUNCT
eajbcs-694	133	1	=	=	VERB
eajbcs-694	133	2	e−(γ+µ+ξ)t	e−(γ+µ+ξ)t	NOUN
eajbcs-694	133	3	(	(	PUNCT
eajbcs-694	133	4	h(0	h(0	PROPN
eajbcs-694	133	5	)	)	PUNCT
eajbcs-694	134	1	+	+	CCONJ
eajbcs-694	134	2	α	α	NUM
eajbcs-694	134	3	∫	∫	PROPN
eajbcs-694	134	4	t	t	PROPN
eajbcs-694	134	5	0	0	NUM
eajbcs-694	134	6	i(s)e(γ+µ+ξ)sds	i(s)e(γ+µ+ξ)sds	PROPN
eajbcs-694	134	7	)	)	PUNCT
eajbcs-694	134	8	≥	≥	NOUN
eajbcs-694	134	9	0	0	NUM
eajbcs-694	134	10	,	,	PUNCT
eajbcs-694	134	11	for	for	ADP
eajbcs-694	134	12	t	t	PROPN
eajbcs-694	134	13	∈	∈	PROPN
eajbcs-694	135	1	[	[	X
eajbcs-694	135	2	0	0	NUM
eajbcs-694	135	3	,	,	PUNCT
eajbcs-694	135	4	t2	t2	NOUN
eajbcs-694	135	5	]	]	PUNCT
eajbcs-694	135	6	.	.	PUNCT
eajbcs-694	136	1	east	east	PROPN
eajbcs-694	136	2	afr	afr	PROPN
eajbcs-694	136	3	.	.	PUNCT
eajbcs-694	137	1	j.	j.	PROPN
eajbcs-694	137	2	biophys	biophys	PROPN
eajbcs-694	137	3	.	.	PUNCT
eajbcs-694	138	1	comput	comput	NOUN
eajbcs-694	138	2	.	.	PUNCT
eajbcs-694	139	1	sci	sci	PROPN
eajbcs-694	139	2	.	.	PUNCT
eajbcs-694	140	1	(	(	PUNCT
eajbcs-694	140	2	2024	2024	NUM
eajbcs-694	140	3	)	)	PUNCT
eajbcs-694	140	4	,	,	PUNCT
eajbcs-694	140	5	vol	vol	NOUN
eajbcs-694	140	6	.	.	PROPN
eajbcs-694	140	7	5	5	NUM
eajbcs-694	140	8	,	,	PUNCT
eajbcs-694	140	9	no	no	INTJ
eajbcs-694	140	10	.	.	NOUN
eajbcs-694	140	11	2	2	NUM
eajbcs-694	140	12	,	,	PUNCT
eajbcs-694	140	13	58	58	NUM
eajbcs-694	140	14	-	-	SYM
eajbcs-694	140	15	86	86	NUM
eajbcs-694	140	16	62	62	NUM
eajbcs-694	140	17	then	then	ADV
eajbcs-694	140	18	e	e	NOUN
eajbcs-694	140	19	′(t2	′(t2	NOUN
eajbcs-694	140	20	)	)	PUNCT
eajbcs-694	140	21	=	=	SYM
eajbcs-694	140	22	β(σ1i(t2)+σ3h(t2))s(t2	β(σ1i(t2)+σ3h(t2))s(t2	PROPN
eajbcs-694	140	23	)	)	PUNCT
eajbcs-694	140	24	≥	≥	NOUN
eajbcs-694	140	25	0	0	NUM
eajbcs-694	140	26	.	.	PUNCT
eajbcs-694	141	1	this	this	PRON
eajbcs-694	141	2	is	be	AUX
eajbcs-694	141	3	a	a	DET
eajbcs-694	141	4	contradiction	contradiction	NOUN
eajbcs-694	141	5	.	.	PUNCT
eajbcs-694	142	1	hence	hence	ADV
eajbcs-694	142	2	h(t	h(t	PROPN
eajbcs-694	142	3	)	)	PUNCT
eajbcs-694	142	4	≥	≥	NOUN
eajbcs-694	142	5	0	0	NUM
eajbcs-694	142	6	for	for	ADP
eajbcs-694	142	7	every	every	DET
eajbcs-694	142	8	t	t	NOUN
eajbcs-694	142	9	∈	∈	PROPN
eajbcs-694	143	1	[	[	X
eajbcs-694	143	2	0	0	NUM
eajbcs-694	143	3	,	,	PUNCT
eajbcs-694	143	4	t1	t1	NOUN
eajbcs-694	143	5	]	]	PUNCT
eajbcs-694	143	6	.	.	PUNCT
eajbcs-694	144	1	integration	integration	NOUN
eajbcs-694	144	2	of	of	ADP
eajbcs-694	144	3	equation	equation	NOUN
eajbcs-694	144	4	(	(	PUNCT
eajbcs-694	144	5	2f	2f	NUM
eajbcs-694	144	6	)	)	PUNCT
eajbcs-694	144	7	leads	lead	VERB
eajbcs-694	144	8	to	to	ADP
eajbcs-694	144	9	r(t	r(t	NOUN
eajbcs-694	144	10	)	)	PUNCT
eajbcs-694	144	11	=	=	SYM
eajbcs-694	144	12	e−(ω+µ)t	e−(ω+µ)t	NOUN
eajbcs-694	144	13	(	(	PUNCT
eajbcs-694	144	14	r(0	r(0	PROPN
eajbcs-694	144	15	)	)	PUNCT
eajbcs-694	144	16	+	+	CCONJ
eajbcs-694	145	1	∫	∫	PROPN
eajbcs-694	145	2	t	t	PROPN
eajbcs-694	145	3	0	0	NUM
eajbcs-694	145	4	(	(	PUNCT
eajbcs-694	145	5	(	(	PUNCT
eajbcs-694	145	6	1−	1−	NUM
eajbcs-694	145	7	τ)δe(s	τ)δe(s	NUM
eajbcs-694	145	8	)	)	PUNCT
eajbcs-694	145	9	+	+	NUM
eajbcs-694	145	10	εi(s	εi(s	PUNCT
eajbcs-694	145	11	)	)	PUNCT
eajbcs-694	145	12	+	+	NUM
eajbcs-694	145	13	γh(s	γh(s	NUM
eajbcs-694	145	14	)	)	PUNCT
eajbcs-694	145	15	)	)	PUNCT
eajbcs-694	145	16	e(ω+µ)sds	e(ω+µ)sds	NOUN
eajbcs-694	145	17	)	)	PUNCT
eajbcs-694	145	18	≥	≥	PROPN
eajbcs-694	145	19	0	0	NUM
eajbcs-694	145	20	,	,	PUNCT
eajbcs-694	145	21	for	for	ADP
eajbcs-694	145	22	t	t	PROPN
eajbcs-694	145	23	∈	∈	PROPN
eajbcs-694	146	1	[	[	X
eajbcs-694	146	2	0	0	NUM
eajbcs-694	146	3	,	,	PUNCT
eajbcs-694	146	4	t1	t1	NOUN
eajbcs-694	146	5	]	]	X
eajbcs-694	146	6	.	.	PUNCT
eajbcs-694	147	1	then	then	ADV
eajbcs-694	147	2	p	p	PROPN
eajbcs-694	147	3	′(t1	′(t1	NOUN
eajbcs-694	147	4	)	)	PUNCT
eajbcs-694	148	1	=	=	SYM
eajbcs-694	148	2	θs(t1	θs(t1	X
eajbcs-694	148	3	)	)	PUNCT
eajbcs-694	149	1	+	+	CCONJ
eajbcs-694	149	2	(	(	PUNCT
eajbcs-694	149	3	1	1	NUM
eajbcs-694	149	4	−	−	NOUN
eajbcs-694	149	5	η)ωr(t1	η)ωr(t1	NOUN
eajbcs-694	149	6	)	)	PUNCT
eajbcs-694	149	7	≥	≥	NOUN
eajbcs-694	149	8	0	0	NUM
eajbcs-694	149	9	.	.	PUNCT
eajbcs-694	150	1	this	this	PRON
eajbcs-694	150	2	is	be	AUX
eajbcs-694	150	3	a	a	DET
eajbcs-694	150	4	contradiction	contradiction	NOUN
eajbcs-694	150	5	.	.	PUNCT
eajbcs-694	151	1	hence	hence	ADV
eajbcs-694	151	2	r(t	r(t	NOUN
eajbcs-694	151	3	)	)	PUNCT
eajbcs-694	151	4	≥	≥	NOUN
eajbcs-694	151	5	0	0	NUM
eajbcs-694	151	6	for	for	ADP
eajbcs-694	151	7	every	every	DET
eajbcs-694	151	8	t	t	NOUN
eajbcs-694	151	9	∈	∈	PROPN
eajbcs-694	152	1	[	[	X
eajbcs-694	152	2	0	0	NUM
eajbcs-694	152	3	,	,	PUNCT
eajbcs-694	152	4	t0	t0	PROPN
eajbcs-694	152	5	]	]	PUNCT
eajbcs-694	152	6	.	.	PUNCT
eajbcs-694	153	1	thus	thus	ADV
eajbcs-694	153	2	s	s	VERB
eajbcs-694	153	3	′(t0	′(t0	NOUN
eajbcs-694	153	4	)	)	PUNCT
eajbcs-694	154	1	=	=	PUNCT
eajbcs-694	155	1	π	π	X
eajbcs-694	155	2	+	+	CCONJ
eajbcs-694	155	3	ηωr(t0	ηωr(t0	NOUN
eajbcs-694	155	4	)	)	PUNCT
eajbcs-694	155	5	+	+	NUM
eajbcs-694	155	6	νp	νp	X
eajbcs-694	155	7	(	(	PUNCT
eajbcs-694	155	8	t0	t0	NOUN
eajbcs-694	155	9	)	)	PUNCT
eajbcs-694	155	10	>	>	X
eajbcs-694	155	11	0	0	NUM
eajbcs-694	155	12	,	,	PUNCT
eajbcs-694	155	13	but	but	CCONJ
eajbcs-694	155	14	this	this	PRON
eajbcs-694	155	15	leads	lead	VERB
eajbcs-694	155	16	to	to	ADP
eajbcs-694	155	17	a	a	DET
eajbcs-694	155	18	contradiction	contradiction	NOUN
eajbcs-694	155	19	to	to	ADP
eajbcs-694	155	20	the	the	DET
eajbcs-694	155	21	assumption	assumption	NOUN
eajbcs-694	155	22	that	that	SCONJ
eajbcs-694	155	23	s	s	VERB
eajbcs-694	155	24	′(t0	′(t0	NOUN
eajbcs-694	155	25	)	)	PUNCT
eajbcs-694	155	26	≤	≤	NOUN
eajbcs-694	155	27	0	0	NUM
eajbcs-694	155	28	.	.	PUNCT
eajbcs-694	156	1	therefore	therefore	ADV
eajbcs-694	156	2	,	,	PUNCT
eajbcs-694	156	3	the	the	DET
eajbcs-694	156	4	solutions	solution	NOUN
eajbcs-694	156	5	s(t	s(t	PROPN
eajbcs-694	156	6	)	)	PUNCT
eajbcs-694	156	7	,	,	PUNCT
eajbcs-694	156	8	p	p	X
eajbcs-694	156	9	(	(	PUNCT
eajbcs-694	156	10	t	t	PROPN
eajbcs-694	156	11	)	)	PUNCT
eajbcs-694	156	12	,	,	PUNCT
eajbcs-694	156	13	e(t	e(t	NOUN
eajbcs-694	156	14	)	)	PUNCT
eajbcs-694	156	15	,	,	PUNCT
eajbcs-694	156	16	i(t	i(t	PROPN
eajbcs-694	156	17	)	)	PUNCT
eajbcs-694	156	18	,	,	PUNCT
eajbcs-694	156	19	h(t	h(t	PROPN
eajbcs-694	156	20	)	)	PUNCT
eajbcs-694	156	21	,	,	PUNCT
eajbcs-694	156	22	r(t	r(t	NOUN
eajbcs-694	156	23	)	)	PUNCT
eajbcs-694	156	24	in	in	ADP
eajbcs-694	156	25	the	the	DET
eajbcs-694	156	26	system	system	NOUN
eajbcs-694	156	27	(	(	PUNCT
eajbcs-694	156	28	2	2	X
eajbcs-694	156	29	)	)	PUNCT
eajbcs-694	156	30	remain	remain	VERB
eajbcs-694	156	31	positive	positive	ADJ
eajbcs-694	156	32	for	for	ADP
eajbcs-694	156	33	all	all	DET
eajbcs-694	156	34	t	t	NOUN
eajbcs-694	156	35	>	>	X
eajbcs-694	156	36	0	0	X
eajbcs-694	156	37	.	.	PUNCT
eajbcs-694	157	1	this	this	PRON
eajbcs-694	157	2	completes	complete	VERB
eajbcs-694	157	3	the	the	DET
eajbcs-694	157	4	proof	proof	NOUN
eajbcs-694	157	5	.	.	PUNCT
eajbcs-694	158	1	theorem	theorem	ADJ
eajbcs-694	158	2	3.2	3.2	NUM
eajbcs-694	158	3	.	.	PUNCT
eajbcs-694	159	1	(	(	PUNCT
eajbcs-694	159	2	boundedness	boundedness	NOUN
eajbcs-694	159	3	)	)	PUNCT
eajbcs-694	159	4	there	there	PRON
eajbcs-694	159	5	exists	exist	VERB
eajbcs-694	159	6	a	a	DET
eajbcs-694	159	7	positively	positively	ADV
eajbcs-694	159	8	invariant	invariant	ADJ
eajbcs-694	159	9	region	region	NOUN
eajbcs-694	159	10	ω	ω	PROPN
eajbcs-694	159	11	in	in	ADP
eajbcs-694	159	12	which	which	PRON
eajbcs-694	159	13	the	the	DET
eajbcs-694	159	14	solution	solution	NOUN
eajbcs-694	159	15	(	(	PUNCT
eajbcs-694	159	16	s(t	s(t	PROPN
eajbcs-694	159	17	)	)	PUNCT
eajbcs-694	159	18	,	,	PUNCT
eajbcs-694	159	19	p	p	X
eajbcs-694	159	20	(	(	PUNCT
eajbcs-694	159	21	t	t	PROPN
eajbcs-694	159	22	)	)	PUNCT
eajbcs-694	159	23	,	,	PUNCT
eajbcs-694	159	24	e(t	e(t	NOUN
eajbcs-694	159	25	)	)	PUNCT
eajbcs-694	159	26	,	,	PUNCT
eajbcs-694	159	27	i(t	i(t	PROPN
eajbcs-694	159	28	)	)	PUNCT
eajbcs-694	159	29	,	,	PUNCT
eajbcs-694	159	30	h(t	h(t	PROPN
eajbcs-694	159	31	)	)	PUNCT
eajbcs-694	159	32	,	,	PUNCT
eajbcs-694	159	33	r(t	r(t	NOUN
eajbcs-694	159	34	)	)	PUNCT
eajbcs-694	159	35	)	)	PUNCT
eajbcs-694	159	36	of	of	ADP
eajbcs-694	159	37	the	the	DET
eajbcs-694	159	38	dynamical	dynamical	ADJ
eajbcs-694	159	39	system	system	NOUN
eajbcs-694	159	40	(	(	PUNCT
eajbcs-694	159	41	2	2	X
eajbcs-694	159	42	)	)	PUNCT
eajbcs-694	159	43	is	be	AUX
eajbcs-694	159	44	bounded	bound	VERB
eajbcs-694	159	45	.	.	PUNCT
eajbcs-694	160	1	proof	proof	NOUN
eajbcs-694	160	2	.	.	PUNCT
eajbcs-694	161	1	the	the	DET
eajbcs-694	161	2	positivity	positivity	NOUN
eajbcs-694	161	3	has	have	AUX
eajbcs-694	161	4	already	already	ADV
eajbcs-694	161	5	been	be	AUX
eajbcs-694	161	6	established	establish	VERB
eajbcs-694	161	7	by	by	ADP
eajbcs-694	161	8	theorem	theorem	NOUN
eajbcs-694	161	9	(	(	PUNCT
eajbcs-694	161	10	3.1	3.1	NUM
eajbcs-694	161	11	)	)	PUNCT
eajbcs-694	161	12	.	.	PUNCT
eajbcs-694	162	1	for	for	ADP
eajbcs-694	162	2	this	this	DET
eajbcs-694	162	3	model	model	NOUN
eajbcs-694	162	4	the	the	DET
eajbcs-694	162	5	total	total	ADJ
eajbcs-694	162	6	population	population	NOUN
eajbcs-694	162	7	is	be	AUX
eajbcs-694	162	8	n(t	n(t	PROPN
eajbcs-694	162	9	)	)	PUNCT
eajbcs-694	163	1	=	=	X
eajbcs-694	163	2	s(t)+p	s(t)+p	X
eajbcs-694	163	3	(	(	PUNCT
eajbcs-694	163	4	t)+e(t)+	t)+e(t)+	ADP
eajbcs-694	163	5	i(t	i(t	PROPN
eajbcs-694	163	6	)	)	PUNCT
eajbcs-694	164	1	+	+	VERB
eajbcs-694	164	2	h(t	h(t	X
eajbcs-694	164	3	)	)	PUNCT
eajbcs-694	164	4	+	+	NOUN
eajbcs-694	164	5	r(t	r(t	NOUN
eajbcs-694	164	6	)	)	PUNCT
eajbcs-694	164	7	.	.	PUNCT
eajbcs-694	165	1	then	then	ADV
eajbcs-694	165	2	,	,	PUNCT
eajbcs-694	165	3	we	we	PRON
eajbcs-694	165	4	obtain	obtain	VERB
eajbcs-694	165	5	:	:	PUNCT
eajbcs-694	166	1	dn	dn	NOUN
eajbcs-694	166	2	dt	dt	NOUN
eajbcs-694	167	1	=	=	PUNCT
eajbcs-694	167	2	π−	π−	PROPN
eajbcs-694	167	3	µn	µn	PROPN
eajbcs-694	167	4	−	−	PROPN
eajbcs-694	167	5	(	(	PUNCT
eajbcs-694	167	6	ρi	ρi	NOUN
eajbcs-694	167	7	+	+	PROPN
eajbcs-694	167	8	ξh	ξh	NOUN
eajbcs-694	167	9	)	)	PUNCT
eajbcs-694	167	10	.	.	PUNCT
eajbcs-694	168	1	this	this	PRON
eajbcs-694	168	2	implies	imply	VERB
eajbcs-694	168	3	that	that	SCONJ
eajbcs-694	168	4	dn	dn	PROPN
eajbcs-694	168	5	dt	dt	NOUN
eajbcs-694	168	6	≤	≤	PROPN
eajbcs-694	169	1	π−	π−	PROPN
eajbcs-694	169	2	µn	µn	PROPN
eajbcs-694	169	3	,	,	PUNCT
eajbcs-694	169	4	since	since	SCONJ
eajbcs-694	169	5	the	the	DET
eajbcs-694	169	6	solution	solution	NOUN
eajbcs-694	169	7	i(t	i(t	NOUN
eajbcs-694	169	8	)	)	PUNCT
eajbcs-694	169	9	and	and	CCONJ
eajbcs-694	169	10	h(t	h(t	PROPN
eajbcs-694	169	11	)	)	PUNCT
eajbcs-694	169	12	are	be	AUX
eajbcs-694	169	13	positive	positive	ADJ
eajbcs-694	169	14	.	.	PUNCT
eajbcs-694	170	1	solving	solve	VERB
eajbcs-694	170	2	the	the	DET
eajbcs-694	170	3	differential	differential	ADJ
eajbcs-694	170	4	inequality	inequality	NOUN
eajbcs-694	170	5	we	we	PRON
eajbcs-694	170	6	get	get	VERB
eajbcs-694	170	7	the	the	DET
eajbcs-694	170	8	relation	relation	NOUN
eajbcs-694	170	9	,	,	PUNCT
eajbcs-694	170	10	n(t	n(t	PROPN
eajbcs-694	170	11	)	)	PUNCT
eajbcs-694	170	12	≤	≤	NUM
eajbcs-694	170	13	π	π	X
eajbcs-694	170	14	µ	µ	X
eajbcs-694	170	15	+	+	CCONJ
eajbcs-694	170	16	(	(	PUNCT
eajbcs-694	170	17	n(0)−	n(0)−	PROPN
eajbcs-694	170	18	π	π	PROPN
eajbcs-694	170	19	µ	µ	X
eajbcs-694	170	20	)	)	PUNCT
eajbcs-694	170	21	e−µt	e−µt	PROPN
eajbcs-694	170	22	,	,	PUNCT
eajbcs-694	170	23	if	if	SCONJ
eajbcs-694	170	24	n(0	n(0	PROPN
eajbcs-694	170	25	)	)	PUNCT
eajbcs-694	170	26	≤	≤	NUM
eajbcs-694	170	27	π	π	PROPN
eajbcs-694	170	28	µ	µ	X
eajbcs-694	170	29	,	,	PUNCT
eajbcs-694	170	30	then	then	ADV
eajbcs-694	170	31	we	we	PRON
eajbcs-694	170	32	obtain	obtain	VERB
eajbcs-694	170	33	0	0	NUM
eajbcs-694	170	34	≤	≤	ADJ
eajbcs-694	170	35	n(t	n(t	PROPN
eajbcs-694	170	36	)	)	PUNCT
eajbcs-694	170	37	≤	≤	NUM
eajbcs-694	171	1	π	π	PROPN
eajbcs-694	171	2	µ	µ	X
eajbcs-694	171	3	,	,	PUNCT
eajbcs-694	171	4	for	for	ADP
eajbcs-694	171	5	all	all	DET
eajbcs-694	171	6	t	t	PROPN
eajbcs-694	171	7	≥	≥	NOUN
eajbcs-694	171	8	0	0	NUM
eajbcs-694	171	9	.	.	PUNCT
eajbcs-694	172	1	if	if	SCONJ
eajbcs-694	172	2	n(0	n(0	PROPN
eajbcs-694	172	3	)	)	PUNCT
eajbcs-694	172	4	≥	≥	NOUN
eajbcs-694	172	5	π	π	PROPN
eajbcs-694	172	6	µ	µ	X
eajbcs-694	172	7	,	,	PUNCT
eajbcs-694	172	8	then	then	ADV
eajbcs-694	172	9	we	we	PRON
eajbcs-694	172	10	have	have	VERB
eajbcs-694	172	11	0	0	NUM
eajbcs-694	172	12	≤	≤	NUM
eajbcs-694	172	13	n(t	n(t	PROPN
eajbcs-694	172	14	)	)	PUNCT
eajbcs-694	172	15	≤	≤	PROPN
eajbcs-694	172	16	n(0	n(0	PROPN
eajbcs-694	172	17	)	)	PUNCT
eajbcs-694	172	18	,	,	PUNCT
eajbcs-694	172	19	for	for	ADP
eajbcs-694	172	20	all	all	DET
eajbcs-694	172	21	t	t	PROPN
eajbcs-694	172	22	≥	≥	NOUN
eajbcs-694	172	23	0	0	NUM
eajbcs-694	172	24	.	.	PUNCT
eajbcs-694	173	1	thus	thus	ADV
eajbcs-694	173	2	,	,	PUNCT
eajbcs-694	173	3	the	the	DET
eajbcs-694	173	4	feasible	feasible	ADJ
eajbcs-694	173	5	solution	solution	NOUN
eajbcs-694	173	6	set	set	VERB
eajbcs-694	173	7	of	of	ADP
eajbcs-694	173	8	the	the	DET
eajbcs-694	173	9	system	system	NOUN
eajbcs-694	173	10	(	(	PUNCT
eajbcs-694	173	11	2	2	X
eajbcs-694	173	12	)	)	PUNCT
eajbcs-694	173	13	remain	remain	VERB
eajbcs-694	173	14	in	in	ADP
eajbcs-694	173	15	the	the	DET
eajbcs-694	173	16	region	region	NOUN
eajbcs-694	173	17	ω	ω	PROPN
eajbcs-694	173	18	=	=	SYM
eajbcs-694	173	19	{	{	PUNCT
eajbcs-694	173	20	(	(	PUNCT
eajbcs-694	173	21	s	s	X
eajbcs-694	173	22	,	,	PUNCT
eajbcs-694	173	23	p	p	X
eajbcs-694	173	24	,	,	PUNCT
eajbcs-694	173	25	e	e	NOUN
eajbcs-694	173	26	,	,	PUNCT
eajbcs-694	173	27	i	i	PRON
eajbcs-694	173	28	,	,	PUNCT
eajbcs-694	173	29	h	h	NOUN
eajbcs-694	173	30	,	,	PUNCT
eajbcs-694	173	31	r	r	NOUN
eajbcs-694	173	32	)	)	PUNCT
eajbcs-694	173	33	∈	∈	NOUN
eajbcs-694	173	34	r6	r6	NOUN
eajbcs-694	173	35	+	+	CCONJ
eajbcs-694	173	36	:	:	SYM
eajbcs-694	173	37	0	0	NUM
eajbcs-694	173	38	≤	≤	NUM
eajbcs-694	173	39	n(t	n(t	PROPN
eajbcs-694	173	40	)	)	PUNCT
eajbcs-694	173	41	≤	≤	NUM
eajbcs-694	174	1	max	max	PROPN
eajbcs-694	174	2	(	(	PUNCT
eajbcs-694	174	3	n(0	n(0	PROPN
eajbcs-694	174	4	)	)	PUNCT
eajbcs-694	174	5	,	,	PUNCT
eajbcs-694	174	6	π	π	PROPN
eajbcs-694	174	7	µ	µ	X
eajbcs-694	174	8	)	)	PUNCT
eajbcs-694	174	9	}	}	PUNCT
eajbcs-694	174	10	.	.	PUNCT
eajbcs-694	175	1	if	if	SCONJ
eajbcs-694	175	2	we	we	PRON
eajbcs-694	175	3	start	start	VERB
eajbcs-694	175	4	with	with	ADP
eajbcs-694	175	5	initial	initial	ADJ
eajbcs-694	175	6	data	datum	NOUN
eajbcs-694	175	7	n(0	n(0	PROPN
eajbcs-694	175	8	)	)	PUNCT
eajbcs-694	175	9	∈	∈	PROPN
eajbcs-694	175	10	ω	ω	PROPN
eajbcs-694	175	11	,	,	PUNCT
eajbcs-694	175	12	then	then	ADV
eajbcs-694	175	13	the	the	DET
eajbcs-694	175	14	solution	solution	NOUN
eajbcs-694	175	15	n(t	n(t	NOUN
eajbcs-694	175	16	)	)	PUNCT
eajbcs-694	175	17	∈	∈	PROPN
eajbcs-694	175	18	ω	ω	PROPN
eajbcs-694	175	19	,	,	PUNCT
eajbcs-694	175	20	for	for	ADP
eajbcs-694	175	21	every	every	DET
eajbcs-694	175	22	t	t	NOUN
eajbcs-694	175	23	>	>	X
eajbcs-694	175	24	0	0	X
eajbcs-694	175	25	.	.	PUNCT
eajbcs-694	176	1	this	this	PRON
eajbcs-694	176	2	shows	show	VERB
eajbcs-694	176	3	the	the	DET
eajbcs-694	176	4	positively	positively	ADV
eajbcs-694	176	5	invariance	invariance	NOUN
eajbcs-694	176	6	of	of	ADP
eajbcs-694	176	7	ω	ω	PROPN
eajbcs-694	176	8	.	.	PUNCT
eajbcs-694	177	1	thus	thus	ADV
eajbcs-694	177	2	,	,	PUNCT
eajbcs-694	177	3	the	the	DET
eajbcs-694	177	4	solution	solution	NOUN
eajbcs-694	177	5	of	of	ADP
eajbcs-694	177	6	the	the	DET
eajbcs-694	177	7	dynamical	dynamical	ADJ
eajbcs-694	177	8	system	system	NOUN
eajbcs-694	177	9	(	(	PUNCT
eajbcs-694	177	10	2	2	X
eajbcs-694	177	11	)	)	PUNCT
eajbcs-694	177	12	is	be	AUX
eajbcs-694	177	13	bounded	bound	VERB
eajbcs-694	177	14	.	.	PUNCT
eajbcs-694	178	1	3.2	3.2	NUM
eajbcs-694	178	2	.	.	PUNCT
eajbcs-694	179	1	steady	steady	ADJ
eajbcs-694	179	2	state	state	NOUN
eajbcs-694	179	3	the	the	DET
eajbcs-694	179	4	steady	steady	ADJ
eajbcs-694	179	5	states	state	NOUN
eajbcs-694	179	6	of	of	ADP
eajbcs-694	179	7	the	the	DET
eajbcs-694	179	8	system	system	NOUN
eajbcs-694	179	9	(	(	PUNCT
eajbcs-694	179	10	2	2	X
eajbcs-694	179	11	)	)	PUNCT
eajbcs-694	179	12	are	be	AUX
eajbcs-694	179	13	solutions	solution	NOUN
eajbcs-694	179	14	of	of	ADP
eajbcs-694	179	15	the	the	DET
eajbcs-694	179	16	following	follow	VERB
eajbcs-694	179	17	equations	equation	NOUN
eajbcs-694	179	18	:	:	PUNCT
eajbcs-694	179	19	0	0	PUNCT
eajbcs-694	180	1	=	=	PUNCT
eajbcs-694	180	2	π	π	X
eajbcs-694	180	3	+	+	CCONJ
eajbcs-694	180	4	ηωr	ηωr	PROPN
eajbcs-694	180	5	+	+	X
eajbcs-694	180	6	νp	νp	ADP
eajbcs-694	180	7	−	−	PROPN
eajbcs-694	180	8	β(σ1i	β(σ1i	NOUN
eajbcs-694	180	9	+	+	NUM
eajbcs-694	180	10	σ2e	σ2e	PROPN
eajbcs-694	180	11	+	+	CCONJ
eajbcs-694	180	12	σ3h)s	σ3h)s	X
eajbcs-694	180	13	−	−	PROPN
eajbcs-694	180	14	(	(	PUNCT
eajbcs-694	180	15	θ	θ	NOUN
eajbcs-694	180	16	+	+	NOUN
eajbcs-694	180	17	µ)s	µ)s	NOUN
eajbcs-694	180	18	0	0	PUNCT
eajbcs-694	181	1	=	=	SYM
eajbcs-694	181	2	θs	θ	NOUN
eajbcs-694	181	3	+	+	X
eajbcs-694	181	4	(	(	PUNCT
eajbcs-694	181	5	1−	1−	NUM
eajbcs-694	181	6	η)ωr−	η)ωr−	PROPN
eajbcs-694	181	7	(	(	PUNCT
eajbcs-694	181	8	ν	ν	X
eajbcs-694	181	9	+	+	X
eajbcs-694	181	10	µ)p	µ)p	NOUN
eajbcs-694	181	11	0	0	PUNCT
eajbcs-694	182	1	=	=	PRON
eajbcs-694	182	2	β(σ1i	β(σ1i	NOUN
eajbcs-694	182	3	+	+	X
eajbcs-694	182	4	σ2e	σ2e	PROPN
eajbcs-694	182	5	+	+	CCONJ
eajbcs-694	182	6	σ3h)s	σ3h)s	X
eajbcs-694	182	7	−	−	PROPN
eajbcs-694	182	8	(	(	PUNCT
eajbcs-694	182	9	δ	δ	NOUN
eajbcs-694	182	10	+	+	NOUN
eajbcs-694	182	11	µ)e	µ)e	NOUN
eajbcs-694	182	12	0	0	PUNCT
eajbcs-694	183	1	=	=	NOUN
eajbcs-694	183	2	τδe	τδe	NOUN
eajbcs-694	183	3	−	−	PROPN
eajbcs-694	184	1	(	(	PUNCT
eajbcs-694	184	2	α	α	PROPN
eajbcs-694	184	3	+	+	CCONJ
eajbcs-694	184	4	ε+	ε+	X
eajbcs-694	184	5	µ+	µ+	X
eajbcs-694	184	6	ρ)i	ρ)i	NOUN
eajbcs-694	184	7	0	0	PUNCT
eajbcs-694	185	1	=	=	NOUN
eajbcs-694	185	2	αi	αi	ADV
eajbcs-694	185	3	−	−	PROPN
eajbcs-694	186	1	(	(	PUNCT
eajbcs-694	186	2	γ	γ	X
eajbcs-694	186	3	+	+	X
eajbcs-694	186	4	µ+	µ+	X
eajbcs-694	186	5	ξ)h	ξ)h	ADJ
eajbcs-694	186	6	0	0	NUM
eajbcs-694	186	7	=(	=(	NOUN
eajbcs-694	186	8	1−	1−	NUM
eajbcs-694	186	9	τ)δe	τ)δe	PROPN
eajbcs-694	187	1	+	+	CCONJ
eajbcs-694	187	2	εi	εi	VERB
eajbcs-694	187	3	+	+	X
eajbcs-694	187	4	γh	γh	ADP
eajbcs-694	187	5	−	−	PROPN
eajbcs-694	187	6	(	(	PUNCT
eajbcs-694	187	7	ω	ω	NOUN
eajbcs-694	187	8	+	+	NOUN
eajbcs-694	187	9	µ)r	µ)r	NOUN
eajbcs-694	187	10	.	.	PUNCT
eajbcs-694	188	1	there	there	PRON
eajbcs-694	188	2	are	be	VERB
eajbcs-694	188	3	at	at	ADP
eajbcs-694	188	4	most	most	ADV
eajbcs-694	188	5	two	two	NUM
eajbcs-694	188	6	steady	steady	ADJ
eajbcs-694	188	7	states	state	NOUN
eajbcs-694	188	8	for	for	ADP
eajbcs-694	188	9	the	the	DET
eajbcs-694	188	10	system	system	NOUN
eajbcs-694	188	11	(	(	PUNCT
eajbcs-694	188	12	2	2	NUM
eajbcs-694	188	13	):	):	PUNCT
eajbcs-694	188	14	the	the	DET
eajbcs-694	188	15	disease	disease	NOUN
eajbcs-694	188	16	free	free	ADJ
eajbcs-694	188	17	equilibrium	equilibrium	NOUN
eajbcs-694	188	18	e0	e0	PROPN
eajbcs-694	188	19	and	and	CCONJ
eajbcs-694	188	20	endemic	endemic	ADJ
eajbcs-694	188	21	equilibrium	equilibrium	NOUN
eajbcs-694	188	22	e1	e1	PROPN
eajbcs-694	188	23	.	.	PUNCT
eajbcs-694	189	1	the	the	DET
eajbcs-694	189	2	disease	disease	NOUN
eajbcs-694	189	3	-	-	PUNCT
eajbcs-694	189	4	free	free	ADJ
eajbcs-694	189	5	equilibrium	equilibrium	NOUN
eajbcs-694	189	6	point	point	NOUN
eajbcs-694	189	7	of	of	ADP
eajbcs-694	189	8	our	our	PRON
eajbcs-694	189	9	model	model	NOUN
eajbcs-694	189	10	is	be	AUX
eajbcs-694	189	11	obtained	obtain	VERB
eajbcs-694	189	12	by	by	ADP
eajbcs-694	189	13	setting	set	VERB
eajbcs-694	189	14	the	the	DET
eajbcs-694	189	15	disease	disease	NOUN
eajbcs-694	189	16	state	state	NOUN
eajbcs-694	189	17	variables	variable	NOUN
eajbcs-694	189	18	e	e	NOUN
eajbcs-694	189	19	=	=	SYM
eajbcs-694	189	20	0	0	PROPN
eajbcs-694	189	21	,	,	PUNCT
eajbcs-694	189	22	i	i	PRON
eajbcs-694	189	23	=	=	NOUN
eajbcs-694	189	24	0	0	NUM
eajbcs-694	189	25	and	and	CCONJ
eajbcs-694	189	26	h	h	NOUN
eajbcs-694	190	1	=	=	NOUN
eajbcs-694	190	2	0	0	NUM
eajbcs-694	190	3	.	.	PUNCT
eajbcs-694	191	1	thus	thus	ADV
eajbcs-694	191	2	,	,	PUNCT
eajbcs-694	191	3	the	the	DET
eajbcs-694	191	4	disease	disease	NOUN
eajbcs-694	191	5	free	free	ADJ
eajbcs-694	191	6	equilibrium	equilibrium	NOUN
eajbcs-694	191	7	point	point	NOUN
eajbcs-694	191	8	is	be	AUX
eajbcs-694	191	9	given	give	VERB
eajbcs-694	191	10	by	by	ADP
eajbcs-694	191	11	e0	e0	PROPN
eajbcs-694	191	12	=	=	SYM
eajbcs-694	191	13	(	(	PUNCT
eajbcs-694	191	14	s0	s0	PROPN
eajbcs-694	191	15	,	,	PUNCT
eajbcs-694	191	16	p	p	NOUN
eajbcs-694	191	17	0	0	NUM
eajbcs-694	191	18	,	,	PUNCT
eajbcs-694	191	19	e0	e0	PROPN
eajbcs-694	191	20	,	,	PUNCT
eajbcs-694	191	21	i0	i0	PROPN
eajbcs-694	191	22	,	,	PUNCT
eajbcs-694	191	23	h0	h0	PROPN
eajbcs-694	191	24	,	,	PUNCT
eajbcs-694	191	25	r0	r0	NOUN
eajbcs-694	191	26	)	)	PUNCT
eajbcs-694	191	27	=	=	PUNCT
eajbcs-694	192	1	(	(	PUNCT
eajbcs-694	192	2	π(ν	π(ν	X
eajbcs-694	192	3	+	+	NUM
eajbcs-694	192	4	µ	µ	X
eajbcs-694	192	5	)	)	PUNCT
eajbcs-694	192	6	µ(θ	µ(θ	ADJ
eajbcs-694	192	7	+	+	NUM
eajbcs-694	192	8	ν	ν	X
eajbcs-694	192	9	+	+	X
eajbcs-694	192	10	µ	µ	X
eajbcs-694	192	11	)	)	PUNCT
eajbcs-694	192	12	,	,	PUNCT
eajbcs-694	192	13	πθ	πθ	ADP
eajbcs-694	192	14	µ(θ	µ(θ	ADJ
eajbcs-694	192	15	+	+	CCONJ
eajbcs-694	192	16	ν	ν	X
eajbcs-694	192	17	+	+	X
eajbcs-694	192	18	µ	µ	X
eajbcs-694	192	19	)	)	PUNCT
eajbcs-694	192	20	,	,	PUNCT
eajbcs-694	192	21	0	0	NUM
eajbcs-694	192	22	,	,	PUNCT
eajbcs-694	192	23	0	0	NUM
eajbcs-694	192	24	,	,	PUNCT
eajbcs-694	192	25	0	0	NUM
eajbcs-694	192	26	,	,	PUNCT
eajbcs-694	192	27	0	0	NUM
eajbcs-694	192	28	)	)	PUNCT
eajbcs-694	192	29	.	.	PUNCT
eajbcs-694	193	1	the	the	DET
eajbcs-694	193	2	existence	existence	NOUN
eajbcs-694	193	3	of	of	ADP
eajbcs-694	193	4	the	the	DET
eajbcs-694	193	5	endemic	endemic	ADJ
eajbcs-694	193	6	equilibrium	equilibrium	NOUN
eajbcs-694	193	7	point	point	NOUN
eajbcs-694	193	8	depends	depend	VERB
eajbcs-694	193	9	on	on	ADP
eajbcs-694	193	10	the	the	DET
eajbcs-694	193	11	basic	basic	ADJ
eajbcs-694	193	12	reproduction	reproduction	NOUN
eajbcs-694	193	13	number	number	NOUN
eajbcs-694	193	14	r0	r0	NOUN
eajbcs-694	193	15	and	and	CCONJ
eajbcs-694	193	16	will	will	AUX
eajbcs-694	193	17	be	be	AUX
eajbcs-694	193	18	presented	present	VERB
eajbcs-694	193	19	later	later	ADV
eajbcs-694	193	20	.	.	PUNCT
eajbcs-694	194	1	3.3	3.3	NUM
eajbcs-694	194	2	.	.	PUNCT
eajbcs-694	194	3	basic	basic	ADJ
eajbcs-694	194	4	reproduction	reproduction	NOUN
eajbcs-694	194	5	number	number	NOUN
eajbcs-694	194	6	the	the	DET
eajbcs-694	194	7	basic	basic	ADJ
eajbcs-694	194	8	reproduction	reproduction	NOUN
eajbcs-694	194	9	number	number	NOUN
eajbcs-694	194	10	,	,	PUNCT
eajbcs-694	194	11	which	which	PRON
eajbcs-694	194	12	is	be	AUX
eajbcs-694	194	13	denoted	denote	VERB
eajbcs-694	194	14	by	by	ADP
eajbcs-694	194	15	r0	r0	NOUN
eajbcs-694	194	16	,	,	PUNCT
eajbcs-694	194	17	and	and	CCONJ
eajbcs-694	194	18	defined	define	VERB
eajbcs-694	194	19	as	as	ADP
eajbcs-694	194	20	the	the	DET
eajbcs-694	194	21	average	average	ADJ
eajbcs-694	194	22	number	number	NOUN
eajbcs-694	194	23	of	of	ADP
eajbcs-694	194	24	secondary	secondary	ADJ
eajbcs-694	194	25	infections	infection	NOUN
eajbcs-694	194	26	produced	produce	VERB
eajbcs-694	194	27	by	by	ADP
eajbcs-694	194	28	a	a	DET
eajbcs-694	194	29	single	single	ADJ
eajbcs-694	194	30	infected	infected	ADJ
eajbcs-694	194	31	individual	individual	NOUN
eajbcs-694	194	32	in	in	ADP
eajbcs-694	194	33	a	a	DET
eajbcs-694	194	34	completely	completely	ADV
eajbcs-694	194	35	suseast	suseast	NOUN
eajbcs-694	194	36	afr	afr	NOUN
eajbcs-694	194	37	.	.	PUNCT
eajbcs-694	195	1	j.	j.	PROPN
eajbcs-694	195	2	biophys	biophys	PROPN
eajbcs-694	195	3	.	.	PUNCT
eajbcs-694	196	1	comput	comput	NOUN
eajbcs-694	196	2	.	.	PUNCT
eajbcs-694	197	1	sci	sci	PROPN
eajbcs-694	197	2	.	.	PUNCT
eajbcs-694	198	1	(	(	PUNCT
eajbcs-694	198	2	2024	2024	NUM
eajbcs-694	198	3	)	)	PUNCT
eajbcs-694	198	4	,	,	PUNCT
eajbcs-694	198	5	vol	vol	NOUN
eajbcs-694	198	6	.	.	PROPN
eajbcs-694	198	7	5	5	NUM
eajbcs-694	198	8	,	,	PUNCT
eajbcs-694	198	9	no	no	INTJ
eajbcs-694	198	10	.	.	NOUN
eajbcs-694	198	11	2	2	NUM
eajbcs-694	198	12	,	,	PUNCT
eajbcs-694	198	13	58	58	NUM
eajbcs-694	198	14	-	-	SYM
eajbcs-694	198	15	86	86	NUM
eajbcs-694	198	16	63	63	NUM
eajbcs-694	198	17	ceptible	ceptible	ADJ
eajbcs-694	198	18	population	population	NOUN
eajbcs-694	198	19	.	.	PUNCT
eajbcs-694	199	1	using	use	VERB
eajbcs-694	199	2	the	the	DET
eajbcs-694	199	3	next	next	ADJ
eajbcs-694	199	4	generation	generation	NOUN
eajbcs-694	199	5	matrix	matrix	NOUN
eajbcs-694	199	6	method	method	NOUN
eajbcs-694	199	7	diekmann	diekmann	PROPN
eajbcs-694	199	8	et	et	PROPN
eajbcs-694	199	9	al	al	PROPN
eajbcs-694	199	10	.	.	PUNCT
eajbcs-694	200	1	(	(	PUNCT
eajbcs-694	200	2	2010	2010	NUM
eajbcs-694	200	3	)	)	PUNCT
eajbcs-694	200	4	,	,	PUNCT
eajbcs-694	200	5	the	the	DET
eajbcs-694	200	6	basic	basic	ADJ
eajbcs-694	200	7	reproduction	reproduction	NOUN
eajbcs-694	200	8	number	number	NOUN
eajbcs-694	200	9	r0	r0	NOUN
eajbcs-694	200	10	can	can	AUX
eajbcs-694	200	11	be	be	AUX
eajbcs-694	200	12	calculated	calculate	VERB
eajbcs-694	200	13	from	from	ADP
eajbcs-694	200	14	the	the	DET
eajbcs-694	200	15	relation	relation	NOUN
eajbcs-694	200	16	r0	r0	NOUN
eajbcs-694	200	17	=	=	NOUN
eajbcs-694	200	18	ρ(fv	ρ(fv	NOUN
eajbcs-694	200	19	−1	−1	NOUN
eajbcs-694	200	20	)	)	PUNCT
eajbcs-694	200	21	.	.	PUNCT
eajbcs-694	201	1	let	let	VERB
eajbcs-694	201	2	f	f	PRON
eajbcs-694	201	3	be	be	AUX
eajbcs-694	201	4	the	the	DET
eajbcs-694	201	5	vector	vector	NOUN
eajbcs-694	201	6	for	for	ADP
eajbcs-694	201	7	the	the	DET
eajbcs-694	201	8	newly	newly	ADV
eajbcs-694	201	9	infected	infect	VERB
eajbcs-694	201	10	and	and	CCONJ
eajbcs-694	201	11	v	v	NOUN
eajbcs-694	201	12	be	be	AUX
eajbcs-694	201	13	the	the	DET
eajbcs-694	201	14	vector	vector	NOUN
eajbcs-694	201	15	for	for	ADP
eajbcs-694	201	16	the	the	DET
eajbcs-694	201	17	transfer	transfer	NOUN
eajbcs-694	201	18	of	of	ADP
eajbcs-694	201	19	individuals	individual	NOUN
eajbcs-694	201	20	into	into	ADP
eajbcs-694	201	21	and	and	CCONJ
eajbcs-694	201	22	out	out	ADP
eajbcs-694	201	23	of	of	ADP
eajbcs-694	201	24	the	the	DET
eajbcs-694	201	25	infected	infected	ADJ
eajbcs-694	201	26	compartments	compartment	NOUN
eajbcs-694	201	27	.	.	PUNCT
eajbcs-694	202	1	let	let	VERB
eajbcs-694	202	2	x	x	PUNCT
eajbcs-694	202	3	=	=	PUNCT
eajbcs-694	202	4	(	(	PUNCT
eajbcs-694	202	5	e	e	NOUN
eajbcs-694	202	6	,	,	PUNCT
eajbcs-694	202	7	i	i	PRON
eajbcs-694	202	8	,	,	PUNCT
eajbcs-694	202	9	h	h	NOUN
eajbcs-694	202	10	)	)	PUNCT
eajbcs-694	202	11	,	,	PUNCT
eajbcs-694	202	12	then	then	ADV
eajbcs-694	202	13	we	we	PRON
eajbcs-694	202	14	obtain	obtain	VERB
eajbcs-694	202	15	:	:	PUNCT
eajbcs-694	202	16	f(x	f(x	PROPN
eajbcs-694	202	17	)	)	PUNCT
eajbcs-694	203	1	=	=	PUNCT
eajbcs-694	203	2			NOUN
eajbcs-694	203	3	β(σ1i	β(σ1i	PUNCT
eajbcs-694	203	4	+	+	NUM
eajbcs-694	203	5	σ2e	σ2e	PROPN
eajbcs-694	203	6	+	+	CCONJ
eajbcs-694	203	7	σ3h)s	σ3h)s	X
eajbcs-694	203	8	0	0	NUM
eajbcs-694	203	9	0	0	NUM
eajbcs-694	203	10			NOUN
eajbcs-694	203	11	and	and	CCONJ
eajbcs-694	203	12	v(x	v(x	NOUN
eajbcs-694	203	13	)	)	PUNCT
eajbcs-694	204	1	=	=	SYM
eajbcs-694	204	2			PROPN
eajbcs-694	204	3	(	(	PUNCT
eajbcs-694	204	4	δ	δ	NOUN
eajbcs-694	204	5	+	+	NOUN
eajbcs-694	204	6	µ)e	µ)e	NOUN
eajbcs-694	204	7	(	(	PUNCT
eajbcs-694	204	8	α	α	NOUN
eajbcs-694	204	9	+	+	X
eajbcs-694	204	10	ε+	ε+	X
eajbcs-694	204	11	µ+	µ+	X
eajbcs-694	204	12	ρ)i	ρ)i	ADJ
eajbcs-694	204	13	−	−	PROPN
eajbcs-694	204	14	τδe	τδe	NOUN
eajbcs-694	204	15	(	(	PUNCT
eajbcs-694	204	16	γ	γ	X
eajbcs-694	204	17	+	+	X
eajbcs-694	204	18	µ+	µ+	X
eajbcs-694	204	19	ξ)h	ξ)h	X
eajbcs-694	204	20	−	−	NOUN
eajbcs-694	204	21	αi	αi	PART
eajbcs-694	204	22			NOUN
eajbcs-694	204	23	.	.	PUNCT
eajbcs-694	205	1	the	the	DET
eajbcs-694	205	2	jacobian	jacobian	ADJ
eajbcs-694	205	3	matrix	matrix	NOUN
eajbcs-694	205	4	to	to	ADP
eajbcs-694	205	5	f	f	PROPN
eajbcs-694	205	6	and	and	CCONJ
eajbcs-694	205	7	v	v	NOUN
eajbcs-694	205	8	are	be	AUX
eajbcs-694	205	9	f	f	X
eajbcs-694	205	10	=	=	X
eajbcs-694	205	11	[	[	PUNCT
eajbcs-694	205	12	∂fi(e0	∂fi(e0	NOUN
eajbcs-694	205	13	)	)	PUNCT
eajbcs-694	205	14	∂xj	∂xj	NOUN
eajbcs-694	205	15	]	]	PUNCT
eajbcs-694	206	1	=	=	PUNCT
eajbcs-694	206	2			X
eajbcs-694	206	3	βσ2s	βσ2s	PROPN
eajbcs-694	206	4	0	0	NUM
eajbcs-694	206	5	βσ1s	βσ1s	NOUN
eajbcs-694	206	6	0	0	NUM
eajbcs-694	207	1	βσ3s	βσ3s	ADJ
eajbcs-694	207	2	0	0	NUM
eajbcs-694	207	3	0	0	NUM
eajbcs-694	207	4	0	0	NUM
eajbcs-694	207	5	0	0	NUM
eajbcs-694	207	6	0	0	NUM
eajbcs-694	207	7	0	0	NUM
eajbcs-694	207	8	0	0	NUM
eajbcs-694	207	9			NOUN
eajbcs-694	207	10	,	,	PUNCT
eajbcs-694	207	11	v	v	NOUN
eajbcs-694	207	12	=	=	SYM
eajbcs-694	207	13	[	[	PUNCT
eajbcs-694	207	14	∂vi(e0	∂vi(e0	ADJ
eajbcs-694	207	15	)	)	PUNCT
eajbcs-694	207	16	∂xj	∂xj	NOUN
eajbcs-694	207	17	]	]	PUNCT
eajbcs-694	208	1	=	=	PUNCT
eajbcs-694	208	2			PROPN
eajbcs-694	208	3	δ	δ	PROPN
eajbcs-694	208	4	+	+	X
eajbcs-694	208	5	µ	µ	X
eajbcs-694	208	6	0	0	NUM
eajbcs-694	208	7	0	0	NUM
eajbcs-694	208	8	−τδ	−τδ	ADP
eajbcs-694	208	9	k1	k1	NOUN
eajbcs-694	208	10	0	0	NUM
eajbcs-694	208	11	0	0	NUM
eajbcs-694	208	12	−α	−α	PROPN
eajbcs-694	208	13	k2	k2	ADJ
eajbcs-694	208	14			NOUN
eajbcs-694	208	15	,	,	PUNCT
eajbcs-694	208	16	where	where	SCONJ
eajbcs-694	208	17	k1	k1	NOUN
eajbcs-694	208	18	=	=	PROPN
eajbcs-694	208	19	α	α	PROPN
eajbcs-694	208	20	+	+	CCONJ
eajbcs-694	208	21	ε+	ε+	X
eajbcs-694	208	22	µ+	µ+	X
eajbcs-694	208	23	ρ	ρ	NOUN
eajbcs-694	208	24	and	and	CCONJ
eajbcs-694	208	25	k2	k2	NOUN
eajbcs-694	208	26	=	=	SYM
eajbcs-694	208	27	γ	γ	PROPN
eajbcs-694	208	28	+	+	X
eajbcs-694	208	29	µ+	µ+	X
eajbcs-694	208	30	ξ	ξ	PROPN
eajbcs-694	208	31	.	.	PUNCT
eajbcs-694	209	1	the	the	DET
eajbcs-694	209	2	next	next	ADJ
eajbcs-694	209	3	-	-	PUNCT
eajbcs-694	209	4	generation	generation	NOUN
eajbcs-694	209	5	matrix	matrix	NOUN
eajbcs-694	209	6	fv	fv	NOUN
eajbcs-694	209	7	−1	−1	NOUN
eajbcs-694	209	8	is	be	AUX
eajbcs-694	209	9	given	give	VERB
eajbcs-694	209	10	by	by	ADP
eajbcs-694	209	11	fv	fv	NUM
eajbcs-694	209	12	−1	−1	NOUN
eajbcs-694	209	13	=	=	SYM
eajbcs-694	209	14			PROPN
eajbcs-694	209	15	r1	r1	NOUN
eajbcs-694	209	16	+	+	PROPN
eajbcs-694	209	17	r2	r2	PROPN
eajbcs-694	209	18	+	+	NOUN
eajbcs-694	209	19	r3	r3	NOUN
eajbcs-694	209	20	βs0(σ1k2+σ3α	βs0(σ1k2+σ3α	NOUN
eajbcs-694	209	21	)	)	PUNCT
eajbcs-694	210	1	k1k2	k1k2	PROPN
eajbcs-694	210	2	βσ3s0	βσ3s0	PROPN
eajbcs-694	210	3	k2	k2	X
eajbcs-694	210	4	0	0	NUM
eajbcs-694	210	5	0	0	NUM
eajbcs-694	210	6	0	0	NUM
eajbcs-694	210	7	0	0	NUM
eajbcs-694	210	8	0	0	NUM
eajbcs-694	210	9	0	0	NUM
eajbcs-694	210	10			NOUN
eajbcs-694	210	11	,	,	PUNCT
eajbcs-694	210	12	where	where	SCONJ
eajbcs-694	210	13	r1	r1	NOUN
eajbcs-694	210	14	=	=	PUNCT
eajbcs-694	210	15	βσ2s0	βσ2s0	PUNCT
eajbcs-694	210	16	δ	δ	PROPN
eajbcs-694	210	17	+	+	X
eajbcs-694	210	18	µ	µ	X
eajbcs-694	210	19	,	,	PUNCT
eajbcs-694	210	20	r2	r2	PROPN
eajbcs-694	210	21	=	=	PUNCT
eajbcs-694	210	22	βσ1s0τδ	βσ1s0τδ	PROPN
eajbcs-694	210	23	k1(δ	k1(δ	PROPN
eajbcs-694	210	24	+	+	CCONJ
eajbcs-694	210	25	µ	µ	X
eajbcs-694	210	26	)	)	PUNCT
eajbcs-694	210	27	and	and	CCONJ
eajbcs-694	210	28	r3	r3	PROPN
eajbcs-694	210	29	=	=	SYM
eajbcs-694	210	30	βσ3s0τδα	βσ3s0τδα	X
eajbcs-694	210	31	k1k2(δ	k1k2(δ	PROPN
eajbcs-694	210	32	+	+	X
eajbcs-694	210	33	µ	µ	X
eajbcs-694	210	34	)	)	PUNCT
eajbcs-694	210	35	.	.	PUNCT
eajbcs-694	211	1	(	(	PUNCT
eajbcs-694	211	2	3	3	X
eajbcs-694	211	3	)	)	PUNCT
eajbcs-694	211	4	we	we	PRON
eajbcs-694	211	5	find	find	VERB
eajbcs-694	211	6	the	the	DET
eajbcs-694	211	7	eigenvalues	eigenvalue	NOUN
eajbcs-694	211	8	of	of	ADP
eajbcs-694	211	9	fv	fv	NOUN
eajbcs-694	211	10	−1	−1	ADV
eajbcs-694	211	11	by	by	ADP
eajbcs-694	211	12	solving	solve	VERB
eajbcs-694	211	13	the	the	DET
eajbcs-694	211	14	characteristic	characteristic	ADJ
eajbcs-694	211	15	equation	equation	NOUN
eajbcs-694	211	16	|fv	|fv	ADP
eajbcs-694	211	17	−1	−1	NOUN
eajbcs-694	211	18	−	−	PROPN
eajbcs-694	212	1	λi|	λi|	NOUN
eajbcs-694	212	2	=	=	SYM
eajbcs-694	212	3	0	0	PUNCT
eajbcs-694	212	4	as	as	ADP
eajbcs-694	212	5	λ1	λ1	PROPN
eajbcs-694	212	6	=	=	SYM
eajbcs-694	212	7	r1	r1	PROPN
eajbcs-694	212	8	+	+	PROPN
eajbcs-694	212	9	r2	r2	PROPN
eajbcs-694	212	10	+	+	PROPN
eajbcs-694	212	11	r3	r3	PROPN
eajbcs-694	212	12	,	,	PUNCT
eajbcs-694	212	13	λ2	λ2	NOUN
eajbcs-694	212	14	=	=	SYM
eajbcs-694	212	15	0	0	NUM
eajbcs-694	212	16	and	and	CCONJ
eajbcs-694	212	17	λ3	λ3	PROPN
eajbcs-694	212	18	=	=	SYM
eajbcs-694	212	19	0	0	PROPN
eajbcs-694	212	20	.	.	PUNCT
eajbcs-694	213	1	the	the	DET
eajbcs-694	213	2	basic	basic	ADJ
eajbcs-694	213	3	reproduction	reproduction	NOUN
eajbcs-694	213	4	number	number	NOUN
eajbcs-694	213	5	r0	r0	NOUN
eajbcs-694	213	6	is	be	AUX
eajbcs-694	213	7	the	the	DET
eajbcs-694	213	8	spectral	spectral	ADJ
eajbcs-694	213	9	radius	radius	NOUN
eajbcs-694	213	10	(	(	PUNCT
eajbcs-694	213	11	the	the	DET
eajbcs-694	213	12	largest	large	ADJ
eajbcs-694	213	13	eigenvalues	eigenvalue	NOUN
eajbcs-694	213	14	in	in	ADP
eajbcs-694	213	15	modulus	modulus	NOUN
eajbcs-694	213	16	)	)	PUNCT
eajbcs-694	213	17	of	of	ADP
eajbcs-694	213	18	fv	fv	PROPN
eajbcs-694	213	19	−1	−1	NOUN
eajbcs-694	213	20	which	which	PRON
eajbcs-694	213	21	is	be	AUX
eajbcs-694	213	22	given	give	VERB
eajbcs-694	213	23	by	by	ADP
eajbcs-694	213	24	r0	r0	NOUN
eajbcs-694	213	25	=	=	SYM
eajbcs-694	213	26	ρ(fv	ρ(fv	NUM
eajbcs-694	213	27	−1	−1	NOUN
eajbcs-694	213	28	)	)	PUNCT
eajbcs-694	214	1	=	=	VERB
eajbcs-694	214	2	r1	r1	PROPN
eajbcs-694	214	3	+	+	PROPN
eajbcs-694	214	4	r2	r2	PROPN
eajbcs-694	214	5	+	+	PROPN
eajbcs-694	214	6	r3	r3	PROPN
eajbcs-694	214	7	.	.	PUNCT
eajbcs-694	215	1	the	the	DET
eajbcs-694	215	2	parts	part	NOUN
eajbcs-694	215	3	r1	r1	PROPN
eajbcs-694	215	4	,	,	PUNCT
eajbcs-694	215	5	r2	r2	PROPN
eajbcs-694	215	6	and	and	CCONJ
eajbcs-694	215	7	r3	r3	PROPN
eajbcs-694	215	8	represent	represent	VERB
eajbcs-694	215	9	the	the	DET
eajbcs-694	215	10	contributions	contribution	NOUN
eajbcs-694	215	11	from	from	ADP
eajbcs-694	215	12	the	the	DET
eajbcs-694	215	13	human	human	NOUN
eajbcs-694	215	14	-	-	PUNCT
eajbcs-694	215	15	to	to	ADP
eajbcs-694	215	16	-	-	PUNCT
eajbcs-694	215	17	human	human	ADJ
eajbcs-694	215	18	transmission	transmission	NOUN
eajbcs-694	215	19	routes	route	NOUN
eajbcs-694	215	20	(	(	PUNCT
eajbcs-694	215	21	exposed	expose	VERB
eajbcs-694	215	22	-	-	PUNCT
eajbcs-694	215	23	to	to	ADP
eajbcs-694	215	24	-	-	PUNCT
eajbcs-694	215	25	susceptible	susceptible	ADJ
eajbcs-694	215	26	,	,	PUNCT
eajbcs-694	215	27	infectedto	infectedto	NOUN
eajbcs-694	215	28	-	-	PUNCT
eajbcs-694	215	29	susceptible	susceptible	ADJ
eajbcs-694	215	30	and	and	CCONJ
eajbcs-694	215	31	hospitalized	hospitalize	VERB
eajbcs-694	215	32	-	-	PUNCT
eajbcs-694	215	33	to	to	ADP
eajbcs-694	215	34	-	-	PUNCT
eajbcs-694	215	35	susceptible	susceptible	ADJ
eajbcs-694	215	36	individuals	individual	NOUN
eajbcs-694	215	37	,	,	PUNCT
eajbcs-694	215	38	respectively	respectively	ADV
eajbcs-694	215	39	)	)	PUNCT
eajbcs-694	215	40	.	.	PUNCT
eajbcs-694	216	1	we	we	PRON
eajbcs-694	216	2	can	can	AUX
eajbcs-694	216	3	rewrite	rewrite	VERB
eajbcs-694	216	4	the	the	DET
eajbcs-694	216	5	basic	basic	ADJ
eajbcs-694	216	6	reproduction	reproduction	NOUN
eajbcs-694	216	7	number	number	NOUN
eajbcs-694	216	8	as	as	SCONJ
eajbcs-694	216	9	follows	follow	VERB
eajbcs-694	216	10	:	:	PUNCT
eajbcs-694	216	11	r0	r0	NOUN
eajbcs-694	216	12	=	=	SYM
eajbcs-694	216	13	πβ(ν	πβ(ν	X
eajbcs-694	216	14	+	+	X
eajbcs-694	216	15	µ	µ	X
eajbcs-694	216	16	)	)	PUNCT
eajbcs-694	216	17	µ(θ	µ(θ	ADJ
eajbcs-694	216	18	+	+	NUM
eajbcs-694	216	19	ν	ν	X
eajbcs-694	216	20	+	+	X
eajbcs-694	216	21	µ)(δ	µ)(δ	X
eajbcs-694	216	22	+	+	CCONJ
eajbcs-694	216	23	µ	µ	X
eajbcs-694	216	24	)	)	PUNCT
eajbcs-694	216	25	(	(	PUNCT
eajbcs-694	216	26	σ2	σ2	NOUN
eajbcs-694	216	27	+	+	PROPN
eajbcs-694	216	28	σ1τδ	σ1τδ	PROPN
eajbcs-694	216	29	α	α	PROPN
eajbcs-694	216	30	+	+	CCONJ
eajbcs-694	216	31	ε+	ε+	X
eajbcs-694	216	32	µ+	µ+	X
eajbcs-694	216	33	ρ	ρ	PROPN
eajbcs-694	216	34	+	+	X
eajbcs-694	216	35	σ3τδα	σ3τδα	PROPN
eajbcs-694	216	36	(	(	PUNCT
eajbcs-694	216	37	α	α	NOUN
eajbcs-694	216	38	+	+	NOUN
eajbcs-694	216	39	ε+	ε+	X
eajbcs-694	216	40	µ+	µ+	X
eajbcs-694	216	41	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	216	42	+	+	CCONJ
eajbcs-694	216	43	µ+	µ+	X
eajbcs-694	216	44	ξ	ξ	NUM
eajbcs-694	216	45	)	)	PUNCT
eajbcs-694	216	46	)	)	PUNCT
eajbcs-694	216	47	.	.	PUNCT
eajbcs-694	217	1	(	(	PUNCT
eajbcs-694	217	2	4	4	X
eajbcs-694	217	3	)	)	PUNCT
eajbcs-694	217	4	3.4	3.4	NUM
eajbcs-694	217	5	.	.	PUNCT
eajbcs-694	218	1	local	local	ADJ
eajbcs-694	218	2	stability	stability	NOUN
eajbcs-694	218	3	of	of	ADP
eajbcs-694	218	4	disease	disease	NOUN
eajbcs-694	218	5	free	free	ADJ
eajbcs-694	218	6	equilibrium	equilibrium	NOUN
eajbcs-694	218	7	theorem	theorem	VERB
eajbcs-694	218	8	3.3	3.3	NUM
eajbcs-694	218	9	.	.	PUNCT
eajbcs-694	219	1	the	the	DET
eajbcs-694	219	2	disease	disease	NOUN
eajbcs-694	219	3	free	free	ADJ
eajbcs-694	219	4	equilibrium	equilibrium	NOUN
eajbcs-694	219	5	point	point	NOUN
eajbcs-694	219	6	e0	e0	PROPN
eajbcs-694	219	7	of	of	ADP
eajbcs-694	219	8	the	the	DET
eajbcs-694	219	9	system	system	NOUN
eajbcs-694	219	10	(	(	PUNCT
eajbcs-694	219	11	2	2	X
eajbcs-694	219	12	)	)	PUNCT
eajbcs-694	219	13	is	be	AUX
eajbcs-694	219	14	locally	locally	ADV
eajbcs-694	219	15	asymptotically	asymptotically	ADV
eajbcs-694	219	16	stable	stable	ADJ
eajbcs-694	219	17	if	if	SCONJ
eajbcs-694	219	18	r0	r0	NOUN
eajbcs-694	219	19	<	<	X
eajbcs-694	219	20	1	1	NUM
eajbcs-694	219	21	and	and	CCONJ
eajbcs-694	219	22	unstable	unstable	ADJ
eajbcs-694	219	23	if	if	SCONJ
eajbcs-694	219	24	r0	r0	NOUN
eajbcs-694	219	25	>	>	X
eajbcs-694	219	26	1	1	X
eajbcs-694	219	27	.	.	PUNCT
eajbcs-694	219	28	proof	proof	NOUN
eajbcs-694	219	29	.	.	PUNCT
eajbcs-694	220	1	the	the	DET
eajbcs-694	220	2	jacobian	jacobian	ADJ
eajbcs-694	220	3	matrix	matrix	NOUN
eajbcs-694	220	4	of	of	ADP
eajbcs-694	220	5	the	the	DET
eajbcs-694	220	6	system	system	NOUN
eajbcs-694	220	7	(	(	PUNCT
eajbcs-694	220	8	2	2	NUM
eajbcs-694	220	9	)	)	PUNCT
eajbcs-694	220	10	at	at	ADP
eajbcs-694	220	11	the	the	DET
eajbcs-694	220	12	disease	disease	NOUN
eajbcs-694	220	13	-	-	PUNCT
eajbcs-694	220	14	free	free	ADJ
eajbcs-694	220	15	equilibrium	equilibrium	NOUN
eajbcs-694	220	16	e0	e0	PROPN
eajbcs-694	220	17	is	be	AUX
eajbcs-694	220	18	given	give	VERB
eajbcs-694	220	19	by	by	ADP
eajbcs-694	220	20	j(e0	j(e0	ADJ
eajbcs-694	220	21	)	)	PUNCT
eajbcs-694	220	22	=	=	SYM
eajbcs-694	220	23			NOUN
eajbcs-694	220	24	−(θ	−(θ	NOUN
eajbcs-694	220	25	+	+	CCONJ
eajbcs-694	220	26	µ	µ	X
eajbcs-694	220	27	)	)	PUNCT
eajbcs-694	220	28	ν	ν	NOUN
eajbcs-694	220	29	−βσ2s	−βσ2	VERB
eajbcs-694	220	30	0	0	NUM
eajbcs-694	220	31	−βσ1s	−βσ1s	X
eajbcs-694	220	32	0	0	X
eajbcs-694	220	33	−βσ3s	−βσ3s	X
eajbcs-694	220	34	0	0	NUM
eajbcs-694	220	35	ηω	ηω	VERB
eajbcs-694	220	36	θ	θ	PROPN
eajbcs-694	220	37	−(ν	−(ν	PROPN
eajbcs-694	220	38	+	+	CCONJ
eajbcs-694	220	39	µ	µ	X
eajbcs-694	220	40	)	)	PUNCT
eajbcs-694	220	41	0	0	NUM
eajbcs-694	220	42	0	0	NUM
eajbcs-694	220	43	0	0	NUM
eajbcs-694	220	44	(	(	PUNCT
eajbcs-694	220	45	1−	1−	NUM
eajbcs-694	220	46	η)ω	η)ω	X
eajbcs-694	220	47	0	0	NUM
eajbcs-694	220	48	0	0	NUM
eajbcs-694	221	1	βσ2s	βσ2s	PROPN
eajbcs-694	221	2	0	0	NUM
eajbcs-694	222	1	−	−	PROPN
eajbcs-694	222	2	(	(	PUNCT
eajbcs-694	222	3	δ	δ	PROPN
eajbcs-694	222	4	+	+	X
eajbcs-694	222	5	µ	µ	X
eajbcs-694	222	6	)	)	PUNCT
eajbcs-694	222	7	βσ1s	βσ1s	PROPN
eajbcs-694	222	8	0	0	NUM
eajbcs-694	222	9	βσ3s	βσ3s	ADJ
eajbcs-694	222	10	0	0	NUM
eajbcs-694	222	11	0	0	NUM
eajbcs-694	222	12	0	0	NUM
eajbcs-694	222	13	0	0	NUM
eajbcs-694	223	1	τδ	τδ	ADP
eajbcs-694	223	2	−k1	−k1	NOUN
eajbcs-694	223	3	0	0	NUM
eajbcs-694	223	4	0	0	NUM
eajbcs-694	223	5	0	0	NUM
eajbcs-694	223	6	0	0	NUM
eajbcs-694	223	7	0	0	NUM
eajbcs-694	224	1	α	α	PRON
eajbcs-694	224	2	−k2	−k2	ADJ
eajbcs-694	224	3	0	0	NUM
eajbcs-694	224	4	0	0	NUM
eajbcs-694	224	5	0	0	NUM
eajbcs-694	225	1	(	(	PUNCT
eajbcs-694	225	2	1−	1−	NUM
eajbcs-694	225	3	τ)δ	τ)δ	X
eajbcs-694	225	4	ε	ε	PROPN
eajbcs-694	225	5	γ	γ	X
eajbcs-694	225	6	−(ω	−(ω	PROPN
eajbcs-694	225	7	+	+	CCONJ
eajbcs-694	225	8	µ	µ	X
eajbcs-694	225	9	)	)	PUNCT
eajbcs-694	225	10			NUM
eajbcs-694	225	11	.	.	PUNCT
eajbcs-694	226	1	east	east	PROPN
eajbcs-694	226	2	afr	afr	PROPN
eajbcs-694	226	3	.	.	PUNCT
eajbcs-694	227	1	j.	j.	PROPN
eajbcs-694	227	2	biophys	biophys	PROPN
eajbcs-694	227	3	.	.	PUNCT
eajbcs-694	228	1	comput	comput	NOUN
eajbcs-694	228	2	.	.	PUNCT
eajbcs-694	229	1	sci	sci	PROPN
eajbcs-694	229	2	.	.	PUNCT
eajbcs-694	230	1	(	(	PUNCT
eajbcs-694	230	2	2024	2024	NUM
eajbcs-694	230	3	)	)	PUNCT
eajbcs-694	230	4	,	,	PUNCT
eajbcs-694	230	5	vol	vol	NOUN
eajbcs-694	230	6	.	.	PROPN
eajbcs-694	230	7	5	5	NUM
eajbcs-694	230	8	,	,	PUNCT
eajbcs-694	230	9	no	no	INTJ
eajbcs-694	230	10	.	.	NOUN
eajbcs-694	230	11	2	2	NUM
eajbcs-694	230	12	,	,	PUNCT
eajbcs-694	230	13	58	58	NUM
eajbcs-694	230	14	-	-	SYM
eajbcs-694	230	15	86	86	NUM
eajbcs-694	230	16	64	64	NUM
eajbcs-694	230	17	the	the	DET
eajbcs-694	230	18	matrix	matrix	NOUN
eajbcs-694	230	19	j(e0	j(e0	NOUN
eajbcs-694	230	20	)	)	PUNCT
eajbcs-694	230	21	is	be	AUX
eajbcs-694	230	22	an	an	DET
eajbcs-694	230	23	upper	upper	ADJ
eajbcs-694	230	24	triangular	triangular	NOUN
eajbcs-694	230	25	block	block	NOUN
eajbcs-694	230	26	matrix	matrix	NOUN
eajbcs-694	230	27	.	.	PUNCT
eajbcs-694	231	1	its	its	PRON
eajbcs-694	231	2	eigenvalues	eigenvalue	NOUN
eajbcs-694	231	3	are	be	AUX
eajbcs-694	231	4	λ1	λ1	ADJ
eajbcs-694	231	5	,	,	PUNCT
eajbcs-694	231	6	λ2	λ2	PROPN
eajbcs-694	231	7	,	,	PUNCT
eajbcs-694	231	8	λ3	λ3	PROPN
eajbcs-694	231	9	,	,	PUNCT
eajbcs-694	231	10	λ4	λ4	PROPN
eajbcs-694	231	11	,	,	PUNCT
eajbcs-694	231	12	λ5	λ5	NOUN
eajbcs-694	231	13	and	and	CCONJ
eajbcs-694	231	14	λ6	λ6	PROPN
eajbcs-694	231	15	.	.	PUNCT
eajbcs-694	232	1	where	where	SCONJ
eajbcs-694	232	2	λ1	λ1	ADJ
eajbcs-694	232	3	,	,	PUNCT
eajbcs-694	232	4	λ2	λ2	NOUN
eajbcs-694	232	5	are	be	AUX
eajbcs-694	232	6	eigenvalues	eigenvalue	NOUN
eajbcs-694	232	7	of	of	ADP
eajbcs-694	232	8	the	the	DET
eajbcs-694	232	9	first	first	ADJ
eajbcs-694	232	10	block	block	NOUN
eajbcs-694	232	11	matrix	matrix	NOUN
eajbcs-694	232	12	of	of	ADP
eajbcs-694	232	13	j(e0	j(e0	ADJ
eajbcs-694	232	14	)	)	PUNCT
eajbcs-694	232	15	and	and	CCONJ
eajbcs-694	232	16	λ3	λ3	PROPN
eajbcs-694	232	17	,	,	PUNCT
eajbcs-694	232	18	λ4	λ4	PROPN
eajbcs-694	232	19	,	,	PUNCT
eajbcs-694	232	20	λ5	λ5	NOUN
eajbcs-694	232	21	,	,	PUNCT
eajbcs-694	232	22	λ6	λ6	NOUN
eajbcs-694	232	23	are	be	AUX
eajbcs-694	232	24	eigenvalues	eigenvalue	NOUN
eajbcs-694	232	25	of	of	ADP
eajbcs-694	232	26	the	the	DET
eajbcs-694	232	27	fourth	fourth	ADJ
eajbcs-694	232	28	block	block	NOUN
eajbcs-694	232	29	matrix	matrix	NOUN
eajbcs-694	232	30	of	of	ADP
eajbcs-694	232	31	j(e0	j(e0	NOUN
eajbcs-694	232	32	)	)	PUNCT
eajbcs-694	232	33	.	.	PUNCT
eajbcs-694	233	1	the	the	DET
eajbcs-694	233	2	first	first	ADJ
eajbcs-694	233	3	block	block	NOUN
eajbcs-694	233	4	matrix	matrix	NOUN
eajbcs-694	233	5	of	of	ADP
eajbcs-694	233	6	j(e0	j(e0	NOUN
eajbcs-694	233	7	)	)	PUNCT
eajbcs-694	233	8	given	give	VERB
eajbcs-694	233	9	by	by	ADP
eajbcs-694	233	10	j1(e0	j1(e0	NOUN
eajbcs-694	233	11	)	)	PUNCT
eajbcs-694	233	12	=	=	PUNCT
eajbcs-694	234	1	[	[	PUNCT
eajbcs-694	234	2	−(θ	−(θ	NOUN
eajbcs-694	234	3	+	+	CCONJ
eajbcs-694	234	4	µ	µ	X
eajbcs-694	234	5	)	)	PUNCT
eajbcs-694	234	6	ν	ν	NOUN
eajbcs-694	234	7	θ	θ	PROPN
eajbcs-694	234	8	−(ν	−(ν	X
eajbcs-694	234	9	+	+	CCONJ
eajbcs-694	234	10	µ	µ	X
eajbcs-694	234	11	)	)	PUNCT
eajbcs-694	234	12	]	]	PUNCT
eajbcs-694	234	13	.	.	PUNCT
eajbcs-694	235	1	we	we	PRON
eajbcs-694	235	2	find	find	VERB
eajbcs-694	235	3	the	the	DET
eajbcs-694	235	4	eigenvalues	eigenvalue	NOUN
eajbcs-694	235	5	of	of	ADP
eajbcs-694	235	6	j1(e0	j1(e0	NOUN
eajbcs-694	235	7	)	)	PUNCT
eajbcs-694	235	8	by	by	ADP
eajbcs-694	235	9	solving	solve	VERB
eajbcs-694	235	10	the	the	DET
eajbcs-694	235	11	characteristic	characteristic	ADJ
eajbcs-694	235	12	equation	equation	NOUN
eajbcs-694	235	13	|j1(e0)−λi|	|j1(e0)−λi|	NOUN
eajbcs-694	235	14	=	=	SYM
eajbcs-694	235	15	0	0	PUNCT
eajbcs-694	235	16	as	as	ADP
eajbcs-694	235	17	λ1	λ1	PROPN
eajbcs-694	235	18	=	=	SYM
eajbcs-694	235	19	−µ	−µ	NOUN
eajbcs-694	235	20	and	and	CCONJ
eajbcs-694	235	21	λ2	λ2	NOUN
eajbcs-694	235	22	=	=	NOUN
eajbcs-694	235	23	−(θ	−(θ	NOUN
eajbcs-694	236	1	+	+	CCONJ
eajbcs-694	236	2	ν	ν	X
eajbcs-694	236	3	+	+	X
eajbcs-694	236	4	µ	µ	NUM
eajbcs-694	236	5	)	)	PUNCT
eajbcs-694	236	6	.	.	PUNCT
eajbcs-694	237	1	the	the	DET
eajbcs-694	237	2	fourth	fourth	ADJ
eajbcs-694	237	3	block	block	NOUN
eajbcs-694	237	4	matrix	matrix	NOUN
eajbcs-694	237	5	of	of	ADP
eajbcs-694	237	6	j(e0	j(e0	NOUN
eajbcs-694	237	7	)	)	PUNCT
eajbcs-694	237	8	is	be	AUX
eajbcs-694	237	9	given	give	VERB
eajbcs-694	237	10	by	by	ADP
eajbcs-694	237	11	j4(e0	j4(e0	NOUN
eajbcs-694	237	12	)	)	PUNCT
eajbcs-694	237	13	=	=	SYM
eajbcs-694	237	14			NOUN
eajbcs-694	238	1	βσ2s	βσ2s	PROPN
eajbcs-694	238	2	0	0	NUM
eajbcs-694	239	1	−	−	PROPN
eajbcs-694	239	2	(	(	PUNCT
eajbcs-694	239	3	δ	δ	PROPN
eajbcs-694	239	4	+	+	X
eajbcs-694	239	5	µ	µ	X
eajbcs-694	239	6	)	)	PUNCT
eajbcs-694	239	7	βσ1s	βσ1s	PROPN
eajbcs-694	239	8	0	0	NUM
eajbcs-694	239	9	βσ3s	βσ3s	ADJ
eajbcs-694	239	10	0	0	NUM
eajbcs-694	239	11	0	0	NUM
eajbcs-694	240	1	τδ	τδ	ADP
eajbcs-694	240	2	−k1	−k1	NOUN
eajbcs-694	240	3	0	0	NUM
eajbcs-694	240	4	0	0	NUM
eajbcs-694	240	5	0	0	NUM
eajbcs-694	241	1	α	α	PRON
eajbcs-694	241	2	−k2	−k2	PROPN
eajbcs-694	241	3	0	0	PUNCT
eajbcs-694	242	1	(	(	PUNCT
eajbcs-694	242	2	1−	1−	NUM
eajbcs-694	242	3	τ)δ	τ)δ	X
eajbcs-694	242	4	ε	ε	PROPN
eajbcs-694	242	5	γ	γ	X
eajbcs-694	242	6	−(ω	−(ω	PROPN
eajbcs-694	242	7	+	+	CCONJ
eajbcs-694	242	8	µ	µ	X
eajbcs-694	242	9	)	)	PUNCT
eajbcs-694	242	10			NOUN
eajbcs-694	242	11	.	.	PUNCT
eajbcs-694	243	1	thus	thus	ADV
eajbcs-694	243	2	the	the	DET
eajbcs-694	243	3	eigenvalues	eigenvalues	PROPN
eajbcs-694	243	4	λ3	λ3	PROPN
eajbcs-694	243	5	,	,	PUNCT
eajbcs-694	243	6	λ4	λ4	PROPN
eajbcs-694	243	7	,	,	PUNCT
eajbcs-694	243	8	λ5	λ5	NOUN
eajbcs-694	243	9	,	,	PUNCT
eajbcs-694	243	10	λ6	λ6	NOUN
eajbcs-694	243	11	are	be	AUX
eajbcs-694	243	12	obtained	obtain	VERB
eajbcs-694	243	13	from	from	ADP
eajbcs-694	243	14	the	the	DET
eajbcs-694	243	15	characteristic	characteristic	ADJ
eajbcs-694	243	16	equation	equation	NOUN
eajbcs-694	243	17	of	of	ADP
eajbcs-694	243	18	j4(e0	j4(e0	NOUN
eajbcs-694	243	19	):	):	PUNCT
eajbcs-694	243	20	(	(	PUNCT
eajbcs-694	243	21	−(ω	−(ω	NOUN
eajbcs-694	243	22	+	+	CCONJ
eajbcs-694	243	23	µ)−	µ)−	NOUN
eajbcs-694	243	24	λ)[λ3	λ)[λ3	PROPN
eajbcs-694	243	25	+	+	CCONJ
eajbcs-694	243	26	(	(	PUNCT
eajbcs-694	243	27	(	(	PUNCT
eajbcs-694	243	28	δ	δ	PROPN
eajbcs-694	243	29	+	+	CCONJ
eajbcs-694	243	30	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	243	31	)	)	PUNCT
eajbcs-694	244	1	+	+	NUM
eajbcs-694	244	2	k1	k1	NOUN
eajbcs-694	244	3	+	+	CCONJ
eajbcs-694	244	4	k2)λ	k2)λ	NOUN
eajbcs-694	244	5	2	2	NUM
eajbcs-694	244	6	+	+	CCONJ
eajbcs-694	245	1	[	[	X
eajbcs-694	245	2	k1(δ	k1(δ	X
eajbcs-694	245	3	+	+	CCONJ
eajbcs-694	245	4	µ)(1−	µ)(1−	PROPN
eajbcs-694	245	5	(	(	PUNCT
eajbcs-694	245	6	r1	r1	NOUN
eajbcs-694	245	7	+	+	NOUN
eajbcs-694	245	8	r2	r2	PROPN
eajbcs-694	245	9	)	)	PUNCT
eajbcs-694	245	10	)	)	PUNCT
eajbcs-694	246	1	+	+	PROPN
eajbcs-694	246	2	k2(δ	k2(δ	PROPN
eajbcs-694	246	3	+	+	NUM
eajbcs-694	246	4	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	246	5	)	)	PUNCT
eajbcs-694	246	6	+	+	CCONJ
eajbcs-694	246	7	k1k2]λ+	k1k2]λ+	PROPN
eajbcs-694	246	8	k1k2(δ	k1k2(δ	NOUN
eajbcs-694	246	9	+	+	CCONJ
eajbcs-694	246	10	µ)(1−r0	µ)(1−r0	PROPN
eajbcs-694	246	11	)	)	PUNCT
eajbcs-694	246	12	]	]	PUNCT
eajbcs-694	247	1	=	=	PUNCT
eajbcs-694	247	2	0	0	X
eajbcs-694	247	3	.	.	PUNCT
eajbcs-694	248	1	from	from	ADP
eajbcs-694	248	2	this	this	DET
eajbcs-694	248	3	equation	equation	NOUN
eajbcs-694	248	4	,	,	PUNCT
eajbcs-694	248	5	we	we	PRON
eajbcs-694	248	6	obtain	obtain	VERB
eajbcs-694	248	7	the	the	DET
eajbcs-694	248	8	values	value	NOUN
eajbcs-694	248	9	for	for	SCONJ
eajbcs-694	248	10	λ	λ	NOUN
eajbcs-694	248	11	to	to	PART
eajbcs-694	248	12	be	be	AUX
eajbcs-694	248	13	λ3	λ3	PROPN
eajbcs-694	248	14	=	=	PROPN
eajbcs-694	248	15	−(ω	−(ω	NOUN
eajbcs-694	248	16	+	+	X
eajbcs-694	248	17	µ	µ	NOUN
eajbcs-694	248	18	)	)	PUNCT
eajbcs-694	248	19	and	and	CCONJ
eajbcs-694	248	20	the	the	DET
eajbcs-694	248	21	eigenvalues	eigenvalue	NOUN
eajbcs-694	248	22	λ4	λ4	PROPN
eajbcs-694	248	23	,	,	PUNCT
eajbcs-694	248	24	λ5	λ5	NOUN
eajbcs-694	248	25	,	,	PUNCT
eajbcs-694	248	26	λ6	λ6	NOUN
eajbcs-694	248	27	are	be	AUX
eajbcs-694	248	28	the	the	DET
eajbcs-694	248	29	roots	root	NOUN
eajbcs-694	248	30	of	of	ADP
eajbcs-694	248	31	the	the	DET
eajbcs-694	248	32	cubic	cubic	ADJ
eajbcs-694	248	33	polynomial	polynomial	ADJ
eajbcs-694	248	34	:	:	PUNCT
eajbcs-694	248	35	p(λ	p(λ	NOUN
eajbcs-694	248	36	)	)	PUNCT
eajbcs-694	249	1	=	=	SYM
eajbcs-694	249	2	a0λ	a0λ	NOUN
eajbcs-694	249	3	3	3	NUM
eajbcs-694	249	4	+	+	CCONJ
eajbcs-694	249	5	a1λ	a1λ	PROPN
eajbcs-694	249	6	2	2	NUM
eajbcs-694	249	7	+	+	CCONJ
eajbcs-694	249	8	a2λ+	a2λ+	NOUN
eajbcs-694	249	9	a3	a3	NOUN
eajbcs-694	249	10	=	=	SYM
eajbcs-694	249	11	0	0	PROPN
eajbcs-694	249	12	,	,	PUNCT
eajbcs-694	250	1	where	where	SCONJ
eajbcs-694	250	2	a0	a0	NOUN
eajbcs-694	250	3	=	=	SYM
eajbcs-694	250	4	1	1	NUM
eajbcs-694	250	5	,	,	PUNCT
eajbcs-694	250	6	a1	a1	NOUN
eajbcs-694	250	7	=	=	SYM
eajbcs-694	250	8	(	(	PUNCT
eajbcs-694	250	9	δ	δ	PROPN
eajbcs-694	250	10	+	+	CCONJ
eajbcs-694	250	11	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	250	12	)	)	PUNCT
eajbcs-694	250	13	+	+	NUM
eajbcs-694	250	14	k1	k1	NOUN
eajbcs-694	250	15	+	+	CCONJ
eajbcs-694	250	16	k2	k2	ADJ
eajbcs-694	250	17	,	,	PUNCT
eajbcs-694	250	18	a2	a2	PROPN
eajbcs-694	250	19	=	=	SYM
eajbcs-694	250	20	k1(δ	k1(δ	PROPN
eajbcs-694	250	21	+	+	CCONJ
eajbcs-694	250	22	µ)(1−	µ)(1−	PROPN
eajbcs-694	250	23	(	(	PUNCT
eajbcs-694	250	24	r1	r1	NOUN
eajbcs-694	250	25	+	+	NOUN
eajbcs-694	250	26	r2	r2	PROPN
eajbcs-694	250	27	)	)	PUNCT
eajbcs-694	250	28	)	)	PUNCT
eajbcs-694	251	1	+	+	CCONJ
eajbcs-694	251	2	k2(δ	k2(δ	PROPN
eajbcs-694	251	3	+	+	NUM
eajbcs-694	251	4	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	251	5	)	)	PUNCT
eajbcs-694	252	1	+	+	CCONJ
eajbcs-694	252	2	k1k2	k1k2	PROPN
eajbcs-694	252	3	,	,	PUNCT
eajbcs-694	252	4	a3	a3	NOUN
eajbcs-694	252	5	=	=	PUNCT
eajbcs-694	252	6	k1k2(δ	k1k2(δ	PROPN
eajbcs-694	252	7	+	+	CCONJ
eajbcs-694	252	8	µ)(1−r0	µ)(1−r0	PROPN
eajbcs-694	252	9	)	)	PUNCT
eajbcs-694	252	10	.	.	PUNCT
eajbcs-694	253	1	furthermore	furthermore	ADV
eajbcs-694	253	2	,	,	PUNCT
eajbcs-694	253	3	a1a2	a1a2	ADP
eajbcs-694	253	4	−	−	PROPN
eajbcs-694	253	5	a3	a3	NOUN
eajbcs-694	253	6	=	=	PUNCT
eajbcs-694	253	7	k1(δ	k1(δ	PROPN
eajbcs-694	254	1	+	+	PUNCT
eajbcs-694	254	2	µ)2(1−r1)(1−	µ)2(1−r1)(1−	PROPN
eajbcs-694	254	3	(	(	PUNCT
eajbcs-694	254	4	r1	r1	NOUN
eajbcs-694	254	5	+	+	NOUN
eajbcs-694	254	6	r2	r2	PROPN
eajbcs-694	254	7	)	)	PUNCT
eajbcs-694	254	8	)	)	PUNCT
eajbcs-694	255	1	+	+	CCONJ
eajbcs-694	255	2	k2(δ	k2(δ	PROPN
eajbcs-694	255	3	+	+	SYM
eajbcs-694	255	4	µ)2(1−r1	µ)2(1−r1	PROPN
eajbcs-694	255	5	)	)	PUNCT
eajbcs-694	255	6	2	2	NUM
eajbcs-694	256	1	+	+	CCONJ
eajbcs-694	256	2	k1k2(δ	k1k2(δ	PROPN
eajbcs-694	256	3	+	+	CCONJ
eajbcs-694	256	4	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	256	5	)	)	PUNCT
eajbcs-694	257	1	+	+	NOUN
eajbcs-694	257	2	k21(δ	k21(δ	PROPN
eajbcs-694	257	3	+	+	X
eajbcs-694	257	4	µ)(1−r0	µ)(1−r0	PROPN
eajbcs-694	257	5	)	)	PUNCT
eajbcs-694	257	6	+	+	CCONJ
eajbcs-694	257	7	k1(k1	k1(k1	PROPN
eajbcs-694	257	8	+	+	CCONJ
eajbcs-694	257	9	k2)[k1(δ	k2)[k1(δ	PROPN
eajbcs-694	257	10	+	+	CCONJ
eajbcs-694	257	11	µ)r3	µ)r3	PROPN
eajbcs-694	258	1	+	+	CCONJ
eajbcs-694	258	2	k2(δ	k2(δ	PROPN
eajbcs-694	258	3	+	+	NUM
eajbcs-694	258	4	µ)(1−r1	µ)(1−r1	PROPN
eajbcs-694	258	5	)	)	PUNCT
eajbcs-694	259	1	+	+	CCONJ
eajbcs-694	259	2	k1k2	k1k2	X
eajbcs-694	259	3	]	]	X
eajbcs-694	259	4	.	.	PUNCT
eajbcs-694	260	1	if	if	SCONJ
eajbcs-694	260	2	r0	r0	NOUN
eajbcs-694	260	3	<	<	X
eajbcs-694	260	4	1	1	NUM
eajbcs-694	260	5	,	,	PUNCT
eajbcs-694	260	6	then	then	ADV
eajbcs-694	260	7	r1	r1	PROPN
eajbcs-694	260	8	,	,	PUNCT
eajbcs-694	260	9	r2	r2	PROPN
eajbcs-694	260	10	,	,	PUNCT
eajbcs-694	260	11	r3	r3	PROPN
eajbcs-694	260	12	and	and	CCONJ
eajbcs-694	260	13	r1	r1	PROPN
eajbcs-694	261	1	+	+	PROPN
eajbcs-694	261	2	r2	r2	PROPN
eajbcs-694	261	3	are	be	AUX
eajbcs-694	261	4	strictly	strictly	ADV
eajbcs-694	261	5	less	less	ADJ
eajbcs-694	261	6	than	than	ADP
eajbcs-694	261	7	one	one	NUM
eajbcs-694	261	8	.	.	PUNCT
eajbcs-694	262	1	since	since	SCONJ
eajbcs-694	262	2	r0	r0	NOUN
eajbcs-694	262	3	=	=	SYM
eajbcs-694	262	4	r1+r2+r3	r1+r2+r3	PROPN
eajbcs-694	262	5	.	.	PUNCT
eajbcs-694	263	1	the	the	DET
eajbcs-694	263	2	coefficients	coefficient	NOUN
eajbcs-694	263	3	a1	a1	PROPN
eajbcs-694	263	4	,	,	PUNCT
eajbcs-694	263	5	a2	a2	PROPN
eajbcs-694	263	6	and	and	CCONJ
eajbcs-694	263	7	a3	a3	NOUN
eajbcs-694	263	8	are	be	AUX
eajbcs-694	263	9	positive	positive	ADJ
eajbcs-694	263	10	and	and	CCONJ
eajbcs-694	263	11	a1a2	a1a2	INTJ
eajbcs-694	263	12	>	>	X
eajbcs-694	263	13	a3	a3	NOUN
eajbcs-694	263	14	if	if	SCONJ
eajbcs-694	263	15	r0	r0	NOUN
eajbcs-694	263	16	<	<	X
eajbcs-694	263	17	1	1	NUM
eajbcs-694	263	18	.	.	PUNCT
eajbcs-694	264	1	thus	thus	ADV
eajbcs-694	264	2	,	,	PUNCT
eajbcs-694	264	3	all	all	DET
eajbcs-694	264	4	the	the	DET
eajbcs-694	264	5	eigenvalues	eigenvalue	NOUN
eajbcs-694	264	6	of	of	ADP
eajbcs-694	264	7	j(e0	j(e0	NOUN
eajbcs-694	264	8	)	)	PUNCT
eajbcs-694	264	9	are	be	AUX
eajbcs-694	264	10	negative	negative	ADJ
eajbcs-694	264	11	.	.	PUNCT
eajbcs-694	265	1	it	it	PRON
eajbcs-694	265	2	follows	follow	VERB
eajbcs-694	265	3	by	by	ADP
eajbcs-694	265	4	routh	routh	PROPN
eajbcs-694	265	5	-	-	PUNCT
eajbcs-694	265	6	hurwitz	hurwitz	PROPN
eajbcs-694	265	7	criteria	criterion	NOUN
eajbcs-694	265	8	that	that	SCONJ
eajbcs-694	265	9	the	the	DET
eajbcs-694	265	10	disease	disease	NOUN
eajbcs-694	265	11	free	free	ADJ
eajbcs-694	265	12	equilibrium	equilibrium	NOUN
eajbcs-694	265	13	e0	e0	PROPN
eajbcs-694	265	14	is	be	AUX
eajbcs-694	265	15	locally	locally	ADV
eajbcs-694	265	16	asymptotically	asymptotically	ADV
eajbcs-694	265	17	stable	stable	ADJ
eajbcs-694	265	18	for	for	ADP
eajbcs-694	265	19	r0	r0	NOUN
eajbcs-694	265	20	<	<	X
eajbcs-694	265	21	1	1	NUM
eajbcs-694	265	22	.	.	PUNCT
eajbcs-694	266	1	if	if	SCONJ
eajbcs-694	266	2	r0	r0	NOUN
eajbcs-694	266	3	>	>	X
eajbcs-694	266	4	1	1	NUM
eajbcs-694	266	5	,	,	PUNCT
eajbcs-694	266	6	then	then	ADV
eajbcs-694	266	7	a3	a3	NOUN
eajbcs-694	266	8	is	be	AUX
eajbcs-694	266	9	negative	negative	ADJ
eajbcs-694	266	10	and	and	CCONJ
eajbcs-694	266	11	the	the	DET
eajbcs-694	266	12	routh	routh	PROPN
eajbcs-694	266	13	-	-	PUNCT
eajbcs-694	266	14	hurwitz	hurwitz	PROPN
eajbcs-694	266	15	criterion	criterion	NOUN
eajbcs-694	266	16	tells	tell	VERB
eajbcs-694	266	17	that	that	SCONJ
eajbcs-694	266	18	the	the	DET
eajbcs-694	266	19	disease	disease	NOUN
eajbcs-694	266	20	free	free	ADJ
eajbcs-694	266	21	equilibrium	equilibrium	NOUN
eajbcs-694	266	22	e0	e0	PROPN
eajbcs-694	266	23	is	be	AUX
eajbcs-694	266	24	unstable	unstable	ADJ
eajbcs-694	266	25	.	.	PUNCT
eajbcs-694	267	1	east	east	PROPN
eajbcs-694	267	2	afr	afr	PROPN
eajbcs-694	267	3	.	.	PUNCT
eajbcs-694	268	1	j.	j.	PROPN
eajbcs-694	268	2	biophys	biophys	PROPN
eajbcs-694	268	3	.	.	PUNCT
eajbcs-694	269	1	comput	comput	NOUN
eajbcs-694	269	2	.	.	PUNCT
eajbcs-694	270	1	sci	sci	PROPN
eajbcs-694	270	2	.	.	PUNCT
eajbcs-694	271	1	(	(	PUNCT
eajbcs-694	271	2	2024	2024	NUM
eajbcs-694	271	3	)	)	PUNCT
eajbcs-694	271	4	,	,	PUNCT
eajbcs-694	271	5	vol	vol	NOUN
eajbcs-694	271	6	.	.	PROPN
eajbcs-694	271	7	5	5	NUM
eajbcs-694	271	8	,	,	PUNCT
eajbcs-694	271	9	no	no	INTJ
eajbcs-694	271	10	.	.	NOUN
eajbcs-694	271	11	2	2	NUM
eajbcs-694	271	12	,	,	PUNCT
eajbcs-694	271	13	58	58	NUM
eajbcs-694	271	14	-	-	SYM
eajbcs-694	271	15	86	86	NUM
eajbcs-694	271	16	65	65	NUM
eajbcs-694	271	17	3.5	3.5	NUM
eajbcs-694	271	18	.	.	PUNCT
eajbcs-694	272	1	global	global	ADJ
eajbcs-694	272	2	stability	stability	NOUN
eajbcs-694	272	3	of	of	ADP
eajbcs-694	272	4	disease	disease	NOUN
eajbcs-694	272	5	free	free	ADJ
eajbcs-694	272	6	equilibrium	equilibrium	NOUN
eajbcs-694	272	7	theorem	theorem	VERB
eajbcs-694	272	8	3.4	3.4	NUM
eajbcs-694	272	9	.	.	PUNCT
eajbcs-694	273	1	for	for	ADP
eajbcs-694	273	2	r0	r0	NOUN
eajbcs-694	273	3	<	<	X
eajbcs-694	273	4	1	1	NUM
eajbcs-694	273	5	,	,	PUNCT
eajbcs-694	273	6	the	the	DET
eajbcs-694	273	7	disease	disease	NOUN
eajbcs-694	273	8	free	free	VERB
eajbcs-694	273	9	equilibrium	equilibrium	NOUN
eajbcs-694	273	10	e0	e0	PROPN
eajbcs-694	273	11	of	of	ADP
eajbcs-694	273	12	the	the	DET
eajbcs-694	273	13	system	system	NOUN
eajbcs-694	273	14	(	(	PUNCT
eajbcs-694	273	15	2	2	X
eajbcs-694	273	16	)	)	PUNCT
eajbcs-694	273	17	is	be	AUX
eajbcs-694	273	18	globally	globally	ADV
eajbcs-694	273	19	asymptotically	asymptotically	ADV
eajbcs-694	273	20	stable	stable	ADJ
eajbcs-694	273	21	if	if	SCONJ
eajbcs-694	273	22	s0	s0	PROPN
eajbcs-694	273	23	≥	≥	PROPN
eajbcs-694	273	24	s.	s.	PROPN
eajbcs-694	273	25	proof	proof	PROPN
eajbcs-694	273	26	.	.	PUNCT
eajbcs-694	274	1	let	let	VERB
eajbcs-694	274	2	us	we	PRON
eajbcs-694	274	3	rewrite	rewrite	VERB
eajbcs-694	274	4	our	our	PRON
eajbcs-694	274	5	model	model	NOUN
eajbcs-694	274	6	system	system	NOUN
eajbcs-694	274	7	(	(	PUNCT
eajbcs-694	274	8	2	2	X
eajbcs-694	274	9	)	)	PUNCT
eajbcs-694	274	10	as	as	ADP
eajbcs-694	274	11	dz1	dz1	ADJ
eajbcs-694	274	12	dt	dt	NOUN
eajbcs-694	274	13	=	=	SYM
eajbcs-694	274	14	f	f	PROPN
eajbcs-694	274	15	(	(	PUNCT
eajbcs-694	274	16	z1	z1	PROPN
eajbcs-694	274	17	,	,	PUNCT
eajbcs-694	274	18	z2	z2	PROPN
eajbcs-694	274	19	)	)	PUNCT
eajbcs-694	274	20	,	,	PUNCT
eajbcs-694	274	21	dz2	dz2	NOUN
eajbcs-694	274	22	dt	dt	NOUN
eajbcs-694	274	23	=	=	SYM
eajbcs-694	274	24	g(z1	g(z1	NOUN
eajbcs-694	274	25	,	,	PUNCT
eajbcs-694	274	26	z2	z2	PROPN
eajbcs-694	274	27	)	)	PUNCT
eajbcs-694	274	28	,	,	PUNCT
eajbcs-694	274	29	g(z1	g(z1	NOUN
eajbcs-694	274	30	,	,	PUNCT
eajbcs-694	274	31	0	0	NUM
eajbcs-694	274	32	)	)	PUNCT
eajbcs-694	274	33	=	=	SYM
eajbcs-694	275	1	0	0	X
eajbcs-694	275	2	.	.	PUNCT
eajbcs-694	275	3	where	where	SCONJ
eajbcs-694	275	4	z1	z1	NOUN
eajbcs-694	275	5	=	=	SYM
eajbcs-694	275	6	(	(	PUNCT
eajbcs-694	275	7	s	s	PROPN
eajbcs-694	275	8	,	,	PUNCT
eajbcs-694	275	9	p	p	X
eajbcs-694	275	10	,	,	PUNCT
eajbcs-694	275	11	r	r	NOUN
eajbcs-694	275	12	)	)	PUNCT
eajbcs-694	275	13	∈	∈	PROPN
eajbcs-694	275	14	r3	r3	PROPN
eajbcs-694	275	15	+	+	CCONJ
eajbcs-694	275	16	represents	represent	VERB
eajbcs-694	275	17	the	the	DET
eajbcs-694	275	18	class	class	NOUN
eajbcs-694	275	19	of	of	ADP
eajbcs-694	275	20	uninfected	uninfected	ADJ
eajbcs-694	275	21	individuals	individual	NOUN
eajbcs-694	275	22	and	and	CCONJ
eajbcs-694	275	23	z2	z2	PROPN
eajbcs-694	275	24	=	=	SYM
eajbcs-694	275	25	(	(	PUNCT
eajbcs-694	275	26	e	e	NOUN
eajbcs-694	275	27	,	,	PUNCT
eajbcs-694	275	28	i	i	PRON
eajbcs-694	275	29	,	,	PUNCT
eajbcs-694	275	30	h	h	NOUN
eajbcs-694	275	31	)	)	PUNCT
eajbcs-694	275	32	∈	∈	PROPN
eajbcs-694	275	33	r3	r3	PROPN
eajbcs-694	275	34	+	+	CCONJ
eajbcs-694	275	35	represents	represent	VERB
eajbcs-694	275	36	the	the	DET
eajbcs-694	275	37	class	class	NOUN
eajbcs-694	275	38	of	of	ADP
eajbcs-694	275	39	infected	infected	ADJ
eajbcs-694	275	40	individuals	individual	NOUN
eajbcs-694	275	41	.	.	PUNCT
eajbcs-694	276	1	the	the	DET
eajbcs-694	276	2	disease	disease	NOUN
eajbcs-694	276	3	free	free	VERB
eajbcs-694	276	4	equilibrium	equilibrium	NOUN
eajbcs-694	276	5	point	point	NOUN
eajbcs-694	276	6	of	of	ADP
eajbcs-694	276	7	the	the	DET
eajbcs-694	276	8	model	model	NOUN
eajbcs-694	276	9	is	be	AUX
eajbcs-694	276	10	denoted	denote	VERB
eajbcs-694	276	11	by	by	ADP
eajbcs-694	276	12	u0	u0	ADJ
eajbcs-694	276	13	=	=	SYM
eajbcs-694	276	14	(	(	PUNCT
eajbcs-694	276	15	z∗	z∗	NOUN
eajbcs-694	276	16	1	1	NUM
eajbcs-694	276	17	,	,	PUNCT
eajbcs-694	276	18	0	0	NUM
eajbcs-694	276	19	)	)	PUNCT
eajbcs-694	276	20	,	,	PUNCT
eajbcs-694	276	21	where	where	SCONJ
eajbcs-694	276	22	z∗	z∗	NOUN
eajbcs-694	276	23	1	1	NUM
eajbcs-694	276	24	=	=	SYM
eajbcs-694	276	25	(	(	PUNCT
eajbcs-694	276	26	π(ν+µ	π(ν+µ	PROPN
eajbcs-694	276	27	)	)	PUNCT
eajbcs-694	276	28	µ(θ+ν+µ	µ(θ+ν+µ	NOUN
eajbcs-694	276	29	)	)	PUNCT
eajbcs-694	276	30	,	,	PUNCT
eajbcs-694	276	31	πθ	πθ	ADP
eajbcs-694	276	32	µ(θ+ν+µ	µ(θ+ν+µ	VERB
eajbcs-694	276	33	)	)	PUNCT
eajbcs-694	276	34	,	,	PUNCT
eajbcs-694	276	35	0	0	NUM
eajbcs-694	276	36	)	)	PUNCT
eajbcs-694	276	37	.	.	PUNCT
eajbcs-694	277	1	since	since	SCONJ
eajbcs-694	277	2	the	the	DET
eajbcs-694	277	3	disease	disease	NOUN
eajbcs-694	277	4	free	free	ADJ
eajbcs-694	277	5	equilibrium	equilibrium	NOUN
eajbcs-694	277	6	point	point	NOUN
eajbcs-694	277	7	is	be	AUX
eajbcs-694	277	8	locally	locally	ADV
eajbcs-694	277	9	asymptotically	asymptotically	ADV
eajbcs-694	277	10	stable	stable	ADJ
eajbcs-694	277	11	(	(	PUNCT
eajbcs-694	277	12	see	see	NOUN
eajbcs-694	277	13	theorem	theorem	NOUN
eajbcs-694	277	14	(	(	PUNCT
eajbcs-694	277	15	3.3	3.3	NUM
eajbcs-694	277	16	)	)	PUNCT
eajbcs-694	277	17	)	)	PUNCT
eajbcs-694	277	18	,	,	PUNCT
eajbcs-694	277	19	to	to	PART
eajbcs-694	277	20	prove	prove	VERB
eajbcs-694	277	21	global	global	ADJ
eajbcs-694	277	22	stability	stability	NOUN
eajbcs-694	277	23	,	,	PUNCT
eajbcs-694	277	24	we	we	PRON
eajbcs-694	277	25	will	will	AUX
eajbcs-694	277	26	apply	apply	VERB
eajbcs-694	277	27	the	the	DET
eajbcs-694	277	28	castillo	castillo	PROPN
eajbcs-694	277	29	-	-	PUNCT
eajbcs-694	277	30	chavez	chavez	PROPN
eajbcs-694	277	31	theorem	theorem	VERB
eajbcs-694	277	32	castillo	castillo	PROPN
eajbcs-694	277	33	-	-	PUNCT
eajbcs-694	277	34	chavez	chavez	PROPN
eajbcs-694	277	35	et	et	PROPN
eajbcs-694	277	36	al	al	PROPN
eajbcs-694	277	37	.	.	PROPN
eajbcs-694	278	1	(	(	PUNCT
eajbcs-694	278	2	2002	2002	NUM
eajbcs-694	278	3	)	)	PUNCT
eajbcs-694	278	4	.	.	PUNCT
eajbcs-694	279	1	from	from	ADP
eajbcs-694	279	2	system	system	NOUN
eajbcs-694	279	3	(	(	PUNCT
eajbcs-694	279	4	2	2	NUM
eajbcs-694	279	5	)	)	PUNCT
eajbcs-694	279	6	,	,	PUNCT
eajbcs-694	279	7	we	we	PRON
eajbcs-694	279	8	have	have	AUX
eajbcs-694	279	9	dz1	dz1	VERB
eajbcs-694	279	10	dt	dt	NOUN
eajbcs-694	280	1	=	=	SYM
eajbcs-694	280	2	f	f	PROPN
eajbcs-694	280	3	(	(	PUNCT
eajbcs-694	280	4	z1	z1	PROPN
eajbcs-694	280	5	,	,	PUNCT
eajbcs-694	280	6	z2	z2	NUM
eajbcs-694	280	7	)	)	PUNCT
eajbcs-694	280	8	=	=	SYM
eajbcs-694	280	9			PROPN
eajbcs-694	280	10	π+	π+	PUNCT
eajbcs-694	280	11	ηωr	ηωr	PROPN
eajbcs-694	280	12	+	+	X
eajbcs-694	280	13	νp	νp	ADP
eajbcs-694	280	14	−	−	PROPN
eajbcs-694	280	15	β(σ1i	β(σ1i	NOUN
eajbcs-694	280	16	+	+	NUM
eajbcs-694	280	17	σ2e	σ2e	PROPN
eajbcs-694	280	18	+	+	CCONJ
eajbcs-694	280	19	σ3h)s	σ3h)s	X
eajbcs-694	280	20	−	−	PROPN
eajbcs-694	280	21	(	(	PUNCT
eajbcs-694	280	22	θ	θ	NOUN
eajbcs-694	280	23	+	+	NOUN
eajbcs-694	280	24	µ)s	µ)s	NOUN
eajbcs-694	280	25	θs	θs	X
eajbcs-694	280	26	+	+	X
eajbcs-694	280	27	(	(	PUNCT
eajbcs-694	280	28	1−	1−	NUM
eajbcs-694	280	29	η)ωr−	η)ωr−	PROPN
eajbcs-694	280	30	(	(	PUNCT
eajbcs-694	280	31	ν	ν	X
eajbcs-694	280	32	+	+	NOUN
eajbcs-694	280	33	µ)p	µ)p	NOUN
eajbcs-694	280	34	(	(	PUNCT
eajbcs-694	280	35	1−	1−	NUM
eajbcs-694	280	36	τ)δe	τ)δe	PROPN
eajbcs-694	280	37	+	+	CCONJ
eajbcs-694	280	38	εi	εi	VERB
eajbcs-694	280	39	+	+	X
eajbcs-694	280	40	γh	γh	ADP
eajbcs-694	280	41	−	−	PROPN
eajbcs-694	280	42	(	(	PUNCT
eajbcs-694	280	43	ω	ω	NOUN
eajbcs-694	280	44	+	+	NOUN
eajbcs-694	280	45	µ)r	µ)r	NOUN
eajbcs-694	280	46			NOUN
eajbcs-694	280	47	,	,	PUNCT
eajbcs-694	280	48	dz2	dz2	NOUN
eajbcs-694	280	49	dt	dt	NOUN
eajbcs-694	280	50	=	=	SYM
eajbcs-694	280	51	g(z1	g(z1	NOUN
eajbcs-694	280	52	,	,	PUNCT
eajbcs-694	280	53	z2	z2	PROPN
eajbcs-694	280	54	)	)	PUNCT
eajbcs-694	280	55	=	=	SYM
eajbcs-694	280	56			NOUN
eajbcs-694	280	57	β(σ1i	β(σ1i	PUNCT
eajbcs-694	280	58	+	+	NUM
eajbcs-694	280	59	σ2e	σ2e	PROPN
eajbcs-694	280	60	+	+	CCONJ
eajbcs-694	280	61	σ3h)s	σ3h)s	X
eajbcs-694	280	62	−	−	PROPN
eajbcs-694	280	63	(	(	PUNCT
eajbcs-694	280	64	δ	δ	NOUN
eajbcs-694	280	65	+	+	NOUN
eajbcs-694	280	66	µ)e	µ)e	NOUN
eajbcs-694	280	67	τδe	τδe	NOUN
eajbcs-694	280	68	−	−	PROPN
eajbcs-694	280	69	(	(	PUNCT
eajbcs-694	280	70	α	α	PROPN
eajbcs-694	280	71	+	+	SYM
eajbcs-694	280	72	ε+	ε+	X
eajbcs-694	280	73	µ+	µ+	X
eajbcs-694	280	74	ρ)i	ρ)i	NOUN
eajbcs-694	280	75	αi	αi	VERB
eajbcs-694	280	76	−	−	PROPN
eajbcs-694	281	1	(	(	PUNCT
eajbcs-694	281	2	γ	γ	X
eajbcs-694	281	3	+	+	X
eajbcs-694	281	4	µ+	µ+	PUNCT
eajbcs-694	281	5	ξ)h	ξ)h	ADJ
eajbcs-694	281	6			NOUN
eajbcs-694	281	7	.	.	PUNCT
eajbcs-694	282	1	i.	i.	NOUN
eajbcs-694	282	2	to	to	PART
eajbcs-694	282	3	show	show	VERB
eajbcs-694	282	4	z∗	z∗	PROPN
eajbcs-694	282	5	1	1	NUM
eajbcs-694	282	6	is	be	AUX
eajbcs-694	282	7	globally	globally	ADV
eajbcs-694	282	8	asymptotically	asymptotically	ADV
eajbcs-694	282	9	stable	stable	ADJ
eajbcs-694	282	10	for	for	ADP
eajbcs-694	282	11	the	the	DET
eajbcs-694	282	12	system	system	NOUN
eajbcs-694	282	13	dz1	dz1	VERB
eajbcs-694	282	14	dt	dt	NOUN
eajbcs-694	282	15	=	=	SYM
eajbcs-694	282	16	f	f	PROPN
eajbcs-694	282	17	(	(	PUNCT
eajbcs-694	282	18	z1	z1	PROPN
eajbcs-694	282	19	,	,	PUNCT
eajbcs-694	282	20	0	0	NUM
eajbcs-694	282	21	)	)	PUNCT
eajbcs-694	282	22	,	,	PUNCT
eajbcs-694	282	23	let	let	VERB
eajbcs-694	282	24	us	we	PRON
eajbcs-694	282	25	consider	consider	VERB
eajbcs-694	282	26	the	the	DET
eajbcs-694	282	27	reduced	reduce	VERB
eajbcs-694	282	28	system	system	NOUN
eajbcs-694	282	29	dz1	dz1	VERB
eajbcs-694	283	1	dt	dt	NOUN
eajbcs-694	283	2	=	=	SYM
eajbcs-694	283	3	f	f	PROPN
eajbcs-694	283	4	(	(	PUNCT
eajbcs-694	283	5	z1	z1	PROPN
eajbcs-694	283	6	,	,	PUNCT
eajbcs-694	283	7	0	0	NUM
eajbcs-694	283	8	)	)	PUNCT
eajbcs-694	283	9	=	=	SYM
eajbcs-694	283	10			PROPN
eajbcs-694	283	11	π+	π+	PUNCT
eajbcs-694	283	12	ηωr	ηωr	PROPN
eajbcs-694	283	13	+	+	X
eajbcs-694	283	14	νp	νp	ADP
eajbcs-694	283	15	−	−	PROPN
eajbcs-694	283	16	(	(	PUNCT
eajbcs-694	283	17	θ	θ	NOUN
eajbcs-694	283	18	+	+	NOUN
eajbcs-694	283	19	µ)s	µ)s	NOUN
eajbcs-694	283	20	θs	θs	X
eajbcs-694	283	21	+	+	X
eajbcs-694	283	22	(	(	PUNCT
eajbcs-694	283	23	1−	1−	NUM
eajbcs-694	283	24	η)ωr−	η)ωr−	PROPN
eajbcs-694	283	25	(	(	PUNCT
eajbcs-694	283	26	ν	ν	X
eajbcs-694	283	27	+	+	NUM
eajbcs-694	283	28	µ)p	µ)p	NOUN
eajbcs-694	283	29	−(ω	−(ω	NOUN
eajbcs-694	283	30	+	+	NOUN
eajbcs-694	283	31	µ)r	µ)r	NOUN
eajbcs-694	283	32			NOUN
eajbcs-694	283	33	.	.	PUNCT
eajbcs-694	284	1	(	(	PUNCT
eajbcs-694	284	2	5	5	X
eajbcs-694	284	3	)	)	PUNCT
eajbcs-694	284	4	we	we	PRON
eajbcs-694	284	5	can	can	AUX
eajbcs-694	284	6	rewrite	rewrite	VERB
eajbcs-694	284	7	the	the	DET
eajbcs-694	284	8	system	system	NOUN
eajbcs-694	284	9	(	(	PUNCT
eajbcs-694	284	10	5	5	NUM
eajbcs-694	284	11	)	)	PUNCT
eajbcs-694	284	12	as	as	ADP
eajbcs-694	284	13	:	:	PUNCT
eajbcs-694	284	14	ds	ds	ADJ
eajbcs-694	284	15	dt	dt	NOUN
eajbcs-694	284	16	=	=	PUNCT
eajbcs-694	284	17	−(θ	−(θ	NOUN
eajbcs-694	284	18	+	+	CCONJ
eajbcs-694	284	19	µ)s	µ)s	NOUN
eajbcs-694	284	20	+	+	CCONJ
eajbcs-694	284	21	νp	νp	X
eajbcs-694	285	1	+	+	CCONJ
eajbcs-694	285	2	ηωr	ηωr	PROPN
eajbcs-694	285	3	+	+	PROPN
eajbcs-694	285	4	π	π	PROPN
eajbcs-694	285	5	,	,	PUNCT
eajbcs-694	285	6	(	(	PUNCT
eajbcs-694	285	7	61	61	NUM
eajbcs-694	285	8	)	)	PUNCT
eajbcs-694	285	9	dp	dp	NOUN
eajbcs-694	285	10	dt	dt	NOUN
eajbcs-694	285	11	=	=	SYM
eajbcs-694	285	12	θs	θs	X
eajbcs-694	285	13	−	−	PROPN
eajbcs-694	285	14	(	(	PUNCT
eajbcs-694	285	15	ν	ν	X
eajbcs-694	285	16	+	+	NOUN
eajbcs-694	285	17	µ)p	µ)p	NOUN
eajbcs-694	285	18	+	+	CCONJ
eajbcs-694	285	19	(	(	PUNCT
eajbcs-694	285	20	1−	1−	NUM
eajbcs-694	285	21	η)ωr	η)ωr	PROPN
eajbcs-694	285	22	,	,	PUNCT
eajbcs-694	285	23	(	(	PUNCT
eajbcs-694	285	24	62	62	NUM
eajbcs-694	285	25	)	)	PUNCT
eajbcs-694	285	26	dr	dr	NOUN
eajbcs-694	285	27	dt	dt	NOUN
eajbcs-694	285	28	=	=	PUNCT
eajbcs-694	285	29	−(ω	−(ω	PROPN
eajbcs-694	285	30	+	+	CCONJ
eajbcs-694	285	31	µ)r	µ)r	NOUN
eajbcs-694	285	32	,	,	PUNCT
eajbcs-694	285	33	(	(	PUNCT
eajbcs-694	285	34	63	63	NUM
eajbcs-694	285	35	)	)	PUNCT
eajbcs-694	285	36	east	east	PROPN
eajbcs-694	285	37	afr	afr	PROPN
eajbcs-694	285	38	.	.	PUNCT
eajbcs-694	286	1	j.	j.	PROPN
eajbcs-694	286	2	biophys	biophys	PROPN
eajbcs-694	286	3	.	.	PUNCT
eajbcs-694	287	1	comput	comput	NOUN
eajbcs-694	287	2	.	.	PUNCT
eajbcs-694	288	1	sci	sci	PROPN
eajbcs-694	288	2	.	.	PUNCT
eajbcs-694	289	1	(	(	PUNCT
eajbcs-694	289	2	2024	2024	NUM
eajbcs-694	289	3	)	)	PUNCT
eajbcs-694	289	4	,	,	PUNCT
eajbcs-694	289	5	vol	vol	NOUN
eajbcs-694	289	6	.	.	PROPN
eajbcs-694	289	7	5	5	NUM
eajbcs-694	289	8	,	,	PUNCT
eajbcs-694	289	9	no	no	INTJ
eajbcs-694	289	10	.	.	NOUN
eajbcs-694	289	11	2	2	NUM
eajbcs-694	289	12	,	,	PUNCT
eajbcs-694	289	13	58	58	NUM
eajbcs-694	289	14	-	-	SYM
eajbcs-694	289	15	86	86	NUM
eajbcs-694	289	16	66	66	NUM
eajbcs-694	289	17	and	and	CCONJ
eajbcs-694	289	18	admits	admit	VERB
eajbcs-694	289	19	as	as	ADP
eajbcs-694	289	20	solutions	solution	NOUN
eajbcs-694	289	21	s(t	s(t	PROPN
eajbcs-694	289	22	)	)	PUNCT
eajbcs-694	289	23	=	=	PUNCT
eajbcs-694	290	1	π(ν	π(ν	PROPN
eajbcs-694	290	2	+	+	NUM
eajbcs-694	290	3	µ	µ	X
eajbcs-694	290	4	)	)	PUNCT
eajbcs-694	290	5	µ(θ	µ(θ	ADJ
eajbcs-694	290	6	+	+	NUM
eajbcs-694	290	7	ν	ν	X
eajbcs-694	290	8	+	+	X
eajbcs-694	290	9	µ	µ	X
eajbcs-694	290	10	)	)	PUNCT
eajbcs-694	290	11	+	+	CCONJ
eajbcs-694	290	12	1	1	NUM
eajbcs-694	290	13	θ	θ	NOUN
eajbcs-694	290	14	+	+	CCONJ
eajbcs-694	290	15	ν	ν	X
eajbcs-694	290	16	[	[	X
eajbcs-694	290	17	(	(	PUNCT
eajbcs-694	290	18	ν(s(0	ν(s(0	ADP
eajbcs-694	290	19	)	)	PUNCT
eajbcs-694	290	20	+	+	X
eajbcs-694	290	21	p	p	X
eajbcs-694	290	22	(	(	PUNCT
eajbcs-694	290	23	0	0	NUM
eajbcs-694	290	24	)	)	PUNCT
eajbcs-694	291	1	+	+	NOUN
eajbcs-694	291	2	r(0))−	r(0))−	NOUN
eajbcs-694	291	3	πν	πν	ADP
eajbcs-694	291	4	µ	µ	NOUN
eajbcs-694	291	5	)	)	PUNCT
eajbcs-694	291	6	e−µt	e−µt	PROPN
eajbcs-694	291	7	+	+	CCONJ
eajbcs-694	291	8	(	(	PUNCT
eajbcs-694	291	9	θs(0)−	θs(0)−	PRON
eajbcs-694	291	10	νp	νp	X
eajbcs-694	291	11	(	(	PUNCT
eajbcs-694	291	12	0	0	NUM
eajbcs-694	291	13	)	)	PUNCT
eajbcs-694	292	1	+	+	CCONJ
eajbcs-694	292	2	ω(ν	ω(ν	ADP
eajbcs-694	292	3	−	−	NOUN
eajbcs-694	292	4	η(θ	η(θ	ADJ
eajbcs-694	292	5	+	+	NUM
eajbcs-694	292	6	ν	ν	NOUN
eajbcs-694	292	7	)	)	PUNCT
eajbcs-694	292	8	)	)	PUNCT
eajbcs-694	293	1	θ	θ	PROPN
eajbcs-694	294	1	+	+	CCONJ
eajbcs-694	294	2	ν	ν	X
eajbcs-694	294	3	−	−	PROPN
eajbcs-694	294	4	ω	ω	NUM
eajbcs-694	294	5	r(0)−	r(0)−	NOUN
eajbcs-694	294	6	πθ	πθ	ADP
eajbcs-694	294	7	θ	θ	PROPN
eajbcs-694	294	8	+	+	CCONJ
eajbcs-694	294	9	ν	ν	X
eajbcs-694	294	10	+	+	CCONJ
eajbcs-694	294	11	µ	µ	X
eajbcs-694	294	12	)	)	PUNCT
eajbcs-694	294	13	e−(θ+ν+µ)t	e−(θ+ν+µ)t	NOUN
eajbcs-694	294	14	]	]	PUNCT
eajbcs-694	294	15	−	−	PROPN
eajbcs-694	294	16	(	(	PUNCT
eajbcs-694	294	17	ν	ν	NOUN
eajbcs-694	294	18	−	−	PROPN
eajbcs-694	294	19	ηω	ηω	NOUN
eajbcs-694	294	20	)	)	PUNCT
eajbcs-694	294	21	θ	θ	PROPN
eajbcs-694	295	1	+	+	CCONJ
eajbcs-694	295	2	ν	ν	X
eajbcs-694	295	3	−	−	PROPN
eajbcs-694	295	4	ω	ω	NUM
eajbcs-694	295	5	r(0)e−(ω+µ)t	r(0)e−(ω+µ)t	PROPN
eajbcs-694	295	6	,	,	PUNCT
eajbcs-694	295	7	p	p	X
eajbcs-694	295	8	(	(	PUNCT
eajbcs-694	295	9	t	t	NOUN
eajbcs-694	295	10	)	)	PUNCT
eajbcs-694	296	1	=	=	PUNCT
eajbcs-694	296	2	πθ	πθ	ADP
eajbcs-694	296	3	µ(θ	µ(θ	ADJ
eajbcs-694	296	4	+	+	CCONJ
eajbcs-694	296	5	ν	ν	X
eajbcs-694	296	6	+	+	X
eajbcs-694	296	7	µ	µ	X
eajbcs-694	296	8	)	)	PUNCT
eajbcs-694	296	9	+	+	CCONJ
eajbcs-694	296	10	1	1	NUM
eajbcs-694	296	11	θ	θ	NOUN
eajbcs-694	297	1	+	+	CCONJ
eajbcs-694	297	2	ν	ν	X
eajbcs-694	297	3	[	[	X
eajbcs-694	297	4	(	(	PUNCT
eajbcs-694	297	5	θ(s(0	θ(s(0	NUM
eajbcs-694	297	6	)	)	PUNCT
eajbcs-694	297	7	+	+	NOUN
eajbcs-694	297	8	p	p	X
eajbcs-694	297	9	(	(	PUNCT
eajbcs-694	297	10	0	0	NUM
eajbcs-694	297	11	)	)	PUNCT
eajbcs-694	297	12	+	+	NOUN
eajbcs-694	297	13	r(0))−	r(0))−	NOUN
eajbcs-694	297	14	πθ	πθ	ADP
eajbcs-694	297	15	µ	µ	X
eajbcs-694	297	16	)	)	PUNCT
eajbcs-694	297	17	e−µt	e−µt	PROPN
eajbcs-694	297	18	−	−	PROPN
eajbcs-694	297	19	(	(	PUNCT
eajbcs-694	297	20	θs(0)−	θs(0)−	NOUN
eajbcs-694	297	21	νp	νp	X
eajbcs-694	297	22	(	(	PUNCT
eajbcs-694	297	23	0	0	NUM
eajbcs-694	297	24	)	)	PUNCT
eajbcs-694	297	25	+	+	CCONJ
eajbcs-694	297	26	ω(ν	ω(ν	ADP
eajbcs-694	297	27	−	−	NOUN
eajbcs-694	297	28	η(θ	η(θ	ADJ
eajbcs-694	297	29	+	+	NUM
eajbcs-694	297	30	ν	ν	NOUN
eajbcs-694	297	31	)	)	PUNCT
eajbcs-694	297	32	)	)	PUNCT
eajbcs-694	298	1	θ	θ	PROPN
eajbcs-694	299	1	+	+	CCONJ
eajbcs-694	299	2	ν	ν	X
eajbcs-694	299	3	−	−	PROPN
eajbcs-694	299	4	ω	ω	NUM
eajbcs-694	299	5	r(0)−	r(0)−	NOUN
eajbcs-694	299	6	πθ	πθ	ADP
eajbcs-694	299	7	θ	θ	PROPN
eajbcs-694	299	8	+	+	CCONJ
eajbcs-694	299	9	ν	ν	X
eajbcs-694	299	10	+	+	CCONJ
eajbcs-694	299	11	µ	µ	X
eajbcs-694	299	12	)	)	PUNCT
eajbcs-694	299	13	e−(θ+ν+µ)t	e−(θ+ν+µ)t	NOUN
eajbcs-694	299	14	]	]	PUNCT
eajbcs-694	299	15	−	−	PROPN
eajbcs-694	300	1	(	(	PUNCT
eajbcs-694	300	2	θ	θ	X
eajbcs-694	300	3	−	−	PROPN
eajbcs-694	300	4	(	(	PUNCT
eajbcs-694	300	5	1−	1−	NUM
eajbcs-694	300	6	η)ω	η)ω	ADJ
eajbcs-694	300	7	)	)	PUNCT
eajbcs-694	300	8	θ	θ	PROPN
eajbcs-694	301	1	+	+	PUNCT
eajbcs-694	301	2	ν	ν	X
eajbcs-694	301	3	−	−	PROPN
eajbcs-694	301	4	ω	ω	NUM
eajbcs-694	301	5	r(0)e−(ω+µ)t	r(0)e−(ω+µ)t	PROPN
eajbcs-694	301	6	,	,	PUNCT
eajbcs-694	301	7	r(t	r(t	NOUN
eajbcs-694	301	8	)	)	PUNCT
eajbcs-694	301	9	=	=	SYM
eajbcs-694	302	1	r(0)e−(ω+µ)t	r(0)e−(ω+µ)t	NOUN
eajbcs-694	302	2	.	.	PUNCT
eajbcs-694	303	1	taking	take	VERB
eajbcs-694	303	2	the	the	DET
eajbcs-694	303	3	limit	limit	NOUN
eajbcs-694	303	4	as	as	SCONJ
eajbcs-694	303	5	t	t	PROPN
eajbcs-694	303	6	goes	go	VERB
eajbcs-694	303	7	to	to	ADP
eajbcs-694	303	8	∞	∞	PROPN
eajbcs-694	303	9	,	,	PUNCT
eajbcs-694	303	10	we	we	PRON
eajbcs-694	303	11	obtain	obtain	VERB
eajbcs-694	303	12	(	(	PUNCT
eajbcs-694	303	13	s(t	s(t	PROPN
eajbcs-694	303	14	)	)	PUNCT
eajbcs-694	303	15	,	,	PUNCT
eajbcs-694	303	16	p	p	X
eajbcs-694	303	17	(	(	PUNCT
eajbcs-694	303	18	t	t	PROPN
eajbcs-694	303	19	)	)	PUNCT
eajbcs-694	303	20	,	,	PUNCT
eajbcs-694	303	21	r(t	r(t	NOUN
eajbcs-694	303	22	)	)	PUNCT
eajbcs-694	303	23	)	)	PUNCT
eajbcs-694	304	1	→	→	PUNCT
eajbcs-694	304	2	(	(	PUNCT
eajbcs-694	304	3	π(ν	π(ν	X
eajbcs-694	304	4	+	+	NUM
eajbcs-694	304	5	µ	µ	X
eajbcs-694	304	6	)	)	PUNCT
eajbcs-694	304	7	µ(θ	µ(θ	ADJ
eajbcs-694	304	8	+	+	NUM
eajbcs-694	304	9	ν	ν	X
eajbcs-694	304	10	+	+	X
eajbcs-694	304	11	µ	µ	X
eajbcs-694	304	12	)	)	PUNCT
eajbcs-694	304	13	,	,	PUNCT
eajbcs-694	304	14	πθ	πθ	ADP
eajbcs-694	304	15	µ(θ	µ(θ	ADJ
eajbcs-694	304	16	+	+	CCONJ
eajbcs-694	304	17	ν	ν	X
eajbcs-694	304	18	+	+	X
eajbcs-694	304	19	µ	µ	X
eajbcs-694	304	20	)	)	PUNCT
eajbcs-694	304	21	,	,	PUNCT
eajbcs-694	304	22	0	0	NUM
eajbcs-694	304	23	)	)	PUNCT
eajbcs-694	304	24	=	=	SYM
eajbcs-694	304	25	z∗	z∗	NOUN
eajbcs-694	304	26	1	1	NUM
eajbcs-694	304	27	.	.	PUNCT
eajbcs-694	305	1	therefore	therefore	ADV
eajbcs-694	305	2	,	,	PUNCT
eajbcs-694	305	3	z∗	z∗	PROPN
eajbcs-694	305	4	1	1	NUM
eajbcs-694	305	5	is	be	AUX
eajbcs-694	305	6	globally	globally	ADV
eajbcs-694	305	7	asymptotically	asymptotically	ADV
eajbcs-694	305	8	stable	stable	ADJ
eajbcs-694	305	9	for	for	ADP
eajbcs-694	305	10	the	the	DET
eajbcs-694	305	11	system	system	NOUN
eajbcs-694	305	12	dz1	dz1	VERB
eajbcs-694	305	13	dt	dt	NOUN
eajbcs-694	305	14	=	=	SYM
eajbcs-694	305	15	f	f	PROPN
eajbcs-694	305	16	(	(	PUNCT
eajbcs-694	305	17	z1	z1	PROPN
eajbcs-694	305	18	,	,	PUNCT
eajbcs-694	305	19	0	0	NUM
eajbcs-694	305	20	)	)	PUNCT
eajbcs-694	305	21	.	.	PUNCT
eajbcs-694	306	1	ii	ii	PROPN
eajbcs-694	306	2	.	.	PUNCT
eajbcs-694	307	1	we	we	PRON
eajbcs-694	307	2	will	will	AUX
eajbcs-694	307	3	show	show	VERB
eajbcs-694	307	4	that	that	DET
eajbcs-694	307	5	g(z1	g(z1	NOUN
eajbcs-694	307	6	,	,	PUNCT
eajbcs-694	307	7	z2	z2	PROPN
eajbcs-694	307	8	)	)	PUNCT
eajbcs-694	307	9	=	=	PUNCT
eajbcs-694	307	10	az2	az2	PROPN
eajbcs-694	308	1	−	−	PROPN
eajbcs-694	308	2	ĝ(z1	ĝ(z1	PROPN
eajbcs-694	308	3	,	,	PUNCT
eajbcs-694	308	4	z2	z2	NOUN
eajbcs-694	308	5	)	)	PUNCT
eajbcs-694	308	6	,	,	PUNCT
eajbcs-694	308	7	ĝ(z1	ĝ(z1	NOUN
eajbcs-694	308	8	,	,	PUNCT
eajbcs-694	308	9	z2	z2	PROPN
eajbcs-694	308	10	)	)	PUNCT
eajbcs-694	308	11	≥	≥	NOUN
eajbcs-694	308	12	0	0	NUM
eajbcs-694	308	13	for	for	ADP
eajbcs-694	308	14	(	(	PUNCT
eajbcs-694	308	15	z1	z1	ADJ
eajbcs-694	308	16	,	,	PUNCT
eajbcs-694	308	17	z2	z2	NUM
eajbcs-694	308	18	)	)	PUNCT
eajbcs-694	308	19	∈	∈	PROPN
eajbcs-694	308	20	ω	ω	NUM
eajbcs-694	308	21	where	where	SCONJ
eajbcs-694	308	22	a	a	DET
eajbcs-694	308	23	=	=	SYM
eajbcs-694	308	24	∂g	∂g	PROPN
eajbcs-694	308	25	∂z2	∂z2	PROPN
eajbcs-694	308	26	(	(	PUNCT
eajbcs-694	308	27	z∗	z∗	NOUN
eajbcs-694	308	28	1	1	NUM
eajbcs-694	308	29	,	,	PUNCT
eajbcs-694	308	30	0	0	NUM
eajbcs-694	308	31	)	)	PUNCT
eajbcs-694	308	32	is	be	AUX
eajbcs-694	308	33	a	a	DET
eajbcs-694	308	34	metzler	metzler	NOUN
eajbcs-694	308	35	matrix	matrix	NOUN
eajbcs-694	308	36	(	(	PUNCT
eajbcs-694	308	37	the	the	DET
eajbcs-694	308	38	off	off	ADJ
eajbcs-694	308	39	diagonal	diagonal	ADJ
eajbcs-694	308	40	elements	element	NOUN
eajbcs-694	308	41	of	of	ADP
eajbcs-694	308	42	a	a	PRON
eajbcs-694	308	43	are	be	AUX
eajbcs-694	308	44	non	non	ADJ
eajbcs-694	308	45	-	-	ADJ
eajbcs-694	308	46	negative	negative	ADJ
eajbcs-694	308	47	)	)	PUNCT
eajbcs-694	308	48	and	and	CCONJ
eajbcs-694	308	49	ω	ω	PROPN
eajbcs-694	308	50	is	be	AUX
eajbcs-694	308	51	the	the	DET
eajbcs-694	308	52	region	region	NOUN
eajbcs-694	308	53	where	where	SCONJ
eajbcs-694	308	54	the	the	DET
eajbcs-694	308	55	model	model	NOUN
eajbcs-694	308	56	makes	make	VERB
eajbcs-694	308	57	biological	biological	ADJ
eajbcs-694	308	58	sense	sense	NOUN
eajbcs-694	308	59	.	.	PUNCT
eajbcs-694	309	1	consider	consider	VERB
eajbcs-694	309	2	a	a	DET
eajbcs-694	309	3	matrix	matrix	NOUN
eajbcs-694	309	4	a	a	DET
eajbcs-694	309	5	=	=	SYM
eajbcs-694	309	6	∂g	∂g	PROPN
eajbcs-694	309	7	∂z2	∂z2	PROPN
eajbcs-694	309	8	(	(	PUNCT
eajbcs-694	309	9	z∗	z∗	NOUN
eajbcs-694	309	10	1	1	NUM
eajbcs-694	309	11	,	,	PUNCT
eajbcs-694	309	12	0	0	NUM
eajbcs-694	309	13	)	)	PUNCT
eajbcs-694	309	14	=	=	PUNCT
eajbcs-694	309	15			NOUN
eajbcs-694	309	16	βσ2s	βσ2s	PROPN
eajbcs-694	309	17	0	0	NUM
eajbcs-694	310	1	−	−	PROPN
eajbcs-694	311	1	(	(	PUNCT
eajbcs-694	311	2	δ	δ	PROPN
eajbcs-694	311	3	+	+	X
eajbcs-694	311	4	µ	µ	X
eajbcs-694	311	5	)	)	PUNCT
eajbcs-694	311	6	βσ1s	βσ1s	PROPN
eajbcs-694	311	7	0	0	NUM
eajbcs-694	311	8	βσ3s	βσ3s	ADJ
eajbcs-694	311	9	0	0	PUNCT
eajbcs-694	312	1	τδ	τδ	ADP
eajbcs-694	312	2	−(α	−(α	ADJ
eajbcs-694	312	3	+	+	CCONJ
eajbcs-694	312	4	ε+	ε+	X
eajbcs-694	312	5	µ+	µ+	X
eajbcs-694	312	6	ρ	ρ	NOUN
eajbcs-694	312	7	)	)	PUNCT
eajbcs-694	312	8	0	0	NUM
eajbcs-694	312	9	0	0	NUM
eajbcs-694	313	1	α	α	PRON
eajbcs-694	313	2	−(γ	−(γ	ADJ
eajbcs-694	313	3	+	+	CCONJ
eajbcs-694	313	4	µ+	µ+	X
eajbcs-694	313	5	ξ	ξ	NOUN
eajbcs-694	313	6	)	)	PUNCT
eajbcs-694	313	7			NOUN
eajbcs-694	313	8	.	.	PUNCT
eajbcs-694	314	1	hence	hence	ADV
eajbcs-694	314	2	,	,	PUNCT
eajbcs-694	314	3	a	a	PRON
eajbcs-694	314	4	is	be	AUX
eajbcs-694	314	5	a	a	DET
eajbcs-694	314	6	metzer	metzer	ADJ
eajbcs-694	314	7	matrix	matrix	NOUN
eajbcs-694	314	8	(	(	PUNCT
eajbcs-694	314	9	off	off	ADP
eajbcs-694	314	10	diagonal	diagonal	ADJ
eajbcs-694	314	11	elements	element	NOUN
eajbcs-694	314	12	are	be	AUX
eajbcs-694	314	13	non	non	ADJ
eajbcs-694	314	14	-	-	ADJ
eajbcs-694	314	15	negative	negative	ADJ
eajbcs-694	314	16	)	)	PUNCT
eajbcs-694	314	17	.	.	PUNCT
eajbcs-694	315	1	here	here	ADV
eajbcs-694	315	2	,	,	PUNCT
eajbcs-694	315	3	ĝ(z1	ĝ(z1	NOUN
eajbcs-694	315	4	,	,	PUNCT
eajbcs-694	315	5	z2	z2	NUM
eajbcs-694	315	6	)	)	PUNCT
eajbcs-694	315	7	=	=	PUNCT
eajbcs-694	316	1	az2	az2	PROPN
eajbcs-694	316	2	−g(z1	−g(z1	PROPN
eajbcs-694	316	3	,	,	PUNCT
eajbcs-694	316	4	z2	z2	PROPN
eajbcs-694	316	5	)	)	PUNCT
eajbcs-694	316	6	.	.	PUNCT
eajbcs-694	317	1	after	after	ADP
eajbcs-694	317	2	some	some	DET
eajbcs-694	317	3	simplification	simplification	NOUN
eajbcs-694	317	4	,	,	PUNCT
eajbcs-694	317	5	we	we	PRON
eajbcs-694	317	6	obtain	obtain	VERB
eajbcs-694	317	7	ĝ(z1	ĝ(z1	NOUN
eajbcs-694	317	8	,	,	PUNCT
eajbcs-694	317	9	z2	z2	NUM
eajbcs-694	317	10	)	)	PUNCT
eajbcs-694	317	11	=	=	PUNCT
eajbcs-694	317	12			PROPN
eajbcs-694	317	13	βσ2e(s	βσ2e(s	NUM
eajbcs-694	317	14	0	0	NUM
eajbcs-694	318	1	−	−	PROPN
eajbcs-694	318	2	s	s	X
eajbcs-694	318	3	)	)	PUNCT
eajbcs-694	318	4	+	+	CCONJ
eajbcs-694	318	5	βσ1i(s	βσ1i(s	SYM
eajbcs-694	318	6	0	0	NUM
eajbcs-694	318	7	−	−	NUM
eajbcs-694	318	8	s	s	PART
eajbcs-694	318	9	)	)	PUNCT
eajbcs-694	318	10	+	+	NUM
eajbcs-694	318	11	βσ3h(s0	βσ3h(s0	NUM
eajbcs-694	318	12	−	−	NUM
eajbcs-694	318	13	s	s	PART
eajbcs-694	318	14	)	)	PUNCT
eajbcs-694	318	15	0	0	NUM
eajbcs-694	318	16	0	0	NUM
eajbcs-694	318	17			NOUN
eajbcs-694	318	18	,	,	PUNCT
eajbcs-694	318	19	ĝ(z1	ĝ(z1	NOUN
eajbcs-694	318	20	,	,	PUNCT
eajbcs-694	318	21	z2	z2	NOUN
eajbcs-694	318	22	)	)	PUNCT
eajbcs-694	318	23	=	=	PUNCT
eajbcs-694	318	24	(	(	PUNCT
eajbcs-694	318	25	s0	s0	PROPN
eajbcs-694	318	26	−	−	PROPN
eajbcs-694	318	27	s	s	PART
eajbcs-694	318	28	)	)	PUNCT
eajbcs-694	318	29			PROPN
eajbcs-694	318	30	β(σ2e	β(σ2e	PUNCT
eajbcs-694	318	31	+	+	CCONJ
eajbcs-694	318	32	σ1i	σ1i	NOUN
eajbcs-694	318	33	+	+	CCONJ
eajbcs-694	318	34	σ3h	σ3h	NOUN
eajbcs-694	318	35	)	)	PUNCT
eajbcs-694	318	36	0	0	NUM
eajbcs-694	318	37	0	0	NUM
eajbcs-694	319	1			PROPN
eajbcs-694	319	2	≥	≥	NOUN
eajbcs-694	319	3	0	0	NUM
eajbcs-694	319	4	.	.	PUNCT
eajbcs-694	320	1	therefore	therefore	ADV
eajbcs-694	320	2	by	by	ADP
eajbcs-694	320	3	castillo	castillo	PROPN
eajbcs-694	320	4	-	-	PUNCT
eajbcs-694	320	5	chavez	chavez	PROPN
eajbcs-694	320	6	theorem	theorem	VERB
eajbcs-694	320	7	castillo	castillo	PROPN
eajbcs-694	320	8	-	-	PUNCT
eajbcs-694	320	9	chavez	chavez	PROPN
eajbcs-694	320	10	et	et	PROPN
eajbcs-694	320	11	al	al	PROPN
eajbcs-694	320	12	.	.	PROPN
eajbcs-694	321	1	(	(	PUNCT
eajbcs-694	321	2	2002	2002	NUM
eajbcs-694	321	3	)	)	PUNCT
eajbcs-694	321	4	,	,	PUNCT
eajbcs-694	321	5	the	the	DET
eajbcs-694	321	6	disease	disease	NOUN
eajbcs-694	321	7	free	free	ADJ
eajbcs-694	321	8	equilibrium	equilibrium	NOUN
eajbcs-694	321	9	point	point	NOUN
eajbcs-694	321	10	e0	e0	PROPN
eajbcs-694	321	11	of	of	ADP
eajbcs-694	321	12	the	the	DET
eajbcs-694	321	13	system	system	NOUN
eajbcs-694	321	14	(	(	PUNCT
eajbcs-694	321	15	2	2	X
eajbcs-694	321	16	)	)	PUNCT
eajbcs-694	321	17	is	be	AUX
eajbcs-694	321	18	globally	globally	ADV
eajbcs-694	321	19	asymptotically	asymptotically	ADV
eajbcs-694	321	20	stable	stable	ADJ
eajbcs-694	321	21	for	for	ADP
eajbcs-694	321	22	r0	r0	NOUN
eajbcs-694	321	23	<	<	X
eajbcs-694	321	24	1	1	NUM
eajbcs-694	321	25	.	.	PUNCT
eajbcs-694	321	26	east	east	PROPN
eajbcs-694	321	27	afr	afr	PROPN
eajbcs-694	321	28	.	.	PUNCT
eajbcs-694	322	1	j.	j.	PROPN
eajbcs-694	322	2	biophys	biophys	PROPN
eajbcs-694	322	3	.	.	PUNCT
eajbcs-694	323	1	comput	comput	NOUN
eajbcs-694	323	2	.	.	PUNCT
eajbcs-694	324	1	sci	sci	PROPN
eajbcs-694	324	2	.	.	PUNCT
eajbcs-694	325	1	(	(	PUNCT
eajbcs-694	325	2	2024	2024	NUM
eajbcs-694	325	3	)	)	PUNCT
eajbcs-694	325	4	,	,	PUNCT
eajbcs-694	325	5	vol	vol	NOUN
eajbcs-694	325	6	.	.	PROPN
eajbcs-694	325	7	5	5	NUM
eajbcs-694	325	8	,	,	PUNCT
eajbcs-694	325	9	no	no	INTJ
eajbcs-694	325	10	.	.	NOUN
eajbcs-694	325	11	2	2	NUM
eajbcs-694	325	12	,	,	PUNCT
eajbcs-694	325	13	58	58	NUM
eajbcs-694	325	14	-	-	SYM
eajbcs-694	325	15	86	86	NUM
eajbcs-694	325	16	67	67	NUM
eajbcs-694	325	17	3.6	3.6	NUM
eajbcs-694	325	18	.	.	PUNCT
eajbcs-694	326	1	endemic	endemic	ADJ
eajbcs-694	326	2	equilibrium	equilibrium	NOUN
eajbcs-694	326	3	point	point	NOUN
eajbcs-694	326	4	endemic	endemic	ADJ
eajbcs-694	326	5	equilibrium	equilibrium	NOUN
eajbcs-694	326	6	point	point	NOUN
eajbcs-694	326	7	is	be	AUX
eajbcs-694	326	8	a	a	DET
eajbcs-694	326	9	steady	steady	ADJ
eajbcs-694	326	10	state	state	NOUN
eajbcs-694	326	11	solution	solution	NOUN
eajbcs-694	326	12	where	where	SCONJ
eajbcs-694	326	13	the	the	DET
eajbcs-694	326	14	disease	disease	NOUN
eajbcs-694	326	15	persists	persist	VERB
eajbcs-694	326	16	in	in	ADP
eajbcs-694	326	17	the	the	DET
eajbcs-694	326	18	population	population	NOUN
eajbcs-694	326	19	.	.	PUNCT
eajbcs-694	327	1	in	in	ADP
eajbcs-694	327	2	the	the	DET
eajbcs-694	327	3	presence	presence	NOUN
eajbcs-694	327	4	of	of	ADP
eajbcs-694	327	5	disease	disease	NOUN
eajbcs-694	327	6	in	in	ADP
eajbcs-694	327	7	the	the	DET
eajbcs-694	327	8	population	population	NOUN
eajbcs-694	327	9	,	,	PUNCT
eajbcs-694	327	10	there	there	PRON
eajbcs-694	327	11	exist	exist	VERB
eajbcs-694	327	12	an	an	DET
eajbcs-694	327	13	equilibrium	equilibrium	NOUN
eajbcs-694	327	14	point	point	NOUN
eajbcs-694	327	15	called	call	VERB
eajbcs-694	327	16	endemic	endemic	ADJ
eajbcs-694	327	17	equilibrium	equilibrium	NOUN
eajbcs-694	327	18	point	point	NOUN
eajbcs-694	327	19	denoted	denote	VERB
eajbcs-694	327	20	by	by	ADP
eajbcs-694	327	21	e1	e1	NOUN
eajbcs-694	327	22	=	=	SYM
eajbcs-694	327	23	(	(	PUNCT
eajbcs-694	327	24	s∗	s∗	PROPN
eajbcs-694	327	25	,	,	PUNCT
eajbcs-694	327	26	p	p	NOUN
eajbcs-694	327	27	∗	∗	NOUN
eajbcs-694	327	28	,	,	PUNCT
eajbcs-694	327	29	e∗	e∗	NOUN
eajbcs-694	327	30	,	,	PUNCT
eajbcs-694	327	31	i∗	i∗	PROPN
eajbcs-694	327	32	,	,	PUNCT
eajbcs-694	327	33	h∗	h∗	PROPN
eajbcs-694	327	34	,	,	PUNCT
eajbcs-694	327	35	r∗	r∗	PROPN
eajbcs-694	327	36	)	)	PUNCT
eajbcs-694	327	37	.	.	PUNCT
eajbcs-694	328	1	it	it	PRON
eajbcs-694	328	2	can	can	AUX
eajbcs-694	328	3	be	be	AUX
eajbcs-694	328	4	obtained	obtain	VERB
eajbcs-694	328	5	by	by	ADP
eajbcs-694	328	6	setting	set	VERB
eajbcs-694	328	7	each	each	DET
eajbcs-694	328	8	equation	equation	NOUN
eajbcs-694	328	9	of	of	ADP
eajbcs-694	328	10	the	the	DET
eajbcs-694	328	11	system	system	NOUN
eajbcs-694	328	12	(	(	PUNCT
eajbcs-694	328	13	2	2	X
eajbcs-694	328	14	)	)	PUNCT
eajbcs-694	328	15	equal	equal	ADJ
eajbcs-694	328	16	to	to	ADP
eajbcs-694	328	17	zero	zero	NUM
eajbcs-694	328	18	.	.	PUNCT
eajbcs-694	329	1	then	then	ADV
eajbcs-694	329	2	we	we	PRON
eajbcs-694	329	3	obtained	obtain	VERB
eajbcs-694	329	4	s∗	s∗	PROPN
eajbcs-694	329	5	=	=	SYM
eajbcs-694	329	6	s0	s0	PROPN
eajbcs-694	329	7	r0	r0	NOUN
eajbcs-694	329	8	,	,	PUNCT
eajbcs-694	329	9	p	p	NOUN
eajbcs-694	329	10	∗	∗	NOUN
eajbcs-694	329	11	=	=	PUNCT
eajbcs-694	329	12	θs0k	θs0k	PROPN
eajbcs-694	329	13	+	+	CCONJ
eajbcs-694	329	14	(	(	PUNCT
eajbcs-694	329	15	1−	1−	NUM
eajbcs-694	329	16	η)ωµδ(θ	η)ωµδ(θ	VERB
eajbcs-694	329	17	+	+	X
eajbcs-694	329	18	ν	ν	X
eajbcs-694	329	19	+	+	CCONJ
eajbcs-694	329	20	µ)[(1−	µ)[(1−	X
eajbcs-694	329	21	τ)k1k2	τ)k1k2	PROPN
eajbcs-694	329	22	+	+	CCONJ
eajbcs-694	329	23	ετk2	ετk2	PROPN
eajbcs-694	329	24	+	+	CCONJ
eajbcs-694	329	25	γατ	γατ	NOUN
eajbcs-694	329	26	]	]	PUNCT
eajbcs-694	329	27	s0(r0	s0(r0	ADJ
eajbcs-694	329	28	−	−	PROPN
eajbcs-694	329	29	1	1	NUM
eajbcs-694	329	30	)	)	PUNCT
eajbcs-694	329	31	(	(	PUNCT
eajbcs-694	329	32	ν	ν	NOUN
eajbcs-694	329	33	+	+	CCONJ
eajbcs-694	329	34	µ)kr0	µ)kr0	NOUN
eajbcs-694	329	35	,	,	PUNCT
eajbcs-694	329	36	e∗	e∗	PROPN
eajbcs-694	329	37	=	=	SYM
eajbcs-694	329	38	k1k2µ(ω	k1k2µ(ω	PROPN
eajbcs-694	329	39	+	+	CCONJ
eajbcs-694	329	40	µ)(θ	µ)(θ	PUNCT
eajbcs-694	329	41	+	+	NUM
eajbcs-694	329	42	ν	ν	X
eajbcs-694	329	43	+	+	CCONJ
eajbcs-694	329	44	µ)s0(r0	µ)s0(r0	ADJ
eajbcs-694	329	45	−	−	NUM
eajbcs-694	329	46	1	1	NUM
eajbcs-694	329	47	)	)	PUNCT
eajbcs-694	329	48	kr0	kr0	NOUN
eajbcs-694	329	49	,	,	PUNCT
eajbcs-694	329	50	i∗	i∗	NOUN
eajbcs-694	329	51	=	=	SYM
eajbcs-694	329	52	k2δµτ(ω	k2δµτ(ω	PUNCT
eajbcs-694	330	1	+	+	PUNCT
eajbcs-694	330	2	µ)(θ	µ)(θ	PUNCT
eajbcs-694	330	3	+	+	NUM
eajbcs-694	330	4	ν	ν	X
eajbcs-694	330	5	+	+	CCONJ
eajbcs-694	330	6	µ)s0(r0	µ)s0(r0	ADJ
eajbcs-694	330	7	−	−	NUM
eajbcs-694	330	8	1	1	NUM
eajbcs-694	330	9	)	)	PUNCT
eajbcs-694	330	10	kr0	kr0	NOUN
eajbcs-694	330	11	,	,	PUNCT
eajbcs-694	330	12	h∗	h∗	PROPN
eajbcs-694	330	13	=	=	PUNCT
eajbcs-694	330	14	αδµτ(ω	αδµτ(ω	PROPN
eajbcs-694	330	15	+	+	CCONJ
eajbcs-694	330	16	µ)(θ	µ)(θ	PUNCT
eajbcs-694	330	17	+	+	CCONJ
eajbcs-694	330	18	ν	ν	X
eajbcs-694	330	19	+	+	CCONJ
eajbcs-694	330	20	µ)s0(r0	µ)s0(r0	ADJ
eajbcs-694	330	21	−	−	NUM
eajbcs-694	330	22	1	1	NUM
eajbcs-694	330	23	)	)	PUNCT
eajbcs-694	330	24	kr0	kr0	NOUN
eajbcs-694	330	25	,	,	PUNCT
eajbcs-694	330	26	r∗	r∗	PROPN
eajbcs-694	330	27	=	=	PUNCT
eajbcs-694	330	28	δµ(θ	δµ(θ	X
eajbcs-694	330	29	+	+	NUM
eajbcs-694	330	30	ν	ν	X
eajbcs-694	330	31	+	+	CCONJ
eajbcs-694	330	32	µ)[(1−	µ)[(1−	X
eajbcs-694	330	33	τ)k1k2	τ)k1k2	PROPN
eajbcs-694	330	34	+	+	CCONJ
eajbcs-694	330	35	ετk2	ετk2	PROPN
eajbcs-694	330	36	+	+	CCONJ
eajbcs-694	330	37	γατ	γατ	NOUN
eajbcs-694	330	38	]	]	PUNCT
eajbcs-694	330	39	s0(r0	s0(r0	ADJ
eajbcs-694	330	40	−	−	PROPN
eajbcs-694	330	41	1	1	X
eajbcs-694	330	42	)	)	PUNCT
eajbcs-694	330	43	kr0	kr0	NOUN
eajbcs-694	330	44	,	,	PUNCT
eajbcs-694	330	45	where	where	SCONJ
eajbcs-694	330	46	k	k	NOUN
eajbcs-694	330	47	=	=	PUNCT
eajbcs-694	330	48	k1k2[µ(ν	k1k2[µ(ν	NOUN
eajbcs-694	330	49	+	+	CCONJ
eajbcs-694	330	50	µ)(ω	µ)(ω	VERB
eajbcs-694	330	51	+	+	CCONJ
eajbcs-694	330	52	δ	δ	PROPN
eajbcs-694	330	53	+	+	X
eajbcs-694	330	54	µ	µ	X
eajbcs-694	330	55	)	)	PUNCT
eajbcs-694	331	1	+	+	CCONJ
eajbcs-694	331	2	δµ(1−	δµ(1−	ADJ
eajbcs-694	331	3	η)ω	η)ω	NOUN
eajbcs-694	331	4	]	]	X
eajbcs-694	331	5	+	+	CCONJ
eajbcs-694	331	6	ωδτ(ηµ+	ωδτ(ηµ+	PROPN
eajbcs-694	331	7	ν)[α(µ+	ν)[α(µ+	PROPN
eajbcs-694	331	8	ξ	ξ	PROPN
eajbcs-694	331	9	)	)	PUNCT
eajbcs-694	332	1	+	+	CCONJ
eajbcs-694	332	2	k2(µ+	k2(µ+	PROPN
eajbcs-694	332	3	ρ	ρ	PROPN
eajbcs-694	332	4	)	)	PUNCT
eajbcs-694	332	5	]	]	PUNCT
eajbcs-694	332	6	,	,	PUNCT
eajbcs-694	332	7	provided	provide	VERB
eajbcs-694	332	8	that	that	DET
eajbcs-694	332	9	r0	r0	NOUN
eajbcs-694	332	10	>	>	X
eajbcs-694	332	11	1	1	NUM
eajbcs-694	332	12	.	.	PUNCT
eajbcs-694	333	1	from	from	ADP
eajbcs-694	333	2	this	this	PRON
eajbcs-694	333	3	we	we	PRON
eajbcs-694	333	4	see	see	VERB
eajbcs-694	333	5	that	that	PRON
eajbcs-694	333	6	for	for	SCONJ
eajbcs-694	333	7	the	the	DET
eajbcs-694	333	8	endemic	endemic	ADJ
eajbcs-694	333	9	equilibrium	equilibrium	NOUN
eajbcs-694	333	10	to	to	PART
eajbcs-694	333	11	exist	exist	VERB
eajbcs-694	333	12	r0	r0	NOUN
eajbcs-694	333	13	>	>	X
eajbcs-694	333	14	1	1	X
eajbcs-694	333	15	.	.	PUNCT
eajbcs-694	334	1	moreover	moreover	ADV
eajbcs-694	334	2	,	,	PUNCT
eajbcs-694	334	3	the	the	DET
eajbcs-694	334	4	force	force	NOUN
eajbcs-694	334	5	of	of	ADP
eajbcs-694	334	6	infection	infection	NOUN
eajbcs-694	334	7	can	can	AUX
eajbcs-694	334	8	be	be	AUX
eajbcs-694	334	9	updated	update	VERB
eajbcs-694	334	10	as	as	ADP
eajbcs-694	334	11	λ∗	λ∗	NOUN
eajbcs-694	334	12	=	=	PUNCT
eajbcs-694	334	13	β(σ1i	β(σ1i	NOUN
eajbcs-694	334	14	∗	∗	NOUN
eajbcs-694	334	15	+	+	NUM
eajbcs-694	335	1	σ2e	σ2e	PROPN
eajbcs-694	335	2	∗	∗	NOUN
eajbcs-694	335	3	+	+	CCONJ
eajbcs-694	335	4	σ3h	σ3h	NOUN
eajbcs-694	335	5	∗	∗	NOUN
eajbcs-694	335	6	)	)	PUNCT
eajbcs-694	335	7	.	.	PUNCT
eajbcs-694	336	1	(	(	PUNCT
eajbcs-694	336	2	7	7	X
eajbcs-694	336	3	)	)	PUNCT
eajbcs-694	336	4	when	when	SCONJ
eajbcs-694	336	5	we	we	PRON
eajbcs-694	336	6	substitute	substitute	VERB
eajbcs-694	336	7	the	the	DET
eajbcs-694	336	8	expression	expression	NOUN
eajbcs-694	336	9	for	for	ADP
eajbcs-694	336	10	e∗	e∗	PROPN
eajbcs-694	336	11	,	,	PUNCT
eajbcs-694	336	12	i∗	i∗	NOUN
eajbcs-694	336	13	and	and	CCONJ
eajbcs-694	336	14	h∗	h∗	NOUN
eajbcs-694	336	15	into	into	ADP
eajbcs-694	336	16	the	the	DET
eajbcs-694	336	17	force	force	NOUN
eajbcs-694	336	18	of	of	ADP
eajbcs-694	336	19	infection	infection	NOUN
eajbcs-694	336	20	λ∗	λ∗	PROPN
eajbcs-694	336	21	,	,	PUNCT
eajbcs-694	336	22	we	we	PRON
eajbcs-694	336	23	obtain	obtain	VERB
eajbcs-694	336	24	λ∗	λ∗	NOUN
eajbcs-694	336	25	=	=	SYM
eajbcs-694	336	26	µk1k2(δ	µk1k2(δ	PROPN
eajbcs-694	336	27	+	+	CCONJ
eajbcs-694	336	28	µ)(ω	µ)(ω	VERB
eajbcs-694	336	29	+	+	NUM
eajbcs-694	336	30	µ)(θ	µ)(θ	SYM
eajbcs-694	336	31	+	+	NUM
eajbcs-694	336	32	ν	ν	X
eajbcs-694	336	33	+	+	CCONJ
eajbcs-694	336	34	µ)(r0	µ)(r0	NOUN
eajbcs-694	336	35	−	−	NOUN
eajbcs-694	336	36	1	1	NUM
eajbcs-694	336	37	)	)	PUNCT
eajbcs-694	336	38	k	k	NOUN
eajbcs-694	336	39	,	,	PUNCT
eajbcs-694	336	40	(	(	PUNCT
eajbcs-694	336	41	8)	8)	NUM
eajbcs-694	336	42	provided	provide	VERB
eajbcs-694	336	43	that	that	DET
eajbcs-694	336	44	r0	r0	NOUN
eajbcs-694	336	45	>	>	X
eajbcs-694	336	46	1	1	NUM
eajbcs-694	336	47	.	.	PUNCT
eajbcs-694	336	48	from	from	ADP
eajbcs-694	336	49	this	this	PRON
eajbcs-694	336	50	,	,	PUNCT
eajbcs-694	336	51	we	we	PRON
eajbcs-694	336	52	see	see	VERB
eajbcs-694	336	53	that	that	SCONJ
eajbcs-694	336	54	,	,	PUNCT
eajbcs-694	336	55	there	there	PRON
eajbcs-694	336	56	is	be	VERB
eajbcs-694	336	57	no	no	DET
eajbcs-694	336	58	endemic	endemic	ADJ
eajbcs-694	336	59	equilibrium	equilibrium	NOUN
eajbcs-694	336	60	of	of	ADP
eajbcs-694	336	61	the	the	DET
eajbcs-694	336	62	system	system	NOUN
eajbcs-694	336	63	(	(	PUNCT
eajbcs-694	336	64	2	2	X
eajbcs-694	336	65	)	)	PUNCT
eajbcs-694	336	66	if	if	SCONJ
eajbcs-694	336	67	r0	r0	NOUN
eajbcs-694	336	68	<	<	X
eajbcs-694	336	69	1	1	NUM
eajbcs-694	336	70	.	.	PUNCT
eajbcs-694	337	1	therefore	therefore	ADV
eajbcs-694	337	2	,	,	PUNCT
eajbcs-694	337	3	this	this	DET
eajbcs-694	337	4	condition	condition	NOUN
eajbcs-694	337	5	shows	show	VERB
eajbcs-694	337	6	that	that	SCONJ
eajbcs-694	337	7	it	it	PRON
eajbcs-694	337	8	is	be	AUX
eajbcs-694	337	9	not	not	PART
eajbcs-694	337	10	possible	possible	ADJ
eajbcs-694	337	11	for	for	ADP
eajbcs-694	337	12	backward	backward	ADJ
eajbcs-694	337	13	bifurcation	bifurcation	NOUN
eajbcs-694	337	14	in	in	ADP
eajbcs-694	337	15	this	this	DET
eajbcs-694	337	16	model	model	NOUN
eajbcs-694	337	17	.	.	PUNCT
eajbcs-694	338	1	hence	hence	ADV
eajbcs-694	338	2	we	we	PRON
eajbcs-694	338	3	have	have	AUX
eajbcs-694	338	4	established	establish	VERB
eajbcs-694	338	5	the	the	DET
eajbcs-694	338	6	following	follow	VERB
eajbcs-694	338	7	result	result	NOUN
eajbcs-694	338	8	.	.	PUNCT
eajbcs-694	339	1	3.7	3.7	NUM
eajbcs-694	339	2	.	.	PUNCT
eajbcs-694	340	1	local	local	ADJ
eajbcs-694	340	2	stability	stability	NOUN
eajbcs-694	340	3	of	of	ADP
eajbcs-694	340	4	endemic	endemic	ADJ
eajbcs-694	340	5	equilibrium	equilibrium	NOUN
eajbcs-694	340	6	theorem	theorem	VERB
eajbcs-694	340	7	3.6	3.6	NUM
eajbcs-694	340	8	.	.	PUNCT
eajbcs-694	341	1	the	the	DET
eajbcs-694	341	2	endemic	endemic	ADJ
eajbcs-694	341	3	equilibrium	equilibrium	NOUN
eajbcs-694	341	4	e1	e1	NOUN
eajbcs-694	341	5	of	of	ADP
eajbcs-694	341	6	the	the	DET
eajbcs-694	341	7	system	system	NOUN
eajbcs-694	341	8	(	(	PUNCT
eajbcs-694	341	9	2	2	X
eajbcs-694	341	10	)	)	PUNCT
eajbcs-694	341	11	is	be	AUX
eajbcs-694	341	12	locally	locally	ADV
eajbcs-694	341	13	asymptotically	asymptotically	ADV
eajbcs-694	341	14	stable	stable	ADJ
eajbcs-694	341	15	theorem	theorem	NOUN
eajbcs-694	341	16	3.5	3.5	NUM
eajbcs-694	341	17	.	.	PUNCT
eajbcs-694	342	1	a	a	DET
eajbcs-694	342	2	unique	unique	ADJ
eajbcs-694	342	3	endemic	endemic	ADJ
eajbcs-694	342	4	equilibrium	equilibrium	NOUN
eajbcs-694	342	5	point	point	NOUN
eajbcs-694	342	6	e1	e1	NOUN
eajbcs-694	342	7	=	=	SYM
eajbcs-694	342	8	(	(	PUNCT
eajbcs-694	342	9	s∗	s∗	PROPN
eajbcs-694	342	10	,	,	PUNCT
eajbcs-694	342	11	p	p	NOUN
eajbcs-694	342	12	∗	∗	NOUN
eajbcs-694	342	13	,	,	PUNCT
eajbcs-694	342	14	e∗	e∗	NOUN
eajbcs-694	342	15	,	,	PUNCT
eajbcs-694	342	16	i∗	i∗	PROPN
eajbcs-694	342	17	,	,	PUNCT
eajbcs-694	342	18	h∗	h∗	PROPN
eajbcs-694	342	19	,	,	PUNCT
eajbcs-694	342	20	r∗	r∗	PROPN
eajbcs-694	342	21	)	)	PUNCT
eajbcs-694	342	22	exists	exist	VERB
eajbcs-694	342	23	and	and	CCONJ
eajbcs-694	342	24	is	be	AUX
eajbcs-694	342	25	positive	positive	ADJ
eajbcs-694	342	26	if	if	SCONJ
eajbcs-694	342	27	r0	r0	NOUN
eajbcs-694	342	28	>	>	X
eajbcs-694	342	29	1	1	NUM
eajbcs-694	342	30	.	.	PUNCT
eajbcs-694	342	31	when	when	SCONJ
eajbcs-694	342	32	we	we	PRON
eajbcs-694	342	33	plot	plot	VERB
eajbcs-694	342	34	the	the	DET
eajbcs-694	342	35	force	force	NOUN
eajbcs-694	342	36	of	of	ADP
eajbcs-694	342	37	infection	infection	NOUN
eajbcs-694	342	38	λ∗	λ∗	PROPN
eajbcs-694	342	39	over	over	ADP
eajbcs-694	342	40	r0	r0	NOUN
eajbcs-694	342	41	by	by	ADP
eajbcs-694	342	42	using	use	VERB
eajbcs-694	342	43	the	the	DET
eajbcs-694	342	44	expression	expression	NOUN
eajbcs-694	342	45	for	for	ADP
eajbcs-694	342	46	λ∗	λ∗	NOUN
eajbcs-694	342	47	we	we	PRON
eajbcs-694	342	48	got	get	VERB
eajbcs-694	342	49	a	a	DET
eajbcs-694	342	50	forward	forward	ADJ
eajbcs-694	342	51	bifurcation	bifurcation	NOUN
eajbcs-694	342	52	in	in	ADP
eajbcs-694	342	53	figure	figure	NOUN
eajbcs-694	342	54	(	(	PUNCT
eajbcs-694	342	55	2	2	NUM
eajbcs-694	342	56	)	)	PUNCT
eajbcs-694	342	57	.	.	PUNCT
eajbcs-694	343	1	if	if	SCONJ
eajbcs-694	343	2	r0	r0	NOUN
eajbcs-694	343	3	>	>	X
eajbcs-694	343	4	1	1	NUM
eajbcs-694	343	5	.	.	PUNCT
eajbcs-694	343	6	east	east	PROPN
eajbcs-694	343	7	afr	afr	PROPN
eajbcs-694	343	8	.	.	PUNCT
eajbcs-694	344	1	j.	j.	PROPN
eajbcs-694	344	2	biophys	biophys	PROPN
eajbcs-694	344	3	.	.	PUNCT
eajbcs-694	345	1	comput	comput	NOUN
eajbcs-694	345	2	.	.	PUNCT
eajbcs-694	346	1	sci	sci	PROPN
eajbcs-694	346	2	.	.	PUNCT
eajbcs-694	347	1	(	(	PUNCT
eajbcs-694	347	2	2024	2024	NUM
eajbcs-694	347	3	)	)	PUNCT
eajbcs-694	347	4	,	,	PUNCT
eajbcs-694	347	5	vol	vol	NOUN
eajbcs-694	347	6	.	.	PROPN
eajbcs-694	347	7	5	5	NUM
eajbcs-694	347	8	,	,	PUNCT
eajbcs-694	347	9	no	no	INTJ
eajbcs-694	347	10	.	.	NOUN
eajbcs-694	347	11	2	2	NUM
eajbcs-694	347	12	,	,	PUNCT
eajbcs-694	347	13	58	58	NUM
eajbcs-694	347	14	-	-	SYM
eajbcs-694	347	15	86	86	NUM
eajbcs-694	347	16	68	68	NUM
eajbcs-694	347	17	proof	proof	NOUN
eajbcs-694	347	18	.	.	PUNCT
eajbcs-694	348	1	to	to	PART
eajbcs-694	348	2	determine	determine	VERB
eajbcs-694	348	3	the	the	DET
eajbcs-694	348	4	local	local	ADJ
eajbcs-694	348	5	stability	stability	NOUN
eajbcs-694	348	6	of	of	ADP
eajbcs-694	348	7	endemic	endemic	ADJ
eajbcs-694	348	8	equilibrium	equilibrium	NOUN
eajbcs-694	348	9	,	,	PUNCT
eajbcs-694	348	10	we	we	PRON
eajbcs-694	348	11	used	use	VERB
eajbcs-694	348	12	the	the	DET
eajbcs-694	348	13	center	center	ADJ
eajbcs-694	348	14	manifold	manifold	ADJ
eajbcs-694	348	15	theory	theory	NOUN
eajbcs-694	348	16	castillo	castillo	PROPN
eajbcs-694	348	17	-	-	PUNCT
eajbcs-694	348	18	chavez	chavez	PROPN
eajbcs-694	348	19	and	and	CCONJ
eajbcs-694	348	20	song	song	NOUN
eajbcs-694	348	21	(	(	PUNCT
eajbcs-694	348	22	2004	2004	NUM
eajbcs-694	348	23	)	)	PUNCT
eajbcs-694	348	24	,	,	PUNCT
eajbcs-694	348	25	by	by	ADP
eajbcs-694	348	26	taking	take	VERB
eajbcs-694	348	27	β	β	PRON
eajbcs-694	348	28	as	as	ADP
eajbcs-694	348	29	a	a	DET
eajbcs-694	348	30	bifurcation	bifurcation	NOUN
eajbcs-694	348	31	parameter	parameter	NOUN
eajbcs-694	348	32	.	.	PUNCT
eajbcs-694	349	1	we	we	PRON
eajbcs-694	349	2	make	make	VERB
eajbcs-694	349	3	the	the	DET
eajbcs-694	349	4	following	follow	VERB
eajbcs-694	349	5	change	change	NOUN
eajbcs-694	349	6	of	of	ADP
eajbcs-694	349	7	variables	variable	NOUN
eajbcs-694	349	8	on	on	ADP
eajbcs-694	349	9	the	the	DET
eajbcs-694	349	10	system	system	NOUN
eajbcs-694	349	11	(	(	PUNCT
eajbcs-694	349	12	2	2	NUM
eajbcs-694	349	13	)	)	PUNCT
eajbcs-694	349	14	.	.	PUNCT
eajbcs-694	350	1	let	let	VERB
eajbcs-694	350	2	s	s	NOUN
eajbcs-694	350	3	=	=	SYM
eajbcs-694	350	4	x1	x1	PROPN
eajbcs-694	350	5	,	,	PUNCT
eajbcs-694	350	6	p	p	X
eajbcs-694	350	7	=	=	SYM
eajbcs-694	350	8	x2	x2	PROPN
eajbcs-694	350	9	,	,	PUNCT
eajbcs-694	350	10	e	e	X
eajbcs-694	350	11	=	=	SYM
eajbcs-694	350	12	x3	x3	PROPN
eajbcs-694	350	13	,	,	PUNCT
eajbcs-694	350	14	i	i	PROPN
eajbcs-694	350	15	=	=	SYM
eajbcs-694	350	16	x4	x4	PROPN
eajbcs-694	350	17	,	,	PUNCT
eajbcs-694	350	18	h	h	NOUN
eajbcs-694	350	19	=	=	SYM
eajbcs-694	350	20	x5	x5	PROPN
eajbcs-694	350	21	and	and	CCONJ
eajbcs-694	350	22	r	r	NOUN
eajbcs-694	350	23	=	=	SYM
eajbcs-694	350	24	x6	x6	PROPN
eajbcs-694	350	25	.	.	PUNCT
eajbcs-694	351	1	moreover	moreover	ADV
eajbcs-694	351	2	,	,	PUNCT
eajbcs-694	351	3	by	by	ADP
eajbcs-694	351	4	using	use	VERB
eajbcs-694	351	5	vector	vector	NOUN
eajbcs-694	351	6	notation	notation	NOUN
eajbcs-694	351	7	x	x	PUNCT
eajbcs-694	351	8	=	=	SYM
eajbcs-694	351	9	(	(	PUNCT
eajbcs-694	351	10	x1	x1	PROPN
eajbcs-694	351	11	,	,	PUNCT
eajbcs-694	351	12	x2	x2	PROPN
eajbcs-694	351	13	,	,	PUNCT
eajbcs-694	351	14	x3	x3	PROPN
eajbcs-694	351	15	,	,	PUNCT
eajbcs-694	351	16	x4	x4	PROPN
eajbcs-694	351	17	,	,	PUNCT
eajbcs-694	351	18	x5	x5	PROPN
eajbcs-694	351	19	,	,	PUNCT
eajbcs-694	351	20	x6	x6	PROPN
eajbcs-694	351	21	)	)	PUNCT
eajbcs-694	351	22	t	t	NOUN
eajbcs-694	351	23	,	,	PUNCT
eajbcs-694	351	24	the	the	DET
eajbcs-694	351	25	system	system	NOUN
eajbcs-694	351	26	(	(	PUNCT
eajbcs-694	351	27	2	2	X
eajbcs-694	351	28	)	)	PUNCT
eajbcs-694	351	29	can	can	AUX
eajbcs-694	351	30	be	be	AUX
eajbcs-694	351	31	written	write	VERB
eajbcs-694	351	32	in	in	ADP
eajbcs-694	351	33	the	the	DET
eajbcs-694	351	34	form	form	NOUN
eajbcs-694	351	35	dx	dx	PROPN
eajbcs-694	351	36	dt	dt	NOUN
eajbcs-694	352	1	=	=	SYM
eajbcs-694	352	2	f	f	PROPN
eajbcs-694	352	3	(	(	PUNCT
eajbcs-694	352	4	x	x	NOUN
eajbcs-694	352	5	)	)	PUNCT
eajbcs-694	352	6	,	,	PUNCT
eajbcs-694	352	7	with	with	ADP
eajbcs-694	352	8	f	f	PROPN
eajbcs-694	352	9	=	=	SYM
eajbcs-694	352	10	(	(	PUNCT
eajbcs-694	352	11	f1	f1	PROPN
eajbcs-694	352	12	,	,	PUNCT
eajbcs-694	352	13	f2	f2	PROPN
eajbcs-694	352	14	,	,	PUNCT
eajbcs-694	352	15	f3	f3	ADJ
eajbcs-694	352	16	,	,	PUNCT
eajbcs-694	352	17	f4	f4	PROPN
eajbcs-694	352	18	,	,	PUNCT
eajbcs-694	352	19	f5	f5	PROPN
eajbcs-694	352	20	,	,	PUNCT
eajbcs-694	352	21	f6	f6	PROPN
eajbcs-694	352	22	)	)	PUNCT
eajbcs-694	352	23	t	t	PROPN
eajbcs-694	352	24	.	.	PUNCT
eajbcs-694	353	1	we	we	PRON
eajbcs-694	353	2	choose	choose	VERB
eajbcs-694	353	3	β	β	X
eajbcs-694	353	4	=	=	PUNCT
eajbcs-694	353	5	β∗	β∗	NOUN
eajbcs-694	353	6	as	as	ADP
eajbcs-694	353	7	a	a	DET
eajbcs-694	353	8	bifurcation	bifurcation	NOUN
eajbcs-694	353	9	parameter	parameter	NOUN
eajbcs-694	353	10	.	.	PUNCT
eajbcs-694	354	1	solving	solve	VERB
eajbcs-694	354	2	for	for	ADP
eajbcs-694	354	3	β∗	β∗	NOUN
eajbcs-694	354	4	from	from	ADP
eajbcs-694	354	5	r0	r0	NOUN
eajbcs-694	354	6	=	=	SYM
eajbcs-694	354	7	1	1	NUM
eajbcs-694	354	8	,	,	PUNCT
eajbcs-694	354	9	we	we	PRON
eajbcs-694	354	10	obtain	obtain	VERB
eajbcs-694	354	11	β∗	β∗	NOUN
eajbcs-694	354	12	=	=	PUNCT
eajbcs-694	355	1	µ(θ	µ(θ	ADJ
eajbcs-694	355	2	+	+	NUM
eajbcs-694	355	3	ν	ν	X
eajbcs-694	355	4	+	+	X
eajbcs-694	355	5	µ)(δ	µ)(δ	PUNCT
eajbcs-694	355	6	+	+	CCONJ
eajbcs-694	355	7	µ)(α	µ)(α	X
eajbcs-694	355	8	+	+	CCONJ
eajbcs-694	355	9	ε+	ε+	X
eajbcs-694	355	10	µ+	µ+	X
eajbcs-694	355	11	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	355	12	+	+	CCONJ
eajbcs-694	355	13	µ+	µ+	X
eajbcs-694	355	14	ξ	ξ	X
eajbcs-694	355	15	)	)	PUNCT
eajbcs-694	355	16	π(ν	π(ν	X
eajbcs-694	356	1	+	+	CCONJ
eajbcs-694	356	2	µ)[σ2(α	µ)[σ2(α	PROPN
eajbcs-694	356	3	+	+	X
eajbcs-694	356	4	ε+	ε+	X
eajbcs-694	356	5	µ+	µ+	X
eajbcs-694	356	6	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	356	7	+	+	CCONJ
eajbcs-694	356	8	µ+	µ+	X
eajbcs-694	356	9	ξ	ξ	X
eajbcs-694	356	10	)	)	PUNCT
eajbcs-694	356	11	+	+	CCONJ
eajbcs-694	356	12	σ1τδ(γ	σ1τδ(γ	PROPN
eajbcs-694	356	13	+	+	CCONJ
eajbcs-694	356	14	µ+	µ+	X
eajbcs-694	356	15	ξ	ξ	X
eajbcs-694	356	16	)	)	PUNCT
eajbcs-694	356	17	+	+	CCONJ
eajbcs-694	356	18	σ3τδα	σ3τδα	PROPN
eajbcs-694	356	19	]	]	PUNCT
eajbcs-694	356	20	.	.	PUNCT
eajbcs-694	357	1	the	the	DET
eajbcs-694	357	2	jacobian	jacobian	ADJ
eajbcs-694	357	3	matrix	matrix	NOUN
eajbcs-694	357	4	of	of	ADP
eajbcs-694	357	5	the	the	DET
eajbcs-694	357	6	system	system	NOUN
eajbcs-694	357	7	(	(	PUNCT
eajbcs-694	357	8	2	2	X
eajbcs-694	357	9	)	)	PUNCT
eajbcs-694	357	10	evaluated	evaluate	VERB
eajbcs-694	357	11	at	at	ADP
eajbcs-694	357	12	the	the	DET
eajbcs-694	357	13	disease	disease	NOUN
eajbcs-694	357	14	free	free	ADJ
eajbcs-694	357	15	equilibrium	equilibrium	NOUN
eajbcs-694	357	16	e0	e0	PROPN
eajbcs-694	357	17	with	with	ADP
eajbcs-694	357	18	β	β	X
eajbcs-694	357	19	=	=	SYM
eajbcs-694	357	20	β∗	β∗	NOUN
eajbcs-694	357	21	is	be	AUX
eajbcs-694	357	22	given	give	VERB
eajbcs-694	357	23	by	by	ADP
eajbcs-694	357	24	j∗	j∗	ADJ
eajbcs-694	357	25	=	=	SYM
eajbcs-694	357	26			NOUN
eajbcs-694	357	27	−(θ	−(θ	NOUN
eajbcs-694	357	28	+	+	CCONJ
eajbcs-694	357	29	µ	µ	X
eajbcs-694	357	30	)	)	PUNCT
eajbcs-694	357	31	ν	ν	NOUN
eajbcs-694	357	32	−β∗σ2s	−β∗σ2s	NOUN
eajbcs-694	357	33	0	0	NUM
eajbcs-694	358	1	−β∗σ1s	−β∗σ1s	NUM
eajbcs-694	358	2	0	0	NUM
eajbcs-694	358	3	−β∗σ3s	−β∗σ3s	PROPN
eajbcs-694	358	4	0	0	NUM
eajbcs-694	358	5	ηω	ηω	VERB
eajbcs-694	358	6	θ	θ	PROPN
eajbcs-694	358	7	−(ν	−(ν	PROPN
eajbcs-694	358	8	+	+	CCONJ
eajbcs-694	358	9	µ	µ	X
eajbcs-694	358	10	)	)	PUNCT
eajbcs-694	358	11	0	0	NUM
eajbcs-694	358	12	0	0	NUM
eajbcs-694	358	13	0	0	NUM
eajbcs-694	359	1	(	(	PUNCT
eajbcs-694	359	2	1−	1−	NUM
eajbcs-694	359	3	η)ω	η)ω	X
eajbcs-694	359	4	0	0	NUM
eajbcs-694	359	5	0	0	X
eajbcs-694	360	1	β∗σ2s	β∗σ2s	NUM
eajbcs-694	360	2	0	0	NUM
eajbcs-694	361	1	−	−	PROPN
eajbcs-694	362	1	(	(	PUNCT
eajbcs-694	362	2	δ	δ	PROPN
eajbcs-694	362	3	+	+	X
eajbcs-694	362	4	µ	µ	X
eajbcs-694	362	5	)	)	PUNCT
eajbcs-694	363	1	β∗σ1s	β∗σ1s	ADP
eajbcs-694	363	2	0	0	NUM
eajbcs-694	364	1	β∗σ3s	β∗σ3s	NUM
eajbcs-694	364	2	0	0	NUM
eajbcs-694	364	3	0	0	NUM
eajbcs-694	364	4	0	0	NUM
eajbcs-694	364	5	0	0	NUM
eajbcs-694	364	6	τδ	τδ	ADP
eajbcs-694	364	7	−k1	−k1	NOUN
eajbcs-694	364	8	0	0	NUM
eajbcs-694	364	9	0	0	NUM
eajbcs-694	364	10	0	0	NUM
eajbcs-694	364	11	0	0	NUM
eajbcs-694	364	12	0	0	NUM
eajbcs-694	365	1	α	α	PRON
eajbcs-694	365	2	−k2	−k2	ADJ
eajbcs-694	365	3	0	0	NUM
eajbcs-694	365	4	0	0	NUM
eajbcs-694	365	5	0	0	NUM
eajbcs-694	366	1	(	(	PUNCT
eajbcs-694	366	2	1−	1−	NUM
eajbcs-694	366	3	τ)δ	τ)δ	X
eajbcs-694	366	4	ε	ε	PROPN
eajbcs-694	366	5	γ	γ	X
eajbcs-694	366	6	−(ω	−(ω	PROPN
eajbcs-694	366	7	+	+	CCONJ
eajbcs-694	366	8	µ	µ	X
eajbcs-694	366	9	)	)	PUNCT
eajbcs-694	366	10			NUM
eajbcs-694	366	11	,	,	PUNCT
eajbcs-694	366	12	where	where	SCONJ
eajbcs-694	366	13	k1	k1	NOUN
eajbcs-694	366	14	=	=	PROPN
eajbcs-694	366	15	α	α	PROPN
eajbcs-694	366	16	+	+	CCONJ
eajbcs-694	366	17	ε+	ε+	X
eajbcs-694	366	18	µ+	µ+	X
eajbcs-694	366	19	ρ	ρ	PROPN
eajbcs-694	366	20	,	,	PUNCT
eajbcs-694	366	21	k2	k2	NOUN
eajbcs-694	366	22	=	=	SYM
eajbcs-694	366	23	γ	γ	PROPN
eajbcs-694	366	24	+	+	X
eajbcs-694	366	25	µ+	µ+	X
eajbcs-694	366	26	ξ	ξ	PROPN
eajbcs-694	366	27	.	.	PUNCT
eajbcs-694	367	1	the	the	DET
eajbcs-694	367	2	jacobian	jacobian	ADJ
eajbcs-694	367	3	matrix	matrix	NOUN
eajbcs-694	367	4	j∗	j∗	NOUN
eajbcs-694	367	5	of	of	ADP
eajbcs-694	367	6	the	the	DET
eajbcs-694	367	7	linearized	linearize	VERB
eajbcs-694	367	8	system	system	NOUN
eajbcs-694	367	9	has	have	VERB
eajbcs-694	367	10	a	a	DET
eajbcs-694	367	11	simple	simple	ADJ
eajbcs-694	367	12	zero	zero	NUM
eajbcs-694	367	13	eigenvalue	eigenvalue	NOUN
eajbcs-694	367	14	with	with	ADP
eajbcs-694	367	15	all	all	DET
eajbcs-694	367	16	other	other	ADJ
eajbcs-694	367	17	eigenvalues	eigenvalue	NOUN
eajbcs-694	367	18	having	have	VERB
eajbcs-694	367	19	negative	negative	ADJ
eajbcs-694	367	20	real	real	ADJ
eajbcs-694	367	21	part	part	NOUN
eajbcs-694	367	22	,	,	PUNCT
eajbcs-694	367	23	hence	hence	ADV
eajbcs-694	367	24	the	the	DET
eajbcs-694	367	25	center	center	ADJ
eajbcs-694	367	26	manifold	manifold	ADJ
eajbcs-694	367	27	theory	theory	NOUN
eajbcs-694	367	28	will	will	AUX
eajbcs-694	367	29	be	be	AUX
eajbcs-694	367	30	used	use	VERB
eajbcs-694	367	31	to	to	PART
eajbcs-694	367	32	analyse	analyse	VERB
eajbcs-694	367	33	the	the	DET
eajbcs-694	367	34	dynamics	dynamic	NOUN
eajbcs-694	367	35	of	of	ADP
eajbcs-694	367	36	the	the	DET
eajbcs-694	367	37	system	system	NOUN
eajbcs-694	367	38	near	near	ADP
eajbcs-694	367	39	β	β	X
eajbcs-694	367	40	=	=	SYM
eajbcs-694	367	41	β∗.	β∗.	PUNCT
eajbcs-694	367	42	thus	thus	ADV
eajbcs-694	367	43	,	,	PUNCT
eajbcs-694	367	44	e0	e0	PROPN
eajbcs-694	367	45	is	be	AUX
eajbcs-694	367	46	a	a	DET
eajbcs-694	367	47	nonhyperbolic	nonhyperbolic	ADJ
eajbcs-694	367	48	equilibrium	equilibrium	NOUN
eajbcs-694	367	49	,	,	PUNCT
eajbcs-694	367	50	when	when	SCONJ
eajbcs-694	367	51	β	β	X
eajbcs-694	367	52	=	=	SYM
eajbcs-694	367	53	β∗.	β∗.	PUNCT
eajbcs-694	367	54	now	now	ADV
eajbcs-694	367	55	,	,	PUNCT
eajbcs-694	367	56	the	the	DET
eajbcs-694	367	57	components	component	NOUN
eajbcs-694	367	58	of	of	ADP
eajbcs-694	367	59	the	the	DET
eajbcs-694	367	60	right	right	ADJ
eajbcs-694	367	61	eigenvector	eigenvector	NOUN
eajbcs-694	367	62	w	w	PROPN
eajbcs-694	367	63	=	=	PUNCT
eajbcs-694	367	64	(	(	PUNCT
eajbcs-694	367	65	w1	w1	NOUN
eajbcs-694	367	66	,	,	PUNCT
eajbcs-694	367	67	w2	w2	NOUN
eajbcs-694	367	68	,	,	PUNCT
eajbcs-694	367	69	w3	w3	PROPN
eajbcs-694	367	70	,	,	PUNCT
eajbcs-694	367	71	w4	w4	NOUN
eajbcs-694	367	72	,	,	PUNCT
eajbcs-694	367	73	w5	w5	PROPN
eajbcs-694	367	74	,	,	PUNCT
eajbcs-694	367	75	w6	w6	PROPN
eajbcs-694	367	76	)	)	PUNCT
eajbcs-694	367	77	t	t	PROPN
eajbcs-694	367	78	of	of	ADP
eajbcs-694	367	79	j∗	j∗	PROPN
eajbcs-694	367	80	associated	associate	VERB
eajbcs-694	367	81	with	with	ADP
eajbcs-694	367	82	the	the	DET
eajbcs-694	367	83	zero	zero	NUM
eajbcs-694	367	84	eigenvalue	eigenvalue	NOUN
eajbcs-694	367	85	are	be	AUX
eajbcs-694	367	86	given	give	VERB
eajbcs-694	367	87	by	by	ADP
eajbcs-694	367	88	w1	w1	NOUN
eajbcs-694	367	89	=	=	SYM
eajbcs-694	367	90	−k1k2[µ(ν	−k1k2[µ(ν	PROPN
eajbcs-694	367	91	+	+	CCONJ
eajbcs-694	367	92	µ)(ω	µ)(ω	VERB
eajbcs-694	367	93	+	+	PUNCT
eajbcs-694	367	94	δ	δ	PROPN
eajbcs-694	367	95	+	+	X
eajbcs-694	367	96	µ	µ	X
eajbcs-694	367	97	)	)	PUNCT
eajbcs-694	368	1	+	+	CCONJ
eajbcs-694	368	2	δµ(1−	δµ(1−	ADJ
eajbcs-694	368	3	η)ω	η)ω	NOUN
eajbcs-694	368	4	]	]	X
eajbcs-694	368	5	+	+	CCONJ
eajbcs-694	368	6	ωδτ(ηµ+	ωδτ(ηµ+	PROPN
eajbcs-694	368	7	ν)[α(µ+	ν)[α(µ+	PROPN
eajbcs-694	368	8	ξ	ξ	PROPN
eajbcs-694	368	9	)	)	PUNCT
eajbcs-694	369	1	+	+	CCONJ
eajbcs-694	369	2	k2(µ+	k2(µ+	PROPN
eajbcs-694	369	3	ρ	ρ	PROPN
eajbcs-694	369	4	)	)	PUNCT
eajbcs-694	369	5	]	]	PUNCT
eajbcs-694	369	6	τδµk2(θ	τδµk2(θ	PUNCT
eajbcs-694	370	1	+	+	CCONJ
eajbcs-694	370	2	ν	ν	X
eajbcs-694	370	3	+	+	CCONJ
eajbcs-694	370	4	µ)(ω	µ)(ω	ADJ
eajbcs-694	370	5	+	+	CCONJ
eajbcs-694	370	6	µ	µ	X
eajbcs-694	370	7	)	)	PUNCT
eajbcs-694	370	8	w4	w4	NOUN
eajbcs-694	370	9	,	,	PUNCT
eajbcs-694	370	10	w2	w2	NOUN
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eajbcs-694	370	12	−ωδµ[τ	−ωδµ[τ	PROPN
eajbcs-694	370	13	(	(	PUNCT
eajbcs-694	370	14	k2(µ+	k2(µ+	PROPN
eajbcs-694	370	15	ρ	ρ	PROPN
eajbcs-694	370	16	)	)	PUNCT
eajbcs-694	370	17	+	+	NUM
eajbcs-694	370	18	α(µ+	α(µ+	X
eajbcs-694	370	19	ξ	ξ	X
eajbcs-694	370	20	)	)	PUNCT
eajbcs-694	370	21	)	)	PUNCT
eajbcs-694	371	1	(	(	PUNCT
eajbcs-694	371	2	2−	2−	NUM
eajbcs-694	371	3	η)−	η)−	PROPN
eajbcs-694	371	4	(	(	PUNCT
eajbcs-694	371	5	1−	1−	NUM
eajbcs-694	371	6	η	η	NOUN
eajbcs-694	371	7	)	)	PUNCT
eajbcs-694	371	8	]	]	PUNCT
eajbcs-694	372	1	+	+	CCONJ
eajbcs-694	372	2	θµ(δ	θµ(δ	X
eajbcs-694	372	3	+	+	X
eajbcs-694	372	4	ω	ω	PROPN
eajbcs-694	372	5	+	+	X
eajbcs-694	372	6	µ	µ	X
eajbcs-694	372	7	)	)	PUNCT
eajbcs-694	372	8	τδµk2(θ	τδµk2(θ	PUNCT
eajbcs-694	373	1	+	+	CCONJ
eajbcs-694	373	2	ν	ν	X
eajbcs-694	373	3	+	+	CCONJ
eajbcs-694	373	4	µ)(ω	µ)(ω	ADJ
eajbcs-694	373	5	+	+	CCONJ
eajbcs-694	373	6	µ	µ	X
eajbcs-694	373	7	)	)	PUNCT
eajbcs-694	373	8	w4	w4	NOUN
eajbcs-694	373	9	,	,	PUNCT
eajbcs-694	373	10	w3	w3	PROPN
eajbcs-694	373	11	=	=	PROPN
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eajbcs-694	373	13	+	+	CCONJ
eajbcs-694	373	14	ε+	ε+	X
eajbcs-694	373	15	µ+	µ+	X
eajbcs-694	373	16	ρ	ρ	PROPN
eajbcs-694	373	17	τδ	τδ	ADP
eajbcs-694	373	18	w4	w4	NOUN
eajbcs-694	373	19	,	,	PUNCT
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eajbcs-694	373	21	=	=	PUNCT
eajbcs-694	373	22	w4	w4	PROPN
eajbcs-694	373	23	>	>	X
eajbcs-694	373	24	0	0	PROPN
eajbcs-694	373	25	,	,	PUNCT
eajbcs-694	373	26	w5	w5	PROPN
eajbcs-694	373	27	=	=	PUNCT
eajbcs-694	373	28	α	α	PROPN
eajbcs-694	373	29	γ	γ	X
eajbcs-694	373	30	+	+	X
eajbcs-694	373	31	µ+	µ+	PROPN
eajbcs-694	373	32	ξ	ξ	PROPN
eajbcs-694	373	33	w4	w4	NOUN
eajbcs-694	373	34	,	,	PUNCT
eajbcs-694	373	35	w6	w6	PROPN
eajbcs-694	373	36	=	=	PUNCT
eajbcs-694	373	37	(	(	PUNCT
eajbcs-694	373	38	1−	1−	NUM
eajbcs-694	373	39	τ)(α	τ)(α	NOUN
eajbcs-694	373	40	+	+	CCONJ
eajbcs-694	373	41	ε+	ε+	X
eajbcs-694	373	42	µ+	µ+	X
eajbcs-694	373	43	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	373	44	+	+	CCONJ
eajbcs-694	373	45	µ+	µ+	X
eajbcs-694	373	46	ξ	ξ	NOUN
eajbcs-694	373	47	)	)	PUNCT
eajbcs-694	373	48	+	+	NUM
eajbcs-694	373	49	ετ(γ	ετ(γ	NOUN
eajbcs-694	373	50	+	+	CCONJ
eajbcs-694	373	51	µ+	µ+	X
eajbcs-694	373	52	ξ	ξ	X
eajbcs-694	373	53	)	)	PUNCT
eajbcs-694	373	54	+	+	NUM
eajbcs-694	373	55	γατ	γατ	NOUN
eajbcs-694	373	56	τ(ω	τ(ω	PUNCT
eajbcs-694	373	57	+	+	PUNCT
eajbcs-694	373	58	µ)(γ	µ)(γ	NOUN
eajbcs-694	373	59	+	+	CCONJ
eajbcs-694	373	60	µ+	µ+	X
eajbcs-694	373	61	ξ	ξ	ADJ
eajbcs-694	373	62	)	)	PUNCT
eajbcs-694	373	63	w4	w4	NOUN
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eajbcs-694	374	1	similarly	similarly	ADV
eajbcs-694	374	2	,	,	PUNCT
eajbcs-694	374	3	the	the	DET
eajbcs-694	374	4	components	component	NOUN
eajbcs-694	374	5	of	of	ADP
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eajbcs-694	374	7	left	left	ADJ
eajbcs-694	374	8	eigenvector	eigenvector	NOUN
eajbcs-694	374	9	v	v	NOUN
eajbcs-694	374	10	=	=	SYM
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eajbcs-694	374	17	,	,	PUNCT
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eajbcs-694	374	19	,	,	PUNCT
eajbcs-694	374	20	v5	v5	PROPN
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eajbcs-694	374	22	v6	v6	NOUN
eajbcs-694	374	23	)	)	PUNCT
eajbcs-694	374	24	t	t	PROPN
eajbcs-694	374	25	of	of	ADP
eajbcs-694	374	26	j∗	j∗	PROPN
eajbcs-694	374	27	associated	associate	VERB
eajbcs-694	374	28	with	with	ADP
eajbcs-694	374	29	the	the	DET
eajbcs-694	374	30	zero	zero	NUM
eajbcs-694	374	31	eigenvalue	eigenvalue	NOUN
eajbcs-694	374	32	are	be	AUX
eajbcs-694	374	33	given	give	VERB
eajbcs-694	374	34	by	by	ADP
eajbcs-694	374	35	v1	v1	NOUN
eajbcs-694	374	36	=	=	SYM
eajbcs-694	374	37	v2	v2	PROPN
eajbcs-694	374	38	=	=	SYM
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eajbcs-694	374	42	,	,	PUNCT
eajbcs-694	374	43	v3	v3	PROPN
eajbcs-694	374	44	=	=	PROPN
eajbcs-694	374	45	v3	v3	PROPN
eajbcs-694	374	46	>	>	X
eajbcs-694	374	47	0	0	PROPN
eajbcs-694	374	48	,	,	PUNCT
eajbcs-694	374	49	v4	v4	PROPN
eajbcs-694	374	50	=	=	SYM
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eajbcs-694	375	2	+	+	PUNCT
eajbcs-694	375	3	µ+	µ+	X
eajbcs-694	375	4	ξ	ξ	NUM
eajbcs-694	375	5	)	)	PUNCT
eajbcs-694	375	6	+	+	NUM
eajbcs-694	375	7	ασ3	ασ3	NOUN
eajbcs-694	375	8	)	)	PUNCT
eajbcs-694	375	9	(	(	PUNCT
eajbcs-694	375	10	α	α	PROPN
eajbcs-694	375	11	+	+	CCONJ
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eajbcs-694	375	13	µ+	µ+	X
eajbcs-694	375	14	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	375	15	+	+	CCONJ
eajbcs-694	375	16	µ+	µ+	X
eajbcs-694	375	17	ξ	ξ	PROPN
eajbcs-694	375	18	)	)	PUNCT
eajbcs-694	375	19	v3	v3	PROPN
eajbcs-694	375	20	,	,	PUNCT
eajbcs-694	375	21	v5	v5	PROPN
eajbcs-694	375	22	=	=	SYM
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eajbcs-694	375	24	γ	γ	PROPN
eajbcs-694	375	25	+	+	NUM
eajbcs-694	375	26	µ+	µ+	PROPN
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eajbcs-694	375	28	v3	v3	PROPN
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eajbcs-694	376	1	east	east	PROPN
eajbcs-694	376	2	afr	afr	PROPN
eajbcs-694	376	3	.	.	PUNCT
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eajbcs-694	377	2	biophys	biophys	PROPN
eajbcs-694	377	3	.	.	PUNCT
eajbcs-694	378	1	comput	comput	NOUN
eajbcs-694	378	2	.	.	PUNCT
eajbcs-694	379	1	sci	sci	PROPN
eajbcs-694	379	2	.	.	PUNCT
eajbcs-694	380	1	(	(	PUNCT
eajbcs-694	380	2	2024	2024	NUM
eajbcs-694	380	3	)	)	PUNCT
eajbcs-694	380	4	,	,	PUNCT
eajbcs-694	380	5	vol	vol	NOUN
eajbcs-694	380	6	.	.	PROPN
eajbcs-694	380	7	5	5	NUM
eajbcs-694	380	8	,	,	PUNCT
eajbcs-694	380	9	no	no	INTJ
eajbcs-694	380	10	.	.	NOUN
eajbcs-694	380	11	2	2	NUM
eajbcs-694	380	12	,	,	PUNCT
eajbcs-694	380	13	58	58	NUM
eajbcs-694	380	14	-	-	SYM
eajbcs-694	380	15	86	86	NUM
eajbcs-694	380	16	69	69	NUM
eajbcs-694	380	17	since	since	SCONJ
eajbcs-694	380	18	the	the	DET
eajbcs-694	380	19	first	first	ADJ
eajbcs-694	380	20	,	,	PUNCT
eajbcs-694	380	21	second	second	ADJ
eajbcs-694	380	22	and	and	CCONJ
eajbcs-694	380	23	six	six	NUM
eajbcs-694	380	24	component	component	NOUN
eajbcs-694	380	25	of	of	ADP
eajbcs-694	380	26	v	v	NOUN
eajbcs-694	380	27	are	be	AUX
eajbcs-694	380	28	zero	zero	NUM
eajbcs-694	380	29	,	,	PUNCT
eajbcs-694	380	30	we	we	PRON
eajbcs-694	380	31	do	do	AUX
eajbcs-694	380	32	n’t	not	PART
eajbcs-694	380	33	need	need	VERB
eajbcs-694	380	34	the	the	DET
eajbcs-694	380	35	partial	partial	ADJ
eajbcs-694	380	36	derivatives	derivative	NOUN
eajbcs-694	380	37	of	of	ADP
eajbcs-694	380	38	f1	f1	NOUN
eajbcs-694	380	39	,	,	PUNCT
eajbcs-694	380	40	f2	f2	PROPN
eajbcs-694	380	41	and	and	CCONJ
eajbcs-694	380	42	f6	f6	PROPN
eajbcs-694	380	43	.	.	PUNCT
eajbcs-694	381	1	from	from	ADP
eajbcs-694	381	2	the	the	DET
eajbcs-694	381	3	partial	partial	ADJ
eajbcs-694	381	4	derivatives	derivative	NOUN
eajbcs-694	381	5	of	of	ADP
eajbcs-694	381	6	f3	f3	ADJ
eajbcs-694	381	7	,	,	PUNCT
eajbcs-694	381	8	f4	f4	ADJ
eajbcs-694	381	9	and	and	CCONJ
eajbcs-694	381	10	f5	f5	NOUN
eajbcs-694	381	11	at	at	ADP
eajbcs-694	381	12	the	the	DET
eajbcs-694	381	13	disease	disease	NOUN
eajbcs-694	381	14	free	free	ADJ
eajbcs-694	381	15	equilibrium	equilibrium	NOUN
eajbcs-694	381	16	point	point	NOUN
eajbcs-694	381	17	,	,	PUNCT
eajbcs-694	381	18	the	the	DET
eajbcs-694	381	19	only	only	ADJ
eajbcs-694	381	20	ones	one	NOUN
eajbcs-694	381	21	that	that	PRON
eajbcs-694	381	22	are	be	AUX
eajbcs-694	381	23	nonzero	nonzero	NOUN
eajbcs-694	381	24	are	be	AUX
eajbcs-694	381	25	the	the	DET
eajbcs-694	381	26	following	following	NOUN
eajbcs-694	381	27	:	:	PUNCT
eajbcs-694	381	28	∂2f3	∂2f3	PUNCT
eajbcs-694	381	29	∂x3∂x1	∂x3∂x1	NOUN
eajbcs-694	382	1	=	=	SYM
eajbcs-694	382	2	∂2f3	∂2f3	PUNCT
eajbcs-694	382	3	∂x1∂x3	∂x1∂x3	ADV
eajbcs-694	382	4	=	=	SYM
eajbcs-694	382	5	β∗σ2	β∗σ2	PROPN
eajbcs-694	382	6	,	,	PUNCT
eajbcs-694	382	7	∂2f3	∂2f3	ADP
eajbcs-694	382	8	∂x4∂x1	∂x4∂x1	NOUN
eajbcs-694	382	9	=	=	PUNCT
eajbcs-694	382	10	∂2f3	∂2f3	PUNCT
eajbcs-694	382	11	∂x1∂x4	∂x1∂x4	PROPN
eajbcs-694	382	12	=	=	SYM
eajbcs-694	382	13	β∗σ1	β∗σ1	NOUN
eajbcs-694	382	14	,	,	PUNCT
eajbcs-694	382	15	∂2f3	∂2f3	ADP
eajbcs-694	382	16	∂x5∂x1	∂x5∂x1	NOUN
eajbcs-694	382	17	=	=	PUNCT
eajbcs-694	382	18	∂2f3	∂2f3	PUNCT
eajbcs-694	382	19	∂x1∂x5	∂x1∂x5	NOUN
eajbcs-694	382	20	=	=	SYM
eajbcs-694	382	21	β∗σ3	β∗σ3	NOUN
eajbcs-694	382	22	,	,	PUNCT
eajbcs-694	382	23	∂2f3	∂2f3	PUNCT
eajbcs-694	382	24	∂x3∂β	∂x3∂β	PROPN
eajbcs-694	382	25	=	=	SYM
eajbcs-694	382	26	σ2s	σ2s	PROPN
eajbcs-694	382	27	0	0	NUM
eajbcs-694	382	28	,	,	PUNCT
eajbcs-694	382	29	∂2f3	∂2f3	ADP
eajbcs-694	382	30	∂x4∂β	∂x4∂β	VERB
eajbcs-694	382	31	=	=	SYM
eajbcs-694	382	32	σ1s	σ1s	PROPN
eajbcs-694	382	33	0	0	NUM
eajbcs-694	382	34	,	,	PUNCT
eajbcs-694	382	35	∂2f3	∂2f3	ADP
eajbcs-694	382	36	∂x5∂β	∂x5∂β	NOUN
eajbcs-694	382	37	=	=	SYM
eajbcs-694	382	38	σ3s	σ3s	PROPN
eajbcs-694	382	39	0	0	NUM
eajbcs-694	382	40	.	.	PUNCT
eajbcs-694	383	1	the	the	DET
eajbcs-694	383	2	direction	direction	NOUN
eajbcs-694	383	3	of	of	ADP
eajbcs-694	383	4	the	the	DET
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eajbcs-694	383	6	at	at	ADP
eajbcs-694	383	7	r0	r0	NOUN
eajbcs-694	383	8	=	=	SYM
eajbcs-694	383	9	1	1	NUM
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eajbcs-694	383	12	by	by	ADP
eajbcs-694	383	13	the	the	DET
eajbcs-694	383	14	signs	sign	NOUN
eajbcs-694	383	15	of	of	ADP
eajbcs-694	383	16	the	the	DET
eajbcs-694	383	17	bifurcation	bifurcation	NOUN
eajbcs-694	383	18	coefficients	coefficient	VERB
eajbcs-694	383	19	a	a	PRON
eajbcs-694	383	20	and	and	CCONJ
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eajbcs-694	383	24	a	a	DET
eajbcs-694	383	25	=	=	PROPN
eajbcs-694	383	26	v3	v3	PROPN
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eajbcs-694	384	3	j=1	j=1	PROPN
eajbcs-694	384	4	wiwj	wiwj	NOUN
eajbcs-694	384	5	∂2f3	∂2f3	PUNCT
eajbcs-694	384	6	∂xi∂xj	∂xi∂xj	NOUN
eajbcs-694	384	7	(	(	PUNCT
eajbcs-694	384	8	s0	s0	PROPN
eajbcs-694	384	9	,	,	PUNCT
eajbcs-694	384	10	p	p	NOUN
eajbcs-694	384	11	0	0	NUM
eajbcs-694	384	12	,	,	PUNCT
eajbcs-694	384	13	0	0	NUM
eajbcs-694	384	14	,	,	PUNCT
eajbcs-694	384	15	0	0	NUM
eajbcs-694	384	16	,	,	PUNCT
eajbcs-694	384	17	0	0	NUM
eajbcs-694	384	18	,	,	PUNCT
eajbcs-694	384	19	0	0	NUM
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eajbcs-694	385	1	=	=	SYM
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eajbcs-694	385	3	(	(	PUNCT
eajbcs-694	385	4	w3β	w3β	PROPN
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eajbcs-694	385	6	+	+	CCONJ
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eajbcs-694	385	9	+	+	CCONJ
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eajbcs-694	386	1	=	=	SYM
eajbcs-694	387	1	2β∗w1	2β∗w1	NUM
eajbcs-694	387	2	(	(	PUNCT
eajbcs-694	387	3	σ2k1	σ2k1	NOUN
eajbcs-694	387	4	τδ	τδ	ADV
eajbcs-694	387	5	+	+	CCONJ
eajbcs-694	387	6	σ1	σ1	NOUN
eajbcs-694	387	7	+	+	CCONJ
eajbcs-694	387	8	ασ3	ασ3	PROPN
eajbcs-694	387	9	k2	k2	PROPN
eajbcs-694	387	10	)	)	PUNCT
eajbcs-694	387	11	v3w4	v3w4	PUNCT
eajbcs-694	388	1	=	=	NOUN
eajbcs-694	388	2	−	−	PROPN
eajbcs-694	388	3	2(δ	2(δ	NUM
eajbcs-694	388	4	+	+	SYM
eajbcs-694	388	5	µ	µ	X
eajbcs-694	388	6	)	)	PUNCT
eajbcs-694	388	7	(	(	PUNCT
eajbcs-694	388	8	τδ)2k2π(ν	τδ)2k2π(ν	PUNCT
eajbcs-694	388	9	+	+	CCONJ
eajbcs-694	388	10	µ)(ω	µ)(ω	ADJ
eajbcs-694	388	11	+	+	CCONJ
eajbcs-694	388	12	µ	µ	X
eajbcs-694	388	13	)	)	PUNCT
eajbcs-694	388	14	[	[	PUNCT
eajbcs-694	388	15	k21k2	k21k2	NOUN
eajbcs-694	388	16	(	(	PUNCT
eajbcs-694	388	17	µ(ν	µ(ν	PROPN
eajbcs-694	388	18	+	+	CCONJ
eajbcs-694	388	19	µ)(ω	µ)(ω	PROPN
eajbcs-694	388	20	+	+	CCONJ
eajbcs-694	388	21	δ	δ	PROPN
eajbcs-694	388	22	+	+	X
eajbcs-694	388	23	µ	µ	X
eajbcs-694	388	24	)	)	PUNCT
eajbcs-694	388	25	+	+	CCONJ
eajbcs-694	388	26	δµ(1−	δµ(1−	NOUN
eajbcs-694	388	27	η)ω	η)ω	NOUN
eajbcs-694	388	28	)	)	PUNCT
eajbcs-694	388	29	+	+	CCONJ
eajbcs-694	388	30	ωδτk1(ηµ+	ωδτk1(ηµ+	PROPN
eajbcs-694	388	31	ν	ν	NOUN
eajbcs-694	388	32	)	)	PUNCT
eajbcs-694	388	33	(	(	PUNCT
eajbcs-694	388	34	α(µ+	α(µ+	X
eajbcs-694	388	35	ξ	ξ	X
eajbcs-694	388	36	)	)	PUNCT
eajbcs-694	388	37	+	+	CCONJ
eajbcs-694	388	38	k2(µ+	k2(µ+	PROPN
eajbcs-694	388	39	ρ	ρ	PROPN
eajbcs-694	388	40	)	)	PUNCT
eajbcs-694	388	41	)	)	PUNCT
eajbcs-694	388	42	]	]	PUNCT
eajbcs-694	389	1	v3w	v3w	PROPN
eajbcs-694	389	2	2	2	NUM
eajbcs-694	389	3	4	4	NUM
eajbcs-694	389	4	<	<	X
eajbcs-694	389	5	0	0	NUM
eajbcs-694	389	6	.	.	PUNCT
eajbcs-694	390	1	and	and	CCONJ
eajbcs-694	390	2	b	b	X
eajbcs-694	390	3	=	=	PROPN
eajbcs-694	390	4	v3	v3	PROPN
eajbcs-694	390	5	6∑	6∑	PROPN
eajbcs-694	390	6	i=1	i=1	PROPN
eajbcs-694	391	1	wi	wi	PROPN
eajbcs-694	391	2	∂2f3	∂2f3	PUNCT
eajbcs-694	391	3	∂xi∂β	∂xi∂β	PROPN
eajbcs-694	391	4	(	(	PUNCT
eajbcs-694	391	5	s0	s0	PROPN
eajbcs-694	391	6	,	,	PUNCT
eajbcs-694	391	7	p	p	NOUN
eajbcs-694	391	8	0	0	NUM
eajbcs-694	391	9	,	,	PUNCT
eajbcs-694	391	10	0	0	NUM
eajbcs-694	391	11	,	,	PUNCT
eajbcs-694	391	12	0	0	NUM
eajbcs-694	391	13	,	,	PUNCT
eajbcs-694	391	14	0	0	NUM
eajbcs-694	391	15	,	,	PUNCT
eajbcs-694	391	16	0	0	NUM
eajbcs-694	391	17	)	)	PUNCT
eajbcs-694	392	1	=	=	SYM
eajbcs-694	392	2	v3	v3	PROPN
eajbcs-694	392	3	(	(	PUNCT
eajbcs-694	392	4	k1w4σ2s	k1w4σ2s	PROPN
eajbcs-694	392	5	0	0	PUNCT
eajbcs-694	393	1	τδ	τδ	ADP
eajbcs-694	394	1	+	+	CCONJ
eajbcs-694	394	2	w4σ1s	w4σ1s	NUM
eajbcs-694	394	3	0	0	NUM
eajbcs-694	395	1	+	+	CCONJ
eajbcs-694	395	2	αw4σ3s	αw4σ3s	NUM
eajbcs-694	395	3	0	0	NUM
eajbcs-694	395	4	k2	k2	NOUN
eajbcs-694	395	5	)	)	PUNCT
eajbcs-694	395	6	=	=	SYM
eajbcs-694	396	1	(	(	PUNCT
eajbcs-694	396	2	δ	δ	PROPN
eajbcs-694	396	3	+	+	CCONJ
eajbcs-694	396	4	µ)(α	µ)(α	X
eajbcs-694	396	5	+	+	CCONJ
eajbcs-694	396	6	ε+	ε+	X
eajbcs-694	396	7	µ+	µ+	X
eajbcs-694	396	8	ρ	ρ	NOUN
eajbcs-694	396	9	)	)	PUNCT
eajbcs-694	396	10	β∗τδ	β∗τδ	X
eajbcs-694	397	1	v3w4	v3w4	INTJ
eajbcs-694	397	2	>	>	PUNCT
eajbcs-694	397	3	0	0	X
eajbcs-694	397	4	.	.	PUNCT
eajbcs-694	398	1	since	since	SCONJ
eajbcs-694	398	2	a	a	DET
eajbcs-694	398	3	<	<	X
eajbcs-694	398	4	0	0	NUM
eajbcs-694	398	5	and	and	CCONJ
eajbcs-694	398	6	b	b	NOUN
eajbcs-694	398	7	>	>	X
eajbcs-694	398	8	0	0	PUNCT
eajbcs-694	398	9	at	at	ADP
eajbcs-694	398	10	β	β	X
eajbcs-694	398	11	=	=	SYM
eajbcs-694	398	12	β∗.	β∗.	PROPN
eajbcs-694	398	13	based	base	VERB
eajbcs-694	398	14	on	on	ADP
eajbcs-694	398	15	the	the	DET
eajbcs-694	398	16	theorem	theorem	ADJ
eajbcs-694	398	17	4.1	4.1	NUM
eajbcs-694	398	18	stated	state	VERB
eajbcs-694	398	19	in	in	ADP
eajbcs-694	398	20	castillo	castillo	PROPN
eajbcs-694	398	21	-	-	PUNCT
eajbcs-694	398	22	chavez	chavez	PROPN
eajbcs-694	398	23	and	and	CCONJ
eajbcs-694	398	24	song	song	NOUN
eajbcs-694	398	25	(	(	PUNCT
eajbcs-694	398	26	2004	2004	NUM
eajbcs-694	398	27	)	)	PUNCT
eajbcs-694	398	28	,	,	PUNCT
eajbcs-694	398	29	the	the	DET
eajbcs-694	398	30	system	system	NOUN
eajbcs-694	398	31	(	(	PUNCT
eajbcs-694	398	32	2	2	X
eajbcs-694	398	33	)	)	PUNCT
eajbcs-694	398	34	undergoes	undergo	VERB
eajbcs-694	398	35	a	a	DET
eajbcs-694	398	36	forward	forward	ADJ
eajbcs-694	398	37	bifurcation	bifurcation	NOUN
eajbcs-694	398	38	at	at	ADP
eajbcs-694	398	39	r0	r0	NOUN
eajbcs-694	398	40	=	=	SYM
eajbcs-694	398	41	1	1	NUM
eajbcs-694	398	42	and	and	CCONJ
eajbcs-694	398	43	the	the	DET
eajbcs-694	398	44	unique	unique	ADJ
eajbcs-694	398	45	endemic	endemic	ADJ
eajbcs-694	398	46	equilibrium	equilibrium	NOUN
eajbcs-694	398	47	e1	e1	NOUN
eajbcs-694	398	48	is	be	AUX
eajbcs-694	398	49	locally	locally	ADV
eajbcs-694	398	50	asymptotically	asymptotically	ADV
eajbcs-694	398	51	stable	stable	ADJ
eajbcs-694	398	52	for	for	ADP
eajbcs-694	398	53	r0	r0	PROPN
eajbcs-694	398	54	>	>	X
eajbcs-694	398	55	1	1	NUM
eajbcs-694	398	56	.	.	X
eajbcs-694	398	57	3.8	3.8	NUM
eajbcs-694	398	58	.	.	PUNCT
eajbcs-694	399	1	bifurcation	bifurcation	NOUN
eajbcs-694	399	2	analysis	analysis	NOUN
eajbcs-694	399	3	we	we	PRON
eajbcs-694	399	4	investigate	investigate	VERB
eajbcs-694	399	5	the	the	DET
eajbcs-694	399	6	nature	nature	NOUN
eajbcs-694	399	7	of	of	ADP
eajbcs-694	399	8	the	the	DET
eajbcs-694	399	9	bifurcation	bifurcation	NOUN
eajbcs-694	399	10	by	by	ADP
eajbcs-694	399	11	using	use	VERB
eajbcs-694	399	12	the	the	DET
eajbcs-694	399	13	center	center	ADJ
eajbcs-694	399	14	manifold	manifold	ADJ
eajbcs-694	399	15	theory	theory	NOUN
eajbcs-694	399	16	castillochavez	castillochavez	PROPN
eajbcs-694	399	17	and	and	CCONJ
eajbcs-694	399	18	song	song	NOUN
eajbcs-694	399	19	(	(	PUNCT
eajbcs-694	399	20	2004	2004	NUM
eajbcs-694	399	21	)	)	PUNCT
eajbcs-694	399	22	.	.	PUNCT
eajbcs-694	400	1	in	in	ADP
eajbcs-694	400	2	short	short	ADJ
eajbcs-694	400	3	,	,	PUNCT
eajbcs-694	400	4	the	the	DET
eajbcs-694	400	5	theory	theory	NOUN
eajbcs-694	400	6	is	be	AUX
eajbcs-694	400	7	summarized	summarize	VERB
eajbcs-694	400	8	by	by	ADP
eajbcs-694	400	9	theorem	theorem	VERB
eajbcs-694	400	10	4.1	4.1	NUM
eajbcs-694	400	11	in	in	ADP
eajbcs-694	400	12	castillochavez	castillochavez	PROPN
eajbcs-694	400	13	and	and	CCONJ
eajbcs-694	400	14	song	song	NOUN
eajbcs-694	400	15	(	(	PUNCT
eajbcs-694	400	16	2004	2004	NUM
eajbcs-694	400	17	)	)	PUNCT
eajbcs-694	400	18	.	.	PUNCT
eajbcs-694	401	1	in	in	ADP
eajbcs-694	401	2	such	such	DET
eajbcs-694	401	3	a	a	DET
eajbcs-694	401	4	theorem	theorem	NOUN
eajbcs-694	401	5	,	,	PUNCT
eajbcs-694	401	6	there	there	PRON
eajbcs-694	401	7	are	be	VERB
eajbcs-694	401	8	two	two	NUM
eajbcs-694	401	9	important	important	ADJ
eajbcs-694	401	10	quantities	quantity	NOUN
eajbcs-694	401	11	:	:	PUNCT
eajbcs-694	401	12	the	the	DET
eajbcs-694	401	13	coefficients	coefficient	NOUN
eajbcs-694	401	14	,	,	PUNCT
eajbcs-694	401	15	say	say	VERB
eajbcs-694	401	16	a	a	PRON
eajbcs-694	401	17	and	and	CCONJ
eajbcs-694	401	18	b	b	NOUN
eajbcs-694	401	19	,	,	PUNCT
eajbcs-694	401	20	of	of	ADP
eajbcs-694	401	21	the	the	DET
eajbcs-694	401	22	normal	normal	ADJ
eajbcs-694	401	23	form	form	NOUN
eajbcs-694	401	24	representing	represent	VERB
eajbcs-694	401	25	the	the	DET
eajbcs-694	401	26	dynamics	dynamic	NOUN
eajbcs-694	401	27	of	of	ADP
eajbcs-694	401	28	the	the	DET
eajbcs-694	401	29	system	system	NOUN
eajbcs-694	401	30	on	on	ADP
eajbcs-694	401	31	the	the	DET
eajbcs-694	401	32	central	central	ADJ
eajbcs-694	401	33	manifold	manifold	NOUN
eajbcs-694	401	34	.	.	PUNCT
eajbcs-694	402	1	these	these	DET
eajbcs-694	402	2	coefficients	coefficient	NOUN
eajbcs-694	402	3	decide	decide	VERB
eajbcs-694	402	4	the	the	DET
eajbcs-694	402	5	bifurcation	bifurcation	NOUN
eajbcs-694	402	6	.	.	PUNCT
eajbcs-694	403	1	in	in	ADP
eajbcs-694	403	2	particular	particular	ADJ
eajbcs-694	403	3	,	,	PUNCT
eajbcs-694	403	4	if	if	SCONJ
eajbcs-694	403	5	a	a	DET
eajbcs-694	403	6	<	<	X
eajbcs-694	403	7	0	0	NUM
eajbcs-694	403	8	and	and	CCONJ
eajbcs-694	403	9	b	b	NOUN
eajbcs-694	403	10	>	>	X
eajbcs-694	403	11	0	0	NUM
eajbcs-694	403	12	,	,	PUNCT
eajbcs-694	403	13	then	then	ADV
eajbcs-694	403	14	the	the	DET
eajbcs-694	403	15	bifurcation	bifurcation	NOUN
eajbcs-694	403	16	is	be	AUX
eajbcs-694	403	17	forward	forward	ADV
eajbcs-694	403	18	.	.	PUNCT
eajbcs-694	404	1	in	in	ADP
eajbcs-694	404	2	the	the	DET
eajbcs-694	404	3	proof	proof	NOUN
eajbcs-694	404	4	of	of	ADP
eajbcs-694	404	5	theorem	theorem	NOUN
eajbcs-694	404	6	(	(	PUNCT
eajbcs-694	404	7	3.6	3.6	NUM
eajbcs-694	404	8	)	)	PUNCT
eajbcs-694	404	9	,	,	PUNCT
eajbcs-694	404	10	we	we	PRON
eajbcs-694	404	11	have	have	AUX
eajbcs-694	404	12	already	already	ADV
eajbcs-694	404	13	justified	justify	VERB
eajbcs-694	404	14	the	the	DET
eajbcs-694	404	15	system	system	NOUN
eajbcs-694	404	16	(	(	PUNCT
eajbcs-694	404	17	2	2	X
eajbcs-694	404	18	)	)	PUNCT
eajbcs-694	404	19	undergoes	undergo	VERB
eajbcs-694	404	20	a	a	DET
eajbcs-694	404	21	forward	forward	ADJ
eajbcs-694	404	22	bifurcation	bifurcation	NOUN
eajbcs-694	404	23	at	at	ADP
eajbcs-694	404	24	r0	r0	NOUN
eajbcs-694	404	25	=	=	NOUN
eajbcs-694	404	26	1	1	NUM
eajbcs-694	404	27	.	.	PUNCT
eajbcs-694	405	1	thus	thus	ADV
eajbcs-694	405	2	the	the	DET
eajbcs-694	405	3	basic	basic	ADJ
eajbcs-694	405	4	reproduction	reproduction	NOUN
eajbcs-694	405	5	number	number	NOUN
eajbcs-694	405	6	r0	r0	NOUN
eajbcs-694	405	7	plays	play	VERB
eajbcs-694	405	8	an	an	DET
eajbcs-694	405	9	important	important	ADJ
eajbcs-694	405	10	role	role	NOUN
eajbcs-694	405	11	in	in	ADP
eajbcs-694	405	12	the	the	DET
eajbcs-694	405	13	disease	disease	NOUN
eajbcs-694	405	14	spread	spread	VERB
eajbcs-694	405	15	.	.	PUNCT
eajbcs-694	406	1	if	if	SCONJ
eajbcs-694	406	2	r0	r0	NOUN
eajbcs-694	406	3	<	<	X
eajbcs-694	406	4	1	1	NUM
eajbcs-694	406	5	,	,	PUNCT
eajbcs-694	406	6	then	then	ADV
eajbcs-694	406	7	its	its	PRON
eajbcs-694	406	8	easy	easy	ADJ
eajbcs-694	406	9	to	to	PART
eajbcs-694	406	10	control	control	VERB
eajbcs-694	406	11	the	the	DET
eajbcs-694	406	12	disease	disease	NOUN
eajbcs-694	406	13	but	but	CCONJ
eajbcs-694	406	14	if	if	SCONJ
eajbcs-694	406	15	r0	r0	NOUN
eajbcs-694	406	16	>	>	X
eajbcs-694	406	17	1	1	NUM
eajbcs-694	406	18	,	,	PUNCT
eajbcs-694	406	19	then	then	ADV
eajbcs-694	406	20	the	the	DET
eajbcs-694	406	21	society	society	NOUN
eajbcs-694	406	22	will	will	AUX
eajbcs-694	406	23	experience	experience	VERB
eajbcs-694	406	24	endemic	endemic	ADJ
eajbcs-694	406	25	disease	disease	NOUN
eajbcs-694	406	26	spreading	spread	VERB
eajbcs-694	406	27	.	.	PUNCT
eajbcs-694	407	1	the	the	DET
eajbcs-694	407	2	forward	forward	ADJ
eajbcs-694	407	3	bifurcation	bifurcation	NOUN
eajbcs-694	407	4	diagram	diagram	NOUN
eajbcs-694	407	5	can	can	AUX
eajbcs-694	407	6	be	be	AUX
eajbcs-694	407	7	seen	see	VERB
eajbcs-694	407	8	in	in	ADP
eajbcs-694	407	9	figure	figure	NOUN
eajbcs-694	407	10	(	(	PUNCT
eajbcs-694	407	11	2	2	NUM
eajbcs-694	407	12	)	)	PUNCT
eajbcs-694	407	13	.	.	PUNCT
eajbcs-694	408	1	east	east	PROPN
eajbcs-694	408	2	afr	afr	PROPN
eajbcs-694	408	3	.	.	PUNCT
eajbcs-694	409	1	j.	j.	PROPN
eajbcs-694	409	2	biophys	biophys	PROPN
eajbcs-694	409	3	.	.	PUNCT
eajbcs-694	410	1	comput	comput	NOUN
eajbcs-694	410	2	.	.	PUNCT
eajbcs-694	411	1	sci	sci	PROPN
eajbcs-694	411	2	.	.	PUNCT
eajbcs-694	412	1	(	(	PUNCT
eajbcs-694	412	2	2024	2024	NUM
eajbcs-694	412	3	)	)	PUNCT
eajbcs-694	412	4	,	,	PUNCT
eajbcs-694	412	5	vol	vol	NOUN
eajbcs-694	412	6	.	.	PROPN
eajbcs-694	412	7	5	5	NUM
eajbcs-694	412	8	,	,	PUNCT
eajbcs-694	412	9	no	no	INTJ
eajbcs-694	412	10	.	.	NOUN
eajbcs-694	412	11	2	2	NUM
eajbcs-694	412	12	,	,	PUNCT
eajbcs-694	412	13	58	58	NUM
eajbcs-694	412	14	-	-	SYM
eajbcs-694	412	15	86	86	NUM
eajbcs-694	412	16	70	70	NUM
eajbcs-694	412	17	figure	figure	NOUN
eajbcs-694	412	18	2	2	NUM
eajbcs-694	412	19	:	:	PUNCT
eajbcs-694	412	20	forward	forward	ADJ
eajbcs-694	412	21	bifurcation	bifurcation	NOUN
eajbcs-694	412	22	diagram	diagram	NOUN
eajbcs-694	412	23	for	for	ADP
eajbcs-694	412	24	the	the	DET
eajbcs-694	412	25	covid-19	covid-19	PROPN
eajbcs-694	412	26	model	model	NOUN
eajbcs-694	412	27	(	(	PUNCT
eajbcs-694	412	28	2	2	NUM
eajbcs-694	412	29	)	)	PUNCT
eajbcs-694	412	30	.	.	PUNCT
eajbcs-694	413	1	from	from	ADP
eajbcs-694	413	2	figure	figure	NOUN
eajbcs-694	413	3	(	(	PUNCT
eajbcs-694	413	4	2	2	NUM
eajbcs-694	413	5	)	)	PUNCT
eajbcs-694	413	6	,	,	PUNCT
eajbcs-694	413	7	it	it	PRON
eajbcs-694	413	8	is	be	AUX
eajbcs-694	413	9	clear	clear	ADJ
eajbcs-694	413	10	that	that	SCONJ
eajbcs-694	413	11	when	when	SCONJ
eajbcs-694	413	12	r0	r0	NOUN
eajbcs-694	413	13	<	<	X
eajbcs-694	413	14	1	1	NUM
eajbcs-694	413	15	,	,	PUNCT
eajbcs-694	413	16	the	the	DET
eajbcs-694	413	17	system	system	NOUN
eajbcs-694	413	18	(	(	PUNCT
eajbcs-694	413	19	2	2	X
eajbcs-694	413	20	)	)	PUNCT
eajbcs-694	413	21	has	have	VERB
eajbcs-694	413	22	no	no	DET
eajbcs-694	413	23	endemic	endemic	ADJ
eajbcs-694	413	24	equilibrium	equilibrium	NOUN
eajbcs-694	413	25	and	and	CCONJ
eajbcs-694	413	26	the	the	DET
eajbcs-694	413	27	disease	disease	NOUN
eajbcs-694	413	28	-	-	PUNCT
eajbcs-694	413	29	free	free	ADJ
eajbcs-694	413	30	equilibrium	equilibrium	NOUN
eajbcs-694	413	31	is	be	AUX
eajbcs-694	413	32	stable	stable	ADJ
eajbcs-694	413	33	.	.	PUNCT
eajbcs-694	414	1	when	when	SCONJ
eajbcs-694	414	2	r0	r0	NOUN
eajbcs-694	414	3	>	>	X
eajbcs-694	414	4	1	1	NUM
eajbcs-694	414	5	,	,	PUNCT
eajbcs-694	414	6	a	a	DET
eajbcs-694	414	7	stable	stable	ADJ
eajbcs-694	414	8	endemic	endemic	ADJ
eajbcs-694	414	9	equilibrium	equilibrium	NOUN
eajbcs-694	414	10	appears	appear	VERB
eajbcs-694	414	11	and	and	CCONJ
eajbcs-694	414	12	the	the	DET
eajbcs-694	414	13	disease	disease	NOUN
eajbcs-694	414	14	free	free	ADJ
eajbcs-694	414	15	equilibrium	equilibrium	NOUN
eajbcs-694	414	16	becomes	become	VERB
eajbcs-694	414	17	unstable	unstable	ADJ
eajbcs-694	414	18	,	,	PUNCT
eajbcs-694	414	19	i.e.	i.e.	X
eajbcs-694	414	20	exchange	exchange	NOUN
eajbcs-694	414	21	of	of	ADP
eajbcs-694	414	22	stability	stability	NOUN
eajbcs-694	414	23	of	of	ADP
eajbcs-694	414	24	the	the	DET
eajbcs-694	414	25	equilibrium	equilibrium	NOUN
eajbcs-694	414	26	’s	’s	PART
eajbcs-694	414	27	(	(	PUNCT
eajbcs-694	414	28	forward	forward	ADJ
eajbcs-694	414	29	bifurcation	bifurcation	NOUN
eajbcs-694	414	30	)	)	PUNCT
eajbcs-694	414	31	occurs	occur	VERB
eajbcs-694	414	32	at	at	ADP
eajbcs-694	414	33	the	the	DET
eajbcs-694	414	34	bifurcation	bifurcation	NOUN
eajbcs-694	414	35	point	point	NOUN
eajbcs-694	414	36	r∗	r∗	NOUN
eajbcs-694	414	37	0	0	NUM
eajbcs-694	415	1	=	=	SYM
eajbcs-694	415	2	1	1	NUM
eajbcs-694	415	3	.	.	NUM
eajbcs-694	415	4	3.9	3.9	NUM
eajbcs-694	415	5	.	.	PUNCT
eajbcs-694	416	1	sensitivity	sensitivity	NOUN
eajbcs-694	416	2	analysis	analysis	NOUN
eajbcs-694	416	3	sensitivity	sensitivity	NOUN
eajbcs-694	416	4	analysis	analysis	NOUN
eajbcs-694	416	5	is	be	AUX
eajbcs-694	416	6	a	a	DET
eajbcs-694	416	7	useful	useful	ADJ
eajbcs-694	416	8	tool	tool	NOUN
eajbcs-694	416	9	in	in	ADP
eajbcs-694	416	10	model	model	NOUN
eajbcs-694	416	11	building	building	NOUN
eajbcs-694	416	12	as	as	ADV
eajbcs-694	416	13	well	well	ADV
eajbcs-694	416	14	as	as	ADP
eajbcs-694	416	15	in	in	ADP
eajbcs-694	416	16	model	model	NOUN
eajbcs-694	416	17	evaluation	evaluation	NOUN
eajbcs-694	416	18	by	by	ADP
eajbcs-694	416	19	showing	show	VERB
eajbcs-694	416	20	how	how	SCONJ
eajbcs-694	416	21	the	the	DET
eajbcs-694	416	22	model	model	NOUN
eajbcs-694	416	23	behavior	behavior	NOUN
eajbcs-694	416	24	responds	respond	VERB
eajbcs-694	416	25	to	to	ADP
eajbcs-694	416	26	changes	change	NOUN
eajbcs-694	416	27	in	in	ADP
eajbcs-694	416	28	parameter	parameter	NOUN
eajbcs-694	416	29	values	value	NOUN
eajbcs-694	416	30	martcheva	martcheva	PROPN
eajbcs-694	416	31	(	(	PUNCT
eajbcs-694	416	32	2015	2015	NUM
eajbcs-694	416	33	)	)	PUNCT
eajbcs-694	416	34	.	.	PUNCT
eajbcs-694	417	1	the	the	DET
eajbcs-694	417	2	threshold	threshold	NOUN
eajbcs-694	417	3	parameter	parameter	PROPN
eajbcs-694	417	4	r0	r0	NOUN
eajbcs-694	417	5	which	which	PRON
eajbcs-694	417	6	determines	determine	VERB
eajbcs-694	417	7	stability	stability	NOUN
eajbcs-694	417	8	is	be	AUX
eajbcs-694	417	9	a	a	DET
eajbcs-694	417	10	function	function	NOUN
eajbcs-694	417	11	of	of	ADP
eajbcs-694	417	12	the	the	DET
eajbcs-694	417	13	parameters	parameter	NOUN
eajbcs-694	417	14	π	π	PROPN
eajbcs-694	417	15	,	,	PUNCT
eajbcs-694	417	16	β	β	PROPN
eajbcs-694	417	17	,	,	PUNCT
eajbcs-694	417	18	σ1	σ1	PROPN
eajbcs-694	417	19	,	,	PUNCT
eajbcs-694	417	20	σ2	σ2	PROPN
eajbcs-694	417	21	,	,	PUNCT
eajbcs-694	417	22	σ3	σ3	PROPN
eajbcs-694	417	23	,	,	PUNCT
eajbcs-694	417	24	ν	ν	PROPN
eajbcs-694	417	25	,	,	PUNCT
eajbcs-694	417	26	µ	µ	NOUN
eajbcs-694	417	27	,	,	PUNCT
eajbcs-694	417	28	θ	θ	PROPN
eajbcs-694	417	29	,	,	PUNCT
eajbcs-694	417	30	δ	δ	PROPN
eajbcs-694	417	31	,	,	PUNCT
eajbcs-694	417	32	γ	γ	PROPN
eajbcs-694	417	33	,	,	PUNCT
eajbcs-694	417	34	ξ	ξ	PROPN
eajbcs-694	417	35	,	,	PUNCT
eajbcs-694	417	36	α	α	PROPN
eajbcs-694	417	37	,	,	PUNCT
eajbcs-694	417	38	ε	ε	PROPN
eajbcs-694	417	39	,	,	PUNCT
eajbcs-694	417	40	ρ	ρ	PROPN
eajbcs-694	417	41	,	,	PUNCT
eajbcs-694	417	42	τ	τ	X
eajbcs-694	417	43	.	.	PUNCT
eajbcs-694	418	1	we	we	PRON
eajbcs-694	418	2	recall	recall	VERB
eajbcs-694	418	3	that	that	SCONJ
eajbcs-694	418	4	the	the	DET
eajbcs-694	418	5	basic	basic	ADJ
eajbcs-694	418	6	reproduction	reproduction	NOUN
eajbcs-694	418	7	number	number	NOUN
eajbcs-694	418	8	r0	r0	NOUN
eajbcs-694	418	9	is	be	AUX
eajbcs-694	418	10	given	give	VERB
eajbcs-694	418	11	by	by	ADP
eajbcs-694	418	12	r0	r0	NOUN
eajbcs-694	418	13	=	=	SYM
eajbcs-694	418	14	πβ(ν	πβ(ν	X
eajbcs-694	418	15	+	+	X
eajbcs-694	418	16	µ	µ	X
eajbcs-694	418	17	)	)	PUNCT
eajbcs-694	419	1	[	[	X
eajbcs-694	419	2	(	(	PUNCT
eajbcs-694	419	3	γ	γ	X
eajbcs-694	419	4	+	+	X
eajbcs-694	419	5	µ+	µ+	X
eajbcs-694	419	6	ξ	ξ	NOUN
eajbcs-694	419	7	)	)	PUNCT
eajbcs-694	420	1	[	[	X
eajbcs-694	420	2	σ2(α	σ2(α	X
eajbcs-694	420	3	+	+	X
eajbcs-694	420	4	ε+	ε+	X
eajbcs-694	420	5	µ+	µ+	X
eajbcs-694	420	6	ρ	ρ	NOUN
eajbcs-694	420	7	)	)	PUNCT
eajbcs-694	420	8	+	+	CCONJ
eajbcs-694	420	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	420	10	]	]	X
eajbcs-694	420	11	+	+	CCONJ
eajbcs-694	420	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	420	13	]	]	X
eajbcs-694	420	14	µ(θ	µ(θ	PROPN
eajbcs-694	420	15	+	+	NUM
eajbcs-694	420	16	ν	ν	X
eajbcs-694	420	17	+	+	X
eajbcs-694	420	18	µ)(δ	µ)(δ	PUNCT
eajbcs-694	420	19	+	+	CCONJ
eajbcs-694	420	20	µ)(α	µ)(α	X
eajbcs-694	420	21	+	+	CCONJ
eajbcs-694	420	22	ε+	ε+	X
eajbcs-694	420	23	µ+	µ+	X
eajbcs-694	420	24	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	420	25	+	+	CCONJ
eajbcs-694	420	26	µ+	µ+	X
eajbcs-694	420	27	ξ	ξ	NOUN
eajbcs-694	420	28	)	)	PUNCT
eajbcs-694	420	29	.	.	PUNCT
eajbcs-694	421	1	thus	thus	ADV
eajbcs-694	421	2	,	,	PUNCT
eajbcs-694	421	3	in	in	ADP
eajbcs-694	421	4	order	order	NOUN
eajbcs-694	421	5	to	to	PART
eajbcs-694	421	6	identify	identify	VERB
eajbcs-694	421	7	the	the	DET
eajbcs-694	421	8	most	most	ADV
eajbcs-694	421	9	sensitive	sensitive	ADJ
eajbcs-694	421	10	parameters	parameter	NOUN
eajbcs-694	421	11	for	for	ADP
eajbcs-694	421	12	model	model	NOUN
eajbcs-694	421	13	(	(	PUNCT
eajbcs-694	421	14	2	2	NUM
eajbcs-694	421	15	)	)	PUNCT
eajbcs-694	421	16	,	,	PUNCT
eajbcs-694	421	17	we	we	PRON
eajbcs-694	421	18	compute	compute	VERB
eajbcs-694	421	19	the	the	DET
eajbcs-694	421	20	relative	relative	ADJ
eajbcs-694	421	21	sensitivity	sensitivity	NOUN
eajbcs-694	421	22	of	of	ADP
eajbcs-694	421	23	r0	r0	NOUN
eajbcs-694	421	24	with	with	ADP
eajbcs-694	421	25	respect	respect	NOUN
eajbcs-694	421	26	to	to	ADP
eajbcs-694	421	27	the	the	DET
eajbcs-694	421	28	above	above	ADJ
eajbcs-694	421	29	parameters	parameter	NOUN
eajbcs-694	421	30	.	.	PUNCT
eajbcs-694	422	1	then	then	ADV
eajbcs-694	422	2	using	use	VERB
eajbcs-694	422	3	the	the	DET
eajbcs-694	422	4	parameter	parameter	NOUN
eajbcs-694	422	5	values	value	NOUN
eajbcs-694	422	6	from	from	ADP
eajbcs-694	422	7	table	table	NOUN
eajbcs-694	422	8	(	(	PUNCT
eajbcs-694	422	9	3	3	NUM
eajbcs-694	422	10	)	)	PUNCT
eajbcs-694	422	11	,	,	PUNCT
eajbcs-694	422	12	we	we	PRON
eajbcs-694	422	13	display	display	VERB
eajbcs-694	422	14	the	the	DET
eajbcs-694	422	15	sensitivity	sensitivity	NOUN
eajbcs-694	422	16	indices	index	NOUN
eajbcs-694	422	17	of	of	ADP
eajbcs-694	422	18	r0	r0	NOUN
eajbcs-694	422	19	with	with	ADP
eajbcs-694	422	20	respect	respect	NOUN
eajbcs-694	422	21	to	to	ADP
eajbcs-694	422	22	the	the	DET
eajbcs-694	422	23	parameters	parameter	NOUN
eajbcs-694	422	24	in	in	ADP
eajbcs-694	422	25	figure	figure	NOUN
eajbcs-694	422	26	(	(	PUNCT
eajbcs-694	422	27	3	3	NUM
eajbcs-694	422	28	)	)	PUNCT
eajbcs-694	422	29	.	.	PUNCT
eajbcs-694	423	1	east	east	PROPN
eajbcs-694	423	2	afr	afr	PROPN
eajbcs-694	423	3	.	.	PUNCT
eajbcs-694	424	1	j.	j.	PROPN
eajbcs-694	424	2	biophys	biophys	PROPN
eajbcs-694	424	3	.	.	PUNCT
eajbcs-694	425	1	comput	comput	NOUN
eajbcs-694	425	2	.	.	PUNCT
eajbcs-694	426	1	sci	sci	PROPN
eajbcs-694	426	2	.	.	PUNCT
eajbcs-694	427	1	(	(	PUNCT
eajbcs-694	427	2	2024	2024	NUM
eajbcs-694	427	3	)	)	PUNCT
eajbcs-694	427	4	,	,	PUNCT
eajbcs-694	427	5	vol	vol	NOUN
eajbcs-694	427	6	.	.	PROPN
eajbcs-694	427	7	5	5	NUM
eajbcs-694	427	8	,	,	PUNCT
eajbcs-694	427	9	no	no	INTJ
eajbcs-694	427	10	.	.	NOUN
eajbcs-694	427	11	2	2	NUM
eajbcs-694	427	12	,	,	PUNCT
eajbcs-694	427	13	58	58	NUM
eajbcs-694	427	14	-	-	SYM
eajbcs-694	427	15	86	86	NUM
eajbcs-694	427	16	71	71	NUM
eajbcs-694	427	17	△	△	NOUN
eajbcs-694	427	18	r0	r0	NOUN
eajbcs-694	427	19	β	β	NOUN
eajbcs-694	427	20	=	=	PUNCT
eajbcs-694	428	1	∂r0	∂r0	ADP
eajbcs-694	428	2	∂β	∂β	PROPN
eajbcs-694	428	3	×	×	NOUN
eajbcs-694	428	4	β	β	X
eajbcs-694	428	5	r0	r0	NOUN
eajbcs-694	428	6	=	=	SYM
eajbcs-694	428	7	1	1	NUM
eajbcs-694	428	8	,	,	PUNCT
eajbcs-694	428	9	△	△	X
eajbcs-694	428	10	r0	r0	NOUN
eajbcs-694	428	11	π	π	NOUN
eajbcs-694	428	12	=	=	PUNCT
eajbcs-694	429	1	∂r0	∂r0	ADP
eajbcs-694	429	2	∂π	∂π	PROPN
eajbcs-694	429	3	×	×	PROPN
eajbcs-694	429	4	π	π	PROPN
eajbcs-694	429	5	r0	r0	NOUN
eajbcs-694	429	6	=	=	SYM
eajbcs-694	429	7	1	1	NUM
eajbcs-694	429	8	,	,	PUNCT
eajbcs-694	429	9	△	△	X
eajbcs-694	429	10	r0	r0	NOUN
eajbcs-694	429	11	σ1	σ1	NOUN
eajbcs-694	429	12	=	=	PUNCT
eajbcs-694	430	1	∂r0	∂r0	ADP
eajbcs-694	430	2	∂σ1	∂σ1	PROPN
eajbcs-694	430	3	×	×	PROPN
eajbcs-694	430	4	σ1	σ1	NOUN
eajbcs-694	430	5	r0	r0	NOUN
eajbcs-694	430	6	=	=	NOUN
eajbcs-694	430	7	σ1τδ(γ	σ1τδ(γ	PROPN
eajbcs-694	430	8	+	+	CCONJ
eajbcs-694	430	9	µ+	µ+	X
eajbcs-694	430	10	ξ	ξ	X
eajbcs-694	430	11	)	)	PUNCT
eajbcs-694	430	12	(	(	PUNCT
eajbcs-694	430	13	γ	γ	X
eajbcs-694	430	14	+	+	X
eajbcs-694	430	15	µ+	µ+	X
eajbcs-694	430	16	ξ	ξ	NOUN
eajbcs-694	430	17	)	)	PUNCT
eajbcs-694	431	1	[	[	X
eajbcs-694	431	2	σ2(α	σ2(α	X
eajbcs-694	431	3	+	+	X
eajbcs-694	431	4	ε+	ε+	X
eajbcs-694	431	5	µ+	µ+	X
eajbcs-694	431	6	ρ	ρ	NOUN
eajbcs-694	431	7	)	)	PUNCT
eajbcs-694	431	8	+	+	CCONJ
eajbcs-694	431	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	431	10	]	]	X
eajbcs-694	431	11	+	+	CCONJ
eajbcs-694	431	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	431	13	,	,	PUNCT
eajbcs-694	431	14	△	△	X
eajbcs-694	431	15	r0	r0	NOUN
eajbcs-694	431	16	σ2	σ2	NOUN
eajbcs-694	431	17	=	=	PUNCT
eajbcs-694	432	1	∂r0	∂r0	PROPN
eajbcs-694	432	2	∂σ2	∂σ2	PROPN
eajbcs-694	432	3	×	×	PROPN
eajbcs-694	432	4	σ2	σ2	PROPN
eajbcs-694	432	5	r0	r0	NOUN
eajbcs-694	432	6	=	=	PUNCT
eajbcs-694	433	1	σ2(α	σ2(α	X
eajbcs-694	433	2	+	+	X
eajbcs-694	433	3	ε+	ε+	X
eajbcs-694	433	4	µ+	µ+	X
eajbcs-694	433	5	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	433	6	+	+	CCONJ
eajbcs-694	433	7	µ+	µ+	X
eajbcs-694	433	8	ξ	ξ	PROPN
eajbcs-694	433	9	)	)	PUNCT
eajbcs-694	433	10	(	(	PUNCT
eajbcs-694	433	11	γ	γ	X
eajbcs-694	433	12	+	+	X
eajbcs-694	433	13	µ+	µ+	X
eajbcs-694	433	14	ξ	ξ	NOUN
eajbcs-694	433	15	)	)	PUNCT
eajbcs-694	434	1	[	[	X
eajbcs-694	434	2	σ2(α	σ2(α	X
eajbcs-694	434	3	+	+	X
eajbcs-694	434	4	ε+	ε+	X
eajbcs-694	434	5	µ+	µ+	X
eajbcs-694	434	6	ρ	ρ	NOUN
eajbcs-694	434	7	)	)	PUNCT
eajbcs-694	434	8	+	+	CCONJ
eajbcs-694	434	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	434	10	]	]	X
eajbcs-694	434	11	+	+	CCONJ
eajbcs-694	434	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	434	13	,	,	PUNCT
eajbcs-694	434	14	△	△	X
eajbcs-694	434	15	r0	r0	NOUN
eajbcs-694	434	16	σ3	σ3	NOUN
eajbcs-694	434	17	=	=	PUNCT
eajbcs-694	435	1	∂r0	∂r0	PROPN
eajbcs-694	435	2	∂σ3	∂σ3	PROPN
eajbcs-694	435	3	×	×	NOUN
eajbcs-694	435	4	σ3	σ3	PROPN
eajbcs-694	435	5	r0	r0	NOUN
eajbcs-694	435	6	=	=	PUNCT
eajbcs-694	435	7	σ3τδα	σ3τδα	PROPN
eajbcs-694	435	8	(	(	PUNCT
eajbcs-694	435	9	γ	γ	X
eajbcs-694	435	10	+	+	X
eajbcs-694	435	11	µ+	µ+	X
eajbcs-694	435	12	ξ	ξ	NOUN
eajbcs-694	435	13	)	)	PUNCT
eajbcs-694	436	1	[	[	X
eajbcs-694	436	2	σ2(α	σ2(α	X
eajbcs-694	436	3	+	+	X
eajbcs-694	436	4	ε+	ε+	X
eajbcs-694	436	5	µ+	µ+	X
eajbcs-694	436	6	ρ	ρ	NOUN
eajbcs-694	436	7	)	)	PUNCT
eajbcs-694	436	8	+	+	CCONJ
eajbcs-694	436	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	436	10	]	]	X
eajbcs-694	436	11	+	+	CCONJ
eajbcs-694	436	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	436	13	,	,	PUNCT
eajbcs-694	436	14	△	△	X
eajbcs-694	436	15	r0	r0	NOUN
eajbcs-694	436	16	τ	τ	X
eajbcs-694	436	17	=	=	PUNCT
eajbcs-694	436	18	∂r0	∂r0	ADP
eajbcs-694	436	19	∂τ	∂τ	PROPN
eajbcs-694	436	20	×	×	NOUN
eajbcs-694	436	21	τ	τ	X
eajbcs-694	436	22	r0	r0	NOUN
eajbcs-694	436	23	=	=	PUNCT
eajbcs-694	436	24	τδ	τδ	X
eajbcs-694	437	1	[	[	X
eajbcs-694	437	2	σ1(γ	σ1(γ	X
eajbcs-694	437	3	+	+	CCONJ
eajbcs-694	437	4	µ+	µ+	X
eajbcs-694	437	5	ξ	ξ	X
eajbcs-694	437	6	)	)	PUNCT
eajbcs-694	438	1	+	+	NUM
eajbcs-694	438	2	σ3α	σ3α	NOUN
eajbcs-694	438	3	]	]	PUNCT
eajbcs-694	438	4	(	(	PUNCT
eajbcs-694	438	5	γ	γ	X
eajbcs-694	438	6	+	+	X
eajbcs-694	438	7	µ+	µ+	X
eajbcs-694	438	8	ξ	ξ	NOUN
eajbcs-694	438	9	)	)	PUNCT
eajbcs-694	439	1	[	[	X
eajbcs-694	439	2	σ2(α	σ2(α	X
eajbcs-694	439	3	+	+	X
eajbcs-694	439	4	ε+	ε+	X
eajbcs-694	439	5	µ+	µ+	X
eajbcs-694	439	6	ρ	ρ	NOUN
eajbcs-694	439	7	)	)	PUNCT
eajbcs-694	439	8	+	+	CCONJ
eajbcs-694	439	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	439	10	]	]	X
eajbcs-694	439	11	+	+	CCONJ
eajbcs-694	439	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	439	13	,	,	PUNCT
eajbcs-694	439	14	△	△	NOUN
eajbcs-694	439	15	r0	r0	NOUN
eajbcs-694	439	16	ν	ν	NOUN
eajbcs-694	439	17	=	=	PUNCT
eajbcs-694	440	1	∂r0	∂r0	ADP
eajbcs-694	440	2	∂ν	∂ν	PRON
eajbcs-694	440	3	×	×	NOUN
eajbcs-694	440	4	ν	ν	NOUN
eajbcs-694	440	5	r0	r0	NOUN
eajbcs-694	440	6	=	=	SYM
eajbcs-694	440	7	θν	θν	PROPN
eajbcs-694	440	8	(	(	PUNCT
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eajbcs-694	440	10	+	+	CCONJ
eajbcs-694	440	11	ν	ν	X
eajbcs-694	440	12	+	+	CCONJ
eajbcs-694	440	13	µ)(ν	µ)(ν	SYM
eajbcs-694	440	14	+	+	CCONJ
eajbcs-694	440	15	µ	µ	X
eajbcs-694	440	16	)	)	PUNCT
eajbcs-694	440	17	,	,	PUNCT
eajbcs-694	440	18	△	△	X
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eajbcs-694	440	20	α	α	NOUN
eajbcs-694	440	21	=	=	PUNCT
eajbcs-694	441	1	∂r0	∂r0	ADP
eajbcs-694	441	2	∂α	∂α	PROPN
eajbcs-694	441	3	×	×	NOUN
eajbcs-694	441	4	α	α	NOUN
eajbcs-694	441	5	r0	r0	NOUN
eajbcs-694	441	6	=	=	SYM
eajbcs-694	441	7	ατδ	ατδ	PROPN
eajbcs-694	442	1	[	[	X
eajbcs-694	442	2	σ3(ε+	σ3(ε+	PROPN
eajbcs-694	442	3	µ+	µ+	PROPN
eajbcs-694	442	4	ρ)−	ρ)−	PROPN
eajbcs-694	442	5	σ1(γ	σ1(γ	PROPN
eajbcs-694	442	6	+	+	CCONJ
eajbcs-694	442	7	µ+	µ+	X
eajbcs-694	442	8	ξ	ξ	PROPN
eajbcs-694	442	9	)	)	PUNCT
eajbcs-694	442	10	]	]	PUNCT
eajbcs-694	442	11	(	(	PUNCT
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eajbcs-694	442	13	+	+	X
eajbcs-694	442	14	ε+	ε+	X
eajbcs-694	442	15	µ+	µ+	X
eajbcs-694	442	16	ρ	ρ	NOUN
eajbcs-694	442	17	)	)	PUNCT
eajbcs-694	443	1	[	[	X
eajbcs-694	443	2	(	(	PUNCT
eajbcs-694	443	3	γ	γ	X
eajbcs-694	443	4	+	+	X
eajbcs-694	443	5	µ+	µ+	X
eajbcs-694	443	6	ξ	ξ	NOUN
eajbcs-694	443	7	)	)	PUNCT
eajbcs-694	444	1	[	[	X
eajbcs-694	444	2	σ2(α	σ2(α	X
eajbcs-694	444	3	+	+	X
eajbcs-694	444	4	ε+	ε+	X
eajbcs-694	444	5	µ+	µ+	X
eajbcs-694	444	6	ρ	ρ	NOUN
eajbcs-694	444	7	)	)	PUNCT
eajbcs-694	444	8	+	+	CCONJ
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eajbcs-694	444	10	]	]	X
eajbcs-694	444	11	+	+	CCONJ
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eajbcs-694	444	13	]	]	PUNCT
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eajbcs-694	444	15	△	△	X
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eajbcs-694	444	18	=	=	PUNCT
eajbcs-694	445	1	∂r0	∂r0	ADP
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eajbcs-694	445	3	×	×	PROPN
eajbcs-694	445	4	δ	δ	NOUN
eajbcs-694	445	5	r0	r0	NOUN
eajbcs-694	445	6	=	=	PUNCT
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eajbcs-694	446	1	[	[	X
eajbcs-694	446	2	(	(	PUNCT
eajbcs-694	446	3	γ	γ	X
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eajbcs-694	446	5	µ+	µ+	X
eajbcs-694	446	6	ξ	ξ	NOUN
eajbcs-694	446	7	)	)	PUNCT
eajbcs-694	447	1	[	[	X
eajbcs-694	447	2	σ1τµ−	σ1τµ−	X
eajbcs-694	447	3	σ2(α	σ2(α	X
eajbcs-694	447	4	+	+	X
eajbcs-694	447	5	ε+	ε+	X
eajbcs-694	447	6	µ+	µ+	X
eajbcs-694	447	7	ρ	ρ	NOUN
eajbcs-694	447	8	)	)	PUNCT
eajbcs-694	447	9	]	]	PUNCT
eajbcs-694	447	10	+	+	PUNCT
eajbcs-694	447	11	σ3ταµ	σ3ταµ	X
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eajbcs-694	447	13	(	(	PUNCT
eajbcs-694	447	14	δ	δ	PROPN
eajbcs-694	447	15	+	+	X
eajbcs-694	447	16	µ	µ	X
eajbcs-694	447	17	)	)	PUNCT
eajbcs-694	448	1	[	[	X
eajbcs-694	448	2	(	(	PUNCT
eajbcs-694	448	3	γ	γ	X
eajbcs-694	448	4	+	+	X
eajbcs-694	448	5	µ+	µ+	X
eajbcs-694	448	6	ξ	ξ	NOUN
eajbcs-694	448	7	)	)	PUNCT
eajbcs-694	449	1	[	[	X
eajbcs-694	449	2	σ2(α	σ2(α	X
eajbcs-694	449	3	+	+	X
eajbcs-694	449	4	ε+	ε+	X
eajbcs-694	449	5	µ+	µ+	X
eajbcs-694	449	6	ρ	ρ	NOUN
eajbcs-694	449	7	)	)	PUNCT
eajbcs-694	449	8	+	+	CCONJ
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eajbcs-694	449	10	]	]	X
eajbcs-694	449	11	+	+	CCONJ
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eajbcs-694	449	13	]	]	PUNCT
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eajbcs-694	449	15	△	△	X
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eajbcs-694	449	18	=	=	PUNCT
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eajbcs-694	450	4	θ	θ	NOUN
eajbcs-694	450	5	r0	r0	NOUN
eajbcs-694	450	6	=	=	PUNCT
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eajbcs-694	451	2	θ	θ	PROPN
eajbcs-694	451	3	θ	θ	PROPN
eajbcs-694	451	4	+	+	CCONJ
eajbcs-694	451	5	ν	ν	X
eajbcs-694	451	6	+	+	X
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eajbcs-694	451	8	,	,	PUNCT
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eajbcs-694	452	5	r0	r0	NOUN
eajbcs-694	452	6	=	=	PUNCT
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eajbcs-694	452	8	ετδ	ετδ	NOUN
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eajbcs-694	453	3	+	+	CCONJ
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eajbcs-694	453	5	ξ	ξ	X
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eajbcs-694	454	1	+	+	NUM
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eajbcs-694	455	7	)	)	PUNCT
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eajbcs-694	457	6	=	=	SYM
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eajbcs-694	457	11	+	+	CCONJ
eajbcs-694	457	12	µ+	µ+	X
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eajbcs-694	457	15	+	+	NUM
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eajbcs-694	457	22	µ+	µ+	X
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eajbcs-694	458	4	+	+	X
eajbcs-694	458	5	µ+	µ+	X
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eajbcs-694	459	7	)	)	PUNCT
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eajbcs-694	460	5	r0	r0	NOUN
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eajbcs-694	461	5	+	+	X
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eajbcs-694	462	1	[	[	X
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eajbcs-694	462	4	+	+	X
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eajbcs-694	463	8	+	+	CCONJ
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eajbcs-694	463	11	+	+	CCONJ
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eajbcs-694	463	15	△	△	X
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eajbcs-694	463	18	=	=	PUNCT
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eajbcs-694	464	4	ξ	ξ	X
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eajbcs-694	464	6	=	=	PUNCT
eajbcs-694	464	7	−	−	NOUN
eajbcs-694	464	8	σ3τδαξ	σ3τδαξ	NOUN
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eajbcs-694	464	10	γ	γ	X
eajbcs-694	464	11	+	+	X
eajbcs-694	464	12	µ+	µ+	X
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eajbcs-694	464	14	)	)	PUNCT
eajbcs-694	465	1	[	[	X
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eajbcs-694	465	3	γ	γ	X
eajbcs-694	465	4	+	+	X
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eajbcs-694	465	7	)	)	PUNCT
eajbcs-694	466	1	[	[	X
eajbcs-694	466	2	σ2(α	σ2(α	X
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eajbcs-694	466	8	+	+	CCONJ
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eajbcs-694	466	11	+	+	CCONJ
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eajbcs-694	466	15	△	△	X
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eajbcs-694	466	17	µ	µ	X
eajbcs-694	466	18	=	=	PUNCT
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eajbcs-694	466	20	∂µ	∂µ	PROPN
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eajbcs-694	466	22	µ	µ	DET
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eajbcs-694	466	32	−	−	PROPN
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eajbcs-694	466	34	σ2a(α	σ2a(α	PROPN
eajbcs-694	466	35	+	+	CCONJ
eajbcs-694	466	36	ε+	ε+	X
eajbcs-694	466	37	µ+	µ+	X
eajbcs-694	466	38	ρ)2(γ	ρ)2(γ	PROPN
eajbcs-694	466	39	+	+	CCONJ
eajbcs-694	466	40	µ+	µ+	X
eajbcs-694	466	41	ξ)2(ν	ξ)2(ν	PRON
eajbcs-694	466	42	+	+	NUM
eajbcs-694	466	43	µ	µ	X
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eajbcs-694	467	1	+	+	NUM
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eajbcs-694	468	1	+	+	CCONJ
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eajbcs-694	468	3	ρ)a+b](γ	ρ)a+b](γ	NOUN
eajbcs-694	468	4	+	+	CCONJ
eajbcs-694	468	5	µ+	µ+	X
eajbcs-694	468	6	ξ)2(ν	ξ)2(ν	ADP
eajbcs-694	468	7	+	+	NUM
eajbcs-694	468	8	µ	µ	X
eajbcs-694	468	9	)	)	PUNCT
eajbcs-694	469	1	+	+	CCONJ
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eajbcs-694	469	6	+	+	ADJ
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eajbcs-694	469	8	)	)	PUNCT
eajbcs-694	469	9	+	+	CCONJ
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eajbcs-694	469	12	+	+	NOUN
eajbcs-694	469	13	ε+	ε+	NOUN
eajbcs-694	469	14	ρ)b	ρ)b	X
eajbcs-694	469	15	+	+	CCONJ
eajbcs-694	469	16	c	c	X
eajbcs-694	469	17	]	]	X
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eajbcs-694	469	20	+	+	X
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eajbcs-694	469	24	]	]	PUNCT
eajbcs-694	469	25	,	,	PUNCT
eajbcs-694	469	26	where	where	SCONJ
eajbcs-694	469	27	a	a	DET
eajbcs-694	469	28	=	=	SYM
eajbcs-694	469	29	δ(θ	δ(θ	PROPN
eajbcs-694	469	30	+	+	CCONJ
eajbcs-694	469	31	ν	ν	NOUN
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eajbcs-694	469	33	+	+	NUM
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eajbcs-694	469	35	+	+	NUM
eajbcs-694	469	36	θ	θ	PROPN
eajbcs-694	469	37	+	+	SYM
eajbcs-694	469	38	ν	ν	X
eajbcs-694	469	39	)	)	PUNCT
eajbcs-694	469	40	+	+	NUM
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eajbcs-694	469	42	,	,	PUNCT
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eajbcs-694	469	44	=	=	SYM
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eajbcs-694	469	46	+	+	CCONJ
eajbcs-694	469	47	ν	ν	NOUN
eajbcs-694	469	48	)	)	PUNCT
eajbcs-694	469	49	+	+	CCONJ
eajbcs-694	469	50	3µ2(δ	3µ2(δ	NUM
eajbcs-694	469	51	+	+	NUM
eajbcs-694	469	52	θ	θ	PROPN
eajbcs-694	469	53	+	+	SYM
eajbcs-694	469	54	ν	ν	X
eajbcs-694	469	55	)	)	PUNCT
eajbcs-694	469	56	+	+	NUM
eajbcs-694	469	57	4µ3	4µ3	NUM
eajbcs-694	469	58	,	,	PUNCT
eajbcs-694	469	59	c	c	NOUN
eajbcs-694	469	60	=	=	SYM
eajbcs-694	469	61	3µ2δ(θ	3µ2δ(θ	NUM
eajbcs-694	469	62	+	+	CCONJ
eajbcs-694	469	63	ν	ν	NOUN
eajbcs-694	469	64	)	)	PUNCT
eajbcs-694	469	65	+	+	NOUN
eajbcs-694	469	66	4µ(δ	4µ(δ	NOUN
eajbcs-694	469	67	+	+	NUM
eajbcs-694	469	68	θ	θ	PROPN
eajbcs-694	469	69	+	+	SYM
eajbcs-694	469	70	ν	ν	X
eajbcs-694	469	71	)	)	PUNCT
eajbcs-694	469	72	+	+	CCONJ
eajbcs-694	469	73	5µ4	5µ4	NUM
eajbcs-694	469	74	,	,	PUNCT
eajbcs-694	469	75	d	d	NOUN
eajbcs-694	469	76	=	=	SYM
eajbcs-694	469	77	(	(	PUNCT
eajbcs-694	469	78	α	α	NOUN
eajbcs-694	469	79	+	+	X
eajbcs-694	469	80	ε+	ε+	X
eajbcs-694	469	81	µ+	µ+	X
eajbcs-694	469	82	ρ)(γ	ρ)(γ	NOUN
eajbcs-694	469	83	+	+	CCONJ
eajbcs-694	469	84	µ+	µ+	X
eajbcs-694	469	85	ξ)(δ	ξ)(δ	NUM
eajbcs-694	469	86	+	+	PUNCT
eajbcs-694	469	87	µ)(θ	µ)(θ	PUNCT
eajbcs-694	469	88	+	+	CCONJ
eajbcs-694	469	89	ν	ν	X
eajbcs-694	469	90	+	+	X
eajbcs-694	469	91	µ	µ	X
eajbcs-694	469	92	)	)	PUNCT
eajbcs-694	469	93	[	[	X
eajbcs-694	469	94	(	(	PUNCT
eajbcs-694	469	95	γ	γ	X
eajbcs-694	469	96	+	+	X
eajbcs-694	469	97	µ+	µ+	X
eajbcs-694	469	98	ξ	ξ	NOUN
eajbcs-694	469	99	)	)	PUNCT
eajbcs-694	470	1	[	[	X
eajbcs-694	470	2	σ2(α	σ2(α	X
eajbcs-694	470	3	+	+	X
eajbcs-694	470	4	ε+	ε+	X
eajbcs-694	470	5	µ+	µ+	X
eajbcs-694	470	6	ρ	ρ	NOUN
eajbcs-694	470	7	)	)	PUNCT
eajbcs-694	470	8	+	+	CCONJ
eajbcs-694	470	9	σ1τδ	σ1τδ	PUNCT
eajbcs-694	470	10	]	]	X
eajbcs-694	470	11	+	+	CCONJ
eajbcs-694	470	12	σ3τδα	σ3τδα	PROPN
eajbcs-694	470	13	]	]	PUNCT
eajbcs-694	470	14	.	.	PUNCT
eajbcs-694	471	1	east	east	PROPN
eajbcs-694	471	2	afr	afr	PROPN
eajbcs-694	471	3	.	.	PUNCT
eajbcs-694	472	1	j.	j.	PROPN
eajbcs-694	472	2	biophys	biophys	PROPN
eajbcs-694	472	3	.	.	PUNCT
eajbcs-694	473	1	comput	comput	NOUN
eajbcs-694	473	2	.	.	PUNCT
eajbcs-694	474	1	sci	sci	PROPN
eajbcs-694	474	2	.	.	PUNCT
eajbcs-694	475	1	(	(	PUNCT
eajbcs-694	475	2	2024	2024	NUM
eajbcs-694	475	3	)	)	PUNCT
eajbcs-694	475	4	,	,	PUNCT
eajbcs-694	475	5	vol	vol	NOUN
eajbcs-694	475	6	.	.	PROPN
eajbcs-694	475	7	5	5	NUM
eajbcs-694	475	8	,	,	PUNCT
eajbcs-694	475	9	no	no	INTJ
eajbcs-694	475	10	.	.	NOUN
eajbcs-694	475	11	2	2	NUM
eajbcs-694	475	12	,	,	PUNCT
eajbcs-694	475	13	58	58	NUM
eajbcs-694	475	14	-	-	SYM
eajbcs-694	475	15	86	86	NUM
eajbcs-694	475	16	72	72	NUM
eajbcs-694	475	17	note	note	NOUN
eajbcs-694	475	18	that	that	SCONJ
eajbcs-694	475	19	the	the	DET
eajbcs-694	475	20	sensitivity	sensitivity	NOUN
eajbcs-694	475	21	index	index	NOUN
eajbcs-694	475	22	may	may	AUX
eajbcs-694	475	23	depend	depend	VERB
eajbcs-694	475	24	on	on	ADP
eajbcs-694	475	25	several	several	ADJ
eajbcs-694	475	26	parameters	parameter	NOUN
eajbcs-694	475	27	of	of	ADP
eajbcs-694	475	28	the	the	DET
eajbcs-694	475	29	system	system	NOUN
eajbcs-694	475	30	,	,	PUNCT
eajbcs-694	475	31	but	but	CCONJ
eajbcs-694	475	32	also	also	ADV
eajbcs-694	475	33	can	can	AUX
eajbcs-694	475	34	be	be	AUX
eajbcs-694	475	35	constant	constant	ADJ
eajbcs-694	475	36	,	,	PUNCT
eajbcs-694	475	37	independent	independent	ADJ
eajbcs-694	475	38	of	of	ADP
eajbcs-694	475	39	any	any	DET
eajbcs-694	475	40	parameter	parameter	NOUN
eajbcs-694	475	41	.	.	PUNCT
eajbcs-694	476	1	for	for	ADP
eajbcs-694	476	2	example	example	NOUN
eajbcs-694	476	3	,	,	PUNCT
eajbcs-694	476	4	△	△	X
eajbcs-694	476	5	r0	r0	NOUN
eajbcs-694	476	6	β	β	X
eajbcs-694	476	7	=	=	SYM
eajbcs-694	476	8	+1	+1	PROPN
eajbcs-694	476	9	means	mean	VERB
eajbcs-694	476	10	that	that	SCONJ
eajbcs-694	476	11	increasing	increase	VERB
eajbcs-694	476	12	(	(	PUNCT
eajbcs-694	476	13	decreasing	decrease	VERB
eajbcs-694	476	14	)	)	PUNCT
eajbcs-694	476	15	β	β	NOUN
eajbcs-694	476	16	by	by	ADP
eajbcs-694	476	17	a	a	DET
eajbcs-694	476	18	given	give	VERB
eajbcs-694	476	19	percentage	percentage	NOUN
eajbcs-694	476	20	increases	increase	NOUN
eajbcs-694	476	21	(	(	PUNCT
eajbcs-694	476	22	decreases	decrease	NOUN
eajbcs-694	476	23	)	)	PUNCT
eajbcs-694	476	24	always	always	ADV
eajbcs-694	476	25	r0	r0	VERB
eajbcs-694	476	26	by	by	ADP
eajbcs-694	476	27	that	that	DET
eajbcs-694	476	28	same	same	ADJ
eajbcs-694	476	29	percentage	percentage	NOUN
eajbcs-694	476	30	.	.	PUNCT
eajbcs-694	477	1	figure	figure	NOUN
eajbcs-694	477	2	3	3	NUM
eajbcs-694	477	3	:	:	PUNCT
eajbcs-694	477	4	the	the	DET
eajbcs-694	477	5	sensitivity	sensitivity	NOUN
eajbcs-694	477	6	indices	indice	VERB
eajbcs-694	477	7	of	of	ADP
eajbcs-694	477	8	r0	r0	NOUN
eajbcs-694	477	9	with	with	ADP
eajbcs-694	477	10	respect	respect	NOUN
eajbcs-694	477	11	to	to	ADP
eajbcs-694	477	12	the	the	DET
eajbcs-694	477	13	parameters	parameter	NOUN
eajbcs-694	477	14	.	.	PUNCT
eajbcs-694	478	1	figure	figure	NOUN
eajbcs-694	478	2	(	(	PUNCT
eajbcs-694	478	3	3	3	X
eajbcs-694	478	4	)	)	PUNCT
eajbcs-694	478	5	shows	show	VERB
eajbcs-694	478	6	that	that	SCONJ
eajbcs-694	478	7	the	the	DET
eajbcs-694	478	8	recruitment	recruitment	NOUN
eajbcs-694	478	9	rate	rate	NOUN
eajbcs-694	478	10	π	π	PROPN
eajbcs-694	478	11	,	,	PUNCT
eajbcs-694	478	12	the	the	DET
eajbcs-694	478	13	contact	contact	NOUN
eajbcs-694	478	14	rate	rate	NOUN
eajbcs-694	478	15	β	β	PROPN
eajbcs-694	478	16	,	,	PUNCT
eajbcs-694	478	17	the	the	DET
eajbcs-694	478	18	modification	modification	NOUN
eajbcs-694	478	19	parameter	parameter	NOUN
eajbcs-694	478	20	σ2	σ2	PROPN
eajbcs-694	478	21	,	,	PUNCT
eajbcs-694	478	22	the	the	DET
eajbcs-694	478	23	exposed	expose	VERB
eajbcs-694	478	24	progression	progression	NOUN
eajbcs-694	478	25	rate	rate	NOUN
eajbcs-694	478	26	δ	δ	PROPN
eajbcs-694	478	27	,	,	PUNCT
eajbcs-694	478	28	the	the	DET
eajbcs-694	478	29	natural	natural	ADJ
eajbcs-694	478	30	death	death	NOUN
eajbcs-694	478	31	rate	rate	NOUN
eajbcs-694	478	32	µ	µ	PROPN
eajbcs-694	478	33	,	,	PUNCT
eajbcs-694	478	34	the	the	DET
eajbcs-694	478	35	protection	protection	NOUN
eajbcs-694	478	36	rate	rate	NOUN
eajbcs-694	478	37	θ	θ	PROPN
eajbcs-694	478	38	and	and	CCONJ
eajbcs-694	478	39	the	the	DET
eajbcs-694	478	40	waning	wane	VERB
eajbcs-694	478	41	rate	rate	NOUN
eajbcs-694	478	42	of	of	ADP
eajbcs-694	478	43	protected	protect	VERB
eajbcs-694	478	44	individuals	individual	NOUN
eajbcs-694	478	45	to	to	PART
eajbcs-694	478	46	susceptible	susceptible	ADJ
eajbcs-694	478	47	class	class	NOUN
eajbcs-694	478	48	ν	ν	NOUN
eajbcs-694	478	49	are	be	AUX
eajbcs-694	478	50	the	the	DET
eajbcs-694	478	51	most	most	ADV
eajbcs-694	478	52	sensitive	sensitive	ADJ
eajbcs-694	478	53	parameters	parameter	NOUN
eajbcs-694	478	54	for	for	ADP
eajbcs-694	478	55	r0	r0	NOUN
eajbcs-694	478	56	.	.	PUNCT
eajbcs-694	479	1	the	the	DET
eajbcs-694	479	2	parameters	parameter	NOUN
eajbcs-694	479	3	τ	τ	PROPN
eajbcs-694	479	4	,	,	PUNCT
eajbcs-694	479	5	σ1	σ1	PROPN
eajbcs-694	479	6	,	,	PUNCT
eajbcs-694	479	7	σ2	σ2	PROPN
eajbcs-694	479	8	,	,	PUNCT
eajbcs-694	479	9	σ3	σ3	PROPN
eajbcs-694	479	10	,	,	PUNCT
eajbcs-694	479	11	ν	ν	PROPN
eajbcs-694	479	12	,	,	PUNCT
eajbcs-694	479	13	β	β	X
eajbcs-694	479	14	and	and	CCONJ
eajbcs-694	479	15	π	π	PROPN
eajbcs-694	479	16	have	have	VERB
eajbcs-694	479	17	positive	positive	ADJ
eajbcs-694	479	18	correlation	correlation	NOUN
eajbcs-694	479	19	with	with	ADP
eajbcs-694	479	20	r0	r0	NOUN
eajbcs-694	479	21	.	.	PUNCT
eajbcs-694	480	1	this	this	PRON
eajbcs-694	480	2	indicates	indicate	VERB
eajbcs-694	480	3	that	that	SCONJ
eajbcs-694	480	4	the	the	DET
eajbcs-694	480	5	spread	spread	NOUN
eajbcs-694	480	6	of	of	ADP
eajbcs-694	480	7	covid-19	covid-19	PROPN
eajbcs-694	480	8	decreases	decrease	VERB
eajbcs-694	480	9	with	with	ADP
eajbcs-694	480	10	decrease	decrease	NOUN
eajbcs-694	480	11	of	of	ADP
eajbcs-694	480	12	these	these	DET
eajbcs-694	480	13	parameters	parameter	NOUN
eajbcs-694	480	14	.	.	PUNCT
eajbcs-694	481	1	the	the	DET
eajbcs-694	481	2	parameters	parameter	NOUN
eajbcs-694	481	3	ξ	ξ	PROPN
eajbcs-694	481	4	,	,	PUNCT
eajbcs-694	481	5	θ	θ	PROPN
eajbcs-694	481	6	,	,	PUNCT
eajbcs-694	481	7	ρ	ρ	PROPN
eajbcs-694	481	8	,	,	PUNCT
eajbcs-694	481	9	µ	µ	NOUN
eajbcs-694	481	10	,	,	PUNCT
eajbcs-694	481	11	γ	γ	PROPN
eajbcs-694	481	12	,	,	PUNCT
eajbcs-694	481	13	ε	ε	PROPN
eajbcs-694	481	14	,	,	PUNCT
eajbcs-694	481	15	δ	δ	PROPN
eajbcs-694	481	16	and	and	CCONJ
eajbcs-694	481	17	α	α	PROPN
eajbcs-694	481	18	have	have	VERB
eajbcs-694	481	19	negative	negative	ADJ
eajbcs-694	481	20	correlation	correlation	NOUN
eajbcs-694	481	21	with	with	ADP
eajbcs-694	481	22	r0	r0	NOUN
eajbcs-694	481	23	.	.	PUNCT
eajbcs-694	482	1	this	this	PRON
eajbcs-694	482	2	implies	imply	VERB
eajbcs-694	482	3	that	that	SCONJ
eajbcs-694	482	4	the	the	DET
eajbcs-694	482	5	spread	spread	NOUN
eajbcs-694	482	6	of	of	ADP
eajbcs-694	482	7	the	the	DET
eajbcs-694	482	8	virus	virus	NOUN
eajbcs-694	482	9	decreases	decrease	VERB
eajbcs-694	482	10	with	with	ADP
eajbcs-694	482	11	an	an	DET
eajbcs-694	482	12	increase	increase	NOUN
eajbcs-694	482	13	of	of	ADP
eajbcs-694	482	14	these	these	DET
eajbcs-694	482	15	parameters	parameter	NOUN
eajbcs-694	482	16	.	.	PUNCT
eajbcs-694	483	1	4	4	X
eajbcs-694	483	2	.	.	X
eajbcs-694	483	3	numerical	numerical	ADJ
eajbcs-694	483	4	simulations	simulation	NOUN
eajbcs-694	483	5	and	and	CCONJ
eajbcs-694	483	6	discussion	discussion	NOUN
eajbcs-694	483	7	in	in	ADP
eajbcs-694	483	8	this	this	DET
eajbcs-694	483	9	section	section	NOUN
eajbcs-694	483	10	,	,	PUNCT
eajbcs-694	483	11	we	we	PRON
eajbcs-694	483	12	perform	perform	VERB
eajbcs-694	483	13	numerical	numerical	ADJ
eajbcs-694	483	14	simulation	simulation	NOUN
eajbcs-694	483	15	to	to	PART
eajbcs-694	483	16	support	support	VERB
eajbcs-694	483	17	our	our	PRON
eajbcs-694	483	18	analytical	analytical	ADJ
eajbcs-694	483	19	results	result	NOUN
eajbcs-694	483	20	.	.	PUNCT
eajbcs-694	484	1	the	the	DET
eajbcs-694	484	2	numerical	numerical	ADJ
eajbcs-694	484	3	simulations	simulation	NOUN
eajbcs-694	484	4	are	be	AUX
eajbcs-694	484	5	carried	carry	VERB
eajbcs-694	484	6	out	out	ADP
eajbcs-694	484	7	with	with	ADP
eajbcs-694	484	8	help	help	NOUN
eajbcs-694	484	9	of	of	ADP
eajbcs-694	484	10	the	the	DET
eajbcs-694	484	11	ode45	ode45	NOUN
eajbcs-694	484	12	matlab	matlab	PROPN
eajbcs-694	484	13	tool	tool	NOUN
eajbcs-694	484	14	.	.	PUNCT
eajbcs-694	485	1	using	use	VERB
eajbcs-694	485	2	the	the	DET
eajbcs-694	485	3	parameter	parameter	NOUN
eajbcs-694	485	4	values	value	NOUN
eajbcs-694	485	5	given	give	VERB
eajbcs-694	485	6	in	in	ADP
eajbcs-694	485	7	table	table	NOUN
eajbcs-694	485	8	(	(	PUNCT
eajbcs-694	485	9	3	3	NUM
eajbcs-694	485	10	)	)	PUNCT
eajbcs-694	485	11	and	and	CCONJ
eajbcs-694	485	12	the	the	DET
eajbcs-694	485	13	initial	initial	ADJ
eajbcs-694	485	14	conditions	condition	NOUN
eajbcs-694	485	15	below	below	ADV
eajbcs-694	485	16	in	in	ADP
eajbcs-694	485	17	the	the	DET
eajbcs-694	485	18	model	model	NOUN
eajbcs-694	485	19	equations	equation	NOUN
eajbcs-694	485	20	(	(	PUNCT
eajbcs-694	485	21	2	2	NUM
eajbcs-694	485	22	)	)	PUNCT
eajbcs-694	485	23	simulation	simulation	NOUN
eajbcs-694	485	24	study	study	NOUN
eajbcs-694	485	25	is	be	AUX
eajbcs-694	485	26	conducted	conduct	VERB
eajbcs-694	485	27	.	.	PUNCT
eajbcs-694	486	1	the	the	DET
eajbcs-694	486	2	parameter	parameter	NOUN
eajbcs-694	486	3	values	value	NOUN
eajbcs-694	486	4	have	have	AUX
eajbcs-694	486	5	been	be	AUX
eajbcs-694	486	6	taken	take	VERB
eajbcs-694	486	7	based	base	VERB
eajbcs-694	486	8	on	on	ADP
eajbcs-694	486	9	the	the	DET
eajbcs-694	486	10	literature	literature	NOUN
eajbcs-694	486	11	and	and	CCONJ
eajbcs-694	486	12	the	the	DET
eajbcs-694	486	13	real	real	ADJ
eajbcs-694	486	14	characteristics	characteristic	NOUN
eajbcs-694	486	15	of	of	ADP
eajbcs-694	486	16	the	the	DET
eajbcs-694	486	17	virus	virus	NOUN
eajbcs-694	486	18	.	.	PUNCT
eajbcs-694	487	1	furthermore	furthermore	ADV
eajbcs-694	487	2	,	,	PUNCT
eajbcs-694	487	3	initial	initial	ADJ
eajbcs-694	487	4	conditions	condition	NOUN
eajbcs-694	487	5	are	be	AUX
eajbcs-694	487	6	determined	determine	VERB
eajbcs-694	487	7	as	as	ADP
eajbcs-694	487	8	follow	follow	NOUN
eajbcs-694	487	9	.	.	PUNCT
eajbcs-694	488	1	the	the	DET
eajbcs-694	488	2	total	total	ADJ
eajbcs-694	488	3	population	population	NOUN
eajbcs-694	488	4	of	of	ADP
eajbcs-694	488	5	ethiopia	ethiopia	PROPN
eajbcs-694	488	6	for	for	ADP
eajbcs-694	488	7	the	the	DET
eajbcs-694	488	8	year	year	NOUN
eajbcs-694	488	9	2021	2021	NUM
eajbcs-694	488	10	is	be	AUX
eajbcs-694	488	11	estimated	estimate	VERB
eajbcs-694	488	12	about	about	ADP
eajbcs-694	488	13	n(0	n(0	PROPN
eajbcs-694	488	14	)	)	PUNCT
eajbcs-694	488	15	=	=	NUM
eajbcs-694	488	16	114	114	NUM
eajbcs-694	488	17	,	,	PUNCT
eajbcs-694	488	18	963	963	NUM
eajbcs-694	488	19	,	,	PUNCT
eajbcs-694	488	20	588	588	NUM
eajbcs-694	488	21	people	people	NOUN
eajbcs-694	488	22	kifle	kifle	VERB
eajbcs-694	488	23	and	and	CCONJ
eajbcs-694	488	24	obsu	obsu	PROPN
eajbcs-694	488	25	(	(	PUNCT
eajbcs-694	488	26	2022	2022	NUM
eajbcs-694	488	27	)	)	PUNCT
eajbcs-694	488	28	.	.	PUNCT
eajbcs-694	489	1	on	on	ADP
eajbcs-694	489	2	december	december	PROPN
eajbcs-694	489	3	31	31	NUM
eajbcs-694	489	4	,	,	PUNCT
eajbcs-694	489	5	2020	2020	NUM
eajbcs-694	489	6	the	the	DET
eajbcs-694	489	7	total	total	ADJ
eajbcs-694	489	8	active	active	ADJ
eajbcs-694	489	9	cases	case	NOUN
eajbcs-694	489	10	(	(	PUNCT
eajbcs-694	489	11	infected	infect	VERB
eajbcs-694	489	12	individuals	individual	NOUN
eajbcs-694	489	13	are	be	AUX
eajbcs-694	489	14	10,245	10,245	NUM
eajbcs-694	489	15	i.e	i.e	PRON
eajbcs-694	489	16	i(0	i(0	PROPN
eajbcs-694	489	17	)	)	PUNCT
eajbcs-694	489	18	=	=	SYM
eajbcs-694	489	19	10	10	NUM
eajbcs-694	489	20	,	,	PUNCT
eajbcs-694	489	21	245	245	NUM
eajbcs-694	489	22	)	)	PUNCT
eajbcs-694	489	23	,	,	PUNCT
eajbcs-694	489	24	and	and	CCONJ
eajbcs-694	489	25	the	the	DET
eajbcs-694	489	26	hospitalized	hospitalize	VERB
eajbcs-694	489	27	are	be	AUX
eajbcs-694	489	28	h(0	h(0	PROPN
eajbcs-694	489	29	)	)	PUNCT
eajbcs-694	490	1	=	=	SYM
eajbcs-694	490	2	5062	5062	NUM
eajbcs-694	490	3	and	and	CCONJ
eajbcs-694	490	4	the	the	DET
eajbcs-694	490	5	total	total	NOUN
eajbcs-694	490	6	recovered	recover	VERB
eajbcs-694	490	7	by	by	ADP
eajbcs-694	490	8	the	the	DET
eajbcs-694	490	9	date	date	NOUN
eajbcs-694	490	10	are	be	AUX
eajbcs-694	490	11	r(0	r(0	PROPN
eajbcs-694	490	12	)	)	PUNCT
eajbcs-694	490	13	=	=	SYM
eajbcs-694	491	1	81	81	NUM
eajbcs-694	491	2	,	,	PUNCT
eajbcs-694	491	3	144	144	NUM
eajbcs-694	491	4	and	and	CCONJ
eajbcs-694	491	5	we	we	PRON
eajbcs-694	491	6	are	be	AUX
eajbcs-694	491	7	assumed	assume	VERB
eajbcs-694	491	8	p	p	X
eajbcs-694	491	9	(	(	PUNCT
eajbcs-694	491	10	0	0	NUM
eajbcs-694	491	11	)	)	PUNCT
eajbcs-694	491	12	=	=	SYM
eajbcs-694	491	13	108076	108076	NUM
eajbcs-694	491	14	,	,	PUNCT
eajbcs-694	491	15	e(0	e(0	NOUN
eajbcs-694	491	16	)	)	PUNCT
eajbcs-694	491	17	=	=	SYM
eajbcs-694	491	18	15000	15000	NUM
eajbcs-694	491	19	and	and	CCONJ
eajbcs-694	491	20	hence	hence	ADV
eajbcs-694	491	21	is	be	AUX
eajbcs-694	491	22	s(0	s(0	PROPN
eajbcs-694	491	23	)	)	PUNCT
eajbcs-694	492	1	=	=	SYM
eajbcs-694	492	2	n(0)−	n(0)−	PROPN
eajbcs-694	492	3	(	(	PUNCT
eajbcs-694	492	4	p	p	X
eajbcs-694	492	5	(	(	PUNCT
eajbcs-694	492	6	0	0	NUM
eajbcs-694	492	7	)	)	PUNCT
eajbcs-694	492	8	+	+	CCONJ
eajbcs-694	492	9	e(0	e(0	NOUN
eajbcs-694	492	10	)	)	PUNCT
eajbcs-694	493	1	+	+	CCONJ
eajbcs-694	493	2	i(0	i(0	PROPN
eajbcs-694	493	3	)	)	PUNCT
eajbcs-694	494	1	+	+	PUNCT
eajbcs-694	494	2	h(0	h(0	NOUN
eajbcs-694	494	3	)	)	PUNCT
eajbcs-694	494	4	+	+	CCONJ
eajbcs-694	494	5	r(0	r(0	PROPN
eajbcs-694	494	6	)	)	PUNCT
eajbcs-694	494	7	)	)	PUNCT
eajbcs-694	495	1	=	=	SYM
eajbcs-694	495	2	114	114	NUM
eajbcs-694	495	3	,	,	PUNCT
eajbcs-694	495	4	744	744	NUM
eajbcs-694	495	5	,	,	PUNCT
eajbcs-694	495	6	061	061	NUM
eajbcs-694	495	7	.	.	PUNCT
eajbcs-694	496	1	east	east	PROPN
eajbcs-694	496	2	afr	afr	PROPN
eajbcs-694	496	3	.	.	PUNCT
eajbcs-694	497	1	j.	j.	PROPN
eajbcs-694	497	2	biophys	biophys	PROPN
eajbcs-694	497	3	.	.	PUNCT
eajbcs-694	498	1	comput	comput	NOUN
eajbcs-694	498	2	.	.	PUNCT
eajbcs-694	499	1	sci	sci	PROPN
eajbcs-694	499	2	.	.	PUNCT
eajbcs-694	500	1	(	(	PUNCT
eajbcs-694	500	2	2024	2024	NUM
eajbcs-694	500	3	)	)	PUNCT
eajbcs-694	500	4	,	,	PUNCT
eajbcs-694	500	5	vol	vol	NOUN
eajbcs-694	500	6	.	.	PROPN
eajbcs-694	500	7	5	5	NUM
eajbcs-694	500	8	,	,	PUNCT
eajbcs-694	500	9	no	no	INTJ
eajbcs-694	500	10	.	.	NOUN
eajbcs-694	500	11	2	2	NUM
eajbcs-694	500	12	,	,	PUNCT
eajbcs-694	500	13	58	58	NUM
eajbcs-694	500	14	-	-	SYM
eajbcs-694	500	15	86	86	NUM
eajbcs-694	500	16	73	73	NUM
eajbcs-694	500	17	table	table	NOUN
eajbcs-694	500	18	3	3	NUM
eajbcs-694	500	19	:	:	PUNCT
eajbcs-694	500	20	the	the	DET
eajbcs-694	500	21	parameter	parameter	NOUN
eajbcs-694	500	22	values	value	NOUN
eajbcs-694	500	23	of	of	ADP
eajbcs-694	500	24	the	the	DET
eajbcs-694	500	25	modified	modify	VERB
eajbcs-694	500	26	model	model	NOUN
eajbcs-694	500	27	(	(	PUNCT
eajbcs-694	500	28	per	per	ADP
eajbcs-694	500	29	day	day	NOUN
eajbcs-694	500	30	)	)	PUNCT
eajbcs-694	500	31	.	.	PUNCT
eajbcs-694	501	1	parameter	parameter	PROPN
eajbcs-694	501	2	value	value	PROPN
eajbcs-694	501	3	source	source	NOUN
eajbcs-694	501	4	π	π	PROPN
eajbcs-694	501	5	1300	1300	NUM
eajbcs-694	501	6	assumed	assume	VERB
eajbcs-694	501	7	θ	θ	PROPN
eajbcs-694	501	8	0.7	0.7	NUM
eajbcs-694	501	9	assumed	assume	VERB
eajbcs-694	501	10	ν	ν	PROPN
eajbcs-694	501	11	0.15	0.15	NUM
eajbcs-694	501	12	assumed	assume	VERB
eajbcs-694	501	13	β	β	PROPN
eajbcs-694	501	14	0.00000058	0.00000058	NUM
eajbcs-694	501	15	assumed	assume	VERB
eajbcs-694	501	16	σ1	σ1	PROPN
eajbcs-694	501	17	0.0001	0.0001	NUM
eajbcs-694	501	18	alemneh	alemneh	NOUN
eajbcs-694	501	19	and	and	CCONJ
eajbcs-694	501	20	telahun	telahun	PROPN
eajbcs-694	501	21	(	(	PUNCT
eajbcs-694	501	22	2020	2020	NUM
eajbcs-694	501	23	)	)	PUNCT
eajbcs-694	501	24	σ2	σ2	NOUN
eajbcs-694	501	25	0.02	0.02	NUM
eajbcs-694	501	26	alemneh	alemneh	NOUN
eajbcs-694	501	27	and	and	CCONJ
eajbcs-694	501	28	telahun	telahun	PROPN
eajbcs-694	501	29	(	(	PUNCT
eajbcs-694	501	30	2020	2020	NUM
eajbcs-694	501	31	)	)	PUNCT
eajbcs-694	501	32	σ3	σ3	PROPN
eajbcs-694	501	33	0.00003	0.00003	PROPN
eajbcs-694	501	34	assumed	assume	VERB
eajbcs-694	501	35	δ	δ	PROPN
eajbcs-694	501	36	1/14	1/14	NUM
eajbcs-694	501	37	kifle	kifle	VERB
eajbcs-694	501	38	and	and	CCONJ
eajbcs-694	501	39	obsu	obsu	PROPN
eajbcs-694	501	40	(	(	PUNCT
eajbcs-694	501	41	2022	2022	NUM
eajbcs-694	501	42	)	)	PUNCT
eajbcs-694	501	43	α	α	PROPN
eajbcs-694	501	44	0.04	0.04	NUM
eajbcs-694	501	45	assumed	assume	VERB
eajbcs-694	501	46	µ	µ	DET
eajbcs-694	501	47	1/(64	1/(64	NOUN
eajbcs-694	501	48	*	*	SYM
eajbcs-694	501	49	12	12	NUM
eajbcs-694	501	50	*	*	PUNCT
eajbcs-694	501	51	30)kifle	30)kifle	NUM
eajbcs-694	501	52	and	and	CCONJ
eajbcs-694	501	53	obsu	obsu	PROPN
eajbcs-694	501	54	(	(	PUNCT
eajbcs-694	501	55	2022	2022	NUM
eajbcs-694	501	56	)	)	PUNCT
eajbcs-694	501	57	ρ	ρ	PROPN
eajbcs-694	501	58	0.0004	0.0004	NUM
eajbcs-694	501	59	alemneh	alemneh	NOUN
eajbcs-694	501	60	and	and	CCONJ
eajbcs-694	501	61	telahun	telahun	PROPN
eajbcs-694	501	62	(	(	PUNCT
eajbcs-694	501	63	2020	2020	NUM
eajbcs-694	501	64	)	)	PUNCT
eajbcs-694	501	65	ξ	ξ	SYM
eajbcs-694	501	66	0.015	0.015	NUM
eajbcs-694	501	67	assumed	assume	VERB
eajbcs-694	501	68	τ	τ	PROPN
eajbcs-694	501	69	0.7	0.7	NUM
eajbcs-694	501	70	alemneh	alemneh	NOUN
eajbcs-694	501	71	and	and	CCONJ
eajbcs-694	501	72	telahun	telahun	PROPN
eajbcs-694	501	73	(	(	PUNCT
eajbcs-694	501	74	2020	2020	NUM
eajbcs-694	501	75	)	)	PUNCT
eajbcs-694	501	76	ε	ε	PROPN
eajbcs-694	501	77	0.0476	0.0476	NUM
eajbcs-694	501	78	assumed	assume	VERB
eajbcs-694	501	79	γ	γ	PROPN
eajbcs-694	501	80	0.033	0.033	NUM
eajbcs-694	501	81	assumed	assume	VERB
eajbcs-694	501	82	η	η	PROPN
eajbcs-694	501	83	0.4	0.4	NUM
eajbcs-694	501	84	assumed	assume	VERB
eajbcs-694	501	85	ω	ω	PROPN
eajbcs-694	501	86	0.011	0.011	NUM
eajbcs-694	501	87	assumed	assume	VERB
eajbcs-694	501	88	east	east	PROPN
eajbcs-694	501	89	afr	afr	PROPN
eajbcs-694	501	90	.	.	PUNCT
eajbcs-694	502	1	j.	j.	PROPN
eajbcs-694	502	2	biophys	biophys	PROPN
eajbcs-694	502	3	.	.	PUNCT
eajbcs-694	503	1	comput	comput	NOUN
eajbcs-694	503	2	.	.	PUNCT
eajbcs-694	504	1	sci	sci	PROPN
eajbcs-694	504	2	.	.	PUNCT
eajbcs-694	505	1	(	(	PUNCT
eajbcs-694	505	2	2024	2024	NUM
eajbcs-694	505	3	)	)	PUNCT
eajbcs-694	505	4	,	,	PUNCT
eajbcs-694	505	5	vol	vol	NOUN
eajbcs-694	505	6	.	.	PROPN
eajbcs-694	505	7	5	5	NUM
eajbcs-694	505	8	,	,	PUNCT
eajbcs-694	505	9	no	no	INTJ
eajbcs-694	505	10	.	.	NOUN
eajbcs-694	505	11	2	2	NUM
eajbcs-694	505	12	,	,	PUNCT
eajbcs-694	505	13	58	58	NUM
eajbcs-694	505	14	-	-	SYM
eajbcs-694	505	15	86	86	NUM
eajbcs-694	505	16	74	74	NUM
eajbcs-694	505	17	figure	figure	NOUN
eajbcs-694	505	18	(	(	PUNCT
eajbcs-694	505	19	4	4	NUM
eajbcs-694	505	20	)	)	PUNCT
eajbcs-694	505	21	shows	show	VERB
eajbcs-694	505	22	the	the	DET
eajbcs-694	505	23	predicted	predict	VERB
eajbcs-694	505	24	total	total	ADJ
eajbcs-694	505	25	confirmed	confirm	VERB
eajbcs-694	505	26	cases	case	NOUN
eajbcs-694	505	27	and	and	CCONJ
eajbcs-694	505	28	the	the	DET
eajbcs-694	505	29	real	real	ADJ
eajbcs-694	505	30	data	datum	NOUN
eajbcs-694	505	31	total	total	NOUN
eajbcs-694	505	32	confirmed	confirm	VERB
eajbcs-694	505	33	cases	case	NOUN
eajbcs-694	505	34	for	for	ADP
eajbcs-694	505	35	ethiopia	ethiopia	PROPN
eajbcs-694	505	36	from	from	ADP
eajbcs-694	505	37	31	31	NUM
eajbcs-694	505	38	december	december	PROPN
eajbcs-694	505	39	2020	2020	NUM
eajbcs-694	505	40	to	to	ADP
eajbcs-694	505	41	30	30	NUM
eajbcs-694	505	42	march	march	NOUN
eajbcs-694	505	43	2022	2022	NUM
eajbcs-694	505	44	(	(	PUNCT
eajbcs-694	505	45	it	it	PRON
eajbcs-694	505	46	the	the	DET
eajbcs-694	505	47	data	datum	NOUN
eajbcs-694	505	48	has	have	AUX
eajbcs-694	505	49	been	be	AUX
eajbcs-694	505	50	taken	take	VERB
eajbcs-694	505	51	covid-19	covid-19	PROPN
eajbcs-694	505	52	pandemic	pandemic	ADJ
eajbcs-694	505	53	(	(	PUNCT
eajbcs-694	505	54	2022	2022	NUM
eajbcs-694	505	55	)	)	PUNCT
eajbcs-694	505	56	)	)	PUNCT
eajbcs-694	505	57	.	.	PUNCT
eajbcs-694	506	1	there	there	PRON
eajbcs-694	506	2	is	be	VERB
eajbcs-694	506	3	some	some	DET
eajbcs-694	506	4	differences	difference	NOUN
eajbcs-694	506	5	between	between	ADP
eajbcs-694	506	6	the	the	DET
eajbcs-694	506	7	prediction	prediction	NOUN
eajbcs-694	506	8	and	and	CCONJ
eajbcs-694	506	9	the	the	DET
eajbcs-694	506	10	real	real	ADJ
eajbcs-694	506	11	data	datum	NOUN
eajbcs-694	506	12	that	that	PRON
eajbcs-694	506	13	should	should	AUX
eajbcs-694	506	14	come	come	VERB
eajbcs-694	506	15	due	due	ADJ
eajbcs-694	506	16	to	to	ADP
eajbcs-694	506	17	the	the	DET
eajbcs-694	506	18	fact	fact	NOUN
eajbcs-694	506	19	that	that	SCONJ
eajbcs-694	506	20	there	there	PRON
eajbcs-694	506	21	were	be	VERB
eajbcs-694	506	22	no	no	DET
eajbcs-694	506	23	enough	enough	ADJ
eajbcs-694	506	24	covid	covid	PROPN
eajbcs-694	506	25	test	test	NOUN
eajbcs-694	506	26	kits	kit	NOUN
eajbcs-694	506	27	during	during	ADP
eajbcs-694	506	28	the	the	DET
eajbcs-694	506	29	early	early	ADJ
eajbcs-694	506	30	time	time	NOUN
eajbcs-694	506	31	not	not	PART
eajbcs-694	506	32	only	only	ADV
eajbcs-694	506	33	in	in	ADP
eajbcs-694	506	34	ethiopia	ethiopia	PROPN
eajbcs-694	506	35	but	but	CCONJ
eajbcs-694	506	36	also	also	ADV
eajbcs-694	506	37	all	all	ADV
eajbcs-694	506	38	over	over	ADP
eajbcs-694	506	39	the	the	DET
eajbcs-694	506	40	world	world	NOUN
eajbcs-694	506	41	.	.	PUNCT
eajbcs-694	507	1	after	after	ADP
eajbcs-694	507	2	wards	ward	NOUN
eajbcs-694	507	3	we	we	PRON
eajbcs-694	507	4	continue	continue	VERB
eajbcs-694	507	5	to	to	PART
eajbcs-694	507	6	validate	validate	VERB
eajbcs-694	507	7	the	the	DET
eajbcs-694	507	8	local	local	ADJ
eajbcs-694	507	9	stability	stability	NOUN
eajbcs-694	507	10	of	of	ADP
eajbcs-694	507	11	dfe	dfe	PROPN
eajbcs-694	507	12	and	and	CCONJ
eajbcs-694	507	13	eep	eep	NOUN
eajbcs-694	507	14	of	of	ADP
eajbcs-694	507	15	model	model	PROPN
eajbcs-694	507	16	.	.	PUNCT
eajbcs-694	508	1	figure	figure	VERB
eajbcs-694	508	2	4	4	NUM
eajbcs-694	508	3	:	:	PUNCT
eajbcs-694	508	4	the	the	DET
eajbcs-694	508	5	prediction	prediction	NOUN
eajbcs-694	508	6	using	use	VERB
eajbcs-694	508	7	model	model	NOUN
eajbcs-694	508	8	(	(	PUNCT
eajbcs-694	508	9	2	2	NUM
eajbcs-694	508	10	)	)	PUNCT
eajbcs-694	508	11	and	and	CCONJ
eajbcs-694	508	12	the	the	DET
eajbcs-694	508	13	real	real	ADJ
eajbcs-694	508	14	data	datum	NOUN
eajbcs-694	508	15	of	of	ADP
eajbcs-694	508	16	confirmed	confirm	VERB
eajbcs-694	508	17	cases	case	NOUN
eajbcs-694	508	18	for	for	ADP
eajbcs-694	508	19	ethiopia	ethiopia	PROPN
eajbcs-694	508	20	from	from	ADP
eajbcs-694	508	21	31	31	NUM
eajbcs-694	508	22	december	december	PROPN
eajbcs-694	508	23	2020	2020	NUM
eajbcs-694	508	24	to	to	ADP
eajbcs-694	508	25	30	30	NUM
eajbcs-694	508	26	march	march	NOUN
eajbcs-694	508	27	2022	2022	NUM
eajbcs-694	508	28	in	in	ADP
eajbcs-694	508	29	figure	figure	NOUN
eajbcs-694	508	30	(	(	PUNCT
eajbcs-694	508	31	5	5	NUM
eajbcs-694	508	32	)	)	PUNCT
eajbcs-694	508	33	with	with	ADP
eajbcs-694	508	34	r0	r0	NOUN
eajbcs-694	508	35	=	=	SYM
eajbcs-694	508	36	0.22950	0.22950	NUM
eajbcs-694	508	37	,	,	PUNCT
eajbcs-694	508	38	we	we	PRON
eajbcs-694	508	39	observe	observe	VERB
eajbcs-694	508	40	that	that	SCONJ
eajbcs-694	508	41	for	for	ADP
eajbcs-694	508	42	the	the	DET
eajbcs-694	508	43	basic	basic	ADJ
eajbcs-694	508	44	reproduction	reproduction	NOUN
eajbcs-694	508	45	number	number	NOUN
eajbcs-694	508	46	r0	r0	NOUN
eajbcs-694	508	47	<	<	X
eajbcs-694	508	48	1	1	NUM
eajbcs-694	508	49	,	,	PUNCT
eajbcs-694	508	50	all	all	DET
eajbcs-694	508	51	solutions	solution	NOUN
eajbcs-694	508	52	curve	curve	NOUN
eajbcs-694	508	53	goes	go	VERB
eajbcs-694	508	54	to	to	ADP
eajbcs-694	508	55	the	the	DET
eajbcs-694	508	56	disease	disease	NOUN
eajbcs-694	508	57	free	free	ADJ
eajbcs-694	508	58	equilibrium	equilibrium	NOUN
eajbcs-694	508	59	point	point	NOUN
eajbcs-694	508	60	.	.	PUNCT
eajbcs-694	509	1	as	as	ADP
eajbcs-694	509	2	a	a	DET
eajbcs-694	509	3	result	result	NOUN
eajbcs-694	509	4	,	,	PUNCT
eajbcs-694	509	5	the	the	DET
eajbcs-694	509	6	disease	disease	NOUN
eajbcs-694	509	7	goes	go	VERB
eajbcs-694	509	8	to	to	ADP
eajbcs-694	509	9	extinct	extinct	ADJ
eajbcs-694	509	10	or	or	CCONJ
eajbcs-694	509	11	the	the	DET
eajbcs-694	509	12	disease	disease	NOUN
eajbcs-694	509	13	dies	die	VERB
eajbcs-694	509	14	out	out	ADP
eajbcs-694	509	15	.	.	PUNCT
eajbcs-694	510	1	in	in	ADP
eajbcs-694	510	2	figure	figure	NOUN
eajbcs-694	510	3	(	(	PUNCT
eajbcs-694	510	4	6	6	NUM
eajbcs-694	510	5	)	)	PUNCT
eajbcs-694	510	6	with	with	ADP
eajbcs-694	510	7	r0	r0	NOUN
eajbcs-694	510	8	=	=	NOUN
eajbcs-694	510	9	1.10791	1.10791	NUM
eajbcs-694	510	10	,	,	PUNCT
eajbcs-694	510	11	we	we	PRON
eajbcs-694	510	12	observe	observe	VERB
eajbcs-694	510	13	that	that	SCONJ
eajbcs-694	510	14	for	for	ADP
eajbcs-694	510	15	the	the	DET
eajbcs-694	510	16	basic	basic	ADJ
eajbcs-694	510	17	reproduction	reproduction	NOUN
eajbcs-694	510	18	number	number	NOUN
eajbcs-694	510	19	r0	r0	NOUN
eajbcs-694	510	20	>	>	X
eajbcs-694	510	21	1	1	NUM
eajbcs-694	510	22	,	,	PUNCT
eajbcs-694	510	23	all	all	DET
eajbcs-694	510	24	solutions	solution	NOUN
eajbcs-694	510	25	curve	curve	NOUN
eajbcs-694	510	26	goes	go	VERB
eajbcs-694	510	27	away	away	ADV
eajbcs-694	510	28	from	from	ADP
eajbcs-694	510	29	the	the	DET
eajbcs-694	510	30	disease	disease	NOUN
eajbcs-694	510	31	free	free	ADJ
eajbcs-694	510	32	equilibrium	equilibrium	NOUN
eajbcs-694	510	33	point	point	NOUN
eajbcs-694	510	34	.	.	PUNCT
eajbcs-694	511	1	these	these	PRON
eajbcs-694	511	2	indicate	indicate	VERB
eajbcs-694	511	3	that	that	SCONJ
eajbcs-694	511	4	the	the	DET
eajbcs-694	511	5	disease	disease	NOUN
eajbcs-694	511	6	-	-	PUNCT
eajbcs-694	511	7	free	free	ADJ
eajbcs-694	511	8	equilibrium	equilibrium	NOUN
eajbcs-694	511	9	point	point	NOUN
eajbcs-694	511	10	is	be	AUX
eajbcs-694	511	11	unstable	unstable	ADJ
eajbcs-694	511	12	for	for	ADP
eajbcs-694	511	13	the	the	DET
eajbcs-694	511	14	values	value	NOUN
eajbcs-694	511	15	of	of	ADP
eajbcs-694	511	16	r0	r0	NOUN
eajbcs-694	511	17	>	>	X
eajbcs-694	511	18	1	1	NUM
eajbcs-694	511	19	,	,	PUNCT
eajbcs-694	511	20	and	and	CCONJ
eajbcs-694	511	21	the	the	DET
eajbcs-694	511	22	solutions	solution	NOUN
eajbcs-694	511	23	will	will	AUX
eajbcs-694	511	24	go	go	VERB
eajbcs-694	511	25	to	to	ADP
eajbcs-694	511	26	the	the	DET
eajbcs-694	511	27	endemic	endemic	ADJ
eajbcs-694	511	28	equilibrium	equilibrium	NOUN
eajbcs-694	511	29	point	point	NOUN
eajbcs-694	511	30	.	.	PUNCT
eajbcs-694	512	1	consequently	consequently	ADV
eajbcs-694	512	2	,	,	PUNCT
eajbcs-694	512	3	the	the	DET
eajbcs-694	512	4	disease	disease	NOUN
eajbcs-694	512	5	invade	invade	VERB
eajbcs-694	512	6	in	in	ADP
eajbcs-694	512	7	a	a	DET
eajbcs-694	512	8	population	population	NOUN
eajbcs-694	512	9	.	.	PUNCT
eajbcs-694	513	1	in	in	ADP
eajbcs-694	513	2	figure	figure	NOUN
eajbcs-694	513	3	(	(	PUNCT
eajbcs-694	513	4	7	7	NUM
eajbcs-694	513	5	)	)	PUNCT
eajbcs-694	513	6	,	,	PUNCT
eajbcs-694	513	7	we	we	PRON
eajbcs-694	513	8	observe	observe	VERB
eajbcs-694	513	9	as	as	ADP
eajbcs-694	513	10	the	the	DET
eajbcs-694	513	11	protection	protection	NOUN
eajbcs-694	513	12	rate	rate	NOUN
eajbcs-694	513	13	θ	θ	PROPN
eajbcs-694	513	14	increases	increase	NOUN
eajbcs-694	513	15	,	,	PUNCT
eajbcs-694	513	16	all	all	DET
eajbcs-694	513	17	infected	infected	ADJ
eajbcs-694	513	18	classes	class	NOUN
eajbcs-694	513	19	will	will	AUX
eajbcs-694	513	20	significantly	significantly	ADV
eajbcs-694	513	21	decrease	decrease	VERB
eajbcs-694	513	22	over	over	ADP
eajbcs-694	513	23	time	time	NOUN
eajbcs-694	513	24	.	.	PUNCT
eajbcs-694	514	1	this	this	PRON
eajbcs-694	514	2	confirmed	confirm	VERB
eajbcs-694	514	3	the	the	DET
eajbcs-694	514	4	result	result	NOUN
eajbcs-694	514	5	from	from	ADP
eajbcs-694	514	6	the	the	DET
eajbcs-694	514	7	fact	fact	NOUN
eajbcs-694	514	8	that	that	SCONJ
eajbcs-694	514	9	strict	strict	ADJ
eajbcs-694	514	10	use	use	NOUN
eajbcs-694	514	11	of	of	ADP
eajbcs-694	514	12	safety	safety	NOUN
eajbcs-694	514	13	(	(	PUNCT
eajbcs-694	514	14	protection	protection	NOUN
eajbcs-694	514	15	)	)	PUNCT
eajbcs-694	514	16	measures	measure	NOUN
eajbcs-694	514	17	within	within	ADP
eajbcs-694	514	18	the	the	DET
eajbcs-694	514	19	population	population	NOUN
eajbcs-694	514	20	plays	play	VERB
eajbcs-694	514	21	a	a	DET
eajbcs-694	514	22	critical	critical	ADJ
eajbcs-694	514	23	role	role	NOUN
eajbcs-694	514	24	in	in	ADP
eajbcs-694	514	25	confine	confine	NOUN
eajbcs-694	514	26	the	the	DET
eajbcs-694	514	27	spread	spread	NOUN
eajbcs-694	514	28	of	of	ADP
eajbcs-694	514	29	the	the	DET
eajbcs-694	514	30	disease	disease	NOUN
eajbcs-694	514	31	as	as	ADP
eajbcs-694	514	32	in	in	ADP
eajbcs-694	514	33	bachar	bachar	PROPN
eajbcs-694	514	34	et	et	PROPN
eajbcs-694	514	35	al	al	PROPN
eajbcs-694	514	36	.	.	PROPN
eajbcs-694	515	1	(	(	PUNCT
eajbcs-694	515	2	2021	2021	NUM
eajbcs-694	515	3	)	)	PUNCT
eajbcs-694	515	4	.	.	PUNCT
eajbcs-694	516	1	it	it	PRON
eajbcs-694	516	2	is	be	AUX
eajbcs-694	516	3	predicted	predict	VERB
eajbcs-694	516	4	that	that	SCONJ
eajbcs-694	516	5	the	the	DET
eajbcs-694	516	6	population	population	NOUN
eajbcs-694	516	7	will	will	AUX
eajbcs-694	516	8	be	be	AUX
eajbcs-694	516	9	disease	disease	NOUN
eajbcs-694	516	10	-	-	PUNCT
eajbcs-694	516	11	free	free	ADJ
eajbcs-694	516	12	.	.	PUNCT
eajbcs-694	517	1	this	this	PRON
eajbcs-694	517	2	is	be	AUX
eajbcs-694	517	3	due	due	ADJ
eajbcs-694	517	4	to	to	ADP
eajbcs-694	517	5	the	the	DET
eajbcs-694	517	6	case	case	NOUN
eajbcs-694	517	7	that	that	SCONJ
eajbcs-694	517	8	when	when	SCONJ
eajbcs-694	517	9	protection	protection	NOUN
eajbcs-694	517	10	rate	rate	NOUN
eajbcs-694	517	11	increases	increase	VERB
eajbcs-694	517	12	the	the	DET
eajbcs-694	517	13	basic	basic	ADJ
eajbcs-694	517	14	reproduction	reproduction	NOUN
eajbcs-694	517	15	number	number	NOUN
eajbcs-694	517	16	decreases	decrease	VERB
eajbcs-694	517	17	.	.	PUNCT
eajbcs-694	518	1	east	east	PROPN
eajbcs-694	518	2	afr	afr	PROPN
eajbcs-694	518	3	.	.	PUNCT
eajbcs-694	519	1	j.	j.	PROPN
eajbcs-694	519	2	biophys	biophys	PROPN
eajbcs-694	519	3	.	.	PUNCT
eajbcs-694	520	1	comput	comput	NOUN
eajbcs-694	520	2	.	.	PUNCT
eajbcs-694	521	1	sci	sci	PROPN
eajbcs-694	521	2	.	.	PUNCT
eajbcs-694	522	1	(	(	PUNCT
eajbcs-694	522	2	2024	2024	NUM
eajbcs-694	522	3	)	)	PUNCT
eajbcs-694	522	4	,	,	PUNCT
eajbcs-694	522	5	vol	vol	NOUN
eajbcs-694	522	6	.	.	PROPN
eajbcs-694	522	7	5	5	NUM
eajbcs-694	522	8	,	,	PUNCT
eajbcs-694	522	9	no	no	INTJ
eajbcs-694	522	10	.	.	NOUN
eajbcs-694	522	11	2	2	NUM
eajbcs-694	522	12	,	,	PUNCT
eajbcs-694	522	13	58	58	NUM
eajbcs-694	522	14	-	-	SYM
eajbcs-694	522	15	86	86	NUM
eajbcs-694	522	16	75	75	NUM
eajbcs-694	522	17	figure	figure	NOUN
eajbcs-694	522	18	5	5	NUM
eajbcs-694	522	19	:	:	PUNCT
eajbcs-694	522	20	the	the	DET
eajbcs-694	522	21	time	time	NOUN
eajbcs-694	522	22	series	series	PROPN
eajbcs-694	522	23	plot	plot	NOUN
eajbcs-694	522	24	of	of	ADP
eajbcs-694	522	25	model	model	NOUN
eajbcs-694	522	26	(	(	PUNCT
eajbcs-694	522	27	2	2	NUM
eajbcs-694	522	28	)	)	PUNCT
eajbcs-694	522	29	when	when	SCONJ
eajbcs-694	522	30	r0	r0	NOUN
eajbcs-694	522	31	=	=	NOUN
eajbcs-694	522	32	0.22950	0.22950	NUM
eajbcs-694	522	33	in	in	ADP
eajbcs-694	522	34	figure	figure	NOUN
eajbcs-694	522	35	(	(	PUNCT
eajbcs-694	522	36	8)	8)	NUM
eajbcs-694	522	37	,	,	PUNCT
eajbcs-694	522	38	we	we	PRON
eajbcs-694	522	39	observe	observe	VERB
eajbcs-694	522	40	that	that	SCONJ
eajbcs-694	522	41	as	as	SCONJ
eajbcs-694	522	42	the	the	DET
eajbcs-694	522	43	contact	contact	NOUN
eajbcs-694	522	44	rate	rate	NOUN
eajbcs-694	522	45	β	β	NOUN
eajbcs-694	522	46	decrease	decrease	NOUN
eajbcs-694	522	47	,	,	PUNCT
eajbcs-694	522	48	r0	r0	NOUN
eajbcs-694	522	49	and	and	CCONJ
eajbcs-694	522	50	also	also	ADV
eajbcs-694	522	51	all	all	DET
eajbcs-694	522	52	infected	infected	ADJ
eajbcs-694	522	53	classes	class	NOUN
eajbcs-694	522	54	are	be	AUX
eajbcs-694	522	55	decreasing	decrease	VERB
eajbcs-694	522	56	.	.	PUNCT
eajbcs-694	523	1	further	far	ADV
eajbcs-694	523	2	this	this	DET
eajbcs-694	523	3	habitual	habitual	NOUN
eajbcs-694	523	4	the	the	DET
eajbcs-694	523	5	result	result	NOUN
eajbcs-694	523	6	obtained	obtain	VERB
eajbcs-694	523	7	from	from	ADP
eajbcs-694	523	8	the	the	DET
eajbcs-694	523	9	fact	fact	NOUN
eajbcs-694	523	10	that	that	SCONJ
eajbcs-694	523	11	a	a	DET
eajbcs-694	523	12	decrease	decrease	NOUN
eajbcs-694	523	13	in	in	ADP
eajbcs-694	523	14	contact	contact	NOUN
eajbcs-694	523	15	among	among	ADP
eajbcs-694	523	16	the	the	DET
eajbcs-694	523	17	population	population	NOUN
eajbcs-694	523	18	plays	play	VERB
eajbcs-694	523	19	a	a	DET
eajbcs-694	523	20	indispensable	indispensable	ADJ
eajbcs-694	523	21	role	role	NOUN
eajbcs-694	523	22	in	in	ADP
eajbcs-694	523	23	curtailing	curtail	VERB
eajbcs-694	523	24	the	the	DET
eajbcs-694	523	25	spread	spread	NOUN
eajbcs-694	523	26	of	of	ADP
eajbcs-694	523	27	the	the	DET
eajbcs-694	523	28	disease	disease	NOUN
eajbcs-694	523	29	as	as	ADP
eajbcs-694	523	30	in	in	ADP
eajbcs-694	523	31	ahmed	ahmed	PROPN
eajbcs-694	523	32	et	et	PROPN
eajbcs-694	523	33	al	al	PROPN
eajbcs-694	523	34	.	.	PROPN
eajbcs-694	524	1	(	(	PUNCT
eajbcs-694	524	2	2021	2021	NUM
eajbcs-694	524	3	)	)	PUNCT
eajbcs-694	524	4	.	.	PUNCT
eajbcs-694	525	1	5	5	X
eajbcs-694	525	2	.	.	X
eajbcs-694	525	3	extension	extension	NOUN
eajbcs-694	525	4	of	of	ADP
eajbcs-694	525	5	the	the	DET
eajbcs-694	525	6	modified	modified	ADJ
eajbcs-694	525	7	model	model	NOUN
eajbcs-694	525	8	into	into	ADP
eajbcs-694	525	9	an	an	DET
eajbcs-694	525	10	optimal	optimal	ADJ
eajbcs-694	525	11	control	control	NOUN
eajbcs-694	525	12	in	in	ADP
eajbcs-694	525	13	this	this	DET
eajbcs-694	525	14	section	section	NOUN
eajbcs-694	525	15	,	,	PUNCT
eajbcs-694	525	16	we	we	PRON
eajbcs-694	525	17	will	will	AUX
eajbcs-694	525	18	use	use	VERB
eajbcs-694	525	19	optimal	optimal	ADJ
eajbcs-694	525	20	control	control	NOUN
eajbcs-694	525	21	theory	theory	NOUN
eajbcs-694	525	22	to	to	PART
eajbcs-694	525	23	find	find	VERB
eajbcs-694	525	24	protection	protection	NOUN
eajbcs-694	525	25	(	(	PUNCT
eajbcs-694	525	26	for	for	ADP
eajbcs-694	525	27	susceptible	susceptible	ADJ
eajbcs-694	525	28	)	)	PUNCT
eajbcs-694	525	29	and	and	CCONJ
eajbcs-694	525	30	hospitalization	hospitalization	NOUN
eajbcs-694	525	31	(	(	PUNCT
eajbcs-694	525	32	for	for	ADP
eajbcs-694	525	33	infected	infected	ADJ
eajbcs-694	525	34	)	)	PUNCT
eajbcs-694	525	35	strategies	strategy	NOUN
eajbcs-694	525	36	that	that	PRON
eajbcs-694	525	37	would	would	AUX
eajbcs-694	525	38	mitigate	mitigate	VERB
eajbcs-694	525	39	the	the	DET
eajbcs-694	525	40	spread	spread	NOUN
eajbcs-694	525	41	of	of	ADP
eajbcs-694	525	42	covid19	covid19	NOUN
eajbcs-694	525	43	in	in	ADP
eajbcs-694	525	44	the	the	DET
eajbcs-694	525	45	population	population	NOUN
eajbcs-694	525	46	.	.	PUNCT
eajbcs-694	526	1	5.1	5.1	NUM
eajbcs-694	526	2	.	.	PUNCT
eajbcs-694	527	1	optimal	optimal	ADJ
eajbcs-694	527	2	protection	protection	NOUN
eajbcs-694	527	3	and	and	CCONJ
eajbcs-694	527	4	hospitalization	hospitalization	NOUN
eajbcs-694	527	5	using	use	VERB
eajbcs-694	527	6	modified	modified	ADJ
eajbcs-694	527	7	model	model	NOUN
eajbcs-694	527	8	in	in	ADP
eajbcs-694	527	9	this	this	DET
eajbcs-694	527	10	subsection	subsection	NOUN
eajbcs-694	527	11	,	,	PUNCT
eajbcs-694	527	12	we	we	PRON
eajbcs-694	527	13	study	study	VERB
eajbcs-694	527	14	the	the	DET
eajbcs-694	527	15	optimal	optimal	ADJ
eajbcs-694	527	16	protection	protection	NOUN
eajbcs-694	527	17	of	of	ADP
eajbcs-694	527	18	susceptible	susceptible	ADJ
eajbcs-694	527	19	population	population	NOUN
eajbcs-694	527	20	and	and	CCONJ
eajbcs-694	527	21	hospitalization	hospitalization	NOUN
eajbcs-694	527	22	of	of	ADP
eajbcs-694	527	23	infected	infected	ADJ
eajbcs-694	527	24	individuals	individual	NOUN
eajbcs-694	527	25	in	in	ADP
eajbcs-694	527	26	order	order	NOUN
eajbcs-694	527	27	to	to	PART
eajbcs-694	527	28	minimize	minimize	VERB
eajbcs-694	527	29	the	the	DET
eajbcs-694	527	30	outbreak	outbreak	NOUN
eajbcs-694	527	31	of	of	ADP
eajbcs-694	527	32	covid-19	covid-19	PROPN
eajbcs-694	527	33	in	in	ADP
eajbcs-694	527	34	the	the	DET
eajbcs-694	527	35	population	population	NOUN
eajbcs-694	527	36	.	.	PUNCT
eajbcs-694	528	1	let	let	VERB
eajbcs-694	528	2	us	we	PRON
eajbcs-694	528	3	define	define	VERB
eajbcs-694	528	4	our	our	PRON
eajbcs-694	528	5	control	control	NOUN
eajbcs-694	528	6	set	set	VERB
eajbcs-694	528	7	u	u	PRON
eajbcs-694	528	8	to	to	PART
eajbcs-694	528	9	be	be	AUX
eajbcs-694	528	10	u	u	NOUN
eajbcs-694	528	11	=	=	PUNCT
eajbcs-694	528	12	{	{	PUNCT
eajbcs-694	528	13	(	(	PUNCT
eajbcs-694	528	14	θ(t	θ(t	PROPN
eajbcs-694	528	15	)	)	PUNCT
eajbcs-694	528	16	,	,	PUNCT
eajbcs-694	528	17	α(t	α(t	PROPN
eajbcs-694	528	18	)	)	PUNCT
eajbcs-694	528	19	)	)	PUNCT
eajbcs-694	528	20	:	:	PUNCT
eajbcs-694	528	21	0	0	NUM
eajbcs-694	528	22	≤	≤	NUM
eajbcs-694	528	23	θ(t	θ(t	NOUN
eajbcs-694	528	24	)	)	PUNCT
eajbcs-694	528	25	,	,	PUNCT
eajbcs-694	528	26	α(t	α(t	PROPN
eajbcs-694	528	27	)	)	PUNCT
eajbcs-694	528	28	≤	≤	NOUN
eajbcs-694	528	29	ϵi	ϵi	NOUN
eajbcs-694	528	30	,	,	PUNCT
eajbcs-694	528	31	0	0	NUM
eajbcs-694	528	32	≤	≤	NUM
eajbcs-694	528	33	t	t	PROPN
eajbcs-694	528	34	≤	≤	PROPN
eajbcs-694	528	35	t	t	PROPN
eajbcs-694	528	36	,	,	PUNCT
eajbcs-694	528	37	0	0	PUNCT
eajbcs-694	528	38	<	<	X
eajbcs-694	528	39	ϵi	ϵi	X
eajbcs-694	528	40	≤	≤	ADV
eajbcs-694	528	41	1	1	NUM
eajbcs-694	528	42	,	,	PUNCT
eajbcs-694	528	43	i	i	PRON
eajbcs-694	528	44	=	=	NOUN
eajbcs-694	528	45	1	1	NUM
eajbcs-694	528	46	,	,	PUNCT
eajbcs-694	528	47	2	2	NUM
eajbcs-694	528	48	}	}	PUNCT
eajbcs-694	528	49	,	,	PUNCT
eajbcs-694	528	50	(	(	PUNCT
eajbcs-694	528	51	9	9	X
eajbcs-694	528	52	)	)	PUNCT
eajbcs-694	528	53	where	where	SCONJ
eajbcs-694	528	54	θ(t	θ(t	NOUN
eajbcs-694	528	55	)	)	PUNCT
eajbcs-694	528	56	and	and	CCONJ
eajbcs-694	528	57	α(t	α(t	PROPN
eajbcs-694	528	58	)	)	PUNCT
eajbcs-694	528	59	are	be	AUX
eajbcs-694	528	60	lebesgue	lebesgue	ADJ
eajbcs-694	528	61	measurable	measurable	ADJ
eajbcs-694	528	62	quantities	quantity	NOUN
eajbcs-694	528	63	bounded	bound	VERB
eajbcs-694	528	64	above	above	ADV
eajbcs-694	528	65	by	by	ADP
eajbcs-694	528	66	ϵ1	ϵ1	ADJ
eajbcs-694	528	67	and	and	CCONJ
eajbcs-694	528	68	ϵ2	ϵ2	VERB
eajbcs-694	528	69	respectively	respectively	ADV
eajbcs-694	528	70	.	.	PUNCT
eajbcs-694	529	1	we	we	PRON
eajbcs-694	529	2	will	will	AUX
eajbcs-694	529	3	minimize	minimize	VERB
eajbcs-694	529	4	the	the	DET
eajbcs-694	529	5	objective	objective	ADJ
eajbcs-694	529	6	functional	functional	ADJ
eajbcs-694	529	7	j	j	PROPN
eajbcs-694	530	1	[	[	X
eajbcs-694	530	2	θ(t	θ(t	PROPN
eajbcs-694	530	3	)	)	PUNCT
eajbcs-694	530	4	,	,	PUNCT
eajbcs-694	530	5	α(t	α(t	PROPN
eajbcs-694	530	6	)	)	PUNCT
eajbcs-694	530	7	]	]	PUNCT
eajbcs-694	531	1	=	=	PUNCT
eajbcs-694	531	2	∫	∫	PROPN
eajbcs-694	531	3	t	t	PROPN
eajbcs-694	531	4	0	0	NUM
eajbcs-694	531	5	(	(	PUNCT
eajbcs-694	531	6	b1e(t	b1e(t	PROPN
eajbcs-694	531	7	)	)	PUNCT
eajbcs-694	531	8	+	+	ADJ
eajbcs-694	531	9	b2i(t	b2i(t	NOUN
eajbcs-694	531	10	)	)	PUNCT
eajbcs-694	532	1	+	+	CCONJ
eajbcs-694	532	2	1	1	NUM
eajbcs-694	532	3	2	2	NUM
eajbcs-694	532	4	(	(	PUNCT
eajbcs-694	532	5	a1θ	a1θ	PROPN
eajbcs-694	532	6	2(t	2(t	NUM
eajbcs-694	532	7	)	)	PUNCT
eajbcs-694	533	1	+	+	ADV
eajbcs-694	533	2	a2α	a2α	X
eajbcs-694	533	3	2(t	2(t	NUM
eajbcs-694	533	4	)	)	PUNCT
eajbcs-694	533	5	)	)	PUNCT
eajbcs-694	533	6	)	)	PUNCT
eajbcs-694	534	1	dt	dt	X
eajbcs-694	534	2	,	,	PUNCT
eajbcs-694	534	3	(	(	PUNCT
eajbcs-694	534	4	10	10	NUM
eajbcs-694	534	5	)	)	PUNCT
eajbcs-694	534	6	where	where	SCONJ
eajbcs-694	534	7	constants	constant	NOUN
eajbcs-694	534	8	b1	b1	NOUN
eajbcs-694	534	9	,	,	PUNCT
eajbcs-694	534	10	b2	b2	NOUN
eajbcs-694	534	11	,	,	PUNCT
eajbcs-694	534	12	a1	a1	NOUN
eajbcs-694	534	13	and	and	CCONJ
eajbcs-694	534	14	a2	a2	PROPN
eajbcs-694	534	15	are	be	AUX
eajbcs-694	534	16	positive	positive	ADJ
eajbcs-694	534	17	.	.	PUNCT
eajbcs-694	535	1	here	here	ADV
eajbcs-694	535	2	,	,	PUNCT
eajbcs-694	535	3	we	we	PRON
eajbcs-694	535	4	want	want	VERB
eajbcs-694	535	5	to	to	PART
eajbcs-694	535	6	find	find	VERB
eajbcs-694	535	7	the	the	DET
eajbcs-694	535	8	optimal	optimal	ADJ
eajbcs-694	535	9	values	value	NOUN
eajbcs-694	535	10	θ(t	θ(t	NOUN
eajbcs-694	535	11	)	)	PUNCT
eajbcs-694	535	12	and	and	CCONJ
eajbcs-694	535	13	α(t	α(t	NOUN
eajbcs-694	535	14	)	)	PUNCT
eajbcs-694	535	15	that	that	PRON
eajbcs-694	535	16	minimizes	minimize	VERB
eajbcs-694	535	17	the	the	DET
eajbcs-694	535	18	objective	objective	ADJ
eajbcs-694	535	19	functional	functional	ADJ
eajbcs-694	535	20	(	(	PUNCT
eajbcs-694	535	21	10	10	NUM
eajbcs-694	535	22	)	)	PUNCT
eajbcs-694	535	23	subject	subject	NOUN
eajbcs-694	535	24	to	to	ADP
eajbcs-694	535	25	the	the	DET
eajbcs-694	535	26	state	state	NOUN
eajbcs-694	535	27	system	system	NOUN
eajbcs-694	535	28	(	(	PUNCT
eajbcs-694	535	29	2	2	NUM
eajbcs-694	535	30	)	)	PUNCT
eajbcs-694	535	31	.	.	PUNCT
eajbcs-694	536	1	the	the	DET
eajbcs-694	536	2	goal	goal	NOUN
eajbcs-694	536	3	is	be	AUX
eajbcs-694	536	4	to	to	PART
eajbcs-694	536	5	find	find	VERB
eajbcs-694	536	6	the	the	DET
eajbcs-694	536	7	optimal	optimal	ADJ
eajbcs-694	536	8	control	control	NOUN
eajbcs-694	536	9	(	(	PUNCT
eajbcs-694	536	10	θ∗(t	θ∗(t	NOUN
eajbcs-694	536	11	)	)	PUNCT
eajbcs-694	536	12	,	,	PUNCT
eajbcs-694	536	13	α∗(t	α∗(t	NOUN
eajbcs-694	536	14	)	)	PUNCT
eajbcs-694	536	15	)	)	PUNCT
eajbcs-694	537	1	such	such	ADJ
eajbcs-694	537	2	that	that	SCONJ
eajbcs-694	537	3	j	j	PROPN
eajbcs-694	537	4	[	[	X
eajbcs-694	537	5	θ∗(t	θ∗(t	NOUN
eajbcs-694	537	6	)	)	PUNCT
eajbcs-694	537	7	,	,	PUNCT
eajbcs-694	537	8	α∗(t	α∗(t	NOUN
eajbcs-694	537	9	)	)	PUNCT
eajbcs-694	537	10	]	]	PUNCT
eajbcs-694	538	1	=	=	SYM
eajbcs-694	538	2	min	min	PROPN
eajbcs-694	538	3	(	(	PUNCT
eajbcs-694	538	4	θ	θ	PROPN
eajbcs-694	538	5	,	,	PUNCT
eajbcs-694	538	6	α)∈u	α)∈u	X
eajbcs-694	538	7	j	j	PROPN
eajbcs-694	539	1	[	[	X
eajbcs-694	539	2	θ(t	θ(t	PROPN
eajbcs-694	539	3	)	)	PUNCT
eajbcs-694	539	4	,	,	PUNCT
eajbcs-694	539	5	α(t	α(t	PROPN
eajbcs-694	539	6	)	)	PUNCT
eajbcs-694	539	7	]	]	PUNCT
eajbcs-694	539	8	.	.	PUNCT
eajbcs-694	540	1	(	(	PUNCT
eajbcs-694	540	2	11	11	NUM
eajbcs-694	540	3	)	)	PUNCT
eajbcs-694	540	4	east	east	PROPN
eajbcs-694	540	5	afr	afr	PROPN
eajbcs-694	540	6	.	.	PUNCT
eajbcs-694	541	1	j.	j.	PROPN
eajbcs-694	541	2	biophys	biophys	PROPN
eajbcs-694	541	3	.	.	PUNCT
eajbcs-694	542	1	comput	comput	NOUN
eajbcs-694	542	2	.	.	PUNCT
eajbcs-694	543	1	sci	sci	PROPN
eajbcs-694	543	2	.	.	PUNCT
eajbcs-694	544	1	(	(	PUNCT
eajbcs-694	544	2	2024	2024	NUM
eajbcs-694	544	3	)	)	PUNCT
eajbcs-694	544	4	,	,	PUNCT
eajbcs-694	544	5	vol	vol	NOUN
eajbcs-694	544	6	.	.	PROPN
eajbcs-694	544	7	5	5	NUM
eajbcs-694	544	8	,	,	PUNCT
eajbcs-694	544	9	no	no	INTJ
eajbcs-694	544	10	.	.	NOUN
eajbcs-694	544	11	2	2	NUM
eajbcs-694	544	12	,	,	PUNCT
eajbcs-694	544	13	58	58	NUM
eajbcs-694	544	14	-	-	SYM
eajbcs-694	544	15	86	86	NUM
eajbcs-694	544	16	76	76	NUM
eajbcs-694	544	17	figure	figure	NOUN
eajbcs-694	544	18	6	6	NUM
eajbcs-694	544	19	:	:	PUNCT
eajbcs-694	544	20	the	the	DET
eajbcs-694	544	21	time	time	NOUN
eajbcs-694	544	22	series	series	PROPN
eajbcs-694	544	23	plot	plot	NOUN
eajbcs-694	544	24	of	of	ADP
eajbcs-694	544	25	model	model	NOUN
eajbcs-694	544	26	(	(	PUNCT
eajbcs-694	544	27	2	2	NUM
eajbcs-694	544	28	)	)	PUNCT
eajbcs-694	544	29	when	when	SCONJ
eajbcs-694	544	30	β	β	X
eajbcs-694	544	31	=	=	PUNCT
eajbcs-694	544	32	0.0000028	0.0000028	NUM
eajbcs-694	544	33	with	with	ADP
eajbcs-694	544	34	r0	r0	NOUN
eajbcs-694	544	35	=	=	SYM
eajbcs-694	544	36	1.11073	1.11073	NUM
eajbcs-694	544	37	figure	figure	NOUN
eajbcs-694	544	38	7	7	NUM
eajbcs-694	544	39	:	:	PUNCT
eajbcs-694	544	40	impact	impact	NOUN
eajbcs-694	544	41	of	of	ADP
eajbcs-694	544	42	protection	protection	NOUN
eajbcs-694	544	43	rate	rate	NOUN
eajbcs-694	544	44	θ	θ	PROPN
eajbcs-694	544	45	on	on	ADP
eajbcs-694	544	46	r0	r0	NOUN
eajbcs-694	544	47	and	and	CCONJ
eajbcs-694	544	48	infected	infected	ADJ
eajbcs-694	544	49	classes	class	NOUN
eajbcs-694	544	50	.	.	PUNCT
eajbcs-694	545	1	5.2	5.2	NUM
eajbcs-694	545	2	.	.	PUNCT
eajbcs-694	545	3	existence	existence	NOUN
eajbcs-694	545	4	of	of	ADP
eajbcs-694	545	5	an	an	DET
eajbcs-694	545	6	optimal	optimal	ADJ
eajbcs-694	545	7	control	control	NOUN
eajbcs-694	545	8	the	the	DET
eajbcs-694	545	9	existence	existence	NOUN
eajbcs-694	545	10	of	of	ADP
eajbcs-694	545	11	the	the	DET
eajbcs-694	545	12	optimal	optimal	ADJ
eajbcs-694	545	13	control	control	NOUN
eajbcs-694	545	14	can	can	AUX
eajbcs-694	545	15	be	be	AUX
eajbcs-694	545	16	showed	show	VERB
eajbcs-694	545	17	by	by	ADP
eajbcs-694	545	18	using	use	VERB
eajbcs-694	545	19	an	an	DET
eajbcs-694	545	20	approach	approach	NOUN
eajbcs-694	545	21	of	of	ADP
eajbcs-694	545	22	fleming	fleming	PROPN
eajbcs-694	545	23	and	and	CCONJ
eajbcs-694	545	24	east	east	PROPN
eajbcs-694	545	25	afr	afr	PROPN
eajbcs-694	545	26	.	.	PUNCT
eajbcs-694	546	1	j.	j.	PROPN
eajbcs-694	546	2	biophys	biophys	PROPN
eajbcs-694	546	3	.	.	PUNCT
eajbcs-694	547	1	comput	comput	NOUN
eajbcs-694	547	2	.	.	PUNCT
eajbcs-694	548	1	sci	sci	PROPN
eajbcs-694	548	2	.	.	PUNCT
eajbcs-694	549	1	(	(	PUNCT
eajbcs-694	549	2	2024	2024	NUM
eajbcs-694	549	3	)	)	PUNCT
eajbcs-694	549	4	,	,	PUNCT
eajbcs-694	549	5	vol	vol	NOUN
eajbcs-694	549	6	.	.	PROPN
eajbcs-694	549	7	5	5	NUM
eajbcs-694	549	8	,	,	PUNCT
eajbcs-694	549	9	no	no	INTJ
eajbcs-694	549	10	.	.	NOUN
eajbcs-694	549	11	2	2	NUM
eajbcs-694	549	12	,	,	PUNCT
eajbcs-694	549	13	58	58	NUM
eajbcs-694	549	14	-	-	SYM
eajbcs-694	549	15	86	86	NUM
eajbcs-694	549	16	77	77	NUM
eajbcs-694	549	17	figure	figure	NOUN
eajbcs-694	549	18	8	8	NUM
eajbcs-694	549	19	:	:	PUNCT
eajbcs-694	549	20	impact	impact	NOUN
eajbcs-694	549	21	of	of	ADP
eajbcs-694	549	22	contact	contact	NOUN
eajbcs-694	549	23	rate	rate	NOUN
eajbcs-694	549	24	β	β	X
eajbcs-694	549	25	on	on	ADP
eajbcs-694	549	26	r0	r0	NOUN
eajbcs-694	549	27	and	and	CCONJ
eajbcs-694	549	28	infected	infected	ADJ
eajbcs-694	549	29	classes	class	NOUN
eajbcs-694	549	30	.	.	PUNCT
eajbcs-694	550	1	rishel	rishel	PROPN
eajbcs-694	550	2	(	(	PUNCT
eajbcs-694	550	3	2012	2012	NUM
eajbcs-694	550	4	)	)	PUNCT
eajbcs-694	550	5	.	.	PUNCT
eajbcs-694	551	1	theorem	theorem	VERB
eajbcs-694	551	2	5.1	5.1	NUM
eajbcs-694	551	3	.	.	PUNCT
eajbcs-694	552	1	given	give	VERB
eajbcs-694	552	2	the	the	DET
eajbcs-694	552	3	objective	objective	ADJ
eajbcs-694	552	4	functional	functional	ADJ
eajbcs-694	552	5	j(θ(t	j(θ(t	PROPN
eajbcs-694	552	6	)	)	PUNCT
eajbcs-694	552	7	,	,	PUNCT
eajbcs-694	552	8	α(t	α(t	PROPN
eajbcs-694	552	9	)	)	PUNCT
eajbcs-694	552	10	)	)	PUNCT
eajbcs-694	552	11	(	(	PUNCT
eajbcs-694	552	12	10	10	NUM
eajbcs-694	552	13	)	)	PUNCT
eajbcs-694	552	14	with	with	ADP
eajbcs-694	552	15	admissible	admissible	ADJ
eajbcs-694	552	16	control	control	NOUN
eajbcs-694	552	17	set	set	VERB
eajbcs-694	552	18	u	u	NOUN
eajbcs-694	552	19	,	,	PUNCT
eajbcs-694	552	20	subject	subject	ADJ
eajbcs-694	552	21	to	to	ADP
eajbcs-694	552	22	the	the	DET
eajbcs-694	552	23	state	state	NOUN
eajbcs-694	552	24	system	system	NOUN
eajbcs-694	552	25	(	(	PUNCT
eajbcs-694	552	26	2	2	NUM
eajbcs-694	552	27	)	)	PUNCT
eajbcs-694	552	28	,	,	PUNCT
eajbcs-694	552	29	then	then	ADV
eajbcs-694	552	30	there	there	PRON
eajbcs-694	552	31	exist	exist	VERB
eajbcs-694	552	32	an	an	DET
eajbcs-694	552	33	optimal	optimal	ADJ
eajbcs-694	552	34	control	control	NOUN
eajbcs-694	552	35	double	double	ADJ
eajbcs-694	552	36	u∗	u∗	NOUN
eajbcs-694	552	37	=	=	PUNCT
eajbcs-694	552	38	(	(	PUNCT
eajbcs-694	552	39	θ∗	θ∗	NOUN
eajbcs-694	552	40	,	,	PUNCT
eajbcs-694	552	41	α∗	α∗	NOUN
eajbcs-694	552	42	)	)	PUNCT
eajbcs-694	552	43	in	in	ADP
eajbcs-694	552	44	u	u	PRON
eajbcs-694	553	1	such	such	ADJ
eajbcs-694	553	2	that	that	SCONJ
eajbcs-694	553	3	j	j	PROPN
eajbcs-694	554	1	[	[	X
eajbcs-694	554	2	θ∗	θ∗	NOUN
eajbcs-694	554	3	,	,	PUNCT
eajbcs-694	554	4	α∗	α∗	VERB
eajbcs-694	554	5	]	]	PUNCT
eajbcs-694	554	6	=	=	SYM
eajbcs-694	554	7	min	min	PROPN
eajbcs-694	554	8	(	(	PUNCT
eajbcs-694	554	9	θ	θ	PROPN
eajbcs-694	554	10	,	,	PUNCT
eajbcs-694	554	11	α)∈u	α)∈u	X
eajbcs-694	554	12	j	j	PROPN
eajbcs-694	555	1	[	[	X
eajbcs-694	555	2	θ(t	θ(t	PROPN
eajbcs-694	555	3	)	)	PUNCT
eajbcs-694	555	4	,	,	PUNCT
eajbcs-694	555	5	α(t	α(t	PROPN
eajbcs-694	555	6	)	)	PUNCT
eajbcs-694	555	7	]	]	PUNCT
eajbcs-694	555	8	.	.	PUNCT
eajbcs-694	556	1	(	(	PUNCT
eajbcs-694	556	2	12	12	NUM
eajbcs-694	556	3	)	)	PUNCT
eajbcs-694	556	4	proof	proof	NOUN
eajbcs-694	556	5	.	.	PUNCT
eajbcs-694	557	1	to	to	PART
eajbcs-694	557	2	prove	prove	VERB
eajbcs-694	557	3	the	the	DET
eajbcs-694	557	4	existence	existence	NOUN
eajbcs-694	557	5	of	of	ADP
eajbcs-694	557	6	optimal	optimal	ADJ
eajbcs-694	557	7	control	control	NOUN
eajbcs-694	557	8	,	,	PUNCT
eajbcs-694	557	9	we	we	PRON
eajbcs-694	557	10	need	need	VERB
eajbcs-694	557	11	to	to	PART
eajbcs-694	557	12	verify	verify	VERB
eajbcs-694	557	13	the	the	DET
eajbcs-694	557	14	following	following	ADJ
eajbcs-694	557	15	conditions	condition	NOUN
eajbcs-694	557	16	.	.	PUNCT
eajbcs-694	558	1	(	(	PUNCT
eajbcs-694	558	2	a	a	X
eajbcs-694	558	3	)	)	PUNCT
eajbcs-694	558	4	the	the	DET
eajbcs-694	558	5	set	set	NOUN
eajbcs-694	558	6	of	of	ADP
eajbcs-694	558	7	solutions	solution	NOUN
eajbcs-694	558	8	to	to	ADP
eajbcs-694	558	9	the	the	DET
eajbcs-694	558	10	state	state	NOUN
eajbcs-694	558	11	system	system	NOUN
eajbcs-694	558	12	(	(	PUNCT
eajbcs-694	558	13	2	2	NUM
eajbcs-694	558	14	)	)	PUNCT
eajbcs-694	558	15	and	and	CCONJ
eajbcs-694	558	16	control	control	NOUN
eajbcs-694	558	17	parameters	parameter	NOUN
eajbcs-694	558	18	in	in	ADP
eajbcs-694	558	19	(	(	PUNCT
eajbcs-694	558	20	9	9	NUM
eajbcs-694	558	21	)	)	PUNCT
eajbcs-694	558	22	are	be	AUX
eajbcs-694	558	23	non	non	ADJ
eajbcs-694	558	24	-	-	ADJ
eajbcs-694	558	25	empty	empty	ADJ
eajbcs-694	558	26	.	.	PUNCT
eajbcs-694	559	1	(	(	PUNCT
eajbcs-694	559	2	b	b	X
eajbcs-694	559	3	)	)	PUNCT
eajbcs-694	559	4	the	the	DET
eajbcs-694	559	5	set	set	NOUN
eajbcs-694	559	6	u	u	NOUN
eajbcs-694	559	7	is	be	AUX
eajbcs-694	559	8	convex	convex	ADJ
eajbcs-694	559	9	and	and	CCONJ
eajbcs-694	559	10	closed	closed	ADJ
eajbcs-694	559	11	.	.	PUNCT
eajbcs-694	560	1	(	(	PUNCT
eajbcs-694	560	2	c	c	X
eajbcs-694	560	3	)	)	PUNCT
eajbcs-694	560	4	the	the	DET
eajbcs-694	560	5	right	right	ADJ
eajbcs-694	560	6	hand	hand	NOUN
eajbcs-694	560	7	side	side	NOUN
eajbcs-694	560	8	of	of	ADP
eajbcs-694	560	9	system	system	NOUN
eajbcs-694	560	10	(	(	PUNCT
eajbcs-694	560	11	2	2	X
eajbcs-694	560	12	)	)	PUNCT
eajbcs-694	560	13	is	be	AUX
eajbcs-694	560	14	bounded	bound	VERB
eajbcs-694	560	15	above	above	ADV
eajbcs-694	560	16	by	by	ADP
eajbcs-694	560	17	sum	sum	NOUN
eajbcs-694	560	18	of	of	ADP
eajbcs-694	560	19	bounded	bounded	ADJ
eajbcs-694	560	20	control	control	NOUN
eajbcs-694	560	21	and	and	CCONJ
eajbcs-694	560	22	state	state	NOUN
eajbcs-694	560	23	and	and	CCONJ
eajbcs-694	560	24	can	can	AUX
eajbcs-694	560	25	be	be	AUX
eajbcs-694	560	26	written	write	VERB
eajbcs-694	560	27	as	as	ADP
eajbcs-694	560	28	a	a	DET
eajbcs-694	560	29	linear	linear	ADJ
eajbcs-694	560	30	function	function	NOUN
eajbcs-694	560	31	of	of	ADP
eajbcs-694	560	32	the	the	DET
eajbcs-694	560	33	control	control	NOUN
eajbcs-694	560	34	variables	variable	NOUN
eajbcs-694	560	35	with	with	ADP
eajbcs-694	560	36	coefficients	coefficient	NOUN
eajbcs-694	560	37	dependent	dependent	ADJ
eajbcs-694	560	38	on	on	ADP
eajbcs-694	560	39	time	time	NOUN
eajbcs-694	560	40	and	and	CCONJ
eajbcs-694	560	41	state	state	NOUN
eajbcs-694	560	42	variables	variable	NOUN
eajbcs-694	560	43	.	.	PUNCT
eajbcs-694	561	1	(	(	PUNCT
eajbcs-694	561	2	d	d	X
eajbcs-694	561	3	)	)	PUNCT
eajbcs-694	561	4	the	the	DET
eajbcs-694	561	5	integrand	integrand	PROPN
eajbcs-694	561	6	function	function	NOUN
eajbcs-694	561	7	l(e	l(e	NOUN
eajbcs-694	561	8	,	,	PUNCT
eajbcs-694	561	9	i	i	PRON
eajbcs-694	561	10	,	,	PUNCT
eajbcs-694	561	11	θ	θ	PROPN
eajbcs-694	561	12	,	,	PUNCT
eajbcs-694	561	13	α	α	PROPN
eajbcs-694	561	14	,	,	PUNCT
eajbcs-694	561	15	t	t	PROPN
eajbcs-694	561	16	)	)	PUNCT
eajbcs-694	561	17	is	be	AUX
eajbcs-694	561	18	convex	convex	ADJ
eajbcs-694	561	19	on	on	ADP
eajbcs-694	561	20	u	u	NOUN
eajbcs-694	561	21	and	and	CCONJ
eajbcs-694	561	22	l(e	l(e	PROPN
eajbcs-694	561	23	,	,	PUNCT
eajbcs-694	561	24	i	i	PROPN
eajbcs-694	561	25	,	,	PUNCT
eajbcs-694	561	26	θ	θ	PROPN
eajbcs-694	561	27	,	,	PUNCT
eajbcs-694	561	28	α	α	PROPN
eajbcs-694	561	29	,	,	PUNCT
eajbcs-694	561	30	t	t	PROPN
eajbcs-694	561	31	)	)	PUNCT
eajbcs-694	561	32	≥	≥	NOUN
eajbcs-694	561	33	h(u	h(u	PROPN
eajbcs-694	561	34	)	)	PUNCT
eajbcs-694	561	35	,	,	PUNCT
eajbcs-694	561	36	where	where	SCONJ
eajbcs-694	561	37	h(u	h(u	PROPN
eajbcs-694	561	38	)	)	PUNCT
eajbcs-694	561	39	is	be	AUX
eajbcs-694	561	40	continuous	continuous	ADJ
eajbcs-694	561	41	and	and	CCONJ
eajbcs-694	561	42	||u||−1h(u	||u||−1h(u	NOUN
eajbcs-694	561	43	)	)	PUNCT
eajbcs-694	561	44	→	→	SYM
eajbcs-694	561	45	∞	∞	NUM
eajbcs-694	561	46	when	when	SCONJ
eajbcs-694	561	47	||u||	||u||	ADV
eajbcs-694	561	48	→	→	SYM
eajbcs-694	561	49	∞.	∞.	PROPN
eajbcs-694	561	50	here	here	ADV
eajbcs-694	561	51	u	u	PROPN
eajbcs-694	561	52	=	=	SYM
eajbcs-694	561	53	(	(	PUNCT
eajbcs-694	561	54	θ(t	θ(t	PROPN
eajbcs-694	561	55	)	)	PUNCT
eajbcs-694	561	56	,	,	PUNCT
eajbcs-694	561	57	α(t	α(t	PROPN
eajbcs-694	561	58	)	)	PUNCT
eajbcs-694	561	59	)	)	PUNCT
eajbcs-694	561	60	.	.	PUNCT
eajbcs-694	562	1	in	in	ADP
eajbcs-694	562	2	theorem	theorem	NOUN
eajbcs-694	562	3	(	(	PUNCT
eajbcs-694	562	4	3.2	3.2	NUM
eajbcs-694	562	5	)	)	PUNCT
eajbcs-694	562	6	,	,	PUNCT
eajbcs-694	562	7	we	we	PRON
eajbcs-694	562	8	have	have	AUX
eajbcs-694	562	9	already	already	ADV
eajbcs-694	562	10	justified	justify	VERB
eajbcs-694	562	11	the	the	DET
eajbcs-694	562	12	boundedness	boundedness	NOUN
eajbcs-694	562	13	of	of	ADP
eajbcs-694	562	14	the	the	DET
eajbcs-694	562	15	solution	solution	NOUN
eajbcs-694	562	16	of	of	ADP
eajbcs-694	562	17	the	the	DET
eajbcs-694	562	18	state	state	NOUN
eajbcs-694	562	19	system	system	NOUN
eajbcs-694	562	20	(	(	PUNCT
eajbcs-694	562	21	2	2	NUM
eajbcs-694	562	22	)	)	PUNCT
eajbcs-694	562	23	.	.	PUNCT
eajbcs-694	563	1	since	since	SCONJ
eajbcs-694	563	2	our	our	PRON
eajbcs-694	563	3	solution	solution	NOUN
eajbcs-694	563	4	for	for	ADP
eajbcs-694	563	5	the	the	DET
eajbcs-694	563	6	model	model	NOUN
eajbcs-694	563	7	is	be	AUX
eajbcs-694	563	8	bounded	bound	VERB
eajbcs-694	563	9	by	by	ADP
eajbcs-694	563	10	n(t	n(t	PROPN
eajbcs-694	563	11	)	)	PUNCT
eajbcs-694	563	12	≤	≤	NUM
eajbcs-694	563	13	max	max	PROPN
eajbcs-694	563	14	(	(	PUNCT
eajbcs-694	563	15	n(0	n(0	PROPN
eajbcs-694	563	16	)	)	PUNCT
eajbcs-694	563	17	,	,	PUNCT
eajbcs-694	563	18	π	π	PROPN
eajbcs-694	563	19	µ	µ	X
eajbcs-694	563	20	)	)	PUNCT
eajbcs-694	563	21	for	for	ADP
eajbcs-694	563	22	all	all	DET
eajbcs-694	563	23	t	t	PROPN
eajbcs-694	563	24	≥	≥	NOUN
eajbcs-694	563	25	0	0	NUM
eajbcs-694	563	26	.	.	PUNCT
eajbcs-694	564	1	this	this	PRON
eajbcs-694	564	2	implies	imply	VERB
eajbcs-694	564	3	that	that	SCONJ
eajbcs-694	564	4	the	the	DET
eajbcs-694	564	5	solutions	solution	NOUN
eajbcs-694	564	6	of	of	ADP
eajbcs-694	564	7	the	the	DET
eajbcs-694	564	8	state	state	NOUN
eajbcs-694	564	9	system	system	NOUN
eajbcs-694	564	10	are	be	AUX
eajbcs-694	564	11	continuous	continuous	ADJ
eajbcs-694	564	12	and	and	CCONJ
eajbcs-694	564	13	bounded	bound	VERB
eajbcs-694	564	14	for	for	ADP
eajbcs-694	564	15	each	each	DET
eajbcs-694	564	16	admissible	admissible	ADJ
eajbcs-694	564	17	control	control	NOUN
eajbcs-694	564	18	functions	function	NOUN
eajbcs-694	564	19	in	in	ADP
eajbcs-694	564	20	u	u	PROPN
eajbcs-694	564	21	.	.	PUNCT
eajbcs-694	565	1	moreover	moreover	ADV
eajbcs-694	565	2	,	,	PUNCT
eajbcs-694	565	3	the	the	DET
eajbcs-694	565	4	right	right	ADJ
eajbcs-694	565	5	hand	hand	NOUN
eajbcs-694	565	6	side	side	NOUN
eajbcs-694	565	7	of	of	ADP
eajbcs-694	565	8	the	the	DET
eajbcs-694	565	9	model	model	NOUN
eajbcs-694	565	10	equations	equation	NOUN
eajbcs-694	565	11	(	(	PUNCT
eajbcs-694	565	12	2	2	X
eajbcs-694	565	13	)	)	PUNCT
eajbcs-694	565	14	satisfes	satisfe	NOUN
eajbcs-694	565	15	the	the	DET
eajbcs-694	565	16	lipschitz	lipschitz	NOUN
eajbcs-694	565	17	condition	condition	NOUN
eajbcs-694	565	18	with	with	ADP
eajbcs-694	565	19	respect	respect	NOUN
eajbcs-694	565	20	to	to	ADP
eajbcs-694	565	21	state	state	NOUN
eajbcs-694	565	22	variables	variable	NOUN
eajbcs-694	565	23	.	.	PUNCT
eajbcs-694	566	1	hence	hence	ADV
eajbcs-694	566	2	,	,	PUNCT
eajbcs-694	566	3	the	the	DET
eajbcs-694	566	4	state	state	NOUN
eajbcs-694	566	5	system	system	NOUN
eajbcs-694	566	6	(	(	PUNCT
eajbcs-694	566	7	2	2	X
eajbcs-694	566	8	)	)	PUNCT
eajbcs-694	566	9	has	have	VERB
eajbcs-694	566	10	a	a	DET
eajbcs-694	566	11	unique	unique	ADJ
eajbcs-694	566	12	solution	solution	NOUN
eajbcs-694	566	13	corresponding	correspond	VERB
eajbcs-694	566	14	to	to	ADP
eajbcs-694	566	15	each	each	DET
eajbcs-694	566	16	admissible	admissible	ADJ
eajbcs-694	566	17	control	control	NOUN
eajbcs-694	566	18	function	function	NOUN
eajbcs-694	566	19	(	(	PUNCT
eajbcs-694	566	20	θ(t	θ(t	PROPN
eajbcs-694	566	21	)	)	PUNCT
eajbcs-694	566	22	,	,	PUNCT
eajbcs-694	566	23	α(t	α(t	PROPN
eajbcs-694	566	24	)	)	PUNCT
eajbcs-694	566	25	)	)	PUNCT
eajbcs-694	567	1	∈	∈	PROPN
eajbcs-694	567	2	u	u	NOUN
eajbcs-694	567	3	.	.	PUNCT
eajbcs-694	568	1	thus	thus	ADV
eajbcs-694	568	2	,	,	PUNCT
eajbcs-694	568	3	condition	condition	NOUN
eajbcs-694	568	4	(	(	PUNCT
eajbcs-694	568	5	a	a	X
eajbcs-694	568	6	)	)	PUNCT
eajbcs-694	568	7	is	be	AUX
eajbcs-694	568	8	achieved	achieve	VERB
eajbcs-694	568	9	.	.	PUNCT
eajbcs-694	569	1	to	to	PART
eajbcs-694	569	2	verify	verify	VERB
eajbcs-694	569	3	condition	condition	NOUN
eajbcs-694	569	4	(	(	PUNCT
eajbcs-694	569	5	b	b	NOUN
eajbcs-694	569	6	)	)	PUNCT
eajbcs-694	569	7	,	,	PUNCT
eajbcs-694	569	8	given	give	VERB
eajbcs-694	569	9	that	that	SCONJ
eajbcs-694	569	10	the	the	DET
eajbcs-694	569	11	control	control	NOUN
eajbcs-694	569	12	set	set	VERB
eajbcs-694	569	13	u	u	NOUN
eajbcs-694	569	14	=	=	PUNCT
eajbcs-694	569	15	{	{	PUNCT
eajbcs-694	569	16	u	u	NOUN
eajbcs-694	569	17	∈	∈	PROPN
eajbcs-694	569	18	r2	r2	PROPN
eajbcs-694	569	19	:	:	PUNCT
eajbcs-694	569	20	||u||∞	||u||∞	VERB
eajbcs-694	569	21	≤	≤	ADV
eajbcs-694	569	22	1	1	NUM
eajbcs-694	569	23	}	}	PUNCT
eajbcs-694	569	24	.	.	PUNCT
eajbcs-694	570	1	let	let	VERB
eajbcs-694	570	2	ψ	ψ	X
eajbcs-694	570	3	∈	∈	PROPN
eajbcs-694	570	4	[	[	X
eajbcs-694	570	5	0	0	NUM
eajbcs-694	570	6	,	,	PUNCT
eajbcs-694	570	7	1	1	NUM
eajbcs-694	570	8	]	]	PUNCT
eajbcs-694	570	9	and	and	CCONJ
eajbcs-694	570	10	v1	v1	NOUN
eajbcs-694	570	11	,	,	PUNCT
eajbcs-694	570	12	v2	v2	PROPN
eajbcs-694	570	13	∈	∈	PROPN
eajbcs-694	570	14	u	u	NOUN
eajbcs-694	570	15	such	such	ADJ
eajbcs-694	570	16	that	that	SCONJ
eajbcs-694	570	17	||v1||∞	||v1||∞	NOUN
eajbcs-694	570	18	≤	≤	NUM
eajbcs-694	570	19	1	1	NUM
eajbcs-694	570	20	and	and	CCONJ
eajbcs-694	570	21	||v2||∞	||v2||∞	VERB
eajbcs-694	570	22	≤	≤	NUM
eajbcs-694	570	23	1	1	NUM
eajbcs-694	570	24	,	,	PUNCT
eajbcs-694	570	25	then	then	ADV
eajbcs-694	570	26	||ψv1+(1−ψ)v2||∞	||ψv1+(1−ψ)v2||∞	VERB
eajbcs-694	570	27	≤	≤	NUM
eajbcs-694	570	28	ψ||v1||∞+(1−ψ)||v2||∞	ψ||v1||∞+(1−ψ)||v2||∞	NOUN
eajbcs-694	570	29	≤	≤	NUM
eajbcs-694	570	30	1	1	NUM
eajbcs-694	570	31	.	.	PUNCT
eajbcs-694	571	1	thus	thus	ADV
eajbcs-694	571	2	,	,	PUNCT
eajbcs-694	571	3	the	the	DET
eajbcs-694	571	4	set	set	NOUN
eajbcs-694	571	5	u	u	NOUN
eajbcs-694	571	6	is	be	AUX
eajbcs-694	571	7	convex	convex	ADJ
eajbcs-694	571	8	and	and	CCONJ
eajbcs-694	571	9	closed	close	VERB
eajbcs-694	571	10	.	.	PUNCT
eajbcs-694	572	1	east	east	PROPN
eajbcs-694	572	2	afr	afr	PROPN
eajbcs-694	572	3	.	.	PUNCT
eajbcs-694	573	1	j.	j.	PROPN
eajbcs-694	573	2	biophys	biophys	PROPN
eajbcs-694	573	3	.	.	PUNCT
eajbcs-694	574	1	comput	comput	NOUN
eajbcs-694	574	2	.	.	PUNCT
eajbcs-694	575	1	sci	sci	PROPN
eajbcs-694	575	2	.	.	PUNCT
eajbcs-694	576	1	(	(	PUNCT
eajbcs-694	576	2	2024	2024	NUM
eajbcs-694	576	3	)	)	PUNCT
eajbcs-694	576	4	,	,	PUNCT
eajbcs-694	576	5	vol	vol	NOUN
eajbcs-694	576	6	.	.	PROPN
eajbcs-694	576	7	5	5	NUM
eajbcs-694	576	8	,	,	PUNCT
eajbcs-694	576	9	no	no	INTJ
eajbcs-694	576	10	.	.	NOUN
eajbcs-694	576	11	2	2	NUM
eajbcs-694	576	12	,	,	PUNCT
eajbcs-694	576	13	58	58	NUM
eajbcs-694	576	14	-	-	SYM
eajbcs-694	576	15	86	86	NUM
eajbcs-694	576	16	78	78	NUM
eajbcs-694	576	17	to	to	PART
eajbcs-694	576	18	verify	verify	VERB
eajbcs-694	576	19	condition	condition	NOUN
eajbcs-694	576	20	(	(	PUNCT
eajbcs-694	576	21	c	c	NOUN
eajbcs-694	576	22	)	)	PUNCT
eajbcs-694	576	23	,	,	PUNCT
eajbcs-694	576	24	let	let	VERB
eajbcs-694	576	25	u	u	PRON
eajbcs-694	576	26	=	=	X
eajbcs-694	576	27	(	(	PUNCT
eajbcs-694	576	28	θ(t	θ(t	PROPN
eajbcs-694	576	29	)	)	PUNCT
eajbcs-694	576	30	,	,	PUNCT
eajbcs-694	576	31	α(t	α(t	PROPN
eajbcs-694	576	32	)	)	PUNCT
eajbcs-694	576	33	)	)	PUNCT
eajbcs-694	577	1	∈	∈	PROPN
eajbcs-694	577	2	u	u	NOUN
eajbcs-694	577	3	,	,	PUNCT
eajbcs-694	577	4	x	x	SYM
eajbcs-694	577	5	=	=	SYM
eajbcs-694	577	6	(	(	PUNCT
eajbcs-694	577	7	s	s	PROPN
eajbcs-694	577	8	,	,	PUNCT
eajbcs-694	577	9	p	p	X
eajbcs-694	577	10	,	,	PUNCT
eajbcs-694	577	11	e	e	NOUN
eajbcs-694	577	12	,	,	PUNCT
eajbcs-694	577	13	i	i	PRON
eajbcs-694	577	14	,	,	PUNCT
eajbcs-694	577	15	h	h	NOUN
eajbcs-694	577	16	,	,	PUNCT
eajbcs-694	577	17	r	r	NOUN
eajbcs-694	577	18	)	)	PUNCT
eajbcs-694	577	19	and	and	CCONJ
eajbcs-694	577	20	the	the	DET
eajbcs-694	577	21	right	right	ADJ
eajbcs-694	577	22	hand	hand	NOUN
eajbcs-694	577	23	side	side	NOUN
eajbcs-694	577	24	of	of	ADP
eajbcs-694	577	25	the	the	DET
eajbcs-694	577	26	state	state	NOUN
eajbcs-694	577	27	system	system	NOUN
eajbcs-694	577	28	(	(	PUNCT
eajbcs-694	577	29	2	2	X
eajbcs-694	577	30	)	)	PUNCT
eajbcs-694	577	31	is	be	AUX
eajbcs-694	577	32	given	give	VERB
eajbcs-694	577	33	by	by	ADP
eajbcs-694	577	34	f(t	f(t	NOUN
eajbcs-694	577	35	,	,	PUNCT
eajbcs-694	577	36	x	x	NOUN
eajbcs-694	577	37	,	,	PUNCT
eajbcs-694	577	38	u	u	NOUN
eajbcs-694	577	39	)	)	PUNCT
eajbcs-694	577	40	=	=	SYM
eajbcs-694	577	41			NOUN
eajbcs-694	577	42	π+	π+	PUNCT
eajbcs-694	577	43	ηωr	ηωr	PROPN
eajbcs-694	577	44	+	+	X
eajbcs-694	577	45	νp	νp	ADP
eajbcs-694	577	46	−	−	PROPN
eajbcs-694	577	47	β(σ1i	β(σ1i	NOUN
eajbcs-694	577	48	+	+	NUM
eajbcs-694	577	49	σ2e	σ2e	PROPN
eajbcs-694	577	50	+	+	CCONJ
eajbcs-694	577	51	σ3h)s	σ3h)s	X
eajbcs-694	577	52	−	−	PROPN
eajbcs-694	577	53	(	(	PUNCT
eajbcs-694	577	54	θ(t	θ(t	PROPN
eajbcs-694	577	55	)	)	PUNCT
eajbcs-694	577	56	+	+	NUM
eajbcs-694	577	57	µ)s	µ)s	NOUN
eajbcs-694	578	1	θ(t)s	θ(t)s	NUM
eajbcs-694	578	2	+	+	CCONJ
eajbcs-694	578	3	(	(	PUNCT
eajbcs-694	578	4	1−	1−	NUM
eajbcs-694	578	5	η)ωr−	η)ωr−	PROPN
eajbcs-694	578	6	(	(	PUNCT
eajbcs-694	578	7	ν	ν	NOUN
eajbcs-694	578	8	+	+	NOUN
eajbcs-694	578	9	µ)p	µ)p	NOUN
eajbcs-694	578	10	β(σ1i	β(σ1i	PUNCT
eajbcs-694	578	11	+	+	NUM
eajbcs-694	578	12	σ2e	σ2e	PROPN
eajbcs-694	578	13	+	+	CCONJ
eajbcs-694	578	14	σ3h)s	σ3h)s	X
eajbcs-694	578	15	−	−	PROPN
eajbcs-694	578	16	(	(	PUNCT
eajbcs-694	578	17	δ	δ	NOUN
eajbcs-694	578	18	+	+	NOUN
eajbcs-694	578	19	µ)e	µ)e	NOUN
eajbcs-694	578	20	τδe	τδe	NOUN
eajbcs-694	578	21	−	−	PROPN
eajbcs-694	578	22	(	(	PUNCT
eajbcs-694	578	23	α(t	α(t	PROPN
eajbcs-694	578	24	)	)	PUNCT
eajbcs-694	578	25	+	+	NUM
eajbcs-694	578	26	ε+	ε+	X
eajbcs-694	578	27	µ+	µ+	X
eajbcs-694	578	28	ρ)i	ρ)i	NOUN
eajbcs-694	578	29	α(t)i	α(t)i	NUM
eajbcs-694	578	30	−	−	PROPN
eajbcs-694	578	31	(	(	PUNCT
eajbcs-694	578	32	γ	γ	X
eajbcs-694	578	33	+	+	X
eajbcs-694	578	34	µ+	µ+	X
eajbcs-694	578	35	ξ)h	ξ)h	X
eajbcs-694	578	36	(	(	PUNCT
eajbcs-694	578	37	1−	1−	NUM
eajbcs-694	578	38	τ)δe	τ)δe	PROPN
eajbcs-694	578	39	+	+	CCONJ
eajbcs-694	578	40	εi	εi	VERB
eajbcs-694	578	41	+	+	X
eajbcs-694	578	42	γh	γh	ADP
eajbcs-694	578	43	−	−	PROPN
eajbcs-694	578	44	(	(	PUNCT
eajbcs-694	578	45	ω	ω	NOUN
eajbcs-694	578	46	+	+	NOUN
eajbcs-694	578	47	µ)r	µ)r	NOUN
eajbcs-694	578	48			NUM
eajbcs-694	578	49	.	.	PUNCT
eajbcs-694	579	1	(	(	PUNCT
eajbcs-694	579	2	13	13	NUM
eajbcs-694	579	3	)	)	PUNCT
eajbcs-694	579	4	then	then	ADV
eajbcs-694	579	5	from	from	ADP
eajbcs-694	579	6	(	(	PUNCT
eajbcs-694	579	7	13	13	NUM
eajbcs-694	579	8	)	)	PUNCT
eajbcs-694	579	9	we	we	PRON
eajbcs-694	579	10	get	get	VERB
eajbcs-694	579	11	,	,	PUNCT
eajbcs-694	579	12	f(t	f(t	PROPN
eajbcs-694	579	13	,	,	PUNCT
eajbcs-694	579	14	x	x	NOUN
eajbcs-694	579	15	,	,	PUNCT
eajbcs-694	579	16	u	u	NOUN
eajbcs-694	579	17	)	)	PUNCT
eajbcs-694	579	18	=	=	SYM
eajbcs-694	579	19	g(t	g(t	PROPN
eajbcs-694	579	20	,	,	PUNCT
eajbcs-694	579	21	x	x	X
eajbcs-694	579	22	)	)	PUNCT
eajbcs-694	580	1	+	+	CCONJ
eajbcs-694	580	2	h(t	h(t	PROPN
eajbcs-694	580	3	,	,	PUNCT
eajbcs-694	580	4	x)ut	x)ut	PROPN
eajbcs-694	580	5	,	,	PUNCT
eajbcs-694	580	6	where	where	SCONJ
eajbcs-694	580	7	g(t	g(t	PROPN
eajbcs-694	580	8	,	,	PUNCT
eajbcs-694	580	9	x	x	NOUN
eajbcs-694	580	10	)	)	PUNCT
eajbcs-694	580	11	=	=	SYM
eajbcs-694	580	12			NOUN
eajbcs-694	580	13	π+	π+	PUNCT
eajbcs-694	580	14	ηωr	ηωr	PROPN
eajbcs-694	580	15	+	+	X
eajbcs-694	580	16	νp	νp	ADP
eajbcs-694	580	17	−	−	PROPN
eajbcs-694	580	18	β(σ1i	β(σ1i	NOUN
eajbcs-694	580	19	+	+	NUM
eajbcs-694	580	20	σ2e	σ2e	PROPN
eajbcs-694	580	21	+	+	CCONJ
eajbcs-694	580	22	σ3h)s	σ3h)s	X
eajbcs-694	580	23	−	−	X
eajbcs-694	580	24	µs	µs	X
eajbcs-694	580	25	(	(	PUNCT
eajbcs-694	580	26	1−	1−	NUM
eajbcs-694	580	27	η)ωr−	η)ωr−	PROPN
eajbcs-694	580	28	(	(	PUNCT
eajbcs-694	580	29	ν	ν	NOUN
eajbcs-694	580	30	+	+	NOUN
eajbcs-694	580	31	µ)p	µ)p	NOUN
eajbcs-694	580	32	β(σ1i	β(σ1i	PUNCT
eajbcs-694	580	33	+	+	NUM
eajbcs-694	580	34	σ2e	σ2e	PROPN
eajbcs-694	581	1	+	+	CCONJ
eajbcs-694	581	2	σ3h)s	σ3h)s	X
eajbcs-694	581	3	−	−	PROPN
eajbcs-694	581	4	(	(	PUNCT
eajbcs-694	581	5	δ	δ	NOUN
eajbcs-694	581	6	+	+	NOUN
eajbcs-694	581	7	µ)e	µ)e	NOUN
eajbcs-694	581	8	τδe	τδe	NOUN
eajbcs-694	581	9	−	−	PROPN
eajbcs-694	581	10	(	(	PUNCT
eajbcs-694	581	11	ε+	ε+	X
eajbcs-694	581	12	µ+	µ+	X
eajbcs-694	581	13	ρ)i	ρ)i	ADJ
eajbcs-694	581	14	−(γ	−(γ	ADJ
eajbcs-694	581	15	+	+	X
eajbcs-694	581	16	µ+	µ+	X
eajbcs-694	581	17	ξ)h	ξ)h	X
eajbcs-694	581	18	(	(	PUNCT
eajbcs-694	581	19	1−	1−	NUM
eajbcs-694	581	20	τ)δe	τ)δe	PROPN
eajbcs-694	581	21	+	+	CCONJ
eajbcs-694	581	22	εi	εi	VERB
eajbcs-694	581	23	+	+	X
eajbcs-694	581	24	γh	γh	ADP
eajbcs-694	581	25	−	−	PROPN
eajbcs-694	581	26	(	(	PUNCT
eajbcs-694	581	27	ω	ω	NOUN
eajbcs-694	581	28	+	+	NOUN
eajbcs-694	581	29	µ)r	µ)r	NOUN
eajbcs-694	581	30			NUM
eajbcs-694	581	31	and	and	CCONJ
eajbcs-694	581	32	h(t	h(t	PROPN
eajbcs-694	581	33	,	,	PUNCT
eajbcs-694	581	34	x	x	X
eajbcs-694	581	35	)	)	PUNCT
eajbcs-694	581	36	=	=	SYM
eajbcs-694	581	37			NOUN
eajbcs-694	581	38	−s	−s	NOUN
eajbcs-694	581	39	0	0	NUM
eajbcs-694	581	40	s	s	NOUN
eajbcs-694	581	41	0	0	NUM
eajbcs-694	581	42	0	0	NUM
eajbcs-694	581	43	0	0	NUM
eajbcs-694	581	44	0	0	NUM
eajbcs-694	582	1	−i	−i	NOUN
eajbcs-694	582	2	0	0	PUNCT
eajbcs-694	583	1	i	i	PRON
eajbcs-694	583	2	0	0	NUM
eajbcs-694	583	3	0	0	NUM
eajbcs-694	583	4			NUM
eajbcs-694	583	5	.	.	PUNCT
eajbcs-694	584	1	since	since	SCONJ
eajbcs-694	584	2	,	,	PUNCT
eajbcs-694	584	3	by	by	ADP
eajbcs-694	584	4	using	use	VERB
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eajbcs-694	584	6	properties	property	NOUN
eajbcs-694	584	7	of	of	ADP
eajbcs-694	584	8	a	a	DET
eajbcs-694	584	9	norm	norm	NOUN
eajbcs-694	584	10	of	of	ADP
eajbcs-694	584	11	a	a	DET
eajbcs-694	584	12	matrix	matrix	NOUN
eajbcs-694	584	13	we	we	PRON
eajbcs-694	584	14	have	have	VERB
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eajbcs-694	584	17	,	,	PUNCT
eajbcs-694	584	18	x	x	NOUN
eajbcs-694	584	19	,	,	PUNCT
eajbcs-694	584	20	u)||	u)||	ADJ
eajbcs-694	584	21	=	=	SYM
eajbcs-694	584	22	||g(t	||g(t	X
eajbcs-694	584	23	,	,	PUNCT
eajbcs-694	584	24	x	x	X
eajbcs-694	584	25	)	)	PUNCT
eajbcs-694	584	26	+	+	CCONJ
eajbcs-694	584	27	h(t	h(t	PROPN
eajbcs-694	584	28	,	,	PUNCT
eajbcs-694	584	29	x)ut	x)ut	PROPN
eajbcs-694	584	30	||	||	NOUN
eajbcs-694	584	31	≤	≤	NUM
eajbcs-694	584	32	||g(t	||g(t	NOUN
eajbcs-694	584	33	,	,	PUNCT
eajbcs-694	584	34	x)||+	x)||+	PROPN
eajbcs-694	584	35	||h(t	||h(t	NOUN
eajbcs-694	584	36	,	,	PUNCT
eajbcs-694	584	37	x)||	x)||	PROPN
eajbcs-694	584	38	||u||	||u||	NOUN
eajbcs-694	584	39	.	.	PUNCT
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eajbcs-694	585	2	,	,	PUNCT
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eajbcs-694	585	4	(	(	PUNCT
eajbcs-694	585	5	c	c	NOUN
eajbcs-694	585	6	)	)	PUNCT
eajbcs-694	585	7	is	be	AUX
eajbcs-694	585	8	proved	prove	VERB
eajbcs-694	585	9	.	.	PUNCT
eajbcs-694	586	1	to	to	PART
eajbcs-694	586	2	verify	verify	VERB
eajbcs-694	586	3	condition	condition	NOUN
eajbcs-694	586	4	(	(	PUNCT
eajbcs-694	586	5	d	d	NOUN
eajbcs-694	586	6	)	)	PUNCT
eajbcs-694	586	7	,	,	PUNCT
eajbcs-694	586	8	the	the	DET
eajbcs-694	586	9	integrand	integrand	NOUN
eajbcs-694	586	10	of	of	ADP
eajbcs-694	586	11	the	the	DET
eajbcs-694	586	12	objective	objective	ADJ
eajbcs-694	586	13	functional	functional	ADJ
eajbcs-694	586	14	(	(	PUNCT
eajbcs-694	586	15	10	10	NUM
eajbcs-694	586	16	)	)	PUNCT
eajbcs-694	586	17	l(e	l(e	NOUN
eajbcs-694	586	18	,	,	PUNCT
eajbcs-694	586	19	i	i	PROPN
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eajbcs-694	586	25	t	t	PROPN
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eajbcs-694	586	27	=	=	SYM
eajbcs-694	586	28	b1e(t	b1e(t	PROPN
eajbcs-694	586	29	)	)	PUNCT
eajbcs-694	587	1	+	+	ADJ
eajbcs-694	587	2	b2i(t	b2i(t	NOUN
eajbcs-694	587	3	)	)	PUNCT
eajbcs-694	588	1	+	+	CCONJ
eajbcs-694	588	2	1	1	NUM
eajbcs-694	588	3	2	2	NUM
eajbcs-694	588	4	(	(	PUNCT
eajbcs-694	588	5	a1θ	a1θ	PROPN
eajbcs-694	588	6	2(t	2(t	NUM
eajbcs-694	588	7	)	)	PUNCT
eajbcs-694	589	1	+	+	CCONJ
eajbcs-694	589	2	a2α	a2α	SYM
eajbcs-694	589	3	2(t	2(t	NUM
eajbcs-694	589	4	)	)	PUNCT
eajbcs-694	589	5	)	)	PUNCT
eajbcs-694	590	1	(	(	PUNCT
eajbcs-694	590	2	14	14	NUM
eajbcs-694	590	3	)	)	PUNCT
eajbcs-694	590	4	is	be	AUX
eajbcs-694	590	5	the	the	DET
eajbcs-694	590	6	sum	sum	NOUN
eajbcs-694	590	7	of	of	ADP
eajbcs-694	590	8	convex	convex	ADJ
eajbcs-694	590	9	function	function	NOUN
eajbcs-694	590	10	and	and	CCONJ
eajbcs-694	590	11	hence	hence	ADV
eajbcs-694	590	12	convex	convex	VERB
eajbcs-694	590	13	with	with	ADP
eajbcs-694	590	14	respect	respect	NOUN
eajbcs-694	590	15	to	to	PART
eajbcs-694	590	16	control	control	VERB
eajbcs-694	590	17	parameters	parameter	NOUN
eajbcs-694	590	18	θ(t	θ(t	NOUN
eajbcs-694	590	19	)	)	PUNCT
eajbcs-694	590	20	and	and	CCONJ
eajbcs-694	590	21	α(t	α(t	NOUN
eajbcs-694	590	22	)	)	PUNCT
eajbcs-694	590	23	.	.	PUNCT
eajbcs-694	591	1	moreover	moreover	ADV
eajbcs-694	591	2	,	,	PUNCT
eajbcs-694	591	3	l(e	l(e	NOUN
eajbcs-694	591	4	,	,	PUNCT
eajbcs-694	591	5	i	i	PROPN
eajbcs-694	591	6	,	,	PUNCT
eajbcs-694	591	7	θ	θ	PROPN
eajbcs-694	591	8	,	,	PUNCT
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eajbcs-694	591	10	,	,	PUNCT
eajbcs-694	591	11	t	t	PROPN
eajbcs-694	591	12	)	)	PUNCT
eajbcs-694	591	13	=	=	SYM
eajbcs-694	591	14	b1e(t	b1e(t	PROPN
eajbcs-694	591	15	)	)	PUNCT
eajbcs-694	592	1	+	+	ADJ
eajbcs-694	592	2	b2i(t	b2i(t	NOUN
eajbcs-694	592	3	)	)	PUNCT
eajbcs-694	593	1	+	+	CCONJ
eajbcs-694	593	2	1	1	NUM
eajbcs-694	593	3	2	2	NUM
eajbcs-694	593	4	(	(	PUNCT
eajbcs-694	593	5	a1θ	a1θ	PROPN
eajbcs-694	593	6	2(t	2(t	NUM
eajbcs-694	593	7	)	)	PUNCT
eajbcs-694	594	1	+	+	CCONJ
eajbcs-694	594	2	a2α	a2α	SYM
eajbcs-694	594	3	2(t	2(t	NUM
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eajbcs-694	594	5	)	)	PUNCT
eajbcs-694	595	1	≥	≥	NOUN
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eajbcs-694	595	3	2	2	NUM
eajbcs-694	595	4	(	(	PUNCT
eajbcs-694	595	5	a1θ	a1θ	PROPN
eajbcs-694	595	6	2(t	2(t	NUM
eajbcs-694	595	7	)	)	PUNCT
eajbcs-694	596	1	+	+	CCONJ
eajbcs-694	596	2	a2α	a2α	SYM
eajbcs-694	596	3	2(t	2(t	NUM
eajbcs-694	596	4	)	)	PUNCT
eajbcs-694	596	5	)	)	PUNCT
eajbcs-694	596	6	.	.	PUNCT
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eajbcs-694	597	2	define	define	VERB
eajbcs-694	597	3	a	a	DET
eajbcs-694	597	4	continuous	continuous	ADJ
eajbcs-694	597	5	function	function	NOUN
eajbcs-694	597	6	h(u	h(u	PROPN
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eajbcs-694	597	8	=	=	VERB
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eajbcs-694	598	4	ϕ	ϕ	NOUN
eajbcs-694	598	5	=	=	SYM
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eajbcs-694	598	11	a2	a2	PROPN
eajbcs-694	598	12	2	2	NUM
eajbcs-694	598	13	)	)	PUNCT
eajbcs-694	598	14	>	>	X
eajbcs-694	598	15	0	0	PUNCT
eajbcs-694	598	16	and	and	CCONJ
eajbcs-694	598	17	u	u	NOUN
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eajbcs-694	598	19	(	(	PUNCT
eajbcs-694	598	20	θ(t	θ(t	PROPN
eajbcs-694	598	21	)	)	PUNCT
eajbcs-694	598	22	,	,	PUNCT
eajbcs-694	598	23	α(t	α(t	PROPN
eajbcs-694	598	24	)	)	PUNCT
eajbcs-694	598	25	)	)	PUNCT
eajbcs-694	598	26	.	.	PUNCT
eajbcs-694	599	1	then	then	ADV
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eajbcs-694	599	4	l(e	l(e	NOUN
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eajbcs-694	599	9	,	,	PUNCT
eajbcs-694	599	10	α	α	PROPN
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eajbcs-694	599	12	t	t	PROPN
eajbcs-694	599	13	)	)	PUNCT
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eajbcs-694	599	15	1	1	NUM
eajbcs-694	599	16	2	2	NUM
eajbcs-694	599	17	(	(	PUNCT
eajbcs-694	599	18	a1θ	a1θ	PROPN
eajbcs-694	599	19	2(t	2(t	NUM
eajbcs-694	599	20	)	)	PUNCT
eajbcs-694	600	1	+	+	ADV
eajbcs-694	600	2	a2α	a2α	X
eajbcs-694	600	3	2(t	2(t	NUM
eajbcs-694	600	4	)	)	PUNCT
eajbcs-694	600	5	)	)	PUNCT
eajbcs-694	601	1	≥	≥	NOUN
eajbcs-694	601	2	ϕ||u||2	ϕ||u||2	ADV
eajbcs-694	601	3	,	,	PUNCT
eajbcs-694	601	4	(	(	PUNCT
eajbcs-694	601	5	15	15	NUM
eajbcs-694	601	6	)	)	PUNCT
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eajbcs-694	601	8	ϕ	ϕ	NOUN
eajbcs-694	601	9	=	=	SYM
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eajbcs-694	601	11	(	(	PUNCT
eajbcs-694	601	12	a1	a1	PROPN
eajbcs-694	601	13	2	2	NUM
eajbcs-694	601	14	,	,	PUNCT
eajbcs-694	601	15	a2	a2	PROPN
eajbcs-694	601	16	2	2	NUM
eajbcs-694	601	17	)	)	PUNCT
eajbcs-694	601	18	>	>	X
eajbcs-694	601	19	0	0	X
eajbcs-694	601	20	.	.	PUNCT
eajbcs-694	602	1	this	this	PRON
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eajbcs-694	602	5	,	,	PUNCT
eajbcs-694	602	6	i	i	PROPN
eajbcs-694	602	7	,	,	PUNCT
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eajbcs-694	602	9	,	,	PUNCT
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eajbcs-694	602	12	t	t	PROPN
eajbcs-694	602	13	)	)	PUNCT
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eajbcs-694	602	15	h(u	h(u	PROPN
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eajbcs-694	603	1	consider	consider	VERB
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eajbcs-694	603	3	||u||−1h(u	||u||−1h(u	NOUN
eajbcs-694	603	4	)	)	PUNCT
eajbcs-694	603	5	=	=	SYM
eajbcs-694	604	1	||u||−1ϕ||u||2	||u||−1ϕ||u||2	PROPN
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eajbcs-694	605	1	ϕ||u||	ϕ||u||	PROPN
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eajbcs-694	606	1	this	this	PRON
eajbcs-694	606	2	gives	give	VERB
eajbcs-694	606	3	that	that	PRON
eajbcs-694	606	4	||u||−1h(u	||u||−1h(u	NOUN
eajbcs-694	606	5	)	)	PUNCT
eajbcs-694	606	6	=	=	SYM
eajbcs-694	607	1	ϕ||u||	ϕ||u||	NOUN
eajbcs-694	607	2	→	→	SYM
eajbcs-694	607	3	∞	∞	NUM
eajbcs-694	607	4	when	when	SCONJ
eajbcs-694	607	5	||u||	||u||	ADJ
eajbcs-694	607	6	→	→	SYM
eajbcs-694	607	7	∞.	∞.	PROPN
eajbcs-694	607	8	thus	thus	ADV
eajbcs-694	607	9	,	,	PUNCT
eajbcs-694	607	10	condition	condition	NOUN
eajbcs-694	607	11	(	(	PUNCT
eajbcs-694	607	12	d	d	X
eajbcs-694	607	13	)	)	PUNCT
eajbcs-694	607	14	is	be	AUX
eajbcs-694	607	15	proved	prove	VERB
eajbcs-694	607	16	.	.	PUNCT
eajbcs-694	608	1	hence	hence	ADV
eajbcs-694	608	2	,	,	PUNCT
eajbcs-694	608	3	all	all	DET
eajbcs-694	608	4	conditions	condition	NOUN
eajbcs-694	608	5	(	(	PUNCT
eajbcs-694	608	6	a)-(d	a)-(d	NOUN
eajbcs-694	608	7	)	)	PUNCT
eajbcs-694	608	8	shows	show	VERB
eajbcs-694	608	9	that	that	SCONJ
eajbcs-694	608	10	there	there	PRON
eajbcs-694	608	11	exists	exist	VERB
eajbcs-694	608	12	an	an	DET
eajbcs-694	608	13	optimal	optimal	ADJ
eajbcs-694	608	14	control	control	NOUN
eajbcs-694	608	15	u∗	u∗	NOUN
eajbcs-694	608	16	=	=	PUNCT
eajbcs-694	608	17	(	(	PUNCT
eajbcs-694	608	18	θ∗	θ∗	NOUN
eajbcs-694	608	19	,	,	PUNCT
eajbcs-694	608	20	α∗	α∗	NOUN
eajbcs-694	608	21	)	)	PUNCT
eajbcs-694	608	22	that	that	PRON
eajbcs-694	608	23	minimizes	minimize	VERB
eajbcs-694	608	24	the	the	DET
eajbcs-694	608	25	cost	cost	NOUN
eajbcs-694	608	26	functional	functional	ADJ
eajbcs-694	608	27	j(θ(t	j(θ(t	NOUN
eajbcs-694	608	28	)	)	PUNCT
eajbcs-694	608	29	,	,	PUNCT
eajbcs-694	608	30	α(t	α(t	PROPN
eajbcs-694	608	31	)	)	PUNCT
eajbcs-694	608	32	)	)	PUNCT
eajbcs-694	608	33	over	over	ADP
eajbcs-694	608	34	u	u	PROPN
eajbcs-694	608	35	.	.	PUNCT
eajbcs-694	609	1	therefore	therefore	ADV
eajbcs-694	609	2	,	,	PUNCT
eajbcs-694	609	3	the	the	DET
eajbcs-694	609	4	existence	existence	NOUN
eajbcs-694	609	5	of	of	ADP
eajbcs-694	609	6	optimal	optimal	ADJ
eajbcs-694	609	7	control	control	NOUN
eajbcs-694	609	8	is	be	AUX
eajbcs-694	609	9	established	establish	VERB
eajbcs-694	609	10	.	.	PUNCT
eajbcs-694	610	1	5.3	5.3	NUM
eajbcs-694	610	2	.	.	PUNCT
eajbcs-694	611	1	the	the	DET
eajbcs-694	611	2	hamiltonian	hamiltonian	ADJ
eajbcs-694	611	3	and	and	CCONJ
eajbcs-694	611	4	optimality	optimality	NOUN
eajbcs-694	611	5	system	system	NOUN
eajbcs-694	611	6	we	we	PRON
eajbcs-694	611	7	used	use	VERB
eajbcs-694	611	8	pontryangin	pontryangin	NOUN
eajbcs-694	611	9	’s	’s	PART
eajbcs-694	611	10	maximum	maximum	ADJ
eajbcs-694	611	11	principle	principle	ADJ
eajbcs-694	611	12	lenhart	lenhart	NOUN
eajbcs-694	611	13	and	and	CCONJ
eajbcs-694	611	14	workman	workman	NOUN
eajbcs-694	611	15	(	(	PUNCT
eajbcs-694	611	16	2007	2007	NUM
eajbcs-694	611	17	)	)	PUNCT
eajbcs-694	611	18	to	to	PART
eajbcs-694	611	19	drive	drive	VERB
eajbcs-694	611	20	the	the	DET
eajbcs-694	611	21	necessary	necessary	ADJ
eajbcs-694	611	22	conditions	condition	NOUN
eajbcs-694	611	23	that	that	SCONJ
eajbcs-694	611	24	an	an	DET
eajbcs-694	611	25	optimal	optimal	ADJ
eajbcs-694	611	26	control	control	NOUN
eajbcs-694	611	27	must	must	AUX
eajbcs-694	611	28	satisfy	satisfy	VERB
eajbcs-694	611	29	.	.	PUNCT
eajbcs-694	612	1	this	this	DET
eajbcs-694	612	2	principle	principle	NOUN
eajbcs-694	612	3	converts	convert	VERB
eajbcs-694	612	4	the	the	DET
eajbcs-694	612	5	objective	objective	ADJ
eajbcs-694	612	6	functional	functional	ADJ
eajbcs-694	612	7	(	(	PUNCT
eajbcs-694	612	8	10	10	NUM
eajbcs-694	612	9	)	)	PUNCT
eajbcs-694	612	10	subject	subject	NOUN
eajbcs-694	612	11	to	to	ADP
eajbcs-694	612	12	the	the	DET
eajbcs-694	612	13	state	state	NOUN
eajbcs-694	612	14	system	system	NOUN
eajbcs-694	612	15	(	(	PUNCT
eajbcs-694	612	16	2	2	NUM
eajbcs-694	612	17	)	)	PUNCT
eajbcs-694	612	18	into	into	ADP
eajbcs-694	612	19	a	a	DET
eajbcs-694	612	20	problem	problem	NOUN
eajbcs-694	612	21	of	of	ADP
eajbcs-694	612	22	minimizing	minimize	VERB
eajbcs-694	612	23	point	point	NOUN
eajbcs-694	612	24	-	-	PUNCT
eajbcs-694	612	25	wise	wise	ADJ
eajbcs-694	612	26	a	a	DET
eajbcs-694	612	27	hamiltonian	hamiltonian	NOUN
eajbcs-694	612	28	(	(	PUNCT
eajbcs-694	612	29	h	h	NOUN
eajbcs-694	612	30	)	)	PUNCT
eajbcs-694	612	31	,	,	PUNCT
eajbcs-694	612	32	with	with	ADP
eajbcs-694	612	33	respect	respect	NOUN
eajbcs-694	612	34	to	to	ADP
eajbcs-694	612	35	θ(t	θ(t	NOUN
eajbcs-694	612	36	)	)	PUNCT
eajbcs-694	612	37	and	and	CCONJ
eajbcs-694	612	38	α(t	α(t	NOUN
eajbcs-694	612	39	)	)	PUNCT
eajbcs-694	612	40	as	as	ADP
eajbcs-694	612	41	:	:	PUNCT
eajbcs-694	612	42	east	east	PROPN
eajbcs-694	612	43	afr	afr	PROPN
eajbcs-694	612	44	.	.	PUNCT
eajbcs-694	613	1	j.	j.	PROPN
eajbcs-694	613	2	biophys	biophys	PROPN
eajbcs-694	613	3	.	.	PUNCT
eajbcs-694	614	1	comput	comput	NOUN
eajbcs-694	614	2	.	.	PUNCT
eajbcs-694	615	1	sci	sci	PROPN
eajbcs-694	615	2	.	.	PUNCT
eajbcs-694	616	1	(	(	PUNCT
eajbcs-694	616	2	2024	2024	NUM
eajbcs-694	616	3	)	)	PUNCT
eajbcs-694	616	4	,	,	PUNCT
eajbcs-694	616	5	vol	vol	NOUN
eajbcs-694	616	6	.	.	PROPN
eajbcs-694	616	7	5	5	NUM
eajbcs-694	616	8	,	,	PUNCT
eajbcs-694	616	9	no	no	INTJ
eajbcs-694	616	10	.	.	NOUN
eajbcs-694	616	11	2	2	NUM
eajbcs-694	616	12	,	,	PUNCT
eajbcs-694	616	13	58	58	NUM
eajbcs-694	616	14	-	-	SYM
eajbcs-694	616	15	86	86	NUM
eajbcs-694	616	16	79	79	NUM
eajbcs-694	616	17	h	h	NOUN
eajbcs-694	616	18	=	=	PUNCT
eajbcs-694	616	19	b1e	b1e	X
eajbcs-694	617	1	+	+	ADJ
eajbcs-694	617	2	b2i	b2i	ADJ
eajbcs-694	617	3	+	+	NOUN
eajbcs-694	617	4	1	1	NUM
eajbcs-694	617	5	2	2	NUM
eajbcs-694	617	6	a1θ	a1θ	ADP
eajbcs-694	617	7	2	2	NUM
eajbcs-694	617	8	+	+	CCONJ
eajbcs-694	617	9	1	1	NUM
eajbcs-694	617	10	2	2	NUM
eajbcs-694	617	11	a2α	a2α	SYM
eajbcs-694	617	12	2	2	NUM
eajbcs-694	617	13	+	+	CCONJ
eajbcs-694	617	14	λ1[π	λ1[π	NOUN
eajbcs-694	617	15	+	+	CCONJ
eajbcs-694	617	16	ηωr	ηωr	NOUN
eajbcs-694	617	17	+	+	CCONJ
eajbcs-694	617	18	νp	νp	ADP
eajbcs-694	617	19	−	−	PROPN
eajbcs-694	617	20	β(σ2e	β(σ2e	PUNCT
eajbcs-694	617	21	+	+	CCONJ
eajbcs-694	617	22	σ1i	σ1i	NOUN
eajbcs-694	618	1	+	+	ADJ
eajbcs-694	618	2	σ3h)s	σ3h)s	X
eajbcs-694	618	3	−	−	PROPN
eajbcs-694	618	4	(	(	PUNCT
eajbcs-694	618	5	θ	θ	NOUN
eajbcs-694	618	6	+	+	NOUN
eajbcs-694	618	7	µ)s	µ)s	NOUN
eajbcs-694	618	8	]	]	PUNCT
eajbcs-694	619	1	+	+	CCONJ
eajbcs-694	619	2	λ2[θs	λ2[θs	ADJ
eajbcs-694	619	3	+	+	CCONJ
eajbcs-694	619	4	(	(	PUNCT
eajbcs-694	619	5	1−	1−	NUM
eajbcs-694	619	6	η)ωr−	η)ωr−	PROPN
eajbcs-694	619	7	(	(	PUNCT
eajbcs-694	619	8	ν	ν	X
eajbcs-694	619	9	+	+	X
eajbcs-694	619	10	µ)p	µ)p	NOUN
eajbcs-694	619	11	]	]	PUNCT
eajbcs-694	620	1	+	+	CCONJ
eajbcs-694	620	2	λ3[β(σ2e	λ3[β(σ2e	X
eajbcs-694	620	3	+	+	CCONJ
eajbcs-694	620	4	σ1i	σ1i	NOUN
eajbcs-694	620	5	+	+	NUM
eajbcs-694	620	6	σ3h)s	σ3h)s	X
eajbcs-694	620	7	−	−	PROPN
eajbcs-694	620	8	(	(	PUNCT
eajbcs-694	620	9	δ	δ	NOUN
eajbcs-694	620	10	+	+	NOUN
eajbcs-694	620	11	µ)e	µ)e	NOUN
eajbcs-694	620	12	]	]	PUNCT
eajbcs-694	621	1	+	+	CCONJ
eajbcs-694	621	2	λ4[τδe	λ4[τδe	ADP
eajbcs-694	621	3	−	−	PROPN
eajbcs-694	621	4	(	(	PUNCT
eajbcs-694	621	5	α	α	NOUN
eajbcs-694	621	6	+	+	CCONJ
eajbcs-694	621	7	ε+	ε+	X
eajbcs-694	621	8	µ+	µ+	X
eajbcs-694	621	9	ρ)i	ρ)i	NOUN
eajbcs-694	621	10	]	]	PUNCT
eajbcs-694	622	1	+	+	CCONJ
eajbcs-694	622	2	λ5[αi	λ5[αi	NOUN
eajbcs-694	622	3	−	−	PROPN
eajbcs-694	622	4	(	(	PUNCT
eajbcs-694	622	5	γ	γ	X
eajbcs-694	622	6	+	+	X
eajbcs-694	622	7	µ+	µ+	X
eajbcs-694	622	8	ξ)h	ξ)h	X
eajbcs-694	622	9	]	]	X
eajbcs-694	622	10	+	+	CCONJ
eajbcs-694	622	11	λ6[(1−	λ6[(1−	PROPN
eajbcs-694	622	12	τ)δe	τ)δe	PROPN
eajbcs-694	622	13	+	+	CCONJ
eajbcs-694	622	14	εi	εi	PROPN
eajbcs-694	622	15	+	+	X
eajbcs-694	622	16	γh	γh	ADP
eajbcs-694	622	17	−	−	PROPN
eajbcs-694	622	18	(	(	PUNCT
eajbcs-694	622	19	ω	ω	NOUN
eajbcs-694	622	20	+	+	CCONJ
eajbcs-694	622	21	µ)r	µ)r	NOUN
eajbcs-694	622	22	]	]	X
eajbcs-694	622	23	,	,	PUNCT
eajbcs-694	622	24	(	(	PUNCT
eajbcs-694	622	25	16	16	NUM
eajbcs-694	622	26	)	)	PUNCT
eajbcs-694	622	27	where	where	SCONJ
eajbcs-694	622	28	λi	λi	X
eajbcs-694	622	29	,	,	PUNCT
eajbcs-694	622	30	i	i	NOUN
eajbcs-694	622	31	=	=	NOUN
eajbcs-694	622	32	1	1	NUM
eajbcs-694	622	33	,	,	PUNCT
eajbcs-694	622	34	2	2	NUM
eajbcs-694	622	35	,	,	PUNCT
eajbcs-694	622	36	3	3	NUM
eajbcs-694	622	37	,	,	PUNCT
eajbcs-694	622	38	4	4	NUM
eajbcs-694	622	39	,	,	PUNCT
eajbcs-694	622	40	5	5	NUM
eajbcs-694	622	41	,	,	PUNCT
eajbcs-694	622	42	6	6	NUM
eajbcs-694	622	43	,	,	PUNCT
eajbcs-694	622	44	represent	represent	VERB
eajbcs-694	622	45	the	the	DET
eajbcs-694	622	46	adjoint	adjoint	NOUN
eajbcs-694	622	47	variables	variable	NOUN
eajbcs-694	622	48	associated	associate	VERB
eajbcs-694	622	49	with	with	ADP
eajbcs-694	622	50	the	the	DET
eajbcs-694	622	51	state	state	NOUN
eajbcs-694	622	52	variables	variable	NOUN
eajbcs-694	622	53	s	s	PROPN
eajbcs-694	622	54	,	,	PUNCT
eajbcs-694	622	55	p	p	X
eajbcs-694	622	56	,	,	PUNCT
eajbcs-694	622	57	e	e	NOUN
eajbcs-694	622	58	,	,	PUNCT
eajbcs-694	622	59	i	i	PRON
eajbcs-694	622	60	,	,	PUNCT
eajbcs-694	622	61	h	h	NOUN
eajbcs-694	622	62	and	and	CCONJ
eajbcs-694	622	63	r	r	NOUN
eajbcs-694	622	64	to	to	PART
eajbcs-694	622	65	be	be	AUX
eajbcs-694	622	66	determined	determine	VERB
eajbcs-694	622	67	suitably	suitably	ADV
eajbcs-694	622	68	by	by	ADP
eajbcs-694	622	69	applying	apply	VERB
eajbcs-694	622	70	pontryagin	pontryagin	NOUN
eajbcs-694	622	71	’s	’s	PART
eajbcs-694	622	72	maximal	maximal	ADJ
eajbcs-694	622	73	principle	principle	ADJ
eajbcs-694	622	74	lenhart	lenhart	NOUN
eajbcs-694	622	75	and	and	CCONJ
eajbcs-694	622	76	workman	workman	NOUN
eajbcs-694	622	77	(	(	PUNCT
eajbcs-694	622	78	2007	2007	NUM
eajbcs-694	622	79	)	)	PUNCT
eajbcs-694	622	80	.	.	PUNCT
eajbcs-694	623	1	theorem	theorem	VERB
eajbcs-694	623	2	5.2	5.2	NUM
eajbcs-694	623	3	.	.	PUNCT
eajbcs-694	624	1	for	for	ADP
eajbcs-694	624	2	an	an	DET
eajbcs-694	624	3	optimal	optimal	ADJ
eajbcs-694	624	4	control	control	NOUN
eajbcs-694	624	5	set	set	VERB
eajbcs-694	624	6	θ	θ	PROPN
eajbcs-694	624	7	,	,	PUNCT
eajbcs-694	624	8	α	α	NOUN
eajbcs-694	624	9	that	that	PRON
eajbcs-694	624	10	minimizes	minimize	VERB
eajbcs-694	624	11	j	j	PROPN
eajbcs-694	624	12	over	over	ADP
eajbcs-694	624	13	u	u	PROPN
eajbcs-694	624	14	,	,	PUNCT
eajbcs-694	624	15	there	there	PRON
eajbcs-694	624	16	are	be	VERB
eajbcs-694	624	17	adjoint	adjoint	NOUN
eajbcs-694	624	18	variables	variable	NOUN
eajbcs-694	624	19	,	,	PUNCT
eajbcs-694	624	20	λ1	λ1	PROPN
eajbcs-694	624	21	,	,	PUNCT
eajbcs-694	624	22	...	...	PUNCT
eajbcs-694	624	23	,	,	PUNCT
eajbcs-694	624	24	λ6	λ6	NOUN
eajbcs-694	624	25	such	such	ADJ
eajbcs-694	624	26	that	that	PRON
eajbcs-694	624	27	:	:	PUNCT
eajbcs-694	625	1	dλ1	dλ1	NOUN
eajbcs-694	625	2	dt	dt	X
eajbcs-694	626	1	=	=	PUNCT
eajbcs-694	627	1	[	[	X
eajbcs-694	627	2	β(σ2e	β(σ2e	PUNCT
eajbcs-694	627	3	+	+	CCONJ
eajbcs-694	627	4	σ1i	σ1i	NOUN
eajbcs-694	627	5	+	+	CCONJ
eajbcs-694	627	6	σ3h	σ3h	NOUN
eajbcs-694	627	7	)	)	PUNCT
eajbcs-694	627	8	+	+	NUM
eajbcs-694	627	9	θ	θ	X
eajbcs-694	628	1	+	+	PUNCT
eajbcs-694	628	2	µ]λ1	µ]λ1	NOUN
eajbcs-694	628	3	−	−	ADP
eajbcs-694	628	4	θλ2	θλ2	NOUN
eajbcs-694	628	5	−	−	NOUN
eajbcs-694	628	6	β(σ2e	β(σ2e	PUNCT
eajbcs-694	628	7	+	+	CCONJ
eajbcs-694	628	8	σ1i	σ1i	NOUN
eajbcs-694	628	9	+	+	CCONJ
eajbcs-694	628	10	σ3h)λ3	σ3h)λ3	PROPN
eajbcs-694	628	11	,	,	PUNCT
eajbcs-694	628	12	dλ2	dλ2	NOUN
eajbcs-694	628	13	dt	dt	NOUN
eajbcs-694	628	14	=	=	PUNCT
eajbcs-694	629	1	−νλ1	−νλ1	X
eajbcs-694	629	2	+	+	X
eajbcs-694	629	3	(	(	PUNCT
eajbcs-694	629	4	ν	ν	X
eajbcs-694	629	5	+	+	X
eajbcs-694	629	6	µ)λ2	µ)λ2	PROPN
eajbcs-694	629	7	,	,	PUNCT
eajbcs-694	629	8	dλ3	dλ3	NOUN
eajbcs-694	629	9	dt	dt	NOUN
eajbcs-694	629	10	=	=	SYM
eajbcs-694	629	11	−b1	−b1	PROPN
eajbcs-694	629	12	+	+	CCONJ
eajbcs-694	629	13	βσ2sλ1	βσ2sλ1	NOUN
eajbcs-694	629	14	−	−	X
eajbcs-694	630	1	[	[	X
eajbcs-694	630	2	βσ2s	βσ2s	X
eajbcs-694	630	3	−	−	PROPN
eajbcs-694	630	4	(	(	PUNCT
eajbcs-694	630	5	δ	δ	PROPN
eajbcs-694	630	6	+	+	CCONJ
eajbcs-694	630	7	µ)]λ3	µ)]λ3	VERB
eajbcs-694	630	8	−	−	X
eajbcs-694	630	9	τδλ4	τδλ4	PROPN
eajbcs-694	630	10	−	−	PROPN
eajbcs-694	631	1	(	(	PUNCT
eajbcs-694	632	1	1−	1−	NUM
eajbcs-694	632	2	τ)δλ6	τ)δλ6	NOUN
eajbcs-694	632	3	,	,	PUNCT
eajbcs-694	632	4	dλ4	dλ4	VERB
eajbcs-694	632	5	dt	dt	NOUN
eajbcs-694	633	1	=	=	PUNCT
eajbcs-694	633	2	−b2	−b2	PROPN
eajbcs-694	633	3	+	+	CCONJ
eajbcs-694	633	4	βσ1sλ1	βσ1sλ1	X
eajbcs-694	633	5	−	−	NOUN
eajbcs-694	633	6	βσ1sλ3	βσ1sλ3	PUNCT
eajbcs-694	634	1	+	+	PUNCT
eajbcs-694	634	2	(	(	PUNCT
eajbcs-694	634	3	α	α	NOUN
eajbcs-694	634	4	+	+	X
eajbcs-694	634	5	ε+	ε+	X
eajbcs-694	634	6	µ+	µ+	VERB
eajbcs-694	634	7	ρ)λ4	ρ)λ4	PROPN
eajbcs-694	634	8	−	−	NOUN
eajbcs-694	634	9	αλ5	αλ5	NOUN
eajbcs-694	634	10	−	−	NOUN
eajbcs-694	634	11	ελ6	ελ6	NOUN
eajbcs-694	634	12	,	,	PUNCT
eajbcs-694	634	13	dλ5	dλ5	INTJ
eajbcs-694	634	14	dt	dt	NOUN
eajbcs-694	634	15	=	=	SYM
eajbcs-694	634	16	βσ3sλ1	βσ3sλ1	NOUN
eajbcs-694	634	17	−	−	NOUN
eajbcs-694	635	1	βσ3sλ3	βσ3sλ3	VERB
eajbcs-694	636	1	+	+	PUNCT
eajbcs-694	636	2	(	(	PUNCT
eajbcs-694	636	3	γ	γ	X
eajbcs-694	636	4	+	+	X
eajbcs-694	636	5	µ+	µ+	PROPN
eajbcs-694	636	6	ξ)λ5	ξ)λ5	PROPN
eajbcs-694	636	7	−	−	PROPN
eajbcs-694	636	8	γλ6	γλ6	NOUN
eajbcs-694	636	9	,	,	PUNCT
eajbcs-694	636	10	dλ6	dλ6	VERB
eajbcs-694	636	11	dt	dt	NOUN
eajbcs-694	636	12	=	=	PUNCT
eajbcs-694	636	13	−ηωλ1	−ηωλ1	NOUN
eajbcs-694	637	1	−	−	NOUN
eajbcs-694	637	2	(	(	PUNCT
eajbcs-694	637	3	1−	1−	NUM
eajbcs-694	637	4	η)ωλ2	η)ωλ2	PROPN
eajbcs-694	637	5	+	+	CCONJ
eajbcs-694	637	6	(	(	PUNCT
eajbcs-694	637	7	ω	ω	NOUN
eajbcs-694	637	8	+	+	X
eajbcs-694	637	9	µ)λ6	µ)λ6	NOUN
eajbcs-694	637	10	.	.	PUNCT
eajbcs-694	638	1	(	(	PUNCT
eajbcs-694	638	2	17	17	NUM
eajbcs-694	638	3	)	)	PUNCT
eajbcs-694	638	4	with	with	ADP
eajbcs-694	638	5	transiversality	transiversality	NOUN
eajbcs-694	638	6	conditions	condition	NOUN
eajbcs-694	638	7	λi(t	λi(t	PRON
eajbcs-694	638	8	)	)	PUNCT
eajbcs-694	639	1	=	=	SYM
eajbcs-694	639	2	0	0	NUM
eajbcs-694	639	3	,	,	PUNCT
eajbcs-694	639	4	i	i	PRON
eajbcs-694	639	5	=	=	NOUN
eajbcs-694	639	6	1	1	NUM
eajbcs-694	639	7	,	,	PUNCT
eajbcs-694	639	8	...	...	PUNCT
eajbcs-694	639	9	,	,	PUNCT
eajbcs-694	639	10	6	6	X
eajbcs-694	639	11	.	.	PUNCT
eajbcs-694	639	12	(	(	PUNCT
eajbcs-694	639	13	18	18	NUM
eajbcs-694	639	14	)	)	PUNCT
eajbcs-694	639	15	moreover	moreover	ADV
eajbcs-694	639	16	,	,	PUNCT
eajbcs-694	639	17	we	we	PRON
eajbcs-694	639	18	obtain	obtain	VERB
eajbcs-694	639	19	the	the	DET
eajbcs-694	639	20	control	control	NOUN
eajbcs-694	639	21	set	set	NOUN
eajbcs-694	639	22	(	(	PUNCT
eajbcs-694	639	23	θ∗	θ∗	NOUN
eajbcs-694	639	24	,	,	PUNCT
eajbcs-694	639	25	α∗	α∗	NOUN
eajbcs-694	639	26	)	)	PUNCT
eajbcs-694	639	27	characterized	characterize	VERB
eajbcs-694	639	28	by	by	ADP
eajbcs-694	639	29	θ∗(t	θ∗(t	NOUN
eajbcs-694	639	30	)	)	PUNCT
eajbcs-694	639	31	=	=	SYM
eajbcs-694	639	32	max	max	PROPN
eajbcs-694	639	33	{	{	PUNCT
eajbcs-694	639	34	0,min	0,min	PROPN
eajbcs-694	639	35	(	(	PUNCT
eajbcs-694	639	36	ϵ1	ϵ1	ADJ
eajbcs-694	639	37	,	,	PUNCT
eajbcs-694	639	38	s(λ1	s(λ1	NOUN
eajbcs-694	639	39	−	−	PROPN
eajbcs-694	639	40	λ2	λ2	SYM
eajbcs-694	639	41	)	)	PUNCT
eajbcs-694	639	42	a1	a1	NOUN
eajbcs-694	639	43	)	)	PUNCT
eajbcs-694	639	44	}	}	PUNCT
eajbcs-694	639	45	,	,	PUNCT
eajbcs-694	639	46	α∗(t	α∗(t	NOUN
eajbcs-694	639	47	)	)	PUNCT
eajbcs-694	639	48	=	=	SYM
eajbcs-694	639	49	max	max	PROPN
eajbcs-694	639	50	{	{	PUNCT
eajbcs-694	639	51	0,min	0,min	PROPN
eajbcs-694	639	52	(	(	PUNCT
eajbcs-694	639	53	ϵ2	ϵ2	ADJ
eajbcs-694	639	54	,	,	PUNCT
eajbcs-694	639	55	i(λ4	i(λ4	ADJ
eajbcs-694	639	56	−	−	NOUN
eajbcs-694	639	57	λ5	λ5	NOUN
eajbcs-694	639	58	)	)	PUNCT
eajbcs-694	639	59	a2	a2	PROPN
eajbcs-694	639	60	)	)	PUNCT
eajbcs-694	639	61	}	}	PUNCT
eajbcs-694	639	62	.	.	PUNCT
eajbcs-694	640	1	(	(	PUNCT
eajbcs-694	640	2	19	19	NUM
eajbcs-694	640	3	)	)	PUNCT
eajbcs-694	640	4	proof	proof	NOUN
eajbcs-694	640	5	.	.	PUNCT
eajbcs-694	641	1	the	the	DET
eajbcs-694	641	2	form	form	NOUN
eajbcs-694	641	3	of	of	ADP
eajbcs-694	641	4	the	the	DET
eajbcs-694	641	5	adjoint	adjoint	PROPN
eajbcs-694	641	6	equations	equation	NOUN
eajbcs-694	641	7	and	and	CCONJ
eajbcs-694	641	8	transversality	transversality	NOUN
eajbcs-694	641	9	conditions	condition	NOUN
eajbcs-694	641	10	are	be	AUX
eajbcs-694	641	11	standard	standard	ADJ
eajbcs-694	641	12	results	result	NOUN
eajbcs-694	641	13	from	from	ADP
eajbcs-694	641	14	pontryagin	pontryagin	NOUN
eajbcs-694	641	15	’s	’s	PART
eajbcs-694	641	16	maximum	maximum	ADJ
eajbcs-694	641	17	principle	principle	ADJ
eajbcs-694	641	18	lenhart	lenhart	NOUN
eajbcs-694	641	19	and	and	CCONJ
eajbcs-694	641	20	workman	workman	NOUN
eajbcs-694	641	21	(	(	PUNCT
eajbcs-694	641	22	2007	2007	NUM
eajbcs-694	641	23	)	)	PUNCT
eajbcs-694	641	24	.	.	PUNCT
eajbcs-694	642	1	we	we	PRON
eajbcs-694	642	2	differentiate	differentiate	VERB
eajbcs-694	642	3	hamiltonian	hamiltonian	NOUN
eajbcs-694	642	4	(	(	PUNCT
eajbcs-694	642	5	h	h	NOUN
eajbcs-694	642	6	)	)	PUNCT
eajbcs-694	642	7	(	(	PUNCT
eajbcs-694	642	8	16	16	NUM
eajbcs-694	642	9	)	)	PUNCT
eajbcs-694	642	10	with	with	ADP
eajbcs-694	642	11	respect	respect	NOUN
eajbcs-694	642	12	to	to	ADP
eajbcs-694	642	13	the	the	DET
eajbcs-694	642	14	state	state	NOUN
eajbcs-694	642	15	variables	variable	NOUN
eajbcs-694	642	16	s	s	PROPN
eajbcs-694	642	17	,	,	PUNCT
eajbcs-694	642	18	p	p	X
eajbcs-694	642	19	,	,	PUNCT
eajbcs-694	642	20	e	e	NOUN
eajbcs-694	642	21	,	,	PUNCT
eajbcs-694	642	22	i	i	PRON
eajbcs-694	642	23	,	,	PUNCT
eajbcs-694	642	24	h	h	NOUN
eajbcs-694	642	25	and	and	CCONJ
eajbcs-694	642	26	r	r	NOUN
eajbcs-694	642	27	,	,	PUNCT
eajbcs-694	642	28	respectively	respectively	ADV
eajbcs-694	642	29	,	,	PUNCT
eajbcs-694	642	30	and	and	CCONJ
eajbcs-694	642	31	then	then	ADV
eajbcs-694	642	32	the	the	DET
eajbcs-694	642	33	adjoint	adjoint	PROPN
eajbcs-694	642	34	east	east	PROPN
eajbcs-694	642	35	afr	afr	PROPN
eajbcs-694	642	36	.	.	PUNCT
eajbcs-694	643	1	j.	j.	PROPN
eajbcs-694	643	2	biophys	biophys	PROPN
eajbcs-694	643	3	.	.	PUNCT
eajbcs-694	644	1	comput	comput	NOUN
eajbcs-694	644	2	.	.	PUNCT
eajbcs-694	645	1	sci	sci	PROPN
eajbcs-694	645	2	.	.	PUNCT
eajbcs-694	646	1	(	(	PUNCT
eajbcs-694	646	2	2024	2024	NUM
eajbcs-694	646	3	)	)	PUNCT
eajbcs-694	646	4	,	,	PUNCT
eajbcs-694	646	5	vol	vol	NOUN
eajbcs-694	646	6	.	.	PROPN
eajbcs-694	646	7	5	5	NUM
eajbcs-694	646	8	,	,	PUNCT
eajbcs-694	646	9	no	no	INTJ
eajbcs-694	646	10	.	.	NOUN
eajbcs-694	646	11	2	2	NUM
eajbcs-694	646	12	,	,	PUNCT
eajbcs-694	646	13	58	58	NUM
eajbcs-694	646	14	-	-	SYM
eajbcs-694	646	15	86	86	NUM
eajbcs-694	646	16	80	80	NUM
eajbcs-694	646	17	system	system	NOUN
eajbcs-694	646	18	can	can	AUX
eajbcs-694	646	19	be	be	AUX
eajbcs-694	646	20	written	write	VERB
eajbcs-694	646	21	as	as	ADP
eajbcs-694	646	22	dλ1	dλ1	PROPN
eajbcs-694	646	23	dt	dt	PUNCT
eajbcs-694	646	24	=	=	PUNCT
eajbcs-694	647	1	−∂h	−∂h	SYM
eajbcs-694	647	2	∂s	∂s	PROPN
eajbcs-694	647	3	=	=	PUNCT
eajbcs-694	648	1	[	[	X
eajbcs-694	648	2	β(σ2e	β(σ2e	PUNCT
eajbcs-694	648	3	+	+	CCONJ
eajbcs-694	648	4	σ1i	σ1i	NOUN
eajbcs-694	648	5	+	+	CCONJ
eajbcs-694	648	6	σ3h	σ3h	NOUN
eajbcs-694	648	7	)	)	PUNCT
eajbcs-694	648	8	+	+	NUM
eajbcs-694	648	9	θ	θ	X
eajbcs-694	649	1	+	+	PUNCT
eajbcs-694	649	2	µ]λ1	µ]λ1	NOUN
eajbcs-694	649	3	−	−	ADP
eajbcs-694	649	4	θλ2	θλ2	NOUN
eajbcs-694	649	5	−	−	NOUN
eajbcs-694	649	6	β(σ2e	β(σ2e	PUNCT
eajbcs-694	649	7	+	+	CCONJ
eajbcs-694	649	8	σ1i	σ1i	NOUN
eajbcs-694	649	9	+	+	CCONJ
eajbcs-694	649	10	σ3h)λ3	σ3h)λ3	PROPN
eajbcs-694	649	11	,	,	PUNCT
eajbcs-694	649	12	dλ2	dλ2	NOUN
eajbcs-694	649	13	dt	dt	NOUN
eajbcs-694	649	14	=	=	PUNCT
eajbcs-694	649	15	−∂h	−∂h	PROPN
eajbcs-694	649	16	∂p	∂p	NOUN
eajbcs-694	649	17	=	=	PUNCT
eajbcs-694	650	1	−νλ1	−νλ1	PROPN
eajbcs-694	650	2	+	+	X
eajbcs-694	650	3	(	(	PUNCT
eajbcs-694	650	4	ν	ν	X
eajbcs-694	650	5	+	+	X
eajbcs-694	650	6	µ)λ2	µ)λ2	PROPN
eajbcs-694	650	7	,	,	PUNCT
eajbcs-694	650	8	dλ3	dλ3	NOUN
eajbcs-694	650	9	dt	dt	NOUN
eajbcs-694	650	10	=	=	PUNCT
eajbcs-694	650	11	−∂h	−∂h	PROPN
eajbcs-694	650	12	∂e	∂e	PROPN
eajbcs-694	650	13	=	=	PROPN
eajbcs-694	650	14	−b1	−b1	PROPN
eajbcs-694	650	15	+	+	CCONJ
eajbcs-694	650	16	βσ2sλ1	βσ2sλ1	NOUN
eajbcs-694	650	17	−	−	X
eajbcs-694	651	1	[	[	X
eajbcs-694	651	2	βσ2s	βσ2s	X
eajbcs-694	651	3	−	−	PROPN
eajbcs-694	651	4	(	(	PUNCT
eajbcs-694	651	5	δ	δ	PROPN
eajbcs-694	651	6	+	+	CCONJ
eajbcs-694	651	7	µ)]λ3	µ)]λ3	VERB
eajbcs-694	651	8	−	−	X
eajbcs-694	651	9	τδλ4	τδλ4	PROPN
eajbcs-694	651	10	−	−	PROPN
eajbcs-694	652	1	(	(	PUNCT
eajbcs-694	653	1	1−	1−	NUM
eajbcs-694	653	2	τ)δλ6	τ)δλ6	NOUN
eajbcs-694	653	3	,	,	PUNCT
eajbcs-694	654	1	dλ4	dλ4	VERB
eajbcs-694	654	2	dt	dt	NOUN
eajbcs-694	654	3	=	=	PUNCT
eajbcs-694	654	4	−∂h	−∂h	PROPN
eajbcs-694	654	5	∂i	∂i	PROPN
eajbcs-694	654	6	=	=	PUNCT
eajbcs-694	654	7	−b2	−b2	PROPN
eajbcs-694	654	8	+	+	CCONJ
eajbcs-694	654	9	βσ1sλ1	βσ1sλ1	X
eajbcs-694	654	10	−	−	NOUN
eajbcs-694	654	11	βσ1sλ3	βσ1sλ3	PUNCT
eajbcs-694	655	1	+	+	PUNCT
eajbcs-694	655	2	(	(	PUNCT
eajbcs-694	655	3	α	α	NOUN
eajbcs-694	655	4	+	+	X
eajbcs-694	655	5	ε+	ε+	X
eajbcs-694	655	6	µ+	µ+	VERB
eajbcs-694	655	7	ρ)λ4	ρ)λ4	PROPN
eajbcs-694	655	8	−	−	NOUN
eajbcs-694	655	9	αλ5	αλ5	NOUN
eajbcs-694	655	10	−	−	NOUN
eajbcs-694	655	11	ελ6	ελ6	NOUN
eajbcs-694	655	12	,	,	PUNCT
eajbcs-694	655	13	dλ5	dλ5	ADJ
eajbcs-694	655	14	dt	dt	X
eajbcs-694	655	15	=	=	SYM
eajbcs-694	655	16	−∂h	−∂h	PROPN
eajbcs-694	655	17	∂h	∂h	PROPN
eajbcs-694	655	18	=	=	SYM
eajbcs-694	655	19	βσ3sλ1	βσ3sλ1	NOUN
eajbcs-694	656	1	−	−	NOUN
eajbcs-694	656	2	βσ3sλ3	βσ3sλ3	VERB
eajbcs-694	657	1	+	+	PUNCT
eajbcs-694	657	2	(	(	PUNCT
eajbcs-694	657	3	γ	γ	X
eajbcs-694	657	4	+	+	X
eajbcs-694	657	5	µ+	µ+	PROPN
eajbcs-694	657	6	ξ)λ5	ξ)λ5	PROPN
eajbcs-694	657	7	−	−	PROPN
eajbcs-694	657	8	γλ6	γλ6	NOUN
eajbcs-694	657	9	,	,	PUNCT
eajbcs-694	657	10	dλ6	dλ6	VERB
eajbcs-694	657	11	dt	dt	NOUN
eajbcs-694	657	12	=	=	PUNCT
eajbcs-694	657	13	−∂h	−∂h	PROPN
eajbcs-694	657	14	∂r	∂r	NOUN
eajbcs-694	657	15	=	=	PUNCT
eajbcs-694	657	16	−ηωλ1	−ηωλ1	NOUN
eajbcs-694	658	1	−	−	NOUN
eajbcs-694	658	2	(	(	PUNCT
eajbcs-694	658	3	1−	1−	NUM
eajbcs-694	658	4	η)ωλ2	η)ωλ2	PROPN
eajbcs-694	658	5	+	+	CCONJ
eajbcs-694	658	6	(	(	PUNCT
eajbcs-694	658	7	ω	ω	NOUN
eajbcs-694	658	8	+	+	X
eajbcs-694	658	9	µ)λ6	µ)λ6	NOUN
eajbcs-694	658	10	.	.	NOUN
eajbcs-694	659	1	with	with	ADP
eajbcs-694	659	2	transversality	transversality	NOUN
eajbcs-694	659	3	conditions	condition	NOUN
eajbcs-694	659	4	λi(t	λi(t	X
eajbcs-694	659	5	)	)	PUNCT
eajbcs-694	659	6	=	=	SYM
eajbcs-694	659	7	0	0	NUM
eajbcs-694	659	8	,	,	PUNCT
eajbcs-694	659	9	i	i	PRON
eajbcs-694	659	10	=	=	NOUN
eajbcs-694	659	11	1	1	NUM
eajbcs-694	659	12	,	,	PUNCT
eajbcs-694	659	13	...	...	PUNCT
eajbcs-694	659	14	,	,	PUNCT
eajbcs-694	659	15	6	6	X
eajbcs-694	659	16	.	.	PUNCT
eajbcs-694	659	17	similarly	similarly	ADV
eajbcs-694	659	18	by	by	ADP
eajbcs-694	659	19	following	follow	VERB
eajbcs-694	659	20	the	the	DET
eajbcs-694	659	21	approach	approach	NOUN
eajbcs-694	659	22	of	of	ADP
eajbcs-694	659	23	pontryagin	pontryagin	NOUN
eajbcs-694	659	24	et	et	PROPN
eajbcs-694	659	25	al	al	PROPN
eajbcs-694	659	26	pontryagin	pontryagin	PROPN
eajbcs-694	659	27	(	(	PUNCT
eajbcs-694	659	28	2018	2018	NUM
eajbcs-694	659	29	)	)	PUNCT
eajbcs-694	659	30	,	,	PUNCT
eajbcs-694	659	31	the	the	DET
eajbcs-694	659	32	characterization	characterization	NOUN
eajbcs-694	659	33	of	of	ADP
eajbcs-694	659	34	optimal	optimal	ADJ
eajbcs-694	659	35	controls	control	NOUN
eajbcs-694	659	36	θ∗(t	θ∗(t	NOUN
eajbcs-694	659	37	)	)	PUNCT
eajbcs-694	659	38	,	,	PUNCT
eajbcs-694	659	39	α∗(t	α∗(t	NOUN
eajbcs-694	659	40	)	)	PUNCT
eajbcs-694	659	41	,	,	PUNCT
eajbcs-694	659	42	that	that	ADV
eajbcs-694	659	43	is	is	ADV
eajbcs-694	659	44	,	,	PUNCT
eajbcs-694	659	45	the	the	DET
eajbcs-694	659	46	optimality	optimality	NOUN
eajbcs-694	659	47	equations	equation	NOUN
eajbcs-694	659	48	are	be	AUX
eajbcs-694	659	49	obtained	obtain	VERB
eajbcs-694	659	50	based	base	VERB
eajbcs-694	659	51	on	on	ADP
eajbcs-694	659	52	the	the	DET
eajbcs-694	659	53	conditions	condition	NOUN
eajbcs-694	659	54	:	:	PUNCT
eajbcs-694	659	55	∂h	∂h	VERB
eajbcs-694	659	56	∂θ	∂θ	NOUN
eajbcs-694	660	1	=	=	SYM
eajbcs-694	660	2	0	0	NUM
eajbcs-694	660	3	and	and	CCONJ
eajbcs-694	660	4	∂h	∂h	PROPN
eajbcs-694	660	5	∂α	∂α	NOUN
eajbcs-694	660	6	=	=	SYM
eajbcs-694	660	7	0	0	PROPN
eajbcs-694	660	8	,	,	PUNCT
eajbcs-694	660	9	which	which	PRON
eajbcs-694	660	10	gives	give	VERB
eajbcs-694	660	11	,	,	PUNCT
eajbcs-694	660	12	θ	θ	PROPN
eajbcs-694	660	13	=	=	SYM
eajbcs-694	660	14	s(λ1	s(λ1	NOUN
eajbcs-694	660	15	−	−	PROPN
eajbcs-694	660	16	λ2	λ2	SYM
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eajbcs-694	662	6	θ∗(t	θ∗(t	NOUN
eajbcs-694	662	7	)	)	PUNCT
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eajbcs-694	662	10	)	)	PUNCT
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eajbcs-694	662	14	θ∗(t	θ∗(t	NOUN
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eajbcs-694	662	24	−	−	PROPN
eajbcs-694	662	25	λ2	λ2	SYM
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eajbcs-694	663	1	=	=	SYM
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eajbcs-694	663	9	−	−	NOUN
eajbcs-694	663	10	λ5	λ5	NOUN
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eajbcs-694	663	15	.	.	PUNCT
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eajbcs-694	664	5	.	.	PUNCT
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eajbcs-694	665	10	(	(	PUNCT
eajbcs-694	665	11	2	2	NUM
eajbcs-694	665	12	)	)	PUNCT
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eajbcs-694	665	19	17	17	NUM
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eajbcs-694	665	24	characterized	characterized	ADJ
eajbcs-694	665	25	control	control	NOUN
eajbcs-694	665	26	set	set	VERB
eajbcs-694	665	27	and	and	CCONJ
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eajbcs-694	665	29	and	and	CCONJ
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eajbcs-694	666	5	following	follow	VERB
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eajbcs-694	666	8	:	:	PUNCT
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eajbcs-694	670	3	)	)	PUNCT
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eajbcs-694	671	5	θ∗	θ∗	NOUN
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eajbcs-694	671	13	+	+	CCONJ
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eajbcs-694	671	15	1−	1−	NUM
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eajbcs-694	671	17	(	(	PUNCT
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eajbcs-694	671	26	+	+	NUM
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eajbcs-694	672	1	+	+	CCONJ
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eajbcs-694	672	3	−	−	PROPN
eajbcs-694	672	4	(	(	PUNCT
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eajbcs-694	672	11	=	=	PUNCT
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eajbcs-694	672	13	−	−	PROPN
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eajbcs-694	673	7	µ+	µ+	PROPN
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eajbcs-694	676	1	[	[	X
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eajbcs-694	676	3	+	+	CCONJ
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eajbcs-694	676	8	+	+	NUM
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eajbcs-694	676	16	+	+	CCONJ
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eajbcs-694	678	6	+	+	CCONJ
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eajbcs-694	678	8	−	−	X
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eajbcs-694	681	3	+	+	CCONJ
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eajbcs-694	682	1	+	+	CCONJ
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eajbcs-694	682	6	µ+	µ+	VERB
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eajbcs-694	684	1	+	+	PUNCT
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eajbcs-694	684	5	µ+	µ+	PROPN
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eajbcs-694	684	7	−	−	PROPN
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eajbcs-694	685	1	−	−	NOUN
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eajbcs-694	685	3	1−	1−	NUM
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eajbcs-694	685	5	+	+	CCONJ
eajbcs-694	685	6	(	(	PUNCT
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eajbcs-694	685	8	+	+	CCONJ
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eajbcs-694	685	10	,	,	PUNCT
eajbcs-694	685	11	λi(t	λi(t	NUM
eajbcs-694	685	12	)	)	PUNCT
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eajbcs-694	686	1	0	0	NUM
eajbcs-694	686	2	,	,	PUNCT
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eajbcs-694	686	5	1	1	NUM
eajbcs-694	686	6	,	,	PUNCT
eajbcs-694	686	7	...	...	PUNCT
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eajbcs-694	686	10	,	,	PUNCT
eajbcs-694	686	11	(	(	PUNCT
eajbcs-694	686	12	s(0	s(0	PROPN
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eajbcs-694	686	14	,	,	PUNCT
eajbcs-694	686	15	p	p	X
eajbcs-694	686	16	(	(	PUNCT
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eajbcs-694	686	19	,	,	PUNCT
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eajbcs-694	686	23	i(0	i(0	PROPN
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eajbcs-694	686	26	h(0	h(0	PROPN
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eajbcs-694	686	30	)	)	PUNCT
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eajbcs-694	687	1	=	=	SYM
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eajbcs-694	687	3	s0	s0	PROPN
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eajbcs-694	687	5	p0	p0	NOUN
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eajbcs-694	687	12	,	,	PUNCT
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eajbcs-694	687	15	.	.	PUNCT
eajbcs-694	688	1	6	6	X
eajbcs-694	688	2	.	.	X
eajbcs-694	688	3	numerical	numerical	ADJ
eajbcs-694	688	4	simulations	simulation	NOUN
eajbcs-694	688	5	of	of	ADP
eajbcs-694	688	6	optimal	optimal	ADJ
eajbcs-694	688	7	control	control	NOUN
eajbcs-694	688	8	problem	problem	NOUN
eajbcs-694	688	9	in	in	ADP
eajbcs-694	688	10	this	this	DET
eajbcs-694	688	11	section	section	NOUN
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eajbcs-694	688	14	perform	perform	VERB
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eajbcs-694	688	16	numerical	numerical	ADJ
eajbcs-694	688	17	solutions	solution	NOUN
eajbcs-694	688	18	on	on	ADP
eajbcs-694	688	19	the	the	DET
eajbcs-694	688	20	modified	modify	VERB
eajbcs-694	688	21	model	model	NOUN
eajbcs-694	688	22	(	(	PUNCT
eajbcs-694	688	23	2	2	NUM
eajbcs-694	688	24	)	)	PUNCT
eajbcs-694	688	25	and	and	CCONJ
eajbcs-694	688	26	the	the	DET
eajbcs-694	688	27	resulting	result	VERB
eajbcs-694	688	28	optimality	optimality	NOUN
eajbcs-694	688	29	system	system	NOUN
eajbcs-694	688	30	consisting	consist	VERB
eajbcs-694	688	31	of	of	ADP
eajbcs-694	688	32	the	the	DET
eajbcs-694	688	33	state	state	NOUN
eajbcs-694	688	34	equations	equation	NOUN
eajbcs-694	688	35	(	(	PUNCT
eajbcs-694	688	36	2	2	NUM
eajbcs-694	688	37	)	)	PUNCT
eajbcs-694	688	38	and	and	CCONJ
eajbcs-694	688	39	the	the	DET
eajbcs-694	688	40	adjoint	adjoint	NOUN
eajbcs-694	688	41	system	system	NOUN
eajbcs-694	688	42	(	(	PUNCT
eajbcs-694	688	43	17	17	NUM
eajbcs-694	688	44	)	)	PUNCT
eajbcs-694	688	45	with	with	ADP
eajbcs-694	688	46	the	the	DET
eajbcs-694	688	47	characterizations	characterization	NOUN
eajbcs-694	688	48	(	(	PUNCT
eajbcs-694	688	49	19	19	NUM
eajbcs-694	688	50	)	)	PUNCT
eajbcs-694	688	51	.	.	PUNCT
eajbcs-694	689	1	we	we	PRON
eajbcs-694	689	2	make	make	VERB
eajbcs-694	689	3	use	use	NOUN
eajbcs-694	689	4	of	of	ADP
eajbcs-694	689	5	the	the	DET
eajbcs-694	689	6	parameter	parameter	NOUN
eajbcs-694	689	7	values	value	NOUN
eajbcs-694	689	8	given	give	VERB
eajbcs-694	689	9	in	in	ADP
eajbcs-694	689	10	table	table	NOUN
eajbcs-694	689	11	(	(	PUNCT
eajbcs-694	689	12	3	3	NUM
eajbcs-694	689	13	)	)	PUNCT
eajbcs-694	689	14	for	for	ADP
eajbcs-694	689	15	the	the	DET
eajbcs-694	689	16	simulation	simulation	NOUN
eajbcs-694	689	17	.	.	PUNCT
eajbcs-694	690	1	an	an	DET
eajbcs-694	690	2	iterative	iterative	NOUN
eajbcs-694	690	3	scheme	scheme	NOUN
eajbcs-694	690	4	is	be	AUX
eajbcs-694	690	5	used	use	VERB
eajbcs-694	690	6	to	to	PART
eajbcs-694	690	7	find	find	VERB
eajbcs-694	690	8	the	the	DET
eajbcs-694	690	9	optimal	optimal	ADJ
eajbcs-694	690	10	solution	solution	NOUN
eajbcs-694	690	11	of	of	ADP
eajbcs-694	690	12	the	the	DET
eajbcs-694	690	13	optimality	optimality	NOUN
eajbcs-694	690	14	system	system	NOUN
eajbcs-694	690	15	.	.	PUNCT
eajbcs-694	691	1	since	since	SCONJ
eajbcs-694	691	2	the	the	DET
eajbcs-694	691	3	state	state	NOUN
eajbcs-694	691	4	system	system	NOUN
eajbcs-694	691	5	(	(	PUNCT
eajbcs-694	691	6	2	2	X
eajbcs-694	691	7	)	)	PUNCT
eajbcs-694	691	8	has	have	VERB
eajbcs-694	691	9	initial	initial	ADJ
eajbcs-694	691	10	conditions	condition	NOUN
eajbcs-694	691	11	and	and	CCONJ
eajbcs-694	691	12	the	the	DET
eajbcs-694	691	13	adjoint	adjoint	PROPN
eajbcs-694	691	14	systems	system	NOUN
eajbcs-694	691	15	(	(	PUNCT
eajbcs-694	691	16	17	17	NUM
eajbcs-694	691	17	)	)	PUNCT
eajbcs-694	691	18	have	have	VERB
eajbcs-694	691	19	final	final	ADJ
eajbcs-694	691	20	conditions	condition	NOUN
eajbcs-694	691	21	,	,	PUNCT
eajbcs-694	691	22	we	we	PRON
eajbcs-694	691	23	solve	solve	VERB
eajbcs-694	691	24	the	the	DET
eajbcs-694	691	25	state	state	NOUN
eajbcs-694	691	26	system	system	NOUN
eajbcs-694	691	27	using	use	VERB
eajbcs-694	691	28	a	a	DET
eajbcs-694	691	29	forward	forward	ADJ
eajbcs-694	691	30	fourth	fourth	ADJ
eajbcs-694	691	31	-	-	PUNCT
eajbcs-694	691	32	order	order	NOUN
eajbcs-694	691	33	runge	runge	NOUN
eajbcs-694	691	34	–	–	PUNCT
eajbcs-694	691	35	kutta	kutta	NOUN
eajbcs-694	691	36	method	method	NOUN
eajbcs-694	691	37	and	and	CCONJ
eajbcs-694	691	38	solve	solve	VERB
eajbcs-694	691	39	the	the	DET
eajbcs-694	691	40	adjoint	adjoint	NOUN
eajbcs-694	691	41	system	system	NOUN
eajbcs-694	691	42	using	use	VERB
eajbcs-694	691	43	a	a	DET
eajbcs-694	691	44	backward	backward	ADJ
eajbcs-694	691	45	fourthorder	fourthorder	NOUN
eajbcs-694	691	46	runge	runge	NOUN
eajbcs-694	691	47	–	–	PUNCT
eajbcs-694	691	48	kutta	kutta	NOUN
eajbcs-694	691	49	method	method	NOUN
eajbcs-694	691	50	?	?	PUNCT
eajbcs-694	691	51	.	.	PUNCT
eajbcs-694	692	1	the	the	DET
eajbcs-694	692	2	solution	solution	NOUN
eajbcs-694	692	3	iterative	iterative	NOUN
eajbcs-694	692	4	scheme	scheme	NOUN
eajbcs-694	692	5	involves	involve	VERB
eajbcs-694	692	6	making	make	VERB
eajbcs-694	692	7	a	a	DET
eajbcs-694	692	8	guess	guess	NOUN
eajbcs-694	692	9	of	of	ADP
eajbcs-694	692	10	the	the	DET
eajbcs-694	692	11	controls	control	NOUN
eajbcs-694	692	12	and	and	CCONJ
eajbcs-694	692	13	using	use	VERB
eajbcs-694	692	14	that	that	DET
eajbcs-694	692	15	guess	guess	NOUN
eajbcs-694	692	16	to	to	PART
eajbcs-694	692	17	solve	solve	VERB
eajbcs-694	692	18	the	the	DET
eajbcs-694	692	19	state	state	NOUN
eajbcs-694	692	20	system	system	NOUN
eajbcs-694	692	21	.	.	PUNCT
eajbcs-694	693	1	the	the	DET
eajbcs-694	693	2	initial	initial	ADJ
eajbcs-694	693	3	guess	guess	NOUN
eajbcs-694	693	4	of	of	ADP
eajbcs-694	693	5	the	the	DET
eajbcs-694	693	6	control	control	NOUN
eajbcs-694	693	7	together	together	ADV
eajbcs-694	693	8	with	with	ADP
eajbcs-694	693	9	the	the	DET
eajbcs-694	693	10	solution	solution	NOUN
eajbcs-694	693	11	of	of	ADP
eajbcs-694	693	12	the	the	DET
eajbcs-694	693	13	state	state	NOUN
eajbcs-694	693	14	systems	system	NOUN
eajbcs-694	693	15	is	be	AUX
eajbcs-694	693	16	used	use	VERB
eajbcs-694	693	17	to	to	PART
eajbcs-694	693	18	solve	solve	VERB
eajbcs-694	693	19	the	the	DET
eajbcs-694	693	20	adjoint	adjoint	NOUN
eajbcs-694	693	21	systems	system	NOUN
eajbcs-694	693	22	.	.	PUNCT
eajbcs-694	694	1	the	the	DET
eajbcs-694	694	2	controls	control	NOUN
eajbcs-694	694	3	are	be	AUX
eajbcs-694	694	4	then	then	ADV
eajbcs-694	694	5	updated	update	VERB
eajbcs-694	694	6	using	use	VERB
eajbcs-694	694	7	a	a	DET
eajbcs-694	694	8	convex	convex	ADJ
eajbcs-694	694	9	combination	combination	NOUN
eajbcs-694	694	10	of	of	ADP
eajbcs-694	694	11	the	the	DET
eajbcs-694	694	12	previous	previous	ADJ
eajbcs-694	694	13	controls	control	NOUN
eajbcs-694	694	14	and	and	CCONJ
eajbcs-694	694	15	the	the	DET
eajbcs-694	694	16	values	value	NOUN
eajbcs-694	694	17	obtained	obtain	VERB
eajbcs-694	694	18	using	use	VERB
eajbcs-694	694	19	the	the	DET
eajbcs-694	694	20	characterizations	characterization	NOUN
eajbcs-694	694	21	.	.	PUNCT
eajbcs-694	695	1	the	the	DET
eajbcs-694	695	2	updated	update	VERB
eajbcs-694	695	3	controls	control	NOUN
eajbcs-694	695	4	are	be	AUX
eajbcs-694	695	5	then	then	ADV
eajbcs-694	695	6	used	use	VERB
eajbcs-694	695	7	to	to	PART
eajbcs-694	695	8	repeat	repeat	VERB
eajbcs-694	695	9	the	the	DET
eajbcs-694	695	10	solution	solution	NOUN
eajbcs-694	695	11	of	of	ADP
eajbcs-694	695	12	the	the	DET
eajbcs-694	695	13	state	state	NOUN
eajbcs-694	695	14	and	and	CCONJ
eajbcs-694	695	15	adjoint	adjoint	PROPN
eajbcs-694	695	16	systems	system	NOUN
eajbcs-694	695	17	.	.	PUNCT
eajbcs-694	696	1	this	this	DET
eajbcs-694	696	2	process	process	NOUN
eajbcs-694	696	3	is	be	AUX
eajbcs-694	696	4	repeated	repeat	VERB
eajbcs-694	696	5	until	until	SCONJ
eajbcs-694	696	6	the	the	DET
eajbcs-694	696	7	values	value	NOUN
eajbcs-694	696	8	in	in	ADP
eajbcs-694	696	9	the	the	DET
eajbcs-694	696	10	current	current	ADJ
eajbcs-694	696	11	iteration	iteration	NOUN
eajbcs-694	696	12	are	be	AUX
eajbcs-694	696	13	close	close	ADJ
eajbcs-694	696	14	enough	enough	ADV
eajbcs-694	696	15	to	to	ADP
eajbcs-694	696	16	the	the	DET
eajbcs-694	696	17	previous	previous	ADJ
eajbcs-694	696	18	iteration	iteration	NOUN
eajbcs-694	696	19	values	value	NOUN
eajbcs-694	696	20	lenhart	lenhart	NOUN
eajbcs-694	696	21	and	and	CCONJ
eajbcs-694	696	22	workman	workman	NOUN
eajbcs-694	696	23	(	(	PUNCT
eajbcs-694	696	24	2007	2007	NUM
eajbcs-694	696	25	)	)	PUNCT
eajbcs-694	696	26	.	.	PUNCT
eajbcs-694	697	1	we	we	PRON
eajbcs-694	697	2	used	use	VERB
eajbcs-694	697	3	b1	b1	NOUN
eajbcs-694	697	4	=	=	SYM
eajbcs-694	697	5	b2	b2	PROPN
eajbcs-694	697	6	=	=	SYM
eajbcs-694	697	7	1	1	NUM
eajbcs-694	697	8	,	,	PUNCT
eajbcs-694	697	9	a1	a1	NOUN
eajbcs-694	697	10	=	=	SYM
eajbcs-694	697	11	40	40	NUM
eajbcs-694	697	12	,	,	PUNCT
eajbcs-694	697	13	a2	a2	PROPN
eajbcs-694	697	14	=	=	SYM
eajbcs-694	697	15	80	80	NUM
eajbcs-694	697	16	and	and	CCONJ
eajbcs-694	697	17	final	final	ADJ
eajbcs-694	697	18	intervention	intervention	NOUN
eajbcs-694	697	19	time	time	NOUN
eajbcs-694	697	20	t	t	PROPN
eajbcs-694	697	21	=	=	SYM
eajbcs-694	697	22	350	350	NUM
eajbcs-694	697	23	days	day	NOUN
eajbcs-694	697	24	for	for	ADP
eajbcs-694	697	25	simulation	simulation	NOUN
eajbcs-694	697	26	of	of	ADP
eajbcs-694	697	27	covid-19	covid-19	PROPN
eajbcs-694	697	28	model	model	NOUN
eajbcs-694	697	29	with	with	ADP
eajbcs-694	697	30	optimal	optimal	ADJ
eajbcs-694	697	31	control	control	NOUN
eajbcs-694	697	32	.	.	PUNCT
eajbcs-694	698	1	additionally	additionally	ADV
eajbcs-694	698	2	,	,	PUNCT
eajbcs-694	698	3	we	we	PRON
eajbcs-694	698	4	used	use	VERB
eajbcs-694	698	5	s(0	s(0	PROPN
eajbcs-694	698	6	)	)	PUNCT
eajbcs-694	698	7	=	=	SYM
eajbcs-694	698	8	110079	110079	NUM
eajbcs-694	698	9	,	,	PUNCT
eajbcs-694	698	10	p	p	X
eajbcs-694	698	11	(	(	PUNCT
eajbcs-694	698	12	0	0	NUM
eajbcs-694	698	13	)	)	PUNCT
eajbcs-694	698	14	=	=	SYM
eajbcs-694	698	15	108076	108076	NUM
eajbcs-694	698	16	,	,	PUNCT
eajbcs-694	698	17	e(0	e(0	NOUN
eajbcs-694	698	18	)	)	PUNCT
eajbcs-694	698	19	=	=	SYM
eajbcs-694	698	20	15000	15000	NUM
eajbcs-694	698	21	,	,	PUNCT
eajbcs-694	698	22	i(0	i(0	PROPN
eajbcs-694	698	23	)	)	PUNCT
eajbcs-694	698	24	=	=	SYM
eajbcs-694	698	25	13813	13813	NUM
eajbcs-694	698	26	,	,	PUNCT
eajbcs-694	698	27	h(0	h(0	PROPN
eajbcs-694	698	28	)	)	PUNCT
eajbcs-694	698	29	=	=	SYM
eajbcs-694	698	30	10050	10050	NUM
eajbcs-694	698	31	,	,	PUNCT
eajbcs-694	698	32	r(0	r(0	PROPN
eajbcs-694	698	33	)	)	PUNCT
eajbcs-694	698	34	=	=	SYM
eajbcs-694	698	35	12156	12156	NUM
eajbcs-694	698	36	as	as	ADP
eajbcs-694	698	37	initial	initial	ADJ
eajbcs-694	698	38	values	value	NOUN
eajbcs-694	698	39	.	.	PUNCT
eajbcs-694	699	1	east	east	PROPN
eajbcs-694	699	2	afr	afr	PROPN
eajbcs-694	699	3	.	.	PUNCT
eajbcs-694	700	1	j.	j.	PROPN
eajbcs-694	700	2	biophys	biophys	PROPN
eajbcs-694	700	3	.	.	PUNCT
eajbcs-694	701	1	comput	comput	NOUN
eajbcs-694	701	2	.	.	PUNCT
eajbcs-694	702	1	sci	sci	PROPN
eajbcs-694	702	2	.	.	PUNCT
eajbcs-694	703	1	(	(	PUNCT
eajbcs-694	703	2	2024	2024	NUM
eajbcs-694	703	3	)	)	PUNCT
eajbcs-694	703	4	,	,	PUNCT
eajbcs-694	703	5	vol	vol	NOUN
eajbcs-694	703	6	.	.	PROPN
eajbcs-694	703	7	5	5	NUM
eajbcs-694	703	8	,	,	PUNCT
eajbcs-694	703	9	no	no	INTJ
eajbcs-694	703	10	.	.	NOUN
eajbcs-694	703	11	2	2	NUM
eajbcs-694	703	12	,	,	PUNCT
eajbcs-694	703	13	58	58	NUM
eajbcs-694	703	14	-	-	SYM
eajbcs-694	703	15	86	86	NUM
eajbcs-694	703	16	82	82	NUM
eajbcs-694	703	17	6.1	6.1	NUM
eajbcs-694	703	18	.	.	PUNCT
eajbcs-694	704	1	optimal	optimal	ADJ
eajbcs-694	704	2	control	control	NOUN
eajbcs-694	704	3	comparisons	comparison	NOUN
eajbcs-694	704	4	and	and	CCONJ
eajbcs-694	704	5	strategies	strategy	NOUN
eajbcs-694	704	6	in	in	ADP
eajbcs-694	704	7	this	this	DET
eajbcs-694	704	8	subsection	subsection	NOUN
eajbcs-694	704	9	,	,	PUNCT
eajbcs-694	704	10	we	we	PRON
eajbcs-694	704	11	compare	compare	VERB
eajbcs-694	704	12	the	the	DET
eajbcs-694	704	13	results	result	NOUN
eajbcs-694	704	14	of	of	ADP
eajbcs-694	704	15	constant	constant	ADJ
eajbcs-694	704	16	and	and	CCONJ
eajbcs-694	704	17	optimal	optimal	ADJ
eajbcs-694	704	18	control	control	NOUN
eajbcs-694	704	19	as	as	SCONJ
eajbcs-694	704	20	did	do	VERB
eajbcs-694	704	21	in	in	ADP
eajbcs-694	704	22	?	?	PUNCT
eajbcs-694	704	23	.	.	PUNCT
eajbcs-694	705	1	we	we	PRON
eajbcs-694	705	2	first	first	ADV
eajbcs-694	705	3	compare	compare	VERB
eajbcs-694	705	4	the	the	DET
eajbcs-694	705	5	cost	cost	NOUN
eajbcs-694	705	6	of	of	ADP
eajbcs-694	705	7	infection	infection	NOUN
eajbcs-694	705	8	for	for	ADP
eajbcs-694	705	9	each	each	DET
eajbcs-694	705	10	strategies	strategy	NOUN
eajbcs-694	705	11	and	and	CCONJ
eajbcs-694	705	12	then	then	ADV
eajbcs-694	705	13	compare	compare	VERB
eajbcs-694	705	14	the	the	DET
eajbcs-694	705	15	exposed	expose	VERB
eajbcs-694	705	16	and	and	CCONJ
eajbcs-694	705	17	infected	infected	ADJ
eajbcs-694	705	18	population	population	NOUN
eajbcs-694	705	19	.	.	PUNCT
eajbcs-694	706	1	the	the	DET
eajbcs-694	706	2	optimal	optimal	ADJ
eajbcs-694	706	3	control	control	NOUN
eajbcs-694	706	4	cost	cost	NOUN
eajbcs-694	706	5	for	for	ADP
eajbcs-694	706	6	each	each	DET
eajbcs-694	706	7	parameter	parameter	NOUN
eajbcs-694	706	8	is	be	AUX
eajbcs-694	706	9	less	less	ADJ
eajbcs-694	706	10	than	than	ADP
eajbcs-694	706	11	the	the	DET
eajbcs-694	706	12	constant	constant	ADJ
eajbcs-694	706	13	control	control	NOUN
eajbcs-694	706	14	at	at	ADP
eajbcs-694	706	15	all	all	DET
eajbcs-694	706	16	times	time	NOUN
eajbcs-694	706	17	as	as	SCONJ
eajbcs-694	706	18	depicted	depict	VERB
eajbcs-694	706	19	in	in	ADP
eajbcs-694	706	20	figures	figure	NOUN
eajbcs-694	706	21	(	(	PUNCT
eajbcs-694	706	22	91	91	NUM
eajbcs-694	706	23	)	)	PUNCT
eajbcs-694	706	24	and	and	CCONJ
eajbcs-694	706	25	(	(	PUNCT
eajbcs-694	706	26	101	101	NUM
eajbcs-694	706	27	)	)	PUNCT
eajbcs-694	706	28	.	.	PUNCT
eajbcs-694	707	1	it	it	PRON
eajbcs-694	707	2	can	can	AUX
eajbcs-694	707	3	be	be	AUX
eajbcs-694	707	4	observed	observe	VERB
eajbcs-694	707	5	that	that	SCONJ
eajbcs-694	707	6	for	for	ADP
eajbcs-694	707	7	protection	protection	NOUN
eajbcs-694	707	8	rate	rate	NOUN
eajbcs-694	707	9	θ	θ	PROPN
eajbcs-694	707	10	the	the	DET
eajbcs-694	707	11	cost	cost	NOUN
eajbcs-694	707	12	reduction	reduction	NOUN
eajbcs-694	707	13	is	be	AUX
eajbcs-694	707	14	significant	significant	ADJ
eajbcs-694	707	15	compared	compare	VERB
eajbcs-694	707	16	to	to	ADP
eajbcs-694	707	17	the	the	DET
eajbcs-694	707	18	other	other	ADJ
eajbcs-694	707	19	parameters	parameter	NOUN
eajbcs-694	707	20	which	which	PRON
eajbcs-694	707	21	is	be	AUX
eajbcs-694	707	22	due	due	ADJ
eajbcs-694	707	23	to	to	ADP
eajbcs-694	707	24	the	the	DET
eajbcs-694	707	25	high	high	ADJ
eajbcs-694	707	26	sensitivity	sensitivity	NOUN
eajbcs-694	707	27	of	of	ADP
eajbcs-694	707	28	parameter	parameter	NOUN
eajbcs-694	707	29	over	over	ADP
eajbcs-694	707	30	the	the	DET
eajbcs-694	707	31	system	system	NOUN
eajbcs-694	707	32	.	.	PUNCT
eajbcs-694	708	1	0	0	NUM
eajbcs-694	708	2	50	50	NUM
eajbcs-694	708	3	100	100	NUM
eajbcs-694	708	4	150	150	NUM
eajbcs-694	708	5	200	200	NUM
eajbcs-694	708	6	250	250	NUM
eajbcs-694	708	7	300	300	NUM
eajbcs-694	708	8	350	350	NUM
eajbcs-694	708	9	time	time	NOUN
eajbcs-694	708	10	in	in	ADP
eajbcs-694	708	11	days	day	NOUN
eajbcs-694	708	12	2	2	NUM
eajbcs-694	708	13	4	4	NUM
eajbcs-694	708	14	6	6	NUM
eajbcs-694	708	15	8	8	NUM
eajbcs-694	708	16	10	10	NUM
eajbcs-694	708	17	12	12	NUM
eajbcs-694	708	18	14	14	NUM
eajbcs-694	708	19	16	16	NUM
eajbcs-694	708	20	co	co	NOUN
eajbcs-694	708	21	st	st	PROPN
eajbcs-694	708	22	of	of	ADP
eajbcs-694	708	23	di	di	X
eajbcs-694	708	24	se	se	PROPN
eajbcs-694	708	25	as	as	ADP
eajbcs-694	708	26	e	e	PROPN
eajbcs-694	708	27	c	c	PROPN
eajbcs-694	708	28	on	on	ADP
eajbcs-694	708	29	tro	tro	X
eajbcs-694	708	30	l	l	NOUN
eajbcs-694	708	31	#	#	SYM
eajbcs-694	708	32	104	104	NUM
eajbcs-694	708	33	optimal	optimal	ADJ
eajbcs-694	708	34	control	control	NOUN
eajbcs-694	709	1	3=0.01	3=0.01	PRON
eajbcs-694	709	2	3=0.24	3=0.24	NUM
eajbcs-694	709	3	3=0.47	3=0.47	NUM
eajbcs-694	710	1	3=0.7	3=0.7	NUM
eajbcs-694	710	2	(	(	PUNCT
eajbcs-694	710	3	1	1	NUM
eajbcs-694	710	4	)	)	PUNCT
eajbcs-694	710	5	0	0	NUM
eajbcs-694	710	6	50	50	NUM
eajbcs-694	710	7	100	100	NUM
eajbcs-694	710	8	150	150	NUM
eajbcs-694	710	9	200	200	NUM
eajbcs-694	710	10	250	250	NUM
eajbcs-694	710	11	300	300	NUM
eajbcs-694	710	12	350	350	NUM
eajbcs-694	710	13	time	time	NOUN
eajbcs-694	710	14	in	in	ADP
eajbcs-694	710	15	days	day	NOUN
eajbcs-694	710	16	0	0	NUM
eajbcs-694	710	17	2	2	NUM
eajbcs-694	710	18	4	4	NUM
eajbcs-694	710	19	6	6	NUM
eajbcs-694	710	20	8	8	NUM
eajbcs-694	710	21	10	10	NUM
eajbcs-694	710	22	12	12	NUM
eajbcs-694	710	23	14	14	NUM
eajbcs-694	710	24	ex	ex	X
eajbcs-694	710	25	po	po	X
eajbcs-694	710	26	se	se	PROPN
eajbcs-694	711	1	d	d	PROPN
eajbcs-694	712	1	p	p	X
eajbcs-694	712	2	op	op	NOUN
eajbcs-694	712	3	ula	ula	PROPN
eajbcs-694	712	4	tio	tio	VERB
eajbcs-694	712	5	n	n	CCONJ
eajbcs-694	712	6	#	#	SYM
eajbcs-694	712	7	104	104	NUM
eajbcs-694	712	8	optimal	optimal	ADJ
eajbcs-694	712	9	control	control	NOUN
eajbcs-694	713	1	3=0.01	3=0.01	PRON
eajbcs-694	713	2	3=0.24	3=0.24	NUM
eajbcs-694	713	3	3=0.47	3=0.47	NUM
eajbcs-694	714	1	3=0.7	3=0.7	NUM
eajbcs-694	714	2	(	(	PUNCT
eajbcs-694	714	3	2	2	NUM
eajbcs-694	714	4	)	)	PUNCT
eajbcs-694	714	5	0	0	NUM
eajbcs-694	714	6	50	50	NUM
eajbcs-694	714	7	100	100	NUM
eajbcs-694	714	8	150	150	NUM
eajbcs-694	714	9	200	200	NUM
eajbcs-694	714	10	250	250	NUM
eajbcs-694	714	11	300	300	NUM
eajbcs-694	714	12	350	350	NUM
eajbcs-694	714	13	time	time	NOUN
eajbcs-694	714	14	in	in	ADP
eajbcs-694	714	15	days	day	NOUN
eajbcs-694	714	16	1	1	NUM
eajbcs-694	714	17	2	2	NUM
eajbcs-694	714	18	3	3	NUM
eajbcs-694	714	19	4	4	NUM
eajbcs-694	714	20	5	5	NUM
eajbcs-694	714	21	6	6	NUM
eajbcs-694	714	22	7	7	NUM
eajbcs-694	714	23	inf	inf	NOUN
eajbcs-694	714	24	ec	ec	PROPN
eajbcs-694	714	25	ted	te	VERB
eajbcs-694	714	26	p	p	PROPN
eajbcs-694	714	27	op	op	NOUN
eajbcs-694	714	28	ula	ula	PROPN
eajbcs-694	714	29	tio	tio	VERB
eajbcs-694	714	30	n	n	CCONJ
eajbcs-694	714	31	#	#	SYM
eajbcs-694	714	32	104	104	NUM
eajbcs-694	714	33	optimal	optimal	ADJ
eajbcs-694	714	34	control	control	NOUN
eajbcs-694	715	1	3=0.01	3=0.01	PRON
eajbcs-694	715	2	3=0.24	3=0.24	NUM
eajbcs-694	715	3	3=0.47	3=0.47	NUM
eajbcs-694	716	1	3=0.7	3=0.7	NUM
eajbcs-694	716	2	(	(	PUNCT
eajbcs-694	716	3	3	3	NUM
eajbcs-694	716	4	)	)	PUNCT
eajbcs-694	716	5	0	0	NUM
eajbcs-694	716	6	50	50	NUM
eajbcs-694	716	7	100	100	NUM
eajbcs-694	716	8	150	150	NUM
eajbcs-694	716	9	200	200	NUM
eajbcs-694	716	10	250	250	NUM
eajbcs-694	716	11	300	300	NUM
eajbcs-694	716	12	350	350	NUM
eajbcs-694	716	13	time	time	NOUN
eajbcs-694	716	14	in	in	ADP
eajbcs-694	716	15	days	day	NOUN
eajbcs-694	716	16	0	0	NUM
eajbcs-694	716	17	0.1	0.1	NUM
eajbcs-694	716	18	0.2	0.2	NUM
eajbcs-694	716	19	0.3	0.3	NUM
eajbcs-694	716	20	0.4	0.4	NUM
eajbcs-694	716	21	0.5	0.5	NUM
eajbcs-694	716	22	0.6	0.6	NUM
eajbcs-694	716	23	0.7	0.7	NUM
eajbcs-694	716	24	0.8	0.8	NUM
eajbcs-694	716	25	0.9	0.9	NUM
eajbcs-694	716	26	1	1	NUM
eajbcs-694	716	27	pr	pr	NOUN
eajbcs-694	716	28	op	op	NOUN
eajbcs-694	716	29	or	or	CCONJ
eajbcs-694	716	30	tio	tio	NOUN
eajbcs-694	716	31	n	n	CCONJ
eajbcs-694	716	32	o	o	NOUN
eajbcs-694	717	1	f	f	X
eajbcs-694	718	1	p	p	X
eajbcs-694	718	2	op	op	NOUN
eajbcs-694	718	3	ula	ula	PROPN
eajbcs-694	718	4	tio	tio	VERB
eajbcs-694	718	5	n	n	CCONJ
eajbcs-694	718	6	p	p	PROPN
eajbcs-694	718	7	ro	ro	X
eajbcs-694	718	8	tec	tec	PROPN
eajbcs-694	719	1	ted	ted	PROPN
eajbcs-694	719	2	-=	-=	NUM
eajbcs-694	719	3	0.0001	0.0001	NUM
eajbcs-694	719	4	-=0.00015	-=0.00015	PROPN
eajbcs-694	719	5	-=	-=	VERB
eajbcs-694	719	6	0.0002	0.0002	NUM
eajbcs-694	719	7	-=0.00025	-=0.00025	NOUN
eajbcs-694	719	8	(	(	PUNCT
eajbcs-694	719	9	d	d	NOUN
eajbcs-694	719	10	)	)	PUNCT
eajbcs-694	719	11	figure	figure	NOUN
eajbcs-694	719	12	9	9	NUM
eajbcs-694	719	13	:	:	PUNCT
eajbcs-694	719	14	optimal	optimal	ADJ
eajbcs-694	719	15	control	control	NOUN
eajbcs-694	719	16	for	for	ADP
eajbcs-694	719	17	θ	θ	PROPN
eajbcs-694	719	18	.	.	PUNCT
eajbcs-694	720	1	(	(	PUNCT
eajbcs-694	720	2	a	a	X
eajbcs-694	720	3	)	)	PUNCT
eajbcs-694	720	4	cost	cost	NOUN
eajbcs-694	720	5	comparison	comparison	NOUN
eajbcs-694	720	6	for	for	ADP
eajbcs-694	720	7	optimal	optimal	ADJ
eajbcs-694	720	8	and	and	CCONJ
eajbcs-694	720	9	constant	constant	ADJ
eajbcs-694	720	10	controls	control	NOUN
eajbcs-694	720	11	for	for	ADP
eajbcs-694	720	12	θ	θ	PROPN
eajbcs-694	720	13	.	.	PUNCT
eajbcs-694	721	1	(	(	PUNCT
eajbcs-694	721	2	b	b	NOUN
eajbcs-694	721	3	)	)	PUNCT
eajbcs-694	721	4	exposed	expose	VERB
eajbcs-694	721	5	population	population	NOUN
eajbcs-694	721	6	comparison	comparison	NOUN
eajbcs-694	721	7	for	for	ADP
eajbcs-694	721	8	optimal	optimal	ADJ
eajbcs-694	721	9	and	and	CCONJ
eajbcs-694	721	10	constant	constant	ADJ
eajbcs-694	721	11	controls	control	NOUN
eajbcs-694	721	12	for	for	ADP
eajbcs-694	721	13	θ	θ	PROPN
eajbcs-694	721	14	.	.	PUNCT
eajbcs-694	722	1	(	(	PUNCT
eajbcs-694	722	2	c	c	X
eajbcs-694	722	3	)	)	PUNCT
eajbcs-694	722	4	infected	infected	ADJ
eajbcs-694	722	5	population	population	NOUN
eajbcs-694	722	6	comparison	comparison	NOUN
eajbcs-694	722	7	for	for	ADP
eajbcs-694	722	8	optimal	optimal	ADJ
eajbcs-694	722	9	and	and	CCONJ
eajbcs-694	722	10	constant	constant	ADJ
eajbcs-694	722	11	controls	control	NOUN
eajbcs-694	722	12	for	for	ADP
eajbcs-694	722	13	θ	θ	PROPN
eajbcs-694	722	14	.	.	PUNCT
eajbcs-694	723	1	(	(	PUNCT
eajbcs-694	723	2	d	d	X
eajbcs-694	723	3	)	)	PUNCT
eajbcs-694	723	4	optimal	optimal	ADJ
eajbcs-694	723	5	strategies	strategy	NOUN
eajbcs-694	723	6	for	for	ADP
eajbcs-694	723	7	θ	θ	PROPN
eajbcs-694	723	8	with	with	ADP
eajbcs-694	723	9	different	different	ADJ
eajbcs-694	723	10	β	β	NOUN
eajbcs-694	723	11	.	.	PUNCT
eajbcs-694	724	1	now	now	ADV
eajbcs-694	724	2	we	we	PRON
eajbcs-694	724	3	compare	compare	VERB
eajbcs-694	724	4	the	the	DET
eajbcs-694	724	5	exposed	expose	VERB
eajbcs-694	724	6	population	population	NOUN
eajbcs-694	724	7	using	use	VERB
eajbcs-694	724	8	constant	constant	ADJ
eajbcs-694	724	9	and	and	CCONJ
eajbcs-694	724	10	optimal	optimal	ADJ
eajbcs-694	724	11	controls	control	NOUN
eajbcs-694	724	12	.	.	PUNCT
eajbcs-694	725	1	in	in	ADP
eajbcs-694	725	2	figure	figure	NOUN
eajbcs-694	725	3	(	(	PUNCT
eajbcs-694	725	4	92	92	NUM
eajbcs-694	725	5	)	)	PUNCT
eajbcs-694	725	6	,	,	PUNCT
eajbcs-694	725	7	we	we	PRON
eajbcs-694	725	8	observe	observe	VERB
eajbcs-694	725	9	that	that	SCONJ
eajbcs-694	725	10	the	the	DET
eajbcs-694	725	11	optimal	optimal	ADJ
eajbcs-694	725	12	time	time	NOUN
eajbcs-694	725	13	dependent	dependent	ADJ
eajbcs-694	725	14	protection	protection	NOUN
eajbcs-694	725	15	strategy	strategy	NOUN
eajbcs-694	725	16	,	,	PUNCT
eajbcs-694	725	17	θ	θ	PROPN
eajbcs-694	725	18	,	,	PUNCT
eajbcs-694	725	19	yields	yield	VERB
eajbcs-694	725	20	a	a	DET
eajbcs-694	725	21	significant	significant	ADJ
eajbcs-694	725	22	drop	drop	NOUN
eajbcs-694	725	23	in	in	ADP
eajbcs-694	725	24	exposed	expose	VERB
eajbcs-694	725	25	population	population	NOUN
eajbcs-694	725	26	when	when	SCONJ
eajbcs-694	725	27	compared	compare	VERB
eajbcs-694	725	28	to	to	ADP
eajbcs-694	725	29	its	its	PRON
eajbcs-694	725	30	constant	constant	ADJ
eajbcs-694	725	31	counterparts	counterpart	NOUN
eajbcs-694	725	32	,	,	PUNCT
eajbcs-694	725	33	similarly	similarly	ADV
eajbcs-694	725	34	using	use	VERB
eajbcs-694	725	35	the	the	DET
eajbcs-694	725	36	optimal	optimal	ADJ
eajbcs-694	725	37	time	time	NOUN
eajbcs-694	726	1	dependent	dependent	ADJ
eajbcs-694	726	2	hospitalization	hospitalization	NOUN
eajbcs-694	726	3	strategy	strategy	NOUN
eajbcs-694	726	4	α	α	PRON
eajbcs-694	726	5	we	we	PRON
eajbcs-694	726	6	see	see	VERB
eajbcs-694	726	7	a	a	DET
eajbcs-694	726	8	significant	significant	ADJ
eajbcs-694	726	9	drop	drop	NOUN
eajbcs-694	726	10	in	in	ADP
eajbcs-694	726	11	the	the	DET
eajbcs-694	726	12	exposed	expose	VERB
eajbcs-694	726	13	population	population	NOUN
eajbcs-694	726	14	as	as	SCONJ
eajbcs-694	726	15	compared	compare	VERB
eajbcs-694	726	16	to	to	ADP
eajbcs-694	726	17	constant	constant	ADJ
eajbcs-694	726	18	rates	rate	NOUN
eajbcs-694	726	19	of	of	ADP
eajbcs-694	726	20	hospitalization	hospitalization	NOUN
eajbcs-694	726	21	over	over	ADP
eajbcs-694	726	22	time	time	NOUN
eajbcs-694	726	23	as	as	SCONJ
eajbcs-694	726	24	depicted	depict	VERB
eajbcs-694	726	25	in	in	ADP
eajbcs-694	726	26	figure	figure	NOUN
eajbcs-694	726	27	(	(	PUNCT
eajbcs-694	726	28	102	102	NUM
eajbcs-694	726	29	)	)	PUNCT
eajbcs-694	726	30	.	.	PUNCT
eajbcs-694	727	1	we	we	PRON
eajbcs-694	727	2	compare	compare	VERB
eajbcs-694	727	3	the	the	DET
eajbcs-694	727	4	infected	infected	ADJ
eajbcs-694	727	5	population	population	NOUN
eajbcs-694	727	6	using	use	VERB
eajbcs-694	727	7	constant	constant	ADJ
eajbcs-694	727	8	and	and	CCONJ
eajbcs-694	727	9	optimal	optimal	ADJ
eajbcs-694	727	10	controls	control	NOUN
eajbcs-694	727	11	.	.	PUNCT
eajbcs-694	728	1	in	in	ADP
eajbcs-694	728	2	figure	figure	NOUN
eajbcs-694	728	3	(	(	PUNCT
eajbcs-694	728	4	93	93	NUM
eajbcs-694	728	5	)	)	PUNCT
eajbcs-694	728	6	,	,	PUNCT
eajbcs-694	728	7	we	we	PRON
eajbcs-694	728	8	observe	observe	VERB
eajbcs-694	728	9	that	that	SCONJ
eajbcs-694	728	10	the	the	DET
eajbcs-694	728	11	optimal	optimal	ADJ
eajbcs-694	728	12	time	time	NOUN
eajbcs-694	728	13	dependent	dependent	ADJ
eajbcs-694	728	14	protection	protection	NOUN
eajbcs-694	728	15	strategy	strategy	NOUN
eajbcs-694	728	16	,	,	PUNCT
eajbcs-694	728	17	θ	θ	PROPN
eajbcs-694	728	18	,	,	PUNCT
eajbcs-694	728	19	yields	yield	VERB
eajbcs-694	728	20	a	a	DET
eajbcs-694	728	21	significant	significant	ADJ
eajbcs-694	728	22	drop	drop	NOUN
eajbcs-694	728	23	in	in	ADP
eajbcs-694	728	24	infected	infected	ADJ
eajbcs-694	728	25	population	population	NOUN
eajbcs-694	728	26	when	when	SCONJ
eajbcs-694	728	27	compared	compare	VERB
eajbcs-694	728	28	to	to	ADP
eajbcs-694	728	29	its	its	PRON
eajbcs-694	728	30	constant	constant	ADJ
eajbcs-694	728	31	counterparts	counterpart	NOUN
eajbcs-694	728	32	,	,	PUNCT
eajbcs-694	728	33	similarly	similarly	ADV
eajbcs-694	728	34	using	use	VERB
eajbcs-694	728	35	the	the	DET
eajbcs-694	728	36	optimal	optimal	ADJ
eajbcs-694	728	37	time	time	NOUN
eajbcs-694	729	1	dependent	dependent	ADJ
eajbcs-694	729	2	hospitalization	hospitalization	NOUN
eajbcs-694	729	3	strategy	strategy	NOUN
eajbcs-694	729	4	α	α	PRON
eajbcs-694	729	5	we	we	PRON
eajbcs-694	729	6	see	see	VERB
eajbcs-694	729	7	a	a	DET
eajbcs-694	729	8	significant	significant	ADJ
eajbcs-694	729	9	drop	drop	NOUN
eajbcs-694	729	10	in	in	ADP
eajbcs-694	729	11	the	the	DET
eajbcs-694	729	12	infected	infected	ADJ
eajbcs-694	729	13	population	population	NOUN
eajbcs-694	729	14	as	as	SCONJ
eajbcs-694	729	15	compared	compare	VERB
eajbcs-694	729	16	to	to	ADP
eajbcs-694	729	17	constant	constant	ADJ
eajbcs-694	729	18	rates	rate	NOUN
eajbcs-694	729	19	of	of	ADP
eajbcs-694	729	20	hospitalization	hospitalization	NOUN
eajbcs-694	729	21	over	over	ADP
eajbcs-694	729	22	time	time	NOUN
eajbcs-694	729	23	as	as	SCONJ
eajbcs-694	729	24	depicted	depict	VERB
eajbcs-694	729	25	in	in	ADP
eajbcs-694	729	26	figure	figure	NOUN
eajbcs-694	729	27	(	(	PUNCT
eajbcs-694	729	28	103	103	NUM
eajbcs-694	729	29	)	)	PUNCT
eajbcs-694	729	30	.	.	PUNCT
eajbcs-694	730	1	as	as	ADP
eajbcs-694	730	2	contact	contact	NOUN
eajbcs-694	730	3	rate	rate	NOUN
eajbcs-694	730	4	β	β	NOUN
eajbcs-694	730	5	increases	increase	NOUN
eajbcs-694	730	6	,	,	PUNCT
eajbcs-694	730	7	so	so	ADV
eajbcs-694	730	8	does	do	VERB
eajbcs-694	730	9	the	the	DET
eajbcs-694	730	10	time	time	NOUN
eajbcs-694	730	11	for	for	ADP
eajbcs-694	730	12	which	which	PRON
eajbcs-694	730	13	the	the	DET
eajbcs-694	730	14	maximum	maximum	ADJ
eajbcs-694	730	15	control	control	NOUN
eajbcs-694	730	16	should	should	AUX
eajbcs-694	730	17	be	be	AUX
eajbcs-694	730	18	applied	apply	VERB
eajbcs-694	730	19	.	.	PUNCT
eajbcs-694	731	1	in	in	ADP
eajbcs-694	731	2	figures	figure	NOUN
eajbcs-694	731	3	(	(	PUNCT
eajbcs-694	731	4	9d	9d	NOUN
eajbcs-694	731	5	)	)	PUNCT
eajbcs-694	731	6	and	and	CCONJ
eajbcs-694	731	7	(	(	PUNCT
eajbcs-694	731	8	10d	10d	NUM
eajbcs-694	731	9	)	)	PUNCT
eajbcs-694	731	10	,	,	PUNCT
eajbcs-694	731	11	we	we	PRON
eajbcs-694	731	12	obeast	obeast	VERB
eajbcs-694	731	13	afr	afr	PROPN
eajbcs-694	731	14	.	.	PUNCT
eajbcs-694	732	1	j.	j.	PROPN
eajbcs-694	732	2	biophys	biophys	PROPN
eajbcs-694	732	3	.	.	PUNCT
eajbcs-694	733	1	comput	comput	NOUN
eajbcs-694	733	2	.	.	PUNCT
eajbcs-694	734	1	sci	sci	PROPN
eajbcs-694	734	2	.	.	PUNCT
eajbcs-694	735	1	(	(	PUNCT
eajbcs-694	735	2	2024	2024	NUM
eajbcs-694	735	3	)	)	PUNCT
eajbcs-694	735	4	,	,	PUNCT
eajbcs-694	735	5	vol	vol	NOUN
eajbcs-694	735	6	.	.	PROPN
eajbcs-694	735	7	5	5	NUM
eajbcs-694	735	8	,	,	PUNCT
eajbcs-694	735	9	no	no	INTJ
eajbcs-694	735	10	.	.	NOUN
eajbcs-694	735	11	2	2	NUM
eajbcs-694	735	12	,	,	PUNCT
eajbcs-694	735	13	58	58	NUM
eajbcs-694	735	14	-	-	SYM
eajbcs-694	735	15	86	86	NUM
eajbcs-694	735	16	83	83	NUM
eajbcs-694	735	17	0	0	NUM
eajbcs-694	735	18	50	50	NUM
eajbcs-694	735	19	100	100	NUM
eajbcs-694	735	20	150	150	NUM
eajbcs-694	735	21	200	200	NUM
eajbcs-694	735	22	250	250	NUM
eajbcs-694	735	23	300	300	NUM
eajbcs-694	735	24	350	350	NUM
eajbcs-694	735	25	time	time	NOUN
eajbcs-694	735	26	in	in	ADP
eajbcs-694	735	27	days	day	NOUN
eajbcs-694	735	28	2	2	NUM
eajbcs-694	735	29	3	3	NUM
eajbcs-694	735	30	4	4	NUM
eajbcs-694	735	31	5	5	NUM
eajbcs-694	735	32	6	6	NUM
eajbcs-694	735	33	7	7	NUM
eajbcs-694	735	34	8	8	NUM
eajbcs-694	735	35	9	9	NUM
eajbcs-694	735	36	10	10	NUM
eajbcs-694	735	37	co	co	NOUN
eajbcs-694	735	38	st	st	PROPN
eajbcs-694	735	39	of	of	ADP
eajbcs-694	735	40	di	di	X
eajbcs-694	735	41	se	se	PROPN
eajbcs-694	735	42	as	as	ADP
eajbcs-694	735	43	e	e	PROPN
eajbcs-694	735	44	c	c	PROPN
eajbcs-694	735	45	on	on	ADP
eajbcs-694	735	46	tro	tro	X
eajbcs-694	735	47	l	l	NOUN
eajbcs-694	735	48	#	#	SYM
eajbcs-694	735	49	104	104	NUM
eajbcs-694	735	50	optimal	optimal	ADJ
eajbcs-694	735	51	control	control	NOUN
eajbcs-694	735	52	,	,	PUNCT
eajbcs-694	735	53	=	=	NOUN
eajbcs-694	735	54	0.1	0.1	NUM
eajbcs-694	735	55	,	,	PUNCT
eajbcs-694	735	56	=	=	NOUN
eajbcs-694	735	57	0.16667	0.16667	NUM
eajbcs-694	735	58	,	,	PUNCT
eajbcs-694	735	59	=	=	NOUN
eajbcs-694	735	60	0.23333	0.23333	NUM
eajbcs-694	735	61	,	,	PUNCT
eajbcs-694	736	1	=	=	NOUN
eajbcs-694	736	2	0.3	0.3	NUM
eajbcs-694	736	3	(	(	PUNCT
eajbcs-694	736	4	1	1	NUM
eajbcs-694	736	5	)	)	PUNCT
eajbcs-694	736	6	0	0	NUM
eajbcs-694	736	7	50	50	NUM
eajbcs-694	736	8	100	100	NUM
eajbcs-694	736	9	150	150	NUM
eajbcs-694	736	10	200	200	NUM
eajbcs-694	736	11	250	250	NUM
eajbcs-694	736	12	300	300	NUM
eajbcs-694	736	13	350	350	NUM
eajbcs-694	736	14	time	time	NOUN
eajbcs-694	736	15	in	in	ADP
eajbcs-694	736	16	days	day	NOUN
eajbcs-694	736	17	1	1	NUM
eajbcs-694	736	18	2	2	NUM
eajbcs-694	736	19	3	3	NUM
eajbcs-694	736	20	4	4	NUM
eajbcs-694	736	21	5	5	NUM
eajbcs-694	736	22	6	6	NUM
eajbcs-694	736	23	7	7	NUM
eajbcs-694	736	24	ex	ex	X
eajbcs-694	736	25	po	po	X
eajbcs-694	736	26	se	se	PROPN
eajbcs-694	737	1	d	d	PROPN
eajbcs-694	738	1	p	p	X
eajbcs-694	738	2	op	op	NOUN
eajbcs-694	738	3	ula	ula	PROPN
eajbcs-694	738	4	tio	tio	VERB
eajbcs-694	738	5	n	n	CCONJ
eajbcs-694	738	6	#	#	SYM
eajbcs-694	738	7	104	104	NUM
eajbcs-694	738	8	optimal	optimal	ADJ
eajbcs-694	738	9	control	control	NOUN
eajbcs-694	738	10	,	,	PUNCT
eajbcs-694	738	11	=	=	NOUN
eajbcs-694	738	12	0.1	0.1	NUM
eajbcs-694	738	13	,	,	PUNCT
eajbcs-694	738	14	=	=	NOUN
eajbcs-694	738	15	0.16667	0.16667	NUM
eajbcs-694	738	16	,	,	PUNCT
eajbcs-694	738	17	=	=	NOUN
eajbcs-694	738	18	0.23333	0.23333	NUM
eajbcs-694	738	19	,	,	PUNCT
eajbcs-694	738	20	=	=	NOUN
eajbcs-694	738	21	0.3	0.3	NUM
eajbcs-694	738	22	(	(	PUNCT
eajbcs-694	738	23	2	2	NUM
eajbcs-694	738	24	)	)	PUNCT
eajbcs-694	738	25	0	0	NUM
eajbcs-694	739	1	50	50	NUM
eajbcs-694	739	2	100	100	NUM
eajbcs-694	739	3	150	150	NUM
eajbcs-694	739	4	200	200	NUM
eajbcs-694	739	5	250	250	NUM
eajbcs-694	739	6	300	300	NUM
eajbcs-694	739	7	350	350	NUM
eajbcs-694	739	8	time	time	NOUN
eajbcs-694	739	9	in	in	ADP
eajbcs-694	739	10	days	day	NOUN
eajbcs-694	739	11	0	0	NUM
eajbcs-694	739	12	0.5	0.5	NUM
eajbcs-694	739	13	1	1	NUM
eajbcs-694	739	14	1.5	1.5	NUM
eajbcs-694	739	15	2	2	NUM
eajbcs-694	739	16	2.5	2.5	NUM
eajbcs-694	739	17	3	3	NUM
eajbcs-694	739	18	inf	inf	NOUN
eajbcs-694	739	19	ec	ec	PROPN
eajbcs-694	739	20	ted	te	VERB
eajbcs-694	739	21	p	p	PROPN
eajbcs-694	739	22	op	op	NOUN
eajbcs-694	739	23	ula	ula	PROPN
eajbcs-694	739	24	tio	tio	VERB
eajbcs-694	739	25	n	n	CCONJ
eajbcs-694	739	26	#	#	SYM
eajbcs-694	739	27	104	104	NUM
eajbcs-694	739	28	optimal	optimal	ADJ
eajbcs-694	739	29	control	control	NOUN
eajbcs-694	739	30	,	,	PUNCT
eajbcs-694	739	31	=	=	NOUN
eajbcs-694	739	32	0.1	0.1	NUM
eajbcs-694	739	33	,	,	PUNCT
eajbcs-694	739	34	=	=	NOUN
eajbcs-694	739	35	0.16667	0.16667	NUM
eajbcs-694	739	36	,	,	PUNCT
eajbcs-694	739	37	=	=	NOUN
eajbcs-694	739	38	0.23333	0.23333	NUM
eajbcs-694	739	39	,	,	PUNCT
eajbcs-694	740	1	=	=	NOUN
eajbcs-694	740	2	0.3	0.3	NUM
eajbcs-694	740	3	(	(	PUNCT
eajbcs-694	740	4	3	3	NUM
eajbcs-694	740	5	)	)	PUNCT
eajbcs-694	740	6	0	0	NUM
eajbcs-694	740	7	50	50	NUM
eajbcs-694	740	8	100	100	NUM
eajbcs-694	740	9	150	150	NUM
eajbcs-694	740	10	200	200	NUM
eajbcs-694	740	11	250	250	NUM
eajbcs-694	740	12	300	300	NUM
eajbcs-694	740	13	350	350	NUM
eajbcs-694	740	14	time	time	NOUN
eajbcs-694	740	15	in	in	ADP
eajbcs-694	740	16	days	day	NOUN
eajbcs-694	740	17	0	0	NUM
eajbcs-694	740	18	0.1	0.1	NUM
eajbcs-694	740	19	0.2	0.2	NUM
eajbcs-694	740	20	0.3	0.3	NUM
eajbcs-694	740	21	0.4	0.4	NUM
eajbcs-694	740	22	0.5	0.5	NUM
eajbcs-694	740	23	0.6	0.6	NUM
eajbcs-694	740	24	0.7	0.7	NUM
eajbcs-694	740	25	0.8	0.8	NUM
eajbcs-694	740	26	pr	pr	NOUN
eajbcs-694	740	27	op	op	NOUN
eajbcs-694	740	28	or	or	CCONJ
eajbcs-694	740	29	tio	tio	NOUN
eajbcs-694	740	30	n	n	CCONJ
eajbcs-694	740	31	o	o	NOUN
eajbcs-694	740	32	f	f	X
eajbcs-694	740	33	p	p	X
eajbcs-694	740	34	op	op	NOUN
eajbcs-694	740	35	ula	ula	PROPN
eajbcs-694	740	36	tio	tio	PROPN
eajbcs-694	740	37	n	n	CCONJ
eajbcs-694	740	38	h	h	NOUN
eajbcs-694	740	39	os	os	NOUN
eajbcs-694	740	40	pit	pit	PROPN
eajbcs-694	741	1	ali	ali	PROPN
eajbcs-694	741	2	ze	ze	PROPN
eajbcs-694	741	3	d	d	PROPN
eajbcs-694	741	4	-=	-=	NOUN
eajbcs-694	741	5	0.0001	0.0001	NUM
eajbcs-694	741	6	-=0.00015	-=0.00015	PROPN
eajbcs-694	741	7	-=	-=	VERB
eajbcs-694	741	8	0.0002	0.0002	NUM
eajbcs-694	741	9	-=0.00025	-=0.00025	NOUN
eajbcs-694	741	10	(	(	PUNCT
eajbcs-694	741	11	d	d	NOUN
eajbcs-694	741	12	)	)	PUNCT
eajbcs-694	741	13	figure	figure	NOUN
eajbcs-694	741	14	10	10	NUM
eajbcs-694	741	15	:	:	PUNCT
eajbcs-694	741	16	optimal	optimal	ADJ
eajbcs-694	741	17	control	control	NOUN
eajbcs-694	741	18	for	for	ADP
eajbcs-694	741	19	α	α	NOUN
eajbcs-694	741	20	.	.	PUNCT
eajbcs-694	742	1	(	(	PUNCT
eajbcs-694	742	2	a	a	X
eajbcs-694	742	3	)	)	PUNCT
eajbcs-694	742	4	cost	cost	NOUN
eajbcs-694	742	5	comparison	comparison	NOUN
eajbcs-694	742	6	for	for	ADP
eajbcs-694	742	7	optimal	optimal	ADJ
eajbcs-694	742	8	and	and	CCONJ
eajbcs-694	742	9	constant	constant	ADJ
eajbcs-694	742	10	controls	control	NOUN
eajbcs-694	742	11	for	for	ADP
eajbcs-694	742	12	α	α	NOUN
eajbcs-694	742	13	.	.	PUNCT
eajbcs-694	743	1	(	(	PUNCT
eajbcs-694	743	2	b	b	NOUN
eajbcs-694	743	3	)	)	PUNCT
eajbcs-694	743	4	exposed	expose	VERB
eajbcs-694	743	5	population	population	NOUN
eajbcs-694	743	6	comparison	comparison	NOUN
eajbcs-694	743	7	for	for	ADP
eajbcs-694	743	8	optimal	optimal	ADJ
eajbcs-694	743	9	and	and	CCONJ
eajbcs-694	743	10	constant	constant	ADJ
eajbcs-694	743	11	controls	control	NOUN
eajbcs-694	743	12	for	for	ADP
eajbcs-694	743	13	α	α	NOUN
eajbcs-694	743	14	.	.	PUNCT
eajbcs-694	744	1	(	(	PUNCT
eajbcs-694	744	2	c	c	X
eajbcs-694	744	3	)	)	PUNCT
eajbcs-694	744	4	infected	infected	ADJ
eajbcs-694	744	5	population	population	NOUN
eajbcs-694	744	6	comparison	comparison	NOUN
eajbcs-694	744	7	for	for	ADP
eajbcs-694	744	8	optimal	optimal	ADJ
eajbcs-694	744	9	and	and	CCONJ
eajbcs-694	744	10	constant	constant	ADJ
eajbcs-694	744	11	controls	control	NOUN
eajbcs-694	744	12	for	for	ADP
eajbcs-694	744	13	α	α	NOUN
eajbcs-694	744	14	.	.	PUNCT
eajbcs-694	745	1	(	(	PUNCT
eajbcs-694	745	2	d	d	X
eajbcs-694	745	3	)	)	PUNCT
eajbcs-694	745	4	optimal	optimal	ADJ
eajbcs-694	745	5	strategies	strategy	NOUN
eajbcs-694	745	6	for	for	ADP
eajbcs-694	745	7	α	α	NOUN
eajbcs-694	745	8	with	with	ADP
eajbcs-694	745	9	different	different	ADJ
eajbcs-694	745	10	β	β	NOUN
eajbcs-694	745	11	.	.	PUNCT
eajbcs-694	745	12	serve	serve	VERB
eajbcs-694	745	13	that	that	PRON
eajbcs-694	745	14	for	for	ADP
eajbcs-694	745	15	both	both	CCONJ
eajbcs-694	745	16	protection	protection	NOUN
eajbcs-694	745	17	and	and	CCONJ
eajbcs-694	745	18	hospitalization	hospitalization	NOUN
eajbcs-694	745	19	the	the	DET
eajbcs-694	745	20	maximum	maximum	ADJ
eajbcs-694	745	21	possible	possible	ADJ
eajbcs-694	745	22	protection	protection	NOUN
eajbcs-694	745	23	and	and	CCONJ
eajbcs-694	745	24	hospitalization	hospitalization	NOUN
eajbcs-694	745	25	rates	rate	NOUN
eajbcs-694	745	26	should	should	AUX
eajbcs-694	745	27	be	be	AUX
eajbcs-694	745	28	maintained	maintain	VERB
eajbcs-694	745	29	to	to	ADP
eajbcs-694	745	30	the	the	DET
eajbcs-694	745	31	first	first	ADJ
eajbcs-694	745	32	few	few	ADJ
eajbcs-694	745	33	days	day	NOUN
eajbcs-694	745	34	of	of	ADP
eajbcs-694	745	35	the	the	DET
eajbcs-694	745	36	disease	disease	NOUN
eajbcs-694	745	37	which	which	PRON
eajbcs-694	745	38	can	can	AUX
eajbcs-694	745	39	then	then	ADV
eajbcs-694	745	40	be	be	AUX
eajbcs-694	745	41	reduced	reduce	VERB
eajbcs-694	745	42	over	over	ADP
eajbcs-694	745	43	time	time	NOUN
eajbcs-694	745	44	.	.	PUNCT
eajbcs-694	746	1	finally	finally	ADV
eajbcs-694	746	2	,	,	PUNCT
eajbcs-694	746	3	from	from	ADP
eajbcs-694	746	4	figures	figure	NOUN
eajbcs-694	746	5	(	(	PUNCT
eajbcs-694	746	6	112	112	NUM
eajbcs-694	746	7	)	)	PUNCT
eajbcs-694	746	8	and	and	CCONJ
eajbcs-694	746	9	(	(	PUNCT
eajbcs-694	746	10	113	113	NUM
eajbcs-694	746	11	)	)	PUNCT
eajbcs-694	746	12	,	,	PUNCT
eajbcs-694	746	13	one	one	PRON
eajbcs-694	746	14	can	can	AUX
eajbcs-694	746	15	easily	easily	ADV
eajbcs-694	746	16	conclude	conclude	VERB
eajbcs-694	746	17	that	that	SCONJ
eajbcs-694	746	18	the	the	DET
eajbcs-694	746	19	combination	combination	NOUN
eajbcs-694	746	20	of	of	ADP
eajbcs-694	746	21	the	the	DET
eajbcs-694	746	22	two	two	NUM
eajbcs-694	746	23	controls	control	NOUN
eajbcs-694	746	24	is	be	AUX
eajbcs-694	746	25	significantly	significantly	ADV
eajbcs-694	746	26	more	more	ADV
eajbcs-694	746	27	effective	effective	ADJ
eajbcs-694	746	28	in	in	ADP
eajbcs-694	746	29	reducing	reduce	VERB
eajbcs-694	746	30	the	the	DET
eajbcs-694	746	31	spread	spread	NOUN
eajbcs-694	746	32	of	of	ADP
eajbcs-694	746	33	the	the	DET
eajbcs-694	746	34	virus	virus	NOUN
eajbcs-694	746	35	than	than	ADP
eajbcs-694	746	36	when	when	SCONJ
eajbcs-694	746	37	each	each	DET
eajbcs-694	746	38	control	control	NOUN
eajbcs-694	746	39	is	be	AUX
eajbcs-694	746	40	singly	singly	ADV
eajbcs-694	746	41	applied	apply	VERB
eajbcs-694	746	42	.	.	PUNCT
eajbcs-694	747	1	hence	hence	ADV
eajbcs-694	747	2	,	,	PUNCT
eajbcs-694	747	3	the	the	DET
eajbcs-694	747	4	best	good	ADJ
eajbcs-694	747	5	choice	choice	NOUN
eajbcs-694	747	6	is	be	AUX
eajbcs-694	747	7	to	to	PART
eajbcs-694	747	8	apply	apply	VERB
eajbcs-694	747	9	two	two	NUM
eajbcs-694	747	10	controls	control	NOUN
eajbcs-694	747	11	all	all	ADV
eajbcs-694	747	12	together	together	ADV
eajbcs-694	747	13	to	to	PART
eajbcs-694	747	14	mitigate	mitigate	VERB
eajbcs-694	747	15	the	the	DET
eajbcs-694	747	16	spread	spread	NOUN
eajbcs-694	747	17	of	of	ADP
eajbcs-694	747	18	covid-19	covid-19	PROPN
eajbcs-694	747	19	in	in	ADP
eajbcs-694	747	20	the	the	DET
eajbcs-694	747	21	population	population	NOUN
eajbcs-694	747	22	.	.	PUNCT
eajbcs-694	748	1	from	from	ADP
eajbcs-694	748	2	figure	figure	NOUN
eajbcs-694	748	3	(	(	PUNCT
eajbcs-694	748	4	111	111	NUM
eajbcs-694	748	5	)	)	PUNCT
eajbcs-694	748	6	,	,	PUNCT
eajbcs-694	748	7	one	one	PRON
eajbcs-694	748	8	can	can	AUX
eajbcs-694	748	9	observe	observe	VERB
eajbcs-694	748	10	that	that	SCONJ
eajbcs-694	748	11	the	the	DET
eajbcs-694	748	12	combined	combined	ADJ
eajbcs-694	748	13	implementation	implementation	NOUN
eajbcs-694	748	14	of	of	ADP
eajbcs-694	748	15	the	the	DET
eajbcs-694	748	16	two	two	NUM
eajbcs-694	748	17	control	control	NOUN
eajbcs-694	748	18	measures	measure	NOUN
eajbcs-694	748	19	is	be	AUX
eajbcs-694	748	20	the	the	DET
eajbcs-694	748	21	most	most	ADV
eajbcs-694	748	22	cost	cost	NOUN
eajbcs-694	748	23	-	-	PUNCT
eajbcs-694	748	24	effective	effective	ADJ
eajbcs-694	748	25	when	when	SCONJ
eajbcs-694	748	26	compared	compare	VERB
eajbcs-694	748	27	with	with	ADP
eajbcs-694	748	28	the	the	DET
eajbcs-694	748	29	single	single	ADJ
eajbcs-694	748	30	implementation	implementation	NOUN
eajbcs-694	748	31	of	of	ADP
eajbcs-694	748	32	each	each	DET
eajbcs-694	748	33	control	control	NOUN
eajbcs-694	748	34	measure	measure	NOUN
eajbcs-694	748	35	.	.	PUNCT
eajbcs-694	749	1	7	7	X
eajbcs-694	749	2	.	.	X
eajbcs-694	749	3	conclusion	conclusion	NOUN
eajbcs-694	749	4	in	in	ADP
eajbcs-694	749	5	this	this	DET
eajbcs-694	749	6	paper	paper	NOUN
eajbcs-694	749	7	,	,	PUNCT
eajbcs-694	749	8	we	we	PRON
eajbcs-694	749	9	proposed	propose	VERB
eajbcs-694	749	10	a	a	DET
eajbcs-694	749	11	deterministic	deterministic	ADJ
eajbcs-694	749	12	compartmental	compartmental	ADJ
eajbcs-694	749	13	model	model	NOUN
eajbcs-694	749	14	to	to	PART
eajbcs-694	749	15	study	study	VERB
eajbcs-694	749	16	the	the	DET
eajbcs-694	749	17	transmission	transmission	NOUN
eajbcs-694	749	18	dynamics	dynamic	NOUN
eajbcs-694	749	19	of	of	ADP
eajbcs-694	749	20	covid-19	covid-19	PROPN
eajbcs-694	749	21	.	.	PUNCT
eajbcs-694	750	1	the	the	DET
eajbcs-694	750	2	model	model	NOUN
eajbcs-694	750	3	was	be	AUX
eajbcs-694	750	4	an	an	DET
eajbcs-694	750	5	extension	extension	NOUN
eajbcs-694	750	6	of	of	ADP
eajbcs-694	750	7	the	the	DET
eajbcs-694	750	8	existing	exist	VERB
eajbcs-694	750	9	seir	seir	NOUN
eajbcs-694	750	10	model	model	NOUN
eajbcs-694	750	11	by	by	ADP
eajbcs-694	750	12	including	include	VERB
eajbcs-694	750	13	protected	protect	VERB
eajbcs-694	750	14	and	and	CCONJ
eajbcs-694	750	15	hospitalized	hospitalize	VERB
eajbcs-694	750	16	individuals	individual	NOUN
eajbcs-694	750	17	.	.	PUNCT
eajbcs-694	751	1	we	we	PRON
eajbcs-694	751	2	established	establish	VERB
eajbcs-694	751	3	the	the	DET
eajbcs-694	751	4	well	well	NOUN
eajbcs-694	751	5	-	-	PUNCT
eajbcs-694	751	6	posedness	posedness	NOUN
eajbcs-694	751	7	of	of	ADP
eajbcs-694	751	8	the	the	DET
eajbcs-694	751	9	modified	modify	VERB
eajbcs-694	751	10	model	model	NOUN
eajbcs-694	751	11	by	by	ADP
eajbcs-694	751	12	proving	prove	VERB
eajbcs-694	751	13	the	the	DET
eajbcs-694	751	14	existence	existence	NOUN
eajbcs-694	751	15	,	,	PUNCT
eajbcs-694	751	16	positivity	positivity	NOUN
eajbcs-694	751	17	,	,	PUNCT
eajbcs-694	751	18	and	and	CCONJ
eajbcs-694	751	19	boundedness	boundedness	NOUN
eajbcs-694	751	20	of	of	ADP
eajbcs-694	751	21	the	the	DET
eajbcs-694	751	22	solutions	solution	NOUN
eajbcs-694	751	23	.	.	PUNCT
eajbcs-694	752	1	we	we	PRON
eajbcs-694	752	2	computed	compute	VERB
eajbcs-694	752	3	the	the	DET
eajbcs-694	752	4	steady	steady	ADJ
eajbcs-694	752	5	states	state	NOUN
eajbcs-694	752	6	and	and	CCONJ
eajbcs-694	752	7	the	the	DET
eajbcs-694	752	8	basic	basic	ADJ
eajbcs-694	752	9	reproduction	reproduction	NOUN
eajbcs-694	752	10	number	number	NOUN
eajbcs-694	752	11	r0	r0	NOUN
eajbcs-694	752	12	.	.	PUNCT
eajbcs-694	753	1	based	base	VERB
eajbcs-694	753	2	on	on	ADP
eajbcs-694	753	3	the	the	DET
eajbcs-694	753	4	reproduction	reproduction	NOUN
eajbcs-694	753	5	number	number	NOUN
eajbcs-694	753	6	r0	r0	NOUN
eajbcs-694	753	7	,	,	PUNCT
eajbcs-694	753	8	it	it	PRON
eajbcs-694	753	9	is	be	AUX
eajbcs-694	753	10	revealed	reveal	VERB
eajbcs-694	753	11	that	that	SCONJ
eajbcs-694	753	12	whenever	whenever	SCONJ
eajbcs-694	753	13	r0	r0	NOUN
eajbcs-694	753	14	<	<	X
eajbcs-694	753	15	1	1	NUM
eajbcs-694	753	16	,	,	PUNCT
eajbcs-694	753	17	the	the	DET
eajbcs-694	753	18	system	system	NOUN
eajbcs-694	753	19	has	have	VERB
eajbcs-694	753	20	only	only	ADV
eajbcs-694	753	21	disease	disease	NOUN
eajbcs-694	753	22	free	free	ADJ
eajbcs-694	753	23	equilibrium	equilibrium	NOUN
eajbcs-694	753	24	e0	e0	PROPN
eajbcs-694	753	25	which	which	PRON
eajbcs-694	753	26	is	be	AUX
eajbcs-694	753	27	locally	locally	ADV
eajbcs-694	753	28	as	as	ADV
eajbcs-694	753	29	well	well	ADV
eajbcs-694	753	30	as	as	ADP
eajbcs-694	753	31	globally	globally	ADV
eajbcs-694	753	32	asymptotically	asymptotically	ADV
eajbcs-694	753	33	stable	stable	ADJ
eajbcs-694	753	34	.	.	PUNCT
eajbcs-694	754	1	when	when	SCONJ
eajbcs-694	754	2	r0	r0	NOUN
eajbcs-694	754	3	>	>	X
eajbcs-694	754	4	1	1	NUM
eajbcs-694	754	5	,	,	PUNCT
eajbcs-694	754	6	the	the	DET
eajbcs-694	754	7	system	system	NOUN
eajbcs-694	754	8	has	have	VERB
eajbcs-694	754	9	a	a	DET
eajbcs-694	754	10	unique	unique	ADJ
eajbcs-694	754	11	endemic	endemic	ADJ
eajbcs-694	754	12	equilibrium	equilibrium	NOUN
eajbcs-694	754	13	e1	e1	NOUN
eajbcs-694	754	14	which	which	PRON
eajbcs-694	754	15	is	be	AUX
eajbcs-694	754	16	locally	locally	ADV
eajbcs-694	754	17	stable	stable	ADJ
eajbcs-694	754	18	and	and	CCONJ
eajbcs-694	754	19	the	the	DET
eajbcs-694	754	20	disease	disease	NOUN
eajbcs-694	754	21	free	free	ADJ
eajbcs-694	754	22	equilibrium	equilibrium	NOUN
eajbcs-694	754	23	e0	e0	PROPN
eajbcs-694	754	24	becomes	become	VERB
eajbcs-694	754	25	unstable	unstable	ADJ
eajbcs-694	754	26	.	.	PUNCT
eajbcs-694	755	1	we	we	PRON
eajbcs-694	755	2	have	have	AUX
eajbcs-694	755	3	observed	observe	VERB
eajbcs-694	755	4	that	that	SCONJ
eajbcs-694	755	5	the	the	DET
eajbcs-694	755	6	outbreak	outbreak	NOUN
eajbcs-694	755	7	of	of	ADP
eajbcs-694	755	8	the	the	DET
eajbcs-694	755	9	disease	disease	NOUN
eajbcs-694	755	10	dies	die	VERB
eajbcs-694	755	11	out	out	ADP
eajbcs-694	755	12	if	if	SCONJ
eajbcs-694	755	13	r0	r0	NOUN
eajbcs-694	755	14	<	<	X
eajbcs-694	755	15	1	1	NUM
eajbcs-694	755	16	and	and	CCONJ
eajbcs-694	755	17	the	the	DET
eajbcs-694	755	18	disease	disease	NOUN
eajbcs-694	755	19	is	be	AUX
eajbcs-694	755	20	endemic	endemic	ADJ
eajbcs-694	755	21	if	if	SCONJ
eajbcs-694	755	22	r0	r0	NOUN
eajbcs-694	755	23	>	>	X
eajbcs-694	756	1	1	1	X
eajbcs-694	756	2	.	.	X
eajbcs-694	756	3	using	use	VERB
eajbcs-694	756	4	center	center	ADJ
eajbcs-694	756	5	manifold	manifold	ADJ
eajbcs-694	756	6	theory	theory	NOUN
eajbcs-694	756	7	,	,	PUNCT
eajbcs-694	756	8	bifurcation	bifurcation	NOUN
eajbcs-694	756	9	analysis	analysis	NOUN
eajbcs-694	756	10	of	of	ADP
eajbcs-694	756	11	the	the	DET
eajbcs-694	756	12	modified	modify	VERB
eajbcs-694	756	13	model	model	NOUN
eajbcs-694	756	14	was	be	AUX
eajbcs-694	756	15	proven	prove	VERB
eajbcs-694	756	16	and	and	CCONJ
eajbcs-694	756	17	the	the	DET
eajbcs-694	756	18	model	model	NOUN
eajbcs-694	756	19	exhibits	exhibit	VERB
eajbcs-694	756	20	forward	forward	ADJ
eajbcs-694	756	21	bifurcation	bifurcation	NOUN
eajbcs-694	756	22	at	at	ADP
eajbcs-694	756	23	r0	r0	NOUN
eajbcs-694	756	24	=	=	NOUN
eajbcs-694	756	25	1	1	X
eajbcs-694	756	26	.	.	PUNCT
eajbcs-694	757	1	in	in	ADP
eajbcs-694	757	2	addition	addition	NOUN
eajbcs-694	757	3	from	from	ADP
eajbcs-694	757	4	sensitivity	sensitivity	NOUN
eajbcs-694	757	5	analysis	analysis	NOUN
eajbcs-694	757	6	of	of	ADP
eajbcs-694	757	7	r0	r0	NOUN
eajbcs-694	757	8	,	,	PUNCT
eajbcs-694	757	9	we	we	PRON
eajbcs-694	757	10	observed	observe	VERB
eajbcs-694	757	11	that	that	SCONJ
eajbcs-694	757	12	the	the	DET
eajbcs-694	757	13	recruitment	recruitment	NOUN
eajbcs-694	757	14	rate	rate	NOUN
eajbcs-694	757	15	π	π	PROPN
eajbcs-694	757	16	and	and	CCONJ
eajbcs-694	757	17	east	east	PROPN
eajbcs-694	757	18	afr	afr	PROPN
eajbcs-694	757	19	.	.	PUNCT
eajbcs-694	758	1	j.	j.	PROPN
eajbcs-694	758	2	biophys	biophys	PROPN
eajbcs-694	758	3	.	.	PUNCT
eajbcs-694	759	1	comput	comput	NOUN
eajbcs-694	759	2	.	.	PUNCT
eajbcs-694	760	1	sci	sci	PROPN
eajbcs-694	760	2	.	.	PUNCT
eajbcs-694	761	1	(	(	PUNCT
eajbcs-694	761	2	2024	2024	NUM
eajbcs-694	761	3	)	)	PUNCT
eajbcs-694	761	4	,	,	PUNCT
eajbcs-694	761	5	vol	vol	NOUN
eajbcs-694	761	6	.	.	PROPN
eajbcs-694	761	7	5	5	NUM
eajbcs-694	761	8	,	,	PUNCT
eajbcs-694	761	9	no	no	INTJ
eajbcs-694	761	10	.	.	NOUN
eajbcs-694	761	11	2	2	NUM
eajbcs-694	761	12	,	,	PUNCT
eajbcs-694	761	13	58	58	NUM
eajbcs-694	761	14	-	-	SYM
eajbcs-694	761	15	86	86	NUM
eajbcs-694	761	16	84	84	NUM
eajbcs-694	761	17	0	0	NUM
eajbcs-694	761	18	50	50	NUM
eajbcs-694	761	19	100	100	NUM
eajbcs-694	761	20	150	150	NUM
eajbcs-694	761	21	200	200	NUM
eajbcs-694	761	22	250	250	NUM
eajbcs-694	761	23	300	300	NUM
eajbcs-694	761	24	350	350	NUM
eajbcs-694	761	25	time	time	NOUN
eajbcs-694	761	26	in	in	ADP
eajbcs-694	761	27	days	day	NOUN
eajbcs-694	761	28	2	2	NUM
eajbcs-694	761	29	2.5	2.5	NUM
eajbcs-694	761	30	3	3	NUM
eajbcs-694	761	31	3.5	3.5	NUM
eajbcs-694	761	32	4	4	NUM
eajbcs-694	761	33	4.5	4.5	NUM
eajbcs-694	761	34	5	5	NUM
eajbcs-694	761	35	5.5	5.5	NUM
eajbcs-694	761	36	6	6	NUM
eajbcs-694	761	37	6.5	6.5	NUM
eajbcs-694	761	38	7	7	NUM
eajbcs-694	761	39	co	co	NOUN
eajbcs-694	761	40	st	st	PROPN
eajbcs-694	761	41	of	of	ADP
eajbcs-694	761	42	di	di	X
eajbcs-694	761	43	se	se	PROPN
eajbcs-694	761	44	as	as	ADP
eajbcs-694	761	45	e	e	PROPN
eajbcs-694	761	46	c	c	PROPN
eajbcs-694	761	47	on	on	ADP
eajbcs-694	761	48	tro	tro	X
eajbcs-694	761	49	l	l	NOUN
eajbcs-694	761	50	#	#	SYM
eajbcs-694	761	51	104	104	NUM
eajbcs-694	761	52	both	both	CCONJ
eajbcs-694	761	53	optimal	optimal	ADJ
eajbcs-694	761	54	only	only	ADV
eajbcs-694	761	55	,	,	PUNCT
eajbcs-694	761	56	optimal	optimal	ADJ
eajbcs-694	761	57	only	only	ADV
eajbcs-694	761	58	3	3	NUM
eajbcs-694	761	59	optimal	optimal	ADJ
eajbcs-694	761	60	(	(	PUNCT
eajbcs-694	761	61	1	1	NUM
eajbcs-694	761	62	)	)	PUNCT
eajbcs-694	761	63	0	0	NUM
eajbcs-694	762	1	50	50	NUM
eajbcs-694	762	2	100	100	NUM
eajbcs-694	762	3	150	150	NUM
eajbcs-694	762	4	200	200	NUM
eajbcs-694	762	5	250	250	NUM
eajbcs-694	762	6	300	300	NUM
eajbcs-694	762	7	350	350	NUM
eajbcs-694	762	8	time	time	NOUN
eajbcs-694	762	9	in	in	ADP
eajbcs-694	762	10	days	day	NOUN
eajbcs-694	762	11	1	1	NUM
eajbcs-694	762	12	2	2	NUM
eajbcs-694	762	13	3	3	NUM
eajbcs-694	762	14	4	4	NUM
eajbcs-694	762	15	5	5	NUM
eajbcs-694	762	16	6	6	NUM
eajbcs-694	762	17	7	7	NUM
eajbcs-694	762	18	ex	ex	X
eajbcs-694	762	19	po	po	X
eajbcs-694	762	20	se	se	PROPN
eajbcs-694	763	1	d	d	PROPN
eajbcs-694	764	1	p	p	X
eajbcs-694	764	2	op	op	NOUN
eajbcs-694	764	3	ula	ula	PROPN
eajbcs-694	764	4	tio	tio	VERB
eajbcs-694	764	5	n	n	CCONJ
eajbcs-694	764	6	#	#	SYM
eajbcs-694	764	7	104	104	NUM
eajbcs-694	764	8	both	both	CCONJ
eajbcs-694	764	9	optimal	optimal	ADJ
eajbcs-694	764	10	only	only	ADV
eajbcs-694	764	11	,	,	PUNCT
eajbcs-694	764	12	optimal	optimal	ADJ
eajbcs-694	764	13	only	only	ADV
eajbcs-694	764	14	3	3	NUM
eajbcs-694	764	15	optimal	optimal	ADJ
eajbcs-694	764	16	(	(	PUNCT
eajbcs-694	764	17	2	2	NUM
eajbcs-694	764	18	)	)	PUNCT
eajbcs-694	764	19	0	0	NUM
eajbcs-694	765	1	50	50	NUM
eajbcs-694	765	2	100	100	NUM
eajbcs-694	765	3	150	150	NUM
eajbcs-694	765	4	200	200	NUM
eajbcs-694	765	5	250	250	NUM
eajbcs-694	765	6	300	300	NUM
eajbcs-694	765	7	350	350	NUM
eajbcs-694	765	8	time	time	NOUN
eajbcs-694	765	9	in	in	ADP
eajbcs-694	765	10	days	day	NOUN
eajbcs-694	765	11	0	0	NUM
eajbcs-694	765	12	0.5	0.5	NUM
eajbcs-694	765	13	1	1	NUM
eajbcs-694	765	14	1.5	1.5	NUM
eajbcs-694	765	15	2	2	NUM
eajbcs-694	765	16	2.5	2.5	NUM
eajbcs-694	765	17	3	3	NUM
eajbcs-694	765	18	3.5	3.5	NUM
eajbcs-694	765	19	inf	inf	NOUN
eajbcs-694	765	20	ec	ec	PROPN
eajbcs-694	765	21	ted	te	VERB
eajbcs-694	765	22	p	p	PROPN
eajbcs-694	765	23	op	op	NOUN
eajbcs-694	765	24	ula	ula	PROPN
eajbcs-694	765	25	tio	tio	VERB
eajbcs-694	765	26	n	n	CCONJ
eajbcs-694	765	27	#	#	SYM
eajbcs-694	765	28	104	104	NUM
eajbcs-694	765	29	both	both	CCONJ
eajbcs-694	765	30	optimal	optimal	ADJ
eajbcs-694	765	31	only	only	ADV
eajbcs-694	765	32	,	,	PUNCT
eajbcs-694	765	33	optimal	optimal	ADJ
eajbcs-694	765	34	only	only	ADV
eajbcs-694	765	35	3	3	NUM
eajbcs-694	765	36	optimal	optimal	ADJ
eajbcs-694	765	37	(	(	PUNCT
eajbcs-694	765	38	3	3	NUM
eajbcs-694	765	39	)	)	PUNCT
eajbcs-694	765	40	figure	figure	NOUN
eajbcs-694	765	41	11	11	NUM
eajbcs-694	765	42	:	:	PUNCT
eajbcs-694	765	43	optimal	optimal	ADJ
eajbcs-694	765	44	control	control	NOUN
eajbcs-694	765	45	for	for	ADP
eajbcs-694	765	46	θ	θ	PROPN
eajbcs-694	765	47	,	,	PUNCT
eajbcs-694	765	48	α	α	NOUN
eajbcs-694	765	49	and	and	CCONJ
eajbcs-694	765	50	both	both	PRON
eajbcs-694	765	51	.	.	PUNCT
eajbcs-694	766	1	(	(	PUNCT
eajbcs-694	766	2	a	a	X
eajbcs-694	766	3	)	)	PUNCT
eajbcs-694	766	4	cost	cost	NOUN
eajbcs-694	766	5	comparison	comparison	NOUN
eajbcs-694	766	6	for	for	ADP
eajbcs-694	766	7	optimal	optimal	ADJ
eajbcs-694	766	8	θ	θ	PROPN
eajbcs-694	766	9	,	,	PUNCT
eajbcs-694	766	10	α	α	PROPN
eajbcs-694	766	11	(	(	PUNCT
eajbcs-694	766	12	b	b	NOUN
eajbcs-694	766	13	)	)	PUNCT
eajbcs-694	766	14	exposedand	exposedand	NOUN
eajbcs-694	766	15	both	both	PRON
eajbcs-694	766	16	.	.	PUNCT
eajbcs-694	767	1	population	population	NOUN
eajbcs-694	767	2	comparison	comparison	NOUN
eajbcs-694	767	3	.	.	PUNCT
eajbcs-694	768	1	(	(	PUNCT
eajbcs-694	768	2	c	c	X
eajbcs-694	768	3	)	)	PUNCT
eajbcs-694	768	4	infected	infected	ADJ
eajbcs-694	768	5	population	population	NOUN
eajbcs-694	768	6	comparison	comparison	NOUN
eajbcs-694	768	7	.	.	PUNCT
eajbcs-694	769	1	references	reference	NOUN
eajbcs-694	769	2	ahmed	ahmed	PROPN
eajbcs-694	769	3	i.	i.	PROPN
eajbcs-694	769	4	,	,	PUNCT
eajbcs-694	769	5	modu	modu	PROPN
eajbcs-694	769	6	g.	g.	PROPN
eajbcs-694	769	7	u.	u.	PROPN
eajbcs-694	769	8	,	,	PUNCT
eajbcs-694	769	9	yusuf	yusuf	PROPN
eajbcs-694	769	10	a.	a.	PROPN
eajbcs-694	769	11	,	,	PUNCT
eajbcs-694	769	12	kumam	kumam	PROPN
eajbcs-694	769	13	p.	p.	PROPN
eajbcs-694	769	14	and	and	CCONJ
eajbcs-694	769	15	yusuf	yusuf	PROPN
eajbcs-694	769	16	i.	i.	PROPN
eajbcs-694	769	17	(	(	PUNCT
eajbcs-694	769	18	2021	2021	NUM
eajbcs-694	769	19	)	)	PUNCT
eajbcs-694	769	20	,	,	PUNCT
eajbcs-694	769	21	‘	'	PUNCT
eajbcs-694	769	22	a	a	DET
eajbcs-694	769	23	mathematical	mathematical	ADJ
eajbcs-694	769	24	model	model	NOUN
eajbcs-694	769	25	of	of	ADP
eajbcs-694	769	26	coronavirus	coronavirus	NOUN
eajbcs-694	769	27	disease	disease	NOUN
eajbcs-694	769	28	(	(	PUNCT
eajbcs-694	769	29	covid-19	covid-19	PROPN
eajbcs-694	769	30	)	)	PUNCT
eajbcs-694	769	31	containing	contain	VERB
eajbcs-694	769	32	asymptomatic	asymptomatic	ADJ
eajbcs-694	769	33	and	and	CCONJ
eajbcs-694	769	34	symptomatic	symptomatic	ADJ
eajbcs-694	769	35	classes	class	NOUN
eajbcs-694	769	36	’	'	PUNCT
eajbcs-694	769	37	,	,	PUNCT
eajbcs-694	769	38	results	result	NOUN
eajbcs-694	769	39	in	in	ADP
eajbcs-694	769	40	physics	physics	NOUN
eajbcs-694	769	41	21	21	NUM
eajbcs-694	769	42	,	,	PUNCT
eajbcs-694	769	43	103776	103776	NUM
eajbcs-694	769	44	.	.	PUNCT
eajbcs-694	770	1	alemneh	alemneh	PROPN
eajbcs-694	770	2	h.	h.	PROPN
eajbcs-694	770	3	t.	t.	PROPN
eajbcs-694	770	4	and	and	CCONJ
eajbcs-694	770	5	telahun	telahun	PROPN
eajbcs-694	770	6	g.	g.	PROPN
eajbcs-694	770	7	t.	t.	PROPN
eajbcs-694	770	8	(	(	PUNCT
eajbcs-694	770	9	2020	2020	NUM
eajbcs-694	770	10	)	)	PUNCT
eajbcs-694	770	11	,	,	PUNCT
eajbcs-694	770	12	‘	'	PUNCT
eajbcs-694	770	13	mathematical	mathematical	ADJ
eajbcs-694	770	14	modeling	modeling	NOUN
eajbcs-694	770	15	and	and	CCONJ
eajbcs-694	770	16	optimal	optimal	ADJ
eajbcs-694	770	17	control	control	NOUN
eajbcs-694	770	18	analysis	analysis	NOUN
eajbcs-694	770	19	of	of	ADP
eajbcs-694	770	20	covid-19	covid-19	PROPN
eajbcs-694	770	21	in	in	ADP
eajbcs-694	770	22	ethiopia	ethiopia	PROPN
eajbcs-694	770	23	’	'	PUNCT
eajbcs-694	770	24	,	,	PUNCT
eajbcs-694	770	25	medrxiv	medrxiv	PROPN
eajbcs-694	770	26	.	.	PUNCT
eajbcs-694	771	1	allen	allen	PROPN
eajbcs-694	771	2	l.	l.	PROPN
eajbcs-694	771	3	,	,	PUNCT
eajbcs-694	771	4	brauer	brauer	PROPN
eajbcs-694	771	5	f.	f.	PROPN
eajbcs-694	771	6	,	,	PUNCT
eajbcs-694	771	7	van	van	PROPN
eajbcs-694	771	8	den	den	PROPN
eajbcs-694	771	9	driessche	driessche	PROPN
eajbcs-694	771	10	p.	p.	PROPN
eajbcs-694	771	11	and	and	CCONJ
eajbcs-694	771	12	wu	wu	PROPN
eajbcs-694	771	13	,	,	PUNCT
eajbcs-694	771	14	j.	j.	PROPN
eajbcs-694	771	15	(	(	PUNCT
eajbcs-694	771	16	2008	2008	NUM
eajbcs-694	771	17	)	)	PUNCT
eajbcs-694	771	18	,	,	PUNCT
eajbcs-694	771	19	‘	'	PUNCT
eajbcs-694	771	20	mathematical	mathematical	ADJ
eajbcs-694	771	21	epidemiology	epidemiology	NOUN
eajbcs-694	771	22	,	,	PUNCT
eajbcs-694	771	23	volume	volume	NOUN
eajbcs-694	771	24	1945	1945	NUM
eajbcs-694	771	25	of	of	ADP
eajbcs-694	771	26	lecture	lecture	NOUN
eajbcs-694	771	27	notes	note	NOUN
eajbcs-694	771	28	in	in	ADP
eajbcs-694	771	29	mathematics	mathematic	NOUN
eajbcs-694	771	30	’	'	PUNCT
eajbcs-694	771	31	,	,	PUNCT
eajbcs-694	771	32	springer	springer	NOUN
eajbcs-694	771	33	,	,	PUNCT
eajbcs-694	771	34	berlin	berlin	PROPN
eajbcs-694	771	35	13	13	NUM
eajbcs-694	771	36	,	,	PUNCT
eajbcs-694	771	37	14	14	NUM
eajbcs-694	771	38	.	.	PUNCT
eajbcs-694	772	1	bachar	bachar	PROPN
eajbcs-694	772	2	m.	m.	PROPN
eajbcs-694	772	3	,	,	PUNCT
eajbcs-694	772	4	khamsi	khamsi	PROPN
eajbcs-694	772	5	m.	m.	NOUN
eajbcs-694	772	6	a.	a.	PROPN
eajbcs-694	772	7	and	and	CCONJ
eajbcs-694	772	8	bounkhel	bounkhel	PROPN
eajbcs-694	772	9	m.	m.	PROPN
eajbcs-694	772	10	(	(	PUNCT
eajbcs-694	772	11	2021	2021	NUM
eajbcs-694	772	12	)	)	PUNCT
eajbcs-694	772	13	,	,	PUNCT
eajbcs-694	772	14	‘	'	PUNCT
eajbcs-694	772	15	a	a	DET
eajbcs-694	772	16	mathematical	mathematical	ADJ
eajbcs-694	772	17	model	model	NOUN
eajbcs-694	772	18	for	for	ADP
eajbcs-694	772	19	the	the	DET
eajbcs-694	772	20	spread	spread	NOUN
eajbcs-694	772	21	of	of	ADP
eajbcs-694	772	22	covid-19	covid-19	PROPN
eajbcs-694	772	23	and	and	CCONJ
eajbcs-694	772	24	control	control	NOUN
eajbcs-694	772	25	mechanisms	mechanism	NOUN
eajbcs-694	772	26	in	in	ADP
eajbcs-694	772	27	saudi	saudi	PROPN
eajbcs-694	772	28	arabia	arabia	PROPN
eajbcs-694	772	29	’	'	PUNCT
eajbcs-694	772	30	,	,	PUNCT
eajbcs-694	772	31	advances	advance	NOUN
eajbcs-694	772	32	in	in	ADP
eajbcs-694	772	33	difference	difference	NOUN
eajbcs-694	772	34	equations	equation	NOUN
eajbcs-694	772	35	2021(1	2021(1	NUM
eajbcs-694	772	36	)	)	PUNCT
eajbcs-694	772	37	,	,	PUNCT
eajbcs-694	772	38	1–18	1–18	PROPN
eajbcs-694	772	39	.	.	PUNCT
eajbcs-694	773	1	brauer	brauer	PROPN
eajbcs-694	773	2	f.	f.	PROPN
eajbcs-694	773	3	,	,	PUNCT
eajbcs-694	773	4	castillo	castillo	PROPN
eajbcs-694	773	5	-	-	PUNCT
eajbcs-694	773	6	chavez	chavez	PROPN
eajbcs-694	773	7	c.	c.	PROPN
eajbcs-694	773	8	and	and	CCONJ
eajbcs-694	773	9	feng	feng	PROPN
eajbcs-694	773	10	z.	z.	PROPN
eajbcs-694	773	11	east	east	PROPN
eajbcs-694	773	12	afr	afr	PROPN
eajbcs-694	773	13	.	.	PUNCT
eajbcs-694	774	1	j.	j.	PROPN
eajbcs-694	774	2	biophys	biophys	PROPN
eajbcs-694	774	3	.	.	PUNCT
eajbcs-694	775	1	comput	comput	NOUN
eajbcs-694	775	2	.	.	PUNCT
eajbcs-694	776	1	sci	sci	PROPN
eajbcs-694	776	2	.	.	PUNCT
eajbcs-694	777	1	(	(	PUNCT
eajbcs-694	777	2	2024	2024	NUM
eajbcs-694	777	3	)	)	PUNCT
eajbcs-694	777	4	,	,	PUNCT
eajbcs-694	777	5	vol	vol	NOUN
eajbcs-694	777	6	.	.	PROPN
eajbcs-694	777	7	5	5	NUM
eajbcs-694	777	8	,	,	PUNCT
eajbcs-694	777	9	no	no	INTJ
eajbcs-694	777	10	.	.	NOUN
eajbcs-694	777	11	2	2	NUM
eajbcs-694	777	12	,	,	PUNCT
eajbcs-694	777	13	58	58	NUM
eajbcs-694	777	14	-	-	SYM
eajbcs-694	777	15	86	86	NUM
eajbcs-694	777	16	85	85	NUM
eajbcs-694	777	17	contact	contact	NOUN
eajbcs-694	777	18	rate	rate	NOUN
eajbcs-694	777	19	β	β	X
eajbcs-694	777	20	are	be	AUX
eajbcs-694	777	21	most	most	ADV
eajbcs-694	777	22	sensitive	sensitive	ADJ
eajbcs-694	777	23	parameters	parameter	NOUN
eajbcs-694	777	24	to	to	ADP
eajbcs-694	777	25	our	our	PRON
eajbcs-694	777	26	model	model	NOUN
eajbcs-694	777	27	.	.	PUNCT
eajbcs-694	778	1	numerical	numerical	PROPN
eajbcs-694	778	2	results	result	NOUN
eajbcs-694	778	3	support	support	VERB
eajbcs-694	778	4	the	the	DET
eajbcs-694	778	5	fact	fact	NOUN
eajbcs-694	778	6	that	that	SCONJ
eajbcs-694	778	7	decrease	decrease	NOUN
eajbcs-694	778	8	in	in	ADP
eajbcs-694	778	9	the	the	DET
eajbcs-694	778	10	contact	contact	NOUN
eajbcs-694	778	11	rate	rate	NOUN
eajbcs-694	778	12	β	β	NOUN
eajbcs-694	778	13	causes	cause	VERB
eajbcs-694	778	14	the	the	DET
eajbcs-694	778	15	decrease	decrease	NOUN
eajbcs-694	778	16	in	in	ADP
eajbcs-694	778	17	the	the	DET
eajbcs-694	778	18	value	value	NOUN
eajbcs-694	778	19	of	of	ADP
eajbcs-694	778	20	r0	r0	NOUN
eajbcs-694	778	21	and	and	CCONJ
eajbcs-694	778	22	after	after	ADP
eajbcs-694	778	23	a	a	DET
eajbcs-694	778	24	certain	certain	ADJ
eajbcs-694	778	25	level	level	NOUN
eajbcs-694	778	26	of	of	ADP
eajbcs-694	778	27	β	β	NOUN
eajbcs-694	778	28	,	,	PUNCT
eajbcs-694	778	29	r0	r0	NOUN
eajbcs-694	778	30	become	become	VERB
eajbcs-694	778	31	less	less	ADJ
eajbcs-694	778	32	than	than	ADP
eajbcs-694	778	33	one	one	NUM
eajbcs-694	778	34	.	.	PUNCT
eajbcs-694	779	1	if	if	SCONJ
eajbcs-694	779	2	the	the	DET
eajbcs-694	779	3	protection	protection	NOUN
eajbcs-694	779	4	rate	rate	NOUN
eajbcs-694	779	5	increases	increase	NOUN
eajbcs-694	779	6	,	,	PUNCT
eajbcs-694	779	7	then	then	ADV
eajbcs-694	779	8	all	all	DET
eajbcs-694	779	9	infected	infected	ADJ
eajbcs-694	779	10	classes	class	NOUN
eajbcs-694	779	11	are	be	AUX
eajbcs-694	779	12	decrease	decrease	NOUN
eajbcs-694	779	13	.	.	PUNCT
eajbcs-694	780	1	it	it	PRON
eajbcs-694	780	2	is	be	AUX
eajbcs-694	780	3	predicted	predict	VERB
eajbcs-694	780	4	that	that	SCONJ
eajbcs-694	780	5	the	the	DET
eajbcs-694	780	6	population	population	NOUN
eajbcs-694	780	7	will	will	AUX
eajbcs-694	780	8	be	be	AUX
eajbcs-694	780	9	disease	disease	NOUN
eajbcs-694	780	10	-	-	PUNCT
eajbcs-694	780	11	free	free	ADJ
eajbcs-694	780	12	.	.	PUNCT
eajbcs-694	781	1	furthermore	furthermore	ADV
eajbcs-694	781	2	,	,	PUNCT
eajbcs-694	781	3	using	use	VERB
eajbcs-694	781	4	optimal	optimal	ADJ
eajbcs-694	781	5	control	control	NOUN
eajbcs-694	781	6	theory	theory	NOUN
eajbcs-694	781	7	we	we	PRON
eajbcs-694	781	8	suggest	suggest	VERB
eajbcs-694	781	9	protection	protection	NOUN
eajbcs-694	781	10	and	and	CCONJ
eajbcs-694	781	11	hospitalization	hospitalization	NOUN
eajbcs-694	781	12	strategies	strategy	NOUN
eajbcs-694	781	13	.	.	PUNCT
eajbcs-694	782	1	pontryagin	pontryagin	NOUN
eajbcs-694	782	2	’s	’s	PART
eajbcs-694	782	3	maximum	maximum	ADJ
eajbcs-694	782	4	principle	principle	NOUN
eajbcs-694	782	5	is	be	AUX
eajbcs-694	782	6	used	use	VERB
eajbcs-694	782	7	to	to	PART
eajbcs-694	782	8	establish	establish	VERB
eajbcs-694	782	9	the	the	DET
eajbcs-694	782	10	existence	existence	NOUN
eajbcs-694	782	11	and	and	CCONJ
eajbcs-694	782	12	characterization	characterization	NOUN
eajbcs-694	782	13	of	of	ADP
eajbcs-694	782	14	optimal	optimal	ADJ
eajbcs-694	782	15	controls	control	NOUN
eajbcs-694	782	16	.	.	PUNCT
eajbcs-694	783	1	the	the	DET
eajbcs-694	783	2	study	study	NOUN
eajbcs-694	783	3	demonstrates	demonstrate	VERB
eajbcs-694	783	4	that	that	SCONJ
eajbcs-694	783	5	the	the	DET
eajbcs-694	783	6	combined	combined	ADJ
eajbcs-694	783	7	application	application	NOUN
eajbcs-694	783	8	of	of	ADP
eajbcs-694	783	9	both	both	DET
eajbcs-694	783	10	controls	control	NOUN
eajbcs-694	783	11	is	be	AUX
eajbcs-694	783	12	much	much	ADV
eajbcs-694	783	13	more	more	ADV
eajbcs-694	783	14	effective	effective	ADJ
eajbcs-694	783	15	in	in	ADP
eajbcs-694	783	16	reducing	reduce	VERB
eajbcs-694	783	17	the	the	DET
eajbcs-694	783	18	spread	spread	NOUN
eajbcs-694	783	19	of	of	ADP
eajbcs-694	783	20	the	the	DET
eajbcs-694	783	21	virus	virus	NOUN
eajbcs-694	783	22	than	than	ADP
eajbcs-694	783	23	when	when	SCONJ
eajbcs-694	783	24	each	each	DET
eajbcs-694	783	25	control	control	NOUN
eajbcs-694	783	26	is	be	AUX
eajbcs-694	783	27	applied	apply	VERB
eajbcs-694	783	28	individually	individually	ADV
eajbcs-694	783	29	.	.	PUNCT
eajbcs-694	784	1	finally	finally	ADV
eajbcs-694	784	2	,	,	PUNCT
eajbcs-694	784	3	our	our	PRON
eajbcs-694	784	4	analysis	analysis	NOUN
eajbcs-694	784	5	indicates	indicate	VERB
eajbcs-694	784	6	that	that	SCONJ
eajbcs-694	784	7	an	an	DET
eajbcs-694	784	8	optimal	optimal	ADJ
eajbcs-694	784	9	control	control	NOUN
eajbcs-694	784	10	is	be	AUX
eajbcs-694	784	11	more	more	ADV
eajbcs-694	784	12	preferable	preferable	ADJ
eajbcs-694	784	13	than	than	ADP
eajbcs-694	784	14	maintaining	maintain	VERB
eajbcs-694	784	15	a	a	DET
eajbcs-694	784	16	high	high	ADJ
eajbcs-694	784	17	constant	constant	ADJ
eajbcs-694	784	18	control	control	NOUN
eajbcs-694	784	19	.	.	PUNCT
eajbcs-694	785	1	(	(	PUNCT
eajbcs-694	785	2	2019	2019	NUM
eajbcs-694	785	3	)	)	PUNCT
eajbcs-694	785	4	,	,	PUNCT
eajbcs-694	785	5	mathematical	mathematical	ADJ
eajbcs-694	785	6	models	model	NOUN
eajbcs-694	785	7	in	in	ADP
eajbcs-694	785	8	epidemiology	epidemiology	NOUN
eajbcs-694	785	9	,	,	PUNCT
eajbcs-694	785	10	vol	vol	NOUN
eajbcs-694	785	11	.	.	PROPN
eajbcs-694	785	12	32	32	NUM
eajbcs-694	785	13	,	,	PUNCT
eajbcs-694	785	14	springer	springer	NOUN
eajbcs-694	785	15	.	.	PUNCT
eajbcs-694	786	1	castillo	castillo	PROPN
eajbcs-694	786	2	-	-	PUNCT
eajbcs-694	786	3	chavez	chavez	PROPN
eajbcs-694	786	4	c.	c.	PROPN
eajbcs-694	786	5	blower	blower	NOUN
eajbcs-694	786	6	,	,	PUNCT
eajbcs-694	786	7	s.	s.	PROPN
eajbcs-694	786	8	van	van	PROPN
eajbcs-694	786	9	den	den	PROPN
eajbcs-694	786	10	driessche	driessche	PROPN
eajbcs-694	786	11	p.	p.	PROPN
eajbcs-694	786	12	,	,	PUNCT
eajbcs-694	786	13	kirschner	kirschner	PROPN
eajbcs-694	786	14	d.	d.	PROPN
eajbcs-694	786	15	and	and	CCONJ
eajbcs-694	786	16	yakubu	yakubu	PROPN
eajbcs-694	786	17	a.-a	a.-a	PROPN
eajbcs-694	786	18	.	.	PUNCT
eajbcs-694	787	1	(	(	PUNCT
eajbcs-694	787	2	2002	2002	NUM
eajbcs-694	787	3	)	)	PUNCT
eajbcs-694	787	4	,	,	PUNCT
eajbcs-694	787	5	mathematical	mathematical	ADJ
eajbcs-694	787	6	approaches	approach	NOUN
eajbcs-694	787	7	for	for	ADP
eajbcs-694	787	8	emerging	emerge	VERB
eajbcs-694	787	9	and	and	CCONJ
eajbcs-694	787	10	reemerging	reemerging	ADJ
eajbcs-694	787	11	infectious	infectious	ADJ
eajbcs-694	787	12	diseases	disease	NOUN
eajbcs-694	787	13	:	:	PUNCT
eajbcs-694	787	14	models	model	NOUN
eajbcs-694	787	15	,	,	PUNCT
eajbcs-694	787	16	methods	method	NOUN
eajbcs-694	787	17	,	,	PUNCT
eajbcs-694	787	18	and	and	CCONJ
eajbcs-694	787	19	theory	theory	NOUN
eajbcs-694	787	20	,	,	PUNCT
eajbcs-694	787	21	vol	vol	NOUN
eajbcs-694	787	22	.	.	PROPN
eajbcs-694	787	23	126	126	NUM
eajbcs-694	787	24	,	,	PUNCT
eajbcs-694	787	25	springer	springer	NOUN
eajbcs-694	787	26	science	science	PROPN
eajbcs-694	787	27	&	&	CCONJ
eajbcs-694	787	28	business	business	NOUN
eajbcs-694	787	29	media	medium	NOUN
eajbcs-694	787	30	.	.	PUNCT
eajbcs-694	788	1	castillo	castillo	PROPN
eajbcs-694	788	2	-	-	PUNCT
eajbcs-694	788	3	chavez	chavez	PROPN
eajbcs-694	788	4	c.	c.	PROPN
eajbcs-694	788	5	and	and	CCONJ
eajbcs-694	788	6	song	song	PROPN
eajbcs-694	788	7	b.	b.	PROPN
eajbcs-694	788	8	(	(	PUNCT
eajbcs-694	788	9	2004	2004	NUM
eajbcs-694	788	10	)	)	PUNCT
eajbcs-694	788	11	,	,	PUNCT
eajbcs-694	788	12	‘	'	PUNCT
eajbcs-694	788	13	dynamical	dynamical	ADJ
eajbcs-694	788	14	models	model	NOUN
eajbcs-694	788	15	of	of	ADP
eajbcs-694	788	16	tuberculosis	tuberculosis	NOUN
eajbcs-694	788	17	and	and	CCONJ
eajbcs-694	788	18	their	their	PRON
eajbcs-694	788	19	applications	application	NOUN
eajbcs-694	788	20	’	'	PUNCT
eajbcs-694	788	21	,	,	PUNCT
eajbcs-694	788	22	mathematical	mathematical	ADJ
eajbcs-694	788	23	biosciences	bioscience	NOUN
eajbcs-694	788	24	&	&	CCONJ
eajbcs-694	788	25	engineering	engineering	PROPN
eajbcs-694	788	26	1(2	1(2	NUM
eajbcs-694	788	27	)	)	PUNCT
eajbcs-694	788	28	,	,	PUNCT
eajbcs-694	788	29	361	361	X
eajbcs-694	788	30	.	.	PUNCT
eajbcs-694	789	1	covid-19	covid-19	PROPN
eajbcs-694	789	2	pandemic	pandemic	ADJ
eajbcs-694	789	3	(	(	PUNCT
eajbcs-694	789	4	2022	2022	NUM
eajbcs-694	789	5	)	)	PUNCT
eajbcs-694	789	6	,	,	PUNCT
eajbcs-694	789	7	‘	'	PUNCT
eajbcs-694	789	8	covid-19	covid-19	PROPN
eajbcs-694	789	9	pandemic	pandemic	NOUN
eajbcs-694	789	10	in	in	ADP
eajbcs-694	789	11	ethiopia	ethiopia	PROPN
eajbcs-694	789	12	—	—	PUNCT
eajbcs-694	789	13	wikipedia	wikipedia	PROPN
eajbcs-694	789	14	,	,	PUNCT
eajbcs-694	789	15	the	the	DET
eajbcs-694	789	16	free	free	ADJ
eajbcs-694	789	17	encyclopedia	encyclopedia	NOUN
eajbcs-694	789	18	’	'	PUNCT
eajbcs-694	789	19	.	.	PUNCT
eajbcs-694	790	1	diekmann	diekmann	PROPN
eajbcs-694	790	2	o.	o.	PROPN
eajbcs-694	790	3	,	,	PUNCT
eajbcs-694	790	4	heesterbeek	heesterbeek	VERB
eajbcs-694	790	5	j.	j.	PROPN
eajbcs-694	790	6	and	and	CCONJ
eajbcs-694	790	7	roberts	roberts	PROPN
eajbcs-694	790	8	m.	m.	PROPN
eajbcs-694	790	9	g.	g.	PROPN
eajbcs-694	790	10	(	(	PUNCT
eajbcs-694	790	11	2010	2010	NUM
eajbcs-694	790	12	)	)	PUNCT
eajbcs-694	790	13	,	,	PUNCT
eajbcs-694	790	14	‘	'	PUNCT
eajbcs-694	790	15	the	the	DET
eajbcs-694	790	16	construction	construction	NOUN
eajbcs-694	790	17	of	of	ADP
eajbcs-694	790	18	nextgeneration	nextgeneration	NOUN
eajbcs-694	790	19	matrices	matrix	NOUN
eajbcs-694	790	20	for	for	ADP
eajbcs-694	790	21	compartmental	compartmental	ADJ
eajbcs-694	790	22	epidemic	epidemic	NOUN
eajbcs-694	790	23	models	model	NOUN
eajbcs-694	790	24	’	'	PUNCT
eajbcs-694	790	25	,	,	PUNCT
eajbcs-694	790	26	journal	journal	NOUN
eajbcs-694	790	27	of	of	ADP
eajbcs-694	790	28	the	the	DET
eajbcs-694	790	29	royal	royal	PROPN
eajbcs-694	790	30	society	society	PROPN
eajbcs-694	790	31	interface	interface	NOUN
eajbcs-694	790	32	7(47	7(47	NUM
eajbcs-694	790	33	)	)	PUNCT
eajbcs-694	790	34	,	,	PUNCT
eajbcs-694	790	35	873–885	873–885	NUM
eajbcs-694	790	36	.	.	PUNCT
eajbcs-694	791	1	fleming	fleme	VERB
eajbcs-694	791	2	w.	w.	PROPN
eajbcs-694	791	3	h.	h.	PROPN
eajbcs-694	791	4	and	and	CCONJ
eajbcs-694	791	5	rishel	rishel	PROPN
eajbcs-694	791	6	r.	r.	PROPN
eajbcs-694	791	7	w.	w.	PROPN
eajbcs-694	791	8	(	(	PUNCT
eajbcs-694	791	9	2012	2012	NUM
eajbcs-694	791	10	)	)	PUNCT
eajbcs-694	791	11	,	,	PUNCT
eajbcs-694	791	12	deterministic	deterministic	ADJ
eajbcs-694	791	13	and	and	CCONJ
eajbcs-694	791	14	stochastic	stochastic	ADJ
eajbcs-694	791	15	optimal	optimal	ADJ
eajbcs-694	791	16	control	control	NOUN
eajbcs-694	791	17	,	,	PUNCT
eajbcs-694	791	18	vol	vol	NOUN
eajbcs-694	791	19	.	.	PROPN
eajbcs-694	791	20	1	1	NUM
eajbcs-694	791	21	,	,	PUNCT
eajbcs-694	791	22	springer	springer	NOUN
eajbcs-694	791	23	science	science	PROPN
eajbcs-694	791	24	&	&	CCONJ
eajbcs-694	791	25	business	business	NOUN
eajbcs-694	791	26	media	medium	NOUN
eajbcs-694	791	27	.	.	PUNCT
eajbcs-694	792	1	gurmu	gurmu	NOUN
eajbcs-694	792	2	e.	e.	PROPN
eajbcs-694	792	3	d.	d.	PROPN
eajbcs-694	792	4	,	,	PUNCT
eajbcs-694	792	5	batu	batu	PROPN
eajbcs-694	792	6	,	,	PUNCT
eajbcs-694	792	7	g.	g.	PROPN
eajbcs-694	792	8	b.	b.	PROPN
eajbcs-694	792	9	and	and	CCONJ
eajbcs-694	792	10	wameko	wameko	PROPN
eajbcs-694	792	11	,	,	PUNCT
eajbcs-694	792	12	m.	m.	NOUN
eajbcs-694	792	13	s.	s.	PROPN
eajbcs-694	792	14	(	(	PUNCT
eajbcs-694	792	15	2020	2020	NUM
eajbcs-694	792	16	)	)	PUNCT
eajbcs-694	792	17	,	,	PUNCT
eajbcs-694	792	18	‘	'	PUNCT
eajbcs-694	792	19	mathematical	mathematical	ADJ
eajbcs-694	792	20	model	model	NOUN
eajbcs-694	792	21	of	of	ADP
eajbcs-694	792	22	novel	novel	NOUN
eajbcs-694	792	23	covid-19	covid-19	PROPN
eajbcs-694	792	24	and	and	CCONJ
eajbcs-694	792	25	its	its	PRON
eajbcs-694	792	26	transmission	transmission	NOUN
eajbcs-694	792	27	dynamics	dynamic	NOUN
eajbcs-694	792	28	’	'	PUNCT
eajbcs-694	792	29	,	,	PUNCT
eajbcs-694	792	30	international	international	ADJ
eajbcs-694	792	31	journal	journal	NOUN
eajbcs-694	792	32	of	of	ADP
eajbcs-694	792	33	mathematical	mathematical	ADJ
eajbcs-694	792	34	modelling	modelling	NOUN
eajbcs-694	792	35	&	&	CCONJ
eajbcs-694	792	36	computations	computation	NOUN
eajbcs-694	792	37	10(2	10(2	PROPN
eajbcs-694	792	38	(	(	PUNCT
eajbcs-694	792	39	spring	spring	NOUN
eajbcs-694	792	40	)	)	PUNCT
eajbcs-694	792	41	)	)	PUNCT
eajbcs-694	792	42	,	,	PUNCT
eajbcs-694	792	43	141–159	141–159	NUM
eajbcs-694	792	44	.	.	PUNCT
eajbcs-694	793	1	kifle	kifle	PROPN
eajbcs-694	793	2	z.	z.	PROPN
eajbcs-694	793	3	s.	s.	PROPN
eajbcs-694	793	4	and	and	CCONJ
eajbcs-694	793	5	obsu	obsu	PROPN
eajbcs-694	793	6	,	,	PUNCT
eajbcs-694	793	7	l.	l.	PROPN
eajbcs-694	793	8	l.	l.	PROPN
eajbcs-694	793	9	(	(	PUNCT
eajbcs-694	793	10	2022	2022	NUM
eajbcs-694	793	11	)	)	PUNCT
eajbcs-694	793	12	,	,	PUNCT
eajbcs-694	793	13	‘	'	PUNCT
eajbcs-694	793	14	mathematical	mathematical	ADJ
eajbcs-694	793	15	modeling	modeling	NOUN
eajbcs-694	793	16	for	for	ADP
eajbcs-694	793	17	covid-19	covid-19	PROPN
eajbcs-694	793	18	transmission	transmission	NOUN
eajbcs-694	793	19	dynamics	dynamic	NOUN
eajbcs-694	793	20	:	:	PUNCT
eajbcs-694	793	21	a	a	DET
eajbcs-694	793	22	case	case	NOUN
eajbcs-694	793	23	study	study	NOUN
eajbcs-694	793	24	in	in	ADP
eajbcs-694	793	25	ethiopia	ethiopia	PROPN
eajbcs-694	793	26	’	'	PUNCT
eajbcs-694	793	27	,	,	PUNCT
eajbcs-694	793	28	results	result	NOUN
eajbcs-694	793	29	in	in	ADP
eajbcs-694	793	30	physics	physics	NOUN
eajbcs-694	793	31	p.	p.	NOUN
eajbcs-694	793	32	105191	105191	NUM
eajbcs-694	793	33	.	.	PUNCT
eajbcs-694	794	1	lemecha	lemecha	ADV
eajbcs-694	794	2	obsu	obsu	PROPN
eajbcs-694	794	3	l.	l.	PROPN
eajbcs-694	794	4	and	and	CCONJ
eajbcs-694	794	5	feyissa	feyissa	PROPN
eajbcs-694	794	6	balcha	balcha	PROPN
eajbcs-694	794	7	s.	s.	PROPN
eajbcs-694	794	8	(	(	PUNCT
eajbcs-694	794	9	2020	2020	NUM
eajbcs-694	794	10	)	)	PUNCT
eajbcs-694	794	11	,	,	PUNCT
eajbcs-694	794	12	‘	'	PUNCT
eajbcs-694	794	13	optimal	optimal	ADJ
eajbcs-694	794	14	control	control	NOUN
eajbcs-694	794	15	strategies	strategy	NOUN
eajbcs-694	794	16	for	for	ADP
eajbcs-694	794	17	the	the	DET
eajbcs-694	794	18	transmission	transmission	NOUN
eajbcs-694	794	19	risk	risk	NOUN
eajbcs-694	794	20	of	of	ADP
eajbcs-694	794	21	covid-19	covid-19	PROPN
eajbcs-694	794	22	’	'	PUNCT
eajbcs-694	794	23	,	,	PUNCT
eajbcs-694	794	24	journal	journal	NOUN
eajbcs-694	794	25	of	of	ADP
eajbcs-694	794	26	biological	biological	ADJ
eajbcs-694	794	27	dynamics	dynamic	NOUN
eajbcs-694	794	28	14(1	14(1	NUM
eajbcs-694	794	29	)	)	PUNCT
eajbcs-694	794	30	,	,	PUNCT
eajbcs-694	794	31	590–607	590–607	NUM
eajbcs-694	794	32	.	.	PUNCT
eajbcs-694	795	1	lenhart	lenhart	PROPN
eajbcs-694	795	2	s.	s.	PROPN
eajbcs-694	795	3	and	and	CCONJ
eajbcs-694	795	4	workman	workman	NOUN
eajbcs-694	795	5	j.	j.	PROPN
eajbcs-694	795	6	t.	t.	PROPN
eajbcs-694	795	7	(	(	PUNCT
eajbcs-694	795	8	2007	2007	NUM
eajbcs-694	795	9	)	)	PUNCT
eajbcs-694	795	10	,	,	PUNCT
eajbcs-694	795	11	optimal	optimal	ADJ
eajbcs-694	795	12	control	control	NOUN
eajbcs-694	795	13	applied	apply	VERB
eajbcs-694	795	14	to	to	ADP
eajbcs-694	795	15	biological	biological	ADJ
eajbcs-694	795	16	models	model	NOUN
eajbcs-694	795	17	,	,	PUNCT
eajbcs-694	795	18	crc	crc	NOUN
eajbcs-694	795	19	press	press	PROPN
eajbcs-694	795	20	.	.	PUNCT
eajbcs-694	796	1	martcheva	martcheva	PROPN
eajbcs-694	796	2	m.	m.	PROPN
eajbcs-694	796	3	(	(	PUNCT
eajbcs-694	796	4	2015	2015	NUM
eajbcs-694	796	5	)	)	PUNCT
eajbcs-694	796	6	,	,	PUNCT
eajbcs-694	796	7	an	an	DET
eajbcs-694	796	8	introduction	introduction	NOUN
eajbcs-694	796	9	to	to	ADP
eajbcs-694	796	10	mathematical	mathematical	ADJ
eajbcs-694	796	11	epidemiology	epidemiology	NOUN
eajbcs-694	796	12	,	,	PUNCT
eajbcs-694	796	13	vol	vol	NOUN
eajbcs-694	796	14	.	.	PROPN
eajbcs-694	796	15	61	61	NUM
eajbcs-694	796	16	,	,	PUNCT
eajbcs-694	796	17	springer	springer	NOUN
eajbcs-694	796	18	.	.	PUNCT
eajbcs-694	797	1	pontryagin	pontryagin	PROPN
eajbcs-694	797	2	l.	l.	PROPN
eajbcs-694	797	3	s.	s.	PROPN
eajbcs-694	797	4	(	(	PUNCT
eajbcs-694	797	5	2018	2018	NUM
eajbcs-694	797	6	)	)	PUNCT
eajbcs-694	797	7	,	,	PUNCT
eajbcs-694	797	8	mathematical	mathematical	ADJ
eajbcs-694	797	9	theory	theory	NOUN
eajbcs-694	797	10	of	of	ADP
eajbcs-694	797	11	optimal	optimal	ADJ
eajbcs-694	797	12	processes	process	NOUN
eajbcs-694	797	13	,	,	PUNCT
eajbcs-694	797	14	routledge	routledge	PROPN
eajbcs-694	797	15	.	.	PUNCT
eajbcs-694	798	1	sohrabi	sohrabi	PROPN
eajbcs-694	798	2	c.	c.	PROPN
eajbcs-694	798	3	,	,	PUNCT
eajbcs-694	798	4	alsafi	alsafi	PROPN
eajbcs-694	798	5	z.	z.	PROPN
eajbcs-694	798	6	,	,	PUNCT
eajbcs-694	798	7	o’neill	o’neill	PROPN
eajbcs-694	798	8	n.	n.	PROPN
eajbcs-694	798	9	,	,	PUNCT
eajbcs-694	798	10	khan	khan	PROPN
eajbcs-694	798	11	,	,	PUNCT
eajbcs-694	798	12	m.	m.	NOUN
eajbcs-694	798	13	,	,	PUNCT
eajbcs-694	798	14	kerwan	kerwan	PROPN
eajbcs-694	798	15	a.	a.	PROPN
eajbcs-694	798	16	,	,	PUNCT
eajbcs-694	798	17	al	al	PROPN
eajbcs-694	798	18	-	-	PUNCT
eajbcs-694	798	19	jabir	jabir	PROPN
eajbcs-694	798	20	a.	a.	PROPN
eajbcs-694	798	21	,	,	PUNCT
eajbcs-694	798	22	iosifidis	iosifidis	PROPN
eajbcs-694	798	23	c.	c.	PROPN
eajbcs-694	798	24	and	and	CCONJ
eajbcs-694	798	25	agha	agha	PROPN
eajbcs-694	798	26	r.	r.	PROPN
eajbcs-694	798	27	(	(	PUNCT
eajbcs-694	798	28	2020	2020	NUM
eajbcs-694	798	29	)	)	PUNCT
eajbcs-694	798	30	,	,	PUNCT
eajbcs-694	798	31	‘	'	PUNCT
eajbcs-694	798	32	world	world	NOUN
eajbcs-694	798	33	health	health	NOUN
eajbcs-694	798	34	organization	organization	NOUN
eajbcs-694	798	35	declares	declare	VERB
eajbcs-694	798	36	global	global	ADJ
eajbcs-694	798	37	emergency	emergency	NOUN
eajbcs-694	798	38	:	:	PUNCT
eajbcs-694	798	39	a	a	DET
eajbcs-694	798	40	review	review	NOUN
eajbcs-694	798	41	of	of	ADP
eajbcs-694	798	42	the	the	DET
eajbcs-694	798	43	2019	2019	NUM
eajbcs-694	798	44	novel	novel	ADJ
eajbcs-694	798	45	coronavirus	coronavirus	NOUN
eajbcs-694	798	46	(	(	PUNCT
eajbcs-694	798	47	covid-19	covid-19	PROPN
eajbcs-694	798	48	)	)	PUNCT
eajbcs-694	798	49	’	'	PUNCT
eajbcs-694	798	50	,	,	PUNCT
eajbcs-694	798	51	international	international	ADJ
eajbcs-694	798	52	journal	journal	NOUN
eajbcs-694	798	53	of	of	ADP
eajbcs-694	798	54	surgery	surgery	NOUN
eajbcs-694	798	55	76	76	NUM
eajbcs-694	798	56	,	,	PUNCT
eajbcs-694	798	57	71	71	NUM
eajbcs-694	798	58	–	–	SYM
eajbcs-694	798	59	76	76	NUM
eajbcs-694	799	1	.	.	PUNCT
eajbcs-694	799	2	toquero	toquero	NOUN
eajbcs-694	799	3	c.	c.	PROPN
eajbcs-694	799	4	m.	m.	NOUN
eajbcs-694	799	5	(	(	PUNCT
eajbcs-694	799	6	2020	2020	NUM
eajbcs-694	799	7	)	)	PUNCT
eajbcs-694	799	8	,	,	PUNCT
eajbcs-694	799	9	‘	'	PUNCT
eajbcs-694	799	10	challenges	challenge	NOUN
eajbcs-694	799	11	and	and	CCONJ
eajbcs-694	799	12	opportunities	opportunity	NOUN
eajbcs-694	799	13	for	for	ADP
eajbcs-694	799	14	higher	high	ADJ
eajbcs-694	799	15	education	education	NOUN
eajbcs-694	799	16	amid	amid	ADP
eajbcs-694	799	17	the	the	DET
eajbcs-694	799	18	covid-19	covid-19	PROPN
eajbcs-694	799	19	pandemic	pandemic	NOUN
eajbcs-694	799	20	:	:	PUNCT
eajbcs-694	799	21	the	the	DET
eajbcs-694	799	22	philippine	philippine	ADJ
eajbcs-694	799	23	context	context	NOUN
eajbcs-694	799	24	.	.	PUNCT
eajbcs-694	799	25	’	'	PUNCT
eajbcs-694	799	26	,	,	PUNCT
eajbcs-694	799	27	pedagogical	pedagogical	ADJ
eajbcs-694	799	28	research	research	NOUN
eajbcs-694	799	29	5(4	5(4	NUM
eajbcs-694	799	30	)	)	PUNCT
eajbcs-694	799	31	.	.	PUNCT
eajbcs-694	800	1	valcher	valcher	PROPN
eajbcs-694	800	2	m.	m.	PROPN
eajbcs-694	800	3	e.	e.	PROPN
eajbcs-694	800	4	(	(	PUNCT
eajbcs-694	800	5	2002	2002	NUM
eajbcs-694	800	6	)	)	PUNCT
eajbcs-694	800	7	,	,	PUNCT
eajbcs-694	800	8	‘	'	PUNCT
eajbcs-694	800	9	positive	positive	ADJ
eajbcs-694	800	10	systems	system	NOUN
eajbcs-694	800	11	in	in	ADP
eajbcs-694	800	12	the	the	DET
eajbcs-694	800	13	behavioral	behavioral	ADJ
eajbcs-694	800	14	approach	approach	NOUN
eajbcs-694	800	15	:	:	PUNCT
eajbcs-694	800	16	main	main	ADJ
eajbcs-694	800	17	issues	issue	NOUN
eajbcs-694	800	18	and	and	CCONJ
eajbcs-694	800	19	recent	recent	ADJ
eajbcs-694	800	20	results	result	NOUN
eajbcs-694	800	21	’	'	PUNCT
eajbcs-694	800	22	,	,	PUNCT
eajbcs-694	800	23	electronic	electronic	ADJ
eajbcs-694	800	24	proceedings	proceeding	NOUN
eajbcs-694	800	25	of	of	ADP
eajbcs-694	800	26	mtns	mtns	PROPN
eajbcs-694	800	27	2002	2002	NUM
eajbcs-694	800	28	.	.	PUNCT
eajbcs-694	801	1	yang	yang	PROPN
eajbcs-694	801	2	c.	c.	PROPN
eajbcs-694	801	3	and	and	CCONJ
eajbcs-694	801	4	wang	wang	PROPN
eajbcs-694	801	5	j.	j.	PROPN
eajbcs-694	801	6	(	(	PUNCT
eajbcs-694	801	7	2020	2020	NUM
eajbcs-694	801	8	)	)	PUNCT
eajbcs-694	801	9	,	,	PUNCT
eajbcs-694	801	10	‘	'	PUNCT
eajbcs-694	801	11	a	a	DET
eajbcs-694	801	12	mathematical	mathematical	ADJ
eajbcs-694	801	13	model	model	NOUN
eajbcs-694	801	14	for	for	ADP
eajbcs-694	801	15	the	the	DET
eajbcs-694	801	16	novel	novel	ADJ
eajbcs-694	801	17	coronavirus	coronavirus	NOUN
eajbcs-694	801	18	epidemic	epidemic	NOUN
eajbcs-694	801	19	in	in	ADP
eajbcs-694	801	20	wuhan	wuhan	PROPN
eajbcs-694	801	21	,	,	PUNCT
eajbcs-694	801	22	china	china	PROPN
eajbcs-694	801	23	’	'	PUNCT
eajbcs-694	801	24	,	,	PUNCT
eajbcs-694	801	25	mathematical	mathematical	ADJ
eajbcs-694	801	26	biosciences	bioscience	NOUN
eajbcs-694	801	27	and	and	CCONJ
eajbcs-694	801	28	engineering	engineering	NOUN
eajbcs-694	801	29	:	:	PUNCT
eajbcs-694	801	30	mbe	mbe	PROPN
eajbcs-694	801	31	17(3	17(3	NUM
eajbcs-694	801	32	)	)	PUNCT
eajbcs-694	801	33	,	,	PUNCT
eajbcs-694	801	34	2708	2708	NUM
eajbcs-694	801	35	.	.	PUNCT
eajbcs-694	802	1	east	east	PROPN
eajbcs-694	802	2	afr	afr	PROPN
eajbcs-694	802	3	.	.	PUNCT
eajbcs-694	803	1	j.	j.	PROPN
eajbcs-694	803	2	biophys	biophys	PROPN
eajbcs-694	803	3	.	.	PUNCT
eajbcs-694	804	1	comput	comput	NOUN
eajbcs-694	804	2	.	.	PUNCT
eajbcs-694	805	1	sci	sci	PROPN
eajbcs-694	805	2	.	.	PUNCT
eajbcs-694	806	1	(	(	PUNCT
eajbcs-694	806	2	2024	2024	NUM
eajbcs-694	806	3	)	)	PUNCT
eajbcs-694	806	4	,	,	PUNCT
eajbcs-694	806	5	vol	vol	NOUN
eajbcs-694	806	6	.	.	PROPN
eajbcs-694	806	7	5	5	NUM
eajbcs-694	806	8	,	,	PUNCT
eajbcs-694	806	9	no	no	INTJ
eajbcs-694	806	10	.	.	NOUN
eajbcs-694	806	11	2	2	NUM
eajbcs-694	806	12	,	,	PUNCT
eajbcs-694	806	13	58	58	NUM
eajbcs-694	806	14	-	-	SYM
eajbcs-694	806	15	86	86	NUM
eajbcs-694	806	16	86	86	NUM
eajbcs-694	806	17	eajbcs__volume_5_issue	eajbcs__volume_5_issue	PROPN
eajbcs-694	806	18	2	2	NUM
eajbcs-694	806	19	_	_	PRON
eajbcs-694	806	20	article	article	NOUN
eajbcs-694	806	21	5	5	NUM
eajbcs-694	806	22	_	_	PUNCT
eajbcs-694	806	23	tinaw	tinaw	NOUN
eajbcs-694	806	24	and	and	CCONJ
eajbcs-694	806	25	kiros	kiro	NOUN
eajbcs-694	806	26	_	_	PUNCT
eajbcs-694	806	27	a	a	DET
eajbcs-694	806	28	mathematical	mathematical	ADJ
eajbcs-694	806	29	model	model	NOUN
eajbcs-694	806	30	for	for	ADP
eajbcs-694	806	31	the	the	DET
eajbcs-694	806	32	transmission	transmission	NOUN
eajbcs-694	806	33	dynamics	dynamic	NOUN
eajbcs-694	806	34	of	of	ADP
eajbcs-694	806	35	covid	covid	PROPN
eajbcs-694	806	36	19	19	NUM
eajbcs-694	806	37	pandemic.pdf	pandemic.pdf	NOUN
eajbcs-694	806	38	(	(	PUNCT
eajbcs-694	806	39	p.64	p.64	PROPN
eajbcs-694	806	40	-	-	ADJ
eajbcs-694	806	41	92	92	NUM
eajbcs-694	806	42	)	)	PUNCT
eajbcs-694	806	43	introduction	introduction	NOUN
eajbcs-694	806	44	mathematical	mathematical	ADJ
eajbcs-694	806	45	model	model	NOUN
eajbcs-694	806	46	formulation	formulation	NOUN
eajbcs-694	806	47	the	the	DET
eajbcs-694	806	48	modified	modify	VERB
eajbcs-694	806	49	mathematical	mathematical	ADJ
eajbcs-694	806	50	model	model	NOUN
eajbcs-694	806	51	model	model	NOUN
eajbcs-694	806	52	formulation	formulation	NOUN
eajbcs-694	806	53	qualitative	qualitative	NOUN
eajbcs-694	806	54	analysis	analysis	NOUN
eajbcs-694	806	55	of	of	ADP
eajbcs-694	806	56	the	the	DET
eajbcs-694	806	57	modified	modify	VERB
eajbcs-694	806	58	model	model	NOUN
eajbcs-694	806	59	well	well	ADV
eajbcs-694	806	60	-	-	PUNCT
eajbcs-694	806	61	posedness	posedness	NOUN
eajbcs-694	806	62	steady	steady	ADJ
eajbcs-694	806	63	state	state	NOUN
eajbcs-694	806	64	basic	basic	ADJ
eajbcs-694	806	65	reproduction	reproduction	NOUN
eajbcs-694	806	66	number	number	NOUN
eajbcs-694	806	67	local	local	ADJ
eajbcs-694	806	68	stability	stability	NOUN
eajbcs-694	806	69	of	of	ADP
eajbcs-694	806	70	disease	disease	NOUN
eajbcs-694	806	71	free	free	ADJ
eajbcs-694	806	72	equilibrium	equilibrium	NOUN
eajbcs-694	806	73	global	global	ADJ
eajbcs-694	806	74	stability	stability	NOUN
eajbcs-694	806	75	of	of	ADP
eajbcs-694	806	76	disease	disease	NOUN
eajbcs-694	806	77	free	free	ADJ
eajbcs-694	806	78	equilibrium	equilibrium	NOUN
eajbcs-694	806	79	endemic	endemic	ADJ
eajbcs-694	806	80	equilibrium	equilibrium	NOUN
eajbcs-694	806	81	point	point	NOUN
eajbcs-694	806	82	local	local	ADJ
eajbcs-694	806	83	stability	stability	NOUN
eajbcs-694	806	84	of	of	ADP
eajbcs-694	806	85	endemic	endemic	ADJ
eajbcs-694	806	86	equilibrium	equilibrium	NOUN
eajbcs-694	806	87	bifurcation	bifurcation	NOUN
eajbcs-694	806	88	analysis	analysis	NOUN
eajbcs-694	806	89	sensitivity	sensitivity	NOUN
eajbcs-694	806	90	analysis	analysis	NOUN
eajbcs-694	806	91	numerical	numerical	ADJ
eajbcs-694	806	92	simulations	simulation	NOUN
eajbcs-694	806	93	and	and	CCONJ
eajbcs-694	806	94	discussion	discussion	NOUN
eajbcs-694	806	95	extension	extension	NOUN
eajbcs-694	806	96	of	of	ADP
eajbcs-694	806	97	the	the	DET
eajbcs-694	806	98	modified	modified	ADJ
eajbcs-694	806	99	model	model	NOUN
eajbcs-694	806	100	into	into	ADP
eajbcs-694	806	101	an	an	DET
eajbcs-694	806	102	optimal	optimal	ADJ
eajbcs-694	806	103	control	control	NOUN
eajbcs-694	806	104	optimal	optimal	ADJ
eajbcs-694	806	105	protection	protection	NOUN
eajbcs-694	806	106	and	and	CCONJ
eajbcs-694	806	107	hospitalization	hospitalization	NOUN
eajbcs-694	806	108	using	use	VERB
eajbcs-694	806	109	modified	modified	ADJ
eajbcs-694	806	110	model	model	NOUN
eajbcs-694	806	111	existence	existence	NOUN
eajbcs-694	806	112	of	of	ADP
eajbcs-694	806	113	an	an	DET
eajbcs-694	806	114	optimal	optimal	ADJ
eajbcs-694	806	115	control	control	NOUN
eajbcs-694	806	116	the	the	DET
eajbcs-694	806	117	hamiltonian	hamiltonian	ADJ
eajbcs-694	806	118	and	and	CCONJ
eajbcs-694	806	119	optimality	optimality	NOUN
eajbcs-694	806	120	system	system	NOUN
eajbcs-694	806	121	numerical	numerical	ADJ
eajbcs-694	806	122	simulations	simulation	NOUN
eajbcs-694	806	123	of	of	ADP
eajbcs-694	806	124	optimal	optimal	ADJ
eajbcs-694	806	125	control	control	NOUN
eajbcs-694	806	126	problem	problem	NOUN
eajbcs-694	806	127	optimal	optimal	ADJ
eajbcs-694	806	128	control	control	NOUN
eajbcs-694	806	129	comparisons	comparison	NOUN
eajbcs-694	806	130	and	and	CCONJ
eajbcs-694	806	131	strategies	strategy	NOUN
eajbcs-694	806	132	conclusion	conclusion	NOUN