RESEARCH Open Access

An epidemiological model for proliferativekidney disease in salmonid populationsLuca Carraro1, Lorenzo Mari2, Hanna Hartikainen3,4, Nicole Strepparava5, Thomas Wahli5, Jukka Jokela3,4,Marino Gatto2, Andrea Rinaldo1,6 and Enrico Bertuzzo1*

Abstract

Background: Proliferative kidney disease (PKD) affects salmonid populations in European and North-Americanrivers. It is caused by the endoparasitic myxozoan Tetracapsuloides bryosalmonae, which exploits freshwaterbryozoans and salmonids as hosts. Incidence and severity of PKD in brown trout populations have recentlyincreased rapidly, causing a decline in fish catches and local extinctions in many river systems. PKD incidence andfish mortality are known to be enhanced by warmer water temperatures. Therefore, environmental change is fearedto increase the severity of PKD outbreaks and extend the disease range to higher latitude and altitude regions. Wepresent the first mathematical model regarding the epidemiology of PKD, including the complex life-cycle of itscausative agent across multiple hosts.

Methods: A dynamical model of PKD epidemiology in riverine host populations is developed. The model accountsfor local demographic and epidemiological dynamics of bryozoans and fish, explicitly incorporates the role oftemperature, and couples intra-seasonal and inter-seasonal dynamics. The former are described in a continuous-time domain, the latter in a discrete-time domain. Stability and sensitivity analyses are performed to investigate thekey processes controlling parasite invasion and persistence.

Results: Stability analysis shows that, for realistic parameter ranges, a disease-free system is highly invasible, whichimplies that the introduction of the parasite in a susceptible community is very likely to trigger a disease outbreak.Sensitivity analysis shows that, when the disease is endemic, the impact of PKD outbreaks is mostly controlled bythe rates of disease development in the fish population.

Conclusions: The developed mathematical model helps further our understanding of the modes of transmission ofPKD in wild salmonid populations, and provides the basis for the design of interventions or mitigation strategies. Itcan also be used to project changes in disease severity and prevalence because of temperature regime shifts, andto guide field and laboratory experiments.

Keywords: Discrete-continuous hybrid model, Climate change, Disease ecology, Fredericella sultana

Abbreviations: DFE, Disease free equilibrium; DFT, Disease free trajectory; PKD, Proliferative kidney disease

BackgroundEpidemics of emerging diseases may have large eco-nomic and ecological impacts, threaten livelihoods, elicitbiodiversity losses and affect key ecosystem services. Un-derstanding causes and patterns of disease emergencebecomes even more topical as causative and correlativelinks with global climate and environmental change are

established [1–3]. Proliferative Kidney Disease (PKD) ofsalmonid fish, caused by Tetracapsuloides bryosalmonae(phylum Cnidaria, class Malacosporea) [4], is highlyproblematic for hatcheries and fish farms, potentiallyreaching 100 % infection prevalence and high mortalities(up to 95 %) in affected fish [5, 6]. Although the impactsof PKD in wild fish populations are still poorly known,PKD is considered a key factor contributing to the de-cline of wild salmonid populations in Switzerland andNorthern Europe [7–9].

* Correspondence: enrico.bertuzzo@epfl.ch1Laboratory of Ecohydrology, École Polytechnique Fédérale de Lausanne,Station 2, Lausanne 1015, SwitzerlandFull list of author information is available at the end of the article

© 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Carraro et al. Parasites & Vectors (2016) 9:487 DOI 10.1186/s13071-016-1759-z

The parasite life-cycle alternates between freshwaterbryozoans, as primary hosts [10–12], and salmonid fish,where infection can cause PKD [4, 13]. The transmissionbetween hosts occurs through spores that are releasedinto water. Development and pathology of PKD aredependent on water temperature, as clinical signs anddisease-related mortality increase with increasing watertemperatures [4, 5, 9, 14–16]. Water temperature is alsokey to the development of T. bryosalmonae in itsbryozoan host. Increased temperatures have been shownto promote the production of spores infective to fish[17, 18]. The connections between temperature, diseasedevelopment and fish mortality suggest that, due to cli-mate change, the prevalence, severity and distribution ofPKD are likely to increase further [19]. Hence, PKDmust be considered as an emerging disease having asizeable and growing impact on the health of salmonidpopulations and, consequently, on the economy of fishfarming industry and hatcheries. In the perspective ofunderstanding drivers and controls of the disease, and inthe search for mitigation strategies, the developmentof a dynamical model of PKD transmission becomescrucial.Epidemiological models incorporating parasites with

simple life-cycles have a long history [20, 21] and havebeen successfully applied to several human diseases andzoonoses. Conversely, models incorporating parasitesand pathogens with complex life-cycles and multiplehosts are less frequently developed, although manymajor human and wildlife diseases are caused by para-sites with such life-cycles. Successful application of epi-demiological models for these diseases fundamentallyrelies on the incorporation of the ecological dynamics ofthe different host populations. In the context of fishdiseases, epidemiological models have been exploited toaddress various issues, such as sea lice infection onsalmon farms [22]; Koi herpes virus of the common carp(Cyprinus carpio) [23]; amyloodinosis, a disease of warmwater mariculture [24]; fish-borne zoonotic trematodesin agriculture-aquaculture farms [25]; Ceratomyxa shasta,a myxozoan parasite endemic to river systems throughoutthe Pacific northwest region of North America [26]; andwhirling disease, a myxozoan disease of farmed and wildsalmonids [27]. As for PKD, the only modeling attemptused a Bayesian probability network to assess the declineof brown trout, citing PKD as a driver of increasedmortality [28].The present work proposes an epidemiological model

capable of describing the intra- and inter-annual dynam-ics of PKD. The model aims to translate the currentknowledge of the modes of transmission of the diseaseand of the life-cycle of its causative agent into amathematical form, making the connection betweentemperature and epidemiological parameters explicit. In

this way, it is possible to state under which conditionsPKD can establish in a fish population. The developmentand the analysis of such a model is also instrumental toguiding further experimental and field studies on PKD.

MethodsIn this section, we first describe the transmission cycleof PKD, and subsequently present the mathematicalmodel that stems from such biological and epidemio-logical evidence.

PKD transmission cycleTetracapsuloides bryosalmonae has a complex life-cyclethat exploits freshwater bryozoans and salmonids ashosts (for review, see [29]). Within bryozoans, the para-site can express either covert or overt infection stages.During covert infections, the parasite exists as non-virulent single-cell stages [30, 31]. The transition toovert infection implies increase in virulence, with theformation of multicellular sacs from which thousands ofT. bryosalmonae spores (approximately 20 μm indiameter) are released into water. Spores are charac-terized by two amoeboid cells and four polar capsules[32, 33]. Parasite transmission from bryozoans to fishcan thus take place only during the overt infectionstage. Overt infection hampers bryozoan growth, whereascovert stages pose low energetic cost to bryozoans [17].Peaks of overt infection have been observed in late springand autumn [34]. In particular, overt stages develop whenbryozoans are undergoing enhanced growth as a result ofoptimal temperatures or food levels [18, 35]. Overt infec-tion also elicits temporary castration [36], with a severereduction in the production of statoblasts (asexually pro-duced dormant propagules). Covertly infected bryozoanscan produce infected statoblasts thus allowing verticaltransmission of T. bryosalmonae [37]. Recovery mecha-nisms for bryozoan colonies are poorly observed andunderstood [37].Parasite spores released into water by overtly infected

bryozoans infect fish through skin and gills [38, 39].Tetracapsuloides bryosalmonae subsequently enters thekidney of fish hosts, where it undergoes multiplicationand differentiation from extrasporogonic to sporogonicstages [40]. Spores developed in the lumen of kidney tu-bules contain one amoebid cell and two polar capsules[41], and are eventually excreted via urine [42]. AlthoughPKD seems to develop in all salmonids to a varyingdegree, life-cycle completion may be highly species-specific. For example, in Europe, parasite spores infectiveto bryozoans develop in the brown trout (Salmo trutta)and the brook trout (Salvelinus fontinalis), but not inthe rainbow trout (Oncorhynchus mykiss) [43–45]. Fishinfected with PKD often die owing to secondary infec-tions [13]; however, PKD alone has been shown to cause

Carraro et al. Parasites & Vectors (2016) 9:487 Page 2 of 16

mortality [14, 15]. Fish that do not die during the acutephase of the infection may become long-term carriers ofthe parasite. Such carriers are reportedly able to infectFredericella sultana, one of the most common bryozoanhosts of T. bryosalmonae, for a period up to 2 years afterexposure [43].

ModelThe proposed model couples PKD transmission andpopulation dynamics of fish and bryozoan populations.When reproduction processes are concentrated in time(as in our case, in which spawning and hatching for fish,and statoblast hatch for bryozoans occur mainly duringthe cold season), population dynamics are traditionallyanalyzed with discrete-time models (see the widelyknown Ricker model [46], introduced to study stock andrecruitment in fisheries). On the other hand, PKD trans-mission occurs continuously throughout the warm sea-son, thus calling for a continuous modelling approach(e.g. the classical Susceptible-Infected-Recovered model[20]). We therefore propose a discrete-continuous hybridframework that couples intra-seasonal and inter-seasonaldynamics. Seasons can be thought of as the periods ofbryozoan proliferation (e.g. from April to November, de-pending on climate). Within-season dynamics are de-scribed by a set of coupled ordinary differential equationsexpressing how bryozoan biomass, fish density, and the

abundances of infective spores and statoblasts change dur-ing the warm season. The transition between the end of awarm season and the beginning of the next one ismodelled as a discrete-time update. Therefore, between-season dynamics are described by a set of difference equa-tions. An interesting framework to formalize discrete-continuous hybrid models is the so-called time scale cal-culus [47]; however, the complexity of the model pre-sented in the following paragraphs prevents an effectiveuse of such formalism. In this work, we focus on a local-scale model, able to mimic PKD dynamics in an isolatedwater body, where fish and bryozoans are well mixed, andthere are no additional inputs or outputs. An outline ofthe model is shown in Fig. 1. The variables of the modelare listed in Table 1; all state variables are referred to asconcentrations.

Within-season modelThe dynamics of the state variables during the warmseason (from time τ0 to τ1) of year y (see Fig. 1c) is de-scribed by the following system of first-order differentialequations:

dBS

dτ¼ gSðBS;BC ;BO;TÞBS−βBZFBS þ ψ BC þ BOð Þ;

ð1aÞ

Fig. 1 Schematic representation of PKD transmission dynamics. a Within-season dynamics. The red-shaded area highlights the bryozoan sub-model.The circular arrows illustrate bryozoan growth. The dotted line indicates that the growth of overtly infected bryozoans is impaired by the parasite. Theblue-shaded area highlights the fish sub-model. The dead-end arrow refers to excess PKD-induced fish mortality. In both sub-models, straight solidlines indicate fluxes among epidemiological classes, dashed arrows represent disease transmission between the two sub-systems. Natural mortality isnot displayed. Parameters driving the transition between classes are displayed in gray. Letters T and τ indicate fluxes that are expected tobe dependent on water temperature T and time τ. b Schematic representation of between-season (overwintering) dynamics relevant to PKDtransmission. Colors as in (a). Continuous lines represent survival over winter. Dashed-dotted lines stand for reproduction processes. c Graphic timelineof the discrete-continuous hybrid model

Carraro et al. Parasites & Vectors (2016) 9:487 Page 3 of 16

dBC

dτ¼ gCðBS;BC ;BO;TÞBC þ βBZFBS− dCO Tð Þ þ ψ½ �BC

þdOC Tð ÞBO;

ð1bÞ

dBO

dτ¼ gOðBS;BC ;BO;TÞBO þ dCO Tð ÞBC− dOC Tð Þ þ ψ½ �BO;

ð1cÞ

dSSdτ

¼ f B τ;Tð ÞBS þ φ f B τ;Tð ÞBC ; ð1dÞ

dSIdτ

¼ 1−φð Þf B τ;Tð ÞBC ; ð1eÞ

dFS

dτ¼ −μFFS−βFZBFS þ ζ FC ; ð1fÞ

dFE

dτ¼ βFZBFS− μF þ h Tð Þ½ �FE ; ð1gÞ

dFI

dτ¼ 1−εð Þh Tð ÞFE− μF þ a Tð Þ þ γ½ �FI ; ð1hÞ

dFC

dτ¼ εh Tð ÞFE þ γ FI− μF þ ζð ÞFC ; ð1iÞ

dZB

dτ¼ πBBO−μZZB; ð1jÞ

dZF

dτ¼ πF FI þ κFCð Þ−μZZF ; ð1kÞ

where the dependence of the parameters on temperatureT and time τ has been explicitly expressed.The first terms in the right-hand sides of Eqs. (1a),

(1b) and (1c) express the growth of the bryozoan

biomass, which is assumed to be logistic-like. Thuswe set

gXðBS;BC ;BO;TÞ ¼ rX Tð Þ 1−ρ BS þ BC þ BOð Þ½ �; ð2Þ

where rX(T) and ρ are the baseline instantaneous growthrate of class X = {S,C,O} and the inverse of the carryingcapacity of the bryozoan population, respectively. We as-sume that susceptible and covertly infected bryozoanshave the same baseline growth rate rS = rC = r, whereasrO << r since the growth of overtly infected bryozoans isstrongly impaired [36]. Both r and rO are assumed to bemonotonically increasing functions of temperature, asexperimental evidence suggests [17]. The term βBBSZF

in Eqs. (1a) and (1b) represents the flux of bryozoan bio-mass from the susceptible to the covertly infected class,with βB being the exposure rate of bryozoans to fish-released spores. The transition from covert to overt in-fection is assumed to occur at a rate dCO [see Eqs. (1b)and (1c)], which is defined as the inverse of the meantime necessary for the development of the overt stage ofinfection in a previously covertly infected bryozoan unit.The rate of transition from overt to covert infection isexpressed by dOC. Both parameters are temperature-dependent: in particular dCO increases with increasingtemperature, while dOC decreases [18]. As previouslystated, the development of an overt infection, whichposes a high energetic cost to the bryozoan colony, alsodepends on host conditions and food availability. How-ever, as a first approximation, temperature is consideredas the only determinant of change of the two transitionrates. Infected bryozoans can possibly clear the infectionand become again susceptible at a rate ψ [Eqs. (1a), (1b)and (1c)]. The production of statoblasts is only achievedby susceptible and covertly infected bryozoans, whereasovertly infected colonies do not produce statoblasts. Inparticular, susceptible bryozoans produce uninfectedstatoblasts SS [Eq. (1d)] at a rate fB that is assumed todepend on both temperature and time; specifically, therelease of statoblasts is enhanced towards the end of theseason. Seemingly, uninfected bryozoans have been ob-served while producing infected statoblasts as well.However, Abd-Elfattah et al. [37] argued that their ob-servation could be interpreted as a sign of a recent lossof infection. Instead, covertly infected bryozoans canproduce both infected SI [Eq. (1e)] and uninfectedstatoblasts [37]. We term φ the probability that infectedbryozoans produce uninfected statoblasts. The totalstatoblast production rate of infected bryozoans is as-sumed to be equal to that of susceptible ones. T. bryo-salmonae spores [Eq. (1j)] are produced by overtlyinfected bryozoans at a constant rate πB. A constantmortality rate for spores (μZ, i.e. the inverse of the time

Table 1 List of model variables

Symbol Variablea Dimension

BS Biomass of susceptible bryozoans [ML-3]

BC Biomass of covertly infected bryozoans [ML-3]

BO Biomass of overtly infected bryozoans [ML-3]

SS Non-infected statoblast abundance [L-3]

SI Infected statoblast abundance [L-3]

FS Susceptible fish abundance [L-3]

FE Exposed fish abundance [L-3]

FI Infected fish abundance [L-3]

FC Carrier fish abundance [L-3]

ZB Abundance of spores released by bryozoans [L-3]

ZF Abundance of spores released by fish [L-3]

Z�B ¼ βFZB Abundance of equivalent spores releasedby bryozoans

[T-1]

Z�F ¼ βBZF Abundance of equivalent spores releasedby fish

[T-1]

aAll model variables are referred to a water volume

Carraro et al. Parasites & Vectors (2016) 9:487 Page 4 of 16

span during which spores are viable) is also accountedfor.As for the fish population, no recruitment or immi-

gration are considered to take place during the warmseason. Therefore, the abundance of susceptible fish[Eq. (1f )] decays monotonically throughout the sea-son, due to natural mortality (at a rate μF, defined asthe inverse of the average lifespan of a fish) and in-fection from bryozoan-released spores ZB. The termβFFSZB in Eqs. (1f ) and (1g) represents the flux offish from the susceptible to the exposed compart-ment, with βF being the exposure rate of fish tobryozoan-released spores. Exposed fish have alreadycontracted the disease but are not yet infective. Thedecrease in the abundance of these fish, besides nat-ural mortality, is ruled by the temperature-dependentrate h [Eq. (1g)]; namely, 1/h is the average time forthe development of the disease in fish. The parameter hincreases with increasing temperature [48]. We assumethat a fraction ε of fish exiting from the exposed class doesnot show clinical symptoms and is not subject to PKD-related mortality, thus directly entering the carrier class.The remaining part (1 − ε) becomes infected. The abun-dance of infected fish [Eq. (1h)] decreases because of bothnatural and PKD-caused mortality. The latter is assumedto occur at a rate a that is positively correlated withtemperature [14, 15]. In addition, infected fish can enterthe carrier compartment at a rate γ [Eq. (1i)], which corre-sponds to the inverse of the average duration of the acutephase of the infection. Fish infected by the parasite thatdid not die in the first year reportedly continue to releaseinfective spores for a period up to two years [43] andno evidence of infection clearing has been observed.However, field and experimental observations on theduration of this carrier stage are still scarce and fur-ther experiments are underway (Strepparava, personalcommunication). Therefore we explore the possibility thatcarrier fish may recover by introducing a recovery rate ζ.Evidence of possible immunity is scant (e.g. Foott &Hedrick observed immunity to second infection in rain-bow trout [49]), thus, as a safe assumption, recovered fishare assumed to enter the susceptible compartment again[Eq. (1f)]. Spores [Eq. (1k)] are released by both infectedand carrier fish at rates πF and κ πF, respectively, with κbeing an appropriate coefficient ranging from 0 to 1.Spore decay is accounted for through the parameter μZ.Since the dynamics of disease transmission between

bryozoan and fish are driven by the product between theconcentration of spores (ZB and ZF) and the rates (βFand βB) at which susceptible organisms are actuallyexposed to the infectious agents, we can introducetwo new state variables ZB

* = βFZB and ZF* = βBZF.

These new quantities are termed equivalent spores:namely, ZF

* (ZB* ) [T−1] is the concentration of spores

needed to infect a unit concentration of susceptiblebryozoan biomass (a single susceptible fish in a unitwater volume) per unit time. With this definition, theexposure rates βF and βB can be discarded and twosynthetic contamination rates πB

* = βFπB [L3 M-1 T-2]and πF

* = βBπF [L3 T-2] are introduced. Note that bothexposure and contamination rates as defined in model(1) are hardly measurable and would most likely needto be calibrated by contrasting model simulationsagainst experimental or field data. The new parameterdefinitions thus reduce the number of parameters ofthe model and make the comparison between dataand model predictions easier and more robust. Thenew set of equations accounting for the rescaled statevariables therefore reads:

dBS

dτ¼ gSðBS;BC ;BO;TÞBS−Z�

FBS þ ψ BC þ BOð Þ;

ð3aÞ

dBC

dτ¼ gCðBS;BC ;BO;TÞBC þ Z�

FBS− dCO Tð Þ þ ψ½ �BC

þdOC Tð ÞBO;

ð3bÞ

dFS

dτ¼ −μFFS−Z�

BFS þ ζFC ; ð3cÞ

dFE

dτ¼ Z�

BFS− μF þ h Tð Þ½ �FE ; ð3dÞ

dZ�B

dτ¼ π�

BBO−μZZ�B; ð3eÞ

dZ�F

dτ¼ π�

F FI þ κFCð Þ−μZZ�F : ð3fÞ

These equations, coupled with Eqs. (1c), (1d), (1e) (1h)and (1i) constitute the within-season model hereafterapplied.

Between-season modelThe following difference equation system relates the stateof the model variables at the end of a season (y; τ1) withthat at the beginning of the following season (y + 1; τ0).

BS yþ 1; τ0ð Þ ¼ σBBS y; τ1ð Þ þ νSS y; τ1ð Þ; ð4aÞ

BC yþ 1; τ0ð Þ ¼ σBBC y; τ1ð Þ þ σOBO y; τ1ð Þ þ νSI y; τ1ð Þ;ð4bÞ

BO yþ 1; τ0ð Þ ¼ 0; ð4cÞ

SS yþ 1; τ0ð Þ ¼ 0; ð4dÞ

SI yþ 1; τ0ð Þ ¼ 0; ð4eÞ

Carraro et al. Parasites & Vectors (2016) 9:487 Page 5 of 16

FS yþ 1; τ0ð Þ ¼ σFFS y; τ1ð Þ þ f FðFS y; τ1ð Þ; FE y; τ1ð Þ;FI y; τ1ð Þ; FC y; τ1ð ÞÞ;

ð4fÞFE yþ 1; τ0ð Þ ¼ 0; ð4gÞ

FI yþ 1; τ0ð Þ ¼ 0; ð4hÞ

FC yþ 1; τ0ð Þ ¼ σF pEFE y; τ1ð Þ þ pIFI y; τ1ð Þ þ FC y; τ1ð Þ½ �;ð4iÞ

ZB yþ 1; τ0ð Þ ¼ 0; ð4jÞ

ZF yþ 1; τ0ð Þ ¼ 0: ð4kÞ

At the beginning of a new season (y + 1; τ0), no overtlyinfected bryozoans are present [Eq. (4c)]; the same con-dition applies for statoblasts [Eqs. (4d) and (4e)] andspores [Eqs. (4j) and (4k)]. Statoblasts surviving for morethan one year are here neglected for the sake of simpli-city. The biomass of susceptible bryozoans [Eq. (4a)] iscomposed of a fraction σB of the susceptible bryozoanbiomass at the end of the previous season (y; τ1) thatmanaged to survive over winter, and of newly hatchedcolonies from the uninfected statoblasts released duringthe previous season. The parameter ν is defined as themean amount of bryozoan biomass produced by a singlestatoblast.The population of covertly infected bryozoan at the

beginning of a new season [Eq. (4b)] is given by the sumof three contributions: the fraction of the covertly in-fected biomass at the end of the previous season thatsurvives over winter (with probability σB), the fraction ofovertly infected bryozoans at time (y; τ1) with survivalprobability σO, the newly established colonies generatedby infected statoblasts, according to the parameter ν.As for fish, we assume that a fraction σF of susceptible

and carrier organisms at the end of the previous seasonsurvives over winter [Eqs. (4f ) and (4i)]. Exposed and in-fected fish either die or enter the carrier class. This ismodelled by computing additional coefficients pE, pI ac-counting for natural and PKD-induced deaths:

pI ¼γ

μF þ a þ γ; pE ¼ h

μF þ hεþ 1−εð Þ γ

μF þ a þ γ

� �:

ð5Þ

Specifically, pE (pI) is the probability that an exposed(infected) fish survives and enters the carrier class in thefirst period of the winter season. In Eq. (5) â and ĥ arethe rates of PKD-caused mortality and disease devel-opment averaged over the first period of the winterseason, respectively. For the sake of simplicity, we assume

a ¼ a T� �

and h ¼ h T� �

, with T being a representativevalue of temperature over the considered period.

Fish reproduction occurs during winter; we assumethat the amount of newborn uninfected fish (termed fF)depends on the total fish population at (y; τ1):

f F ~F� �

¼ η ~F exp −ξ ~F� �

; ð6Þ

where ~F ¼ FS y; τ1ð Þ þ pEFE y; τ1ð Þ þ pIFI y; τ1ð Þ þ FC

y; τ1ð Þ. Eq. (6) assumes a Ricker growth model [46]. Theparameter η is the baseline fertility rate, i.e. the averagenumber of offspring produced by a single fish when thetotal fish population size is low, while ξ determines thestrength of density dependence. For the aforementionedreasons, at the beginning of a season there are neitherexposed nor infected fish [Eqs. (4g) and (4h)]. Table 2summarizes all parameters of systems (1) and (4).

Model simulationIn order to understand possible patterns of disease dy-namics, we ran model simulations. The feasible range ofseveral model parameters was estimated based on litera-ture values and experts’ knowledge. Reasonable valueswere assumed for the remaining ones. Reference param-eter values and feasible ranges are reported in Table 3.In the absence of detailed information about recoverymechanisms for fish and bryozoans, we assumed slowrecovery rates as reference values (average recovery timeequal to half of the lifetime for fish and half of the yearlyproliferation period for bryozoan) and explored awide range of possible values, including the absenceof infection-clearing mechanisms (ψ = ζ = 0). As fortemperature-dependent parameters, linear (r, rO) orparabolic (dCO, dOC, h, a) relationships have been as-sumed; a superlinear dependence on T for the latterparameters was deduced from literature review (seereferences in Table 3). The functional forms used areillustrated in Fig. 2. Note that, for these parameters,the corresponding reference values are obtained fromthe relationships of Fig. 2 by assuming T = 15 °C. Forthe numerical simulations, we use a time series ofstream water temperature measured in River Langeten,Switzerland, in the period 2002–2013 (data provided bythe Swiss Federal Office for the Environment - FOEN).

Analysis of the within-season modelParasite invasion in a previously PKD-free system mostlikely occurs during the warm season. For this reason,we first focus our analysis on the continuous, within-season model and investigate the conditions underwhich the introduction of the parasite in a fully suscep-tible population of fish and bryozoans (say through theintroduction of spores, infected fish or infected bryo-zoans) leads to an outbreak of PKD. To this end, weanalyze the linear stability of the disease-free equilibrium(DFE) in a simplified model that disregards population

Carraro et al. Parasites & Vectors (2016) 9:487 Page 6 of 16

dynamics of both fish and bryozoans (i.e. both popula-tion sizes are treated as model parameters). Technicaldetails are provided in Additional file 1. The instabilityof the within-season DFE would indicate long-term per-sistence of the parasite only if model parameters werekept constant at the value used for the computationof the stability criterion. However, in our system,population dynamics and water temperature variations

constantly affect parameters values. Also, the alterna-tion of warm and cold seasons, each endowed withdifferent eco-epidemiological processes, de facto pre-vents a stable, nontrivial steady-state from beingestablished in the system. Therefore, the instability ofthe within-season DFE can be seen as a useful indica-tor only of the short-term invasibility of the system.To investigate long-term parasite persistence and theestablishment of endemic transmission conditions, amore complex analysis that accounts for both sea-sonal forcing and intra- and inter-seasonal populationand epidemiological dynamics is required, as illus-trated in the following section.

Stability analysis of the full modelAs the seasonal cycle of water temperature criticallycontrols PKD transmission and bryozoan population dy-namics, we study the stability of the disease-free trajec-tory (DFT), a succession of system states in which thedisease is not present and endemic transmission is notpossible. To achieve this aim, we define a Poincaré map

Table 3 Parameter ranges and reference values

Parameter Range Reference value Unit Source

r 0.01 − 0.1 0.06 d-1 [17]

rO 0.005 − 0.05 0.03 d-1 [17]

ψ 0.01 d-1

ρ− 1 20 gm-3

dOC 0.02 − 0.2 0.032 d-1 [18]

dCO 0.02 − 0.2 0.072 d-1 [18]

a 0.015 − 0.1 0.036 d-1 [14, 15]

h 0.01 − 0.075 0.027 d-1 [6, 19, 48]

γ 0.01 − 0.05 0.02 d-1 [43]

ζ 0.001 d-1 [43]

ε 0.1 –

πB* 0.005 m3 g-1 d-2

πF* 0.1 m3 d-2

κ 0.2 –

μF−1 ~ 5 years 2000 d

μZ−1 ≤ 24 h 0.75 d [19, 62]

σB 0 − 0.3 0.1 –

σO 0 − 0.1 0.05 –

σF 0.9 –

η 1 –

ξ− 1 ≤ 1.5 0.5 m-3 [63]

ν 0.035 g

fB 0.1 g-1 d-1

φ 0.55 − 0.8 0.7 – [37]

T 10 °C

Table 2 List of parameters

Parameter Definition Dimension

Bryozoans

Constant

ρ Inverse of carrying capacity [M-1L-3]

βB Exposure rate [L3T-1]

φ Fraction of BC generating uninfected statoblasts [−]

ψ Recovery rate [T-1]

πB Rate of contamination operated by BO [M-1T-1]

σB Probability of survival over winter for BS, BC [−]

σO Probability of survival over winter for BO [−]

ν Biomass generated by one statoblast [M]

Temperature-dependent

r Baseline growth rate of BS, BC [T-1]

rO Baseline growth rate of BO [T-1]

dCO Rate of covert-to-overt transition [T-1]

dOC Rate of overt-to-covert transition [T-1]

Time and temperature-dependent

fB Statoblast production rate [M-1T-1]

Fish

Constant

μF Natural mortality rate [T-1]

βF Exposure rate [L3T-1]

ε Fraction of acute infections [−]

γ Rate of recovery from acute infection [T-1]

ζ Rate of complete recovery [T-1]

πF Rate of contamination operated by FI [T-1]

κ Relative rate of contamination operated by FC [−]

σF Probability of survival over winter [−]

η Baseline reproduction rate [−]

ξ Strength of density dependence [L-3]

Temperature-dependent

h Rate of development of the disease [T-1]

a PKD-caused mortality rate [T-1]

Parasite

μZ Spore decay rate [T-1]

Rescaled parameters

π�B ¼ βFπB Synthetic rate of contamination operated by BO [L3M-1T-2]

π�F ¼ βBπF Synthetic rate of contamination operated by FI [L3T-2]

Carraro et al. Parasites & Vectors (2016) 9:487 Page 7 of 16

coupling between-season update with the within-seasondynamics, with the latter being described through an in-tegral operator derived from Floquet theory [50–52], amathematical framework that allows the analysis of sys-tems of periodically forced differential equations. As thistheory assumes periodic forcing, the time evolution ofwater temperature is approximated via a sinusoidal sig-nal with yearly period.

Disease-free trajectoryThe DFT is defined as a discrete-continuous trajectorydescribing the yearly periodic evolution of biomass orabundances for the uninfected classes (BS, SS, FS) in theabsence of PKD. Therefore, we first focus on a reducedsystem where only susceptible compartments are takeninto account. Let xS(y; τ) = {BS(y; τ); SS(y; τ); FS(y; τ)} bethe state of this system at season y and time τ. Thetime-evolution of this system can be represented asfollows:

xS y; τ0ð Þ→v ⋅ð ÞxS y; τ1ð Þ→w ⋅ð Þ

xS yþ 1; τ0ð Þ→v ⋅ð ÞxS yþ 1; τ1ð Þ

where v(⋅) is an operator describing the continuous-timedynamics of the system from τ0 to τ1, while w(⋅) is anoperator describing the discrete-time update of the sys-tem between the end of season y and the beginning ofseason y + 1. Therefore, we obtain:

xS yþ 1; τ0ð Þ ¼ w v xS y; τ0ð Þð Þð Þ ¼ w∘vð Þ xS y; τ0ð Þð Þ;

where w ∘ v defines a so-called Poincaré map [53]. Afixed point of the Poincaré map is defined as an equilib-rium xS y; τð Þ such that xS y; τ0ð Þ ¼ xS yþ 1; τ0ð Þ ¼ w∘vð ÞxS y; τ0ð Þð Þ.The disease-free trajectory xdf y; τð Þ for systems (1)

and (4) is defined under the hypothesis that water

temperature can be approximated by a sinusoidal func-tion of time, with period equal to one year. We thereforeobtain

xS y; τð Þ ¼ BS;0

ρBS;0 þ 1−ρBS;0� �

exp −R τð Þð Þ; 0; FS;0exp −μF τ−τ0ð Þð Þ

( );

ð7Þ

where

R τð Þ ¼Z τ

τ0

rdt

and BS,0 (FS,0) is the susceptible bryozoan (fish) populationat τ0 along the DFT. Note in fact that xS τ0ð Þ ¼BS;0; 0; FS;0

� �. BS,0 is the solution of the following trascen-

dental equation:

1−σB

ρBS;0 þ 1−ρBS;0� �

exp −R τ1ð Þð Þ

−Z τ1

τ0

νf BρBS;0 þ 1−ρBS;0

� �exp −R τð Þð Þ

dτ ¼ 0;

The value of BS,0 can be obtained numerically for agiven set of parameters. Instead, an analytical expressioncan be found for FS,0:

FS;0 ¼ ξ−1lnη

exp μF τ1−τ0ð Þð Þ−σF

� �exp μF τ1−τ0ð Þð Þ:

ð8Þ

Therefore, xdf y; τð Þ is the trajectory such that its sus-ceptible components are equal to (7), while its infectedcomponents xI = {BC; BO; SI; FE; FI; FC; ZB

* ; ZF* } are null.

Note that Eq. (8) requires η > ηmin = exp(μF(τ1 − τ0)) − σFin order to avoid extinction of fish. For the referenceparameter set of Table 3, one gathers ηmin ≈ 0.205.

Fig. 2 Functional forms for temperature-dependent parameters. The maximum values of these parameters are set to match the upper limits proposedin Table 3. Furthermore, we assume f B ¼ 0:005 T ½oC� þ 0:00025 τ ½d�, where τ is the time elapsed since January 1

Carraro et al. Parasites & Vectors (2016) 9:487 Page 8 of 16

Stability of the disease-free trajectoryIn order to study the stability of the DFT xdf y; τð Þ ofmodel (1) and (4) with respect to small perturbations,we refer to the reduced system where only infected com-partments are considered. Hence, xI(y; τ) denotes thestate of such a system, whose time evolution is repre-sented as follows:

xI y; τ0ð Þ→V ⋅ð ÞxI y; τ1ð Þ→W ⋅ð Þ

xI yþ 1; τ0ð Þ→V ⋅ð ÞxI yþ 1; τ1ð Þ:

ð9Þ

V and W are operators describing the evolution of thelinearized system either within or between seasons. Notethat the fixed point for this system is the null vector.The propagation matrix V is given by the solution of

the following matrix ODE system:

dZdτ

¼ A τð ÞZ; ð10Þ

where A(τ) is a matrix obtained from the Jacobian of thewithin-season system (1), evaluated along the trajectoryxdf y; τð Þ, where all rows and columns related to the sus-ceptible compartments have been removed:

A τð Þ ¼

A11 dOC 0 0 0 0 0 BS

dCO A22 0 0 0 0 0 01−φð Þf B 0 0 0 0 0 0 0

0 0 0 −μF−h 0 0 FS 00 0 0 1−εð Þh A55 0 0 00 0 0 εh γ −μF−ζ 0 00 π�

B 0 0 0 0 −μZ 00 0 0 0 π�

F κ π�F 0 −μZ

2666666666664

3777777777775;

where A11 = r(1 − ρ Bs) − (dCO + ψ), A22 = rO(1 − ρ Bs) −(dOC + ψ), A55 = − μF − a − γ. Equation (10) must be in-tegrated from τ0 to τ1, with initial condition Z(τ0) = I(identity matrix). We thus have V = Z(τ1).Matrix W can be obtained from the Jacobian matrix

of the between-season system (4), by disregarding allrows and columns related to susceptible compart-ments. W is a sparse matrix of order 8 with the fol-lowing non-null entries: W11 = σB, W12 = σO, W13 = ν,W64 = pEσF, W65 = pIσF, W66 = σF.The stability of the disease-free trajectory of system

(1) and (4) corresponds to the stability of the fixedpoint xI = 0 of the Poincaré map defined in (9), i.e.xI(y + 1; τ1) = VW xI(y; τ1). Therefore, the necessaryand sufficient condition for the exponential instabilityof the DFT reads

λmax ¼ maxjλðVWÞj > 1; ð11Þ

where λ are the eigenvalues of matrix VW. When condi-tion (11) is met, the parasite can invade the system. Wetherefore focus our analysis on λmax, i.e. the maximummodulus of the eigenvalues of VW. For a given set of

parameters, the corresponding value of λmax can becomputed numerically.

Sensitivity analysesTo understand how model parameters affect parasiteinvasibility and long-term PKD impact, we performedsensitivity analyses. Specifically, we investigate the effectof parameters on the value of λmax and on the PKD-induced fish loss, here estimated as the percentage offish at the end of the season with respect to the fishpopulation size if the disease were absent. Computationsare run by varying two focus parameters at a time whilekeeping the others at their reference value, as specifiedin Table 3. Temperature-dependent parameters areexpressed via the functional forms of Fig. 2. The effectof these parameters is explored by varying their value at15 °C, while keeping their minimum value at 0 °C (at25 °C for dOC) constant. For parabolic functional forms,the null derivative at their minimum is also kept con-stant. The sinusoidal signal of water temperature is de-rived from the time series shown in the top panel ofFig. 3. Seasons are assumed to start on April 1 and lastfor 200 days. The effect of water temperature is alsoinvestigated by varying both the yearly mean value andthe relative mid-amplitude of the sinusoidal signal, whileusing the relationships of Fig. 2.

ResultsModel simulationFigure 3 shows an example of model simulation where itis assumed that at the beginning of the first season 1 %of the bryozoans are covertly infected and 1 % of the fishbelong to the carrier class. The remaining fractions ofthe host populations are susceptible. This model settingmimics the invasion of the parasite in a fully susceptiblesystem. The total bryozoan population tends to reachthe carrying capacity ρ-1 towards the end of the season.Overt stages of infection are absent in early spring butpeaks are observed in summer. The irregular shape ofthe curves for BC and BO mirrors the variability of watertemperature. Regarding fish, peaks of infection are ob-served during summer. After a few years, prevalencereaches about 80 % at the end of the summer, where fewsusceptible fish are present and most of the survived in-dividuals belong to the carrier class. This is in agreementwith the fact that, although young-of-the-year fish sur-viving the infection are not likely to develop clinicalPKD in the following year, the parasite can remain viablein the host for several seasons after initial exposure[16, 43, 54]. In this model simulation, some 28 % ofthe fish population which is alive at τ0 dies duringthe season because of PKD. About ten years after theinitial invasion, the disease becomes endemic and theseasonal state variables’ trajectories remain virtually

Carraro et al. Parasites & Vectors (2016) 9:487 Page 9 of 16

unchanged. Overall, the model reproduces patterns ofdisease spread in bryozoan and fish populations whichare in good qualitative agreement with evidence fromthe literature and field observations (see [19]).

Analysis of the within-season modelThe analysis of the within-season model (Additional file 1)reveals two types of instability of the DFE. The first is re-lated to the sign of the quantity

T ¼ ψ2 þ dCO þ dOC− r þ rOð Þ 1−ρBSð Þ½ �ψ− 1−ρBSð Þ� rOdCO þ rdOC þ rrO 1−ρBSð Þ½ �;

which only depends on parameters regarding the bryo-zoan sub-model (note that the biomass of susceptiblebryozoan BS is assumed as a model parameter for thisanalysis). When T < 0 , the DFE is unstable and theparasite can spread in the bryozoan population even inthe absence of the fish host. Notably, if bryozoan cannotrecover (ψ = 0), T is always negative, as long as the bryo-zoan biomass is lower than the carrying capacity. When

T > 0, the parasite needs to cycle between the two hoststo possibly invade the system and the instability of theDFE occurs when the reproductive number

R ¼ BSFSdCOhπ�Bπ

�F εκaþ κγ þ 1−εð Þζ þ 1þ εκ−εð ÞμF½ �

T hþ μFð Þ aþ γ þ μFð Þ ζ þ μFð Þμ2Z

is greater than unity. Starting from the reference param-eter set, R increases as parameters FS, dCO, h, πB

* , πF* , in-

crease, and decreases as parameters ρBS, dOC, a, ψincrease (see Additional file 1). Figure 4 illustrates howR evolves during a season, as temperature affectsparameter values, and the population sizes of fish andbryozoan follow the DFT. The effect of temperaturedominates and maximizes the reproductive number,and the related risk of an outbreak, during the warm-est period. With the reference parameter set, T > 0and R > 1 throughout the season: the parasite requiresthe fish host to spread into the system and the DFEis always unstable. Figure 4 also shows how reducingone of the transmission parameters (πB

* or πF* ) or the

Fig. 3 Model simulation. Evolution of the state variables of model (1) and (4) in a 12-years-long simulation. Seasons start on April 1 and last for200 days. Black solid lines represent the time evolution of the overall bryozoan and fish populations; the dashed line indicates fish abundance ifPKD is absent. Initial condition: BS = 0.495ρ-1, BC = 0.005ρ-1, FS = 0.99ξ-1, FC = 0.01ξ-1; other state variables are null. Temperature-dependentparameters are taken from Fig. 2; other parameters are set to their reference values (reported in Table 3). The water temperature time series usedfor the simulation is reported in the top subplot

Carraro et al. Parasites & Vectors (2016) 9:487 Page 10 of 16

fish population size FS by a factor 10 and 20 can curbthe reproductive number below unity during the cool-est periods and throughout the season, respectively.

Sensitivity analyses of the full modelParasite invasionAn exploration of the values of λmax over wide ranges ofthe parameter space is reported in Fig. 5. Numerical re-sults show high invasibility of the system (λmax > 1) forwide ranges of realistic parameters. The DFT becomesstable if one of the contamination rates (πB

* and πF* ,

Fig. 5a) or if the characteristic sizes of bryozoan or fishpopulations (ρ-1 and ξ-1, Fig. 5b) are small. Recoverymechanisms (ψ and ζ, Fig. 5f ) can promote the stabilityof the DFT: in particular, the DFT is stable if infection-clearing in bryozoans is fast enough, whereas if only ζ isincreased, the persistence of the infection in bryozoanshinders the stability of the DFT. The DFT is predictedto be stable also if the rate of transition from overt tocovert infection dOC is considerably higher than the anti-thetic rate dCO (Fig. 5c), although this circumstance oc-curs for unlikely values of these parameters. Finally,stability of the DFT is observed for extremely low valuesof the fish reproduction and recovery rates (η and γ,Fig. 5e).When the DFT is unstable, the maximum modulus

λmax of the eigenvalues of matrix VW can be interpretedas the rate at which model trajectories depart from the

disease-free trajectory. λmax thus represents a measureof how fast the parasite spreads into a susceptible popu-lation. When λmax is slightly greater than unity, the out-break develops slowly and stochastic effects (e.g. due tothe demographic stochasticity of fish and bryozoan in-fected populations), which are not accounted for in thecurrent model formulation, may lead to the extinction ofthe disease. When the DFT is stable, the closer λmax isto unity, the slower perturbations to the DFT fade outand the system returns to a disease-free state. Therefore,λmax can also be seen as an approximate estimate of theactual risk of invasion. In this perspective, Fig. 5 pro-vides information on the role of the model parametersin promoting the establishment of PKD in fish popula-tions. πB

* and πF* have analogous effects in enhancing the

risk for an outbreak. ρ-1, ξ-1 and the fish baseline fertilityrate η are also positively correlated with λmax. Concern-ing the effects of transmission and mortality rates, thehighest values of λmax are found when the disease devel-opment rate h is maximum and PKD-caused mortality ais minimum (Fig. 5d). This condition maximizes thenumber of non-fatal infection in fish, thereby causing anincrease in the release of spores ZF. As for the thermalregime, the mean water temperature during the warmseason stands as the main factor controlling λmax

(Fig. 5g).

PKD-induced fish lossFigure 6 shows the results of the sensitivity analysis ofPKD impact on fish population size. The residual popu-lation size is larger (i.e. PKD impact is reduced) for lowvalues of the contamination rates and the bryozoan car-rying capacity (Fig. 6a and b), whereas no sensitivity tohigh values of these parameters is observed. The bryo-zoan baseline growth rate (Fig. 6b) has no effect onPKD-induced fish loss. Fish population is preservedwhen dOC is high and dCO is low, while the opposite casedoes not result in a severe population decay (Fig. 6c).Expectedly, there is a positive correlation between PKD-induced fish loss and PKD-induced mortality rate a(Fig. 6d). However, if a is extremely high, the immediatedeath of the host limits the transmission of the diseaseand thus its impact, a common feature of epidemio-logical models. PKD impact is enhanced by the rate ofdisease development h. Fish population decay is also en-hanced when both ε and γ are low, as this conditionmaximizes the abundance of acutely infected fish(Fig. 6e); as previously pointed out, if γ is extremely low,this effect is balanced by the augmented relative import-ance of a, which weakens PKD impact. As the recoverytime of bryozoans, ψ-1, decreases, PKD impact mono-tonically decreases (Fig. 6f ). On the other hand, PKDimpact reaches a peak for intermediate rates of fish re-covery (ζ ≈ 0.005 d-1), presumably due to concomitant

Fig. 4 Short-term risk of invasibility. Reproductive number R as afunction of time. Water temperature is approximated as a sinusoidalsignal (as for the computation of the DFT) and temperature-dependent parameters are taken as in Fig. 2. FS(τ) and BS(τ) follow theDFT. The red line refers to the reference parameter set reported inTable 3. The blue and the green lines show the effect of reducing oneof the transmission parameters (πB* or πF*) or the fish population size FSby a factor 10 and 20, respectively

Carraro et al. Parasites & Vectors (2016) 9:487 Page 11 of 16

Fig. 6 Sensitivity analysis: PKD-induced fish loss as a function of model parameters πB* vs. πF* (a); ρ vs. r (b); dOC vs. dCO (c); a vs. h (d); γ vs. ε (e); ψvs. ζ (f); temperature (g). Simulations are run until convergence (100 seasons), with each season lasting for 200 days. Colors refer to the percent-ages of the population size at the end of the last season with respect to the same quantity calculated along the disease-free trajectory [computedas in Eqs. (7) and (8)]. For example, 50 % means that at the end of the last season the population size is half of the population that would have survivedif the disease were absent. State variables at the beginning of the first season are set as in the model simulation of Fig. 3. Pink dashed lines identifyfeasible parameter ranges. Black dots refer to the reference parameter set. All rates are represented as mean times. With regards to temperature-dependent parameters, their value at 15 °C is displayed. Black solid lines in g identify levels of mean temperature during the warm season

Fig. 5 Sensitivity analysis: Parasite invasion. Values of λmax as a function of model parameters πB* vs. πF

* (a); ρ vs. ξ (b); dOC vs. dCO (c); a vs. h (d); γvs. η (e); ψ vs. ζ (f); temperature (g). Pink dashed lines identify feasible parameter ranges. Black dots refer to the reference parameter set. For thesake of clarity, all rates are expressed as mean times (i.e. by their inverse). With regards to temperature-dependent parameters, their value at 15 °Cis displayed. Black solid lines in g identify levels of mean temperature during the warm season

Carraro et al. Parasites & Vectors (2016) 9:487 Page 12 of 16

high level of contamination and increased availability ofsusceptible fish.As expected, higher water temperatures promote

higher fish mortality (Fig. 6g). In particular, PKD impactis mostly related to the mean temperature during thewarm season, whereas the amplitude of the temperaturesinusoidal signal plays a lesser role. For instance, assum-ing a relative mid-amplitude of 0.5, an increase of 5 °C(from 10 to 15 °C) in the seasonal mean watertemperature would produce an overall reduction of 26 %of the total fish population.

DiscussionAccording to the modelling framework developed in thiswork, the disease-free trajectory is found to be unstableover wide ranges of the ecological and epidemiologicalparameters. This property means that the introductionof the parasite in a fully susceptible community of sal-monids and bryozoans would very likely lead to a PKDoutbreak and to long-term parasite establishment. Theability of the parasite to cycle between covert, non-virulent, and overt, transmissive phases in the bryozoanhost allows the parasite to produce large numbers oftransmissive stages without compromising the suscep-tible host population. Similarly, long-term infections introut are also responsible for continuous infections ofnaive bryozoan hosts and promote parasite persistence.This result holds for the deterministic model (1) and (4).Stochastic effects, currently not accounted for in themodel formulation, could actually prevent the invasion.Our findings obviously depend on the current know-ledge of the transmission modes of PKD (and on how ithas been translated into mathematical terms), and thelack thereof. Indeed, the analyses of the within-seasonand the full models show how recovery mechanisms forboth bryozoans and fish reduce the risk of outbreak andcan make the disease-free system stable. However, know-ledge and investigations on these critical processes arestill scant. While the existence of a recovery dynamic forbryozoans is currently unknown (but see [37]), someexperiments conducted on rainbow trout [55] actuallyrevealed hints of a possible recovery from infection insalmonids. Further studies on this topic are needed tobetter elucidate the mechanisms underlying the per-sistence of T. bryosalmonae in fish and bryozoancommunities. Overall, these results underline the rea-sons for the emerging status of PKD throughoutEurope - the extremely successful invasion mechan-ism of the parasite may have facilitated its historicalspread in European streams, and the current changesin climate and other environmental conditions arecausing it to proliferate.Our results highlight how both outbreak risk and the

long-term persistence of the parasite are critically

enhanced by warmer water temperature. The design ofpossible control strategies (e.g. the control of the popu-lation of one of the host or the interactions amongthem) should thus take into account that warmer pe-riods pose a higher risk of a PKD outbreak. On the otherhand, the effect of temperature offers the opportunity todesign alternative intervention strategies aimed at con-trolling stream water temperature through e.g. tree shad-ing or the selective release of cold water from upstreamreservoirs. The analysis of the within-season model alsoshows that, for certain parameter combinations, theparasite can initially spread even in the absence of thefish host. This possible behavior is determined by thefact that the model assumes that infected bryozoan growand that T. bryosalmonae can simultaneously proliferateinside them. This specific dynamics should be betterscrutinized with additional experimental studies becausethe possibility of epidemics hosted solely by the bryo-zoan population has relevant implications for controlstrategies. It implies, for instance, that strategies focusingonly on the fish population or on limiting the interac-tions between the two hosts might not be effective,under certain conditions, to prevent the invasion of theparasite.Simulations and sensitivity analyses provided insights

about the role and importance of the different parame-ters in parasite invasion and outbreak severity. In par-ticular, the analysis of PKD-induced fish loss highlightsthe crucial role of the parameters that govern the transi-tion between epidemiological classes in determining theimpact of PKD outbreaks when the disease is endemic.While the carrying capacity of bryozoans and the con-tamination rates are the main factors controlling the sta-bility of the disease-free trajectory, they appear lessrelevant in determining PKD impact in endemic settings.The underlying reason for this result is that these pa-rameters solely control the rate at which susceptible fishare exposed to the parasite, whereas the infection de-velops over time scales ruled by the parameters govern-ing the transition between epidemiological classes. Apromising aspect is the fact that such parameters andtheir temperature dependence could be estimated in la-boratory experiments by exposing hosts to the parasiteand by monitoring the development of their infectionstatus. Similar studies have already been carried out (seecorresponding references in Table 3). More are needed,however, to better elucidate the role of temperature onPKD dynamics. It should be noted, in fact, that most ofthe available literature on PKD is based on rainbow trout(Oncorhynchus mykiss), which is known to be a dead-end host for T. bryosalmonae, as this species does notallow the completion of the parasite cycle by further in-fecting bryozoans [56]. Conversely, only few studies havefocused on brown trout [43, 56, 57].

Carraro et al. Parasites & Vectors (2016) 9:487 Page 13 of 16

While it could be relevant to experimentally assesshow the release of spores (by both infected fish andbryozoans) varies with temperature, the estimation ofthe actual value of the release rate may be less im-portant. Indeed, the analysis of the model reveals thattransmission dynamics are controlled by the productbetween contamination and exposure rates. The lattercan hardly be estimated under field conditions as theydepend on the probability of a successful encounterbetween viable spores and hosts. Therefore, efforts toprecisely estimate contamination rates would be frus-trated by the large uncertainties associated to expos-ure rates. The proposed rescaled model (3) featurestwo parameters (one per host) that represent theproduct between contamination and exposure rates.Such parameters are key to disease dynamics, as dis-cussed above. However, reasonable values can hardlybe estimated a priori, and would need to be cali-brated for each case study by contrasting model sim-ulations with experimental or field data.Knowledge and literature on bryozoans are rather

scant compared to the vast and traditional literature onpopulation dynamics and habitat distribution of salmo-nids. To improve our understanding of PKD as an emer-ging disease, such knowledge gap must be filled. Inparticular, knowledge of bryozoan habitat suitabilityneeds be improved for an effective mapping on the riskof PKD invasion. This task could be achieved by usingspecies distribution models [58] to relate the presence/absence of bryozoans to environmental variables (e.g.hydrological conditions, characteristics of the river-bedand banks, water quality, temperature). The few existingstudies (e.g. [59]) focus on lakes, while large-scale stud-ies of habitat suitability in streams and rivers are notavailable yet. Furthermore, many aspects of the relation-ship between infection status and bryozoan proliferationare still to be elucidated: for example, population geneticdiversity of bryozoans may be linked to their ability toresist to infection; infective stages may propagate in par-tially infected bryozoan zooids even in the absence offurther exposure to T. bryosalmonae.Our results need be accompanied by an assessment of

the limits of the model. It is acknowledged that young-of-the-year fish are highly likely to contract PKD whenexposed to the parasite for the first time [60]. However,for the sake of simplicity, the age structure of the fishpopulation has not been accounted for in the currentmodel formulation. The fish population can indeed besplit into age-specific classes (e.g. juveniles and adults),each with its own epidemiological compartments withdifferent mortality, recovery and infection rates. To betterelucidate this aspect, experimental investigations are cur-rently being conducted to assess possible age-structure ef-fects on T. bryosalmonae spore load in freshwater systems

(Strepparava, personal communication). Our epidemio-logical model also relies on the simplifying hypothesisthat all statoblasts hatch at the beginning of the fol-lowing season. Hence, another aspect needing furtherinvestigation is the possibility of the existence of astatoblast bank (as suggested by Freeland et al. [61]).A statoblast bank could form on the bed of the waterbody if statoblasts were able to survive and maintaintheir infection status for more than one season. Dor-mant infected statoblasts could thus trigger new PKDoutbreaks even if the disease were no longer presentin fish and bryozoans.Although the described transmission processes imply

proximity between spores and hosts, several mechanismsallow long-distance spreading of PKD along river net-works and lakes, among which bryozoan fragmentation,buoyancy of surfaces with attached colonies, fish migra-tion, and hydrodynamic dispersal of T. bryosalmonaespores and bryozoan statoblasts may play a remarkablerole [19]. To understand the role played by spatial pro-cesses in PKD transmission, the local model presentedherein should be extended by considering the hydro-logical connectivity among different river reaches. Inparticular, modelling efforts at the river network scales(corroborated by field analyses) should try to understandwhether PKD transmission is a spatially diffuse processor rather concentrated in transmission hot-spots. Suchhot-spots could be represented by favorable habitats forboth salmonids and bryozoans. The proximity of thehosts in these habitats could promote high transmissionrates that may in turn be able to sustain the infection atthe network scale through dispersal of fish and hydro-dynamic transport of parasite spores. The possible iden-tification of such hot-spots could offer alternativestrategies to control the disease in the wild, like the con-finement or the removal of one of the hosts in the iso-lated hot-spots.

ConclusionsWe have developed a discrete-continuous hybrid epi-demiological model that incorporates the currentknowledge about T. bryosalmonae life-cycle and PKDtransmission in wild salmonid populations. Our ana-lysis showed that, for realistic parameter ranges, para-site invasion and long-term establishment are verylikely, leading to a high risk of PKD outbreaks in sus-ceptible environments. When the disease is endemic,the impact of PKD outbreaks is shown to be mostlycontrolled by the rates at which the disease developsin the fish population. These parameters can be esti-mated via experimental and field studies. Furtherstudies are needed to gain additional insights on keydisease dynamics, in particular concerning possible re-covery in both hosts.

Carraro et al. Parasites & Vectors (2016) 9:487 Page 14 of 16

Additional files

Additional file 1: Linear stability analysis of the within-season system.(PDF 1644 kb)

Additional file 2: MATLAB script performing simulations of theepidemiological model. It consists of 4 files. (ZIP 24 kb)

AcknowledgementsTwo anonymous Reviewers are acknowledged for their constructivecomments and suggestions.

FundingLC, HH, NS, TW, JJ, AR and EB acknowledge the support provided by theSwiss National Science Foundation through the Sinergia projectCRSII3_147649: “Temperature driven emergence of Proliferative KidneyDisease in salmonid fish - role of ecology, evolution and immunology foraquatic diseases in riverine landscapes”.

Availability of data and materialsThe data supporting the conclusions of this article are included within thearticle and its additional files. A code that performs simulations of theepidemiological model is available in Additional file 2.

Authors’ contributionsAll authors conceived the study. LC, LM, MG, AR and EB developed theepidemiological model. LC, LM, MG performed the stability analysis. HH, NS,TW and JJ provided feedbacks on T. bryosalmonae life cycle and PKDtransmission. LC, LM, AR and EB wrote the first draft. All authors providedcomments and corrections on the draft and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Consent for publicationNot applicable

Ethics approval and consent to participateNot applicable

Author details1Laboratory of Ecohydrology, École Polytechnique Fédérale de Lausanne,Station 2, Lausanne 1015, Switzerland. 2Dipartimento di Elettronica,Informazione e Bioingegneria, Politecnico di Milano, Via Ponzio 34/5, Milan20133, Italy. 3Eawag, Swiss Federal Institute of Aquatic Science andTechnology, Überlandstrasse 133, Dübendorf, 8600, Switzerland. 4Institute ofIntegrative Biology, ETH Zürich, Universitätstrasse 16, Zürich 8092,Switzerland. 5Centre for Fish and Wildlife Health, Universität Bern,Länggassstrasse 122, Bern 3012, Switzerland. 6Dipartimento di IngegneriaCivile, Edile ed Ambientale, Università di Padova, Via Marzolo 9, Padova35131, Italy.

Received: 2 August 2016 Accepted: 15 August 2016

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