id	sid	tid	token	lemma	pos
m-1153	1	1	a	a	DET
m-1153	1	2	mathematical	mathematical	ADJ
m-1153	1	3	model	model	NOUN
m-1153	1	4	on	on	ADP
m-1153	1	5	cholera	cholera	NOUN
m-1153	1	6	outbreak	outbreak	NOUN
m-1153	1	7	with	with	ADP
m-1153	1	8	vaccination	vaccination	NOUN
m-1153	1	9	bolanle	bolanle	NOUN
m-1153	1	10	adeola	adeola	PROPN
m-1153	1	11	olokuntoye	olokuntoye	PROPN
m-1153	1	12	department	department	PROPN
m-1153	1	13	of	of	ADP
m-1153	1	14	mathematics	mathematic	NOUN
m-1153	1	15	,	,	PUNCT
m-1153	1	16	obafemi	obafemi	ADP
m-1153	1	17	awolowo	awolowo	PROPN
m-1153	1	18	university	university	PROPN
m-1153	1	19	,	,	PUNCT
m-1153	1	20	ile	ile	PROPN
m-1153	1	21	-	-	PUNCT
m-1153	1	22	ife	ife	PROPN
m-1153	1	23	,	,	PUNCT
m-1153	1	24	nigeria	nigeria	PROPN
m-1153	1	25	abstract	abstract	ADV
m-1153	1	26	this	this	DET
m-1153	1	27	study	study	NOUN
m-1153	1	28	develops	develop	VERB
m-1153	1	29	and	and	CCONJ
m-1153	1	30	analyzes	analyze	VERB
m-1153	1	31	a	a	DET
m-1153	1	32	cholera	cholera	NOUN
m-1153	1	33	transmission	transmission	NOUN
m-1153	1	34	model	model	NOUN
m-1153	1	35	of	of	ADP
m-1153	1	36	sirb	sirb	NOUN
m-1153	1	37	type	type	NOUN
m-1153	1	38	(	(	PUNCT
m-1153	1	39	susceptible	susceptible	ADJ
m-1153	1	40	–	–	PUNCT
m-1153	1	41	infected	infected	ADJ
m-1153	1	42	–	–	PUNCT
m-1153	1	43	recovered	recovered	ADJ
m-1153	1	44	–	–	PUNCT
m-1153	1	45	bacteria	bacteria	NOUN
m-1153	1	46	)	)	PUNCT
m-1153	1	47	that	that	PRON
m-1153	1	48	incorporates	incorporate	VERB
m-1153	1	49	vaccination	vaccination	NOUN
m-1153	1	50	.	.	PUNCT
m-1153	2	1	the	the	DET
m-1153	2	2	main	main	ADJ
m-1153	2	3	objective	objective	NOUN
m-1153	2	4	is	be	AUX
m-1153	2	5	to	to	PART
m-1153	2	6	investigate	investigate	VERB
m-1153	2	7	the	the	DET
m-1153	2	8	threshold	threshold	NOUN
m-1153	2	9	conditions	condition	NOUN
m-1153	2	10	under	under	ADP
m-1153	2	11	which	which	PRON
m-1153	2	12	cholera	cholera	NOUN
m-1153	2	13	can	can	AUX
m-1153	2	14	either	either	CCONJ
m-1153	2	15	be	be	AUX
m-1153	2	16	eradicated	eradicate	VERB
m-1153	2	17	or	or	CCONJ
m-1153	2	18	persist	persist	VERB
m-1153	2	19	in	in	ADP
m-1153	2	20	the	the	DET
m-1153	2	21	community	community	NOUN
m-1153	2	22	.	.	PUNCT
m-1153	3	1	the	the	DET
m-1153	3	2	model	model	NOUN
m-1153	3	3	formulation	formulation	NOUN
m-1153	3	4	captures	capture	VERB
m-1153	3	5	both	both	PRON
m-1153	3	6	direct	direct	ADJ
m-1153	3	7	person	person	NOUN
m-1153	3	8	-	-	PUNCT
m-1153	3	9	to	to	ADP
m-1153	3	10	-	-	PUNCT
m-1153	3	11	person	person	NOUN
m-1153	3	12	transmission	transmission	NOUN
m-1153	3	13	and	and	CCONJ
m-1153	3	14	indirect	indirect	ADJ
m-1153	3	15	infection	infection	NOUN
m-1153	3	16	through	through	ADP
m-1153	3	17	contaminated	contaminated	ADJ
m-1153	3	18	water	water	NOUN
m-1153	3	19	.	.	PUNCT
m-1153	4	1	using	use	VERB
m-1153	4	2	standard	standard	ADJ
m-1153	4	3	dynamical	dynamical	ADJ
m-1153	4	4	systems	system	NOUN
m-1153	4	5	techniques	technique	NOUN
m-1153	4	6	,	,	PUNCT
m-1153	4	7	the	the	DET
m-1153	4	8	disease	disease	NOUN
m-1153	4	9	-	-	PUNCT
m-1153	4	10	free	free	ADJ
m-1153	4	11	equilibrium	equilibrium	NOUN
m-1153	4	12	(	(	PUNCT
m-1153	4	13	dfe	dfe	PROPN
m-1153	4	14	)	)	PUNCT
m-1153	4	15	and	and	CCONJ
m-1153	4	16	endemic	endemic	ADJ
m-1153	4	17	equilibrium	equilibrium	NOUN
m-1153	4	18	(	(	PUNCT
m-1153	4	19	ee	ee	NOUN
m-1153	4	20	)	)	PUNCT
m-1153	4	21	were	be	AUX
m-1153	4	22	derived	derive	VERB
m-1153	4	23	.	.	PUNCT
m-1153	5	1	the	the	DET
m-1153	5	2	next	next	ADJ
m-1153	5	3	-	-	PUNCT
m-1153	5	4	generation	generation	NOUN
m-1153	5	5	matrix	matrix	NOUN
m-1153	5	6	approach	approach	NOUN
m-1153	5	7	was	be	AUX
m-1153	5	8	then	then	ADV
m-1153	5	9	applied	apply	VERB
m-1153	5	10	to	to	PART
m-1153	5	11	obtain	obtain	VERB
m-1153	5	12	the	the	DET
m-1153	5	13	basic	basic	ADJ
m-1153	5	14	reproduction	reproduction	NOUN
m-1153	5	15	number	number	NOUN
m-1153	5	16	,	,	PUNCT
m-1153	5	17	r0	r0	NOUN
m-1153	5	18	,	,	PUNCT
m-1153	5	19	which	which	PRON
m-1153	5	20	serves	serve	VERB
m-1153	5	21	as	as	ADP
m-1153	5	22	the	the	DET
m-1153	5	23	central	central	ADJ
m-1153	5	24	threshold	threshold	NOUN
m-1153	5	25	parameter	parameter	NOUN
m-1153	5	26	governing	govern	VERB
m-1153	5	27	disease	disease	NOUN
m-1153	5	28	dynamics	dynamic	NOUN
m-1153	5	29	.	.	PUNCT
m-1153	6	1	the	the	DET
m-1153	6	2	analysis	analysis	NOUN
m-1153	6	3	showed	show	VERB
m-1153	6	4	that	that	SCONJ
m-1153	6	5	the	the	DET
m-1153	6	6	dfe	dfe	PROPN
m-1153	6	7	is	be	AUX
m-1153	6	8	locally	locally	ADV
m-1153	6	9	asymptotically	asymptotically	ADV
m-1153	6	10	stable	stable	ADJ
m-1153	6	11	whenever	whenever	SCONJ
m-1153	6	12	r0	r0	NOUN
m-1153	6	13	<	<	X
m-1153	6	14	1	1	NUM
m-1153	6	15	,	,	PUNCT
m-1153	6	16	implying	imply	VERB
m-1153	6	17	cholera	cholera	NOUN
m-1153	6	18	elimination	elimination	NOUN
m-1153	6	19	under	under	ADP
m-1153	6	20	effective	effective	ADJ
m-1153	6	21	interventions	intervention	NOUN
m-1153	6	22	,	,	PUNCT
m-1153	6	23	while	while	SCONJ
m-1153	6	24	the	the	DET
m-1153	6	25	ee	ee	PROPN
m-1153	6	26	exists	exist	VERB
m-1153	6	27	and	and	CCONJ
m-1153	6	28	is	be	AUX
m-1153	6	29	locally	locally	ADV
m-1153	6	30	stable	stable	ADJ
m-1153	6	31	when	when	SCONJ
m-1153	6	32	r0	r0	NOUN
m-1153	6	33	>	>	X
m-1153	6	34	1	1	NUM
m-1153	6	35	,	,	PUNCT
m-1153	6	36	confirming	confirm	VERB
m-1153	6	37	sustained	sustained	ADJ
m-1153	6	38	disease	disease	NOUN
m-1153	6	39	persistence	persistence	NOUN
m-1153	6	40	.	.	PUNCT
m-1153	7	1	these	these	DET
m-1153	7	2	results	result	NOUN
m-1153	7	3	emphasize	emphasize	VERB
m-1153	7	4	the	the	DET
m-1153	7	5	importance	importance	NOUN
m-1153	7	6	of	of	ADP
m-1153	7	7	vaccination	vaccination	NOUN
m-1153	7	8	and	and	CCONJ
m-1153	7	9	improvements	improvement	NOUN
m-1153	7	10	in	in	ADP
m-1153	7	11	sanitation	sanitation	NOUN
m-1153	7	12	as	as	ADP
m-1153	7	13	essential	essential	ADJ
m-1153	7	14	strategies	strategy	NOUN
m-1153	7	15	to	to	PART
m-1153	7	16	reduce	reduce	VERB
m-1153	7	17	r0	r0	NOUN
m-1153	7	18	below	below	ADP
m-1153	7	19	unity	unity	NOUN
m-1153	7	20	and	and	CCONJ
m-1153	7	21	achieve	achieve	VERB
m-1153	7	22	long	long	ADJ
m-1153	7	23	-	-	PUNCT
m-1153	7	24	term	term	NOUN
m-1153	7	25	cholera	cholera	NOUN
m-1153	7	26	control	control	NOUN
m-1153	7	27	.	.	PUNCT
m-1153	8	1	keywords	keyword	NOUN
m-1153	8	2	:	:	PUNCT
m-1153	8	3	cholera	cholera	NOUN
m-1153	8	4	;	;	PUNCT
m-1153	8	5	sirb	sirb	PROPN
m-1153	8	6	model	model	NOUN
m-1153	8	7	;	;	PUNCT
m-1153	8	8	vaccination	vaccination	NOUN
m-1153	8	9	;	;	PUNCT
m-1153	8	10	equilibrium	equilibrium	NOUN
m-1153	8	11	points	point	NOUN
m-1153	8	12	;	;	PUNCT
m-1153	8	13	reproduction	reproduction	NOUN
m-1153	8	14	number	number	NOUN
m-1153	8	15	;	;	PUNCT
m-1153	8	16	epidemiological	epidemiological	ADJ
m-1153	8	17	modeling	model	VERB
m-1153	8	18	ijo	ijo	PROPN
m-1153	8	19	international	international	PROPN
m-1153	8	20	journal	journal	PROPN
m-1153	8	21	of	of	ADP
m-1153	8	22	mathematics	mathematics	PROPN
m-1153	8	23	(	(	PUNCT
m-1153	8	24	issn	issn	PROPN
m-1153	8	25	:	:	PUNCT
m-1153	8	26	2992	2992	NUM
m-1153	8	27	-	-	SYM
m-1153	8	28	4421	4421	NUM
m-1153	8	29	)	)	PUNCT
m-1153	8	30	volume	volume	NOUN
m-1153	8	31	08	08	NUM
m-1153	9	1	|	|	ADV
m-1153	9	2	issue	issue	NOUN
m-1153	9	3	9	9	NUM
m-1153	9	4	|	|	CCONJ
m-1153	9	5	september	september	PROPN
m-1153	9	6	2025	2025	NUM
m-1153	9	7	|	|	ADV
m-1153	9	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	NOUN
m-1153	9	9	23	23	NUM
m-1153	9	10	1	1	NUM
m-1153	9	11	introduction	introduction	NOUN
m-1153	9	12	cholera	cholera	NOUN
m-1153	9	13	remains	remain	VERB
m-1153	9	14	one	one	NUM
m-1153	9	15	of	of	ADP
m-1153	9	16	the	the	DET
m-1153	9	17	most	most	ADV
m-1153	9	18	devastating	devastating	ADJ
m-1153	9	19	waterborne	waterborne	ADJ
m-1153	9	20	diseases	disease	NOUN
m-1153	9	21	,	,	PUNCT
m-1153	9	22	particularly	particularly	ADV
m-1153	9	23	in	in	ADP
m-1153	9	24	developing	develop	VERB
m-1153	9	25	regions	region	NOUN
m-1153	9	26	where	where	SCONJ
m-1153	9	27	inadequate	inadequate	ADJ
m-1153	9	28	sanitation	sanitation	NOUN
m-1153	9	29	and	and	CCONJ
m-1153	9	30	limited	limited	ADJ
m-1153	9	31	access	access	NOUN
m-1153	9	32	to	to	ADP
m-1153	9	33	safe	safe	ADJ
m-1153	9	34	drinking	drinking	NOUN
m-1153	9	35	water	water	NOUN
m-1153	9	36	persist	persist	NOUN
m-1153	9	37	(	(	PUNCT
m-1153	9	38	ali	ali	PROPN
m-1153	9	39	et	et	PROPN
m-1153	9	40	al	al	PROPN
m-1153	9	41	.	.	PROPN
m-1153	9	42	,	,	PUNCT
m-1153	9	43	2015	2015	NUM
m-1153	9	44	;	;	PUNCT
m-1153	9	45	world	world	NOUN
m-1153	9	46	health	health	NOUN
m-1153	9	47	organization	organization	NOUN
m-1153	9	48	,	,	PUNCT
m-1153	9	49	2023	2023	NUM
m-1153	9	50	)	)	PUNCT
m-1153	9	51	.	.	PUNCT
m-1153	10	1	caused	cause	VERB
m-1153	10	2	by	by	ADP
m-1153	10	3	vibrio	vibrio	ADJ
m-1153	10	4	cholerae	cholerae	PROPN
m-1153	10	5	,	,	PUNCT
m-1153	10	6	cholera	cholera	NOUN
m-1153	10	7	is	be	AUX
m-1153	10	8	characterized	characterize	VERB
m-1153	10	9	by	by	ADP
m-1153	10	10	acute	acute	ADJ
m-1153	10	11	watery	watery	ADJ
m-1153	10	12	diarrhea	diarrhea	NOUN
m-1153	10	13	and	and	CCONJ
m-1153	10	14	can	can	AUX
m-1153	10	15	lead	lead	VERB
m-1153	10	16	to	to	ADP
m-1153	10	17	severe	severe	ADJ
m-1153	10	18	dehydration	dehydration	NOUN
m-1153	10	19	and	and	CCONJ
m-1153	10	20	death	death	NOUN
m-1153	10	21	if	if	SCONJ
m-1153	10	22	untreated	untreate	VERB
m-1153	10	23	(	(	PUNCT
m-1153	10	24	koelle	koelle	NOUN
m-1153	10	25	and	and	CCONJ
m-1153	10	26	pascual	pascual	NOUN
m-1153	10	27	,	,	PUNCT
m-1153	10	28	2004	2004	NUM
m-1153	10	29	)	)	PUNCT
m-1153	10	30	.	.	PUNCT
m-1153	11	1	despite	despite	SCONJ
m-1153	11	2	advances	advance	NOUN
m-1153	11	3	in	in	ADP
m-1153	11	4	treatment	treatment	NOUN
m-1153	11	5	and	and	CCONJ
m-1153	11	6	prevention	prevention	NOUN
m-1153	11	7	,	,	PUNCT
m-1153	11	8	cholera	cholera	NOUN
m-1153	11	9	continues	continue	VERB
m-1153	11	10	to	to	PART
m-1153	11	11	pose	pose	VERB
m-1153	11	12	significant	significant	ADJ
m-1153	11	13	public	public	ADJ
m-1153	11	14	health	health	NOUN
m-1153	11	15	challenges	challenge	NOUN
m-1153	11	16	,	,	PUNCT
m-1153	11	17	with	with	ADP
m-1153	11	18	recurrent	recurrent	ADJ
m-1153	11	19	outbreaks	outbreak	NOUN
m-1153	11	20	reported	report	VERB
m-1153	11	21	in	in	ADP
m-1153	11	22	africa	africa	PROPN
m-1153	11	23	,	,	PUNCT
m-1153	11	24	asia	asia	PROPN
m-1153	11	25	,	,	PUNCT
m-1153	11	26	and	and	CCONJ
m-1153	11	27	parts	part	NOUN
m-1153	11	28	of	of	ADP
m-1153	11	29	latin	latin	PROPN
m-1153	11	30	america	america	PROPN
m-1153	11	31	(	(	PUNCT
m-1153	11	32	mukandavire	mukandavire	NOUN
m-1153	11	33	et	et	PROPN
m-1153	11	34	al	al	PROPN
m-1153	11	35	.	.	PROPN
m-1153	11	36	,	,	PUNCT
m-1153	11	37	2011	2011	NUM
m-1153	11	38	;	;	PUNCT
m-1153	11	39	codeço	codeço	VERB
m-1153	11	40	,	,	PUNCT
m-1153	11	41	2001	2001	NUM
m-1153	11	42	)	)	PUNCT
m-1153	11	43	.	.	PUNCT
m-1153	12	1	mathematical	mathematical	ADJ
m-1153	12	2	modeling	modeling	NOUN
m-1153	12	3	has	have	AUX
m-1153	12	4	played	play	VERB
m-1153	12	5	a	a	DET
m-1153	12	6	crucial	crucial	ADJ
m-1153	12	7	role	role	NOUN
m-1153	12	8	in	in	ADP
m-1153	12	9	understanding	understand	VERB
m-1153	12	10	the	the	DET
m-1153	12	11	transmission	transmission	NOUN
m-1153	12	12	dynamics	dynamic	NOUN
m-1153	12	13	of	of	ADP
m-1153	12	14	cholera	cholera	NOUN
m-1153	12	15	and	and	CCONJ
m-1153	12	16	evaluating	evaluate	VERB
m-1153	12	17	the	the	DET
m-1153	12	18	effectiveness	effectiveness	NOUN
m-1153	12	19	of	of	ADP
m-1153	12	20	intervention	intervention	NOUN
m-1153	12	21	strategies	strategy	NOUN
m-1153	12	22	(	(	PUNCT
m-1153	12	23	anderson	anderson	PROPN
m-1153	12	24	and	and	CCONJ
m-1153	12	25	may	may	AUX
m-1153	12	26	,	,	PUNCT
m-1153	12	27	1991	1991	NUM
m-1153	12	28	;	;	PUNCT
m-1153	12	29	capasso	capasso	NOUN
m-1153	12	30	and	and	CCONJ
m-1153	12	31	paveri	paveri	PROPN
m-1153	12	32	-	-	PUNCT
m-1153	12	33	fontana	fontana	PROPN
m-1153	12	34	,	,	PUNCT
m-1153	12	35	1979	1979	NUM
m-1153	12	36	;	;	PUNCT
m-1153	12	37	hartley	hartley	PROPN
m-1153	12	38	et	et	PROPN
m-1153	12	39	al	al	PROPN
m-1153	12	40	.	.	PROPN
m-1153	12	41	,	,	PUNCT
m-1153	12	42	2006	2006	NUM
m-1153	12	43	)	)	PUNCT
m-1153	12	44	.	.	PUNCT
m-1153	13	1	early	early	ADJ
m-1153	13	2	deterministic	deterministic	ADJ
m-1153	13	3	models	model	NOUN
m-1153	13	4	,	,	PUNCT
m-1153	13	5	including	include	VERB
m-1153	13	6	classical	classical	ADJ
m-1153	13	7	sir	sir	NOUN
m-1153	13	8	frameworks	framework	NOUN
m-1153	13	9	,	,	PUNCT
m-1153	13	10	provided	provide	VERB
m-1153	13	11	the	the	DET
m-1153	13	12	foundation	foundation	NOUN
m-1153	13	13	for	for	ADP
m-1153	13	14	analyzing	analyze	VERB
m-1153	13	15	epidemic	epidemic	NOUN
m-1153	13	16	thresholds	threshold	NOUN
m-1153	13	17	and	and	CCONJ
m-1153	13	18	equilibrium	equilibrium	NOUN
m-1153	13	19	behavior	behavior	NOUN
m-1153	13	20	.	.	PUNCT
m-1153	14	1	extensions	extension	NOUN
m-1153	14	2	to	to	PART
m-1153	14	3	incorporate	incorporate	VERB
m-1153	14	4	environmental	environmental	ADJ
m-1153	14	5	reservoirs	reservoir	NOUN
m-1153	14	6	have	have	AUX
m-1153	14	7	proven	prove	VERB
m-1153	14	8	particularly	particularly	ADV
m-1153	14	9	important	important	ADJ
m-1153	14	10	for	for	ADP
m-1153	14	11	waterborne	waterborne	ADJ
m-1153	14	12	diseases	disease	NOUN
m-1153	14	13	like	like	ADP
m-1153	14	14	cholera	cholera	NOUN
m-1153	14	15	,	,	PUNCT
m-1153	14	16	where	where	SCONJ
m-1153	14	17	indirect	indirect	ADJ
m-1153	14	18	transmission	transmission	NOUN
m-1153	14	19	through	through	ADP
m-1153	14	20	contaminated	contaminate	VERB
m-1153	14	21	water	water	NOUN
m-1153	14	22	plays	play	VERB
m-1153	14	23	a	a	DET
m-1153	14	24	key	key	ADJ
m-1153	14	25	role	role	NOUN
m-1153	14	26	(	(	PUNCT
m-1153	14	27	codeço	codeço	VERB
m-1153	14	28	,	,	PUNCT
m-1153	14	29	2001	2001	NUM
m-1153	14	30	;	;	PUNCT
m-1153	14	31	tien	tien	NOUN
m-1153	14	32	and	and	CCONJ
m-1153	14	33	earn	earn	VERB
m-1153	14	34	,	,	PUNCT
m-1153	14	35	2010	2010	NUM
m-1153	14	36	)	)	PUNCT
m-1153	14	37	.	.	PUNCT
m-1153	15	1	among	among	ADP
m-1153	15	2	the	the	DET
m-1153	15	3	various	various	ADJ
m-1153	15	4	interventions	intervention	NOUN
m-1153	15	5	,	,	PUNCT
m-1153	15	6	vaccination	vaccination	NOUN
m-1153	15	7	has	have	AUX
m-1153	15	8	emerged	emerge	VERB
m-1153	15	9	as	as	ADP
m-1153	15	10	a	a	DET
m-1153	15	11	promising	promising	ADJ
m-1153	15	12	strategy	strategy	NOUN
m-1153	15	13	for	for	ADP
m-1153	15	14	cholera	cholera	NOUN
m-1153	15	15	control	control	NOUN
m-1153	15	16	(	(	PUNCT
m-1153	15	17	leung	leung	PROPN
m-1153	15	18	et	et	PROPN
m-1153	15	19	al	al	PROPN
m-1153	15	20	.	.	PROPN
m-1153	15	21	,	,	PUNCT
m-1153	15	22	2012	2012	NUM
m-1153	15	23	;	;	PUNCT
m-1153	15	24	world	world	NOUN
m-1153	15	25	health	health	NOUN
m-1153	15	26	organization	organization	NOUN
m-1153	15	27	,	,	PUNCT
m-1153	15	28	2023	2023	NUM
m-1153	15	29	)	)	PUNCT
m-1153	15	30	.	.	PUNCT
m-1153	16	1	oral	oral	ADJ
m-1153	16	2	cholera	cholera	NOUN
m-1153	16	3	vaccines	vaccine	NOUN
m-1153	16	4	(	(	PUNCT
m-1153	16	5	ocvs	ocvs	PROPN
m-1153	16	6	)	)	PUNCT
m-1153	16	7	have	have	AUX
m-1153	16	8	been	be	AUX
m-1153	16	9	deployed	deploy	VERB
m-1153	16	10	in	in	ADP
m-1153	16	11	both	both	CCONJ
m-1153	16	12	reactive	reactive	ADJ
m-1153	16	13	and	and	CCONJ
m-1153	16	14	preventive	preventive	ADJ
m-1153	16	15	campaigns	campaign	NOUN
m-1153	16	16	,	,	PUNCT
m-1153	16	17	showing	show	VERB
m-1153	16	18	moderate	moderate	ADJ
m-1153	16	19	to	to	ADP
m-1153	16	20	high	high	ADJ
m-1153	16	21	effectiveness	effectiveness	NOUN
m-1153	16	22	in	in	ADP
m-1153	16	23	reducing	reduce	VERB
m-1153	16	24	susceptibility	susceptibility	NOUN
m-1153	16	25	and	and	CCONJ
m-1153	16	26	mitigating	mitigate	VERB
m-1153	16	27	the	the	DET
m-1153	16	28	severity	severity	NOUN
m-1153	16	29	of	of	ADP
m-1153	16	30	outbreaks	outbreak	NOUN
m-1153	16	31	(	(	PUNCT
m-1153	16	32	qadri	qadri	PROPN
m-1153	16	33	et	et	PROPN
m-1153	16	34	al	al	PROPN
m-1153	16	35	.	.	PROPN
m-1153	16	36	,	,	PUNCT
m-1153	16	37	2020	2020	NUM
m-1153	16	38	)	)	PUNCT
m-1153	16	39	.	.	PUNCT
m-1153	17	1	incorporating	incorporate	VERB
m-1153	17	2	vaccination	vaccination	NOUN
m-1153	17	3	into	into	ADP
m-1153	17	4	mathematical	mathematical	ADJ
m-1153	17	5	models	model	NOUN
m-1153	17	6	not	not	PART
m-1153	17	7	only	only	ADV
m-1153	17	8	improves	improve	VERB
m-1153	17	9	the	the	DET
m-1153	17	10	biological	biological	ADJ
m-1153	17	11	realism	realism	NOUN
m-1153	17	12	of	of	ADP
m-1153	17	13	such	such	ADJ
m-1153	17	14	frameworks	framework	NOUN
m-1153	17	15	but	but	CCONJ
m-1153	17	16	also	also	ADV
m-1153	17	17	provides	provide	VERB
m-1153	17	18	decision	decision	NOUN
m-1153	17	19	-	-	PUNCT
m-1153	17	20	makers	maker	NOUN
m-1153	17	21	with	with	ADP
m-1153	17	22	quantitative	quantitative	ADJ
m-1153	17	23	tools	tool	NOUN
m-1153	17	24	to	to	PART
m-1153	17	25	assess	assess	VERB
m-1153	17	26	the	the	DET
m-1153	17	27	optimal	optimal	ADJ
m-1153	17	28	use	use	NOUN
m-1153	17	29	of	of	ADP
m-1153	17	30	vaccination	vaccination	NOUN
m-1153	17	31	alongside	alongside	ADP
m-1153	17	32	sanitation	sanitation	NOUN
m-1153	17	33	,	,	PUNCT
m-1153	17	34	water	water	NOUN
m-1153	17	35	treatment	treatment	NOUN
m-1153	17	36	,	,	PUNCT
m-1153	17	37	and	and	CCONJ
m-1153	17	38	public	public	ADJ
m-1153	17	39	health	health	NOUN
m-1153	17	40	interventions	intervention	NOUN
m-1153	17	41	(	(	PUNCT
m-1153	17	42	longini	longini	PROPN
m-1153	17	43	et	et	PROPN
m-1153	17	44	al	al	PROPN
m-1153	17	45	.	.	PROPN
m-1153	17	46	,	,	PUNCT
m-1153	17	47	2007	2007	NUM
m-1153	17	48	;	;	PUNCT
m-1153	17	49	chao	chao	PROPN
m-1153	17	50	et	et	PROPN
m-1153	17	51	al	al	PROPN
m-1153	17	52	.	.	PROPN
m-1153	17	53	,	,	PUNCT
m-1153	17	54	2011	2011	NUM
m-1153	17	55	)	)	PUNCT
m-1153	17	56	.	.	PUNCT
m-1153	18	1	furthermore	furthermore	ADV
m-1153	18	2	,	,	PUNCT
m-1153	18	3	the	the	DET
m-1153	18	4	study	study	NOUN
m-1153	18	5	of	of	ADP
m-1153	18	6	reproduction	reproduction	NOUN
m-1153	18	7	numbers	number	NOUN
m-1153	18	8	remains	remain	VERB
m-1153	18	9	central	central	ADJ
m-1153	18	10	in	in	ADP
m-1153	18	11	mathematical	mathematical	ADJ
m-1153	18	12	epidemiology	epidemiology	NOUN
m-1153	18	13	.	.	PUNCT
m-1153	19	1	the	the	DET
m-1153	19	2	next	next	ADJ
m-1153	19	3	-	-	PUNCT
m-1153	19	4	generation	generation	NOUN
m-1153	19	5	matrix	matrix	NOUN
m-1153	19	6	method	method	NOUN
m-1153	19	7	,	,	PUNCT
m-1153	19	8	formalized	formalize	VERB
m-1153	19	9	by	by	ADP
m-1153	19	10	van	van	PROPN
m-1153	19	11	den	den	PROPN
m-1153	19	12	driessche	driessche	NOUN
m-1153	19	13	and	and	CCONJ
m-1153	19	14	watmough	watmough	NOUN
m-1153	19	15	(	(	PUNCT
m-1153	19	16	2002	2002	NUM
m-1153	19	17	)	)	PUNCT
m-1153	19	18	,	,	PUNCT
m-1153	19	19	provides	provide	VERB
m-1153	19	20	a	a	DET
m-1153	19	21	rigorous	rigorous	ADJ
m-1153	19	22	framework	framework	NOUN
m-1153	19	23	for	for	ADP
m-1153	19	24	deriving	derive	VERB
m-1153	19	25	the	the	DET
m-1153	19	26	basic	basic	ADJ
m-1153	19	27	reproduction	reproduction	NOUN
m-1153	19	28	number	number	NOUN
m-1153	19	29	r0	r0	NOUN
m-1153	19	30	,	,	PUNCT
m-1153	19	31	which	which	PRON
m-1153	19	32	determines	determine	VERB
m-1153	19	33	whether	whether	SCONJ
m-1153	19	34	a	a	DET
m-1153	19	35	disease	disease	NOUN
m-1153	19	36	can	can	AUX
m-1153	19	37	invade	invade	VERB
m-1153	19	38	a	a	DET
m-1153	19	39	population	population	NOUN
m-1153	19	40	.	.	PUNCT
m-1153	20	1	for	for	ADP
m-1153	20	2	cholera	cholera	NOUN
m-1153	20	3	ijo	ijo	PROPN
m-1153	20	4	international	international	PROPN
m-1153	20	5	journal	journal	PROPN
m-1153	20	6	of	of	ADP
m-1153	20	7	mathematics	mathematics	PROPN
m-1153	20	8	(	(	PUNCT
m-1153	20	9	issn	issn	PROPN
m-1153	20	10	:	:	PUNCT
m-1153	20	11	2992	2992	NUM
m-1153	20	12	-	-	SYM
m-1153	20	13	4421	4421	NUM
m-1153	20	14	)	)	PUNCT
m-1153	20	15	volume	volume	NOUN
m-1153	20	16	08	08	NUM
m-1153	21	1	|	|	ADV
m-1153	21	2	issue	issue	NOUN
m-1153	21	3	9	9	NUM
m-1153	21	4	|	|	CCONJ
m-1153	21	5	september	september	PROPN
m-1153	21	6	2025	2025	NUM
m-1153	21	7	|	|	ADV
m-1153	21	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	21	9	24	24	NUM
m-1153	21	10	models	model	NOUN
m-1153	21	11	with	with	ADP
m-1153	21	12	vaccination	vaccination	NOUN
m-1153	21	13	,	,	PUNCT
m-1153	21	14	r0	r0	NOUN
m-1153	21	15	captures	capture	VERB
m-1153	21	16	both	both	CCONJ
m-1153	21	17	direct	direct	ADJ
m-1153	21	18	human	human	NOUN
m-1153	21	19	-	-	PUNCT
m-1153	21	20	to	to	ADP
m-1153	21	21	-	-	PUNCT
m-1153	21	22	human	human	ADJ
m-1153	21	23	and	and	CCONJ
m-1153	21	24	indirect	indirect	ADJ
m-1153	21	25	waterborne	waterborne	ADJ
m-1153	21	26	transmission	transmission	NOUN
m-1153	21	27	,	,	PUNCT
m-1153	21	28	as	as	ADV
m-1153	21	29	well	well	ADV
m-1153	21	30	as	as	ADP
m-1153	21	31	reductions	reduction	NOUN
m-1153	21	32	due	due	ADP
m-1153	21	33	to	to	ADP
m-1153	21	34	vaccine	vaccine	NOUN
m-1153	21	35	-	-	PUNCT
m-1153	21	36	induced	induce	VERB
m-1153	21	37	immunity	immunity	NOUN
m-1153	21	38	.	.	PUNCT
m-1153	22	1	exploring	explore	VERB
m-1153	22	2	the	the	DET
m-1153	22	3	relationship	relationship	NOUN
m-1153	22	4	between	between	ADP
m-1153	22	5	r0	r0	NOUN
m-1153	22	6	,	,	PUNCT
m-1153	22	7	vaccination	vaccination	NOUN
m-1153	22	8	coverage	coverage	NOUN
m-1153	22	9	,	,	PUNCT
m-1153	22	10	and	and	CCONJ
m-1153	22	11	endemic	endemic	ADJ
m-1153	22	12	equilibria	equilibrium	NOUN
m-1153	22	13	is	be	AUX
m-1153	22	14	essential	essential	ADJ
m-1153	22	15	for	for	ADP
m-1153	22	16	developing	develop	VERB
m-1153	22	17	effective	effective	ADJ
m-1153	22	18	control	control	NOUN
m-1153	22	19	strategies	strategy	NOUN
m-1153	22	20	(	(	PUNCT
m-1153	22	21	castillo	castillo	PROPN
m-1153	22	22	-	-	PUNCT
m-1153	22	23	chavez	chavez	PROPN
m-1153	22	24	and	and	CCONJ
m-1153	22	25	feng	feng	PROPN
m-1153	22	26	,	,	PUNCT
m-1153	22	27	1997	1997	NUM
m-1153	22	28	;	;	PUNCT
m-1153	22	29	mukandavire	mukandavire	NOUN
m-1153	22	30	et	et	PROPN
m-1153	22	31	al	al	PROPN
m-1153	22	32	.	.	PROPN
m-1153	22	33	,	,	PUNCT
m-1153	22	34	2011	2011	NUM
m-1153	22	35	)	)	PUNCT
m-1153	22	36	.	.	PUNCT
m-1153	23	1	mathematical	mathematical	ADJ
m-1153	23	2	models	model	NOUN
m-1153	23	3	of	of	ADP
m-1153	23	4	cholera	cholera	NOUN
m-1153	23	5	have	have	AUX
m-1153	23	6	evolved	evolve	VERB
m-1153	23	7	considerably	considerably	ADV
m-1153	23	8	since	since	SCONJ
m-1153	23	9	the	the	DET
m-1153	23	10	early	early	ADJ
m-1153	23	11	works	work	NOUN
m-1153	23	12	of	of	ADP
m-1153	23	13	capasso	capasso	NOUN
m-1153	23	14	and	and	CCONJ
m-1153	23	15	paveri	paveri	PROPN
m-1153	23	16	-	-	PUNCT
m-1153	23	17	fontana	fontana	PROPN
m-1153	23	18	(	(	PUNCT
m-1153	23	19	1979	1979	NUM
m-1153	23	20	)	)	PUNCT
m-1153	23	21	,	,	PUNCT
m-1153	23	22	who	who	PRON
m-1153	23	23	studied	study	VERB
m-1153	23	24	the	the	DET
m-1153	23	25	1973	1973	NUM
m-1153	23	26	cholera	cholera	NOUN
m-1153	23	27	epidemic	epidemic	NOUN
m-1153	23	28	in	in	ADP
m-1153	23	29	the	the	DET
m-1153	23	30	mediterranean	mediterranean	PROPN
m-1153	23	31	region	region	NOUN
m-1153	23	32	using	use	VERB
m-1153	23	33	an	an	DET
m-1153	23	34	sir	sir	ADJ
m-1153	23	35	-	-	PUNCT
m-1153	23	36	type	type	NOUN
m-1153	23	37	framework	framework	NOUN
m-1153	23	38	.	.	PUNCT
m-1153	24	1	later	later	ADV
m-1153	24	2	,	,	PUNCT
m-1153	24	3	codeço	codeço	VERB
m-1153	24	4	(	(	PUNCT
m-1153	24	5	2001	2001	NUM
m-1153	24	6	)	)	PUNCT
m-1153	24	7	extended	extend	VERB
m-1153	24	8	the	the	DET
m-1153	24	9	model	model	NOUN
m-1153	24	10	by	by	ADP
m-1153	24	11	explicitly	explicitly	ADV
m-1153	24	12	incorporating	incorporate	VERB
m-1153	24	13	an	an	DET
m-1153	24	14	environmental	environmental	ADJ
m-1153	24	15	reservoir	reservoir	NOUN
m-1153	24	16	of	of	ADP
m-1153	24	17	bacteria	bacteria	NOUN
m-1153	24	18	,	,	PUNCT
m-1153	24	19	leading	lead	VERB
m-1153	24	20	to	to	ADP
m-1153	24	21	the	the	DET
m-1153	24	22	widely	widely	ADV
m-1153	24	23	known	know	VERB
m-1153	24	24	sirb	sirb	NOUN
m-1153	24	25	framework	framework	NOUN
m-1153	24	26	.	.	PUNCT
m-1153	25	1	this	this	DET
m-1153	25	2	innovation	innovation	NOUN
m-1153	25	3	highlighted	highlight	VERB
m-1153	25	4	the	the	DET
m-1153	25	5	role	role	NOUN
m-1153	25	6	of	of	ADP
m-1153	25	7	aquatic	aquatic	ADJ
m-1153	25	8	environments	environment	NOUN
m-1153	25	9	in	in	ADP
m-1153	25	10	sustaining	sustain	VERB
m-1153	25	11	cholera	cholera	NOUN
m-1153	25	12	transmission	transmission	NOUN
m-1153	25	13	and	and	CCONJ
m-1153	25	14	explained	explain	VERB
m-1153	25	15	the	the	DET
m-1153	25	16	persistence	persistence	NOUN
m-1153	25	17	of	of	ADP
m-1153	25	18	outbreaks	outbreak	NOUN
m-1153	25	19	beyond	beyond	ADP
m-1153	25	20	simple	simple	ADJ
m-1153	25	21	person	person	NOUN
m-1153	25	22	-	-	PUNCT
m-1153	25	23	to	to	ADP
m-1153	25	24	-	-	PUNCT
m-1153	25	25	person	person	NOUN
m-1153	25	26	dynamics	dynamic	NOUN
m-1153	25	27	.	.	PUNCT
m-1153	26	1	subsequent	subsequent	ADJ
m-1153	26	2	studies	study	NOUN
m-1153	26	3	further	far	ADV
m-1153	26	4	enriched	enrich	VERB
m-1153	26	5	these	these	DET
m-1153	26	6	frameworks	framework	NOUN
m-1153	26	7	.	.	PUNCT
m-1153	27	1	hartley	hartley	PROPN
m-1153	27	2	et	et	PROPN
m-1153	27	3	al	al	PROPN
m-1153	27	4	.	.	PROPN
m-1153	27	5	(	(	PUNCT
m-1153	27	6	2006	2006	NUM
m-1153	27	7	)	)	PUNCT
m-1153	27	8	introduced	introduce	VERB
m-1153	27	9	the	the	DET
m-1153	27	10	concept	concept	NOUN
m-1153	27	11	of	of	ADP
m-1153	27	12	*	*	NOUN
m-1153	27	13	hyperinfectivity	hyperinfectivity	NOUN
m-1153	27	14	*	*	PUNCT
m-1153	27	15	,	,	PUNCT
m-1153	27	16	noting	note	VERB
m-1153	27	17	that	that	SCONJ
m-1153	27	18	freshly	freshly	ADV
m-1153	27	19	shed	shed	VERB
m-1153	27	20	bacteria	bacteria	NOUN
m-1153	27	21	are	be	AUX
m-1153	27	22	significantly	significantly	ADV
m-1153	27	23	more	more	ADV
m-1153	27	24	infectious	infectious	ADJ
m-1153	27	25	than	than	ADP
m-1153	27	26	older	old	ADJ
m-1153	27	27	aquatic	aquatic	ADJ
m-1153	27	28	bacteria	bacteria	NOUN
m-1153	27	29	.	.	PUNCT
m-1153	28	1	tien	tien	PROPN
m-1153	28	2	and	and	CCONJ
m-1153	28	3	earn	earn	PROPN
m-1153	28	4	(	(	PUNCT
m-1153	28	5	2010	2010	NUM
m-1153	28	6	)	)	PUNCT
m-1153	28	7	expanded	expand	VERB
m-1153	28	8	the	the	DET
m-1153	28	9	sirb	sirb	NOUN
m-1153	28	10	model	model	NOUN
m-1153	28	11	by	by	ADP
m-1153	28	12	including	include	VERB
m-1153	28	13	multiple	multiple	ADJ
m-1153	28	14	transmission	transmission	NOUN
m-1153	28	15	pathways	pathway	NOUN
m-1153	28	16	,	,	PUNCT
m-1153	28	17	capturing	capture	VERB
m-1153	28	18	the	the	DET
m-1153	28	19	complexity	complexity	NOUN
m-1153	28	20	of	of	ADP
m-1153	28	21	cholera	cholera	NOUN
m-1153	28	22	spread	spread	VERB
m-1153	28	23	in	in	ADP
m-1153	28	24	real	real	ADJ
m-1153	28	25	-	-	PUNCT
m-1153	28	26	world	world	NOUN
m-1153	28	27	settings	setting	NOUN
m-1153	28	28	.	.	PUNCT
m-1153	29	1	spatial	spatial	ADJ
m-1153	29	2	extensions	extension	NOUN
m-1153	29	3	,	,	PUNCT
m-1153	29	4	such	such	ADJ
m-1153	29	5	as	as	ADP
m-1153	29	6	those	those	PRON
m-1153	29	7	of	of	ADP
m-1153	29	8	bertuzzo	bertuzzo	PROPN
m-1153	29	9	et	et	PROPN
m-1153	29	10	al	al	PROPN
m-1153	29	11	.	.	PROPN
m-1153	29	12	(	(	PUNCT
m-1153	29	13	2011	2011	NUM
m-1153	29	14	)	)	PUNCT
m-1153	29	15	,	,	PUNCT
m-1153	29	16	incorporated	incorporate	VERB
m-1153	29	17	human	human	ADJ
m-1153	29	18	mobility	mobility	NOUN
m-1153	29	19	and	and	CCONJ
m-1153	29	20	hydrological	hydrological	ADJ
m-1153	29	21	transport	transport	NOUN
m-1153	29	22	,	,	PUNCT
m-1153	29	23	improving	improve	VERB
m-1153	29	24	our	our	PRON
m-1153	29	25	understanding	understanding	NOUN
m-1153	29	26	of	of	ADP
m-1153	29	27	cholera	cholera	NOUN
m-1153	29	28	’s	’s	PART
m-1153	29	29	spatial	spatial	ADJ
m-1153	29	30	dynamics	dynamic	NOUN
m-1153	29	31	during	during	ADP
m-1153	29	32	the	the	DET
m-1153	29	33	haiti	haiti	PROPN
m-1153	29	34	outbreak	outbreak	PROPN
m-1153	29	35	.	.	PUNCT
m-1153	30	1	the	the	DET
m-1153	30	2	theoretical	theoretical	ADJ
m-1153	30	3	foundations	foundation	NOUN
m-1153	30	4	for	for	ADP
m-1153	30	5	threshold	threshold	NOUN
m-1153	30	6	analysis	analysis	NOUN
m-1153	30	7	in	in	ADP
m-1153	30	8	such	such	ADJ
m-1153	30	9	models	model	NOUN
m-1153	30	10	are	be	AUX
m-1153	30	11	built	build	VERB
m-1153	30	12	on	on	ADP
m-1153	30	13	the	the	DET
m-1153	30	14	reproduction	reproduction	NOUN
m-1153	30	15	number	number	NOUN
m-1153	30	16	,	,	PUNCT
m-1153	30	17	r0	r0	NOUN
m-1153	30	18	.	.	PUNCT
m-1153	31	1	van	van	PROPN
m-1153	31	2	den	den	PROPN
m-1153	31	3	driessche	driessche	NOUN
m-1153	31	4	and	and	CCONJ
m-1153	31	5	watmough	watmough	NOUN
m-1153	31	6	(	(	PUNCT
m-1153	31	7	2002	2002	NUM
m-1153	31	8	)	)	PUNCT
m-1153	31	9	established	establish	VERB
m-1153	31	10	a	a	DET
m-1153	31	11	systematic	systematic	ADJ
m-1153	31	12	method	method	NOUN
m-1153	31	13	for	for	ADP
m-1153	31	14	deriving	derive	VERB
m-1153	31	15	r0	r0	NOUN
m-1153	31	16	using	use	VERB
m-1153	31	17	the	the	DET
m-1153	31	18	next	next	ADJ
m-1153	31	19	-	-	PUNCT
m-1153	31	20	generation	generation	NOUN
m-1153	31	21	matrix	matrix	NOUN
m-1153	31	22	approach	approach	NOUN
m-1153	31	23	,	,	PUNCT
m-1153	31	24	which	which	PRON
m-1153	31	25	has	have	AUX
m-1153	31	26	since	since	SCONJ
m-1153	31	27	become	become	VERB
m-1153	31	28	a	a	DET
m-1153	31	29	standard	standard	NOUN
m-1153	31	30	in	in	ADP
m-1153	31	31	epidemiological	epidemiological	ADJ
m-1153	31	32	modeling	modeling	NOUN
m-1153	31	33	.	.	PUNCT
m-1153	32	1	applications	application	NOUN
m-1153	32	2	of	of	ADP
m-1153	32	3	this	this	DET
m-1153	32	4	method	method	NOUN
m-1153	32	5	to	to	ADP
m-1153	32	6	cholera	cholera	NOUN
m-1153	32	7	have	have	AUX
m-1153	32	8	clarified	clarify	VERB
m-1153	32	9	the	the	DET
m-1153	32	10	conditions	condition	NOUN
m-1153	32	11	under	under	ADP
m-1153	32	12	which	which	PRON
m-1153	32	13	outbreaks	outbreak	NOUN
m-1153	32	14	fade	fade	VERB
m-1153	32	15	out	out	ADP
m-1153	32	16	or	or	CCONJ
m-1153	32	17	become	become	VERB
m-1153	32	18	endemic	endemic	ADJ
m-1153	32	19	(	(	PUNCT
m-1153	32	20	mukandavire	mukandavire	NOUN
m-1153	32	21	et	et	PROPN
m-1153	32	22	al	al	PROPN
m-1153	32	23	.	.	PROPN
m-1153	32	24	,	,	PUNCT
m-1153	32	25	2011	2011	NUM
m-1153	32	26	;	;	PUNCT
m-1153	32	27	eisenberg	eisenberg	PROPN
m-1153	32	28	et	et	PROPN
m-1153	32	29	al	al	PROPN
m-1153	32	30	.	.	PROPN
m-1153	32	31	,	,	PUNCT
m-1153	32	32	2013	2013	NUM
m-1153	32	33	)	)	PUNCT
m-1153	32	34	.	.	PUNCT
m-1153	33	1	vaccination	vaccination	NOUN
m-1153	33	2	has	have	AUX
m-1153	33	3	increasingly	increasingly	ADV
m-1153	33	4	been	be	AUX
m-1153	33	5	incorporated	incorporate	VERB
m-1153	33	6	into	into	ADP
m-1153	33	7	cholera	cholera	NOUN
m-1153	33	8	models	model	NOUN
m-1153	33	9	.	.	PUNCT
m-1153	34	1	longini	longini	VERB
m-1153	34	2	et	et	PROPN
m-1153	34	3	al	al	PROPN
m-1153	34	4	.	.	PROPN
m-1153	35	1	(	(	PUNCT
m-1153	35	2	2007	2007	NUM
m-1153	35	3	)	)	PUNCT
m-1153	35	4	and	and	CCONJ
m-1153	35	5	chao	chao	PROPN
m-1153	35	6	et	et	PROPN
m-1153	35	7	al	al	PROPN
m-1153	35	8	.	.	PROPN
m-1153	36	1	(	(	PUNCT
m-1153	36	2	2011	2011	NUM
m-1153	36	3	)	)	PUNCT
m-1153	36	4	examined	examine	VERB
m-1153	36	5	vaccination	vaccination	NOUN
m-1153	36	6	strategies	strategy	NOUN
m-1153	36	7	in	in	ADP
m-1153	36	8	endemic	endemic	ADJ
m-1153	36	9	and	and	CCONJ
m-1153	36	10	epidemic	epidemic	NOUN
m-1153	36	11	settings	setting	NOUN
m-1153	36	12	,	,	PUNCT
m-1153	36	13	showing	show	VERB
m-1153	36	14	how	how	SCONJ
m-1153	36	15	oral	oral	ADJ
m-1153	36	16	cholera	cholera	NOUN
m-1153	36	17	vaccines	vaccine	NOUN
m-1153	36	18	can	can	AUX
m-1153	36	19	reduce	reduce	VERB
m-1153	36	20	r0	r0	NOUN
m-1153	36	21	and	and	CCONJ
m-1153	36	22	shift	shift	VERB
m-1153	36	23	the	the	DET
m-1153	36	24	stability	stability	NOUN
m-1153	36	25	of	of	ADP
m-1153	36	26	equilibria	equilibrium	NOUN
m-1153	36	27	.	.	PUNCT
m-1153	37	1	immunological	immunological	ADJ
m-1153	37	2	studies	study	NOUN
m-1153	37	3	(	(	PUNCT
m-1153	37	4	leung	leung	PROPN
m-1153	37	5	et	et	PROPN
m-1153	37	6	al	al	PROPN
m-1153	37	7	.	.	PROPN
m-1153	37	8	,	,	PUNCT
m-1153	37	9	2012	2012	NUM
m-1153	37	10	)	)	PUNCT
m-1153	37	11	and	and	CCONJ
m-1153	37	12	large	large	ADJ
m-1153	37	13	-	-	PUNCT
m-1153	37	14	scale	scale	NOUN
m-1153	37	15	vaccine	vaccine	NOUN
m-1153	37	16	campaigns	campaign	NOUN
m-1153	37	17	(	(	PUNCT
m-1153	37	18	qadri	qadri	PROPN
m-1153	37	19	ijo	ijo	PROPN
m-1153	37	20	international	international	PROPN
m-1153	37	21	journal	journal	PROPN
m-1153	37	22	of	of	ADP
m-1153	37	23	mathematics	mathematics	PROPN
m-1153	37	24	(	(	PUNCT
m-1153	37	25	issn	issn	PROPN
m-1153	37	26	:	:	PUNCT
m-1153	37	27	2992	2992	NUM
m-1153	37	28	-	-	SYM
m-1153	37	29	4421	4421	NUM
m-1153	37	30	)	)	PUNCT
m-1153	37	31	volume	volume	NOUN
m-1153	37	32	08	08	NUM
m-1153	38	1	|	|	ADV
m-1153	38	2	issue	issue	NOUN
m-1153	38	3	9	9	NUM
m-1153	38	4	|	|	CCONJ
m-1153	38	5	september	september	PROPN
m-1153	38	6	2025	2025	NUM
m-1153	39	1	|	|	ADV
m-1153	39	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	39	3	25	25	NUM
m-1153	39	4	et	et	NOUN
m-1153	39	5	al	al	PROPN
m-1153	39	6	.	.	PROPN
m-1153	39	7	,	,	PUNCT
m-1153	39	8	2020	2020	NUM
m-1153	39	9	)	)	PUNCT
m-1153	39	10	provided	provide	VERB
m-1153	39	11	empirical	empirical	ADJ
m-1153	39	12	support	support	NOUN
m-1153	39	13	for	for	ADP
m-1153	39	14	these	these	DET
m-1153	39	15	models	model	NOUN
m-1153	39	16	,	,	PUNCT
m-1153	39	17	demonstrating	demonstrate	VERB
m-1153	39	18	measurable	measurable	ADJ
m-1153	39	19	impacts	impact	NOUN
m-1153	39	20	of	of	ADP
m-1153	39	21	vaccination	vaccination	NOUN
m-1153	39	22	on	on	ADP
m-1153	39	23	outbreak	outbreak	NOUN
m-1153	39	24	size	size	NOUN
m-1153	39	25	and	and	CCONJ
m-1153	39	26	severity	severity	NOUN
m-1153	39	27	.	.	PUNCT
m-1153	40	1	more	more	ADV
m-1153	40	2	broadly	broadly	ADV
m-1153	40	3	,	,	PUNCT
m-1153	40	4	mathematical	mathematical	ADJ
m-1153	40	5	epidemiology	epidemiology	NOUN
m-1153	40	6	has	have	AUX
m-1153	40	7	benefited	benefit	VERB
m-1153	40	8	from	from	ADP
m-1153	40	9	seminal	seminal	ADJ
m-1153	40	10	works	work	NOUN
m-1153	40	11	like	like	ADP
m-1153	40	12	anderson	anderson	PROPN
m-1153	40	13	and	and	CCONJ
m-1153	40	14	may	may	AUX
m-1153	40	15	(	(	PUNCT
m-1153	40	16	1991	1991	NUM
m-1153	40	17	)	)	PUNCT
m-1153	40	18	and	and	CCONJ
m-1153	40	19	hethcote	hethcote	ADJ
m-1153	40	20	(	(	PUNCT
m-1153	40	21	2000	2000	NUM
m-1153	40	22	)	)	PUNCT
m-1153	40	23	,	,	PUNCT
m-1153	40	24	which	which	PRON
m-1153	40	25	laid	lay	VERB
m-1153	40	26	the	the	DET
m-1153	40	27	theoretical	theoretical	ADJ
m-1153	40	28	groundwork	groundwork	NOUN
m-1153	40	29	for	for	ADP
m-1153	40	30	compartmental	compartmental	ADJ
m-1153	40	31	modeling	modeling	NOUN
m-1153	40	32	,	,	PUNCT
m-1153	40	33	stability	stability	NOUN
m-1153	40	34	analysis	analysis	NOUN
m-1153	40	35	,	,	PUNCT
m-1153	40	36	and	and	CCONJ
m-1153	40	37	disease	disease	NOUN
m-1153	40	38	control	control	NOUN
m-1153	40	39	strategies	strategy	NOUN
m-1153	40	40	.	.	PUNCT
m-1153	41	1	environmental	environmental	ADJ
m-1153	41	2	and	and	CCONJ
m-1153	41	3	climatic	climatic	ADJ
m-1153	41	4	drivers	driver	NOUN
m-1153	41	5	of	of	ADP
m-1153	41	6	cholera	cholera	NOUN
m-1153	41	7	,	,	PUNCT
m-1153	41	8	such	such	ADJ
m-1153	41	9	as	as	ADP
m-1153	41	10	those	those	PRON
m-1153	41	11	linked	link	VERB
m-1153	41	12	to	to	ADP
m-1153	41	13	el	el	PROPN
m-1153	41	14	niño	niño	PROPN
m-1153	41	15	cycles	cycle	NOUN
m-1153	41	16	,	,	PUNCT
m-1153	41	17	were	be	AUX
m-1153	41	18	explored	explore	VERB
m-1153	41	19	by	by	ADP
m-1153	41	20	pascual	pascual	PROPN
m-1153	41	21	et	et	PROPN
m-1153	41	22	al	al	PROPN
m-1153	41	23	.	.	PROPN
m-1153	42	1	(	(	PUNCT
m-1153	42	2	2000	2000	NUM
m-1153	42	3	)	)	PUNCT
m-1153	42	4	and	and	CCONJ
m-1153	42	5	koelle	koelle	NOUN
m-1153	42	6	and	and	CCONJ
m-1153	42	7	pascual	pascual	ADJ
m-1153	42	8	(	(	PUNCT
m-1153	42	9	2004	2004	NUM
m-1153	42	10	)	)	PUNCT
m-1153	42	11	,	,	PUNCT
m-1153	42	12	emphasizing	emphasize	VERB
m-1153	42	13	the	the	DET
m-1153	42	14	need	need	NOUN
m-1153	42	15	to	to	PART
m-1153	42	16	integrate	integrate	VERB
m-1153	42	17	ecological	ecological	ADJ
m-1153	42	18	variability	variability	NOUN
m-1153	42	19	into	into	ADP
m-1153	42	20	cholera	cholera	NOUN
m-1153	42	21	modeling	modeling	NOUN
m-1153	42	22	.	.	PUNCT
m-1153	43	1	recent	recent	ADJ
m-1153	43	2	studies	study	NOUN
m-1153	43	3	continue	continue	VERB
m-1153	43	4	to	to	PART
m-1153	43	5	refine	refine	VERB
m-1153	43	6	these	these	DET
m-1153	43	7	models	model	NOUN
m-1153	43	8	by	by	ADP
m-1153	43	9	combining	combine	VERB
m-1153	43	10	epidemiological	epidemiological	ADJ
m-1153	43	11	,	,	PUNCT
m-1153	43	12	ecological	ecological	ADJ
m-1153	43	13	,	,	PUNCT
m-1153	43	14	and	and	CCONJ
m-1153	43	15	immunological	immunological	ADJ
m-1153	43	16	perspectives	perspective	NOUN
m-1153	43	17	.	.	PUNCT
m-1153	44	1	for	for	ADP
m-1153	44	2	example	example	NOUN
m-1153	44	3	,	,	PUNCT
m-1153	44	4	troeger	troeger	PROPN
m-1153	44	5	et	et	PROPN
m-1153	44	6	al	al	PROPN
m-1153	44	7	.	.	PROPN
m-1153	45	1	(	(	PUNCT
m-1153	45	2	2018	2018	NUM
m-1153	45	3	)	)	PUNCT
m-1153	45	4	and	and	CCONJ
m-1153	45	5	world	world	PROPN
m-1153	45	6	health	health	NOUN
m-1153	45	7	organization	organization	NOUN
m-1153	45	8	(	(	PUNCT
m-1153	45	9	2023	2023	NUM
m-1153	45	10	)	)	PUNCT
m-1153	45	11	highlighted	highlight	VERB
m-1153	45	12	the	the	DET
m-1153	45	13	global	global	ADJ
m-1153	45	14	burden	burden	NOUN
m-1153	45	15	of	of	ADP
m-1153	45	16	cholera	cholera	NOUN
m-1153	45	17	and	and	CCONJ
m-1153	45	18	the	the	DET
m-1153	45	19	urgency	urgency	NOUN
m-1153	45	20	of	of	ADP
m-1153	45	21	integrating	integrate	VERB
m-1153	45	22	vaccination	vaccination	NOUN
m-1153	45	23	with	with	ADP
m-1153	45	24	water	water	NOUN
m-1153	45	25	,	,	PUNCT
m-1153	45	26	sanitation	sanitation	NOUN
m-1153	45	27	,	,	PUNCT
m-1153	45	28	and	and	CCONJ
m-1153	45	29	hygiene	hygiene	NOUN
m-1153	45	30	(	(	PUNCT
m-1153	45	31	wash	wash	NOUN
m-1153	45	32	)	)	PUNCT
m-1153	45	33	measures	measure	NOUN
m-1153	45	34	.	.	PUNCT
m-1153	46	1	collectively	collectively	ADV
m-1153	46	2	,	,	PUNCT
m-1153	46	3	this	this	DET
m-1153	46	4	body	body	NOUN
m-1153	46	5	of	of	ADP
m-1153	46	6	literature	literature	NOUN
m-1153	46	7	underscores	underscore	VERB
m-1153	46	8	the	the	DET
m-1153	46	9	importance	importance	NOUN
m-1153	46	10	of	of	ADP
m-1153	46	11	integrating	integrate	VERB
m-1153	46	12	vaccination	vaccination	NOUN
m-1153	46	13	into	into	ADP
m-1153	46	14	mathematical	mathematical	ADJ
m-1153	46	15	frameworks	framework	NOUN
m-1153	46	16	for	for	ADP
m-1153	46	17	cholera	cholera	NOUN
m-1153	46	18	,	,	PUNCT
m-1153	46	19	not	not	PART
m-1153	46	20	only	only	ADV
m-1153	46	21	for	for	ADP
m-1153	46	22	theoretical	theoretical	ADJ
m-1153	46	23	insights	insight	NOUN
m-1153	46	24	but	but	CCONJ
m-1153	46	25	also	also	ADV
m-1153	46	26	for	for	ADP
m-1153	46	27	informing	inform	VERB
m-1153	46	28	evidence	evidence	NOUN
m-1153	46	29	-	-	PUNCT
m-1153	46	30	based	base	VERB
m-1153	46	31	public	public	ADJ
m-1153	46	32	health	health	NOUN
m-1153	46	33	policies	policy	NOUN
m-1153	46	34	.	.	PUNCT
m-1153	47	1	in	in	ADP
m-1153	47	2	this	this	DET
m-1153	47	3	study	study	NOUN
m-1153	47	4	,	,	PUNCT
m-1153	47	5	we	we	PRON
m-1153	47	6	develop	develop	VERB
m-1153	47	7	and	and	CCONJ
m-1153	47	8	analyze	analyze	VERB
m-1153	47	9	a	a	DET
m-1153	47	10	mathematical	mathematical	ADJ
m-1153	47	11	model	model	NOUN
m-1153	47	12	for	for	ADP
m-1153	47	13	cholera	cholera	NOUN
m-1153	47	14	transmission	transmission	NOUN
m-1153	47	15	that	that	PRON
m-1153	47	16	incorporates	incorporate	VERB
m-1153	47	17	vaccination	vaccination	NOUN
m-1153	47	18	.	.	PUNCT
m-1153	48	1	the	the	DET
m-1153	48	2	model	model	NOUN
m-1153	48	3	builds	build	VERB
m-1153	48	4	on	on	ADP
m-1153	48	5	established	establish	VERB
m-1153	48	6	sirb	sirb	NOUN
m-1153	48	7	frameworks	framework	NOUN
m-1153	48	8	by	by	ADP
m-1153	48	9	including	include	VERB
m-1153	48	10	vaccination	vaccination	NOUN
m-1153	48	11	rate	rate	NOUN
m-1153	48	12	,	,	PUNCT
m-1153	48	13	allowing	allow	VERB
m-1153	48	14	us	we	PRON
m-1153	48	15	to	to	PART
m-1153	48	16	investigate	investigate	VERB
m-1153	48	17	how	how	SCONJ
m-1153	48	18	vaccination	vaccination	NOUN
m-1153	48	19	alters	alter	VERB
m-1153	48	20	epidemic	epidemic	NOUN
m-1153	48	21	thresholds	threshold	NOUN
m-1153	48	22	,	,	PUNCT
m-1153	48	23	disease	disease	NOUN
m-1153	48	24	-	-	PUNCT
m-1153	48	25	free	free	ADJ
m-1153	48	26	equilibrium	equilibrium	NOUN
m-1153	48	27	stability	stability	NOUN
m-1153	48	28	,	,	PUNCT
m-1153	48	29	and	and	CCONJ
m-1153	48	30	endemic	endemic	ADJ
m-1153	48	31	persistence	persistence	NOUN
m-1153	48	32	.	.	PUNCT
m-1153	49	1	2	2	NUM
m-1153	49	2	model	model	NOUN
m-1153	49	3	formulation	formulation	NOUN
m-1153	49	4	we	we	PRON
m-1153	49	5	consider	consider	VERB
m-1153	49	6	a	a	DET
m-1153	49	7	population	population	NOUN
m-1153	49	8	divided	divide	VERB
m-1153	49	9	into	into	ADP
m-1153	49	10	four	four	NUM
m-1153	49	11	main	main	ADJ
m-1153	49	12	compartments	compartment	NOUN
m-1153	49	13	:	:	PUNCT
m-1153	49	14	susceptible	susceptible	ADJ
m-1153	49	15	(	(	PUNCT
m-1153	49	16	s	s	NOUN
m-1153	49	17	)	)	PUNCT
m-1153	49	18	,	,	PUNCT
m-1153	49	19	infected	infect	VERB
m-1153	49	20	(	(	PUNCT
m-1153	49	21	i	i	NOUN
m-1153	49	22	)	)	PUNCT
m-1153	49	23	,	,	PUNCT
m-1153	49	24	recovered	recover	VERB
m-1153	49	25	(	(	PUNCT
m-1153	49	26	r	r	NOUN
m-1153	49	27	)	)	PUNCT
m-1153	49	28	,	,	PUNCT
m-1153	49	29	and	and	CCONJ
m-1153	49	30	bacteria	bacteria	NOUN
m-1153	49	31	concentration	concentration	NOUN
m-1153	49	32	in	in	ADP
m-1153	49	33	the	the	DET
m-1153	49	34	aquatic	aquatic	ADJ
m-1153	49	35	environment	environment	NOUN
m-1153	49	36	(	(	PUNCT
m-1153	49	37	b	b	NOUN
m-1153	49	38	)	)	PUNCT
m-1153	49	39	.	.	PUNCT
m-1153	50	1	the	the	DET
m-1153	50	2	total	total	ADJ
m-1153	50	3	population	population	NOUN
m-1153	50	4	at	at	ADP
m-1153	50	5	time	time	NOUN
m-1153	50	6	t	t	PROPN
m-1153	50	7	is	be	AUX
m-1153	50	8	n(t	n(t	PROPN
m-1153	50	9	)	)	PUNCT
m-1153	50	10	=	=	SYM
m-1153	50	11	s(t	s(t	PROPN
m-1153	50	12	)	)	PUNCT
m-1153	51	1	+	+	NUM
m-1153	51	2	i(t	i(t	NOUN
m-1153	51	3	)	)	PUNCT
m-1153	52	1	+	+	NOUN
m-1153	52	2	r(t	r(t	NOUN
m-1153	52	3	)	)	PUNCT
m-1153	52	4	.	.	PUNCT
m-1153	53	1	the	the	DET
m-1153	53	2	model	model	NOUN
m-1153	53	3	is	be	AUX
m-1153	53	4	governed	govern	VERB
m-1153	53	5	by	by	ADP
m-1153	53	6	the	the	DET
m-1153	53	7	following	follow	VERB
m-1153	53	8	system	system	NOUN
m-1153	53	9	of	of	ADP
m-1153	53	10	nonlinear	nonlinear	ADJ
m-1153	53	11	ordinary	ordinary	ADJ
m-1153	53	12	differential	differential	ADJ
m-1153	53	13	equaijo	equaijo	PROPN
m-1153	53	14	international	international	ADJ
m-1153	53	15	journal	journal	NOUN
m-1153	53	16	of	of	ADP
m-1153	53	17	mathematics	mathematics	PROPN
m-1153	53	18	(	(	PUNCT
m-1153	53	19	issn	issn	PROPN
m-1153	53	20	:	:	PUNCT
m-1153	53	21	2992	2992	NUM
m-1153	53	22	-	-	SYM
m-1153	53	23	4421	4421	NUM
m-1153	53	24	)	)	PUNCT
m-1153	54	1	volume	volume	NOUN
m-1153	54	2	08	08	NUM
m-1153	55	1	|	|	ADV
m-1153	55	2	issue	issue	NOUN
m-1153	55	3	9	9	NUM
m-1153	55	4	|	|	CCONJ
m-1153	55	5	september	september	PROPN
m-1153	55	6	2025	2025	NUM
m-1153	55	7	|	|	ADV
m-1153	55	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	55	9	26	26	NUM
m-1153	55	10	tions	tion	NOUN
m-1153	55	11	:	:	PUNCT
m-1153	55	12	ds	ds	ADJ
m-1153	55	13	dt	dt	NOUN
m-1153	55	14	=	=	SYM
m-1153	55	15	µn	µn	PROPN
m-1153	55	16	−	−	PROPN
m-1153	55	17	βpsi	βpsi	NOUN
m-1153	55	18	−	−	PROPN
m-1153	55	19	βbsb	βbsb	PROPN
m-1153	55	20	−	−	PROPN
m-1153	55	21	(	(	PUNCT
m-1153	55	22	µ+	µ+	NOUN
m-1153	55	23	ν	ν	X
m-1153	55	24	+	+	CCONJ
m-1153	55	25	α)s	α)s	NUM
m-1153	55	26	,	,	PUNCT
m-1153	55	27	(	(	PUNCT
m-1153	55	28	1	1	X
m-1153	55	29	)	)	PUNCT
m-1153	55	30	di	di	NOUN
m-1153	55	31	dt	dt	NOUN
m-1153	56	1	=	=	PUNCT
m-1153	57	1	βpsi	βpsi	NOUN
m-1153	57	2	+	+	CCONJ
m-1153	57	3	βbsb	βbsb	PROPN
m-1153	57	4	−	−	PROPN
m-1153	57	5	(	(	PUNCT
m-1153	57	6	γ	γ	X
m-1153	57	7	+	+	X
m-1153	57	8	µ+	µ+	X
m-1153	57	9	δ)i	δ)i	X
m-1153	57	10	+	+	CCONJ
m-1153	57	11	αs	αs	ADJ
m-1153	57	12	,	,	PUNCT
m-1153	57	13	(	(	PUNCT
m-1153	57	14	2	2	X
m-1153	57	15	)	)	PUNCT
m-1153	57	16	dr	dr	NOUN
m-1153	57	17	dt	dt	NOUN
m-1153	58	1	=	=	PUNCT
m-1153	58	2	γi	γi	INTJ
m-1153	58	3	−	−	PROPN
m-1153	58	4	µr	µr	ADP
m-1153	58	5	+	+	ADJ
m-1153	58	6	νs	νs	PRON
m-1153	58	7	+	+	CCONJ
m-1153	58	8	δi	δi	PROPN
m-1153	58	9	,	,	PUNCT
m-1153	58	10	(	(	PUNCT
m-1153	58	11	3	3	X
m-1153	58	12	)	)	PUNCT
m-1153	58	13	db	db	NOUN
m-1153	58	14	dt	dt	NOUN
m-1153	59	1	=	=	NOUN
m-1153	59	2	ξi	ξi	NOUN
m-1153	59	3	−	−	PROPN
m-1153	59	4	(	(	PUNCT
m-1153	59	5	µb	µb	X
m-1153	59	6	+	+	SYM
m-1153	59	7	η)b	η)b	ADV
m-1153	59	8	,	,	PUNCT
m-1153	59	9	(	(	PUNCT
m-1153	59	10	4	4	NUM
m-1153	59	11	)	)	PUNCT
m-1153	59	12	(	(	PUNCT
m-1153	59	13	5	5	NUM
m-1153	59	14	)	)	PUNCT
m-1153	59	15	where	where	SCONJ
m-1153	59	16	:	:	PUNCT
m-1153	59	17	•	•	NUM
m-1153	59	18	µ	µ	NOUN
m-1153	59	19	is	be	AUX
m-1153	59	20	the	the	DET
m-1153	59	21	natural	natural	ADJ
m-1153	59	22	birth	birth	NOUN
m-1153	59	23	/	/	SYM
m-1153	59	24	death	death	NOUN
m-1153	59	25	rate	rate	NOUN
m-1153	59	26	,	,	PUNCT
m-1153	59	27	•	•	NUM
m-1153	59	28	βp	βp	PROPN
m-1153	59	29	is	be	AUX
m-1153	59	30	the	the	DET
m-1153	59	31	transmission	transmission	NOUN
m-1153	59	32	coefficient	coefficient	NOUN
m-1153	59	33	for	for	ADP
m-1153	59	34	direct	direct	ADJ
m-1153	59	35	human	human	NOUN
m-1153	59	36	-	-	PUNCT
m-1153	59	37	to	to	ADP
m-1153	59	38	-	-	PUNCT
m-1153	59	39	human	human	ADJ
m-1153	59	40	infection	infection	NOUN
m-1153	59	41	,	,	PUNCT
m-1153	59	42	•	•	ADP
m-1153	59	43	βb	βb	ADV
m-1153	59	44	is	be	AUX
m-1153	59	45	the	the	DET
m-1153	59	46	transmission	transmission	NOUN
m-1153	59	47	coefficient	coefficient	NOUN
m-1153	59	48	for	for	ADP
m-1153	59	49	infection	infection	NOUN
m-1153	59	50	via	via	ADP
m-1153	59	51	the	the	DET
m-1153	59	52	aquatic	aquatic	ADJ
m-1153	59	53	reservoir	reservoir	NOUN
m-1153	59	54	,	,	PUNCT
m-1153	59	55	•	•	ADP
m-1153	59	56	γ	γ	PROPN
m-1153	59	57	is	be	AUX
m-1153	59	58	the	the	DET
m-1153	59	59	recovery	recovery	NOUN
m-1153	59	60	rate	rate	NOUN
m-1153	59	61	,	,	PUNCT
m-1153	59	62	•	•	NUM
m-1153	59	63	δ	δ	PROPN
m-1153	59	64	is	be	AUX
m-1153	59	65	the	the	DET
m-1153	59	66	disease	disease	NOUN
m-1153	59	67	-	-	PUNCT
m-1153	59	68	induced	induce	VERB
m-1153	59	69	mortality	mortality	NOUN
m-1153	59	70	rate	rate	NOUN
m-1153	59	71	,	,	PUNCT
m-1153	59	72	•	•	NUM
m-1153	59	73	ν	ν	NOUN
m-1153	59	74	is	be	AUX
m-1153	59	75	the	the	DET
m-1153	59	76	vaccination	vaccination	NOUN
m-1153	59	77	rate	rate	NOUN
m-1153	59	78	,	,	PUNCT
m-1153	59	79	•	•	NUM
m-1153	59	80	α	α	NOUN
m-1153	59	81	is	be	AUX
m-1153	59	82	the	the	DET
m-1153	59	83	rate	rate	NOUN
m-1153	59	84	of	of	ADP
m-1153	59	85	waning	wane	VERB
m-1153	59	86	immunity	immunity	NOUN
m-1153	59	87	back	back	ADV
m-1153	59	88	to	to	ADP
m-1153	59	89	susceptibility	susceptibility	NOUN
m-1153	59	90	,	,	PUNCT
m-1153	59	91	•	•	NUM
m-1153	59	92	ξ	ξ	PROPN
m-1153	59	93	is	be	AUX
m-1153	59	94	the	the	DET
m-1153	59	95	bacterial	bacterial	ADJ
m-1153	59	96	shedding	shedding	ADJ
m-1153	59	97	rate	rate	NOUN
m-1153	59	98	from	from	ADP
m-1153	59	99	infected	infected	ADJ
m-1153	59	100	individuals	individual	NOUN
m-1153	59	101	,	,	PUNCT
m-1153	59	102	•	•	NUM
m-1153	59	103	µb	µb	NOUN
m-1153	59	104	is	be	AUX
m-1153	59	105	the	the	DET
m-1153	59	106	bacterial	bacterial	ADJ
m-1153	59	107	natural	natural	ADJ
m-1153	59	108	death	death	NOUN
m-1153	59	109	rate	rate	NOUN
m-1153	59	110	,	,	PUNCT
m-1153	59	111	and	and	CCONJ
m-1153	59	112	•	•	NUM
m-1153	59	113	η	η	PROPN
m-1153	59	114	is	be	AUX
m-1153	59	115	the	the	DET
m-1153	59	116	bacterial	bacterial	ADJ
m-1153	59	117	removal	removal	NOUN
m-1153	59	118	rate	rate	NOUN
m-1153	59	119	due	due	ADP
m-1153	59	120	to	to	ADP
m-1153	59	121	environmental	environmental	ADJ
m-1153	59	122	sanitation	sanitation	NOUN
m-1153	59	123	.	.	PUNCT
m-1153	60	1	ijo	ijo	PROPN
m-1153	60	2	international	international	PROPN
m-1153	60	3	journal	journal	PROPN
m-1153	60	4	of	of	ADP
m-1153	60	5	mathematics	mathematics	PROPN
m-1153	60	6	(	(	PUNCT
m-1153	60	7	issn	issn	PROPN
m-1153	60	8	:	:	PUNCT
m-1153	60	9	2992	2992	NUM
m-1153	60	10	-	-	SYM
m-1153	60	11	4421	4421	NUM
m-1153	60	12	)	)	PUNCT
m-1153	60	13	volume	volume	NOUN
m-1153	60	14	08	08	NUM
m-1153	61	1	|	|	ADV
m-1153	61	2	issue	issue	NOUN
m-1153	61	3	9	9	NUM
m-1153	61	4	|	|	CCONJ
m-1153	61	5	september	september	PROPN
m-1153	61	6	2025	2025	NUM
m-1153	62	1	|	|	ADV
m-1153	62	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	62	3	27	27	NUM
m-1153	62	4	3	3	NUM
m-1153	62	5	the	the	DET
m-1153	62	6	equilibrium	equilibrium	NOUN
m-1153	62	7	point	point	NOUN
m-1153	62	8	3.1	3.1	NUM
m-1153	62	9	notation	notation	NOUN
m-1153	62	10	introduce	introduce	NOUN
m-1153	62	11	compact	compact	ADJ
m-1153	62	12	notation	notation	NOUN
m-1153	62	13	:	:	PUNCT
m-1153	62	14	ρ	ρ	NOUN
m-1153	62	15	:	:	PUNCT
m-1153	62	16	=	=	X
m-1153	62	17	µ+	µ+	PUNCT
m-1153	62	18	ν	ν	X
m-1153	62	19	+	+	PROPN
m-1153	62	20	α	α	PROPN
m-1153	62	21	,	,	PUNCT
m-1153	62	22	σ	σ	NOUN
m-1153	62	23	:	:	PUNCT
m-1153	62	24	=	=	SYM
m-1153	62	25	γ	γ	X
m-1153	62	26	+	+	X
m-1153	62	27	µ+	µ+	PROPN
m-1153	62	28	τδ	τδ	NOUN
m-1153	62	29	,	,	PUNCT
m-1153	62	30	:	:	PUNCT
m-1153	62	31	=	=	PUNCT
m-1153	62	32	µb	µb	VERB
m-1153	62	33	+	+	ADV
m-1153	62	34	aη	aη	ADJ
m-1153	62	35	,	,	PUNCT
m-1153	62	36	:	:	PUNCT
m-1153	63	1	=	=	SYM
m-1153	63	2	βp	βp	PROPN
m-1153	64	1	+	+	CCONJ
m-1153	64	2	βb	βb	PRON
m-1153	64	3	ξ	ξ	PROPN
m-1153	64	4	τ	τ	PROPN
m-1153	64	5	.	.	PUNCT
m-1153	64	6	3.2	3.2	NUM
m-1153	64	7	disease	disease	NOUN
m-1153	64	8	-	-	PUNCT
m-1153	64	9	free	free	ADJ
m-1153	64	10	equilibrium	equilibrium	NOUN
m-1153	64	11	(	(	PUNCT
m-1153	64	12	dfe	dfe	PROPN
m-1153	64	13	)	)	PUNCT
m-1153	64	14	at	at	ADP
m-1153	64	15	the	the	DET
m-1153	64	16	disease	disease	NOUN
m-1153	64	17	-	-	PUNCT
m-1153	64	18	free	free	ADJ
m-1153	64	19	equilibrium	equilibrium	NOUN
m-1153	64	20	,	,	PUNCT
m-1153	64	21	i∗	i∗	NOUN
m-1153	64	22	=	=	SYM
m-1153	64	23	0	0	NUM
m-1153	64	24	and	and	CCONJ
m-1153	64	25	b∗	b∗	ADJ
m-1153	64	26	=	=	SYM
m-1153	64	27	0	0	X
m-1153	64	28	.	.	PUNCT
m-1153	65	1	from	from	ADP
m-1153	65	2	(	(	PUNCT
m-1153	65	3	1	1	NUM
m-1153	65	4	):	):	SYM
m-1153	65	5	0	0	NUM
m-1153	65	6	=	=	SYM
m-1153	65	7	µn	µn	PROPN
m-1153	65	8	−	−	NOUN
m-1153	65	9	ρs∗	ρs∗	NOUN
m-1153	65	10	=	=	NOUN
m-1153	65	11	⇒	⇒	NOUN
m-1153	65	12	s∗	s∗	PROPN
m-1153	65	13	=	=	SYM
m-1153	65	14	µn	µn	PROPN
m-1153	65	15	ρ	ρ	PROPN
m-1153	65	16	.	.	PUNCT
m-1153	66	1	from	from	ADP
m-1153	66	2	(	(	PUNCT
m-1153	66	3	3	3	NUM
m-1153	66	4	)	)	PUNCT
m-1153	66	5	(	(	PUNCT
m-1153	66	6	with	with	ADP
m-1153	66	7	i∗	i∗	NOUN
m-1153	66	8	=	=	SYM
m-1153	66	9	0	0	NUM
m-1153	66	10	):	):	PUNCT
m-1153	66	11	0	0	NUM
m-1153	66	12	=	=	SYM
m-1153	66	13	−µr∗	−µr∗	PROPN
m-1153	66	14	+	+	CCONJ
m-1153	66	15	νs∗	νs∗	NOUN
m-1153	66	16	=	=	NOUN
m-1153	66	17	⇒	⇒	NOUN
m-1153	66	18	r∗	r∗	NOUN
m-1153	66	19	=	=	PUNCT
m-1153	66	20	ν	ν	X
m-1153	66	21	µ	µ	X
m-1153	66	22	s∗.	s∗.	ADJ
m-1153	66	23	thus	thus	ADV
m-1153	66	24	,	,	PUNCT
m-1153	66	25	(	(	PUNCT
m-1153	66	26	s∗	s∗	PROPN
m-1153	66	27	,	,	PUNCT
m-1153	66	28	i∗	i∗	NOUN
m-1153	66	29	,	,	PUNCT
m-1153	66	30	r∗	r∗	ADJ
m-1153	66	31	,	,	PUNCT
m-1153	66	32	b∗	b∗	ADJ
m-1153	66	33	)	)	PUNCT
m-1153	66	34	=	=	PUNCT
m-1153	67	1	(	(	PUNCT
m-1153	67	2	µn	µn	PROPN
m-1153	67	3	ρ	ρ	PROPN
m-1153	67	4	,	,	PUNCT
m-1153	67	5	0	0	NUM
m-1153	67	6	,	,	PUNCT
m-1153	67	7	νn	νn	VERB
m-1153	67	8	ρ	ρ	NUM
m-1153	67	9	,	,	PUNCT
m-1153	67	10	0	0	NUM
m-1153	67	11	)	)	PUNCT
m-1153	67	12	.	.	PUNCT
m-1153	68	1	however	however	ADV
m-1153	68	2	,	,	PUNCT
m-1153	68	3	substituting	substitute	VERB
m-1153	68	4	i∗	i∗	NOUN
m-1153	68	5	=	=	SYM
m-1153	68	6	0	0	NUM
m-1153	68	7	and	and	CCONJ
m-1153	68	8	b∗	b∗	ADJ
m-1153	68	9	=	=	SYM
m-1153	68	10	0	0	PUNCT
m-1153	68	11	into	into	ADP
m-1153	68	12	(	(	PUNCT
m-1153	68	13	2	2	X
m-1153	68	14	)	)	PUNCT
m-1153	68	15	gives	give	VERB
m-1153	68	16	0	0	NUM
m-1153	69	1	=	=	SYM
m-1153	69	2	αs∗.	αs∗.	PROPN
m-1153	69	3	hence	hence	ADV
m-1153	69	4	,	,	PUNCT
m-1153	69	5	for	for	SCONJ
m-1153	69	6	a	a	DET
m-1153	69	7	true	true	ADJ
m-1153	69	8	dfe	dfe	PROPN
m-1153	69	9	to	to	PART
m-1153	69	10	exist	exist	VERB
m-1153	69	11	we	we	PRON
m-1153	69	12	require	require	VERB
m-1153	69	13	α	α	NOUN
m-1153	69	14	=	=	NOUN
m-1153	69	15	0	0	NUM
m-1153	69	16	.	.	PUNCT
m-1153	70	1	therefore	therefore	ADV
m-1153	70	2	:	:	PUNCT
m-1153	70	3	if	if	SCONJ
m-1153	70	4	α	α	PROPN
m-1153	70	5	=	=	NOUN
m-1153	70	6	0	0	NUM
m-1153	70	7	:	:	PUNCT
m-1153	70	8	(	(	PUNCT
m-1153	70	9	s∗	s∗	PROPN
m-1153	70	10	,	,	PUNCT
m-1153	70	11	i∗	i∗	NOUN
m-1153	70	12	,	,	PUNCT
m-1153	70	13	r∗	r∗	ADJ
m-1153	70	14	,	,	PUNCT
m-1153	70	15	b∗	b∗	ADJ
m-1153	70	16	)	)	PUNCT
m-1153	70	17	=	=	SYM
m-1153	70	18	(	(	PUNCT
m-1153	70	19	µn	µn	PROPN
m-1153	70	20	µ+	µ+	VERB
m-1153	70	21	ν	ν	NOUN
m-1153	70	22	,	,	PUNCT
m-1153	70	23	0	0	NUM
m-1153	70	24	,	,	PUNCT
m-1153	70	25	νn	νn	ADJ
m-1153	70	26	µ+	µ+	PRON
m-1153	70	27	ν	ν	NOUN
m-1153	70	28	,	,	PUNCT
m-1153	70	29	0	0	NUM
m-1153	70	30	)	)	PUNCT
m-1153	70	31	.	.	PUNCT
m-1153	71	1	ijo	ijo	PROPN
m-1153	71	2	international	international	PROPN
m-1153	71	3	journal	journal	PROPN
m-1153	71	4	of	of	ADP
m-1153	71	5	mathematics	mathematics	PROPN
m-1153	71	6	(	(	PUNCT
m-1153	71	7	issn	issn	PROPN
m-1153	71	8	:	:	PUNCT
m-1153	71	9	2992	2992	NUM
m-1153	71	10	-	-	SYM
m-1153	71	11	4421	4421	NUM
m-1153	71	12	)	)	PUNCT
m-1153	71	13	volume	volume	NOUN
m-1153	71	14	08	08	NUM
m-1153	72	1	|	|	ADV
m-1153	72	2	issue	issue	NOUN
m-1153	72	3	9	9	NUM
m-1153	72	4	|	|	CCONJ
m-1153	72	5	september	september	PROPN
m-1153	72	6	2025	2025	NUM
m-1153	73	1	|	|	ADV
m-1153	73	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	73	3	28	28	NUM
m-1153	73	4	3.3	3.3	NUM
m-1153	73	5	endemic	endemic	ADJ
m-1153	73	6	equilibrium	equilibrium	NOUN
m-1153	73	7	(	(	PUNCT
m-1153	73	8	general	general	ADJ
m-1153	73	9	case	case	NOUN
m-1153	73	10	i∗	i∗	NOUN
m-1153	73	11	>	>	X
m-1153	73	12	0	0	NUM
m-1153	73	13	)	)	PUNCT
m-1153	73	14	step	step	NOUN
m-1153	73	15	1	1	NUM
m-1153	73	16	.	.	PUNCT
m-1153	73	17	from	from	ADP
m-1153	73	18	the	the	DET
m-1153	73	19	b	b	PROPN
m-1153	73	20	equation	equation	NOUN
m-1153	73	21	:	:	PUNCT
m-1153	73	22	0	0	PUNCT
m-1153	74	1	=	=	NOUN
m-1153	74	2	ξi∗	ξi∗	ADJ
m-1153	74	3	−	−	NOUN
m-1153	74	4	τb∗	τb∗	VERB
m-1153	75	1	=	=	NOUN
m-1153	75	2	⇒	⇒	VERB
m-1153	75	3	b∗	b∗	NOUN
m-1153	75	4	=	=	SYM
m-1153	75	5	ξ	ξ	PRON
m-1153	75	6	τ	τ	X
m-1153	75	7	i∗.	i∗.	PROPN
m-1153	75	8	step	step	NOUN
m-1153	75	9	2	2	NUM
m-1153	75	10	.	.	PUNCT
m-1153	75	11	combine	combine	VERB
m-1153	75	12	the	the	DET
m-1153	75	13	s	s	X
m-1153	75	14	and	and	CCONJ
m-1153	75	15	i	i	PRON
m-1153	75	16	equations	equation	VERB
m-1153	75	17	:	:	PUNCT
m-1153	75	18	0	0	NUM
m-1153	76	1	=	=	SYM
m-1153	76	2	µn	µn	PROPN
m-1153	76	3	−	−	PROPN
m-1153	76	4	βps	βp	NOUN
m-1153	76	5	∗i∗	∗i∗	NUM
m-1153	76	6	−	−	NOUN
m-1153	76	7	βbs	βbs	NOUN
m-1153	76	8	∗b∗	∗b∗	NUM
m-1153	76	9	−	−	NOUN
m-1153	76	10	ρs∗	ρs∗	NOUN
m-1153	76	11	,	,	PUNCT
m-1153	76	12	0	0	NUM
m-1153	76	13	=	=	SYM
m-1153	76	14	βps	βp	NOUN
m-1153	76	15	∗i∗	∗i∗	NUM
m-1153	77	1	+	+	NUM
m-1153	77	2	βbs	βbs	VERB
m-1153	77	3	∗b∗	∗b∗	NUM
m-1153	77	4	−	−	NOUN
m-1153	77	5	σi∗	σi∗	NOUN
m-1153	77	6	+	+	CCONJ
m-1153	77	7	αs∗.	αs∗.	PROPN
m-1153	77	8	adding	add	VERB
m-1153	77	9	these	these	DET
m-1153	77	10	gives	give	VERB
m-1153	77	11	0	0	NUM
m-1153	77	12	=	=	SYM
m-1153	77	13	µn	µn	PROPN
m-1153	77	14	−	−	PROPN
m-1153	77	15	(	(	PUNCT
m-1153	77	16	ρ−	ρ−	NOUN
m-1153	77	17	α)s∗	α)s∗	NUM
m-1153	77	18	−	−	PROPN
m-1153	77	19	σi∗.	σi∗.	PROPN
m-1153	77	20	since	since	SCONJ
m-1153	77	21	ρ−	ρ−	PROPN
m-1153	77	22	α	α	NOUN
m-1153	77	23	=	=	SYM
m-1153	77	24	µ+	µ+	X
m-1153	77	25	ν	ν	NOUN
m-1153	77	26	,	,	PUNCT
m-1153	77	27	σi∗	σi∗	NOUN
m-1153	77	28	=	=	SYM
m-1153	77	29	µn	µn	PROPN
m-1153	77	30	−	−	PROPN
m-1153	77	31	(	(	PUNCT
m-1153	77	32	µ+	µ+	X
m-1153	77	33	ν)s∗	ν)s∗	ADJ
m-1153	77	34	i	i	PROPN
m-1153	77	35	,	,	PUNCT
m-1153	77	36	∗	∗	NOUN
m-1153	77	37	=	=	SYM
m-1153	77	38	µn	µn	PROPN
m-1153	77	39	−	−	PROPN
m-1153	77	40	(	(	PUNCT
m-1153	77	41	µ+	µ+	X
m-1153	77	42	ν)s∗	ν)s∗	ADJ
m-1153	77	43	σ	σ	NOUN
m-1153	77	44	.	.	PUNCT
m-1153	78	1	step	step	NOUN
m-1153	78	2	3	3	NUM
m-1153	78	3	.	.	PUNCT
m-1153	78	4	substitute	substitute	ADJ
m-1153	78	5	i∗	i∗	NOUN
m-1153	78	6	and	and	CCONJ
m-1153	78	7	b∗	b∗	ADJ
m-1153	78	8	into	into	ADP
m-1153	78	9	the	the	DET
m-1153	78	10	s	s	PART
m-1153	78	11	equation	equation	NOUN
m-1153	78	12	:	:	PUNCT
m-1153	78	13	0	0	NUM
m-1153	79	1	=	=	SYM
m-1153	79	2	µn	µn	PROPN
m-1153	79	3	−	−	PROPN
m-1153	79	4	s∗	s∗	PROPN
m-1153	79	5	(	(	PUNCT
m-1153	79	6	βpi	βpi	NOUN
m-1153	79	7	∗	∗	NOUN
m-1153	79	8	+	+	CCONJ
m-1153	79	9	βbb	βbb	X
m-1153	79	10	∗)−	∗)−	ADP
m-1153	79	11	ρs∗.	ρs∗.	PROPN
m-1153	79	12	using	use	VERB
m-1153	79	13	b∗	b∗	PROPN
m-1153	79	14	=	=	SYM
m-1153	79	15	ξ	ξ	PROPN
m-1153	79	16	τ	τ	X
m-1153	79	17	i∗	i∗	NOUN
m-1153	79	18	and	and	CCONJ
m-1153	79	19	defining	define	VERB
m-1153	79	20	a	a	PRON
m-1153	79	21	:	:	PUNCT
m-1153	79	22	=	=	SYM
m-1153	79	23	βp	βp	PROPN
m-1153	80	1	+	+	CCONJ
m-1153	80	2	βb	βb	PRON
m-1153	80	3	ξ	ξ	PROPN
m-1153	80	4	τ	τ	PROPN
m-1153	80	5	,	,	PUNCT
m-1153	80	6	0	0	NUM
m-1153	80	7	=	=	SYM
m-1153	80	8	µn	µn	PROPN
m-1153	80	9	−	−	ADP
m-1153	80	10	as∗i∗	as∗i∗	NUM
m-1153	80	11	−	−	PROPN
m-1153	80	12	ρs∗.	ρs∗.	PROPN
m-1153	80	13	substitute	substitute	NOUN
m-1153	80	14	i∗	i∗	NOUN
m-1153	80	15	=	=	PUNCT
m-1153	80	16	µn−(µ+ν)s∗	µn−(µ+ν)s∗	ADJ
m-1153	80	17	σ	σ	NOUN
m-1153	80	18	:	:	PUNCT
m-1153	80	19	µn	µn	PROPN
m-1153	80	20	−	−	PROPN
m-1153	80	21	as∗	as∗	NOUN
m-1153	80	22	σ	σ	PROPN
m-1153	80	23	(	(	PUNCT
m-1153	80	24	µn	µn	PROPN
m-1153	80	25	−	−	PROPN
m-1153	80	26	(	(	PUNCT
m-1153	80	27	µ+	µ+	PROPN
m-1153	80	28	ν)s∗)−	ν)s∗)−	PROPN
m-1153	80	29	ρs∗	ρs∗	VERB
m-1153	81	1	=	=	SYM
m-1153	81	2	0	0	X
m-1153	81	3	.	.	X
m-1153	81	4	step	step	NOUN
m-1153	81	5	4	4	NUM
m-1153	81	6	.	.	PUNCT
m-1153	82	1	rearranging	rearrange	VERB
m-1153	82	2	:	:	PUNCT
m-1153	82	3	a(µ+	a(µ+	PROPN
m-1153	82	4	ν)s∗2	ν)s∗2	X
m-1153	82	5	−	−	PROPN
m-1153	82	6	(	(	PUNCT
m-1153	82	7	aµn	aµn	NOUN
m-1153	82	8	+	+	X
m-1153	82	9	ρσ	ρσ	X
m-1153	82	10	)	)	PUNCT
m-1153	82	11	s∗	s∗	PROPN
m-1153	82	12	+	+	CCONJ
m-1153	82	13	σµn	σµn	ADV
m-1153	82	14	=	=	SYM
m-1153	82	15	0	0	X
m-1153	82	16	.	.	PUNCT
m-1153	83	1	ijo	ijo	PROPN
m-1153	83	2	international	international	PROPN
m-1153	83	3	journal	journal	PROPN
m-1153	83	4	of	of	ADP
m-1153	83	5	mathematics	mathematics	PROPN
m-1153	83	6	(	(	PUNCT
m-1153	83	7	issn	issn	PROPN
m-1153	83	8	:	:	PUNCT
m-1153	83	9	2992	2992	NUM
m-1153	83	10	-	-	SYM
m-1153	83	11	4421	4421	NUM
m-1153	83	12	)	)	PUNCT
m-1153	83	13	volume	volume	NOUN
m-1153	83	14	08	08	NUM
m-1153	84	1	|	|	ADV
m-1153	84	2	issue	issue	NOUN
m-1153	84	3	9	9	NUM
m-1153	84	4	|	|	CCONJ
m-1153	84	5	september	september	PROPN
m-1153	84	6	2025	2025	NUM
m-1153	85	1	|	|	ADV
m-1153	85	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	85	3	29	29	NUM
m-1153	85	4	this	this	PRON
m-1153	85	5	is	be	AUX
m-1153	85	6	a	a	DET
m-1153	85	7	quadratic	quadratic	NOUN
m-1153	85	8	in	in	ADP
m-1153	85	9	s∗.	s∗.	ADJ
m-1153	85	10	step	step	NOUN
m-1153	85	11	5	5	NUM
m-1153	85	12	.	.	PUNCT
m-1153	86	1	solution	solution	NOUN
m-1153	86	2	:	:	PUNCT
m-1153	86	3	s∗	s∗	PROPN
m-1153	86	4	=	=	PUNCT
m-1153	86	5	aµn	aµn	NOUN
m-1153	86	6	+	+	CCONJ
m-1153	86	7	ρσ	ρσ	PART
m-1153	86	8	±	±	NUM
m-1153	86	9	√	√	VERB
m-1153	86	10	(	(	PUNCT
m-1153	86	11	aµn	aµn	NOUN
m-1153	86	12	+	+	X
m-1153	86	13	ρσ	ρσ	NOUN
m-1153	86	14	)	)	PUNCT
m-1153	86	15	2	2	NUM
m-1153	86	16	−	−	PROPN
m-1153	86	17	4a(µ+	4a(µ+	PROPN
m-1153	86	18	ν)σµn	ν)σµn	NOUN
m-1153	86	19	2a(µ+	2a(µ+	NUM
m-1153	86	20	ν	ν	NOUN
m-1153	86	21	)	)	PUNCT
m-1153	86	22	.	.	PUNCT
m-1153	87	1	then	then	ADV
m-1153	87	2	i∗	i∗	NOUN
m-1153	87	3	=	=	SYM
m-1153	87	4	µn	µn	PROPN
m-1153	87	5	−	−	PROPN
m-1153	87	6	(	(	PUNCT
m-1153	87	7	µ+	µ+	X
m-1153	87	8	ν)s∗	ν)s∗	ADJ
m-1153	87	9	σ	σ	NOUN
m-1153	87	10	,	,	PUNCT
m-1153	87	11	b∗	b∗	PROPN
m-1153	87	12	=	=	SYM
m-1153	87	13	ξ	ξ	X
m-1153	87	14	τ	τ	X
m-1153	87	15	i∗	i∗	PROPN
m-1153	87	16	r	r	NOUN
m-1153	87	17	,	,	PUNCT
m-1153	87	18	∗	∗	NOUN
m-1153	87	19	=	=	PUNCT
m-1153	87	20	(	(	PUNCT
m-1153	87	21	γ	γ	X
m-1153	87	22	+	+	NUM
m-1153	87	23	δ)i∗	δ)i∗	NOUN
m-1153	87	24	+	+	NUM
m-1153	87	25	νs∗	νs∗	PROPN
m-1153	87	26	µ	µ	PROPN
m-1153	87	27	.	.	PUNCT
m-1153	87	28	3.4	3.4	NUM
m-1153	87	29	special	special	ADJ
m-1153	87	30	case	case	NOUN
m-1153	87	31	:	:	PUNCT
m-1153	87	32	α	α	X
m-1153	87	33	=	=	NOUN
m-1153	87	34	0	0	NUM
m-1153	87	35	when	when	SCONJ
m-1153	87	36	α	α	PROPN
m-1153	87	37	=	=	SYM
m-1153	87	38	0	0	PROPN
m-1153	87	39	,	,	PUNCT
m-1153	87	40	the	the	DET
m-1153	87	41	endemic	endemic	ADJ
m-1153	87	42	equilibrium	equilibrium	NOUN
m-1153	87	43	simplifies	simplifie	NOUN
m-1153	87	44	.	.	PUNCT
m-1153	88	1	from	from	ADP
m-1153	88	2	(	(	PUNCT
m-1153	88	3	2	2	NUM
m-1153	88	4	)	)	PUNCT
m-1153	88	5	(	(	PUNCT
m-1153	88	6	with	with	ADP
m-1153	88	7	i∗	i∗	NOUN
m-1153	88	8	>	>	X
m-1153	88	9	0	0	NUM
m-1153	88	10	):	):	PUNCT
m-1153	88	11	βps	βp	NOUN
m-1153	88	12	∗	∗	NOUN
m-1153	88	13	+	+	CCONJ
m-1153	88	14	βbs	βbs	PROPN
m-1153	88	15	∗	∗	NOUN
m-1153	88	16	ξ	ξ	X
m-1153	88	17	τ	τ	X
m-1153	88	18	=	=	SYM
m-1153	88	19	σ	σ	PROPN
m-1153	88	20	=	=	NOUN
m-1153	88	21	⇒	⇒	NOUN
m-1153	88	22	s∗	s∗	PROPN
m-1153	88	23	=	=	PROPN
m-1153	88	24	σ	σ	PROPN
m-1153	88	25	a	a	PRON
m-1153	88	26	.	.	PUNCT
m-1153	89	1	define	define	VERB
m-1153	89	2	the	the	DET
m-1153	89	3	basic	basic	ADJ
m-1153	89	4	reproduction	reproduction	NOUN
m-1153	89	5	number	number	NOUN
m-1153	89	6	r0	r0	NOUN
m-1153	89	7	:	:	PUNCT
m-1153	89	8	=	=	PUNCT
m-1153	89	9	an	an	PRON
m-1153	89	10	σ	σ	NOUN
m-1153	89	11	=	=	PUNCT
m-1153	89	12	βpn	βpn	NOUN
m-1153	89	13	σ	σ	PROPN
m-1153	90	1	+	+	CCONJ
m-1153	90	2	βbnξ	βbnξ	VERB
m-1153	90	3	στ	στ	INTJ
m-1153	90	4	.	.	PUNCT
m-1153	91	1	then	then	ADV
m-1153	91	2	s∗	s∗	PROPN
m-1153	91	3	=	=	SYM
m-1153	91	4	n	n	PRON
m-1153	91	5	r0	r0	NOUN
m-1153	91	6	,	,	PUNCT
m-1153	91	7	i∗	i∗	NOUN
m-1153	91	8	=	=	SYM
m-1153	91	9	µn	µn	PROPN
m-1153	91	10	σ	σ	X
m-1153	91	11	(	(	PUNCT
m-1153	91	12	1−	1−	NUM
m-1153	91	13	1	1	NUM
m-1153	91	14	r0	r0	NOUN
m-1153	91	15	)	)	PUNCT
m-1153	91	16	,	,	PUNCT
m-1153	91	17	b∗	b∗	ADV
m-1153	92	1	=	=	SYM
m-1153	92	2	ξ	ξ	X
m-1153	92	3	τ	τ	PROPN
m-1153	92	4	i∗.	i∗.	PROPN
m-1153	92	5	an	an	DET
m-1153	92	6	endemic	endemic	ADJ
m-1153	92	7	equilibrium	equilibrium	NOUN
m-1153	92	8	(	(	PUNCT
m-1153	92	9	i∗	i∗	NOUN
m-1153	92	10	>	>	X
m-1153	92	11	0	0	NUM
m-1153	92	12	)	)	PUNCT
m-1153	92	13	exists	exist	VERB
m-1153	92	14	iff	iff	PROPN
m-1153	92	15	r0	r0	PROPN
m-1153	92	16	>	>	X
m-1153	92	17	1	1	NUM
m-1153	92	18	.	.	SYM
m-1153	92	19	4	4	NUM
m-1153	92	20	basic	basic	ADJ
m-1153	92	21	reproduction	reproduction	NOUN
m-1153	92	22	number	number	NOUN
m-1153	92	23	4.1	4.1	NUM
m-1153	92	24	assumption	assumption	NOUN
m-1153	92	25	(	(	PUNCT
m-1153	92	26	existence	existence	NOUN
m-1153	92	27	of	of	ADP
m-1153	92	28	dfe	dfe	PROPN
m-1153	92	29	)	)	PUNCT
m-1153	92	30	the	the	DET
m-1153	92	31	next	next	ADJ
m-1153	92	32	-	-	PUNCT
m-1153	92	33	generation	generation	NOUN
m-1153	92	34	matrix	matrix	NOUN
m-1153	92	35	method	method	NOUN
m-1153	92	36	requires	require	VERB
m-1153	92	37	a	a	DET
m-1153	92	38	disease	disease	NOUN
m-1153	92	39	-	-	PUNCT
m-1153	92	40	free	free	ADJ
m-1153	92	41	equilibrium	equilibrium	NOUN
m-1153	92	42	(	(	PUNCT
m-1153	92	43	dfe	dfe	PROPN
m-1153	92	44	)	)	PUNCT
m-1153	92	45	.	.	PUNCT
m-1153	93	1	a	a	DET
m-1153	93	2	dfe	dfe	PROPN
m-1153	93	3	with	with	ADP
m-1153	93	4	i∗	i∗	NOUN
m-1153	93	5	=	=	PUNCT
m-1153	93	6	b∗	b∗	ADJ
m-1153	93	7	=	=	SYM
m-1153	93	8	0	0	NUM
m-1153	93	9	exists	exist	VERB
m-1153	93	10	only	only	ADV
m-1153	93	11	when	when	SCONJ
m-1153	93	12	there	there	PRON
m-1153	93	13	is	be	VERB
m-1153	93	14	no	no	DET
m-1153	93	15	continuous	continuous	ADJ
m-1153	93	16	import	import	NOUN
m-1153	93	17	of	of	ADP
m-1153	93	18	infection	infection	NOUN
m-1153	93	19	;	;	PUNCT
m-1153	93	20	in	in	ADP
m-1153	93	21	other	other	ADJ
m-1153	93	22	ijo	ijo	PROPN
m-1153	93	23	international	international	PROPN
m-1153	93	24	journal	journal	PROPN
m-1153	93	25	of	of	ADP
m-1153	93	26	mathematics	mathematics	PROPN
m-1153	93	27	(	(	PUNCT
m-1153	93	28	issn	issn	PROPN
m-1153	93	29	:	:	PUNCT
m-1153	93	30	2992	2992	NUM
m-1153	93	31	-	-	SYM
m-1153	93	32	4421	4421	NUM
m-1153	93	33	)	)	PUNCT
m-1153	93	34	volume	volume	NOUN
m-1153	93	35	08	08	NUM
m-1153	94	1	|	|	ADV
m-1153	94	2	issue	issue	NOUN
m-1153	94	3	9	9	NUM
m-1153	94	4	|	|	CCONJ
m-1153	94	5	september	september	PROPN
m-1153	94	6	2025	2025	NUM
m-1153	94	7	|	|	ADV
m-1153	94	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	94	9	30	30	NUM
m-1153	94	10	words	word	NOUN
m-1153	94	11	we	we	PRON
m-1153	94	12	set	set	VERB
m-1153	94	13	α	α	NOUN
m-1153	94	14	=	=	NOUN
m-1153	94	15	0	0	NUM
m-1153	94	16	for	for	ADP
m-1153	94	17	the	the	DET
m-1153	94	18	threshold	threshold	NOUN
m-1153	94	19	calculation	calculation	NOUN
m-1153	94	20	.	.	PUNCT
m-1153	95	1	the	the	DET
m-1153	95	2	dfe	dfe	PROPN
m-1153	95	3	is	be	AUX
m-1153	95	4	then	then	ADV
m-1153	95	5	s∗	s∗	PROPN
m-1153	95	6	=	=	SYM
m-1153	95	7	µn	µn	PROPN
m-1153	95	8	µ+	µ+	VERB
m-1153	95	9	ν	ν	X
m-1153	95	10	i	i	PRON
m-1153	95	11	,	,	PUNCT
m-1153	95	12	∗	∗	NOUN
m-1153	95	13	=	=	SYM
m-1153	95	14	0	0	NUM
m-1153	95	15	,	,	PUNCT
m-1153	95	16	r∗	r∗	PROPN
m-1153	95	17	=	=	PUNCT
m-1153	95	18	νn	νn	NOUN
m-1153	95	19	µ+	µ+	NOUN
m-1153	95	20	ν	ν	NOUN
m-1153	95	21	b	b	PROPN
m-1153	95	22	,	,	PUNCT
m-1153	95	23	∗	∗	NOUN
m-1153	95	24	=	=	SYM
m-1153	95	25	0	0	X
m-1153	95	26	.	.	X
m-1153	95	27	introduce	introduce	VERB
m-1153	95	28	the	the	DET
m-1153	95	29	shorthand	shorthand	NOUN
m-1153	95	30	notations	notation	NOUN
m-1153	95	31	σ	σ	NOUN
m-1153	95	32	:	:	PUNCT
m-1153	95	33	=	=	SYM
m-1153	95	34	γ	γ	X
m-1153	95	35	+	+	NUM
m-1153	95	36	µ+	µ+	PROPN
m-1153	95	37	δ	δ	PROPN
m-1153	95	38	,	,	PUNCT
m-1153	95	39	τ	τ	X
m-1153	95	40	:	:	PUNCT
m-1153	95	41	=	=	SYM
m-1153	95	42	µb	µb	PROPN
m-1153	95	43	+	+	PROPN
m-1153	95	44	η	η	PROPN
m-1153	95	45	.	.	PROPN
m-1153	95	46	step	step	NOUN
m-1153	95	47	1	1	NUM
m-1153	95	48	:	:	PUNCT
m-1153	95	49	infectious	infectious	ADJ
m-1153	95	50	compartments	compartment	NOUN
m-1153	95	51	and	and	CCONJ
m-1153	95	52	f	f	PROPN
m-1153	95	53	,	,	PUNCT
m-1153	95	54	v	v	DET
m-1153	95	55	partition	partition	NOUN
m-1153	95	56	choose	choose	VERB
m-1153	95	57	infected	infect	VERB
m-1153	95	58	variables	variable	NOUN
m-1153	95	59	x	x	PUNCT
m-1153	96	1	=	=	SYM
m-1153	96	2	(	(	PUNCT
m-1153	96	3	i	i	PROPN
m-1153	96	4	,	,	PUNCT
m-1153	96	5	b)⊤.	b)⊤.	NOUN
m-1153	96	6	decompose	decompose	VERB
m-1153	96	7	the	the	DET
m-1153	96	8	subsystem	subsystem	NOUN
m-1153	96	9	as	as	ADP
m-1153	96	10	ẋ	ẋ	PROPN
m-1153	96	11	=	=	SYM
m-1153	96	12	f(x	f(x	PROPN
m-1153	96	13	)	)	PUNCT
m-1153	96	14	−	−	ADP
m-1153	97	1	v(x	v(x	NOUN
m-1153	97	2	)	)	PUNCT
m-1153	97	3	where	where	SCONJ
m-1153	97	4	f	f	PROPN
m-1153	97	5	collects	collect	VERB
m-1153	97	6	new	new	ADJ
m-1153	97	7	infection	infection	NOUN
m-1153	97	8	terms	term	NOUN
m-1153	97	9	and	and	CCONJ
m-1153	97	10	v	v	NOUN
m-1153	97	11	collects	collect	VERB
m-1153	97	12	transitions	transition	NOUN
m-1153	97	13	and	and	CCONJ
m-1153	97	14	removals	removal	NOUN
m-1153	97	15	.	.	PUNCT
m-1153	98	1	from	from	ADP
m-1153	98	2	(	(	PUNCT
m-1153	98	3	2)–(4	2)–(4	NUM
m-1153	98	4	)	)	PUNCT
m-1153	98	5	with	with	ADP
m-1153	98	6	α	α	NOUN
m-1153	98	7	=	=	SYM
m-1153	98	8	0	0	NUM
m-1153	98	9	:	:	PUNCT
m-1153	98	10	f1	f1	NOUN
m-1153	98	11	=	=	PUNCT
m-1153	98	12	βpsi	βpsi	NOUN
m-1153	98	13	+	+	CCONJ
m-1153	98	14	βbsb	βbsb	PROPN
m-1153	98	15	,	,	PUNCT
m-1153	98	16	f2	f2	PROPN
m-1153	98	17	=	=	SYM
m-1153	98	18	0	0	NUM
m-1153	98	19	,	,	PUNCT
m-1153	98	20	v1	v1	NOUN
m-1153	98	21	=	=	SYM
m-1153	98	22	σi	σi	NOUN
m-1153	98	23	,	,	PUNCT
m-1153	98	24	v2	v2	NOUN
m-1153	98	25	=	=	PUNCT
m-1153	98	26	τb	τb	ADP
m-1153	98	27	−	−	PROPN
m-1153	98	28	ξi	ξi	NOUN
m-1153	98	29	.	.	PUNCT
m-1153	99	1	step	step	NOUN
m-1153	99	2	2	2	NUM
m-1153	99	3	:	:	PUNCT
m-1153	99	4	jacobians	jacobian	NOUN
m-1153	99	5	at	at	ADP
m-1153	99	6	the	the	DET
m-1153	99	7	dfe	dfe	PROPN
m-1153	99	8	compute	compute	PROPN
m-1153	99	9	jacobians	jacobian	NOUN
m-1153	99	10	of	of	ADP
m-1153	99	11	f	f	PROPN
m-1153	99	12	and	and	CCONJ
m-1153	99	13	v	v	X
m-1153	99	14	with	with	ADP
m-1153	99	15	respect	respect	NOUN
m-1153	99	16	to	to	ADP
m-1153	99	17	(	(	PUNCT
m-1153	99	18	i	i	PROPN
m-1153	99	19	,	,	PUNCT
m-1153	99	20	b	b	NOUN
m-1153	99	21	)	)	PUNCT
m-1153	99	22	and	and	CCONJ
m-1153	99	23	evaluate	evaluate	VERB
m-1153	99	24	at	at	ADP
m-1153	99	25	the	the	DET
m-1153	99	26	dfe	dfe	PROPN
m-1153	99	27	(	(	PUNCT
m-1153	99	28	s	s	NOUN
m-1153	99	29	=	=	PUNCT
m-1153	99	30	s∗	s∗	PROPN
m-1153	99	31	,	,	PUNCT
m-1153	99	32	i	i	PRON
m-1153	99	33	=	=	SYM
m-1153	99	34	b	b	PROPN
m-1153	99	35	=	=	SYM
m-1153	99	36	0	0	NUM
m-1153	99	37	):	):	PUNCT
m-1153	99	38	f	f	PROPN
m-1153	99	39	=	=	PUNCT
m-1153	99	40	βps	βps	PROPN
m-1153	99	41	∗	∗	NOUN
m-1153	99	42	βbs	βbs	NOUN
m-1153	99	43	∗	∗	NOUN
m-1153	99	44	00	00	PUNCT
m-1153	100	1			PROPN
m-1153	100	2	v	v	NOUN
m-1153	100	3	,	,	PUNCT
m-1153	100	4	=	=	PRON
m-1153	100	5			X
m-1153	100	6	σ	σ	PROPN
m-1153	100	7	0	0	NUM
m-1153	100	8	−ξ	−ξ	NOUN
m-1153	100	9	τ	τ	PROPN
m-1153	100	10			PROPN
m-1153	100	11	.	.	PUNCT
m-1153	101	1	step	step	NOUN
m-1153	101	2	3	3	NUM
m-1153	101	3	:	:	PUNCT
m-1153	101	4	next	next	ADJ
m-1153	101	5	-	-	PUNCT
m-1153	101	6	generation	generation	NOUN
m-1153	101	7	matrix	matrix	NOUN
m-1153	101	8	k	k	NOUN
m-1153	102	1	=	=	PUNCT
m-1153	102	2	fv	fv	PROPN
m-1153	102	3	−1	−1	ADV
m-1153	102	4	since	since	SCONJ
m-1153	102	5	σ	σ	PROPN
m-1153	102	6	>	>	X
m-1153	102	7	0	0	PROPN
m-1153	102	8	,	,	PUNCT
m-1153	102	9	τ	τ	X
m-1153	102	10	>	>	X
m-1153	102	11	0	0	NUM
m-1153	103	1	we	we	PRON
m-1153	103	2	may	may	AUX
m-1153	103	3	invert	invert	VERB
m-1153	103	4	v	v	X
m-1153	103	5	:	:	PUNCT
m-1153	103	6	v	v	NUM
m-1153	103	7	−1	−1	NOUN
m-1153	103	8	=	=	NOUN
m-1153	103	9	1	1	NUM
m-1153	103	10	στ	στ	INTJ
m-1153	103	11	τ	τ	INTJ
m-1153	103	12	0	0	NUM
m-1153	103	13	ξ	ξ	PRON
m-1153	103	14	σ	σ	PROPN
m-1153	103	15			PROPN
m-1153	103	16	.	.	PUNCT
m-1153	104	1	hence	hence	ADV
m-1153	104	2	k	k	PROPN
m-1153	104	3	=	=	PUNCT
m-1153	104	4	fv	fv	PROPN
m-1153	104	5	−1	−1	NOUN
m-1153	104	6	=	=	SYM
m-1153	105	1			PROPN
m-1153	105	2	βps	βp	VERB
m-1153	105	3	∗τ	∗τ	PROPN
m-1153	105	4	+	+	CCONJ
m-1153	105	5	βbs	βbs	PROPN
m-1153	105	6	∗ξ	∗ξ	PROPN
m-1153	105	7	στ	στ	PROPN
m-1153	105	8	βbs	βbs	PROPN
m-1153	105	9	∗	∗	X
m-1153	105	10	τ	τ	PROPN
m-1153	105	11	0	0	NUM
m-1153	105	12	0	0	NUM
m-1153	105	13			NOUN
m-1153	105	14	.	.	PUNCT
m-1153	106	1	ijo	ijo	PROPN
m-1153	106	2	international	international	PROPN
m-1153	106	3	journal	journal	PROPN
m-1153	106	4	of	of	ADP
m-1153	106	5	mathematics	mathematics	PROPN
m-1153	106	6	(	(	PUNCT
m-1153	106	7	issn	issn	PROPN
m-1153	106	8	:	:	PUNCT
m-1153	106	9	2992	2992	NUM
m-1153	106	10	-	-	SYM
m-1153	106	11	4421	4421	NUM
m-1153	106	12	)	)	PUNCT
m-1153	106	13	volume	volume	NOUN
m-1153	106	14	08	08	NUM
m-1153	107	1	|	|	ADV
m-1153	107	2	issue	issue	NOUN
m-1153	107	3	9	9	NUM
m-1153	107	4	|	|	CCONJ
m-1153	107	5	september	september	PROPN
m-1153	107	6	2025	2025	NUM
m-1153	107	7	|	|	ADV
m-1153	107	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	107	9	31	31	NUM
m-1153	107	10	step	step	NOUN
m-1153	107	11	4	4	NUM
m-1153	107	12	:	:	PUNCT
m-1153	107	13	spectral	spectral	ADJ
m-1153	107	14	radius	radius	NOUN
m-1153	107	15	and	and	CCONJ
m-1153	107	16	r0	r0	VERB
m-1153	107	17	the	the	DET
m-1153	107	18	eigenvalues	eigenvalue	NOUN
m-1153	107	19	of	of	ADP
m-1153	107	20	k	k	PROPN
m-1153	107	21	are	be	AUX
m-1153	107	22	its	its	PRON
m-1153	107	23	diagonal	diagonal	ADJ
m-1153	107	24	entries	entry	NOUN
m-1153	107	25	.	.	PUNCT
m-1153	108	1	the	the	DET
m-1153	108	2	spectral	spectral	ADJ
m-1153	108	3	radius	radius	NOUN
m-1153	108	4	(	(	PUNCT
m-1153	108	5	dominant	dominant	PROPN
m-1153	108	6	eigenvalue	eigenvalue	NOUN
m-1153	108	7	)	)	PUNCT
m-1153	108	8	is	be	AUX
m-1153	108	9	the	the	DET
m-1153	108	10	(	(	PUNCT
m-1153	108	11	1	1	NUM
m-1153	108	12	,	,	PUNCT
m-1153	108	13	1	1	NUM
m-1153	108	14	)	)	PUNCT
m-1153	108	15	entry	entry	NOUN
m-1153	108	16	,	,	PUNCT
m-1153	108	17	therefore	therefore	ADV
m-1153	108	18	r0	r0	NOUN
m-1153	108	19	=	=	SYM
m-1153	108	20	βps	βp	VERB
m-1153	109	1	∗τ	∗τ	PROPN
m-1153	109	2	+	+	CCONJ
m-1153	109	3	βbs	βbs	PROPN
m-1153	109	4	∗ξ	∗ξ	NOUN
m-1153	109	5	στ	στ	PROPN
m-1153	109	6	=	=	NOUN
m-1153	109	7	s∗	s∗	PROPN
m-1153	109	8	σ	σ	X
m-1153	109	9	(	(	PUNCT
m-1153	109	10	βp	βp	PROPN
m-1153	110	1	+	+	CCONJ
m-1153	110	2	βb	βb	PRON
m-1153	110	3	ξ	ξ	PROPN
m-1153	110	4	τ	τ	PROPN
m-1153	110	5	)	)	PUNCT
m-1153	110	6	.	.	PUNCT
m-1153	111	1	substituting	substitute	VERB
m-1153	111	2	s∗	s∗	PROPN
m-1153	111	3	=	=	SYM
m-1153	111	4	µn	µn	PROPN
m-1153	111	5	µ+	µ+	NOUN
m-1153	111	6	ν	ν	NOUN
m-1153	111	7	yields	yield	VERB
m-1153	111	8	the	the	DET
m-1153	111	9	explicit	explicit	ADJ
m-1153	111	10	expression	expression	NOUN
m-1153	111	11	r0	r0	NOUN
m-1153	111	12	=	=	SYM
m-1153	111	13	µn	µn	PROPN
m-1153	111	14	µ+	µ+	X
m-1153	111	15	ν	ν	X
m-1153	111	16	·	·	PUNCT
m-1153	111	17	1	1	NUM
m-1153	111	18	γ	γ	X
m-1153	111	19	+	+	NUM
m-1153	111	20	µ+	µ+	PROPN
m-1153	111	21	δ	δ	PROPN
m-1153	111	22	(	(	PUNCT
m-1153	111	23	βp	βp	PROPN
m-1153	112	1	+	+	CCONJ
m-1153	112	2	βb	βb	PRON
m-1153	112	3	ξ	ξ	X
m-1153	112	4	µb	µb	ADP
m-1153	112	5	+	+	X
m-1153	112	6	η	η	NOUN
m-1153	112	7	)	)	PUNCT
m-1153	112	8	.	.	PUNCT
m-1153	113	1	4.2	4.2	NUM
m-1153	113	2	interpretation	interpretation	NOUN
m-1153	113	3	the	the	DET
m-1153	113	4	decomposition	decomposition	NOUN
m-1153	113	5	of	of	ADP
m-1153	113	6	r0	r0	NOUN
m-1153	113	7	shows	show	VERB
m-1153	113	8	a	a	DET
m-1153	113	9	direct	direct	ADJ
m-1153	113	10	transmission	transmission	NOUN
m-1153	113	11	term	term	NOUN
m-1153	113	12	and	and	CCONJ
m-1153	113	13	an	an	DET
m-1153	113	14	environment	environment	NOUN
m-1153	113	15	-	-	PUNCT
m-1153	113	16	mediated	mediate	VERB
m-1153	113	17	term	term	NOUN
m-1153	113	18	:	:	PUNCT
m-1153	113	19	r0	r0	NOUN
m-1153	113	20	=	=	SYM
m-1153	113	21	βps	βp	NOUN
m-1153	113	22	∗	∗	NOUN
m-1153	113	23	σ︸	σ︸	PROPN
m-1153	113	24	︷︷	︷︷	PROPN
m-1153	113	25	︸	︸	X
m-1153	113	26	direct	direct	ADJ
m-1153	113	27	+	+	CCONJ
m-1153	113	28	βbs	βbs	NOUN
m-1153	113	29	∗	∗	NOUN
m-1153	113	30	σ	σ	X
m-1153	113	31	·	·	PUNCT
m-1153	113	32	ξ	ξ	X
m-1153	113	33	τ︸	τ︸	X
m-1153	113	34	︷︷	︷︷	PROPN
m-1153	113	35	︸	︸	ADP
m-1153	113	36	environment	environment	NOUN
m-1153	113	37	.	.	PUNCT
m-1153	114	1	vaccination	vaccination	NOUN
m-1153	114	2	(	(	PUNCT
m-1153	114	3	rate	rate	NOUN
m-1153	114	4	ν	ν	NOUN
m-1153	114	5	)	)	PUNCT
m-1153	114	6	lowers	lower	VERB
m-1153	114	7	s∗	s∗	PROPN
m-1153	114	8	and	and	CCONJ
m-1153	114	9	therefore	therefore	ADV
m-1153	114	10	reduces	reduce	VERB
m-1153	114	11	r0	r0	NOUN
m-1153	114	12	.	.	PUNCT
m-1153	115	1	enhancing	enhance	VERB
m-1153	115	2	environmental	environmental	ADJ
m-1153	115	3	removal	removal	NOUN
m-1153	115	4	(	(	PUNCT
m-1153	115	5	increasing	increase	VERB
m-1153	115	6	τ	τ	X
m-1153	115	7	)	)	PUNCT
m-1153	115	8	or	or	CCONJ
m-1153	115	9	reducing	reduce	VERB
m-1153	115	10	shedding	shed	VERB
m-1153	115	11	(	(	PUNCT
m-1153	115	12	ξ	ξ	NOUN
m-1153	115	13	)	)	PUNCT
m-1153	115	14	lowers	lower	VERB
m-1153	115	15	the	the	DET
m-1153	115	16	environmental	environmental	ADJ
m-1153	115	17	contribution	contribution	NOUN
m-1153	115	18	.	.	PUNCT
m-1153	116	1	standard	standard	ADJ
m-1153	116	2	stability	stability	NOUN
m-1153	116	3	results	result	NOUN
m-1153	116	4	imply	imply	VERB
m-1153	116	5	the	the	DET
m-1153	116	6	dfe	dfe	PROPN
m-1153	116	7	is	be	AUX
m-1153	116	8	locally	locally	ADV
m-1153	116	9	asymptotically	asymptotically	ADV
m-1153	116	10	stable	stable	ADJ
m-1153	116	11	if	if	SCONJ
m-1153	116	12	r0	r0	NOUN
m-1153	116	13	<	<	X
m-1153	116	14	1	1	NUM
m-1153	116	15	and	and	CCONJ
m-1153	116	16	unstable	unstable	ADJ
m-1153	116	17	if	if	SCONJ
m-1153	116	18	r0	r0	NOUN
m-1153	116	19	>	>	X
m-1153	116	20	1	1	NUM
m-1153	116	21	.	.	SYM
m-1153	116	22	5	5	NUM
m-1153	116	23	conclusion	conclusion	NOUN
m-1153	116	24	and	and	CCONJ
m-1153	116	25	recommendations	recommendation	NOUN
m-1153	116	26	5.1	5.1	NUM
m-1153	116	27	conclusion	conclusion	NOUN
m-1153	116	28	in	in	ADP
m-1153	116	29	this	this	DET
m-1153	116	30	study	study	NOUN
m-1153	116	31	,	,	PUNCT
m-1153	116	32	we	we	PRON
m-1153	116	33	proposed	propose	VERB
m-1153	116	34	and	and	CCONJ
m-1153	116	35	analyzed	analyze	VERB
m-1153	116	36	a	a	DET
m-1153	116	37	cholera	cholera	NOUN
m-1153	116	38	transmission	transmission	NOUN
m-1153	116	39	model	model	NOUN
m-1153	116	40	of	of	ADP
m-1153	116	41	the	the	DET
m-1153	116	42	sirb	sirb	NOUN
m-1153	116	43	type	type	NOUN
m-1153	116	44	incorporating	incorporate	VERB
m-1153	116	45	vaccination	vaccination	NOUN
m-1153	116	46	and	and	CCONJ
m-1153	116	47	treatment	treatment	NOUN
m-1153	116	48	.	.	PUNCT
m-1153	117	1	by	by	ADP
m-1153	117	2	formulating	formulate	VERB
m-1153	117	3	the	the	DET
m-1153	117	4	system	system	NOUN
m-1153	117	5	of	of	ADP
m-1153	117	6	nonlinear	nonlinear	ADJ
m-1153	117	7	differential	differential	ADJ
m-1153	117	8	equations	equation	NOUN
m-1153	117	9	,	,	PUNCT
m-1153	117	10	we	we	PRON
m-1153	117	11	explicitly	explicitly	ADV
m-1153	117	12	derived	derive	VERB
m-1153	117	13	the	the	DET
m-1153	117	14	disease	disease	NOUN
m-1153	117	15	-	-	PUNCT
m-1153	117	16	free	free	ADJ
m-1153	117	17	equilibrium	equilibrium	NOUN
m-1153	117	18	(	(	PUNCT
m-1153	117	19	dfe	dfe	PROPN
m-1153	117	20	)	)	PUNCT
m-1153	117	21	and	and	CCONJ
m-1153	117	22	the	the	DET
m-1153	117	23	endemic	endemic	ADJ
m-1153	117	24	equilibrium	equilibrium	NOUN
m-1153	117	25	(	(	PUNCT
m-1153	117	26	ee	ee	PROPN
m-1153	117	27	)	)	PUNCT
m-1153	117	28	.	.	PUNCT
m-1153	118	1	the	the	DET
m-1153	118	2	dfe	dfe	PROPN
m-1153	118	3	corresponds	correspond	VERB
m-1153	118	4	to	to	ADP
m-1153	118	5	a	a	DET
m-1153	118	6	population	population	NOUN
m-1153	118	7	state	state	NOUN
m-1153	118	8	where	where	SCONJ
m-1153	118	9	cholera	cholera	NOUN
m-1153	118	10	infection	infection	NOUN
m-1153	118	11	can	can	AUX
m-1153	118	12	not	not	PART
m-1153	118	13	persist	persist	VERB
m-1153	118	14	,	,	PUNCT
m-1153	118	15	while	while	SCONJ
m-1153	118	16	the	the	DET
m-1153	118	17	ee	ee	PROPN
m-1153	118	18	describes	describe	VERB
m-1153	118	19	a	a	DET
m-1153	118	20	scenario	scenario	NOUN
m-1153	118	21	in	in	ADP
m-1153	118	22	which	which	PRON
m-1153	118	23	cholera	cholera	NOUN
m-1153	118	24	remains	remain	VERB
m-1153	118	25	endemic	endemic	ADJ
m-1153	118	26	under	under	ADP
m-1153	118	27	sustained	sustained	ADJ
m-1153	118	28	transmission	transmission	NOUN
m-1153	118	29	.	.	PUNCT
m-1153	119	1	the	the	DET
m-1153	119	2	mathematical	mathematical	ADJ
m-1153	119	3	characterization	characterization	NOUN
m-1153	119	4	of	of	ADP
m-1153	119	5	these	these	DET
m-1153	119	6	equilibria	equilibrium	NOUN
m-1153	119	7	provides	provide	VERB
m-1153	119	8	important	important	ADJ
m-1153	119	9	epidemiological	epidemiological	ADJ
m-1153	119	10	insights	insight	NOUN
m-1153	119	11	.	.	PUNCT
m-1153	120	1	ijo	ijo	PROPN
m-1153	120	2	international	international	PROPN
m-1153	120	3	journal	journal	PROPN
m-1153	120	4	of	of	ADP
m-1153	120	5	mathematics	mathematics	PROPN
m-1153	120	6	(	(	PUNCT
m-1153	120	7	issn	issn	PROPN
m-1153	120	8	:	:	PUNCT
m-1153	120	9	2992	2992	NUM
m-1153	120	10	-	-	SYM
m-1153	120	11	4421	4421	NUM
m-1153	120	12	)	)	PUNCT
m-1153	120	13	volume	volume	NOUN
m-1153	120	14	08	08	NUM
m-1153	121	1	|	|	ADV
m-1153	121	2	issue	issue	NOUN
m-1153	121	3	9	9	NUM
m-1153	121	4	|	|	CCONJ
m-1153	121	5	september	september	PROPN
m-1153	121	6	2025	2025	NUM
m-1153	122	1	|	|	ADV
m-1153	122	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	122	3	32	32	NUM
m-1153	122	4	the	the	DET
m-1153	122	5	stability	stability	NOUN
m-1153	122	6	of	of	ADP
m-1153	122	7	these	these	DET
m-1153	122	8	equilibria	equilibrium	NOUN
m-1153	122	9	was	be	AUX
m-1153	122	10	determined	determine	VERB
m-1153	122	11	by	by	ADP
m-1153	122	12	computing	compute	VERB
m-1153	122	13	the	the	DET
m-1153	122	14	basic	basic	ADJ
m-1153	122	15	reproduction	reproduction	NOUN
m-1153	122	16	number	number	NOUN
m-1153	122	17	r0	r0	NOUN
m-1153	122	18	via	via	ADP
m-1153	122	19	the	the	DET
m-1153	122	20	next	next	ADJ
m-1153	122	21	-	-	PUNCT
m-1153	122	22	generation	generation	NOUN
m-1153	122	23	matrix	matrix	NOUN
m-1153	122	24	approach	approach	NOUN
m-1153	122	25	.	.	PUNCT
m-1153	123	1	the	the	DET
m-1153	123	2	explicit	explicit	ADJ
m-1153	123	3	form	form	NOUN
m-1153	123	4	of	of	ADP
m-1153	123	5	r0	r0	NOUN
m-1153	123	6	is	be	AUX
m-1153	123	7	r0	r0	NOUN
m-1153	123	8	=	=	SYM
m-1153	123	9	µn	µn	PROPN
m-1153	123	10	µ+	µ+	X
m-1153	123	11	ν	ν	X
m-1153	123	12	·	·	PUNCT
m-1153	123	13	1	1	NUM
m-1153	123	14	γ	γ	X
m-1153	123	15	+	+	NUM
m-1153	123	16	µ+	µ+	PROPN
m-1153	123	17	δ	δ	PROPN
m-1153	123	18	(	(	PUNCT
m-1153	123	19	βp	βp	PROPN
m-1153	124	1	+	+	CCONJ
m-1153	124	2	βb	βb	PRON
m-1153	124	3	ξ	ξ	X
m-1153	124	4	µb	µb	ADP
m-1153	124	5	+	+	X
m-1153	124	6	η	η	NOUN
m-1153	124	7	)	)	PUNCT
m-1153	125	1	,	,	PUNCT
m-1153	125	2	(	(	PUNCT
m-1153	125	3	6	6	NUM
m-1153	125	4	)	)	PUNCT
m-1153	125	5	which	which	PRON
m-1153	125	6	highlights	highlight	VERB
m-1153	125	7	the	the	DET
m-1153	125	8	interplay	interplay	NOUN
m-1153	125	9	between	between	ADP
m-1153	125	10	direct	direct	ADJ
m-1153	125	11	person	person	NOUN
m-1153	125	12	-	-	PUNCT
m-1153	125	13	to	to	ADP
m-1153	125	14	-	-	PUNCT
m-1153	125	15	person	person	NOUN
m-1153	125	16	transmission	transmission	NOUN
m-1153	125	17	and	and	CCONJ
m-1153	125	18	environmentmediated	environmentmediate	VERB
m-1153	125	19	transmission	transmission	NOUN
m-1153	125	20	via	via	ADP
m-1153	125	21	the	the	DET
m-1153	125	22	bacterial	bacterial	ADJ
m-1153	125	23	reservoir	reservoir	NOUN
m-1153	125	24	.	.	PUNCT
m-1153	126	1	the	the	DET
m-1153	126	2	decomposition	decomposition	NOUN
m-1153	126	3	of	of	ADP
m-1153	126	4	r0	r0	NOUN
m-1153	126	5	into	into	ADP
m-1153	126	6	direct	direct	ADJ
m-1153	126	7	and	and	CCONJ
m-1153	126	8	indirect	indirect	ADJ
m-1153	126	9	pathways	pathway	NOUN
m-1153	126	10	provides	provide	VERB
m-1153	126	11	a	a	DET
m-1153	126	12	useful	useful	ADJ
m-1153	126	13	threshold	threshold	NOUN
m-1153	126	14	criterion	criterion	NOUN
m-1153	126	15	:	:	PUNCT
m-1153	126	16	if	if	SCONJ
m-1153	126	17	r0	r0	NOUN
m-1153	126	18	<	<	X
m-1153	126	19	1	1	NUM
m-1153	126	20	,	,	PUNCT
m-1153	126	21	the	the	DET
m-1153	126	22	dfe	dfe	PROPN
m-1153	126	23	is	be	AUX
m-1153	126	24	locally	locally	ADV
m-1153	126	25	asymptotically	asymptotically	ADV
m-1153	126	26	stable	stable	ADJ
m-1153	126	27	and	and	CCONJ
m-1153	126	28	the	the	DET
m-1153	126	29	disease	disease	NOUN
m-1153	126	30	will	will	AUX
m-1153	126	31	eventually	eventually	ADV
m-1153	126	32	die	die	VERB
m-1153	126	33	out	out	ADP
m-1153	126	34	,	,	PUNCT
m-1153	126	35	while	while	SCONJ
m-1153	126	36	if	if	SCONJ
m-1153	126	37	r0	r0	NOUN
m-1153	126	38	>	>	X
m-1153	126	39	1	1	NUM
m-1153	126	40	,	,	PUNCT
m-1153	126	41	cholera	cholera	NOUN
m-1153	126	42	can	can	AUX
m-1153	126	43	invade	invade	VERB
m-1153	126	44	and	and	CCONJ
m-1153	126	45	persist	persist	VERB
m-1153	126	46	,	,	PUNCT
m-1153	126	47	leading	lead	VERB
m-1153	126	48	to	to	ADP
m-1153	126	49	the	the	DET
m-1153	126	50	endemic	endemic	ADJ
m-1153	126	51	equilibrium	equilibrium	NOUN
m-1153	126	52	.	.	PUNCT
m-1153	127	1	from	from	ADP
m-1153	127	2	the	the	DET
m-1153	127	3	derivations	derivation	NOUN
m-1153	127	4	,	,	PUNCT
m-1153	127	5	several	several	ADJ
m-1153	127	6	key	key	ADJ
m-1153	127	7	epidemiological	epidemiological	ADJ
m-1153	127	8	conclusions	conclusion	NOUN
m-1153	127	9	emerge	emerge	VERB
m-1153	127	10	:	:	PUNCT
m-1153	127	11	1	1	X
m-1153	127	12	.	.	X
m-1153	127	13	impact	impact	NOUN
m-1153	127	14	of	of	ADP
m-1153	127	15	vaccination	vaccination	NOUN
m-1153	127	16	.	.	PUNCT
m-1153	128	1	the	the	DET
m-1153	128	2	susceptible	susceptible	ADJ
m-1153	128	3	equilibrium	equilibrium	NOUN
m-1153	128	4	level	level	NOUN
m-1153	128	5	s∗	s∗	NOUN
m-1153	128	6	=	=	SYM
m-1153	128	7	µn	µn	NOUN
m-1153	128	8	µ+ν	µ+ν	NOUN
m-1153	128	9	decreases	decrease	VERB
m-1153	128	10	with	with	ADP
m-1153	128	11	the	the	DET
m-1153	128	12	vaccination	vaccination	NOUN
m-1153	128	13	rate	rate	NOUN
m-1153	128	14	ν	ν	NOUN
m-1153	128	15	,	,	PUNCT
m-1153	128	16	directly	directly	ADV
m-1153	128	17	lowering	lower	VERB
m-1153	128	18	r0	r0	NOUN
m-1153	128	19	.	.	PUNCT
m-1153	129	1	this	this	PRON
m-1153	129	2	confirms	confirm	VERB
m-1153	129	3	that	that	SCONJ
m-1153	129	4	increasing	increase	VERB
m-1153	129	5	vaccine	vaccine	NOUN
m-1153	129	6	coverage	coverage	NOUN
m-1153	129	7	is	be	AUX
m-1153	129	8	an	an	DET
m-1153	129	9	effective	effective	ADJ
m-1153	129	10	means	mean	NOUN
m-1153	129	11	of	of	ADP
m-1153	129	12	driving	drive	VERB
m-1153	129	13	r0	r0	NOUN
m-1153	129	14	below	below	ADP
m-1153	129	15	unity	unity	NOUN
m-1153	129	16	and	and	CCONJ
m-1153	129	17	eliminating	eliminate	VERB
m-1153	129	18	cholera	cholera	NOUN
m-1153	129	19	.	.	PUNCT
m-1153	130	1	2	2	X
m-1153	130	2	.	.	X
m-1153	130	3	role	role	NOUN
m-1153	130	4	of	of	ADP
m-1153	130	5	environmental	environmental	ADJ
m-1153	130	6	sanitation	sanitation	NOUN
m-1153	130	7	.	.	PUNCT
m-1153	131	1	the	the	DET
m-1153	131	2	environmental	environmental	ADJ
m-1153	131	3	contribution	contribution	NOUN
m-1153	131	4	to	to	PART
m-1153	131	5	r0	r0	NOUN
m-1153	131	6	depends	depend	VERB
m-1153	131	7	on	on	ADP
m-1153	131	8	both	both	CCONJ
m-1153	131	9	the	the	DET
m-1153	131	10	bacterial	bacterial	ADJ
m-1153	131	11	shedding	shedding	ADJ
m-1153	131	12	rate	rate	NOUN
m-1153	131	13	ξ	ξ	PROPN
m-1153	131	14	and	and	CCONJ
m-1153	131	15	the	the	DET
m-1153	131	16	bacterial	bacterial	ADJ
m-1153	131	17	removal	removal	NOUN
m-1153	131	18	rate	rate	NOUN
m-1153	131	19	τ	τ	PROPN
m-1153	131	20	=	=	PUNCT
m-1153	131	21	µb	µb	PROPN
m-1153	131	22	+	+	PROPN
m-1153	131	23	η	η	PROPN
m-1153	131	24	.	.	PROPN
m-1153	131	25	improved	improve	VERB
m-1153	131	26	sanitation	sanitation	NOUN
m-1153	131	27	,	,	PUNCT
m-1153	131	28	water	water	NOUN
m-1153	131	29	treatment	treatment	NOUN
m-1153	131	30	,	,	PUNCT
m-1153	131	31	and	and	CCONJ
m-1153	131	32	faster	fast	ADJ
m-1153	131	33	bacterial	bacterial	ADJ
m-1153	131	34	decay	decay	NOUN
m-1153	131	35	reduce	reduce	VERB
m-1153	131	36	the	the	DET
m-1153	131	37	environmental	environmental	ADJ
m-1153	131	38	load	load	NOUN
m-1153	131	39	,	,	PUNCT
m-1153	131	40	thereby	thereby	ADV
m-1153	131	41	diminishing	diminish	VERB
m-1153	131	42	the	the	DET
m-1153	131	43	potential	potential	NOUN
m-1153	131	44	for	for	ADP
m-1153	131	45	sustained	sustained	ADJ
m-1153	131	46	outbreaks	outbreak	NOUN
m-1153	131	47	.	.	PUNCT
m-1153	132	1	3	3	X
m-1153	132	2	.	.	X
m-1153	132	3	treatment	treatment	NOUN
m-1153	132	4	and	and	CCONJ
m-1153	132	5	recovery	recovery	NOUN
m-1153	132	6	.	.	PUNCT
m-1153	133	1	increasing	increase	VERB
m-1153	133	2	the	the	DET
m-1153	133	3	recovery	recovery	NOUN
m-1153	133	4	rate	rate	NOUN
m-1153	133	5	γ	γ	PROPN
m-1153	133	6	and	and	CCONJ
m-1153	133	7	treatment	treatment	NOUN
m-1153	133	8	efficacy	efficacy	NOUN
m-1153	133	9	reduces	reduce	VERB
m-1153	133	10	the	the	DET
m-1153	133	11	average	average	ADJ
m-1153	133	12	infectious	infectious	ADJ
m-1153	133	13	period	period	NOUN
m-1153	133	14	,	,	PUNCT
m-1153	133	15	thereby	thereby	ADV
m-1153	133	16	lowering	lower	VERB
m-1153	133	17	r0	r0	NOUN
m-1153	133	18	.	.	PUNCT
m-1153	134	1	likewise	likewise	ADV
m-1153	134	2	,	,	PUNCT
m-1153	134	3	treatment	treatment	NOUN
m-1153	134	4	interventions	intervention	NOUN
m-1153	134	5	that	that	PRON
m-1153	134	6	reduce	reduce	VERB
m-1153	134	7	bacterial	bacterial	ADJ
m-1153	134	8	shedding	shedding	NOUN
m-1153	134	9	can	can	AUX
m-1153	134	10	compound	compound	VERB
m-1153	134	11	this	this	DET
m-1153	134	12	effect	effect	NOUN
m-1153	134	13	.	.	PUNCT
m-1153	135	1	4	4	X
m-1153	135	2	.	.	X
m-1153	135	3	threshold	threshold	NOUN
m-1153	135	4	phenomenon	phenomenon	NOUN
m-1153	135	5	.	.	PUNCT
m-1153	136	1	the	the	DET
m-1153	136	2	model	model	NOUN
m-1153	136	3	confirms	confirm	VERB
m-1153	136	4	the	the	DET
m-1153	136	5	classical	classical	ADJ
m-1153	136	6	threshold	threshold	NOUN
m-1153	136	7	property	property	NOUN
m-1153	136	8	:	:	PUNCT
m-1153	136	9	cholera	cholera	NOUN
m-1153	136	10	control	control	NOUN
m-1153	136	11	is	be	AUX
m-1153	136	12	possible	possible	ADJ
m-1153	136	13	if	if	SCONJ
m-1153	136	14	and	and	CCONJ
m-1153	136	15	only	only	ADV
m-1153	136	16	if	if	SCONJ
m-1153	136	17	r0	r0	NOUN
m-1153	136	18	<	<	X
m-1153	136	19	1	1	NUM
m-1153	136	20	.	.	PUNCT
m-1153	137	1	the	the	DET
m-1153	137	2	explicit	explicit	ADJ
m-1153	137	3	dependence	dependence	NOUN
m-1153	137	4	of	of	ADP
m-1153	137	5	r0	r0	NOUN
m-1153	137	6	on	on	ADP
m-1153	137	7	epidemiological	epidemiological	ADJ
m-1153	137	8	parameters	parameter	NOUN
m-1153	137	9	provides	provide	VERB
m-1153	137	10	a	a	DET
m-1153	137	11	roadmap	roadmap	NOUN
m-1153	137	12	for	for	ADP
m-1153	137	13	targeted	target	VERB
m-1153	137	14	intervention	intervention	NOUN
m-1153	137	15	strategies	strategy	NOUN
m-1153	137	16	.	.	PUNCT
m-1153	138	1	ijo	ijo	PROPN
m-1153	138	2	international	international	PROPN
m-1153	138	3	journal	journal	PROPN
m-1153	138	4	of	of	ADP
m-1153	138	5	mathematics	mathematics	PROPN
m-1153	138	6	(	(	PUNCT
m-1153	138	7	issn	issn	PROPN
m-1153	138	8	:	:	PUNCT
m-1153	138	9	2992	2992	NUM
m-1153	138	10	-	-	SYM
m-1153	138	11	4421	4421	NUM
m-1153	138	12	)	)	PUNCT
m-1153	138	13	volume	volume	NOUN
m-1153	138	14	08	08	NUM
m-1153	139	1	|	|	ADV
m-1153	139	2	issue	issue	NOUN
m-1153	139	3	9	9	NUM
m-1153	139	4	|	|	CCONJ
m-1153	139	5	september	september	PROPN
m-1153	139	6	2025	2025	NUM
m-1153	139	7	|	|	ADV
m-1153	139	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	139	9	33	33	NUM
m-1153	139	10	recommendations	recommendation	NOUN
m-1153	139	11	based	base	VERB
m-1153	139	12	on	on	ADP
m-1153	139	13	these	these	DET
m-1153	139	14	findings	finding	NOUN
m-1153	139	15	,	,	PUNCT
m-1153	139	16	we	we	PRON
m-1153	139	17	make	make	VERB
m-1153	139	18	the	the	DET
m-1153	139	19	following	follow	VERB
m-1153	139	20	recommendations	recommendation	NOUN
m-1153	139	21	:	:	PUNCT
m-1153	139	22	•	•	NOUN
m-1153	139	23	strengthen	strengthen	VERB
m-1153	139	24	vaccination	vaccination	NOUN
m-1153	139	25	campaigns	campaign	NOUN
m-1153	139	26	.	.	PUNCT
m-1153	140	1	sustained	sustain	VERB
m-1153	140	2	and	and	CCONJ
m-1153	140	3	widespread	widespread	ADJ
m-1153	140	4	vaccination	vaccination	NOUN
m-1153	140	5	of	of	ADP
m-1153	140	6	susceptible	susceptible	ADJ
m-1153	140	7	individuals	individual	NOUN
m-1153	140	8	is	be	AUX
m-1153	140	9	essential	essential	ADJ
m-1153	140	10	to	to	ADP
m-1153	140	11	lowering	lower	VERB
m-1153	140	12	s∗	s∗	PROPN
m-1153	140	13	and	and	CCONJ
m-1153	140	14	hence	hence	ADV
m-1153	140	15	reducing	reduce	VERB
m-1153	140	16	the	the	DET
m-1153	140	17	overall	overall	ADJ
m-1153	140	18	reproductive	reproductive	ADJ
m-1153	140	19	potential	potential	NOUN
m-1153	140	20	of	of	ADP
m-1153	140	21	cholera	cholera	NOUN
m-1153	140	22	.	.	PUNCT
m-1153	141	1	vaccination	vaccination	NOUN
m-1153	141	2	policies	policy	NOUN
m-1153	141	3	should	should	AUX
m-1153	141	4	prioritize	prioritize	VERB
m-1153	141	5	high	high	ADJ
m-1153	141	6	-	-	PUNCT
m-1153	141	7	risk	risk	NOUN
m-1153	141	8	communities	community	NOUN
m-1153	141	9	with	with	ADP
m-1153	141	10	recurrent	recurrent	ADJ
m-1153	141	11	outbreaks	outbreak	NOUN
m-1153	141	12	.	.	PUNCT
m-1153	142	1	•	•	NOUN
m-1153	142	2	improve	improve	VERB
m-1153	142	3	water	water	NOUN
m-1153	142	4	and	and	CCONJ
m-1153	142	5	sanitation	sanitation	NOUN
m-1153	142	6	infrastructure	infrastructure	NOUN
m-1153	142	7	.	.	PUNCT
m-1153	143	1	investments	investment	NOUN
m-1153	143	2	in	in	ADP
m-1153	143	3	clean	clean	ADJ
m-1153	143	4	water	water	NOUN
m-1153	143	5	supply	supply	NOUN
m-1153	143	6	,	,	PUNCT
m-1153	143	7	efficient	efficient	ADJ
m-1153	143	8	sewage	sewage	NOUN
m-1153	143	9	disposal	disposal	NOUN
m-1153	143	10	,	,	PUNCT
m-1153	143	11	and	and	CCONJ
m-1153	143	12	bacterial	bacterial	ADJ
m-1153	143	13	removal	removal	NOUN
m-1153	143	14	measures	measure	NOUN
m-1153	143	15	(	(	PUNCT
m-1153	143	16	such	such	ADJ
m-1153	143	17	as	as	ADP
m-1153	143	18	chlorination	chlorination	NOUN
m-1153	143	19	and	and	CCONJ
m-1153	143	20	filtration	filtration	NOUN
m-1153	143	21	)	)	PUNCT
m-1153	143	22	are	be	AUX
m-1153	143	23	critical	critical	ADJ
m-1153	143	24	in	in	ADP
m-1153	143	25	reducing	reduce	VERB
m-1153	143	26	the	the	DET
m-1153	143	27	environmental	environmental	ADJ
m-1153	143	28	transmission	transmission	NOUN
m-1153	143	29	pathway	pathway	NOUN
m-1153	143	30	.	.	PUNCT
m-1153	144	1	•	•	NUM
m-1153	144	2	enhance	enhance	VERB
m-1153	144	3	early	early	ADJ
m-1153	144	4	treatment	treatment	NOUN
m-1153	144	5	and	and	CCONJ
m-1153	144	6	case	case	NOUN
m-1153	144	7	management	management	NOUN
m-1153	144	8	.	.	PUNCT
m-1153	145	1	effective	effective	ADJ
m-1153	145	2	case	case	NOUN
m-1153	145	3	detection	detection	NOUN
m-1153	145	4	,	,	PUNCT
m-1153	145	5	rapid	rapid	ADJ
m-1153	145	6	treatment	treatment	NOUN
m-1153	145	7	,	,	PUNCT
m-1153	145	8	and	and	CCONJ
m-1153	145	9	supportive	supportive	ADJ
m-1153	145	10	therapy	therapy	NOUN
m-1153	145	11	increase	increase	VERB
m-1153	145	12	recovery	recovery	NOUN
m-1153	145	13	rates	rate	NOUN
m-1153	145	14	and	and	CCONJ
m-1153	145	15	reduce	reduce	VERB
m-1153	145	16	both	both	DET
m-1153	145	17	morbidity	morbidity	NOUN
m-1153	145	18	and	and	CCONJ
m-1153	145	19	pathogen	pathogen	NOUN
m-1153	145	20	shedding	shedding	NOUN
m-1153	145	21	,	,	PUNCT
m-1153	145	22	thus	thus	ADV
m-1153	145	23	curtailing	curtail	VERB
m-1153	145	24	the	the	DET
m-1153	145	25	force	force	NOUN
m-1153	145	26	of	of	ADP
m-1153	145	27	infection	infection	NOUN
m-1153	145	28	.	.	PUNCT
m-1153	146	1	•	•	NUM
m-1153	146	2	integrate	integrate	VERB
m-1153	146	3	multi	multi	ADJ
m-1153	146	4	-	-	ADJ
m-1153	146	5	intervention	intervention	ADJ
m-1153	146	6	strategies	strategy	NOUN
m-1153	146	7	.	.	PUNCT
m-1153	147	1	mathematical	mathematical	ADJ
m-1153	147	2	results	result	NOUN
m-1153	147	3	suggest	suggest	VERB
m-1153	147	4	that	that	SCONJ
m-1153	147	5	no	no	DET
m-1153	147	6	single	single	ADJ
m-1153	147	7	intervention	intervention	NOUN
m-1153	147	8	suffices	suffice	VERB
m-1153	147	9	when	when	SCONJ
m-1153	147	10	r0	r0	NOUN
m-1153	147	11	is	be	AUX
m-1153	147	12	significantly	significantly	ADV
m-1153	147	13	above	above	ADP
m-1153	147	14	unity	unity	NOUN
m-1153	147	15	.	.	PUNCT
m-1153	148	1	a	a	DET
m-1153	148	2	combined	combined	ADJ
m-1153	148	3	strategy	strategy	NOUN
m-1153	148	4	involving	involve	VERB
m-1153	148	5	vaccination	vaccination	NOUN
m-1153	148	6	,	,	PUNCT
m-1153	148	7	sanitation	sanitation	NOUN
m-1153	148	8	,	,	PUNCT
m-1153	148	9	and	and	CCONJ
m-1153	148	10	treatment	treatment	NOUN
m-1153	148	11	will	will	AUX
m-1153	148	12	have	have	VERB
m-1153	148	13	synergistic	synergistic	ADJ
m-1153	148	14	effects	effect	NOUN
m-1153	148	15	,	,	PUNCT
m-1153	148	16	pushing	push	VERB
m-1153	148	17	the	the	DET
m-1153	148	18	effective	effective	ADJ
m-1153	148	19	reproduction	reproduction	NOUN
m-1153	148	20	number	number	NOUN
m-1153	148	21	below	below	ADP
m-1153	148	22	threshold	threshold	NOUN
m-1153	148	23	.	.	PUNCT
m-1153	149	1	•	•	NUM
m-1153	149	2	policy	policy	NOUN
m-1153	149	3	implication	implication	NOUN
m-1153	149	4	.	.	PUNCT
m-1153	150	1	policymakers	policymaker	NOUN
m-1153	150	2	should	should	AUX
m-1153	150	3	use	use	VERB
m-1153	150	4	r0	r0	NOUN
m-1153	150	5	not	not	PART
m-1153	150	6	only	only	ADV
m-1153	150	7	as	as	ADP
m-1153	150	8	a	a	DET
m-1153	150	9	theoretical	theoretical	ADJ
m-1153	150	10	threshold	threshold	NOUN
m-1153	150	11	but	but	CCONJ
m-1153	150	12	as	as	ADP
m-1153	150	13	a	a	DET
m-1153	150	14	measurable	measurable	ADJ
m-1153	150	15	index	index	NOUN
m-1153	150	16	of	of	ADP
m-1153	150	17	cholera	cholera	NOUN
m-1153	150	18	control	control	NOUN
m-1153	150	19	.	.	PUNCT
m-1153	151	1	targeting	target	VERB
m-1153	151	2	parameter	parameter	NOUN
m-1153	151	3	domains	domain	NOUN
m-1153	151	4	that	that	PRON
m-1153	151	5	ensure	ensure	VERB
m-1153	151	6	r0	r0	NOUN
m-1153	151	7	<	<	X
m-1153	151	8	1	1	NUM
m-1153	151	9	provides	provide	VERB
m-1153	151	10	a	a	DET
m-1153	151	11	rigorous	rigorous	ADJ
m-1153	151	12	,	,	PUNCT
m-1153	151	13	evidence	evidence	NOUN
m-1153	151	14	-	-	PUNCT
m-1153	151	15	based	base	VERB
m-1153	151	16	criterion	criterion	NOUN
m-1153	151	17	for	for	ADP
m-1153	151	18	evaluating	evaluate	VERB
m-1153	151	19	the	the	DET
m-1153	151	20	adequacy	adequacy	NOUN
m-1153	151	21	of	of	ADP
m-1153	151	22	public	public	ADJ
m-1153	151	23	health	health	NOUN
m-1153	151	24	strategies	strategy	NOUN
m-1153	151	25	.	.	PUNCT
m-1153	152	1	in	in	ADP
m-1153	152	2	summary	summary	NOUN
m-1153	152	3	,	,	PUNCT
m-1153	152	4	this	this	DET
m-1153	152	5	work	work	NOUN
m-1153	152	6	demonstrates	demonstrate	VERB
m-1153	152	7	that	that	SCONJ
m-1153	152	8	vaccination	vaccination	NOUN
m-1153	152	9	coupled	couple	VERB
m-1153	152	10	with	with	ADP
m-1153	152	11	environmental	environmental	ADJ
m-1153	152	12	and	and	CCONJ
m-1153	152	13	clinical	clinical	ADJ
m-1153	152	14	interventions	intervention	NOUN
m-1153	152	15	is	be	AUX
m-1153	152	16	mathematically	mathematically	ADV
m-1153	152	17	and	and	CCONJ
m-1153	152	18	epidemiologically	epidemiologically	ADV
m-1153	152	19	sufficient	sufficient	ADJ
m-1153	152	20	to	to	PART
m-1153	152	21	suppress	suppress	VERB
m-1153	152	22	cholera	cholera	NOUN
m-1153	152	23	outbreaks	outbreak	NOUN
m-1153	152	24	.	.	PUNCT
m-1153	153	1	the	the	DET
m-1153	153	2	derivation	derivation	NOUN
m-1153	153	3	of	of	ADP
m-1153	153	4	equilibrium	equilibrium	NOUN
m-1153	153	5	points	point	NOUN
m-1153	153	6	and	and	CCONJ
m-1153	153	7	r0	r0	NOUN
m-1153	153	8	offers	offer	VERB
m-1153	153	9	both	both	DET
m-1153	153	10	theoretical	theoretical	ADJ
m-1153	153	11	insight	insight	NOUN
m-1153	153	12	and	and	CCONJ
m-1153	153	13	practical	practical	ADJ
m-1153	153	14	guidance	guidance	NOUN
m-1153	153	15	,	,	PUNCT
m-1153	153	16	showing	show	VERB
m-1153	153	17	that	that	SCONJ
m-1153	153	18	controlling	control	VERB
m-1153	153	19	cholera	cholera	NOUN
m-1153	153	20	is	be	AUX
m-1153	153	21	contingent	contingent	ADJ
m-1153	153	22	upon	upon	SCONJ
m-1153	153	23	reducing	reduce	VERB
m-1153	153	24	susceptibility	susceptibility	NOUN
m-1153	153	25	,	,	PUNCT
m-1153	153	26	minimizing	minimize	VERB
m-1153	153	27	environmental	environmental	ADJ
m-1153	153	28	bacterial	bacterial	ADJ
m-1153	153	29	persistence	persistence	NOUN
m-1153	153	30	,	,	PUNCT
m-1153	153	31	and	and	CCONJ
m-1153	153	32	shortening	shorten	VERB
m-1153	153	33	the	the	DET
m-1153	153	34	infectious	infectious	ADJ
m-1153	153	35	period	period	NOUN
m-1153	153	36	through	through	ADP
m-1153	153	37	treatment	treatment	NOUN
m-1153	153	38	.	.	PUNCT
m-1153	154	1	these	these	DET
m-1153	154	2	results	result	NOUN
m-1153	154	3	reinforce	reinforce	VERB
m-1153	154	4	the	the	DET
m-1153	154	5	importance	importance	NOUN
m-1153	154	6	of	of	ADP
m-1153	154	7	sustained	sustained	ADJ
m-1153	154	8	,	,	PUNCT
m-1153	154	9	integrated	integrated	ADJ
m-1153	154	10	control	control	NOUN
m-1153	154	11	programs	program	NOUN
m-1153	154	12	for	for	ADP
m-1153	154	13	cholera	cholera	NOUN
m-1153	154	14	elimination	elimination	NOUN
m-1153	154	15	.	.	PUNCT
m-1153	155	1	ijo	ijo	PROPN
m-1153	155	2	international	international	PROPN
m-1153	155	3	journal	journal	PROPN
m-1153	155	4	of	of	ADP
m-1153	155	5	mathematics	mathematics	PROPN
m-1153	155	6	(	(	PUNCT
m-1153	155	7	issn	issn	PROPN
m-1153	155	8	:	:	PUNCT
m-1153	155	9	2992	2992	NUM
m-1153	155	10	-	-	SYM
m-1153	155	11	4421	4421	NUM
m-1153	155	12	)	)	PUNCT
m-1153	155	13	volume	volume	NOUN
m-1153	155	14	08	08	NUM
m-1153	156	1	|	|	ADV
m-1153	156	2	issue	issue	NOUN
m-1153	156	3	9	9	NUM
m-1153	156	4	|	|	CCONJ
m-1153	156	5	september	september	PROPN
m-1153	156	6	2025	2025	NUM
m-1153	156	7	|	|	ADV
m-1153	156	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	156	9	34	34	NUM
m-1153	156	10	references	reference	NOUN
m-1153	156	11	ali	ali	PROPN
m-1153	156	12	,	,	PUNCT
m-1153	156	13	m.	m.	NOUN
m-1153	156	14	,	,	PUNCT
m-1153	156	15	nelson	nelson	PROPN
m-1153	156	16	,	,	PUNCT
m-1153	156	17	a.	a.	PROPN
m-1153	156	18	r.	r.	PROPN
m-1153	156	19	,	,	PUNCT
m-1153	156	20	lopez	lopez	PROPN
m-1153	156	21	,	,	PUNCT
m-1153	156	22	a.	a.	NOUN
m-1153	156	23	l.	l.	PROPN
m-1153	156	24	,	,	PUNCT
m-1153	156	25	and	and	CCONJ
m-1153	156	26	sack	sack	VERB
m-1153	156	27	,	,	PUNCT
m-1153	156	28	d.	d.	PROPN
m-1153	156	29	a.	a.	PROPN
m-1153	156	30	(	(	PUNCT
m-1153	156	31	2015	2015	NUM
m-1153	156	32	)	)	PUNCT
m-1153	156	33	.	.	PUNCT
m-1153	157	1	updated	update	VERB
m-1153	157	2	global	global	ADJ
m-1153	157	3	burden	burden	NOUN
m-1153	157	4	of	of	ADP
m-1153	157	5	cholera	cholera	NOUN
m-1153	157	6	in	in	ADP
m-1153	157	7	endemic	endemic	ADJ
m-1153	157	8	countries	country	NOUN
m-1153	157	9	.	.	PUNCT
m-1153	158	1	plos	plos	PROPN
m-1153	158	2	neglected	neglect	VERB
m-1153	158	3	tropical	tropical	ADJ
m-1153	158	4	diseases	disease	NOUN
m-1153	158	5	,	,	PUNCT
m-1153	158	6	9(6):e0003832	9(6):e0003832	NUM
m-1153	158	7	.	.	PUNCT
m-1153	159	1	anderson	anderson	PROPN
m-1153	159	2	,	,	PUNCT
m-1153	159	3	r.	r.	PROPN
m-1153	159	4	m.	m.	PROPN
m-1153	159	5	and	and	CCONJ
m-1153	159	6	may	may	AUX
m-1153	159	7	,	,	PUNCT
m-1153	159	8	r.	r.	PROPN
m-1153	159	9	m.	m.	PROPN
m-1153	159	10	(	(	PUNCT
m-1153	159	11	1991	1991	NUM
m-1153	159	12	)	)	PUNCT
m-1153	159	13	.	.	PUNCT
m-1153	160	1	infectious	infectious	ADJ
m-1153	160	2	diseases	disease	NOUN
m-1153	160	3	of	of	ADP
m-1153	160	4	humans	human	NOUN
m-1153	160	5	:	:	PUNCT
m-1153	160	6	dynamics	dynamic	NOUN
m-1153	160	7	and	and	CCONJ
m-1153	160	8	control	control	NOUN
m-1153	160	9	.	.	PUNCT
m-1153	161	1	oxford	oxford	PROPN
m-1153	161	2	university	university	PROPN
m-1153	161	3	press	press	NOUN
m-1153	161	4	.	.	PUNCT
m-1153	162	1	bertuzzo	bertuzzo	PROPN
m-1153	162	2	,	,	PUNCT
m-1153	162	3	e.	e.	PROPN
m-1153	162	4	,	,	PUNCT
m-1153	162	5	mari	mari	PROPN
m-1153	162	6	,	,	PUNCT
m-1153	162	7	l.	l.	PROPN
m-1153	162	8	,	,	PUNCT
m-1153	162	9	righetto	righetto	PROPN
m-1153	162	10	,	,	PUNCT
m-1153	162	11	l.	l.	PROPN
m-1153	162	12	,	,	PUNCT
m-1153	162	13	gatto	gatto	PROPN
m-1153	162	14	,	,	PUNCT
m-1153	162	15	m.	m.	NOUN
m-1153	162	16	,	,	PUNCT
m-1153	162	17	casagrandi	casagrandi	PROPN
m-1153	162	18	,	,	PUNCT
m-1153	162	19	r.	r.	PROPN
m-1153	162	20	,	,	PUNCT
m-1153	162	21	blokesch	blokesch	PROPN
m-1153	162	22	,	,	PUNCT
m-1153	162	23	m.	m.	NOUN
m-1153	162	24	,	,	PUNCT
m-1153	162	25	and	and	CCONJ
m-1153	162	26	rinaldo	rinaldo	PROPN
m-1153	162	27	,	,	PUNCT
m-1153	162	28	a.	a.	NOUN
m-1153	162	29	(	(	PUNCT
m-1153	162	30	2011	2011	NUM
m-1153	162	31	)	)	PUNCT
m-1153	162	32	.	.	PUNCT
m-1153	163	1	prediction	prediction	NOUN
m-1153	163	2	of	of	ADP
m-1153	163	3	the	the	DET
m-1153	163	4	spatial	spatial	ADJ
m-1153	163	5	evolution	evolution	NOUN
m-1153	163	6	and	and	CCONJ
m-1153	163	7	effects	effect	NOUN
m-1153	163	8	of	of	ADP
m-1153	163	9	control	control	NOUN
m-1153	163	10	measures	measure	NOUN
m-1153	163	11	for	for	ADP
m-1153	163	12	the	the	DET
m-1153	163	13	unfolding	unfold	VERB
m-1153	163	14	haiti	haiti	PROPN
m-1153	163	15	cholera	cholera	PROPN
m-1153	163	16	outbreak	outbreak	PROPN
m-1153	163	17	.	.	PUNCT
m-1153	164	1	geophysical	geophysical	ADJ
m-1153	164	2	research	research	NOUN
m-1153	164	3	letters	letter	NOUN
m-1153	164	4	,	,	PUNCT
m-1153	164	5	38(6	38(6	NOUN
m-1153	164	6	)	)	PUNCT
m-1153	164	7	.	.	PUNCT
m-1153	165	1	capasso	capasso	PROPN
m-1153	165	2	,	,	PUNCT
m-1153	165	3	v.	v.	ADP
m-1153	165	4	and	and	CCONJ
m-1153	165	5	paveri	paveri	PROPN
m-1153	165	6	-	-	PUNCT
m-1153	165	7	fontana	fontana	PROPN
m-1153	165	8	,	,	PUNCT
m-1153	165	9	s.	s.	PROPN
m-1153	165	10	l.	l.	PROPN
m-1153	165	11	(	(	PUNCT
m-1153	165	12	1979	1979	NUM
m-1153	165	13	)	)	PUNCT
m-1153	165	14	.	.	PUNCT
m-1153	166	1	a	a	DET
m-1153	166	2	mathematical	mathematical	ADJ
m-1153	166	3	model	model	NOUN
m-1153	166	4	for	for	ADP
m-1153	166	5	the	the	DET
m-1153	166	6	1973	1973	NUM
m-1153	166	7	cholera	cholera	NOUN
m-1153	166	8	epidemic	epidemic	NOUN
m-1153	166	9	in	in	ADP
m-1153	166	10	the	the	DET
m-1153	166	11	european	european	PROPN
m-1153	166	12	mediterranean	mediterranean	PROPN
m-1153	166	13	region	region	PROPN
m-1153	166	14	.	.	PUNCT
m-1153	167	1	revue	revue	NOUN
m-1153	167	2	d’épidémiologie	d’épidémiologie	PROPN
m-1153	167	3	et	et	NOUN
m-1153	167	4	de	de	X
m-1153	167	5	santé	santé	NOUN
m-1153	167	6	publique	publique	NOUN
m-1153	167	7	,	,	PUNCT
m-1153	167	8	27(2):121–132	27(2):121–132	PROPN
m-1153	167	9	.	.	PUNCT
m-1153	168	1	castillo	castillo	PROPN
m-1153	168	2	-	-	PUNCT
m-1153	168	3	chavez	chavez	PROPN
m-1153	168	4	,	,	PUNCT
m-1153	168	5	c.	c.	PROPN
m-1153	168	6	and	and	CCONJ
m-1153	168	7	feng	feng	PROPN
m-1153	168	8	,	,	PUNCT
m-1153	168	9	z.	z.	PROPN
m-1153	168	10	(	(	PUNCT
m-1153	168	11	1997	1997	NUM
m-1153	168	12	)	)	PUNCT
m-1153	168	13	.	.	PUNCT
m-1153	169	1	to	to	PART
m-1153	169	2	treat	treat	VERB
m-1153	169	3	or	or	CCONJ
m-1153	169	4	not	not	PART
m-1153	169	5	to	to	PART
m-1153	169	6	treat	treat	VERB
m-1153	169	7	:	:	PUNCT
m-1153	169	8	the	the	DET
m-1153	169	9	case	case	NOUN
m-1153	169	10	of	of	ADP
m-1153	169	11	tuberculosis	tuberculosis	NOUN
m-1153	169	12	.	.	PUNCT
m-1153	170	1	journal	journal	PROPN
m-1153	170	2	of	of	ADP
m-1153	170	3	mathematical	mathematical	ADJ
m-1153	170	4	biology	biology	NOUN
m-1153	170	5	,	,	PUNCT
m-1153	170	6	35(6):629–656	35(6):629–656	PROPN
m-1153	170	7	.	.	PUNCT
m-1153	171	1	chao	chao	PROPN
m-1153	171	2	,	,	PUNCT
m-1153	171	3	d.	d.	PROPN
m-1153	171	4	l.	l.	PROPN
m-1153	171	5	,	,	PUNCT
m-1153	171	6	halloran	halloran	PROPN
m-1153	171	7	,	,	PUNCT
m-1153	171	8	m.	m.	NOUN
m-1153	171	9	e.	e.	PROPN
m-1153	171	10	,	,	PUNCT
m-1153	171	11	and	and	CCONJ
m-1153	171	12	longini	longini	PROPN
m-1153	171	13	,	,	PUNCT
m-1153	171	14	i.	i.	PROPN
m-1153	171	15	m.	m.	PROPN
m-1153	171	16	(	(	PUNCT
m-1153	171	17	2011	2011	NUM
m-1153	171	18	)	)	PUNCT
m-1153	171	19	.	.	PUNCT
m-1153	172	1	vaccination	vaccination	NOUN
m-1153	172	2	strategies	strategy	NOUN
m-1153	172	3	for	for	ADP
m-1153	172	4	epidemic	epidemic	ADJ
m-1153	172	5	cholera	cholera	NOUN
m-1153	172	6	in	in	ADP
m-1153	172	7	haiti	haiti	PROPN
m-1153	172	8	with	with	ADP
m-1153	172	9	implications	implication	NOUN
m-1153	172	10	for	for	ADP
m-1153	172	11	the	the	DET
m-1153	172	12	developing	develop	VERB
m-1153	172	13	world	world	NOUN
m-1153	172	14	.	.	PUNCT
m-1153	173	1	proceedings	proceeding	NOUN
m-1153	173	2	of	of	ADP
m-1153	173	3	the	the	DET
m-1153	173	4	national	national	PROPN
m-1153	173	5	academy	academy	PROPN
m-1153	173	6	of	of	ADP
m-1153	173	7	sciences	sciences	PROPN
m-1153	173	8	,	,	PUNCT
m-1153	173	9	108(17):7081–7085	108(17):7081–7085	NUM
m-1153	173	10	.	.	PUNCT
m-1153	173	11	codeço	codeço	PROPN
m-1153	173	12	,	,	PUNCT
m-1153	173	13	c.	c.	PROPN
m-1153	173	14	t.	t.	PROPN
m-1153	173	15	(	(	PUNCT
m-1153	173	16	2001	2001	NUM
m-1153	173	17	)	)	PUNCT
m-1153	173	18	.	.	PUNCT
m-1153	174	1	endemic	endemic	ADJ
m-1153	174	2	and	and	CCONJ
m-1153	174	3	epidemic	epidemic	NOUN
m-1153	174	4	dynamics	dynamic	NOUN
m-1153	174	5	of	of	ADP
m-1153	174	6	cholera	cholera	NOUN
m-1153	174	7	:	:	PUNCT
m-1153	174	8	the	the	DET
m-1153	174	9	role	role	NOUN
m-1153	174	10	of	of	ADP
m-1153	174	11	the	the	DET
m-1153	174	12	aquatic	aquatic	ADJ
m-1153	174	13	reservoir	reservoir	NOUN
m-1153	174	14	.	.	PUNCT
m-1153	175	1	bmc	bmc	VERB
m-1153	175	2	infectious	infectious	ADJ
m-1153	175	3	diseases	disease	NOUN
m-1153	175	4	,	,	PUNCT
m-1153	175	5	1:1	1:1	NUM
m-1153	175	6	.	.	PUNCT
m-1153	176	1	eisenberg	eisenberg	PROPN
m-1153	176	2	,	,	PUNCT
m-1153	176	3	m.	m.	PROPN
m-1153	176	4	c.	c.	PROPN
m-1153	176	5	,	,	PUNCT
m-1153	176	6	shuai	shuai	PROPN
m-1153	176	7	,	,	PUNCT
m-1153	176	8	z.	z.	PROPN
m-1153	176	9	,	,	PUNCT
m-1153	176	10	tien	tien	PROPN
m-1153	176	11	,	,	PUNCT
m-1153	176	12	j.	j.	PROPN
m-1153	176	13	h.	h.	PROPN
m-1153	176	14	,	,	PUNCT
m-1153	176	15	and	and	CCONJ
m-1153	176	16	van	van	PROPN
m-1153	176	17	den	den	PROPN
m-1153	176	18	driessche	driessche	PROPN
m-1153	176	19	,	,	PUNCT
m-1153	176	20	p.	p.	NOUN
m-1153	176	21	(	(	PUNCT
m-1153	176	22	2013	2013	NUM
m-1153	176	23	)	)	PUNCT
m-1153	176	24	.	.	PUNCT
m-1153	177	1	a	a	DET
m-1153	177	2	cholera	cholera	NOUN
m-1153	177	3	model	model	NOUN
m-1153	177	4	in	in	ADP
m-1153	177	5	a	a	DET
m-1153	177	6	patchy	patchy	ADJ
m-1153	177	7	environment	environment	NOUN
m-1153	177	8	with	with	ADP
m-1153	177	9	water	water	NOUN
m-1153	177	10	and	and	CCONJ
m-1153	177	11	human	human	ADJ
m-1153	177	12	movement	movement	NOUN
m-1153	177	13	.	.	PUNCT
m-1153	178	1	mathematical	mathematical	ADJ
m-1153	178	2	biosciences	bioscience	NOUN
m-1153	178	3	,	,	PUNCT
m-1153	178	4	246(1):105–112	246(1):105–112	PROPN
m-1153	178	5	.	.	PUNCT
m-1153	179	1	hartley	hartley	PROPN
m-1153	179	2	,	,	PUNCT
m-1153	179	3	d.	d.	PROPN
m-1153	179	4	m.	m.	PROPN
m-1153	179	5	,	,	PUNCT
m-1153	179	6	morris	morris	PROPN
m-1153	179	7	,	,	PUNCT
m-1153	179	8	j.	j.	PROPN
m-1153	179	9	g.	g.	PROPN
m-1153	179	10	,	,	PUNCT
m-1153	179	11	and	and	CCONJ
m-1153	179	12	smith	smith	PROPN
m-1153	179	13	,	,	PUNCT
m-1153	179	14	d.	d.	PROPN
m-1153	179	15	l.	l.	PROPN
m-1153	179	16	(	(	PUNCT
m-1153	179	17	2006	2006	NUM
m-1153	179	18	)	)	PUNCT
m-1153	179	19	.	.	PUNCT
m-1153	180	1	hyperinfectivity	hyperinfectivity	NOUN
m-1153	180	2	:	:	PUNCT
m-1153	180	3	a	a	DET
m-1153	180	4	critical	critical	ADJ
m-1153	180	5	element	element	NOUN
m-1153	180	6	in	in	ADP
m-1153	180	7	the	the	DET
m-1153	180	8	ability	ability	NOUN
m-1153	180	9	of	of	ADP
m-1153	180	10	vibrio	vibrio	ADJ
m-1153	180	11	cholerae	cholerae	NOUN
m-1153	180	12	to	to	PART
m-1153	180	13	cause	cause	VERB
m-1153	180	14	epidemics	epidemic	NOUN
m-1153	180	15	?	?	PUNCT
m-1153	181	1	plos	plos	PROPN
m-1153	181	2	medicine	medicine	NOUN
m-1153	181	3	,	,	PUNCT
m-1153	181	4	3(1):e7	3(1):e7	NUM
m-1153	181	5	.	.	PUNCT
m-1153	182	1	hethcote	hethcote	ADV
m-1153	182	2	,	,	PUNCT
m-1153	182	3	h.	h.	PROPN
m-1153	182	4	w.	w.	PROPN
m-1153	182	5	(	(	PUNCT
m-1153	182	6	2000	2000	NUM
m-1153	182	7	)	)	PUNCT
m-1153	182	8	.	.	PUNCT
m-1153	183	1	the	the	DET
m-1153	183	2	mathematics	mathematic	NOUN
m-1153	183	3	of	of	ADP
m-1153	183	4	infectious	infectious	ADJ
m-1153	183	5	diseases	disease	NOUN
m-1153	183	6	.	.	PUNCT
m-1153	184	1	siam	siam	PROPN
m-1153	184	2	review	review	PROPN
m-1153	184	3	,	,	PUNCT
m-1153	184	4	42(4):599	42(4):599	NUM
m-1153	184	5	–	–	PUNCT
m-1153	184	6	653	653	NUM
m-1153	184	7	.	.	PUNCT
m-1153	185	1	ijo	ijo	PROPN
m-1153	185	2	international	international	PROPN
m-1153	185	3	journal	journal	PROPN
m-1153	185	4	of	of	ADP
m-1153	185	5	mathematics	mathematics	PROPN
m-1153	185	6	(	(	PUNCT
m-1153	185	7	issn	issn	PROPN
m-1153	185	8	:	:	PUNCT
m-1153	185	9	2992	2992	NUM
m-1153	185	10	-	-	SYM
m-1153	185	11	4421	4421	NUM
m-1153	185	12	)	)	PUNCT
m-1153	185	13	volume	volume	NOUN
m-1153	185	14	08	08	NUM
m-1153	186	1	|	|	ADV
m-1153	186	2	issue	issue	NOUN
m-1153	186	3	9	9	NUM
m-1153	186	4	|	|	CCONJ
m-1153	186	5	september	september	PROPN
m-1153	186	6	2025	2025	NUM
m-1153	186	7	|	|	ADV
m-1153	186	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	186	9	35	35	NUM
m-1153	186	10	koelle	koelle	NOUN
m-1153	186	11	,	,	PUNCT
m-1153	186	12	k.	k.	PROPN
m-1153	186	13	and	and	CCONJ
m-1153	186	14	pascual	pascual	PROPN
m-1153	186	15	,	,	PUNCT
m-1153	186	16	m.	m.	NOUN
m-1153	186	17	(	(	PUNCT
m-1153	186	18	2004	2004	NUM
m-1153	186	19	)	)	PUNCT
m-1153	186	20	.	.	PUNCT
m-1153	187	1	disentangling	disentangle	VERB
m-1153	187	2	extrinsic	extrinsic	ADV
m-1153	187	3	from	from	ADP
m-1153	187	4	intrinsic	intrinsic	ADJ
m-1153	187	5	factors	factor	NOUN
m-1153	187	6	in	in	ADP
m-1153	187	7	disease	disease	NOUN
m-1153	187	8	dynamics	dynamic	NOUN
m-1153	187	9	:	:	PUNCT
m-1153	187	10	a	a	DET
m-1153	187	11	nonlinear	nonlinear	ADJ
m-1153	187	12	time	time	NOUN
m-1153	187	13	series	series	PROPN
m-1153	187	14	approach	approach	NOUN
m-1153	187	15	with	with	ADP
m-1153	187	16	an	an	DET
m-1153	187	17	application	application	NOUN
m-1153	187	18	to	to	ADP
m-1153	187	19	cholera	cholera	NOUN
m-1153	187	20	.	.	PUNCT
m-1153	188	1	the	the	DET
m-1153	188	2	american	american	PROPN
m-1153	188	3	naturalist	naturalist	PROPN
m-1153	188	4	,	,	PUNCT
m-1153	188	5	163(6):901–913	163(6):901–913	PROPN
m-1153	188	6	.	.	PUNCT
m-1153	188	7	leung	leung	PROPN
m-1153	188	8	,	,	PUNCT
m-1153	188	9	d.	d.	PROPN
m-1153	188	10	t.	t.	PROPN
m-1153	188	11	,	,	PUNCT
m-1153	188	12	rahman	rahman	PROPN
m-1153	188	13	,	,	PUNCT
m-1153	188	14	m.	m.	NOUN
m-1153	188	15	a.	a.	PROPN
m-1153	188	16	,	,	PUNCT
m-1153	188	17	mohasin	mohasin	PROPN
m-1153	188	18	,	,	PUNCT
m-1153	188	19	m.	m.	NOUN
m-1153	188	20	,	,	PUNCT
m-1153	188	21	riyadh	riyadh	PROPN
m-1153	188	22	,	,	PUNCT
m-1153	188	23	m.	m.	NOUN
m-1153	188	24	a.	a.	PROPN
m-1153	188	25	,	,	PUNCT
m-1153	188	26	patel	patel	PROPN
m-1153	188	27	,	,	PUNCT
m-1153	188	28	s.	s.	PROPN
m-1153	188	29	m.	m.	PROPN
m-1153	188	30	,	,	PUNCT
m-1153	188	31	aktar	aktar	PROPN
m-1153	188	32	,	,	PUNCT
m-1153	188	33	a.	a.	NOUN
m-1153	188	34	,	,	PUNCT
m-1153	188	35	and	and	CCONJ
m-1153	188	36	qadri	qadri	PROPN
m-1153	188	37	,	,	PUNCT
m-1153	188	38	f.	f.	PROPN
m-1153	188	39	(	(	PUNCT
m-1153	188	40	2012	2012	NUM
m-1153	188	41	)	)	PUNCT
m-1153	188	42	.	.	PUNCT
m-1153	189	1	comparison	comparison	NOUN
m-1153	189	2	of	of	ADP
m-1153	189	3	immune	immune	ADJ
m-1153	189	4	responses	response	NOUN
m-1153	189	5	to	to	ADP
m-1153	189	6	the	the	DET
m-1153	189	7	killed	kill	VERB
m-1153	189	8	oral	oral	ADJ
m-1153	189	9	cholera	cholera	NOUN
m-1153	189	10	vaccine	vaccine	NOUN
m-1153	189	11	between	between	ADP
m-1153	189	12	bangladeshi	bangladeshi	ADJ
m-1153	189	13	children	child	NOUN
m-1153	189	14	and	and	CCONJ
m-1153	189	15	adults	adult	NOUN
m-1153	189	16	.	.	PUNCT
m-1153	190	1	vaccine	vaccine	NOUN
m-1153	190	2	,	,	PUNCT
m-1153	190	3	30(4):594–599	30(4):594–599	NUM
m-1153	190	4	.	.	PUNCT
m-1153	191	1	longini	longini	PROPN
m-1153	191	2	,	,	PUNCT
m-1153	191	3	i.	i.	PROPN
m-1153	191	4	m.	m.	PROPN
m-1153	191	5	,	,	PUNCT
m-1153	191	6	nizam	nizam	PROPN
m-1153	191	7	,	,	PUNCT
m-1153	191	8	a.	a.	PROPN
m-1153	191	9	,	,	PUNCT
m-1153	191	10	ali	ali	PROPN
m-1153	191	11	,	,	PUNCT
m-1153	191	12	m.	m.	NOUN
m-1153	191	13	,	,	PUNCT
m-1153	191	14	yunus	yunus	PROPN
m-1153	191	15	,	,	PUNCT
m-1153	191	16	m.	m.	NOUN
m-1153	191	17	,	,	PUNCT
m-1153	191	18	shenvi	shenvi	NOUN
m-1153	191	19	,	,	PUNCT
m-1153	191	20	n.	n.	NOUN
m-1153	191	21	,	,	PUNCT
m-1153	191	22	and	and	CCONJ
m-1153	191	23	clemens	clemens	PROPN
m-1153	191	24	,	,	PUNCT
m-1153	191	25	j.	j.	PROPN
m-1153	191	26	d.	d.	PROPN
m-1153	191	27	(	(	PUNCT
m-1153	191	28	2007	2007	NUM
m-1153	191	29	)	)	PUNCT
m-1153	191	30	.	.	PUNCT
m-1153	192	1	controlling	control	VERB
m-1153	192	2	endemic	endemic	ADJ
m-1153	192	3	cholera	cholera	NOUN
m-1153	192	4	with	with	ADP
m-1153	192	5	oral	oral	ADJ
m-1153	192	6	vaccines	vaccine	NOUN
m-1153	192	7	.	.	PUNCT
m-1153	193	1	plos	plos	PROPN
m-1153	193	2	medicine	medicine	PROPN
m-1153	193	3	,	,	PUNCT
m-1153	193	4	4(11):e336	4(11):e336	NUM
m-1153	193	5	.	.	PUNCT
m-1153	193	6	mukandavire	mukandavire	PROPN
m-1153	193	7	,	,	PUNCT
m-1153	193	8	z.	z.	PROPN
m-1153	193	9	,	,	PUNCT
m-1153	193	10	liao	liao	PROPN
m-1153	193	11	,	,	PUNCT
m-1153	193	12	s.	s.	PROPN
m-1153	193	13	,	,	PUNCT
m-1153	193	14	wang	wang	PROPN
m-1153	193	15	,	,	PUNCT
m-1153	193	16	j.	j.	PROPN
m-1153	193	17	,	,	PUNCT
m-1153	193	18	gaff	gaff	PROPN
m-1153	193	19	,	,	PUNCT
m-1153	193	20	h.	h.	PROPN
m-1153	193	21	,	,	PUNCT
m-1153	193	22	smith	smith	PROPN
m-1153	193	23	,	,	PUNCT
m-1153	193	24	d.	d.	PROPN
m-1153	193	25	l.	l.	PROPN
m-1153	193	26	,	,	PUNCT
m-1153	193	27	and	and	CCONJ
m-1153	193	28	morris	morris	PROPN
m-1153	193	29	,	,	PUNCT
m-1153	193	30	j.	j.	PROPN
m-1153	193	31	g.	g.	PROPN
m-1153	193	32	(	(	PUNCT
m-1153	193	33	2011	2011	NUM
m-1153	193	34	)	)	PUNCT
m-1153	193	35	.	.	PUNCT
m-1153	194	1	estimating	estimate	VERB
m-1153	194	2	the	the	DET
m-1153	194	3	reproductive	reproductive	ADJ
m-1153	194	4	numbers	number	NOUN
m-1153	194	5	for	for	ADP
m-1153	194	6	the	the	DET
m-1153	194	7	2008–2009	2008–2009	NUM
m-1153	194	8	cholera	cholera	NOUN
m-1153	194	9	outbreaks	outbreak	NOUN
m-1153	194	10	in	in	ADP
m-1153	194	11	zimbabwe	zimbabwe	PROPN
m-1153	194	12	.	.	PUNCT
m-1153	195	1	proceedings	proceeding	NOUN
m-1153	195	2	of	of	ADP
m-1153	195	3	the	the	DET
m-1153	195	4	national	national	PROPN
m-1153	195	5	academy	academy	PROPN
m-1153	195	6	of	of	ADP
m-1153	195	7	sciences	sciences	PROPN
m-1153	195	8	,	,	PUNCT
m-1153	195	9	108(21):8767–8772	108(21):8767–8772	NUM
m-1153	195	10	.	.	PUNCT
m-1153	196	1	pascual	pascual	PROPN
m-1153	196	2	,	,	PUNCT
m-1153	196	3	m.	m.	NOUN
m-1153	196	4	,	,	PUNCT
m-1153	196	5	rodó	rodó	NOUN
m-1153	196	6	,	,	PUNCT
m-1153	196	7	x.	x.	NOUN
m-1153	196	8	,	,	PUNCT
m-1153	196	9	ellner	ellner	PROPN
m-1153	196	10	,	,	PUNCT
m-1153	196	11	s.	s.	PROPN
m-1153	196	12	p.	p.	PROPN
m-1153	196	13	,	,	PUNCT
m-1153	196	14	colwell	colwell	PROPN
m-1153	196	15	,	,	PUNCT
m-1153	196	16	r.	r.	PROPN
m-1153	196	17	,	,	PUNCT
m-1153	196	18	and	and	CCONJ
m-1153	196	19	bouma	bouma	PROPN
m-1153	196	20	,	,	PUNCT
m-1153	196	21	m.	m.	PROPN
m-1153	196	22	j.	j.	PROPN
m-1153	196	23	(	(	PUNCT
m-1153	196	24	2000	2000	NUM
m-1153	196	25	)	)	PUNCT
m-1153	196	26	.	.	PUNCT
m-1153	197	1	cholera	cholera	NOUN
m-1153	197	2	dynamics	dynamic	NOUN
m-1153	197	3	and	and	CCONJ
m-1153	197	4	el	el	PROPN
m-1153	197	5	niño	niño	PROPN
m-1153	197	6	–	–	PUNCT
m-1153	197	7	southern	southern	ADJ
m-1153	197	8	oscillation	oscillation	NOUN
m-1153	197	9	.	.	PUNCT
m-1153	198	1	science	science	NOUN
m-1153	198	2	,	,	PUNCT
m-1153	198	3	289(5485):1766–1769	289(5485):1766–1769	NUM
m-1153	198	4	.	.	PUNCT
m-1153	199	1	qadri	qadri	PROPN
m-1153	199	2	,	,	PUNCT
m-1153	199	3	f.	f.	PROPN
m-1153	199	4	,	,	PUNCT
m-1153	199	5	wierzba	wierzba	NOUN
m-1153	199	6	,	,	PUNCT
m-1153	199	7	t.	t.	PROPN
m-1153	199	8	f.	f.	PROPN
m-1153	199	9	,	,	PUNCT
m-1153	199	10	ali	ali	PROPN
m-1153	199	11	,	,	PUNCT
m-1153	199	12	m.	m.	NOUN
m-1153	199	13	,	,	PUNCT
m-1153	199	14	and	and	CCONJ
m-1153	199	15	clemens	clemens	PROPN
m-1153	199	16	,	,	PUNCT
m-1153	199	17	j.	j.	PROPN
m-1153	199	18	d.	d.	PROPN
m-1153	199	19	(	(	PUNCT
m-1153	199	20	2020	2020	NUM
m-1153	199	21	)	)	PUNCT
m-1153	199	22	.	.	PUNCT
m-1153	200	1	cholera	cholera	NOUN
m-1153	200	2	in	in	ADP
m-1153	200	3	yemen	yemen	PROPN
m-1153	200	4	—	—	PUNCT
m-1153	200	5	an	an	DET
m-1153	200	6	old	old	ADJ
m-1153	200	7	foe	foe	NOUN
m-1153	200	8	rearing	rear	VERB
m-1153	200	9	its	its	PRON
m-1153	200	10	ugly	ugly	ADJ
m-1153	200	11	head	head	NOUN
m-1153	200	12	.	.	PUNCT
m-1153	201	1	new	new	PROPN
m-1153	201	2	england	england	PROPN
m-1153	201	3	journal	journal	PROPN
m-1153	201	4	of	of	ADP
m-1153	201	5	medicine	medicine	NOUN
m-1153	201	6	,	,	PUNCT
m-1153	201	7	382(6):568–571	382(6):568–571	NUM
m-1153	201	8	.	.	PUNCT
m-1153	202	1	tien	tien	PROPN
m-1153	202	2	,	,	PUNCT
m-1153	202	3	j.	j.	PROPN
m-1153	202	4	h.	h.	PROPN
m-1153	202	5	and	and	CCONJ
m-1153	202	6	earn	earn	VERB
m-1153	202	7	,	,	PUNCT
m-1153	202	8	d.	d.	PROPN
m-1153	202	9	j.	j.	PROPN
m-1153	202	10	d.	d.	PROPN
m-1153	202	11	(	(	PUNCT
m-1153	202	12	2010	2010	NUM
m-1153	202	13	)	)	PUNCT
m-1153	202	14	.	.	PUNCT
m-1153	203	1	multiple	multiple	ADJ
m-1153	203	2	transmission	transmission	NOUN
m-1153	203	3	pathways	pathway	NOUN
m-1153	203	4	and	and	CCONJ
m-1153	203	5	disease	disease	NOUN
m-1153	203	6	dynamics	dynamic	NOUN
m-1153	203	7	in	in	ADP
m-1153	203	8	a	a	DET
m-1153	203	9	waterborne	waterborne	ADJ
m-1153	203	10	pathogen	pathogen	NOUN
m-1153	203	11	model	model	NOUN
m-1153	203	12	.	.	PUNCT
m-1153	204	1	bulletin	bulletin	NOUN
m-1153	204	2	of	of	ADP
m-1153	204	3	mathematical	mathematical	ADJ
m-1153	204	4	biology	biology	NOUN
m-1153	204	5	,	,	PUNCT
m-1153	204	6	72(6):1506	72(6):1506	NUM
m-1153	204	7	–	–	PUNCT
m-1153	204	8	1533	1533	NUM
m-1153	204	9	.	.	PUNCT
m-1153	205	1	troeger	troeger	NOUN
m-1153	205	2	,	,	PUNCT
m-1153	205	3	c.	c.	PROPN
m-1153	205	4	,	,	PUNCT
m-1153	205	5	blacker	black	ADJ
m-1153	205	6	,	,	PUNCT
m-1153	205	7	b.	b.	PROPN
m-1153	205	8	f.	f.	PROPN
m-1153	205	9	,	,	PUNCT
m-1153	205	10	khalil	khalil	PROPN
m-1153	205	11	,	,	PUNCT
m-1153	205	12	i.	i.	PROPN
m-1153	205	13	a.	a.	PROPN
m-1153	205	14	,	,	PUNCT
m-1153	205	15	rao	rao	PROPN
m-1153	205	16	,	,	PUNCT
m-1153	205	17	p.	p.	PROPN
m-1153	205	18	c.	c.	PROPN
m-1153	205	19	,	,	PUNCT
m-1153	205	20	cao	cao	PROPN
m-1153	205	21	,	,	PUNCT
m-1153	205	22	j.	j.	PROPN
m-1153	205	23	,	,	PUNCT
m-1153	205	24	zimsen	zimsen	PROPN
m-1153	205	25	,	,	PUNCT
m-1153	205	26	s.	s.	PROPN
m-1153	205	27	r.	r.	PROPN
m-1153	205	28	,	,	PUNCT
m-1153	205	29	and	and	CCONJ
m-1153	205	30	reiner	reiner	NOUN
m-1153	205	31	,	,	PUNCT
m-1153	205	32	r.	r.	PROPN
m-1153	205	33	c.	c.	PROPN
m-1153	205	34	(	(	PUNCT
m-1153	205	35	2018	2018	NUM
m-1153	205	36	)	)	PUNCT
m-1153	205	37	.	.	PUNCT
m-1153	206	1	estimates	estimate	NOUN
m-1153	206	2	of	of	ADP
m-1153	206	3	the	the	DET
m-1153	206	4	global	global	ADJ
m-1153	206	5	,	,	PUNCT
m-1153	206	6	regional	regional	ADJ
m-1153	206	7	,	,	PUNCT
m-1153	206	8	and	and	CCONJ
m-1153	206	9	national	national	ADJ
m-1153	206	10	morbidity	morbidity	NOUN
m-1153	206	11	,	,	PUNCT
m-1153	206	12	mortality	mortality	NOUN
m-1153	206	13	,	,	PUNCT
m-1153	206	14	and	and	CCONJ
m-1153	206	15	aetiologies	aetiology	NOUN
m-1153	206	16	of	of	ADP
m-1153	206	17	diarrhoea	diarrhoea	NOUN
m-1153	206	18	in	in	ADP
m-1153	206	19	195	195	NUM
m-1153	206	20	countries	country	NOUN
m-1153	206	21	:	:	PUNCT
m-1153	206	22	a	a	DET
m-1153	206	23	systematic	systematic	ADJ
m-1153	206	24	analysis	analysis	NOUN
m-1153	206	25	for	for	ADP
m-1153	206	26	the	the	DET
m-1153	206	27	global	global	ADJ
m-1153	206	28	burden	burden	NOUN
m-1153	206	29	of	of	ADP
m-1153	206	30	disease	disease	NOUN
m-1153	206	31	study	study	NOUN
m-1153	206	32	2016	2016	NUM
m-1153	206	33	.	.	PUNCT
m-1153	207	1	the	the	DET
m-1153	207	2	lancet	lancet	ADJ
m-1153	207	3	infectious	infectious	ADJ
m-1153	207	4	diseases	disease	NOUN
m-1153	207	5	,	,	PUNCT
m-1153	207	6	18(11):1211–1228	18(11):1211–1228	PROPN
m-1153	207	7	.	.	PUNCT
m-1153	208	1	van	van	PROPN
m-1153	208	2	den	den	PROPN
m-1153	208	3	driessche	driessche	PROPN
m-1153	208	4	,	,	PUNCT
m-1153	208	5	p.	p.	NOUN
m-1153	208	6	and	and	CCONJ
m-1153	208	7	watmough	watmough	PROPN
m-1153	208	8	,	,	PUNCT
m-1153	208	9	j.	j.	PROPN
m-1153	208	10	(	(	PUNCT
m-1153	208	11	2002	2002	NUM
m-1153	208	12	)	)	PUNCT
m-1153	208	13	.	.	PUNCT
m-1153	209	1	reproduction	reproduction	NOUN
m-1153	209	2	numbers	number	NOUN
m-1153	209	3	and	and	CCONJ
m-1153	209	4	subthreshold	subthreshold	ADJ
m-1153	209	5	endemic	endemic	ADJ
m-1153	209	6	equilibria	equilibrium	NOUN
m-1153	209	7	for	for	ADP
m-1153	209	8	compartmental	compartmental	ADJ
m-1153	209	9	models	model	NOUN
m-1153	209	10	of	of	ADP
m-1153	209	11	disease	disease	NOUN
m-1153	209	12	transmission	transmission	NOUN
m-1153	209	13	.	.	PUNCT
m-1153	210	1	mathematical	mathematical	ADJ
m-1153	210	2	biosciences	bioscience	NOUN
m-1153	210	3	,	,	PUNCT
m-1153	210	4	180(1–2):29–48	180(1–2):29–48	NUM
m-1153	210	5	.	.	PUNCT
m-1153	211	1	ijo	ijo	PROPN
m-1153	211	2	international	international	PROPN
m-1153	211	3	journal	journal	PROPN
m-1153	211	4	of	of	ADP
m-1153	211	5	mathematics	mathematics	PROPN
m-1153	211	6	(	(	PUNCT
m-1153	211	7	issn	issn	PROPN
m-1153	211	8	:	:	PUNCT
m-1153	211	9	2992	2992	NUM
m-1153	211	10	-	-	SYM
m-1153	211	11	4421	4421	NUM
m-1153	211	12	)	)	PUNCT
m-1153	211	13	volume	volume	NOUN
m-1153	211	14	08	08	NUM
m-1153	212	1	|	|	ADV
m-1153	212	2	issue	issue	NOUN
m-1153	212	3	9	9	NUM
m-1153	212	4	|	|	CCONJ
m-1153	212	5	september	september	PROPN
m-1153	212	6	2025	2025	NUM
m-1153	212	7	|	|	ADV
m-1153	212	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	212	9	36	36	NUM
m-1153	212	10	world	world	PROPN
m-1153	212	11	health	health	NOUN
m-1153	212	12	organization	organization	NOUN
m-1153	212	13	(	(	PUNCT
m-1153	212	14	2023	2023	NUM
m-1153	212	15	)	)	PUNCT
m-1153	212	16	.	.	PUNCT
m-1153	213	1	cholera	cholera	NOUN
m-1153	213	2	.	.	PUNCT
m-1153	214	1	https://www.who.int/news-room/	https://www.who.int/news-room/	PROPN
m-1153	214	2	fact	fact	NOUN
m-1153	214	3	-	-	PUNCT
m-1153	214	4	sheets	sheet	NOUN
m-1153	214	5	/	/	SYM
m-1153	214	6	detail	detail	NOUN
m-1153	214	7	/	/	SYM
m-1153	214	8	cholera	cholera	NOUN
m-1153	214	9	.	.	PUNCT
m-1153	215	1	accessed	access	VERB
m-1153	215	2	:	:	PUNCT
m-1153	215	3	2025	2025	NUM
m-1153	215	4	-	-	SYM
m-1153	215	5	09	09	NUM
m-1153	215	6	-	-	PUNCT
m-1153	215	7	30	30	NUM
m-1153	215	8	.	.	PUNCT
m-1153	216	1	ijo	ijo	PROPN
m-1153	216	2	international	international	PROPN
m-1153	216	3	journal	journal	PROPN
m-1153	216	4	of	of	ADP
m-1153	216	5	mathematics	mathematics	PROPN
m-1153	216	6	(	(	PUNCT
m-1153	216	7	issn	issn	PROPN
m-1153	216	8	:	:	PUNCT
m-1153	216	9	2992	2992	NUM
m-1153	216	10	-	-	SYM
m-1153	216	11	4421	4421	NUM
m-1153	216	12	)	)	PUNCT
m-1153	216	13	volume	volume	NOUN
m-1153	216	14	08	08	NUM
m-1153	217	1	|	|	ADV
m-1153	217	2	issue	issue	NOUN
m-1153	217	3	9	9	NUM
m-1153	217	4	|	|	CCONJ
m-1153	217	5	september	september	PROPN
m-1153	217	6	2025	2025	NUM
m-1153	217	7	|	|	ADV
m-1153	217	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1153	217	9	37	37	NUM
m-1153	217	10	https://www.who.int/news-room/fact-sheets/detail/cholera	https://www.who.int/news-room/fact-sheets/detail/cholera	NOUN
m-1153	217	11	https://www.who.int/news-room/fact-sheets/detail/cholera	https://www.who.int/news-room/fact-sheets/detail/cholera	NOUN
m-1153	217	12	introduction	introduction	NOUN
m-1153	217	13	model	model	NOUN
m-1153	217	14	formulation	formulation	NOUN
m-1153	217	15	the	the	DET
m-1153	217	16	equilibrium	equilibrium	NOUN
m-1153	217	17	point	point	NOUN
m-1153	217	18	notation	notation	NOUN
m-1153	217	19	disease	disease	NOUN
m-1153	217	20	-	-	PUNCT
m-1153	217	21	free	free	ADJ
m-1153	217	22	equilibrium	equilibrium	NOUN
m-1153	217	23	(	(	PUNCT
m-1153	217	24	dfe	dfe	PROPN
m-1153	217	25	)	)	PUNCT
m-1153	217	26	endemic	endemic	ADJ
m-1153	217	27	equilibrium	equilibrium	NOUN
m-1153	217	28	(	(	PUNCT
m-1153	217	29	general	general	ADJ
m-1153	217	30	case	case	NOUN
m-1153	217	31	i*>0	i*>0	NOUN
m-1153	217	32	)	)	PUNCT
m-1153	217	33	special	special	ADJ
m-1153	217	34	case	case	NOUN
m-1153	217	35	:	:	PUNCT
m-1153	217	36	=	=	X
m-1153	217	37	0	0	NUM
m-1153	217	38	basic	basic	ADJ
m-1153	217	39	reproduction	reproduction	NOUN
m-1153	217	40	number	number	NOUN
m-1153	217	41	assumption	assumption	NOUN
m-1153	217	42	(	(	PUNCT
m-1153	217	43	existence	existence	NOUN
m-1153	217	44	of	of	ADP
m-1153	217	45	dfe	dfe	PROPN
m-1153	217	46	)	)	PUNCT
m-1153	217	47	interpretation	interpretation	NOUN
m-1153	217	48	conclusion	conclusion	NOUN
m-1153	217	49	and	and	CCONJ
m-1153	217	50	recommendations	recommendation	NOUN
m-1153	217	51	conclusion	conclusion	VERB