Microparasitic disease dynamics in benthic suspension feeders: Infective dose, non-focal hosts, and particle diffusion

Microparasitic disease dynamics in benthic suspension feeders:1 infective dose, non-focal hosts, and particle di↵usion2 G. Bidegain a,⇤, E.N. Powell a, J.M. Klinck b, T. Ben-Horin c,d, E.E. Hofmann b 3 aGulf Coast Research Laboratory, University of Southern Mississippi, 703 East Beach Drive, Ocean Springs, Mississippi 3954 USA4 bCenter of Coastal Physical Oceanography, Old Dominon University, 4111 Monarch Way, Norfolk, Virginia 23529 USA5 cHaskin Shellfish Research Laboratory, Rutgers University, 6959 Miller Avenue, Port Norris, New Jersey 08349 USA6 dDepartment of Fisheries, Animal and Veterinary Science, University of Rhode Island, 20A Woodward Hall, 9 East Alumni Avenue, Kingston,7 Rhode Island 02881, USA8 Abstract9 Benthic suspension-feeders can accumulate substantial numbers of microparasitic pathogens by contacting or filtering particles10 while feeding, thus making them highly vulnerable to infectious diseases. The study of disease dynamics in these marine organisms11 requires an innovative approach to modeling. To do so, we developed a single-population deterministic compartmental model12 adapted from the mathematical theory of epidemics. The model is a continuous-time model, unstructured in spatial or age terms,13 and configured to simulate the dynamics of diverse dose (body burden)–dependent infectious disease transmission processes in14 suspension feeders caused by susceptible individuals contacting or absorbing (filtering) infectious waterborne pathogens. Di↵erent15 scenarios were simulated to explore the e↵ect of recruitment, filtration rate, particle loss, di↵usion-like processes in the water16 column and non-focal hosts (i.e. non-susceptible in terms of disease) on disease incidence. An increase in recruitment (i.e. new17 disease free susceptibles) can reduce the prevalence of infection due to the dilution e↵ect of adding more susceptibles, but the18 disease can spread faster for the same reason. Lower infective particle accumulation rates or increasing particle loss rates in the19 environment reduce the prevalence of infection. This e↵ect is trivial when the water is saturated with infective particles released20 by infected and/or dead animals. Di↵usion of particles from the local pool available to suspension feeders to the adjacent remote21 pool, prompted by a large remote volume and high particle exchange, limits epizootic development. Similarly, the likelihood of an22 epizootic can be constrained in a large susceptible population when competition for pathogens, more ’active’ in active filter feeders23 than in passive suspension feeders, reduces the per capita infective particle accumulation rate. In passive suspension feeders,24 decreasing the area of the feeding surface has the same e↵ect in constraining disease development. The e↵ect of competition for25 infective particles in essence diluting the infective particle concentration in the water column is magnified when the susceptible26 population is part of a community with non-focal filter feeders, and is particularly e↵ective in limiting disease development in27 high infective dose systems. At the same time, this active foraging strategy makes filter feeders more vulnerable to epizootics.28 The model is a suitable framework for studying the disease dynamics and determinants of disease outbreaks in benthic suspension29 feeders.30 c 2015 Published by Elsevier Ltd. Keywords: 31 disease model, epizootic, transmission, suspension feeders, filter feeders, infective-dose, overfiltration, non–focal host32 © Revised 2016. Manuscript This manuscript version is made available under the CC-BY-NC-ND 4.0 license Click here to view linked References https://creativecommons.org/licenses/by-nc-nd/4.0/ Bidegain, G., Powell, E. N., Klinck, J. M., Ben-Horin, T., & Hofmann, E. E. (2016). Microparasitic disease dynamics in benthic suspension feeders: infective dose, non-focal hosts, and particle diffusion. Ecological Modelling, 328, 44-61 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 2 1. Introduction33 34 Benthic suspension-feeders are among the optimal foragers in the marine context because the energetic cost for35 capturing prey is almost nil in passive sessile animals and very low in active filter feeders (Riisgård et al.,1993;36 Riisgård and Larsen,1995). Suspension feeders play a major role in the structure and function of marine ecosys-37 tems (Dame,1993;Newell,2004) by transferring energy from the pelagic zone to the benthos (Newell,2004;Porter38 et al.,2004). This feeding mode as a foraging strategy has a downside in terms of the transmission and spread of39 waterborne diseases. In addition to food particles, suspension-feeders can also accumulate a substantial number of40 infectious pathogens such as bacteria, fungi, protozoans, and viruses from the water. For instance, the scleractinian41 coral Madracis mirabilis can accumulate 1 x 107bacterial cells cm2h1and clearance rates in bivalves can be much42 higher (e.g., 8 L h1in oysters, Powell et al. (1992)). This potential to accumulate particles, in turn can catalyze major43 epizootics and massive mortalities causing substantial changes to ecosystem structure and serious losses in shellfish-44 eries and aquaculture (La↵erty et al.,2015). Among the best characterized, geographically wide-spread, and virulent45 infectious diseases in suspension feeders are MSX and Dermo in oysters (Villalba et al.,2004), white plague disease46 and black band disease (BBD) in corals (Sokolow et al.,2009;Zvuloni et al.,2009), and brown ring disease BRD,47 and Perkinsosis in clams (Paillard,2004;Dang et al.,2010).48 Proliferation–based disease models have been considered sufficient to describe disease impact in benthic suspen-49 sion feeder populations characterized by rapid non–point–source transmission, such as bivalves (Calvo et al.,2001;50 Powell et al.,2011,2012b;Powell and Hofmann,2015). Host proliferation models alone or together with hydro-51 dynamic models can explore the e↵ect of environmental factors, such as temperature and salinity, on the process52 of pathogen proliferation and, consequently, on the infection intensity. These models assume high prevalence and53 simulate epizootic development and host morbidity and mortality based upon population infection intensity as the54 pathogen proliferates within the host. For these diseases the dynamics of transmission are typically poorly described55 (Ford,1992;Ford and Smolowitz,2007;Gray et al.,2009) and cursorily integrated into disease models (Powell et al.,56 1996,1999). Only a few studies have adapted the epidemic SIRmodels (Kermack and McKendrick,1927;Anderson57 and May,1981) to benthic marine organisms, despite the fact that the structure of these models as commonly used for58 terrestrial diseases is equally applicable and adaptable to describe the epizootiology in benthic animals (McCallum59 et al.,2004). For this purpose, however, some marine system-specific features such as the buoyancy and the high60 potential of dispersion and dilution of waterborne pathogens (Strathmann,1990;McCallum et al.,2003) and certain61 unique feeding modes such as suspension feeding may need to be incorporated. Kuris and La↵erty (1992) developed a62 model incorporating both recruitment of the parasite and the host to compare the e↵ect of various management strate-63 gies on a hypothetical crustacean parasitized by a parasitic castrator. In this non-point-source host-pathogen system,64 the number of nearby infected animals is relatively unimportant in comparison with the number of infective pathogens65 in the water. McCallum et al. (2005) modeled the dynamics of withering syndrome in abalones by incorporating the66 free-living pathogen stage and disease transmission through contact between this stage and the host. More recently,67 Sokolow et al. (2009) and Yakob and Mumby (2011) formulated the dynamics of disease in corals describing the68 transmission of disease by contact between the host and free-living pathogens. (Bidegain et al.,2016)formulated sim-69 ple compartmental models to yield the basic reproduction number R0for a variety of marine host-pathogen systems70 to explore the relative importance of the host and pathogen traits that determine transmission.71 Most compartmental disease models (e.g., SI, SIR, SEIR, etc.) follow the classic mass action approach where72 disease transmission is a function of the number of contacts between susceptible individuals and infective particles73 in the water (Regoes et al.,2002;Sokolow et al.,2009;Yakob and Mumby,2011). However, it may be important to74 include the well-known dose-e↵ect in disease transmission for some suspension feeders such as oysters (Bushek et al.,75 1997;Ford et al.,1999;Powell et al.,1999); e↵ective dose is a characteristic with potentially interesting e↵ects on76 disease transmission that seems crucial to take into account. Active filter feeders can be sufficiently dense to compete77 for food (e.g., Fr´ echette et al.,1992;Wilson–Ormond et al.,1997;Widdows et al.,2002) particularly under conditions78 of slow flow and limited vertical advection. Similarly, they may ‘compete’ for pathogens, presumably reducing the79 concentration of infective particles sufficiently to limit body burden below the infective dose and, in turn, limiting80 epizootic development (i.e. the overfiltration scenario – Bidegain et al. in press). Passive filter feeders may show this81 mechanism less obviously since they more importantly depend on the ambient flow and the body surface exposed to82 that flow, rather than on an active pumping of water from the environment through a filter (Sebens et al.,1996,1998).83 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 3 When host diversity increases, the disease risk can decrease in what is known as the dilution e↵ect. This well-84 studied e↵ect in Lyme disease (Ostfeld and Keesing,2000) appears to be a more general phenomenon that relies on the85 idea that a certain community with high species diversity will contain a proportion of incompetent hosts, in terms of86 disease, that compete for particles and deflect infectious pathogens away from the susceptible hosts, thereby reducing87 infection prevalence and disease risk. This marine suspension feeder community could be composed, for instance,88 by the scleractinian coral Madracis mirabilis and the colonial ascidian Trididemnum solidun, since data show that89 both organisms are e↵ective bacterial suspension feeders (Bak et al.,1998). Other examples may by a reef composed90 by oysters and mussels, since mussels that attach to oyster reefs and shells can double the reef’s filtration capacity91 (Gedan et al.,2014), or by oysters and ascidian tunicates (T. Ben-Horin, unpublished data).92 Another interesting characteristic of some diseases of suspension feeders is the importance of dead infected ani-93 mals as a source of infective particles and, hence, the role of mortality and subsequent scavenging and tissue decay in94 disease transmission. For instance, the pathogen body burden in dead oysters infected by Dermo disease is commonly95 higher and the potential release rate upon death is relatively much faster than that of infected live animals (Bushek96 et al.,2002). Dead corals infected by the black band disease may also release pathogens through breakdown of decay-97 ing tissue (Richardson,2004). Once released, di↵usion processes in the water column may have an e↵ect in pathogen98 dynamics and consequently, in disease transmission. The population turnover rate (i.e. high mortality and recruitment99 rates) thus is another mechanism of controlling epizootics in suspension feeders. Yakob and Mumby (2011) found that100 allowing for a more dynamic population turnover in an epizoological model of coral disease not only gives a superior101 fit to empirical data, but also suggests that emerging coral assemblages could be far less prone to epizootics. The role102 of predation in the context of SIR models has been considered (Su and Hui,2011;Wang et al.,2011), but has rarely103 been applied in the marine context (Liao et al.,2008). In filter feeders such as oysters the important contribution of104 dead animals releasing infective particles may counterweigh the e↵ect of the population turnover rate as a restraining105 influence on epizootics (Bidegain et al., unpublished).106 A theoretical transmission-based model or a host-population model that incorporates all of these features and de-107 scribes the dynamics of the host in all possible stages (i.e. susceptible, live infected, dead infected) and the waterborne108 pathogens does not exist for active or passive suspension feeders. In this paper, we develop an SI model capable of109 reproducing these processes. For this purpose, we adapt the Kermack and McKendrick (1927) epidemiological the-110 ory and the microparasitic epidemic model of Anderson and May (1981). The model reproduces the dynamics of111 both susceptible and infective stages of the host, whether alive or dead, and the waterborne infective pathogens. Our112 study system includes dose-dependent (i.e. body burden dependent) disease transmission wherein suspension feeders113 contact or absorb (filtering) waterborne infective pathogens released by live or dead infected animals. The model con-114 templates the e↵ect on disease transmission of non-point sources of pathogens and di↵usion processes in the water115 column. Moreover, the model permits the study of potential mechanisms by which suspension feeders might dilute116 the disease risk, such as competitive interactions among and between species, and the e↵ect of recruitment of new117 susceptible individuals. Finally, the model yields the formulation of the basic reproduction number R0for both active118 and passive filter feeders, to comprehensively describe the relative importance of some of these processes driving119 epizootics.120 2. The model121 2.1. Mathematical theory and model structure122 123 The model is a one-population deterministic compartmental model continuous in time, unstructured in spatial or124 age terms (Cuddington and Beisner,2005), and configured to simulate the dynamics of a diversity of infectious disease125 transmission processes in suspension feeders caused by susceptible individuals contacting or absorbing (filtering)126 infectious waterborne pathogens such as bacteria, fungi, protozoans, and viruses from the water column.127 As a compartmental model, which is the most frequently used class of model in epidemiology (Diekmann et al.,128 2013), individuals and pathogens can take on a finite number of discrete states. Each state is representative of a129 subpopulation of individuals or pathogens at a given time (Table 1) which, in turn, together with the corresponding130 parameters (Table 2) satisfy a system of nonlinear ordinary di↵erential equations (ODEs) describing the dynamics131 of the host-pathogen association. The incorporation of a dead infected subpopulation stems form the fact that some132 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 4 suspension feeders such as oysters or corals may also transmit the disease by releasing pathogens upon death (Bushek133 et al.,2002;Richardson,2004). A variant of the model incorporates another variable representing an alternate suspen-134 sion feeder host population which is incompetent in terms of disease (H), that is, the alternate host does not develop135 the disease and pathogens are inactivated inside the animals acting as an important mechanism of controlling the136 concentration of infective particles in the water column and as a sink of pathogens.137 The population of waterborne pathogens is divided into three classes depending upon their location in the system138 based on the division of the environment into two volumes of water: The first is defined as the ‘local pool’ of pathogens139 P. This pool is the pathogen concentration in the ‘local volume’ defined as Vl, adjacent to the bottom, within which140 pathogens are released by infected and dead infected individuals and remain free floating. Pathogens in this volume141 are susceptible to contact with or being filtered by hosts, and can lose their infective properties after some time or142 otherwise be lost (i.e advection, di↵usion, predation). For example, for oysters, the height of the local volume or the143 volume available for filtration is considered to be around 10 cm (Wilson–Ormond et al.,1997). This is a typical height144 for an oyster clump (Powell et al.,1987,2012a), but this height would exaggerate the e↵ect of an infaunal filter feeder145 (Ertman and Jumars,1988;Monismith et al.,1990;Widdows et al.,1998). The second state variable for pathogens146 is the remote pool U. This second pool is the concentration of pathogens in a second volume contiguous to the local147 volume, defined as the remote volume Vr, where a remote pool of particles can accumulate without direct interaction148 with hosts. The size of the remote volume would depend on how deep the water column is above the near-bottom layer149 directly a↵ected by the filtering benthic population. Di↵usional exchange of particles between these two volumes is150 assumed. The third subpopulation of pathogens is that accumulated in the susceptible population through contact or151 filtration (F).152 The infection state in the model is presence/absence, that is, the animals are infected or are not, and if they are153 infected, they have the ’average’ parasite load. We assume this because microparasites are usually (but not always)154 unicellular microorganisms, such as viruses, bacteria and protozoans that can multiply rapidly within a host (McCal-155 lum et al.,2004) relative to a one year time step. The transfer of individuals from absence to presence of disease is by156 a dose-dependent or body burden dependent transmission process detailed in section 2.4.1.157 2.2. Miscellany158 159 Variables related to the host population are defined with respect to the concentration of individuals in a given160 surface area of the bottom, whereas variables related to the pathogens are defined in number of particles in a given161 water volume or the number of particles in vivo. Note that for simplicity in rendering equations, the reciprocals of the162 local and remote volumes Vland Vr,sl and sr respectively, are often used.163 Variable Definition Unit SSusceptible hosts in the population Number of individuals IInfected individuals in the population Number of individuals DI Dead infected individuals in the population Number of individuals DS Dead susceptible individuals in the population Number of individuals PFree-living pathogens in the environment, local pool Number of particles FTotal number of pathogens absorbed or filtered by the population Number of particles UPathogens from a remote volume, remote pool Number of particles HAlternate non–competent reservoir host Number of individuals Table 1: Variables of the model. Note that the model has an implicit surface area or volume for the host subpopulations and pathogens, respectively. Fis the exception and the corresponding unit is internal particles in the susceptible population. The mathematical theory of this model contemplates that disease transmission requires the pathogen to exist164 outside the host and remain infectious for sometime finite period. Thus, the susceptible and infected individuals are165 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 5 not necessarily in contact or even components of the same local population, since the free-living pathogen stage can166 drift in the fluid and contact or be filtered by susceptible animals located near or far away from the particle source.167 Natural mortality of the host and mortality due to disease are also integrated into the formulation of the model. Thus,168 the infected individuals can die due to both natural causes and disease. The model also assumes an open population169 since it contemplates demographic turnover; that is, the recruitment process is integrated in the dynamics of the170 host. The model does not incorporate the process of recovery from disease because only a few examples of marine171 wild populations recovering from disease are known in marine systems (Gilmour et al.,2013;Paillard et al.,2014;172 Vega Thurber et al.,2014). Consequently, individuals never recover from the disease and infected individuals remain173 infected until they die.174 2.3. Model scheme175 176 The flow diagram of the model (Fig. 1) shows the most important processes in the disease transmission process in177 suspension feeders. Disease is transmitted to the susceptible population Sthorough filtration (active suspension feed-178 ers) or contact (passive suspension feeders) at a rate by a dose-dependent transmission. Infected animals Idie due179 to both natural mortality mSand disease mortality mI, while susceptible individuals Sonly die due to natural causes.180 The alternate incompetent (i.e. not susceptible to the disease) suspension feeder host Hcompeting for waterborne181 pathogens with the susceptible host is also located in the bottom. This alternate host only dies by natural mortality182 assuming that it is resistant to the disease and does not release particles to the water; that is, they are assumed to be183 inactivated by the immune system or by diapedesis. An exchange between the local near–bottom pathogen pool P184 and the remote pool Uoccurs at a certain exchange rate by a di↵usion-like process proportional to the di↵erence in185 concentration between the two pools.186 mS S! d bDI DI! a F! c bI I! fI P! ! F! fS P! ! D" SI!DI! P β S !mI I! r P! U" σ U ! Remote&Volume& Vr# Local&Volume& Vl# γ (sr U – sl P)! H! fH P! DS! Figure 1: Model flow diagram. The model variables are represented by capital letters: susceptible animals (S), infected animals (I), dead susceptible (DS ) and dead infected animals (DI), waterborne pathogens (P), internal pool of pathogens (F), remote pool of pathogens (U) and alternate host population (H). Dashed arrows represent the main processes in the model. The parameters involved in these transmission are described in Table 2. 2.4. Model equations187 188 The equations that follow represent the disease transmission process and the model variables. The host and189 pathogen states or subpopulations satisfy a system of ODEs describing the dynamic of the host-pathogen association.190 The numerical model for this ODE system is programmed in Matlab 8.1. The set of coupled di↵erential equations191 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 6 is solved with a 4th–order predictor corrector scheme using the Adams-Bashforth predictor and the Adams-Moulton192 corrector.193 2.4.1. Infective dose and disease transmission194 195 Disease transmission in suspension feeders, at least in filter feeders, most likely occurs via an infective dose196 (Bushek et al.,1997;Ford et al.,1999;Powell et al.,1999) rather than by unique contact between a single pathogen197 or fomite and host. Consequently, the transmission process in the model is based on the fact that susceptible animals198 require some minimum level of body burden of infectious particles, the infective dose, to become infectious. Further-199 more, the model assumes that as hosts absorb or filter particles, some susceptible individuals will have a relatively200 large body burden whereas most will have a smaller body burden. The distribution of the number of susceptible201 animals (S) with each level of body burden (b) (Fig. 2) is assumed to have the form202 S(b)=S0e⇢b,(1) which has a simpler expression but similar behavior to the negative binomial distribution, very common in benthic203 infected hosts (Anderson and Gordon,1982;Ford et al.,1999)), in having a long tail describing the declining frequency204 of individuals of increasing infection in the population distribution. S0is the total number of susceptible individuals.205 For body burdens up to several hundred, ⇢is a well approximated by ˆ S/F, where ˆ Sis the number of susceptible206 animals and Fis the number of infectious particles housed in the susceptible subpopulation (Eq. (2)). Thus F/S207 represents the average concentration of infectious particles in susceptible individuals.208 dF dt =(F+mS+a+cS)F+fSASSP.(2) Infectious particles are filtered out (active filter feeders) by or come into contact (passive filter feeders) with209 susceptible individuals at the filtration or contact rate fS, being the contact rate proportional to the feeding surface AS 210 for passive filter feeders (see section 2.4.2 for a more detailed description of particle uptake). Infectious particles can211 be removed from the susceptible population by four processes: (i) the reduction of the internal pool of particles in the212 susceptible population due to removal of susceptible individuals that become infected tF(see details in Eq. (6)), (ii)213 the background mortality mSas it removes individuals with incorporated infected particles, (iii) the inactivation inside214 susceptible animals (a), controlled by some component of the immune response such as phagocytosis, inactivation by215 oxygen radicles, binding by lectins, etc. (Renwrantz,1986;Villalba et al.,2004), and (iv) the release of particles from216 susceptibles through, for instance, faeces. Whether a susceptible becomes infected is a balance between a, the release217 rate of particles cS, the filtration of particles by susceptibles fsSP, and the concentration of infective particles that218 needs to be reached to generate the infective dose (bmin, see Fig. 2).219 The distribution of infective particle body burden in susceptible hosts in Fig. 2has a very long tail so that a few220 uninfected animals have a high body burden. The portion of the susceptible pool eligible to transition to the infected221 pool is circumscribed by bmax, the maximum body burden for any animal considered to be uninfected and bmin, the222 minimum body burden for animals eligible to transition to the infected pool. At any time, there will be some number223 of susceptible animals (S) with a total absorbed pool of infectious particles (F). These values are used to determine224 how many susceptible animals with a body burden between bmin and bmax should become infected.225 The total number of susceptible animals at any time is226 S=Zbmax 0 ˆ S0e⇢bdb =ˆ S0 ⇢(1 ebmax ) (3) where ˆ S0is the number of susceptibles per body burden. The total body burden for the population is F=Zbmax 0 ˆ S0be ⇢bdb =ˆ S0 ⇢2(1 (1 +⇢bmax)e⇢bmax .(4) Fand ˆ Sare both known at any step in the model. Consequently, the values of S0and ⇢can be calculated easily.227 The total number of susceptible individuals moved to the infected population is228 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 7 0! 50! 100! 150! 200! 250! 300! 0!50!100!150!200!250!300!350!400! Susceptible population ! (individuals/area)! Body burden (pathogens/individual)! bmin bmax Figure 2: A theoretical distribution of the population with each level of infective particle body burden (b). The minimum body burden at which susceptible animals are considered to be infected and thus transfered to the infected population is bmin. The maximum body burden for any susceptible animal considered to be uninfected is bmax. tS=Zbmax bmin ˆ S0e⇢bdb =ˆ S0 ⇢(e⇢bmin e⇢bmax ).(5) The total number of infectious particles that should be removed from the susceptible internal pool as a consequence is229 tF=Zbmax bmin bˆ S0e⇢bdb =ˆ S0 ⇢2((1 +⇢bmin)e⇢bmin )(1 +⇢bmax)e⇢bmax ).(6) Note that since the heavily infected susceptibles are the ones that are moved to the infected pool, a large number230 of internal infectious particles will be removed from the internal pool of the susceptibles. Based on these calculations,231 a fraction tS/Sof the susceptibles need to be moved to the infected pool and tF/Fof the internal infectious particles232 need to be removed from the susceptible internal pool. These changes are assumed to occur at a certain rate ↵[1/day]233 depending on the host-pathogen system. For instance, for a change that occurs at a rate of 10% per day, ↵is 2.3. The234 terms in the governing equations representing disease transmission thus have the form235 S=↵tS S(7) where ↵is a specific rate to move a fraction of the individuals from the susceptible population Sto the infected236 population I. Similarly, the rate at which the internal particles are removed from the internal pool Fis237 F=↵tF F.(8) Note that the standard mass action model with infective particles with an instantaneous dose response, typically238 represented as PS (Bidegain et al., in press), is a specific case of the dose-response model presented here. For ⇢=0,239 Eq. (5) becomes240 tS=ˆ S0(bmax bmin),(9) and consequently, S=ˆ S0bmax and S=↵(1 bmin bmax ). The standard mass action model does not explicitly specify241 particle accumulation or body burden, so that bmin =0; thus, Eq. 7simplifies for the case of an instantaneous dose242 response transmission S=↵.243 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 8 2.4.2. Uptake of particles by filtration or contact: active and passive suspension feeders244 245 The model is formulated to permit the study of both active (e.g., bivalves) and passive (e.g. corals) suspension246 feeders. The main feeding di↵erence between these organisms is that active filter feeders pump water actively through247 a sieve while passive suspension feeders are infected by passively contacting particles transported by water currents.248 Thus, disease transmission in filter feeders depends on the specific filtration rate (Powell et al.,1999) whereas, for249 passive suspension feeders, the transmission of disease is a function of the exposed surface area of the individual or250 colony devoted to food collection and the water flow speed [Sebens et al. (1996) and Sebens et al. (1998) demonstrated251 that food capture is higher in corals with high polyp and colony exposed surface area in relation to their biomass, and252 is limited at low flow conditions]. The model incorporates the term fto represent these two foraging strategies. In253 both cases, we consider that the capture rate of food and infective particles is similar. Consequently, for active filter254 feeders frepresents the rate at which particles are filtered by one individual per unit of time, whereas for passive255 suspension feeders frepresents the rate at which the particles contact the exposed surface area of an individual per256 unit of time (see Table 2).257 258 Active f ilter f eeders259 260 Particle removal from the water due to active filtration by susceptible individuals (see Eq. (14)) has the form261 fSSP, where fSis the filtration rate (volume filtered) per susceptible individual, Sis the number of benthic sus-262 ceptible individuals, and Pis the concentration. Similarly, the term to represent the particle removal due to active263 filtration of infected individuals (I) and non-focal or alternate incompetent hosts in terms of disease (H) would be264 fIIPand fHHP, respectively. The incorporation of specific filtration rates for susceptible and infected individuals265 permits simulation of disease dynamics for cases where infected or severely diseased individuals show a reduction in266 clearance rate. Similarly, the non-focal host has a specific filtration rate.267 268 Passive suspension f eeders269 270 For passive suspension feeders, the particle removal from the water by passive contact (see Eq. (15)) has the form271 fSA SP, where fS is the adrift particle contact rate per susceptible individual and ASis the exposed surface area272 per individual. Similarly, the term to represent the particle removal due to passive contact of infected individuals (I)273 and non-focal or alternate incompetent hosts (H) with particles would be fIIA IPand fHHA sP, respectively. The274 model also specifies specific contact rates for susceptible and infected individuals, and for non-focal hosts. However,275 considering that (i) the water movement is the primary mechanism bringing zooplankton into contact with passive276 suspension feeders and that exposed surface is crucial (Sebens et al.,1998), and (ii) the model already incorporates a277 term for individual exposed surface area, the term fmay be a function of the water flow speed rather than a disease-278 a↵ected characteristic and, thus, may likely be considered the same for di↵erent subpopulations of the same host.279 2.4.3. Susceptible Individuals280 281 Sis the number of susceptible animals in a given surface area of the bottom. Susceptible individuals are lost by282 two processes: infection and natural mortality. The disease transmission rate is controlled by Swhich is a function283 of body burden, allowing for a dose-response to infection, and it is an estimate of the rate at which individuals become284 infected (see details in section 2.4.1), whereas the natural mortality rate is mS(Eq. (10)).285 The model allows recruitment of new susceptible individuals when the population density (S+I) falls below the286 carrying capacity K. Susceptible and infected animals can produce recruits at di↵erent rates nSand nI, respectively287 allowing for reduced recruitment associated with infected individuals (Yakob and Mumby,2011). Finally, the model288 permits an external source of recruits (SRCS).289 dS dt =(S+mS)S+⇣1S+I K⌘(nSS+nII+SRCS) (10) where, if 1 S+I K>0, then the carrying capacity is not reached and new individuals can be recruited to the susceptible290 population as a function of the recruitment rates nS,nI, and the potential external source of recruits Src S.291 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 9 2.4.4. Infected Individuals292 293 Infected individuals are transferred from the susceptible subpopulation to the infected subpopulation at the rate S (Eq. (11)). Infected individuals die at a rate controlled by natural mortality mSand disease mortality mI. dI dt =SS(mS+mI)I(11) 2.4.5. Dead Susceptible Individuals294 295 Susceptible individuals die at a rate mSwhich is a background mortality not related to disease processes (Eqs.296 (10) and (12)). The subpopulation of dead individuals is reduced at a rate d, which represents bacterial decomposition297 or scavenging processes. Although true scavengers do not exist in the marine world, many predators scavenge adven-298 titiously (Hoese,1962;Veale et al.,2000;Morello et al.,2005). These dead susceptible individuals are a diagnostic299 and do not a↵ect the behavior of the model.300 d DS dt =mSSd DS (12) 2.4.6. Dead Infected Individuals301 302 Infected individuals die from disease at a mortality rate mIand su↵er background mortality at rate mS(Eqs. (11)303 and (13)). Similarly to dead susceptible animals, these dead infected individuals decay or are scavenged at a rate d.304 dDI dt =(mS+mI)IdDI (13) 2.4.7. Infectious Particles in the local volume305 306 The benthic community comprises susceptible hosts, infected hosts, and alternate hosts, filtering out or contacting307 particles. The model analyses two situations in terms of the type of benthic system contributing to the removal of308 particles from the system.309 2.4.7.1. Susceptible and infected animals removing and releasing particles.310 311 Pis the number of infectious particles in the volume of water immediately accesible to the suspension feeder312 population (Eqs. (14) and (15) ). Infectious particles are mainly added to the water by release from infected individuals313 at a rate cI(e.g. by faeces) and from dead infected individuals at a rate cDI. Susceptibles and dead susceptibles can314 release a small amount of particles through faeces at a rate cSand upon death through decomposition and scavenging315 processes at a rate cDS . If dead infected animals release their entire body burden of particles, then cDI =d, or they316 may release a much smaller number depending on the characteristics of the decay or scavenging process removing317 dead animals, which may inactivate a substantial proportion of or all of the infective particles before they can be318 released into the water column. Infectious particles are removed from the local volume by one of two processes: (i)319 inactivation at a rate r, by dilution, transport downstream, or by reduction of infectiousness by inactivation or death,320 and (ii) filtration by or contact with susceptible and infected individuals.321 The local water volume can exchange particles with the vertically adjacent volume if the concentrations are dif-322 ferent, through a di↵usion-like process controlled by parameter (see section 2.2) (Eqs. (14) and (15)). Finally, an323 unspecified source of infectious particles from another benthic community (SRCP) is also permitted.324 325 Active f ilter f eeders326 327 At this point, the model di↵erentiates between active filter feeders (e.g., bivalves) (Eq. (14)) and passive suspen-328 sion feeders (e.g., corals) (Equation 15) (see also section 2.4.2) to represent the distinct removal of particles from the329 water. Thus, the change in the number of infective particles in a given water volume for active filter feeders is330 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 16 We examined the potential importance of infective particle loss influencing infection prevalence. For this purpose,473 we varied the rate of particle loss in the local pool, that is the rate of inactivation of pathogens due to dilution, advec-474 tion, and mortality. Increasing particle loss limits the prevalence of infection (Fig. 7c) due to the reduced availability475 of particles to be filtered or contacted. The progressive saturation of particles in the water volume accessible to the476 benthic population (Fig. 6c) cancels the particle loss e↵ect reducing the infection prevalence. Late in the simulation,477 the release rate from infected animals clearly overwhelms the inactivation rate of pathogens in the water due to dilu-478 tion, advection, and natural mortality for the two lower particle loss scenarios tested (see solid and dashed black lines479 in Fig. 7c). In contrast, at the highest particle loss rate tested (see gray solid line in Fig. 7c), the inactivation rate480 of pathogens exceeds the release rate, resulting in the absence of infection for the first 800 days and a relatively low481 prevalence (30%) in the last days of the simulation.482 3.3.4. Di↵usion of particles483 484 Three rates of particle di↵usion were simulated, by means of three remote volume Vr/local volume Vlratios (Fig.485 7d). For most suspension feeders, the volume directly influenced by filter or suspension feeding Vlwill be small (e.g.,486 a volume with a height of 0-15 cm for oyster populations (Wilson–Ormond et al.,1997)). Consequently, the size of487 the remote volume Vrmay be a primary determinant of the prevalence of infection. While the local volume (0.1m3)488 was maintained constant, increasing the size of the remote volume from 0.1m3to 1 m3resulted in a decrease in the489 prevalence of infection of 40% at the beginning of the simulation. Initially, the two lower remote volumes tested490 showed no infections (see black dashed and gray solid lines in Fig. 7d), but as the simulation progressed only the491 larger remote volume limited disease transmission (see gray solid line in Fig. 7d). Similarly to the previous case,492 where particle loss from the local pool was varied, the progressive saturation of particle concentration (Fig. 6a,b)493 resulted, at the end of simulation, in a suppression of the di↵usion e↵ect initially observed for increasing remote494 volumes.495 3.4. Epizootiology and R0 496 497 The generation of an epizootic is regulated by rates involved in the dynamics of both the host and the pathogen,498 and factors associated with the particle di↵usion processes. R0increases linearly with increasing transmission rate S,499 and nonlinearly with increasing initial population N(Fig. 9a, black solid line), pathogen body burden and release rate500 from infected animals cIbIand dead infected animals cDIbDI, and with increasing filtration or contact rate of particles501 fSand local volume sizes Vl(Eq. (17)). The likelihood of an epizootic decreases linearly with increasing inactivation502 rate of particles inside the body of the animal a, and nonlinearly with increasing disease mortality mI, the removal of503 dead individuals from the system d, the remote volume size Vr, the exchange rate of particles between local volume504 and the remote volume , and the loss of particles in the local volume rand the remote volume (Eq. (16)). For505 passive suspension feeders, in addition to filtration or contact rate of particles fS, transmission of disease increases506 nonlinearly with the surface area of the susceptible individual exposed to contact with waterborne pathogens AS.507 508 Host and non-focal host density509 510 In a scenario where an infected animal is introduced into a totally susceptible population, once the population den-511 sity rises sufficiently to compete for pathogens and lower the per capita body burden in the susceptible subpopulation,512 the probability of an epizootic developing remains relatively constant with increasing N (Fig. 9a, black solid line).513 In a scenario with relatively low pathogen body burden, this competition, more likely to occur in active filter feeders514 than in suspension feeders, results in a reduction of disease risk (R0<1) (Fig. 9a, black dashed line).515 In a system with non-focal suspension-feeding hosts H(Equation 20), with a relatively high filtration/contact rate516 of particles fH, the competition e↵ect between host types is intensified and, consequently, the likelihood of an epi-517 zootic decreases with increasing initial abundance of the non-focal host (Fig. 9a, gray solid line). Thus, the threshold518 of the initial host population Nfor an epizootic to occur may be substantially increased with increasing abundance of519 the non-focal host population. For instance, in the Fig. 9a scenario, the epizootic threshold for the initial host popula-520 tion in a system without non-focal filter feeders (N=30) is increased to N=80 after adding the same non-focal host521 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 17 0 500 1000 0 100 200 300 Time (days) Susceptible individuals 0 500 1000 0 500 1000 1500 Time (days) Particles in local pool (number m−3) 0 500 1000 0 100 200 300 Time (days) Infected individuals 0 500 1000 0 5 10 15 Time (days) Particle uptake rate (particles ind−1 day−1) a) !b) ! c) !d) ! Recruitment rate ! (time -1) ! 1 x 10-2 ! 1 x 10-3 ! 1 x 10-4! Recruitment rate ! (time -1) ! 1 x 10-2 ! 1 x 10-3 ! 1 x 10-4! Recruitment rate ! (time -1) ! 1 x 10-2 ! 1 x 10-3 ! 1 x 10-4! Recruitment rate ! (time -1) ! 1 x 10-2 ! 1 x 10-3 ! 1 x 10-4! Figure 8: Comparison of the e↵ect of varying recruitment rate on (a) the susceptible population, (b) the infected population, (c) the particle concentration in the local volume, and (d) particle uptake rate. population (H=30) .522 523 Remote volume size and particle loss524 525 In a scenario with a relatively high exchange rate of particles between the local and remote pools (=1), an in-526 creasing remote volume size, resulting in a di↵usion-like transfer of particles to the remote pool, reduces the particle527 concentration in the local volume available for suspension-feeders and, thus, the likelihood of an epizootic (Fig. 9b,528 dashed line). The e↵ect of increasing particle loss rate in the remote pool on disease development has a similar529 trend and intensity (Fig. 9b, dash-dot line). Note for comparison that the curve is for a remote volume size Vr=1.530 The e↵ect of increasing internal in vivo inactivation rate aalso has a similar trend making R0estimate much more531 sensitive to small changes in this parameter (Fig. 9b).532 533 Infective dose534 535 Finally, we explore the similarity of increasing remote volumes and non-focal host densities, specifically focusing536 on the influence of the infective dose on disease development. When comparing the e↵ect of increasing the remote537 volume size and non-focal host density on disease development for a high infective dose (200 particles), the R0esti-538 mate shows that an increase in non-focal host population density of 100 individuals m2has the same limiting e↵ect539 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 18 0!0.002!0.004!0.006!0.008!0.01! 0.0! 0.5! 1.0! 1.5! 0.0!1.0!2.0!3.0! R0! Remote reservoir ! Particle loss in remote reservoir! Internal inactivation of particles! Particle internal inactivation rate a (time-1)! VΓ σ a σ! Remote reservoir VΓ (m3) Particle loss rate (time-1) " b) " 0.0! 0.5! 1.0! 1.5! 2.0! 0!100!200!300!400!500!600! R0! Initial population, host N and non-focal host H ! Host N! Non-Focal Host, H (N=80)! Host N - Overfiltration! a) " (VΓ =1) Figure 9: R0for increasing (a) initial susceptible Nand non-focal host population density (individuals m2) with the overfiltration scenario for N (with half of the particles), and (b) remote volume size Vr, particle loss rate in the remote pool , and particle internal inactivation rate a. The black dotted line represents the outbreak threshold level R0=1; above this level the likelihood of an epizootic is high; below this level, an outbreak is not expected. on epizootics as a remote volume of 0.1 m3(Fig. 10). This relationship is nonlinear; thus, a non-focal host population540 of 500 individuals m2may have the same limiting e↵ect on epizootics as a remote volume of 2.5 m3. For lower541 infective doses (50 particles), the change in non-focal host population abundance or remote volume is much smaller542 to obtain the same e↵ect on reducing R0. For instance, an increase in non-focal host population from 0 to 200 individ-543 uals m2or in remote volume from 0 to 0.3 m3produces a drop in R0from 6 to 3, whereas for a high infective dose544 (200 particles), the same increase in non-focal host population abundance or remote volume produces a much smaller545 decrease (R0from 1.5 to 0.8). However, if the lower dose disease system has a higher R0than the high dose system, a546 smaller drop in R0in the high dose system may contribute more importantly limiting an epizootic than the substantial547 drop produced by the same change in non-focal host population abundance or remote volume in the low dose system548 (Fig. 10).549 550 Passive suspension feeders and the exposed surface area551 552 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 19 The transmission of disease in passive suspension-feeders not only depends on the contact rate with particles, but553 also the surface area of the susceptible individual exposed to contact with waterborne pathogens (As) (Eq. (17)). First,554 we conducted a simulation to evaluate the e↵ect of varying Ason the accumulation of particles. The simulation was555 run with the conditions for parameters given in Table 2for Case 2, with a carrying capacity Kof 300 individuals.556 Susceptible individuals accumulate particles (Fig. 11a) with time as disease progresses and the water becomes satu-557 rated with particles. The simulation shows an increase in the average host body burden of infective particles as the558 exposed surface area increases. At the higher exposed surface area tested, the average internal number of particles per559 susceptible increases to the infective dose (200 particles) by day 150.560 Second, we introduced an infected animal in a susceptible population of 300 individuals and calculated R0for561 a range of values for of As, including possible realistic feeding surfaces in suspension feeders such as corals (Fig.562 11b). These simulations were conducted for low and high infective doses (50 and 200 particles). The likelihood of an563 epizootic increases nonlinearly with the surface area exposed by the individual to waterborne pathogens AS, resulting564 in the requirement for 3 times larger exposed surface area for an epiozootic to occur in the higher infective dose case565 (Fig. 11b).566 0!0.5!1!1.5!2!2.5!3! 0.0! 1.0! 2.0! 3.0! 4.0! 5.0! 6.0! 0!100!200!300!400!500!600! R0! Initial non-focal host population, H! Non focal host H - High dose! Non focal host H - Low dose! Remote reservoir - High dose! Remote reservoir - Low dose! Remote reservoir VΓ (m3)" Figure 10: R0for increasing non-focal host population density (individuals m2) and remote volume size (m3) for a high infective dose (200 infective particles) and for a low infective dose (50 particles). The gray dotted line represents the outbreak threshold level (R0=1). 4. Discussion567 568 The deterministic compartmental model accounts for the dynamics and epizootiology of waterborne microparasitc569 infectious diseases in suspension-feeders, focusing on both the host and the pathogen dynamics. In this model,570 transmission occurs via particle uptake by contact for passive filter feeders or by filtration for active filter feeders,571 of waterborne infective pathogens under a body-burden based dose-response mechanism. The model yields a set the572 basic reproduction numbers R0which give insight about the relative importance of the parameters determining the573 outbreaks of epizootics. The generation of an epizootic is regulated by rates involved in the dynamics of both the host574 and the pathogen, and factors associated with the particle di↵usion processes.575 4.1. Host dynamics576 577 Simulation results indicate that a relatively high initial concentration of infective particles available for suspension578 feeders, where particle inactivation is slow, the prevalence of disease increases rapidly in the first six months to a high579 and steady level (Fig. 4). Mortality is also relatively high, increasing to a 60% of the population at the end of the580 simulation. Disease mortality rates in bivalve filter feeders can be 50% or more per year. Rates this high are reported581 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 20 b) ! a) ! 0 200 400 600 800 1000 0 50 100 150 200 250 300 350 400 Time (days) Average Particle accumulation per S As=1 cm2 As=0.1cm2 As=0.01 cm2 0.0! 1.0! 2.0! 3.0! 4.0! 5.0! 0.0!0.1!0.2!0.3!0.4!0.5! R0! Exposed surface per individual As (cm-2) ! High dose! Low dose! Figure 11: E↵ect of increasing surface area of susceptible individuals exposed to contact with waterborne pathogens Ason (a) particle accumulation with time, and (b) disease risk (R0) for high (200 infective particles) and low (50 infective particles) infective doses. The gray dotted line represents the outbreak threshold level (R0=1). in oysters in the Gulf of Mexico (Soniat and Brody,1988). In Delaware Bay, the mortality rate is lower; however,582 dermo disease at least doubles the natural mortality rate of the market-size animals in epizootic years (Ford et al.,583 2006;Powell et al.,2008). The fungal disease Aspergillosis can impact sea fan corals with mortalities resulting in the584 loss of >50% of the sea fan colony area in six years (Kim and Harvell,2004). Worldwide distributed clam species585 such as Ruditpes philippinarum show similar mortality rates associated to brown ring disease (Paillard et al.,2014).586 The recruitment of new susceptible individuals to the system (Case 1, Fig. 4) permits levels of prevalence lower587 than 100% (e.g., 90%, Fig. 4). This simulation with a low particle loss rate in the local volume of water saturated with588 infective particles, likely reproduces a ”water-tank” system where eventually all the particles in the water are coming589 into contact with or filtered out by susceptible individuals and so prevalence in the population reaches high levels and590 remains high, even with recruitment. The development and maintenance of high particle concentrations in the water591 column (Fig. 6d) resulting in pandemic infection must be one of the reasons for the rapid transmission of Dermo592 disease and the high mortalities in Eastern oyster populations observed in the early 1990s in Delaware Bay (Ford593 et al.,2006;Bushek et al.,2012) and in other bays where most newly settled juveniles become infected well within594 the first year of life (Powell et al.,1996;Ragone Calvo et al.,2003;McCollough et al.,2007;Soniat et al.,2012).595 Geographically expansive regions with high concentrations of infective particles leading to pandemic infection must596 occur commonly for diseases like Dermo and other Perkinsus-caused diseases in order to explain the continuously597 high prevalence of infection (e.g., Park and Choi,2001;Powell and Hofmann,2015).598 The number of dead animals is small (<1m 2) in both simulations (Case 1 and 2) due to the high removal rate of599 these animals from the system. In general, in marine systems, the bacterial decomposition of organic matter (Allison,600 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 21 1990;Lotz and Soto,2002;Smith,1953) or the action of scavengers removing dead animals (Veale et al.,2000;601 Morello et al.,2005) is a relatively fast process and thus, dead animal tissue is removed quickly from the environment.602 The initial increase of dead infected individuals to a maximum is related to the increase of infected individuals as the603 disease transmission progresses in the population. The infected animals and their decay or scavenging can contribute604 pivotally to transmission by the releasing of particles to the local volume and ultimately supplying the remote pool605 (Fig. 6c) or they can be inconsequential if the act of decay or scavenging results in the loss of infectivity. Few studies606 have examined the importance of post-mortem processes in disease transmission.607 Simulations show the e↵ect of varying transmission rate and carrying capacity on the host population (Case 1,608 Fig. 3vs Case 2, Fig. 4). When the population is below carrying capacity, the susceptible population initially609 increases (Fig. 4) as the recruitment of new susceptible individuals is faster than the rate of infection. As the disease610 transmission progresses in the population with a substantial increase of particles in the environment, the reduction611 of susceptible individuals due to infection exceeds the incorporation of new individuals. The slower transmission612 rate in Case 2 (Fig. 4) importantly reduces the spread of the disease in the population, lowering the prevalence of613 infection. Overall, increasing transmission rate leads to higher prevalence of infection in filter feeders, especially614 at low carrying capacities or at low population densities where the carrying capacity is high. Increasing population615 density, and particularly in cases where carrying capacity is high, can reduce reduce prevalence of infection even for616 relatively high disease transmission rates (Fig. 5). In a contact–based density–dependent disease, a population at its617 carrying capacity has a higher likelihood of disease transmission than when the population is below carrying capacity618 (Gao and Hethcote,1992). However, dense filter feeder populations at carrying capacity may be less vulnerable to619 disease; individual hosts may ‘compete’ for pathogens reducing the concentration of infective particles sufficiently to620 limit body burden below the infective dose and, in turn, limiting the probability of epizootic development (i.e. the621 overfiltration scenario, Bidegain et al., in press).622 4.2. Particle dynamics623 624 Many disease models for marine filter-feeders have addressed the proliferation of infection and subsequent disease625 impact, accepting widespread and rapid transmission to be the norm (Ford et al.,1999;Powell et al.,1999,1996).626 Others have examined the e↵ects of diseases on host population dynamics (Kuris and La↵erty,1992;Yakob and627 Mumby,2011). However, models of marine diseases that incorporate particles as a variable, permitting the water628 column to act as a ‘reservoir’, conduit, or sink for infective particles, are few and none also include an infective629 dose e↵ect based on an in vivo compartment of infective particles sequestered in the susceptible pool of hosts. The630 model that we propose here tracks pathogen dynamics in (i) the local volume with a particle pool interacting with the631 host population, (ii) a remote volume that can act as a source or sink for infective particles depending on the relative632 concentration between the two volumes, and (iii) a transient pool of infective particles in the susceptible individuals633 that may ultimately be inactivated or transition the individual to the infected pool.634 Simulations show that initial infections of a few susceptible individuals initiates the process (Fig. 3a,b) by which635 substantial numbers of infective particles are released into the water thereby becoming available to susceptibles and636 saturating the water with particles (Fig. 6c). This results in an initial di↵usion-like transfer of particles to the adjacent637 remote volume and, eventually, in a ’balance’ in particle concentration between the two water compartments (Fig. 6d).638 A system with a high release rate of pathogens from infected animals or a high rate of release of infective particles upon639 death, and a limited inactivation rate of infective particles either in the water column or in the susceptible host, might640 easily lead to be a highly transmissible system characterized by a relatively high incidence of disease outbreaks. This641 is the case, for instance, for oysters and the pathogen Perkinsus marinus (Bushek et al.,2002), but might also occur in642 corals, where the decay of tissue of infected colonies infected by black-band disease or aspergillosis releases infectious643 fomites or individual particles into the water which then drift to nearby corals (Jolles et al.,2002;Richardson,2004;644 Zvuloni et al.,2009) or in the case of withering syndrome in abalone where infective elements or fomites released645 by infected individuals before or after death spread infection over wide expanses (La↵erty et al.,2015). We suspect646 a similar scenario for the disease of long-spined sea urchins that ravaged the Caribbean populations late in the last647 century (Lessios et al.,1984). Consequently, in these systems, parameters incorporated into the model, such as the648 body burden of pathogens in infected (bI) or dead animals (bDI) and the relative importance of the release rate (c)649 and the removal rate of pathogens in the tissue (a) or water column (rand sigma) might have profound influence in650 determining the frequency and geographic extensiveness of epizootics.651 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 22 The model present here also tracks particle dynamics inside the susceptible population. The dynamics of the652 infective dose e↵ect in filter and suspension feeders is poorly understood (e.g., Chu and Volety,1997;Wang et al.,653 2010). Our model suggest that the accumulation of particles increases to a maximum in the initial phase of the disease654 process as susceptible animals take up particles without reaching an infective dose and that a number of in vivo and655 external processes determine the degree of this accumulation and the probability that sufficient infective particles656 will be accumulated to produce an infection. Among these is the immune response, but equally important are the657 processes controlling uptake, namely the particle acquisition rate (filtering or surface area impingement) and particle658 concentration which itself is controlled by the rate of addition and loss of particles from the local water volume. This659 dynamic is likely enormously important, yet to date has been little studied. Assuming that the disease transmission in660 filter feeders (Bushek et al.,1997;Ford et al.,1999;Powell et al.,1999), and probably in other suspension feeders,661 occurs via an infective dose so that susceptible animals need to have some minimum level of body burden of infectious662 particles before they become infected, some susceptible individuals will then have a relatively high body burden while663 most will have a small body burden (see section 2.4.1). Presumably, these animals will have some unique attributes664 either genetically or environmentally that has resulted in their higher body burdens. Understanding the contributing665 factors that result in a few animals initiating infections than then can propagate by supporting increased particle666 concentration in the water column is a critical need. In this regard, simulations show that the average per capita667 dose in susceptible population is 150 particles (Fig. 6b). This demonstrate the adequate performance of dose-based668 transmission term, considering that the average per capita dose is lower than the minimum dose for susceptibles to669 become infected bmin (200 particles).670 4.3. Recruitment671 672 Recruitment introduces new, healthy individuals into the population which have the net e↵ect of reducing disease673 prevalence. However, in this case, this e↵ect of recruitment on reducing prevalence does not directly limit disease; a674 decrease in prevalence of infection is not a consequence, for instance, of new individuals increasing population particle675 filtration rate and then reducing the local pool of infective particles sufficiently to decrease per capita accumulation676 below the infective dose. The prevalence is lower only due to the addition of more susceptibles to the system. The677 influence of these new susceptible individuals will depend upon their role as new produces of infective particles or as678 new ‘inactivators’ of infective particles. Often these new recruits will promote the disease as the number of individuals679 becoming infected increases (Fig. 8b) despite the reduction of (i) particles in the water column due to increasing680 population particle filtration (Fig 8c) and (ii) the particle uptake rate per animal (Fig. 8d). Rather, these individuals681 provide increased production of infective particles, raising their concentration in the local pool and ultimately the682 remote pool. This explains the occurrence of pandemic infections in marine filter feeders such as oysters regardless683 of increasing recruitment intensities (Ford,1996;Bushek et al.,2012).684 However, the e↵ect of recruitment in limiting the occurrence or the duration of disease epizootics can be substantial685 in both active and passive suspension feeders when enough recruits enter the population to reduce the infective particle686 concentration in the water column such that the particle uptake rate per animal drops to a point where an increasing687 number of animals retain a transient body burden below the infective dose. This e↵ect has been suggested to occur in688 oyster populations following high intensity recruitment events. Even at high salinity, suitable for disease transmission,689 an oyster population might be able to resist epizootic development as long as recruitment sustains the increase in690 population density (Hofmann et al.,1995;Powell et al.,1996).691 4.4. Filtration/contact rate692 693 Filtration rate in filter feeders, such as bivalves, can be a↵ected by diverse characteristics of the particles filtered,694 such as the concentration of phytoplankton and suspended solids and the quality and size of food particles (Hofmann695 et al.,1994;Khalil,1996). The physical parameters of the natural habitat such as temperature, salinity, and water696 flow (MacDonald and Thompson,1986) and the size of the animal can also a↵ect the filtration rate (Dame,2011).697 In passive suspension feeders, such as corals, the contact rate depends on the polyp or colony exposed surface area698 and the flow conditions (Sebens et al.,1996,1998). Such factors a↵ect the incorporation of pathogens by susceptible699 animals and, thus, the disease incidence.700 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 23 We varied filtration/contact rates to explore this e↵ect (Fig. 7b). A relatively low filtration or particle contact701 rate limits disease transmission, due to lower per capita exposure to pathogens. Increasing filtration rate has an e↵ect702 on disease transmission when the concentration of particles in the water is relatively low. In this case, as filtration703 or contact rate increases, the rate of disease transmission increases correspondingly. However, a change in filtration704 or contact rate ceases to exert a significance influence on transmission when the local volume becomes saturated705 with infective particles. At this point, higher filtration rates (Fig. 7b) result in the same high prevalence; that is, the706 per capita dose received by individuals is above the infectious dose over a wide range of population filtration rates.707 However, active filter feeders in particular, and suspension feeders possibly have two mechanisms, explored here,708 to reduce the incidence of infection despite having high filtration rates and contact rates with substantive numbers709 of infective particles in the water column. The first obviously is a higher infective dose which might result from710 selection for more disease resistant genotypes, for example (Munroe et al.,2015) (Fig. 10) and the second is a dense711 assemblage of hosts competing for pathogens and reducing the per capita dose (i.e. the overfiltration scenario –712 Bidegain et al. in press) (Fig. 9a, black dashed line). Because it is well established that dense concentrations of713 bivalves can become food limited (Heasman et al.,1998;Jiang and Gibbs,2005;Powell et al.,1995), it stands to714 reason that the same process would limit per-capita incorporation rates of infective particles. In passive suspension715 feeder hosts in environments with high particle concentrations and contact rates, reduced exposed feeding areas may716 result in lowering the accumulation of particles (Fig. 11a), and thus the prevalence of infection. Whether passive717 suspension feeders can become sufficiently dense to limit per capita particle uptake rate is unclear.718 4.5. Particles loss and di↵usion719 720 The model incorporates two parameters for particle loss in the water column. The parameter raccounts for the721 particle loss within the local volume Vldue to natural mortality or other processes of inactivation such as sedimenta-722 tion, and advection and di↵usion to the remote volume Vr. The parameter accounts for the subsequent inactivation723 through natural mortality or other sources of inactivation such as sedimentation in this remote pool or perhaps loss724 from this compartment through, for example, outwelling. These two volumes in the model are formulated in order to725 account for particle di↵usion by means of concentration gradients between them. Initially, in the simulation when the726 local volume is not saturated with particles, increasing particle loss through di↵usion of particles to the remote pool727 can limit epizootic development. When, the environment is saturated with particles, this e↵ect becomes inconsequen-728 tial (Fig. 7c,d) unless the inactivation rate in the remote pool is high (Fig. 9b). A large remote volume together with a729 high exchange rate of particles between pools and a relatively high inactivation rate of pathogens in the remote pool730 ais an e↵ective mechanism to reduce particle concentration locally and prevent transmission (Bidegain et al., in press)731 (Fig. 9b). Thus, a scenario similar to what is shown in Fig. 7d (gray line), in which the remote volume is substantially732 larger than the local volume, thereby leading to a rapid transfer of pathogens from the local pool to the remote pool,733 is particularly e↵ective at limiting the incidence of infection. This scenario could easily occur in estuaries with short734 water residence times (Armstrong,1982;Marshall and Alden,1997)during periods of increased freshwater inflow or735 during spring tides where the larger tidal volume results in increased outwelling of particles, thereby reducing particle736 concentration in the water column (Ellien et al.,2004;Banas et al.,2007;Gray et al.,2009;Narv´ aez et al.,2012).737 On the other hand, a high concentration of infective particles in the remote volume could overcome any local loss738 mechanisms such as overfiltration or the removal by non-focal hosts, and thus maintain high prevalence. The dynamic739 interchange between local and remote pools if not unique to the marine realm is at least particularly characteristic of740 the disease transmission processes taking place there. This model is a theoretical model that explores the processes741 that initiate and terminate an epizootic. The simplification of hydrodynamics to two volumes, the volume directly742 impacted by the filter feeder and the volume overlying it, with particle di↵usion occurring across the interface is743 sufficient to explain these processes theoretically. The study of a specific case such as the disease transmission on744 a reef or in an estuary would require the use of similar disease models coupled to a hydrodynamic model. While745 complex models of larval dispersal using location-specific hydrodynamics have been employed often (North et al.,746 2008;Ayata et al.,2009;Bidegain et al.,2013), the transmission of disease particles in such applications has been747 little investigated (Murray and Jackson,1992).748 4.6. Epizootiology and R0 749 750 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 24 Simulation results of host and particle dynamics showed that dead infected animals, likely with higher body751 burden and release rates of infective particles than most live infected animals (Bushek et al.,2002;Richardson,2004),752 are determinant in the generation of epizootics in benthic filter and suspension feeders. A relatively small external753 source of infective particles initially available to a benthic suspension–feeder population may infect few individuals.754 Although this external source is consumed rapidly or lost and inactivated in the water column, these initially infected755 individuals can trigger the generation of the epizootic by means of particle release in the live-infected stage and,756 particularly, upon death.757 However, mechanisms potentially exist that might lead to a lowering of the epizootic likelihood in a benthic758 population of suspension feeders even if a relatively large number of particles is available. High densities of sus-759 pension feeders, particularly filter-feeders, may compete for food, but also for waterborne pathogens. Moreover, a760 non-focal host population ingesting or otherwise collecting infective particles (e.g., bivalves, sponges or tunicates),761 may enhance the competition e↵ect, which could lead to a lower per capita exposure to pathogens (Peterson and762 Andre,1980;Beukema and Cad´ ee,1996;Powell et al.,2012c). Thus, the threshold of the initial host population763 for an epizootic to occur may be substantially increased with increasing numbers of non-focal hosts (Fig. 9). This764 intraor inter-specific competition (Fig. 9a and 8) by itself or together with particle di↵usion (Fig. 9b) may reduce765 substantially the likelihood of an epizootic.766 The e↵ect of increasing particle loss on limiting the epizootic development is similar to that produced by an767 increase in remote volume size (Fig. 9b). This means that the remote volume is an e↵ective mechanism for limiting768 an epizootic as it can remove particles from the local pool by di↵usion. Thus, an increase in the remote volume size769 may exert a limiting e↵ect on epizootic development comparable to that produced by inter-specific competition with770 a non-focal host population. Moreover, an increase in these two disease limiting factors will have a higher impact on771 disease control in hosts with a relatively high infective dose (Fig. 10). However, an increase in the remote volume772 needs to be accompanied by particle inactivation or loss (i.e. ) in order to have an e↵ect on epizootic development.773 In absence of this particle removal, the remote volume may become saturated with particles and di↵use them back to774 the local pool thereby making them available to susceptible hosts. In contrast, in vivo inactivation rate of particles,775 by limiting particle body burden below the infective dose, has a more intense e↵ect on disease development than the776 remote volume size or the particle loss in this volume (Fig. 9b).777 Overall, these disease limiting mechanisms apply similarly, but no in equivalent measure, to active and passive778 suspension feeders. In the case of passive suspension feeders, the competition for particles may be less e↵ective than779 in active filter-feeders. The exposed surface area of the individuals Asin a colony of passive suspension feeders (e.g.780 corals) is as important as the density of individuals; thus, high density populations with a small exposed surface area781 per individual, resulting in a reduced particle accumulation, may limit the disease outbreak (Fig. 11) just as well as782 fewer individuals with a large surface area for particle capture. Thus, it seems that the active foraging strategy of filter783 feeders permits these organisms to e↵ectively limit disease outbreaks in comparison to passive feeders, but the same784 attribute makes them more vulnerable to epizootics when population densities are inadequate to control local particle785 concentrations beneath the threshold yielding an e↵ective dose (Fig. 9a and Fig. 11).786 5. Conclusions787 788 A large number of marine diseases are transmitted through the water column from one host to another. This789 water column provides a ‘reservoir’ for infective particles and the mechanisms by which particles are added to it or790 lost from it exert an important influence on the prevalence of disease and more importantly the di↵erence between a791 disease exerting a local impact on a host population and pandemic disease a↵ecting the host over large geographic792 regions. The local population modulates this e↵ect through genetic characteristics that a↵ect the infective dose and793 through varying local availability by modulating particle incorporation and release rates. The dynamic imposed by794 this complex interaction between population and water column, potentially over metapopulation scales, is relatively795 unique to the marine world. Understanding the details of this dynamic is critical to understanding the disease process796 in host populations and to improving management responses to marine disease challenges. Models that incorporate797 these processes, such as the one described here, are important tools, little used heretofore in the marine context, to798 explain the epizootiology of marine diseases. Only when a disease can be understood at the in vivo scale of the799 G. Bidegain et al. /Ecological Modelling XXX (2022) 1–28 25 individual, the local scale of the population, and the geographic scale of the metapopulation will e↵ective approaches800 to management become routinely achievable.801 Acknowledgements802 This investigation was funded by the NSF Ecology and of Infectious Diseases Program Grant # OCE-1216220.803 We appreciate this support.804 References805 Allison, P., 1990. Variation in rates of decay and disarticulation of Echinodermata: implications for the application of actualistic data. Palaios 5,806 432–440.807 Anderson, R., Gordon, D., 1982. Processes influencing the distribution of parasite numbers within host populations with special emphasis on808 parasite-induced host mortalities. Parasitology 85, 373–398.809 Anderson, R. M., May, R. M., 1981. The population dynamics of microparasites and their invertebrate hosts. Philosophical Transactions of the810 Royal Society of London. Series B, Biological Sciences 291, 451–524.811 Armstrong, N. E., 1982. Response of Texas estuaries to freshwater inflows. In: Kennedy, V. S. (Ed.), Proc. 6th Biennial International Estuarine812 Research Conference. Academic Press, pp. 103–120.813 Audemard, C., Ragone Calvo, L., Paynter, K., Reece, K., Burreson, E., 2006. Real-time PCR investigation of parasite ecology: in situ determination814 of oyster parasite Perkinsus marinus transmission dynamics in lower Chesapeake Bay. Parasitology 132, 827–842.815 Ayata, S.-D., Ellien, C., Dumas, F., Dubois, S., Thi´ ebaut, ´ E., 2009. Modelling larval dispersal and settlement of the reef-building polychaete816 Sabellaria alveolata: role of hydroclimatic processes on the sustainability of biogenic reefs. Continental Shelf Research 29, 1605–1623.817 Bak, R., Joenje, M., De Jong, I., Lambrechts, D., Nieuwland, G., 1998. Bacterial suspension feeding by coral reef benthic organisms. Marine818 Ecology Progress Series 175, 285–288.819 Banas, N. S., Hickey, B. M., Newton, J. A., Ruesink, J. L., 2007. Tidal exchange, bivalve grazing, and patterns of primary production in Willapa820 Bay, Washington, USA. Marine Ecology Progress Series 341, 123–139.821 Beukema, J. J., Cad´ ee, G. C., 1996. Consequences of the sudden removal of nearly all mussels and cockles from the Dutch Wadden Sea. Marine822 Ecology 17, 279–289.823 Bidegain, G., B´ arcena, J. F., Garc´ ıa, A., Juanes, J. A., 2013. LARVAHS: predicting clam larval dispersal and recruitment using habitat suitability-824 based particle tracking model. Ecological Modelling 268, 78–92.825 Bidegain, G., Powell, E. E., Klinck, J. M., Ben-Horin, T., Hofmann, E. E., 2016. Marine infectious disease dynamics and outbreak thresholds:826 contact, transmission, pandemic infection, and the potential role of filter feeders. Ecosphere , in press.827 Bushek, D., Ford, S. E., Alcox, K. A., Gustafson, R., Allen Jr., S. K., 1997. Response of the eastern oyster, Crassostrea virginica to in vitro cultured828 Perkinsus marinus and the early fate of parasites delivered via three dosing methods. Journal of Shellfish Research 16, 479–485.829 Bushek, D., Ford, S. E., Burt, I., 2012. Long-term patterns of an estuarine pathogen along a salinity gradient. Journal of Marine Research 70,830 225–251.831 Bushek, D., Ford, S. E., Chintala, M. M., 2002. Comparison of in vitro-cultured and wild-type Perkinsus marinus. III. Fecal elimination and its role832 in transmission. Diseases of Aquatic Organisms 51, 217–225.833 Calvo, L. M. R., Wetzel, R. L., Burreson, E. M., 2001. Development and verification of a model for the population dynamics of the protistan834 parasite, Perkinsus marinus, within its host, the eastern oyster, Crassostrea virginica, in Chesapeake Bay. Journal of Shellfish Research 20,835 231–241.836 Choi, K.-S., Powell, E. N., Lewis, D. H., Ray, S. M., 1994. Instantaneous reproductive e↵ort in female American oysters, Crassostrea virginica,837 measured by a new immunoprecipitation assay. The Biological Bulletin 186, 41–61.838 Choi, K.-S., Wilson, E. A., Lewis, D. H., Powell, E. N., Ray, S. M., 1989. The energetic cost of Perkinsus marinus parasitism in oysters: quantifi-839 cation of the thioglycollate method. Journal of Shellfish Research 8, 125–131.840 Chu, F.-L. E., Volety, A. K., 1997. Disease processes of the parasite Perkinsus marinus in eastern oyster Crassostrea virginica: minimum dose for841 infection initiation, and interaction of temperature, salinity and infective cell dose. Diseases of Aquatic Organisms 28, 61–68.842 Cuddington, K., Beisner, B., 2005. Ecological paradigms lost: routes of theory change. Theoretical Ecology Series. Elsevier Academic, Burlington,843 Massachusetts.844 Dame, R. F., 1993. The role of bivalve filter feeder material fluxes in estuarine ecosystems. In: Bivalve filter feeders, in Estuarine and Coastal845 Ecosystem Processes. Vol. 33 of NATO ASI Series. Springer-Verlag, Berlin, pp. 245–269.846 Dame, R. F., 2011. Ecology of marine bivalves: an ecosystem approach. CRC Marine Science. CRC Press, Boca Raton, FL.847 Dang, C., de Montaudouin, X., Caill-Milly, N., Trumbic, Z., 2010. Spatio-temporal patterns of perkinsosis in the Manila clam Ruditapes philip-848 pinarum from Arcachon Bay (SW France). Diseases of Aquatic Organisms 91, 151–159.849 Diamond, E. A., 2012. Do scavengers influence dermo disease (Perkinsus marinus) transmission?: experiments in oyster parasite trophic interac-850 tions. MSc Thesis, Rutgers University-Graduate School-New Brunswick.851 Diekmann, O., Heesterbeek, H., Britton, T., 2013. Mathematical tools for understanding infectious disease dynamics. Princeton University Press,852 Woodstock, Oxfordshire.853 Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J., 1990. On the definition and the computation of the basic reproduction ratio R0in models for854 infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28, 365–382.855 Diekmann, O., Heesterbeek, J. A. P., Roberts, M. G., 2010. The construction of next-generation matrices for compartmental epidemic models.856 Journal of the Royal Society Interface 7, 873–885.857