Ap il 2016 Volume 7(4) A icle e012861 www.esajou nals.o g
Ma ine in ec ious disease dynamics and ou b eak
h esholds: con ac ansmission, pandemic in ec ion,
and he po en ial ole o il e eede s
Go ka Bidegain,1,† E ic N. Powell,1 John M. Klinck,2
Tal Ben-Ho in,3, 4 and Eileen E. Ho mann2
1Gul Coas Resea ch Labo a o y, Uni e si y o Sou he n Mississippi, 703 Eas Beach D i e, Ocean Sp ings,
Mississippi 39564 USA
2Cen e o Coas al Physical Oceanog aphy, Old Dominon Uni e si y, 4111 Mona ch Way, No olk, Vi ginia 23529 USA
3Haskin Shell ish Resea ch Labo a o y, Ru ge s Uni e si y, 6959 Mille A enue, Po No is, New Je sey 08349 USA
4Depa men o Fishe ies, Animal and Ve e ina y Science, Uni e si y o Rhode Island, 20A Woodwa d Hall,
9 Eas Alumni A enue, Kings on, Rhode Island 02881 USA
Ci a ion: Bidegain, G., E. N. Powell, J. M. Klinck, T. Ben-Ho in, and E. E. Ho mann. 2016. Ma ine in ec ious disease
dynamics and ou b eak h esholds: con ac ansmission, pandemic in ec ion, and he po en ial ole o il e eede s.
Ecosphe e 7(4):e01286. 10.1002/ecs2.1286
Abs ac . Disease-causing o ganisms can ha e signi ican impac s on ma ine species and communi ies.
Howe e , he dynamics ha unde lie he eme gence o disease ou b eaks in ma ine ecosys ems s ill lack
he equi alen le el o desc ip ion, concep ual unde s anding, and modeling con ex ou inely p esen
in he e es ial sys ems. He e, we p opose a heo e ical basis o modeling he ansmission o ma ine
in ec ious diseases (MIDs) de eloped om simple models o he sp ead o in ec ious disease. The models
ep esen he dynamics o a a ie y o hos –pa hogen sys ems including hose unique o ma ine sys ems
whe e ansmission o disease is by con ac wi h wa e bo ne pa hogens bo h di ec ly and h ough il-
e - eeding p ocesses. O e all, he analysis o he epizoo iological models ocused on he mos ele an
p ocesses ha in e ac o d i e he ini ia ion and e mina ion o epizoo ics. A p io i, sys ems wi h mul i-
s ep disease in ec ions (e.g., in ec ion-dea h-pa icle elease- il a ion- ansmission) educed dependence
on indi idual pa ame e s esul ing in inhe en ly slowe ansmissions a es. This is demons ably no he
case; hus, hese al e na i e ansmission pa hways mus also conside ably inc ease he a es o p ocesses
in ol ed in ansmission. Sca enge s emo ing dead in ec ed animals may inhibi disease sp ead in bo h
con ac -based and wa e bo ne pa hogen-based diseases. The capaci y o highly in ec ed animals, bo h
ali e and dead, o elease a subs an ial numbe o in ec i e elemen s in o he wa e column, making hem
a ailable o suspension eede s esul s in such diseases being highly in ec i e wi h a e y small “low-abun-
dance e uge”. In hese sys ems, he body bu den o pa hogens and he ela i e impo ance be ween he
elease and he emo al a e o pa hogens in he hos issue o wa e column becomes pa amoun . Two
p ocesses a e o po en ial consequence inhibi ing epizoo ics. Fi s , la ge wa e olumes abo e he ben hic
suscep ible popula ions can unc ion as a sink o pa hogens. Second, unlike con ac -based disease models
in which an inc ease in he numbe o suscep ible indi iduals in he popula ion inc eases he likelihood
o ansmission and epizoo ic de elopmen , la ge popula ions o il e eede s can educe his likelihood
h ough he o e il a ion o in ec i e pa icles.
Key wo ds: basic ep oduc ion numbe ; epizoo iology; disease ecology; hos –pa hogen models; wa e bo ne pa hogens.
Recei ed 24 July 2015; accep ed 19 Oc obe 2015. Co esponding Edi o : A. Pa k.
† E-mail: [email p o ec ed]
Copy igh : © 2016 Bidegain e al. This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion
License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
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BIDEGAIN ET AL.
In oduc Ion
P oli e a ion o ma ine in ec ious diseases
(MID) subs an ially impac s he s uc u e and
unc ion o di e se ecosys ems by causing sig-
ni ican mo ali ies in ecologically ele an pop-
ula ions o a wide ange o ma ine o ganisms
including mammals, co als, shell ish, in ish, and
sea g ass (Wa d and La e y 2004, Bu ge e al.
2014, La e y e al. 2015). This, in u n, h ea ens
ecologically aluable habi a s such as co al ee s,
oys e beds, sea g ass beds, and he di e si y o
he ocky sho e, and esul s in subs an ial eco-
nomic losses in aquacul u e (Walke and Win on
2010, La e y e al. 2015). Despi e he inc easing
ecogni ion o he impo ance o MIDs, in pa
due o he po en ial o clima e change o ex end
he ange and impac o pa asi es and pa ho-
gens (Ha ell e al. 2002, Bu ge e al. 2014), he
unde s anding o he dynamics ha unde lie he
gene a ion o ou b eaks and associa ed epidemi-
ological concep s lags behind ha o e es ial
ecosys ems (Ha ell e al. 2004).
How epizoo ics a e ini ia ed and e mina ed
in e es ial o ganisms has been desc ibed and
modeled ex ensi ely (e.g., Gill 1928, Acke man
e al. 1984, Ande son 1991). Typically, he con-
ac -based and ec o -bo ne in ec ious diseases
o e es ial e eb a es and hei epidemiology
a e modeled using some adap a ion o he Ke -
mack and McKend ick (1927) (as ep in ed in
Ke mack and McKend ick (1991a, b, c)) o mu-
la ion. In hese models, he ini ia ion o an ep-
idemic e en begins wi h one o a ew in ec ed
indi iduals and a la ge numbe o suscep ible
neighbo s wi h whom con ac is possible (An-
de son and May 1991). Thus, i is assumed ha
ela i ely close con ac be ween he in ec ed indi-
idual, o he ec o , and he hos is equi ed o
ansmission (Hassell 2000, Mund e al. 2009).
Con ac -based diseases also exis in he ma-
ine en i onmen , mos equen ly in ishes (e.g.,
Lo z and So o 2002, Lø dal and Enge 2002,
Ogu e al. 2005), being common in he case o
he ansmission o mul icellula pa asi es such
as ema odes o ces odes (Huspeni and La e y
2004). Al hough some au ho s (Dobson and May
1987, Ogu e al. 2005, K košek 2010) o mula -
ed con ac -based MID models based on he Ke -
mack and McKend ick (1927) model, o he MID
ansmission p ocesses a e di e en in na u e
om hose on land, and adap ing he Ke mack–
McKend ick models equi es he app ecia ion
and inco po a ion o hese undamen al di e -
ences (Ha ell e al. 2004, McCallum e al. 2004).
In MIDs, in addi ion o li e in ec ed animals,
dead in ec ed animals a e an impo an sou ce o
pa hogens. Fo ins ance, he pa hogen body bu -
den in dead oys e s in ec ed by De mo disease
and he po en ial elease a e upon dea h is much
highe han hose o in ec ed li e animals (Bushek
e al. 2002). Simila ly, ish ha died o disease can
be a sou ce o in ec ion by eleasing pa hogen
pa icles o he su ounding wa e (So o and Lo z
2001, Lo z e al. 2003, Vike e al. 2014). In he e es-
ial en i onmen , his ansmission ou e is less
well ep esen ed, al hough one well known exam-
ple is he mic opa asi e Bacillus an h acis, which
in ec s bo h humans and animals and whe e he
in ec ious agen is spo es ha en e he en i on-
men soon a e he dea h o a hos (Ge z 2011).
One o he mos dis inc i e ea u es o MIDs,
pa icula ly o ma ine in e eb a es, is he im-
po ance o spa ial ac o s in de e mining he
sp ead o disease. The di e ences in physical
p ope ies be ween seawa e and ai , such as den-
si y, esul in g ea e buoyancy, longe li e spans,
and long-dis ance dispe sion o aqua ic o gan-
isms including pa hogens (S a hmann 1990).
This, in u n, can esul in impo an pa hogen
dispe sion, concen a ion, and a ailabili y issues
o some in e eb a es such as sessile il e and
suspension eede s (e.g., bi al es and co als).
Such species can accumula e pa hogens om a
dilu e solu ion ha may ha e been eleased nea -
by o om many kilome e s away, hus he num-
be o neighbo ing in ec ed indi iduals may be
ela i ely unimpo an in compa ison wi h he
numbe o in ec i e pa hogens being supplied by
wa e anspo . This suspension- o il e - eede
li e s yle, highly ulne able o disease ansmis-
sion and widesp ead, is a a e condi ion in e es-
ial animals, and apa om swallows who snag
insec s on he wings when lying, he nea es
app oach o his condi ion a e he web-spinning
spide s. No mechanism has e ol ed o concen-
a ing pa icles om he a mosphe e in su icien
quan i y o p o ide an adequa e ood supply o
a e es ial il e eede (S a hmann 1990).
Disease ansmission in il e - eede s p obably
occu s ia an in ec i e dose (Bushek e al. 1997,
Fo d e al. 1999, Powell e al. 1999) a he han by
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BIDEGAIN ET AL.
unique con ac be ween pa hogen and hos . The
phenomenon o he in ec i e dose may be pa ic-
ula ly impo an o il e - eede s because o e -
il a ion (i.e., he wa e is il e ed mo e han once
as i passes h ough he popula ion (O ice e al.
1982)) can occu when he densi y o animals
is high enough, he eby educing he pa hogen
concen a ion a ailable su icien ly o pe mi he
compe i ion o pa hogens and he in e nal inac-
i a ion mechanisms o limi body bu den below
he in ec i e dose le el.
Model adap a ions o long-dis ance in ec ion
o en assume ha in ec ed indi iduals c oss dis-
ance ba ie s a some a e o make con ac wi h
suscep ible hos s o de ine con ac -based dis-
ance c i e ia [Rod íguez and To es-So ando
(2001), bu see also Hassell (2000) o al e na i e
app oaches]. No wi hs anding ha he de el-
opmen o an ai bo ne disease in, o ins ance,
a plan me apopula ion in ol es a p ocess o
dispe sion as well as local dynamics, he ans-
mission p ocess i sel can be modeled as a con-
ac -based and poin -sou ce p ocess (B own and
Ho mølle 2002). The e ec o pa hogen dilu ion
on nonpoin -sou ce ma ine diseases ansmission
common in suspension- eede s (Ho mann e al.
1995), has no ye been in es iga ed heo e ically.
The dis inc i e cha ac e is ics o MIDs oge h-
e wi h he limi ed ba ie s o dispe sal (Mc-
Callum e al. 2003) po en ially makes oceans
a much mo e a o able medium han land o
nonpoin -sou ce p ocesses o con ol he ans-
mission p ocess and he gene a ion o epizoo ics.
These cha ac e is ics a e a p ima y eason why
adap a ion o e es ial epidemiological mod-
els o ma ine diseases emains one o he poo ly
add essed p oblems in MIDs, wi h li le ad ance
(e.g., McCallum e al. 2005, Sokolow e al. 2009,
Yakob and Mumby 2011) as Ha ell e al. (2004)
s a ed i as a p io i y o u u e esea ch. In con-
as , p oli e a ion-based disease models ha e
ecei ed conside able a en ion as unde s and-
ing o p oli e a ion o in ec ion was su icien o
desc ibe he disease impac in popula ions cha -
ac e ized by apid nonpoin -sou ce ansmission
(Cal o e al. 2001, Powell e al. 2011, 2012).
This pape ocuses on he o mula ion o a se-
ies o models exempli ying he dynamics o a
a ie y o MIDs ep esen a i e o a di e si y o
hos , pa hogen, and ansmission p ocesses p es-
en in ma ine ecosys ems. Thus, we s udy disease
ansmission by ei he di ec con ac be ween
suscep ible and in ec i e animals, by con ac wi h
wa e bo ne pa hogens eleased by li e o dead
in ec ed animals h ough passi e impingemen o
in ec i e pa icles ia wa e cu en s o h ough
ac i e il a ion o in ec i e pa icles du ing il e
eeding. The o mula ion and desc ip ion o each
model is p esen ed oge he wi h examples o
ma ine hos –pa hogen sys ems which migh be
app op ia e examples o he gi en ansmission
model. Fo each modeled MID sys em, we ana-
lyze he basic ep oduc ion numbe
R0
and con-
side how changes in model pa ame e s a y he
ou come o he ansmission p ocess ela i e o
he h eshold condi ion o
R0=1
.
Models and BasIc ep oduc Ion nuMBe s
R0
Theo e ical basis o he models
A se ies o models (Fig. 1, Tables 1 and 2),
adap ing o a g ea e o lesse ex en he ma h-
ema ical heo y o epidemics ap opos Ke mack
and McKend ick (1927), a e o mula ed o ep-
esen in ec ious disease ansmission p ocesses
and dynamics in ma ine sys ems (Resul s). Fo
his pu pose, he mo e complex sessile in e e-
b a e disease models, including con ac wi h o
il a ion o wa e bo ne pa hogens and pa icle
di usion p ocesses, a e buil up om hose sim-
ple con ac -based SI models applied o ish and
mammal diseases. The models p esen ed he e do
no co e acul a i e bac e ial pa asi es (Kazama
and Fulle 1977) o complex li e cycles o p o-
ozoan (Robe son 2007) o me azoan pa asi es
(Gam e al. 2008) equi ing in e media e hos s.
We es ic his pape o compa men al mod-
els, he mos equen ly used class o models in
epidemiology (Diekmann e al. 2013). The dynam-
ics o he hos –pa hogen associa ion is desc ibed
by a sys em o o dina y di e en ial equa ions
(ODEs) which ep oduce he change wi h ime
(in days) de e minis ically o all subpopula ion
componen s. Mo e speci ically, we implici ly as-
sume a cons an a ea (in
m
−
2
) o olume (in
m
−
3
)
o he models, in o de o desc ibe he popula ion
in e ms o densi y o indi iduals o concen a ion
o pa hogens ins ead o simply he numbe o in-
di iduals o pa icles. We assume he absence o
mig a ion o ec ui men ; we igno e nondisease
mo ali y. In addi ion, in ec ed indi iduals always
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BIDEGAIN ET AL.
Fig. 1. Flow diag ams o he se ies o models (Table 1, Resul s). The a iables (compa men s) o each model
a e ep esen ed by uppe le e s (suscep ible animals S), in ec ed animals I, dead animals D, wa e bo ne
pa hogens P, il e ed pool o pa hogens in he suscep ible popula ion F, emo e pool o pa hogens Γ). The model
pa ame e s a e ep esen ed by lowe le e s desc ibed in Table 2. O ange solid a ows ep esen he ansmission
p ocesses and dashed black a ows ep esen he o he main p ocesses in he models desc ibed in Resul s.
(a)(b)
(c)(d)
(e) ( )
(g)(h)
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BIDEGAIN ET AL.
die om disease. Tha is, indi iduals do no e-
co e om he disease and, hence, also do no be-
come immune o he disease. This is ou inely he
case o MIDs in in e eb a es (Fo d 1985, Pow-
ell e al. 1996, Cu is 2003) because in e eb a es
do no ha e adap i e immune sys ems (e.g., Chu
and Lapey e 1993, Fo d and T ipp 1996, Allam
and Pailla d 1998) excep ing some pos epizoo ic
co al popula ions wi h adap i e immunological
esis ance in su i ing indi iduals (Mydla z e al.
2010, Reed e al. 2010) and ecupe a ion o aqua-
cul u e species a e an ibio ic ea men .
R0
es ima ion and sensi i i y analysis
R0
ep esen s he numbe o new cases o
in ec ion caused by one in ec ed indi idual in
a popula ion o only suscep ible indi iduals.
Usually, he de ini ion o
R0
in an epidemio-
logical con ex includes he h eshold alue o
1, whe ein, i
R0
>
1
, he disease can in ade
and an epidemic can occu and i
R0
<
1
, he
disease canno in ade and an ou b eak is no
expec ed (Diekmann e al. 1990, Die z 1993).
The o mula ions o
R0
o he se ies o models
p esen ed he e a e ob ained using he nex -gen-
e a ion ma ices (NGM) me hod (Diekmann
e al. 2010, Diekmann e al. 2013).
We analyze he local sensi i i y o
R0
o each
model h ough he sensi i i y index Ω (Ca ibo-
ni e al. 2007). The no malized sensi i i y index
o
R0
wi h espec o any pa ame e
pi
a a ixed
alue
p0
is
(1)
The baseline pa ame e alues (see cap ion in
Fig. 3) we e selec ed using as examples ma ine
diseases desc ibed in each model in oduc ion.
We selec ed pa ame e alues uni o mly dis ib-
u ed (i.e., a inc emen s o 10%) o e he pa ame-
e s ull o a leas wide ange o easible alues.
Ω
R0
pi=
𝜕R
0
𝜕p
i
×
p
i
R
0||||pi=p0
Table 1. Models, model cha ac e is ics, and example disease po en ially applicable. The disease lis is no
mean o be comp ehensi e, no does a unique men ion o a disease imply es ic ion o he disease o ha
pa icula model.
Model T ansmission Applicable sys ems
SI Con ac wi h in ec ed indi iduals Diseases in ish (e.g., salmons) (e.g., Lø dal and Enge 2002, Ogu e al.
2005) and mammals such as seals (Beche e al. 2002) whe e he disease
is ansmi ed h ough ubbing. In co als, con ac be ween sea ans
when g owing close oge he (Smi h e al. 1996)
SID Con ac wi h dead in ec ed
indi iduals
Pola bea s, ish, sh imps, and amphipods ge in ec ed by con ac ing o
eeding on dead ca cases (Lo z and So o 2002, Lo z e al. 2003, Rudol
and An ono ics 2007)
SIP Con ac wi h in ec i e pa icles o
eleased by in ec ed indi iduals
Black-band disease (Richa dson 2004, Z uloni e al. 2009) and
Aspe gillosis (Jolles e al. 2002) in co als; Wi he ing synd ome (WS) in
abalone (Moo e , e al. , 2001, 2002); ansmission o ema ode
ce ca iae (De Mon audouin e al. 1998)
SIPD Con ac wi h in ec i e pa icles
o omi es eleased by dead
in ec ed indi iduals
Black-band disease (Richa dson 2004, Z uloni e al. 2009) and
Aspe gillosis (Jolles e al. 2002) in co als h ough b eakdown o
decaying issue; abalone wi h WS (Moo e , e al. , 2001, 2002) and
sh imp wi h Whi e spo disease (Rudol and An ono ics 2007) shed
pa icles du ing decay and sca enging p ocesses
SIP-F Fil a ion o in ec i e pa icles
eleased by in ec ed indi iduals;
dose dependence
OsHV1 in paci ic oys e s (Schiko ski e al. 2011); MSX (Haskin e al. 1966)
and De mo (Mackin e al. 1950) diseases in oys e s; Pe kinsosis in
clams (Pailla d 2004, Dang e al. 2010)
SIPD-F Fil a ion o in ec i e pa icles
eleased by dead in ec ed
indi iduals; dose dependence
Oys e s in ec ed by De mo disease (Pe kinsus ma inus) elease pa ho-
gens in o he wa e by na u al decomposi ion o he ac ion o sca en-
ge s, hen o be il e ed by he popula ion (Choi e al. 1989, Bushek
e al. 2002). This is a likely ou e o many o he molluscan diseases
SIP-FV Fil a ion o in ec i e pa icles
eleased by in ec ed indi iduals;
dose dependence; dilu ion ia
olume
Sys ems wi h nonpoin sou ces o pa hogens and di usion p ocesses o
wa e bo ne pa hogens, whe e a wa e olume can ac as a ese oi o
pa icles
SIPD-FV Fil a ion o in ec i e pa icles
eleased by dead in ec ed
indi iduals; dose dependence;
dilu ion ia olume
Sys ems wi h nonpoin sou ces o pa hogens and di usion p ocesses o
wa e bo ne pa hogens, whe e a wa e olume can ac as a ese oi o
pa icles
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BIDEGAIN ET AL.
esul s
SI model
We begin wi h he s anda d SI model
(Suscep ible–In ec ed) model in which con ac
wi h an in ec ed indi idual sp eads he in ec-
ion. T ansmission o he disease is con olled
by he ansmission a e
𝛽con ac
(Eqs. 2 and 3).
The numbe o in ec ed indi iduals,
𝛽con ac IS
,
is linea ly p opo ional o he p oduc o he
spa ial densi ies o S and I. Besides ansmis-
sion, he dynamics o he in ec ed subpopula ion
I is con olled by disease mo ali y (mI), whe e
m is he mo ali y a e (Eq. 3): hus,
(2)
(3)
The basic ep oduc ion numbe is:
(4)
whe e N is he ini ial popula ion o suscep ible
indi iduals S.
R0
inc eases linea ly wi h espec
o N (Fig. 2a). Rela i ely la ge popula ions a e
mo e likely o inhibi epizoo ics i disease mo -
ali y a e m is high (i.e., in ec ed hos s emain in
he sys em o a sho e ime and a e less likely o
sp ead he disease) and ansmission a e is el-
a i ely low (i.e., suscep ible hos s a e less easily
in ec ed). The sensi i i y analysis demons a es
ha all pa ame e s ha e he same impac on
R0
(Fig. 3).
SID model
The dis inc i eness o his second model, wi h
espec o p e ious SI model, is ha he SID
dS
d
=−𝛽con ac IS
,
dI
d
=𝛽con ac IS−mI
.
R
0=
𝛽
con ac
N
m,
Table 2. Desc ip ion o a iables and pa ame e s. The las column iden i ies he models in which he a iable
o pa ame e is used. An as e isk iden i ies he use o he a iable in he
R0
o mula ion o ha model. No e
ha all models ha e an implici su ace a ea (
m
−
2
) o olume (
m
−
3
) o indi iduals and wa e bo ne pa hogens
espec i ely.
Va iables,
Pa ame e s De ini ion Uni s
S Suscep ible hos s in he popula ion Numbe o indi iduals
I In ec ed hos s in he popula ion Numbe o indi iduals
DDead in ec ed hos s in he popula ion Numbe o indi iduals
P Wa e bo ne pa hogens in he en i onmen (i.e., local pool) Numbe o pa icles
F To al numbe o pa hogens abso bed o il e ed by he popula ion Numbe o pa icles
Γ Wa e bo ne pa hogens in a emo e pool Numbe o pa icles
N Suscep ible hos s in he ini ial popula ion Numbe o indi iduals
R0
Basic ep oduc ion numbe Nondimensional
𝛽con ac
Disease ansmission a e by di ec con ac be ween suscep ible and in ec ed
indi iduals.
Indi idual
−1
day
−1
𝛽pa icle
Disease ansmission a e by con ac be ween suscep ibles and wa e bo ne
pa hogens.
Pa icle (wa e )
−1
day
−1
𝛽 il a ion
Disease ansmission a e by il a ion o wa e bo ne pa hogens by suscep ibles. Pa icle (in e nal)
−1
day
−1
mDisease mo ali y a e day
−1
d Remo al a e o dead indi iduals by sca enge s o bac e ia (decay) day
−1
cRelease a e o pa hogens om in ec ed o dead animals day
−1
b A e age body bu den o pa hogens in in ec ed o dead animals Numbe o pa icles
Loss a e o wa e bo ne pa hogens om he local pool day
−1
Fil a ion o abso p ion a e o in ec i e pa icles by hos s Indi idual
−1
day
−1
a Reduc ion a e o pa hogens inside hos s by diapedesis, phagocy osis, apop osis,
e c.
day
−1
γ Exchange a e o wa e bo ne pa hogens be ween emo e and local pools.
Exchange is assumed o be di usion-like and hus p opo ional o he
di e ence in concen a ion be ween he wo pools
day
−1
σ Loss a e o wa e bo ne pa hogen om he emo e pool day
−1
V
l
,sl
Vl
, he local olume, and i s ecip ocal sl m
−3
V
Γ
,s
VΓ
, he emo e olume, and i s ecip ocal s m
−3
Ap il 2016 Volume 7(4) A icle e012867 www.esajou nals.o g
BIDEGAIN ET AL.
model inco po a es he dead in ec ed indi iduals
(D) as a sou ce o in ec i e pa icles. SD
(Suscep ibles–Deads) models a guably a e less
common in e es ial habi a s. The in ec ion a e
o he S popula ion is con olled again by he
ansmission a e
𝛽con ac
, and is linea ly p opo -
ional o he spa ial densi y o S and, in his
case, D ins ead o I (Eqs 5 and 6). Eq. 7 desc ibes
he in oduc ion o dead animals o he sys em
a e in ec ed indi iduals die om in ec ion (mI)
and hei disappea ance by na u al decay o
consump ion by sca enge s including conspeci ics
(dD), whe e d ep esen s he emo al a e. Thus,
(5)
(6)
(7)
which yields a basic ep oduc ion numbe :
(8)
dS
d
=−𝛽con ac DS
,
dI
d
=𝛽con ac DS−mI
,
dD
d
=mI −dD
,
R
0=
√
𝛽con ac N
d.
Fig. 2. Theo e ical es ima ions o
R0
o a se ies o models, o inc easing popula ion densi y N. Using as
examples ma ine hos –pa hogen sys ems desc ibed in Table 1, he ollowing alues o he pa ame e s we e used: SI
and SID models (
𝛽con ac
=
1
×
10−3
,
m=d=1×10−1
), SIP and SIPD models (
𝛽pa icle
=1×10
−5
,
m=d=1×10−2
,
c=1×10−3
,
b=1×104
,
=8×10−1
), SIP-F and SIPD models (
𝛽 il a ion
=
1
×
10−5
,
=
2
×
10−3
,
a=1×10−3
; o
he o e il a ion cases (
=
5
×
10−2
,
a=5×10−3
), SIP-FV and SIPD-FV models o educed emo e olume (
VΓ
)
cases (γ = 1, σ = 0.8, sl = 10 o
Vl=0.1
, s = 20 o
VΓ=0.05
) and o he la ge emo e olume cases (s = 1 o
VΓ=1
).
Pa ame e s a e desc ibed and uni s a e p esen ed in Table 2. The o ange do ed line a
R0=1
ep esen s he c i ical
alue o he epizoo ic o occu .
R0
Ini ial popula ion densi y, N
0.0
0.5
1.0
1.5
2.0
0100 200300 400500 600
SI model
SID model
R0
Ini ial popula ion densi y, N
0.0
0.5
1.0
1.5
2.0
0 100 200 300 400 500 600
SIP model
SIPD model
R0
Ini ial popula ion densi y, NIni ial popula ion densi y, NIni ial popula ion densi y, N
0.0
0.5
1.0
1.5
2.0
0100 200300 400500 600
SIP F model
SIPD F model
SIP F o e il a ion
SIPD F o e il a ion
R0
Ini ial popula ion densi y, NIni ial popula ion densi y, NIni ial popula ion densi y, N
0.0
0.5
1.0
1.5
2.0
0 100 200 300 400 500 600
SIP FV Small Remo e Vol
SIPD FV Small Remo e Vol
SIP FV La ge Remo e Vol
SIPD FV La ge Remo e Vol
(a) (b)
(c) (d)
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BIDEGAIN ET AL.
In his sys em, he gene a ion o an epizoo ic, in
addi ion o he ini ial popula ion size and he
disease ansmission a e, is egula ed by he e-
mo al o decay a e o dead animals d, no he
mo ali y a e o in ec ed animals m. The p oba-
bili y o an ou b eak (
R0
>
1
) is lowe o he SID
model a a gi en N han o he SI model (Fig. 2a)
due o he ex a s ep in he ansmission p ocess
(i.e., in ec ion ia dead animals); he impac o
pa ame e s on
R0
is hal ha obse ed in he SI
model (Fig. 3b). Mo eo e , commonly, in na u e,
sca enging a es (Veale e al. 2000, Mo ello e al.
2005) o decay a es (Smi h 1953, Allison 1990,
Lo z and So o 2002) o dead in ec ed animals a e
ma kedly highe han disease mo ali y a es.
This, oge he wi h he ac ha he p ocess is in-
he en ly slowe , make a suscep ible popula ion
less ulne able o an epizoo ic i ansmission
occu s ia di ec con ac wi h dead in ec ed in-
di iduals, assuming ha he sca enge s a e no
in ec ed by he pa hogen and become ese oi s
o he disease (Hoese 1962).
Fig. 3. Sensi i i y analysis (SA) o
R0
o he pa ame e s o a se ies o models. The sensi i i y index ep esen s
he uni
R0
change pe uni change in he gi en pa ame e . The analysis o each pa ame e was compu ed a a 0–1
pa ame e ange o all pa ame e s excep o b (0–10,000), and N (0–200), while he es o he pa ame e s we e
held cons an wi h hese baseline alues: β = 0.001, m = 0.1, d = 0.1, c = 0.1, b = 10,000, = 0.1, a = 0.1, = 0.001, γ = 1,
σ = 0.1, sl = 10 (
Vl
=
0.1
), s = 1 (
VΓ
=
1
), N = 100. The as e isks ma k pa ame e s o which he sensi i i y index was
no cons an o e he e alua ed ange. Fo hese pa ame e s, he sensi i i y index ob ained o he baseline alue
o he pa ame e is shown. The a iabili y o he sensi i i y index o hese pa ame e s is p esen ed in Fig. 6.
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
-1.0
-0.5
0.0
0.5
1.0
N m d c b a sl s
Sensi i i y Index
Pa ame e s
Pa ame e s
SI model SID model
SIP model SIPD model
SIPD-F model
SIP-FV model
* * *
*
*
* *
SIP-F model
*
*
*
*
*
SIPD-FV model
*
* * *
*
*
* *
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BIDEGAIN ET AL.
SIP model
In his model, he disease is no ansmi ed
om in ec ed animals o suscep ible animals by
con ac be ween indi iduals. In ec ed animals
elease in ec ious pa icles in o he en i onmen
(P) and hese wa e bo ne pa hogens can con ac
he suscep ible animals he eby ansmi ing he
disease. We conside a e sion o a model p o-
posed o s udy he popula ion dynamics o mi-
c opa asi ic in ec ions (Ande son and May 1981).
Thus, he model assumes ha suscep ibles a e
in ec ed wi h a a e
𝛽pa iclePS
(Eqs. 9 and 10).
The elease a e o in ec i e pa icles by in ec ed
indi iduals occu s a a e c and he pa hogens
in he wa e a e inac i a ed a a a e (Eq. 11)
by dilu ion, anspo downs eam, o by educ-
ion o in ec iousness by inac i a ion o dea h.
The model can be desc ibed by he ollowing
sys em:
(9)
(10)
(11)
The basic ep oduc ion numbe is de ined as:
(12)
The esponse on
R0
due o changes in ini ial popu-
la ion N has a nonlinea inc easing end (Fig. 2b).
All pa ame e s ha e he same cons an e ec
in
R0
, ega dless o he alue o he pa ame e s
(Fig. 3). In his model (Eq. 12), la ge popula ions
a e less ulne able o epizoo ics in condi ions o
ela i ely high (i.e., a sho pa hogen li e span
in he wa e and/o apid dilu ion) wi h espec
o he pa icle elease a e c (low /c in Fig. 4). As
he pa hogen elease a e ises wi h espec o
he inac i a ion a e o in ec i e pa icles in he
wa e , o a gi en m, he p obabili y o a dis-
ease ou b eak inc eases subs an ially e en a low
ansmission a es o small popula ions. Hence,
ela i ely small popula ions can suppo disease
epizoo ics when pa icle inac i a ion a es a e low
enough ha pa icles accumula e locally o when
he pa icle elease a e o e whelms he a ious
modes o pa icle inac i a ion (Fig. 4). Dea h o
in ec ed indi iduals e ec i ely e mina es pa i-
cle elease; hus, a high mo ali y a e m can limi
epizoo ic de elopmen e en i he body bu den
o pa hogens in he in ec ed indi iduals is high.
SIPD model
A guably, in ma ine sys ems, wa e bo ne
pa hogens (P) a e eleased mo e commonly by
dead in ec ed animals (D) ins ead o li e in-
ec ed indi iduals (I), and he disease is
dS
d
=−𝛽pa iclePS
;
dI
d
=𝛽pa iclePS−mI
;
dP
d
=cbI − P
.
R
0=
√
𝛽pa icleN
m
cb
.
Fig. 4. Epizoo ic h eshold alues o he SIP model
(mo ali y a e m on he y-axis) and he SIPD model
(decay a e o dead in ec ed animals d on he y-axis).
3-D su ace plo s ep esen he le el su ace o
R0=1
,
wi h
𝛽pa icle =0.005
, 0.01 and 0.05, o e a ange o
alues o m, d, and he ecip ocal a io o pa hogen
elease a e c and inac i a ion a e . Abo e he su ace,
R0>1
and he p obabili y o an epizoo ic inc eases.
Below he su ace,
R0<1
and an epizoo ic canno
de elop. Fo a be e isualiza ion and easie
in e p e a ion o he ela i e impo ance o he
pa ame e s, wo di e en iews o he same su aces
a e p esen ed.
0.2
0.4
0.6
0.8
0.2 0.4 0.6 0.8
50
100
150
200
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
50
100
150
200
pa icle = 0.05
pa icle = 0.01
pa icle= 0.005
pa icle = 0.05
pa icle = 0.01
pa icle = 0.005
1.0
1.0 1.0
1.0
Disease mo ali y a e m
Dead
decay
a e d
Pa icle elease a e a
Pa icle loss a e
Pa icle elease a e
a
Pa icle loss a e
Disease mo ali y a e m
Dead decay a e d
Ini ial popula ion densi y, N
Ini ial popula ion densi y, N
(a)
(b)
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BIDEGAIN ET AL.
emo al o dead animals d in he denomina o ,
ins ead o he disease mo ali y a e m, and being
speci ied as he ou h oo ins ead o he hi d
oo . The sensi i i y o
R0
o he pa ame e s is
iden ical o he SIP-FV model wi h he excep ion
ha o he SIPD-FV model, he maximum alues
o he sensi i i y index o a ying pa ame e s is
sligh ly lowe due o he inhe en ly slowe in ec-
ion p ocess implied by he addi ion o one addi-
ional p ocess in Eq. 39 (Fig. 3).
dIscussIon
This con ibu ion co e s he ma hema ical
basis o he dynamics and epizoo iology o a
di e se a ay o ma ine in ec ious diseases,
speci ically ocusing on he mos ele an p o-
cesses ha in e ac o d i e he ini ia ion and
e mina ion o epizoo ics. We adap ed he
Ke mack and McKend ick (1927) epide mi-
ological heo y and he model p oposed by
Ande son and May (1981) o comp ehensi ely
build disease dynamics models o sessile ma-
ine in e eb a es ha con ac o il e wa e -
bo ne pa hogens.
T ansmission o ma ine diseases includes a
numbe o p ocesses ei he a ely o ne e ob-
se ed in he e es ial wo ld. Thus, he o mu-
la ions p oposed include ansmission by di ec
con ac no only be ween li e animals (SI model)
bu also be ween dead animals and li ing sus-
cep ible hos s (SID model). We also explo e cas-
es whe e ansmission occu s by en i onmen al
con ac , ha is, ia pa icle anspo h ough he
wa e column and up ake by con ac o il a-
ion o wa e bo ne in ec i e pa hogens eleased
o he wa e column by li e o dead in ec ed
animals. We inally explo e he in luence o a
dose– esponse mechanism known o be p esen
in il e - eeding mollusks and he po en ial o a
emo e olume o modula e he in ec ion p ocess
h ough di usi e exchange o pa icles wi h he
local pool (Table 1). In each case, we conside
he epizoo ic h esholds o he s udied sys ems
by o mula ing hei speci ic basic ep oduc ion
numbe s
R0
.
Some ela ionships exempli ied by he basic
ep oduc ion numbe o mula ions o he mod-
els p esen ed he e dese e pa icula a en ion.
In ma ine diseases ansmi ed by close con ac
be ween suscep ible and dead in ec ed indi id-
uals (Rudol and An ono ics 2007) (SID model)
a he han con ac be ween suscep ible and li e
in ec ed indi iduals (s anda d SI model), ans-
mission is egula ed by he decay o emo al by
sca enge s o dead animals. T ue sca enge s do
no exis in he ma ine wo ld; howe e , many
p eda o s sca enge ad en i iously (Hoese 1962,
Veale e al. 2000, Mo ello e al. 2005). Pa icula
a en ion has been paid o his p ocess o whi e
spo synd ome in c us aceans in which ansmis-
sion in ol es in ec ed ca casses which ha e died
om in ec ion bu emain in ec ious (So o and
Lo z 2001, Lo z and So o 2002, Lo z e al. 2003).
Ce ain ad en i ious sca enge s emo e he ca -
casses wi hou becoming in ec ed; such ac i i y
may limi he sp ead o disease; an inc ease in
sca enge s o in empe a u e and oxygen condi-
ions (Allison 1988, Kidwell and Baumille 1990,
Pa sons-Hubba d e al. 2008) dec eases he in ec-
ious pe iod and hence, he numbe o second-
a y in ec ions caused by a dead animal. Ra es o
sca enging a e p obably mos eadily modi ied
by he numbe o sca enge s. An inc ease in
sca enge s is p oposed as an impo an ou come
o comme cial ishing (Collie e al. 1997, Veale
e al. 2000), bu whe he his in luences any ma-
ine disease is unknown.
Be ha as i may, mos ma ine diseases ha
equen ly gene a e epizoo ics a e p oli e a-
i e diseases (e.g., Powell e al. 1996, Fo d e al.
1999, Kleeman e al. 2002), ha is, he pa hogen
mul iplies wi hin he hos , equen ly eaching
high cell coun s pe g am o hos issue. Highly
in ec ed animals can elease many in ec i e ele-
men s and his capaci y is exace ba ed upon he
animal's dea h (Bushek e al. 2002). Thus he e,
we ocus on he heo y o ansmission o p o-
li e a i e diseases in he ma ine wo ld, empha-
sizing he cases o ansmission ia wa e bo ne
in ec i e pa icles in popula ions o sessile hos s
o hos s wi h limi ed mobili y, dominan ly in-
e eb a es such as bi al es, co als, abalone, o
some c us aceans (see pa icle-based models,
Table 1 and sec ions SIP model, SIPD model, SIP-F
model, SIPD-F model, SIP-FV model, and SIPD-FV
model). Wi hin he hos , he possibili y ha b, he
hos body bu den, is high, and hus, ha cb, he
numbe o pa icles eleased by li e o dead an-
imals is high, would esul in such diseases be-
ing highly in ec i e e en a anishingly low hos
abundance, ha is, he low-abundance e uge o
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BIDEGAIN ET AL.
Ke mack and McKend ick (1927) may be e y
low o such diseases.
The scena ios explo ed in his pape demon-
s a e ha , heo e ically, unde simila condi ions
o ini ial popula ion densi y, ansmission a e, o
disease mo ali y, he mo e p ocesses in ol ed in
ansmission he less likely ha a ma ine disease
will gene a e an epizoo ic (Figs. 2 and 3). This
is likely one eason why mul icellula pa asi es
a ely p oduce epizoo ics, as mos o hem ha e
complex li e cycles and hus ha e many s eps in
he ansmission p ocess. Consequen ly, a p io i,
in sys ems whe e ansmission in ol es a a ie y
o p ocesses, such as he dea h o in ec ed ani-
mals, dead animals eleasing pa hogens in he
wa e , o il e eede s accumula ing hem (i.e.,
SIPD-F model), an epizoo ic should be less p ob-
able han o con ac -based diseases (SI o SID
models) o he same popula ion densi y. This
is demons ably no he case; hus, he a es o
p ocesses mus also be inc eased conside ably by
hese al e na i e ansmission pa hways. Thus,
a sys em wi h a high elease a e o pa hogens
om animals upon dea h and a limi ed inac i a-
ion a e o in ec i e pa icles ei he in he wa e
column o in he suscep ible hos , could easily
be highly ansmissible and be cha ac e ized by
a high incidence o epizoo ics. This is he case,
o ins ance, o oys e s and he pa hogen Pe k-
insus ma inus.
The pa icle-based models p oposed explici ly
decouple he a e o he in ec ed animal om he
a e o he in ec i e pa icles. The a e o elease
o in ec i e pa icles c is inhe en ly decoupled
om disease mo ali y a e m and he a e o de-
cay o issue d, such ha i m > c o d > c, espec-
i ely, hen some in ec i e pa icles a e ne e e-
leased in o he wa e o in ec o he hos s. Thus,
in hese scena ios, he p obabili y o an epizoo -
ic can be limi ed when disease mo ali y a e m
is high (SIP model) o he emo al a e o dead
animals d is as (SIPD model) and he elease o
pa hogens om li e o dead animals c is slow
compa ed o he pa icle loss in he en i onmen
(Fig. 4). Rega ding mo ali y in he SIP model,
o ins ance, co al species wi h high popula ion
u no e a es a e na u ally mo e esis an o
epizoo ics because o he di ec p opo ionali-
y be ween he ini ial popula ion N o co e age
necessa y o an ou b eak and he mo ali y a e
m (Yakob and Mumby 2011). Fo an epizoo ic o
occu , popula ions wi h a high u no e equi e
highe N o balance he ac ha indi iduals do
no o exis long enough o become in ec ed and
o sp ead he in ec ion (Yakob and Mumby 2011).
The unique aspec o he SIPD model is he na-
u e o he p ocess o o ganic ma e des uc ion
as i con ols he elease a e o pa hogens o he
wa e . A “clean” and as emo al o dead ani-
mals by sca enge s (i.e., pa hogens a e no e-
leased in o he wa e du ing he p ocess and a e
inac i a ed inside sca enge s) leads o a dec ease
in he pa icle elease a e c and an inc ease in
he decay a e o dead animals d, es aining he
p obabili y o an epizoo ic, whe eas apid de-
composi ion may elease a subs an ial numbe o
pa icles o he wa e acili a ing epizoo ic de el-
opmen .
This is a pa icula ly impo an issue o ce -
ain diseases in il e eede s in ol ing a much
la ge numbe o in ec i e pa hogens eleased
by dead animals han by in ec ed li e animals
(SIPD-F model). Fo some molluscan diseases
such as De mo (pa hogen Pe kinsus ma inus),
he in e ence om obse a ion is ha he elease
o pa icles om dead animals occu s apidly
du ing he decay p ocess, ha is, c ≥ d (Bushek
e al. 2002). In hese sys ems, he body bu den o
pa hogens in in ec ed o dead animals and he
ela i e impo ance be ween he elease and he
emo al a e o pa hogens in he issue o wa e
column becomes pa amoun (Fig. 5). The il a-
ion-based models p oposed (SIP-F and SIPD-F)
assume a dose– esponse mechanism. Al hough
some models o disease in il e eede s assume
in ec ion by a single in ec i e elemen o con-
enience (e.g., Powell e al. 1996), he concep o
an in ec i e dose has ecei ed a en ion, pa icu-
la ly o molluscan diseases (Chu 1996, Chu and
Vole y 1997) because hese hos s do ha e some,
albei o en inadequa e, abili y o discha ge o in-
ac i a e accumula ed pa hogens. Diapedesis and
apop osis a e ob ious examples (Kleeman e al.
2002, Sunila and LaBanca 2003). Wha seems
clea is ha he abili y o mos il e eede s o ac-
cumula e in ec i e elemen s o en a exceeds he
abili y o deac i a e hem a he concen a ions
ypically obse ed in he ield (e.g., Audema d
e al. 2006). Thus, il a ion a e may be a domi-
nan de e minan o ansmissi i y. None heless,
he albei limi ed abili y o inac i a e il e ed
in ec i e pa icles may be consequen ial unde
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BIDEGAIN ET AL.
ce ain ci cums ances when he concen a ion o
in ec i e elemen s is low and, he e o e, he ac-
quisi ion a e is slow (Fig. 5).
A dense assemblage o il e eede s can e ec-
i ely educe he concen a ion o pa icles in
he wa e column. Once abundance ises abo e
a ce ain le el, each animal acqui es on a e age
a educed numbe o pa icles om he wa e
(Pe e son and Black 1987, F eche e e al. 1992,
Wilson-O mond e al. 1997). Tha il e eede s
can be su icien ly dense as o compe e o ood
is well desc ibed (e.g., F eche e e al. 1992, Wil-
son-O mond e al. 1997, Widdows e al. 2002).
Such dense assemblages may educe he concen-
a ion o in ec i e pa icles su icien ly o pe mi
he in e nal inac i a ion mechanisms o limi
body bu den below he in ec i e dose le el. This
is he case in he SIP-F model (Eq. 22) whe e high
il a ion a e educes he basic ep oduc ion
numbe
R0
o he o mula o he SIP model, ha
is, he con olling pa ame e becomes in e nal
pa icle inac i a ion a e a. This si ua ion is he
o e il a ion scena io, whe e all he pa hogens
in he wa e a e il e ed. He e, once he popula-
ion densi y ises su icien ly, he p obabili y o
epizoo ic de elopmen emains low e en wi h
inc easing N p o iding ha he numbe o pa -
icles il e ed by each animal is lowe han he
in ec i e dose (Fig. 2c, dashed lines). Unlike mos
disease models in which inc easing N inc eases
he likelihood o ansmission and epizoo ic de-
elopmen , o il e eede s, he p obabili y o an
epizoo ic is low a e y low N and also can be
low a in e media e o high N, he exac p obabil-
i y dis ibu ion being a unc ion o il a ion a e
and in i o inac i a ion a e. This case is shown
in Fig. 5: a low c/a and high popula ion il a ion,
he e is essen ially no scena io exis s o which
R0
is abo e 1.
The SIP-FV and SIPD-FV models emphasize
o he impo an mechanisms con olling he
concen a ion o in ec i e pa icles in he wa e
column. On he addi ion side is he bu e ing
capaci y o a emo e pool eple e wi h in ec-
i e pa icles. On he dilu i e side is a emo e
pool ha ope a es as an in ec i e pa icle sink.
A la ge emo e olume in he SIP-FV and SIPD-
FV models oge he wi h a high exchange a e
o pa icles be ween pools and a ela i ely high
inac i a ion a e o pa hogens in he emo e pool
is an e ec i e mechanism o educe pa icle con-
cen a ion in he local pool. Al hough wa e low
has been conside ed in he con ex o pa asi e
ansmission (De Mon audouin e al. 1998), he
e ec o dilu ion is bes demons a ed by he li -
e a u e on e iliza ion e iciency (Le i an 1991,
Babcock e al. 1994, Thomas 1994). Pa icle con-
cen a ion d ops apidly wi h dis ance om a
poin sou ce due o bo h di usi e and ad ec i e
p ocesses. In he case o in ec i e pa icles, he
exchange a e γ becomes he con olling pa am-
e e (Eq. 36). Gi en a su icien exchange a e
o main ain low pa icle concen a ion locally,
a modes in e nal inac i a ion a e (a in Eq. 36)
may be su icien o p e en ansmission. This
case is shown in Fig. 7, whe e ew pa icles a e
e ained in he local pool due o a apid ans e
o pa hogens o a la ge emo e pool wi h a high
pa icle loss a e.
One applica ion o ansmission models is he
use o he basic ep oduc ion numbe o es ima e
he hos densi y leading o e ec i e local ex inc-
ion o he pa hogen. Ce ainly, his p ocess mus
be e ec i e bo h in he e es ial wo ld, whe e
i is well desc ibed (e.g., Ba le 1960, Hasibede
e al. 1992, Hu nagel e al. 2004), and he ma ine
wo ld. Ce ain ma ine diseases a e cha ac e ized
by widesp ead high p e alence and apid in ec-
ion o newly ec ui ed hos s. De mo in oys e s
is an exempla . Such diseases can be e med pan-
demic in he sense ha hei in ec ion dynamics
is li le in luenced by he local sou ce o in ec i e
pa icles. The SIP-FV and SIPD-FV models o e
insigh in o hese diseases. A disease can become
pandemic only i a emo e pool ha bo s a con-
cen a ion o in ec i e pa icles ha con inuously
bu e s he local emo al o pa hogens. Such an
ou come equi es ei he con inual eplacemen
om sou ce popula ions wi h high mixing such
as migh occu in idally domina ed es ua ies o
limi ed loss om he emo e pool such as migh
occu in es ua ies wi h long wa e esidence
imes. Rega dless, i he emo e pool does no
ope a e as a sink, ha is i he inac i a ion o loss
a e σ in he emo e pool is small, he inal e m
in Eq. 35 becomes
𝜎V
Γ
Vl
and, consequen ly, he ol-
ume
VΓ
becomes he bu e ing agen assu ing a
con inual concen a ion o in ec i e pa icles lo-
cally ha can be expec ed o o e ide any pa icle
sink o loss p ocess in he local pool unde mos
ci cums ances. A ques ion a ises as o he mech-
anism by which a p ocess domina ed by a local
Ap il 2016 Volume 7(4) A icle e0128619 www.esajou nals.o g
BIDEGAIN ET AL.
sou ce becomes pandemic, as mus ha e hap-
pened in he ea ly 1990s in Delawa e Bay (Fo d
1996, Bushek e al. 2012). The SIP-FV and SIPD-
FV models may p o ide he con ex o e alua e
he p obabili y o pandemic disease in a gi en
es ua y o ma ine wa e body.
conclusIons
The ele ance o hese esul s lies in he ac
ha hey suppo esea ch ega ding he ela i e
impo ance o he ac o s in ol ed in he ini-
ia ion and e mina ion o ma ine epizoo ics.
The pa ame e space leading o he local demise
o ma ine diseases o hei expansion depends
on he speci ic pa ame e s o p ima y impo -
ance ha de ine he ansmission p ocess and
hei ela i e alues. He e, he no malized sen-
si i i y index o
R0
wi h espec o pa ame e s
was used o de e mine how sensi i e model
esul s a e o each pa ame e , o e a pa ame e 's
ull o a leas wide ange o easible alues.
The o mula ions we p esen iden i y a numbe
o peculia i ies ha a e ela i ely unique o
ma ine diseases. O impo ance is he ac ha
high abundance does no always enable epi-
zoo ic de elopmen . Ra he , o il e eede s,
he e ec o abundance is bimodal, wi h bo h
low and high abundances mi iga ing agains
disease de elopmen . The deg ee o which he
emo e pool ac s as a sink o bu e ep esen s
a second example. Pandemic disease depends
on a emo e pool ha bu e s he local pool,
whe eas a emo e pool ha ope a es as a sink
limi s he p og ession o disease. In bo h o
hese cases, in a sense, he local popula ion
dynamics is ci cum en ed in ha he po en ial
o epizoo ic de elopmen depends upon ac o s
beyond he ansmission, mo ali y, and pa icle
elease a es wi hin he local popula ion. These
unique ea u es equi e a di e en app ecia ion
o he disease p ocess in ma ine sys ems o
many ma ine diseases ela i e o he classic
e es ial model exempli ied by he Ke mak–
McKend ick o mula ion.
acknowledgMen s
This in es iga ion was unded by he NSF E olu ion
and Ecology o In ec ious Diseases (EEID) P og am
G an # OCE-1216220. We app ecia e his suppo .
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