LETTER Resistance, tolerance and environmental transmission dynamics

determine host extinction risk in a load-dependent amphibian

disease

Mark Q. Wilber,1* Roland A.

Knapp,2 Mary Toothman1 and

Cheryl J. Briggs1

Abstract

While disease-induced extinction is generally considered rare, a number of recently emerging infec-tious diseases with load-dependent pathology have led to extinction in wildlife populations. Trans-mission is a critical factor affecting disease-induced extinction, but the relative importance oftransmission compared to load-dependent host resistance and tolerance is currently unknown.Using a combination of models and experiments on an amphibian species suffering extirpationsfrom the fungal pathogen Batrachochytrium dendrobatidis (Bd), we show that while transmissionfrom an environmental Bd reservoir increased the ability of Bd to invade an amphibian popula-tion and the extinction risk of that population, Bd-induced extinction dynamics were far moresensitive to host resistance and tolerance than to Bd transmission. We demonstrate that this is ageneral result for load-dependent pathogens, where non-linear resistance and tolerance functionscan interact such that small changes in these functions lead to drastic changes in extinctiondynamics.

Keywords

Batrachochytrium dendrobatidis, chytridiomycosis, density-dependent transmission, frequency-dependent transmission, integral projection models, macroparasite, microparasite, Rana muscosa,Rana sierrae.

Ecology Letters (2017) 20: 1169–1181

INTRODUCTION

Disease-induced extinction of a host population is consideredrare (De Castro & Bolker 2005; Smith et al. 2006; McCallum2012). This is because in many systems disease transmission isan increasing function of the density of infected hosts (i.e.density-dependent transmission) such that decreasing hostdensity reduces disease transmission and prevents disease-induced extinctions (McCallum & Dobson 1995; Gerber et al.2005). Theoretical models suggest that to drive a host popula-tion extinct, a disease needs to have alternative transmissiondynamics such that declining host density does not preventfurther disease transmission (De Castro & Bolker 2005; Smithet al. 2006; McCallum et al. 2009). For example, frequency-dependent transmission, in which hosts have a density-inde-pendent number of contacts with other hosts per unit time(McCallum et al. 2001), or abiotic/biotic reservoirs for thepathogen are two transmission scenarios that can lead to dis-ease-induced host extinction. In both cases, decreasing hostdensity does not necessarily lead to a decrease in diseasetransmission. Despite the rarity of disease-induced host extinc-tion (Smith et al. 2006), a number of wildlife diseases, such aschytridiomycosis in amphibians, white-nose syndrome in bats,and facial tumour disease in Tasmanian devils, have recentlybeen identified as the causative agents of host declines andpopulation extinctions (Skerratt et al. 2007; Blehert et al.2008; McCallum et al. 2009).

While the characteristics of the transmission function inthese emerging infectious diseases may ultimately determinewhether a population experiences disease-induced extinction(McCallum 2012), extinction dynamics will also be influencedby how resistant (i.e. the ability of a host to reduce or elimi-nate a pathogen conditional on exposure; Boots et al. 2009;Medzhitov et al. 2012) and/or tolerant (i.e. the ability of ahost to persist with a pathogen load that is typically lethal fornon-tolerant individuals; Roy & Kirchner 2000; Medzhitovet al. 2012) a host is to the pathogen. Therefore, when manag-ing disease-induced declines and extinctions, it may be impor-tant to manage not only for the transmission dynamics, butalso the level of host tolerance and resistance in a population(Kilpatrick 2006; Langwig et al. 2015, 2017; Epstein et al.2016). However, the conditions under which it might be moreeffective to manage for resistance and tolerance instead oftransmission are currently unknown.Understanding the relative importance of resistance and tol-

erance compared to transmission in driving extinction dynam-ics has important implications for managing the emergingamphibian disease chytridiomycosis. Chytridiomycosis iscaused by the amphibian chytrid fungus Batrachochytriumdendrobatidis (Bd) and has resulted in widespread amphibiandeclines and extinctions (Daszak et al. 2003; Skerratt et al.2007). Bd is a cutaneous fungus that disrupts the osmoregula-tory ability of amphibian skin, leading to the potentially fataldisease chytridiomycosis (Voyles et al. 2007, 2009). While a

1Department of Ecology, Evolution and Marine Biology, University of Califor-

nia, Santa Barbara, CA 93106,USA

2Sierra Nevada Aquatic Research Laboratory, University of California, Mam-

moth Lakes, CA 93546,USA

* Correspondence: E-mail: mark.wilber@lifesci.ucsb.edu

© 2017 John Wiley & Sons Ltd/CNRS

Ecology Letters, (2017) 20: 1169–1181 doi: 10.1111/ele.12814

number of transmission-related factors, including an environ-mental pool of Bd zoospores and biotic reservoirs, arehypothesised to contribute to disease-induced extinction ofamphibian populations (Rachowicz & Briggs 2007; Mitchellet al. 2008; Briggs et al. 2010; McCallum 2012; Doddingtonet al. 2013), few studies have attempted to quantify the trans-mission function itself (but see Rachowicz & Briggs (2007)and Bielby et al. (2015)). Quantifying the transmission func-tion is important when considering disease-induced extinctionbecause it determines the ability of a pathogen to invade apopulation as well as drive a host population extinct (De Cas-tro & Bolker 2005; Gerber et al. 2005).In addition to the transmission function, extinction dynam-

ics in amphibian-Bd systems also depend on the dynamics ofBd load on an amphibian host (Briggs et al. 2010). VaryingBd load dynamics among amphibian populations, potentiallyinduced by varying resistance and tolerance mechanisms, canpromote population-level persistence following epizootics(Retallick et al. 2004; Briggs et al. 2010; Grogan et al. 2016;Savage & Zamudio 2016). This variation in resistance and tol-erance could be due to a number of different mechanismsincluding innate and acquired immune responses (Ellisonet al. 2015), differences in host susceptibility (Knapp et al.2016), variation in the amphibian microbiome (Harris et al.2009; Jani & Briggs 2014), temperature-dependent Bd growth(Forrest & Schlaepfer 2011; Knapp et al. 2011), and variablevirulence in Bd strains (Rosenblum et al. 2013; Jenkinsonet al. 2016). Load dynamics are important in Bd systemsbecause disease-induced mortality of amphibians is highlyload-dependent, with survival probability sometimes decreas-ing rapidly at high Bd loads (Stockwell et al. 2010; Vreden-burg et al. 2010). This attribute of load-dependent survivalleads to a simple, but largely untested hypothesis in host-pathogen systems: host populations that are either able to pre-vent large increases in pathogen load (via resistance mecha-nisms) or tolerate high pathogen loads (via tolerancemechanisms), will experience reduced disease-induced extinc-tion risk, even when the transmission rate is high. This is ageneral hypothesis for load-dependent wildlife diseases andamphibian-Bd interactions provide an ideal system in whichto test it.The above hypothesis can be phrased as the following ques-

tion: How important is transmission compared to host toler-ance and resistance for mitigating disease-induced extinction?Answering this question requires quantifying the transmissionfunction, something rarely done in amphibian-Bd systems(Kilpatrick et al. 2010). Moreover, understanding the role ofthis transmission function in the ability of Bd to invade anamphibian population will be important for accurately under-standing any subsequent Bd-induced extinctions (Gerber et al.2005). Therefore, we also ask two additional questions: Whatis the nature of the transmission function in amphibian-Bdsystems? How does this transmission function affect the abil-ity of Bd to invade an amphibian population? We use a com-bination of experiments and dynamical models to show thatempirical patterns of Bd transmission are best modelled usingan environmental Bd pool and that this pool significantlyincreases the ability of Bd to invade a population and thepopulation-level extinction risk. However, despite this large

effect of the environmental pool, we show that host resistanceand tolerance are far more influential on Bd-induced extinc-tion dynamics than transmission. This is likely a general prop-erty of load-dependent diseases in which non-linear resistanceand/or tolerance functions interact such that managing forresistance and tolerance can more effectively mitigate disease-induced extinction than managing for transmission.

METHODS

To answer the questions posed above, we focused on Bd-induced extinction dynamics in the mountain yellow-leggedfrog complex (Rana muscosa and Rana sierrae, henceforth R.muscosa). R. muscosa are native to California’s Sierra Nevadamountains and have experienced significant Bd-induced popu-lation declines and extinctions over the last four decades(Briggs et al. 2005; Vredenburg et al. 2010; Briggs et al. 2010;Knapp et al. 2016). Using this host-parasite system, the Meth-ods section is organised as follows.First, we used a laboratory experiment to quantify the well-

known importance of temperature-dependent Bd growthdynamics on R. muscosa (Andre et al. 2008; Wilber et al.2016). Second, we used a mesocosm experiment to quantifythe nature of the transmission function in the R. muscosa-Bdsystem, testing for both density-dependent transmission, fre-quency-dependent transmission, and transmission from anenvironmental zoospore pool. Third, we used the results fromthese experiments to build a discrete-time, host-parasite Inte-gral Projection Model (IPM) and derived R0 with an environ-mental zoospore pool. We used this result to explore theeffects of the zoospore pool on Bd invasion. Finally, toanswer our primary question regarding the relative impor-tance of transmission compared to resistance and tolerance onextinction risk, we extended our model to consist of a within-year component in which Bd transmission and disease-inducedamphibian mortality occurred and a between-year componentin which R. muscosa demographic transitions occurred(Fig. 1). Using this hybrid model, we explored the sensitivityof Bd-induced extinction to transmission, resistance and toler-ance.

Laboratory and mesocosm experiments

We used laboratory and mesocosm experiments to quantifythe temperature-dependent load dynamics and the transmis-sion dynamics in the R. muscosa-Bd system. The laboratoryexperiment is fully described in Wilber et al. (2016) and con-sisted of 15 adult frogs housed at three different temperatures(4, 12, 20 ∘C; 5 frogs per temperature). Wilber et al. (2016)used this experiment to parameterise four functions relating toBd load dynamics: a load-dependent host survival functionand a temperature-dependent Bd growth function, loss ofinfection function, and initial infection function(Appendix Fig. S1, see Table 1 for function descriptions).To quantify the nature of the transmission function in this

system, we performed a mesocosm experiment that consistedof four different density treatments: 1, 4, 8 and 16 uninfectedadult frogs per mesocosm (volume � 1 m3). Each treatmentwas replicated four times for a total of 16 mesocosms (see

© 2017 John Wiley & Sons Ltd/CNRS

1170 M. Q. Wilber et al. Letter

Appendix S1). In addition to the uninfected adults, eachmesocosm started with five infected tadpoles, which couldrelease Bd zoospores into the environment and subsequentlyinfected adults. All of the adults in a mesocosm were uniquelyidentifiable by pit tags, but the five tadpoles were not. Theexperiment ran for 74 days and every 4–8 days all adults andtadpoles in a mesocosm were swabbed and the zoospore loadon each was determined using quantitative PCR (Boyle et al.2004). Frogs and tadpoles within a mesocosm were alwaysswabbed on the same day.To estimate the transmission function, we measured the

load transitions on all adult frogs from time t to t + Dt overthe first 32 days of the experiment (Dt = 4–8 days depending

on the time between swabs in the experiment). We only usedthe first 32 days of the experiment as after this time point allamphibians experienced an unexplained decline in zoosporeloads (see Appendix Fig. S2). However, because the load tra-jectories over these first 32 days were consistent with otherexperiments (e.g. Wilber et al. 2016) and transitions fromuninfected to infected tended to occur before day 32, we feltconfident in estimating transmission dynamics from only thefirst 32 days. Moreover, we replicated the experiment in silicoto demonstrate that we could recover known transmissionfunctions over the first 32 days of the experiment(Appendix S1). As transmission is the probability of an unin-fected individual gaining an infection in a time step, we only

0

1

2

3

0 5 10 15 20Time (years)

ln (a

dults

+ 1

)

0

1

2

3

4

5

0 5 10 15 20Time (years)

ln (t

adpo

les

+ 1)

−5

0

5

10

15

0 5 10 15 20Time (years)

Mea

n ln

Bd

load

0

5

10

15

0 5 10 15 20Time (years)

ln (z

oosp

ores

+ 1

)

(a)

(c) (d)

(b)

Figure 1 (a) Diagram showing the temporal dynamics of the hybrid model of R. muscosa-Bd. Reproduction and demographic transitions occur once a year

in the spring (red dot and (c).). Disease dynamics are temperature-dependent and are updated every 3 days (vertical dashes and (b).) over the course of the

entire year. (b) The within-season dynamics of R. muscosa and Bd. (c) The between season demography of R. muscosa. Note that survival probability of

susceptible adults (SA), infected adults (IA) and Bd zoospores in the environmental pool (Z) is one because their survival probabilities are already

accounted for in the within-season model. (d) Representative trajectories of adult, tadpole and zoospore pool population sizes from the hybrid model with

a density-dependent transmission function and infection from a dynamic zoospore pool. Five stochastic trajectories are shown from the hybrid model

(coloured lines). The mean ln Bd load, conditional on infection, is also shown. Gaps in trajectories for the mean ln Bd load indicate that no infected

individuals were in the population at those time points.

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1171

Table

1Parametersusedin

theRanamuscosa-Bdhybridmodel

Description

Functionalform

Parameters

Detailsofparameterisation

Infected

survivalfunction,s(x):

Probabilityofhost

survivalfrom

tto

t+1withloadx

logit[s(x)]=b0,0+b1,0x

b0,0=5.295;b1,0=�2

.595

LikelihoodBernoulli(s(x)),xwasz-transform

ed

Appendix

Fig.S1b.

Growth

function,G(x

0 ,x):Probability

density

ofhost

transitioningfrom

load

xto

loadx

0attimet+1

l(x,T)=b0,1+b1,1x+b2,1T

r2(x)=m 0

,1exp(2c 0

,1x)

b0,1=0.012;b1,1=0.799;b2,1=0.092;

m 0,1=5.92;c 0

,1=�0

.049

LikelihoodNorm

al(l(x,T),r2(x))

Appendix

Fig.S1a.

Loss

ofinfectionfunction,l(x):The

probabilityofhost

havinginfectionof

loadxattandlosingitbyt+1

logit[l(x,T)]=b0,2+b1,2x+b2,2T

b0,2=1.213;b1,2=�0

.472;b2,2=�0

.151

LikelihoodBernoulli(l(x,T))Appendix

Fig.S1c.

Initialinfectionburden

function,G0(x

0 ):

Probabilitydensity

ofbeinguninfected

attandgaininganinfectionofx

0at

t+1

l(T)=b0,3+b1,3T

r2(T)=m 0

,3exp(2c 0

,3T)

b0,3=0.642;b1,3=0.137;m 0

,3=0.59;

c 0,3=0.063

LikelihoodNorm

al(l(T),r2(T))

Appendix

Fig.S1d.

Transm

issionfunction,/:The

probabilityofanuninfected

host

gaininganinfectionattimet+1

Functionalform

svary

See

Table

2See

Table

2

Uninfected

adultsurvivalprobability

over

threedaytimestep,s 0

Constant

s 0=0.999

Yearlysurvivalfrom

Briggset

al.(2005)

convertedto

athreedaytimescale

Tem

perature-dependentzoospore

survivalprobability,m

m=f(T)

Non-parametric

Cubic

smoothingsplinebasedondata

from

Woodhamset

al.(2008)Appendix

Fig.S1e.

Proportionofzoosporescontributedto

Zbyinfected

adult

Constant

l A=1

Likelyaconservativeestimate

ofhow

infected

adultscontribute

tothezoospore

pool

Meantadpole

zoospore

load,l T

iConstant

l Ti¼

1487:036

Meanzoospore

loadoftadpolesin

mesocosm

experim

entdescribed

intext

Probabilityoftadpole

isurvivingathree

daytimestep

Constant

s T1¼

0:987;s T

2¼

0:997;s T

3¼

0:997

Yearlysurvivalfrom

Briggset

al.(2005)

convertedto

athreedaytimescale

Probabilityoftadpole

inot

metamorphosing,pTi

Constant

pT1¼

1;pT2¼

0:5;pT3¼

0Briggset

al.(2005)

Probabilityoftadpole

isurviving

metamorphosis,m

Ti

Constant

mT1¼

0:9;m

T2¼

0:9;m

T3¼

1:0

Briggset

al.(2005)

Probabilityofadultreproducing,pA

Constant

pA=0.25

Briggset

al.(2005)

Meanadultreproductiveoutput,λ

Constant

λ(x)=λ 0

=λ=50

Leadsto

realistic

population-level

growth

rate

ofλ

growth

rate�1

.46in

thedisease-free,

density-

independentmodel

(Briggset

al.2005)

Carryingcapacity

parameter,K

Constant

K=5

Leadsto

equilibrium

adultdensities

between10-

15adultsper

m3in

disease-freemodel

AllbandcparametershadaNorm

alpriorwithmean0andstandard

deviation5.Allmparametershadahalf-C

auchypriorfrom

0to

∞withscale

parameter

1.Forallstatisticalmodels,

con-

vergence

wasassessedusingtraceplots

andensuringthattheGelman-R

ubin

statistic

was<1.05(G

elmanet

al.2014).logitspecifies

alogisticlink,xis

lnzoospore

load,andT

istemperature.

Allprobabilitiesare

over

athreedaytimestep.

© 2017 John Wiley & Sons Ltd/CNRS

1172 M. Q. Wilber et al. Letter

included transitions where Bd load was 0 at time t (n = 333).If the load at time t + Dt was positive we assigned this datapoint a value of 1 (infected) and if the load was still zero weassigned it a value of 0 (uninfected).Uninfected frogs can acquire Bd infection through contact

with other infected frogs and through contact with zoosporesin an environmental Bd pool (Courtois et al. 2017). Toaccount for these different pathways, we fit two sets of trans-mission models to the data. The first set of models assumed aconstant level of infection from the zoospore pool and eitherdensity-dependent or frequency-dependent transmission fromconspecifics (Table 2). The second set of models allowedtransmission to be a function of how many zoosporeswere in the zoospore pool at time t in addition to eitherdensity-dependent or frequency-dependent transmission. InAppendix S1, we describe how we defined and fit our trans-mission models with a dynamic zoospore pool. In short, weused the transmission function /(t) = 1� exp (�K(Z(t), I(t))Dt) and allowed for the zoospore pool Z(t) at time t to be anunobserved, dynamic variable that lost zoospores due to envi-ronmental decay and gained zoospores due to productionfrom infected adults and tadpoles at every time step. I(t) isthe number of infected adults at time t. Table 2 gives thetransmission functions and the resulting fits to the data fromthe mesocosm experiment.

The host-parasite IPM and R0

The host-parasite IPMUsing the aforementioned laboratory and mesocosm experi-ments, we parameterised a host-parasite Integral ProjectionModel (IPM) where Bd load on an individual frog was the con-tinuous trait being modelled (Fig. 1b; Metcalf et al. 2015; Wil-ber et al. 2016). Bd load on a frog is estimated as the number ofcopies of Bd DNA detected on standardised skin swabs viaquantitative PCR (Boyle et al. 2004) and provides a continuousmeasure of infection intensity between 0 (uninfected) and anarbitrarily large Bd infection. The IPM is a discrete time modeland here a single time step is three days. This time step is on thesame scale as the generation of time of Bd, which rangesbetween 4–10 days depending on temperature (Piotrowski et al.2004; Woodhams et al. 2008). We used the discrete-time IPMbecause it is easily parameterised from laboratory data which iscollected at discrete time intervals.This IPM tracks two discrete stages at time t: the density of

susceptible adults SA(t) in the population and the density ofzoospores in the environment Z(t). This model also tracks acontinuous, infected stage IA(x, t) where x is ln Bd load andRUx

LxIAðx; tÞdx gives the density of adult frogs with a ln Bd

load between a lower bound Lx and an upper bound Ux attime t. This continuous, infected stage tracks the distributionof Bd loads at any time t in the population.Considering these discrete and continuous stages, the

amphibian-Bd IPM can be written as follows (Fig. 1b)

SAðtþ 1Þ ¼ SAðtÞs0ð1� /Þ þZ Ux

Lx

IAðx; tÞsðxÞlðxÞdx ð1Þ

IAðx0; t þ 1Þ ¼Z Ux

Lx

IAðx; tÞsðxÞð1 � lðxÞÞGðx0; xÞdxþ SAðtÞs0/G0ðx0Þ ð2Þ

Zðt þ 1Þ ¼ ZðtÞm þ lA

Z Ux

Lx

expðxÞIAðx; tÞdx� wðSAðtÞ;ZðtÞÞ

ð3ÞSA(t + 1) describes the density of susceptible adults at timet + 1 and is determined by the number of adults that surviveand do not become infected in a time step (first term in eqn 1)and the number of infected adults that survive and lose theirinfection in a time step (second term in eqn 1). IA(x

0,t + 1)

describes the density of infected adults with a ln Bd load x0at

time t + 1 and is determined by infected adults who survivewith load x, do not lose their load x, and experience a changein load from x to x

0in a time step (first term in eqn 2) and

from uninfected adults who survive, become infected, andgain an initial Bd load of x

0in a time step (second term in

eqn 2). The vital rate functions contained in eqns 1–3 aredescribed in Fig. 1b and Table 1.The equation Z(t + 1) gives the density of zoospores in the

zoospore pool at time t + 1. Z(t + 1) depends on three dis-tinct terms: the survival probability of the zoospores in theenvironment from time t to t + 1 (m), contribution of zoos-pores from infected adults where lA is the proportion of totalzoospores on adults contributed to the zoospore pool over atime step, and removal of zoospores from the zoospore poolby frogs transitioning from uninfected to infected. Thisremoval term w(SA(t),Z(t)) had very little effect on thedynamics of the system and we do not consider it further.Based on the laboratory experiment described above and

known Bd life history (Woodhams et al. 2008), we allowedvarious vital rate functions to be temperature-dependent (e.g.the survival of free-living zoospores, Table 1,Appendix Fig. S1).

R0 with an environmental reservoirUsing the IPM described in eqns 1–3, we calculated R0 toquantify how temperature and the transmission dynamicsaffected the ability of Bd to invade a R. muscosa population –a necessary condition for Bd-induced population declines andextinction. R0 describes the average number of secondaryinfections produced over the lifetime of an average infectedagent (Diekmann et al. 1990; Dietz 1993). When R0 ≤1, apathogen cannot invade a fully susceptible host population.When R0>1, a pathogen can invade a fully susceptible hostpopulation with probability 1�(1/R0) (Gerber et al. 2005;Allen 2015).In Appendix S2 we show that, consistent with continuous-

time disease models (Rohani et al. 2009), R0 for discrete-timeIPMs with an environmental reservoir is composed of trans-mission from host contact and the environment. We use thisresult in combination with our fully parameterised host-para-site IPM to calculate the temperature-dependent R0 for R.muscosa-Bd systems with and without infection from the envi-ronmental zoospore pool.

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1173

The hybrid model

The model described by eqns 1–3 is sufficient to describe thedynamics of an initial epizootic, but to examine Bd-inducedextinction dynamics in R. muscosa populations a number ofadditions need to be made. We briefly describe the hybridmodel that accounts for the within-year Bd dynamics as wellas the between year demography of R. muscosa. Fig. 1 gives avisual representation of the hybrid model and Appendix S3gives a full description.The within-year component (Fig. 1b.), is identical to the

IPM given in eqns 1–3 with the addition of three tadpolestages. The tadpole stage of R. muscosa is likely important ingenerating enzootic dynamics in R. muscosa populations(Briggs et al. 2005, 2010). We assumed all tadpoles wereimmediately infected with Bd and had a constant mean contri-bution to the zoospore pool (Table 1). This is justified by theobservation that most tadpoles in R. muscosa populationscarry high fungal loads, even in enzootic populations (Briggset al. 2010). R. muscosa tadpole survival is not affected by Bdinfection. Therefore, the within-season dynamics of the tad-pole stages were simply given by the probability of a tadpolesurviving from time t to t + 1. Infected tadpoles also con-tributed to the zoospore pool at each time step (Fig. 1b).R. muscosa populations also experience seasonal tempera-

ture fluctuations in which winter lake temperatures drop to c.4 ∘C in the winter (in the unfrozen portion of a lake where thefrogs overwinter) and reach c.20 ∘C in the summer (Knappet al. 2011). We accounted for this seasonal variability byimposing a deterministically fluctuating environment on theR. muscosa-Bd IPM (Fig. 1a). At each discrete time pointwithin a season, a new temperature was calculated and thetemperature-dependent vital rate functions were updatedaccordingly.The between-year component of the hybrid model

accounted for yearly maturation and metamorphosis of thetadpole stages as well as reproduction of adults (Fig. 1c). Weassumed that the recruitment of metamorphed tadpoles intothe adult stage was density-dependent and that all tadpolesentered the adult stage as uninfected (i.e. all individuals areinfected as tadpoles, but lose their infection at metamorphosis,Briggs et al. 2010). Because we have no empirical evidence forBd-induced fertility reduction in R. muscosa, we assumed thatreproduction in uninfected and infected adults was the same.

Simulating the hybrid model

After parameterising the hybrid model using the above experi-ments, we used the model to make predictions about theprobability of disease-induced host extinctions. Because demo-graphic stochasticity is important when predicting extinctionfor small populations (Lande et al. 2003), we included it intothe hybrid model (Caswell 2001; Schreiber & Ross 2016). Todo this, we first discretised the within-season IPM using themid-point rule and 30 mesh points (Easterling et al. 2000) andthen determined the transition of an individual frog or zoos-pore to another state (including death) as a draw from amultinomial distribution with probabilities given by the discre-tised hybrid model at that time step (Appendix S4). In

addition, we assumed that both the production of tadpolesthat occurs once a year in the spring and the number of zoos-pores shed into the zoospore pool at each time step followeda Poisson distribution (Appendix S4).To answer our question regarding the importance of trans-

mission, resistance, and tolerance on Bd-induced extinction,we performed two analyses. First, we examined how differenttransmission functions parameterised from our mesocosmexperiment affected extinction risk. Using the three transmis-sion functions with a dynamic zoospore pool described inTable 2 and the parameter values given in Table 1, we per-formed 500 stochastic simulations of the hybrid model to gen-erate time-dependent extinction curves over a 25 year period.All simulations were started with 10 uninfected adult frogs,85 year-one tadpoles (T1), 12 year-two tadpoles (T2), and 3year-three tadpoles (T3). The relative proportions of adultfrogs and tadpoles were assigned based on the stable stagedistribution in the Bd-free model. While the initial conditionsnecessarily affect the absolute time to extinction, they do notaffect the shapes of the extinction curves for different trans-mission functions relative to each other. All simulationsstarted in the winter at 4 ∘C, with reproduction occurring inthe first spring at 12 ∘C (Fig. 1a). Given our analysis of R0

in the previous section, we assumed that Bd could invadewith a probability of one, such that tadpoles were immedi-ately infected with Bd and began contributing to the zoosporepool Z(t) (see Appendix S4). For each simulation, we calcu-lated time-dependent extinction curves as the mean probabil-ity of going extinct in a given time step over all 500simulations.Second, we performed a sensitivity analysis on the

hybrid model to assess the relative importance of transmis-sion compared to resistance and tolerance. Transmissionwas determined by the parameters in the transmissionfunction /, resistance was determined by the parameters inthe growth function G(x

0,x), the loss of infection function l

(x), and the initial infection burden function G0(x0), and

tolerance was determined by the parameters in the survivalfunction s(x). To perform the sensitivity analysis, we ran1000 simulations using the parameter values given inTable 1 and the initial conditions described above. Foreach simulation we recorded whether or not a R. muscosapopulation went extinct in ≤ 8 years, as this was whereextinction probability was c.50% with the default parame-ter values.On each run of the simulation we perturbed sixteen

lower-level transmission, resistance and tolerance parame-ters by, for each parameter, drawing a random numberfrom a lognormal distribution with median 1 and disper-sion parameter rsensitivity = 0.3 and multiplying the givenparameter by this random number (Sobie 2009). Ourresults were robust to our choice of r and our method ofperturbation (Appendix Fig. S3). For each simulation, wesaved the perturbed parameter values and stored them in a1000 by 16 parameter matrix. Upon completion of thesimulation, we used both regularised logistic regression anda Random Forest classifier in which our response variablewas whether or not a given simulation went extinct andour predictor variables were the scaled (i.e. z-transformed)

© 2017 John Wiley & Sons Ltd/CNRS

1174 M. Q. Wilber et al. Letter

matrix of perturbed predictors (Harper et al. 2011; Pedre-gosa et al. 2011; Lee et al. 2013). Using this approach, wecould then identify the relative importance of each parame-ter in the vital rate functions in predicting whether or notextinction occurred (Sobie 2009). Moreover, we also builta pruned regression tree to visualise the interactive effectsof transmission, resistance, and tolerance parameters onhost extinction risk (Harper et al. 2011). All the code nec-essary to replicate the analyses is provided at https://github.com/mqwilber/host_extinction.

RESULTS

Question 1: What is the structure of the transmission function?

Accounting for the dynamics of the environmental zoosporepool resulted in significantly better transmission modelscompared to those that did not. The transmission modelwith density-dependent host to host transmission as well astransmission from a dynamic zoospore pool was a bettermodel than all other transmission models considered(Table 2). In addition to being a better model in terms ofWAIC, the density-dependent transmission model also cap-tured the marginal pattern of increasing probability ofinfection with increasing density of infected hosts(Appendix Fig. S4).

Question 2: How does the transmission function affect the ability of

Bd to invade?

Using the temperature-dependent vital rate functionsdescribed in Table 1 and the best-fitting density-dependenttransmission function with an environmental zoospore pool(Table 2), we examined how host density and temperatureaffected the ability of Bd to invade a R. muscosa population.When transmission was density-dependent, but did notdepend on the environmental zoospore pool, Bd was able toinvade R. muscosa populations over a large range of densitiesand temperatures, though there was a slight protective effectof low temperatures and low densities (Fig. 2a). Includingtransmission from the environmental zoospore pool substan-tially increased the region in which Bd could invade and inva-sion was highly probable for most temperatures and hostdensities (Fig. 2b and c).

Question 3: How sensitive is disease-induced extinction to

transmission, resistance and tolerance?

The time-dependent probability of extinction was similarbetween the three transmission functions that included adynamic zoospore pool (Appendix Fig. S5, Table 2). The sim-ilarity between these curves was due to the overwhelminginfluence of the infection probability from the zoospore pool,which swamped out the well-known differences between fre-quency-dependent and density-dependent transmission func-tions. Drastically decreasing zoospore survival probabilitybelow what has been observed in laboratory experiments(Woodhams et al. 2008), led to an expected reduction inextinction risk as the transmission probability then declinedwith decreasing host density (given density-dependent trans-mission, Appendix Fig. S5).A sensitivity analysis of Bd-induced host extinction to

transmission, host resistance, and host tolerance showedthat, regardless of the transmission function used, R. mus-cosa extinction was more sensitive to the parameters relatingto host resistance and tolerance than to parameters relatingto transmission (Fig. 3). In particular, the most importantparameter across all transmission functions was the slope ofthe growth function b1,1, which is a parameter affecting hostresistance. For a given temperature, the logistic regressionanalysis showed that decreasing this parameter, whichroughly corresponds to decreasing the mean Bd load on ahost for a given temperature, decreased the probability ofdisease-induced extinction for all transmission functions(Fig. 3a–c). Bd-induced R. muscosa extinction was also sen-sitive to the parameters of the survival function, particularlythe intercept of the survival function b0,0. This parametercan be thought of as the threshold at which Bd-inducedmortality begins to occur given a fixed slope in the survivalfunction. The logistic regression analysis showed thatincreasing this parameter, which corresponds to increasingthe threshold at which R. muscosa begins to suffer load-dependent Bd mortality, decreased the probability of extinc-tion (Fig. 3a–c).Resistance and tolerance parameters also showed significant

interactions when affecting host extinction risk. Random for-ests and pruned regression trees showed the importance of theslope of the growth function as well as the importance ofthe interaction between this parameter and the intercept of

Table 2 The results of fitting transmission functions of the form /=1� exp (�K) to the R. muscosa-Bd mesocosm experiment. I is the total number of

infected adults in a mesocosm at the beginning of a time interval, A is the total number of adults in a tank, Z is the number of zoospores in the mesocosm

at the beginning of the time interval as estimated from a latent zoospore pool model (Appendix S1), and Dt is the time between swabbing events in the

experiment (between 4 and 8 days). All b parameters had a half-Cauchy prior from 0 to ∞ with scale parameter equal to 1. Models with lower WAICs

and higher weights are better models. The bold value indicates the lowest WAIC.

Name Function Parameters WAIC (weight)

Constant zoospore pool K = (b0)Dt b0 = 8.07 9 10�2 day�1 401.1 (0)

Density-dependent w/ constant zoospore pool K = (b0+b1I)Dt b0 = 4.18 9 10�2, b1 = 5.25 9 10�2 336.7 (0)

Frequency-dependent w/ constant zoospore pool K ¼ ðb0 þ b1IAÞDt b0 = 4.28 9 10�2, b1 = 0.551 336.6 (0)

Dynamic zoospore pool K = (b0 ln (Z + 1))Dt b0 = 1.09 9 10�2 345.32 (0)

Density-dependent w/ dynamic zoospore pool K = (b0 ln (Z + 1)+b1I)Dt b0 = 5.29 9 10�3, b1 = 7.52 9 10�2301.18 (0.99)

Frequency-dependent w/ dynamic zoospore pool K ¼ ðb0 lnðZþ 1Þ þ b1IAÞDt b0 = 5.77 9 10�3, b1 = 0.627 311.06 (0.01)

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1175

the survival function (a tolerance parameter) and the tempera-ture-dependency in the growth function (a resistance parame-ter) (Fig. 3a–c; Appendix Fig. S6a–c).

DISCUSSION

Wildlife conservation in the face of disease emphasises theimportance of the transmission function in extinction risk(McCallum 2012). This is a reasonable emphasis as the trans-mission function is ultimately the most important aspect ofdisease-induced extinction: if a host does not get infected witha disease it will not suffer disease-induced mortality. Inamphibian-Bd systems it has been hypothesised that bothamphibian density and an environmental pool of zoosporescan affect transmission (Rachowicz & Briggs 2007; Briggset al. 2010; Courtois et al. 2017), but the nature of this trans-mission has rarely been quantified. We experimentally quanti-fied the transmission function in the R. muscosa-Bd systemand used these results, in combination with a dynamic model,to predict how the environmental zoospore reservoir affectedthe ability of Bd to invade an amphibian population. Consis-tent with previous theory (Godfray et al. 1999; Rohani et al.2009), we found that including an environmental zoosporepool substantially increased R0 for R. muscosa-Bd systems,

such that Bd was able to invade a R. muscosa population formost realistic temperatures and host densities. To the best ofour knowledge, this is the first estimation of R0 in an amphib-ian-Bd system (but see Woodhams et al. 2011, for a discus-sion of R0 in amphibian-Bd systems), and the large value ofR0 across all temperatures and densities is consistent with fieldobservations that temperature and density have little protec-tive effect in the R. muscosa/sierrae system (Knapp et al.2011, R. A. Knapp et al., unpublished). These results suggestthat attempting to prevent Bd invasion into a system may belargely futile and management should be focused on mitigat-ing post-invasion Bd impacts (Langwig et al. 2015).Conditional on Bd invasion, we used our parameterised

model to explore the importance of the transmission functionon Bd-induced amphibian extinction and found that theextinction dynamics were similar between all transmissionmodels with a dynamic zoospore pool. This was due to thelarge number of zoospores shed by infected amphibians com-bined with the laboratory-estimated decay rate of zoosporesoutside the host, leading to a zoospore pool that remainedlarge even for rapidly declining host populations (Fig. 1d).Only considering this result, we would then expect R. muscosapopulations to be at substantial risk of disease-induced extinc-tion given that the persisting zoospore pool prevents a

W/out zoospore pool With zoospore pool

5 10 15 20 5 10 15 20

4

8

12

16

Temperature,°C

Initi

al s

usce

ptib

le fr

og d

ensi

ty,

m− 3

0.00

0.25

0.50

0.75

Invasion probability

With zoopore pool

W/out zoopore pool

0

1

2

3

4

4 8 12 16

Initial susceptible frog density, m−3

ln (R

0)

With zoopore pool

W/out zoopore pool

−1

0

1

2

3

5 10 15 20

Temperature,°C

ln (R

0)

(a)

(b) (c)

Figure 2 R0 and the invasion probability of Bd (1� 1R0) for different temperatures and host densities with and without an environmental zoospore pool.

This calculation of R0 uses the parameters given in Table 1 and the ‘Density dependent w/ dynamic zoospore pool’ transmission function in Table 2. While

R0 will inevitably decrease without transmission from a zoospore pool (i.e. when b0=0; see Appendix S2), the magnitude of that decrease depends on the

estimated transmission coefficient from the zoospore pool. (a) Gives the invasion probability of Bd. The dashed, vertical lines in a. correspond to the

curves shown in (b), where ln (R0) is plotted against initial adult density when temperature is 15 ∘C. The solid, horizontal lines in a. correspond to the

curves shown in (c) where ln (R0) is plotted against temperature when initial adult density is four adults per m3. The grey regions give the 95% credible

intervals. The dashed lines in (b) and (c) correspond to R0 = 1 (ln (R0) = 0), below which Bd cannot invade.

© 2017 John Wiley & Sons Ltd/CNRS

1176 M. Q. Wilber et al. Letter

decrease in transmission rate with declining host density(Anderson & May 1981; Godfray et al. 1999; De Castro &Bolker 2005). This finding is consistent with a number ofother studies that have found that the dynamics of the Bdzoospore pool are critical for determining Bd-inducedamphibian extinctions (Mitchell et al. 2008; Briggs et al. 2010;Doddington et al. 2013). If abiotic or biotic factors such astemperature, stream flow, water chemistry, and/or zoosporeconsumption by aquatic organisms are able to substantiallyincrease zoospore death rate beyond the values seen in the lab(Tunstall 2012; Strauss & Smith 2013; Venesky et al. 2013;Heard et al. 2014; Schmeller et al. 2014), then we mightexpect a reduction, though not an elimination, of Bd invasionprobability and amphibian extinction risk.However, considering only the transmission function

ignores the fact that, conditional on infection, increasingresistance or tolerance to a disease can also reduce disease-induced mortality and thus provide alternative mechanismsby which to manage disease-induced extinction risk (Kil-patrick 2006; Vander Wal et al. 2014; Langwig et al. 2015).Using our model, we found that Bd-induced extinction riskwas far more sensitive to host resistance and tolerance thanto the transmission dynamics of Bd. In particular, extinctionrisk of R. muscosa populations was most sensitive to the vital

rate functions dictating the growth rate of Bd on a host, theload-dependent survival probability, and the interactionbetween these two functions.This result highlights the importance of accounting for the

load-dependent nature of vital rate functions when consider-ing extinction risk in load-dependent diseases such as chytrid-iomycosis. In this study, consistent with results observed inthe field (Vredenburg et al. 2010), the survival function of Bdwas strongly non-linear such that above �9 ln zoospores=8103 zoospores the survival probability of R. muscosarapidly declined (Appendix Fig. S1b). When the survivalfunction is load-dependent and highly non-linear, as observedin some amphibian-Bd systems (Stockwell et al. 2010; Vre-denburg et al. 2010, but see Clare et al. 2016), small changesin host resistance or tolerance can lead to abrupt changes insurvival probability. In particular, non-linearities in the sur-vival function (i.e. tolerance) need to be considered in thecontext of the shape of the growth function (i.e. resistance) ofa parasite on its host.To illustrate the generality of this result, consider the fol-

lowing graphical argument. Take a pathogen growth function(G(x

0, x)) that predicts a static mean pathogen load near the

threshold at which the survival function predicts a drasticdecrease in survival probability (Fig. 4, i.e. a non-linear dose-

Toleranceparameters

Resistanceparameters

Transmissionparameters

(a)

(b)

(c)

Figure 3 (a–c) The sensitivity of R. muscosa extinction probability to parameters dictating transmission, resistance and tolerance of Bd for the three

transmission functions used in the hybrid model. The dark grey bars give the weights of the various parameters when logistic regression is used to classify

whether or not a simulation trajectory experienced extinction. The absolute height of the bar shows the relative importance of that parameter and the

direction specifies what happens to the extinction probability when that parameter is increased. For example, increasing the G(x0,x) parameter b1,1 increases

the probability of extinction. The light grey bars give the relative importances of each parameter in predicting the extinction of a simulation when a

Random Forest classifier was used to account the interactions between parameters on extinction probability. These values are all between zero and one and

the height of a bar indicates the relative importance of a transmission, resistance, or tolerance parameter.

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1177

response curve; Dwyer et al. 1997; Handel & Rohani 2015;Louie et al. 2016). Slightly shifting the slope (or the intercept)of the growth function (i.e. changing resistance) up or downwill increase or decrease the static mean pathogen load andmove a host into the region of the survival curve where eithermortality or survival is almost certain (Fig. 4). Similarly, hold-ing the growth function constant and altering the survival func-tion (i.e. changing tolerance) will change how close the staticmean pathogen load is to the survival function threshold(Fig. 4). This suggests that identifying how the growth functionof a pathogen changes in resistant host populations, whether bydecreasing the slope, decreasing the intercept or by transition-ing from a linear to a non-linear function (Langwig et al. 2017),is important for understanding the sensitivity of extinction riskto both the growth function (resistance) and the survival func-tion (tolerance). Identifying whether or not a disease systemshows this strong interaction between resistance and tolerancecan help determine whether disease mitigation should focus onreducing parasite loads via strategies such as inducing acquiredimmunity, microbial treatments, or selecting for resistance(Harris et al. 2009; Langwig et al. 2015) or decreasing parasitetransmission via strategies such as culling and/or treating thedisease reservoir (Cleaveland et al. 2001).The importance of host resistance and tolerance for our

model’s predictions of disease-induced extinction indicatesthat these host strategies could promote population persis-tence in R. muscosa-Bd populations, as they have in other

species of amphibians suffering from chytridiomycosis, batssuffering from white-nose syndrome, and Tasmanian devilssuffering from facial tumour disease (Hoyt et al. 2016; Savage& Zamudio 2016; Epstein et al. 2016; Langwig et al. 2017). Infact, a recent study showed that many populations of R. sier-rae that experienced Bd-induced population declines over thelast four decades are recovering in the presence of Bd, butwith reduced Bd loads relative to naive populations (Knappet al. 2016). This result is consistent with resistance mecha-nisms reducing Bd load and thus increasing host survivalprobability. While changes in resistance mechanisms, and nottolerance mechanisms, are putatively responsible for persis-tence in a number of load-dependent diseases (Savage &Zamudio 2011; Epstein et al. 2016; Langwig et al. 2017), ourresults show that the shape of the tolerance function is criticalfor understanding the effects of resistance on host persistence(Fig. 4). An important future direction will be to explore howheterogeneities in host resistance, tolerance, and/or transmis-sion functions can promote host persistence (Boots et al.2009; Langwig et al. 2015; Brunner et al. 2017).While our study suggests that an interaction between resis-

tance and tolerance promotes population persistence in R.musocsa-Bd populations, additional mechanisms that we do notconsider here are likely important in other amphibian-Bd sys-tems. For example, laboratory studies have shown that Bd canevolve increased or reduced virulence in as few as 50 genera-tions (less than one year, Langhammer et al. 2013; Voyles et al.2014; Refsnider et al. 2015). If Bd virulence attenuates over thecourse of an epizootic, these changes could augment host per-sistence, without any changes in host resistance or tolerance. Inreality, the evolution of both the pathogen and the host affectspopulation persistence (Vander Wal et al. 2014) and developingload-dependent models that incorporate both of these processesis an open challenge in wildlife epidemiology.The emergence of a number of diseases of conservation con-

cern have highlighted the importance of considering disease-induced extinction in the context of load-dependent parasitedynamics. We show that simultaneously considering transmis-sion, resistance, and tolerance, in conjunction with load-dependent dynamics, provides novel insight regarding how tobest manage emerging infectious diseases.

ACKNOWLEDGMENTS

We would like to acknowledge the Sierra Nevada AquaticResearch Laboratory for logistical support for the experi-ments described in this work. We also acknowledge W. Mur-doch, R. Nisbet, and three anonymous reviewers for helpfulfeedback on this manuscript and S. Weinstein for her artisticcontribution. Funding was provided by the National ScienceFoundation, USA (NSF EEID grant DEB-0723563 and NSFLTREB grant DEB-1557190). M. Wilber was supported by anNSF Graduate Research Fellowship (Grant No. DGE1144085) and the University of California Regents (USA).

AUTHORSHIP

C.B., M.T. and R.K. designed and implemented the labora-tory and mesocosm experiments. M.W. and C.B. designed

Figure 4 The interaction between the growth function determining host

resistance (blue lines) and the survival function determining host tolerance

(red lines) has large impacts on host survival probability. The solid blue

lines of different shades give three different growth functions that describe

how Bd load changes from time step t to time step t + 1. The growth

functions have different slopes representing different levels of resistance.

When one of the growth functions crosses the dashed 1:1 line, this

indicates that the Bd load at time t will be the same as the Bd load at

time t + 1 (i.e. stasis). The red lines of different colours specify different

survival functions with different levels of tolerance. The corresponding

blue coloured circle on one of the red survival curves gives the probability

of a host surviving a time step with the given Bd load at stasis. Each red

survival curve has three different blue dots for each of the three growth

functions. Because of the strong non-linearity of the survival function,

changes in the Bd load at stasis can lead to disproportionate changes in

survival probability. However, the direction and magnitude of these

changes depends on the degree of tolerance represented in the survival

curve.

© 2017 John Wiley & Sons Ltd/CNRS

1178 M. Q. Wilber et al. Letter

and performed the analyses. M.W. drafted the manuscriptand all authors contributed to revisions.

DATA ACCESSIBILITY

Data supporting the results are available from the DryadDigital Repository: http://dx.doi.org/10.5061/dryad.p614h

REFERENCES

Allen, L.J.S. (2015). Stochastic Population and Epidemic Models:

Persistence and Extinction. Springer International Publishing, London.

Anderson, R.M. & May, R.M. (1981). The population dynamics of

microparasites and their invertebrate hosts. Philos. Trans. R. Soc. Lond.

B Biol. Sci., 291, 451–524.Andre, S.E., Parker, J. & Briggs, C.J. (2008). Effect of temperature on

host response to Batra- chochytrium dendrobatidis infection in the

mountain yellow-legged frog (Rana muscosa). J. Wildl. Dis., 44, 716–720.

Bielby, J., Fisher, M.C., Clare, F.C., Rosa, G.M. & Garner, T.W.J.

(2015). Host species vary in infection probability, sub-lethal effects, and

costs of immune response when exposed to an amphibian parasite. Sci.

Rep., 5, 1–8.Blehert, D.S., Hicks, A.C., Behr, M., Meteyer, C.U., Berlowski-Zier,

B.M., Buckles, E.L. et al. (2008). Bat white-nose syndrome: an

emerging fungal pathogen? Science, 323, 227.

Boots, M., Best, A., Miller, M.R. & White, A. (2009). The role of

ecological feedbacks in the evolution of host defence: what does theory

tell us? Philos. Trans. R. Soc. Lond. B Biol. Sci., 364, 27–36.Boyle, D.G., Boyle, D.B., Olsen, V., Morgan, J.A.T. & Hyatt, A.D.

(2004). Rapid quantitative detection of chytridiomycosis

(Batrachochytrium dendrobatidis) in amphibian samples using real-time

Taqman PCR assay. Dis. Aquat. Organ., 60, 141–8.Briggs, C.J., Knapp, R.A. & Vredenburg, V.T. (2010). Enzootic and

epizootic dynamics of the chytrid fungal pathogen of amphibians. Proc.

Natl Acad. Sci. USA, 107, 9695–9700.Briggs, C.J., Vredenburg, V.T., Knapp, R.A. & Rachowicz, L.J. (2005).

Investigating the populationlevel effects of chytridiomycosis: an

emerging infectious disease of amphibians. Ecology, 86, 3149–3159.Brunner, J.L., Beaty, L., Guitard, A. & Russell, D. (2017).

Heterogeneities in the infection process drive ranavirus transmission.

Ecology, 98, 576–582.Caswell, H. (2001). Matrix Population Models: Construction, Analysis, and

Interpretation. 2nd edn. Sinauer, Sunderland, MA.

Clare, F., Daniel, O., Garner, T. & Fisher, M. (2016). Assessing the

ability of swab data to determine the true burden of infection for the

amphibian pathogen Batrachochytrium dendrobatidis. EcoHealth, 13,

360–367.Cleaveland, S., Hess, G.R., Dobson, A.P., Laurenson, M.K., McCallum,

H.I., Roberts, M.G. & Woodroffe, R. (2001). The role of pathogens in

biological conservation. In: The Ecology of Wildlife Diseases (eds

Hudson, P.J., Rizzoli, A., Grenfell, B., Heesterbeek, H. & Dobson, A.).

Oxford University Press, Oxford, pp. 139–150.Courtois, E.A., Loyau, A., Bourgoin, M. & Schmeller, D.S. (2017).

Initiation of Batrachochytrium dendrobatidis infection in the absence of

physical contact with infected hosts - a field study in a high altitude

lake. Oikos, 126, 843–851.Daszak, P., Cunningham, A.A. & Hyatt, A.D. (2003). Infectious disease

and amphibian population declines. Divers. Distrib., 9, 141–150.De Castro, F. & Bolker, B. (2005). Mechanisms of disease-induced

extinction. Ecol. Lett., 8, 117–126.Diekmann, O., Heesterbeek, J.A.P. & Metz, J.A.J. (1990). On the

definition and the computation of the basic reproduction ratio R0 in

models for infectious diseases in heterogeneous populations. J. Math.

Biol., 28, 365–382.

Dietz, K. (1993). The estimation of the basic reproduction number for

infectious diseases. Stat. Methods Med. Res., 2, 23–41.Doddington, B.J., Bosch, J., Oliver, J.A., Grassly, N.C., Garcia, G.,

Schmidt, B.R. et al. (2013). Context-dependent amphibian host

population response to an invading pathogen. Ecology, 94, 1795–1804.

Dwyer, G., Elkinton, J.S. & Buonaccorsi, J.P. (1997). Host heterogeneity

in susceptibility and disease dynamics: tests of a mathematical model.

Am. Nat., 150, 685–707.Easterling, M.R., Ellner, S.P. & Dixon, P.M. (2000). Size-specific

sensitivity: applying a new structured population model. Ecology, 81,

694–708.Ellison, A.R., Tunstall, T., Direnzo, G.V., Hughey, M.C., Rebollar, E.A.,

Belden, L.K. et al. (2015). More than skin deep: functional genomic

basis for resistance to amphibian chytridiomycosis. Genome Biol. Evol.,

7, 286–298.Epstein, B., Jones, M., Hamede, R., Hendricks, S., McCallum, H.,

Murchison, E.P. et al. (2016). Rapid evolutionary response to a

transmissible cancer in Tasmanian devils. Nat. Commun., 7, 12684.

Forrest, M.J. & Schlaepfer, M.A. (2011). Nothing a hot bath won’t

cure: infection rates of amphibian chytrid fungus correlate negatively

with water temperature under natural field settings. PLoS ONE, 6,

e28444.

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A. &

Rubin, D.B. (2014). Bayesian Data Analysis. 3rd edn. Taylor & Francis

Group, LLC, Boca Raton.

Gerber, L.R., McCallum, H., Lafferty, K.D., Sabo, J.L. & Dobson,

A. (2005). Exposing extinction risk analysis to pathogens: is disease

just another form of density dependence? Ecol. Appl., 15, 1402–1414.

Godfray, H.C.J., Briggs, C.J., Barlow, N.D., O’Callaghan, M., Glare,

T.R. & Jackson, T.A. (1999). A model of insect-pathogen dynamics in

which a pathogenic bacterium can also reproduce saprophytically. Proc.

Royal Society B: Biol. Sci., 266, 233–240.Grogan, L.F., Phillott, A.D., Scheele, B.C., Berger, L., Cashins, S.D.,

Bell, S.C.et al. (2016). Endemicity of chytridiomycosis features

pathogen overdispersion. J. Anim. Ecol., 85, 806–816.Handel, A. & Rohani, P. (2015). Crossing the scale from within-host

infection dynamics to between-host transmission fitness: a discussion of

current assumptions and knowledge. Philos. Trans. R. Soc. Lond. B

Biol. Sci., 370, 20140302.

Harper, E.B., Stella, J.C. & Fremier, A.K. (2011). Global sensitivity

analysis for complex ecological models: a case study of riparian

cottonwood population dynamics. Ecol. Appl., 21, 1225–1240Harris, R.N., Brucker, R.M., Walke, J.B., Becker, M.H., Schwantes,

C.R., Flaherty, D.C. , (2009). Skin microbes on frogs prevent

morbidity and mortality caused by a lethal skin fungus. ISME J., 3,

818–824.Heard, G.W., Scroggie, M.P., Clemann, N. & Ramsey, D.S.L. (2014).

Wetland characteristics influence disease r Wetland characteristics

influence disease risk for a threatened amphibian. Ecol. Appl., 24, 650–662.

Hoyt, J.R., Langwig, K.E., Sun, K., Lu, G., Parise, K.L., Jiang, T. et al.

(2016). Host persistence or extinction from emerging infectious disease:

insights from white-nose syndrome in endemic and invading regions.

Proc. Biol. Sci., 283, 20152861.

Jani, A.J. & Briggs, C.J. (2014). The pathogen Batrachochytrium

dendrobatidis disturbs the frog skin microbiome during a natural

epidemic and experimental infection. Proc. Natl Acad. Sci., 111, E5049–E5058.

Jenkinson, T.S., Betancourt Rom�an, C.M., Lambertini, C., Valencia-

Aguilar, A., Rodriguez, D., Nunes- De-Almeida, C.H.L. et al. (2016).

Amphibian-killing chytrid in Brazil comprises both locally

endemic and globally expanding populations. Mol. Ecol., 25, 2978–2996.

Kilpatrick, A.M. (2006). Facilitating the evolution of resistance to avian

malaria in Hawaiian birds. Biol. Cons., 128, 475–485.

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1179

Kilpatrick, A.M., Briggs, C.J. & Daszak, P. (2010). The ecology and

impact of chytridiomycosis: an emerging disease of amphibians. Trends

Ecol. Evol., 25, 109–118.Knapp, R.A., Briggs, C.J., Smith, T.C. & Maurer, J.R. (2011).

Nowhere to hide: impact of a temperature-sensitive amphibian

pathogen along an elevation gradient in the temperate zone.

Ecosphere, 2, art93.

Knapp, R.A., Fellers, G.M., Kleeman, P.M., Miller, D.A.W.,

Vredenburg, V.T., Rosenblum, E.B. et al. (2016). Large-scale recovery

of an endangered amphibian despite ongoing exposure to multiple

stressors. Proc. Natl Acad. Sci., 113, 11889–11894.Lande, R., Engen, S. & Saether, B.-E. (2003). Stochastic Population

Dynamics in Ecology and Conser- vation. Oxford University Press,

Oxford.

Langhammer, P.F., Lips, K.R., Burrowes, P.A., Tunstall, T., Palmer,

C.M. & Collins, J.P. (2013). A fungal pathogen of amphibians,

Batrachochytrium dendrobatidis, attenuates in pathogenicity with

in vitro passages. PLoS ONE, 8, 1–9.Langwig, K.E., Hoyt, J., Parise, K., Frick, W., Foster, J. & Kilpatrick,

A.M. (2017). Resistance in persisting bat populations after white-nose

syndrome invasion. Philos. Trans. R. Soc. Lond., B, Biol. Sci., 372,

20160044.

Langwig, K.E., Voyles, J., Wilber, M.Q., Frick, W.F., Murray, K.A.,

Bolker, B.M. et al. (2015). Context-dependent conservation

responses to emerging wildlife diseases. Front. Ecol. Environ., 13,

195–202.Lee, Y.-S., Liu, O.Z., Hwang, H.S., Knollmann, B.C. & Sobie, E.A.

(2013). Parameter sensitivity analysis of stochastic models provides

insights into cardiac calcium sparks. Biophys. J., 104, 1142–1150.Louie, A., Song, K.H., Hotson, A., Thomas Tate, A. & Schneider, D.S.

(2016). How many parameters does it take to describe disease

tolerance? PLoS Biology, 14, e1002435.

McCallum, H. (2012). Disease and the dynamics of extinction. Philos.

Trans. R. Soc. Lond., B, Biol. Sci., 367, 2828–39.McCallum, H., Barlow, N. & Hone, J. (2001). How should pathogen

transmission be modelled? Trends Ecol. Evol., 16, 295–300.McCallum, H. & Dobson, A. (1995). Detecting disease and parasite

threats to endangered species and ecosystems. Trends Ecol. Evol., 10,

190–194.McCallum, H., Jones, M., Hawkins, C., Hamede, R., Lachish, S., Sinn,

D.L. et al. (2009). Transmission dynamics of Tasmanian devil facial

tumor disease may lead to disease-induced extinction. Ecology, 90,

3379–3392.Medzhitov, R., Schneider, D.S. & Soares, M.P. (2012). Disease tolerance

as a defense strategy. Science, 335, 936–941.Metcalf, C.J.E., Graham, A.L., Martinez-Bakker, M. & Childs, D.Z.

(2015). Opportunities and challenges of Integral Projection Models for

modelling host-parasite dynamics. J. Anim. Ecol., 85, 343–355.Mitchell, K.M., Churcher, T.S., Garner, T.W.J. & Fisher, M.C. (2008).

Persistence of the emerging pathogen Batrachochytrium dendrobatidis

outside the amphibian host greatly increases the probability of host

extinction. Proc. R. Soc. B, 275, 329–334.Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B.,

Grisel, O. et al. (2011). Scikit-learn: machine learning in python. J.

Mach. Learn. Res., 12, 2825–2830.Piotrowski, J.S., Annis, S.L. & Longcore, J.E. (2004). Physiology of

Batrachochytrium dendrobatidis, a chytrid pathogen of amphibians.

Mycologia, 96, 9–15.Rachowicz, L.J. & Briggs, C.J. (2007). Quantifying the disease

transmission function: effects of density on Batrachochytrium

dendrobatidis transmission in the mountain yellow-legged frog Rana

muscosa. J. Anim. Ecol., 76, 711–21.Refsnider, J.M., Poorten, T.J., Langhammer, P.F., Burrowes, P.A. &

Rosenblum, E.B. (2015). Genomic correlates of virulence attenuation in

the deadly amphibian chytrid fungus, Batrachochytrium dendrobatidis.

G3 (Bethesda), 5, 2291–2298.

Retallick, R.W.R., McCallum, H. & Speare, R. (2004). Endemic infection

of the amphibian chytrid fungus in a frog community post-decline.

PLoS Biol., 2, e351.

Rohani, P., Breban, R., Stallknecht, D.E. & Drake, J.M. (2009).

Environmental transmission of low pathogenicity avian influenza

viruses and its implications for pathogen invasion. Proc. Natl Acad.

Sci., 106, 10365–10369.Rosenblum, E.B., James, T.Y., Zamudio, K.R., Poorten, T.J., Ilut, D.,

Rodriguez, D. et al. (2013). Complex history of the amphibian-killing

chytrid fungus revealed with genome resequencing data. Proc. Natl

Acad. Sci., 110, 9385–90.Roy, B.A. & Kirchner, J.W. (2000). Evolutionary eynamics of pathogen

resistance and tolerance. Evolution, 54, 51–63.Savage, A.E. & Zamudio, K.R. (2011). MHC genotypes associate with

resistance to a frog-killing fungus. Proc. Natl Acad. Sci., 108, 16705–16710.

Savage, A.E. & Zamudio, K.R. (2016). Adaptive tolerance to a

pathogenic fungus drives major histocompatibility complex

evolution in natural amphibian populations. Proc. R. Soc. B, 283,

20153115.

Schmeller,D.S.,Blooi,M.,Martel,A.,Garner,T.W.J.,Fisher,M.C.,Azemar,

F. et al. (2014). Microscopic aquatic predators strongly affect infection

dynamicsofagloballyemergedpathogen.Curr.Biol., 24,176–180.Schreiber, S.J. & Ross, N. (2016). Individual-based integral projection

models: the role of size-structure on extinction risk and establishment

success. Methods Ecol. Evol., 7, 867–874.Skerratt, L.F., Berger, L., Speare, R., Cashins, S., McDonald, K.R.,

Phillott, A.D. et al. (2007). Spread of chytridiomycosis has caused

the rapid global decline and extinction of frogs. EcoHealth, 4, 125–134.

Smith, K.F., Sax, D.F. & Lafferty, K.D. (2006). Evidence for the role of

infectious disease in species extinction and endangerment. Conserv.

Biol., 20, 1349–1357.Sobie, E.A. (2009). Parameter sensitivity analysis in electrophysiological

models using multivariable regression. Biophys. J., 96, 1264–1274.Stockwell, M.P., Clulow, J. & Mahony, M.J. (2010). Host species

determines whether infection load increases beyond disease-causing

thresholds following exposure to the amphibian chytrid fungus. Anim.

Conserv., 13, 62–71.Strauss, A. & Smith, K.G. (2013). Why does amphibian chytrid

(Batrachochytrium dendrobatidis) not occur everywhere? An exploratory

study in missouri ponds. PLoS ONE, 8, e76035.

Tunstall, T.S. (2012). Characteristics of the emergent disease

Batrachochytrium dendrobatidis in Rana muscosa and Rana sierrae

species complex. Ph.D. thesis, University of California, Berkeley.

Vander Wal, E., Garant, D., Calm�e, S., Chapman, C.A., Festa-Bianchet,

M., Millien, V. et al. (2014). Applying evolutionary concepts to wildlife

disease ecology and management. Evol. Appl., 7, 856–868.Venesky, M.D., Liu, X., Sauer, E.L. & Rohr, J.R. (2013). Linking

manipulative experiments to field data to test the dilution effect. J.

Anim. Ecol., 83, 557–565.Voyles, J., Berger, L., Young, S., Speare, R., Webb, R., Warner, J.,

Rudd, D., Campbell, R. & Skerratt, L. (2007). Electrolyte depletion

and osmotic imbalance in amphibians with chytridiomycosis. Dis.

Aquat. Organ., 77, 113–118.Voyles, J., Johnson, L.R., Briggs, C.J., Cashins, S.D., Alford, R.A.,

Berger, L. et al. (2014). Experimental evolution alters the rate and

temporal pattern of population growth in Batrachochytrium

dendrobatidis, a lethal fungal pathogen of amphibians. Ecol. Evol., 4,

3633–3641.Voyles, J., Young, S., Berger, L., Campbell, C., Voyles, W.F. &

Dinudom, A. (2009). Pathogenesis of chytridiomycosis, a cause of

catastrophic amphibian declines. Science, 326, 5–8.Vredenburg, V.T., Knapp, R.A., Tunstall, T.S. & Briggs, C.J. (2010).

Dynamics of an emerging disease drive large-scale amphibian

population extinctions. Proc. Natl Acad. Sci., 107, 9689–94.

© 2017 John Wiley & Sons Ltd/CNRS

1180 M. Q. Wilber et al. Letter

Wilber, M.Q., Langwig, K.E., Kilpatrick, A.M., Mccallum, H.I. &

Briggs, C.J. (2016). Integral Projection Models for host-parasite systems

with an application to amphibian chytrid fungus. Methods Ecol. Evol.,

7, 1182–1194.Woodhams, D.C., Alford, R.A., Briggs, C.J., Johnson, M. & Rollins-

Smith, L.A. (2008). Life-history trade-offs influence disease in

changing climates: strategies of an amphibian pathogen. Ecology, 89,

1627–1639.Woodhams, D.C., Bosch, J., Briggs, C.J., Cashins, S., Davis, L.R.,

Lauer, A. et al. (2011). Mitigating amphibian disease: strategies to

maintain wild populations and control chytridiomycosis. Front. Zool.,

8, 8.

SUPPORTING INFORMATION

Additional Supporting Information may be found online inthe supporting information tab for this article.

Editor, Jos�e Mar�ıa GomezManuscript received 3 April 2016First decision made 12 May 2017Manuscript accepted 21 June 2017

© 2017 John Wiley & Sons Ltd/CNRS

Letter Resistance, tolerance and host extinction 1181

Appendices1

1 The mesocosm experiment and the latent zoospore pool2

transmission model3

1.1 The mesocosm experiment4

We performed an outdoor mesocosm experiment in summer 2008 to determine if adult frog den-5

sity influenced Bd infection prevalence and the rate of increase in the fungal load on individuals6

through time in Sierra Nevada yellow legged frogs, Rana sierrae. The experiment was performed7

in four 1.2 m wide x 4.8 m long x 1.2 m deep concrete channels at the Sierra Nevada Aquatic8

Research Laboratory (University of California Natural Reserve System, Mammoth Lakes, CA,9

N37.614176 W118.833452, elevation 2167m). Channels were subdivided with plywood into 410

isolated tanks measuring 1.2 m3 in each channel, for a total of 16 tanks in four blocks. Tanks11

were each filled with approximately 1400 liters of water from adjacent Convict Creek, a Bd-12

negative stream originating from nearby Convict Lake. Sloping shelves (approximately 30cm13

wide) extending from below water to a few cm above the water surface were provided on the14

south-facing wall of each tank for the frogs to bask on during the daylight hours. Lids were15

constructed from wood frames covered in fiberglass screen and wire mesh.16

Infected Rana sierrae tadpoles were collected from Conness Pond, a large, Bd-positive site17

in Yosemite National Park (N37.97485 W119.31134, elevation 3193m). The tadpoles were used18

as the initial source of infection in each tank. Uninfected (Bd-naıve) adults were collected from19

Marmot Lake, (Sierra National Forest, N37.259860 W118.683379, elevation 3589m). All animals20

were swabbed to confirm initial Bd infection status (infected for the tadpoles, and uninfected for21

the adults). Adult frogs were individually marked with Passive Integrative Transponder (PIT)22

tags, which allowed each adult to be uniquely identified. On July 7, 2008, five infected tadpoles23

were added to each tank. Nine days later, tanks were drained 90% and re-filled with fresh24

water from Convict Creek to remove any tannins leached from the plywood and to remove extra25

tadpole food. On July 20, 2008, adult frogs were added to each tank in one of four densities:26

1, 4, 8, and 16 (this is counted as Day 0 of the experiment for all of the analyses in the main27

text). The four tanks in each block received one of each of the four density treatments, assigned28

randomly within each block. Preliminary analyses did not detect a significant effect of block and29

thus the statistical models presented below do not include a block effect.30

Skin swabs were collected from all animals once per week to determine Bd infection status,31

1

using a standardized swabbing protocol (Briggs et al. 2010; Vredenburg et al. 2010). For post-32

metamorphic individuals, a sterile synthetic swab was brushed across the hind feet (concentrating33

on the toe webbing), hind legs, and each side of the abdomen, 5 times each, for a total of 3034

strokes. For tadpoles, the swab was brushed across the mouthparts for 30 strokes. Swabs were35

allowed to air dry, and then were placed in individual tubes, and frozen as soon as possible.36

The number of Bd DNA copies on each swab was determined using quantitative real-time PCR,37

following the protocol of Boyle et al. (2004), as in Briggs et al. (2010). During each weekly38

swabbing event, the weight and snout-to-vent length were recorded for each adult. Tadpoles39

were fed rabbit chow ad libitum and adults were fed Phoenix worms ad libitum during the40

swabbing events.41

1.2 The latent zoospore pool transmission model42

Using the mesocosm transmission experiment described above and in the main text, we sought43

to characterize the transmission function for Bd and R. muscosa. In particular, we wanted to44

quantify how the environmental zoospore pool affected the probability of an amphibian tran-45

sitioning from uninfected to infected in a time step. However, we were unable to measure the46

total number of zoospores in the environmental zoospore pool at each time step and therefore47

assumed that the zoospore pool was a latent variable that needed to be estimated. We estimated48

it by assuming that the zoospore pool obeyed the following dynamics49

Zj(t+ ∆t) = Zj(t) exp(−d∗∆t) + Zj,tadpoles(t)f∗T∆t+ Zj,adults(t)f

∗A∆t (1)

This equation assumes that the dynamics of the unobserved zoospore pool in mesocosm j are50

governed by the survival probability of zoospores in the pool at time t (first term, d∗ is zoospore51

death rate) and contribution of zoospores into the pool based on the total number of zoospores52

on tadpoles (Zj,tadpoles(t)) and adults (Zj,adults(t)) at time t in mesocosm j. f∗T and f∗A are rates53

that relate the total number of zoospores on tadpoles and frogs at time t (as estimated from54

qPCR) to the total number of zoospores that enter the pool. This equation does not include55

the reduction of zoospores in the pool due to frogs acquiring them upon initial infection as we56

found that the small number acquired is inconsequential for the dynamics of the zoospore pool.57

For the ith individual frog in mesocosm j at time t+ 1, the likelihood of gaining an infection58

is given by59

yij(t+ ∆t) ∼ Bernoulli(φj(t)) if yij(t) = 0 (2)

2

Each individual frog i in tank j was swabbed six times over the course of 32 days and yij(t)60

specifies whether frog i in mesocosm j was infected (1) or uninfected (0) at time t+ ∆t, having61

been uninfected at time t. Only observations where an animal was uninfected in the previous62

time step directly contributed to the likelihood (n = 333), but the entire time series for all63

individuals within a mesocosm contributed to the dynamics of the zoospore pool.64

φj(t) is the probability of infection in mesocosm j at time t and takes the general functional65

form66

φj(t) = 1 − exp(−Λ(Zj(t), Aj(t), Ij(t))∆t) (3)

where the probability of infection depends on the total number of zoospores in the pool (Zj(t)),67

the total number of adults (Aj(t)), and the total number of infecteds (Ij(t)) in tank j at time68

t. The number of adults and total number of infecteds are both observed (i.e. measured in the69

experiment) at each time t. In contrast, Zj(t) is unobserved at each time t. The specific forms70

of Λ(Zj(t), Aj(t), Ij(t)) that we considered are given in Table 2 in the main text. We assumed71

that transmission was temperature-independent because we were not able to simultaneously72

manipulate host density and temperature in the mesocosm experiment.73

We assumed that the total number of zoospores Zj(t) is a random variable with a lognormal74

distribution such that75

ln(Zj(t+ ∆t)) ∼ Normal(ln(µj(t+ ∆t)) − σ2

2), σ2) (4)

µj(t+ ∆t) = Z(t)j exp(−d∗∆t) + Ztadpoles(t)jf∗T∆t+ Zadults(t)jf

∗A∆t (5)

Ztadpoles(t)j and Zadults(t)j were both observed at each time t. The σ2

2 terms is a result of76

converting from the expected value of the lognormal distribution to the mean of the normal77

distribution on the log scale.78

To fit these models, we assumed that the initial zoospore load in the pool was given by79

lnZj(t = 0) ∼ Normal(ln(2000), σ2). We specified this prior distribution based on the average80

number zoospores on the tadpoles at the beginning of the experiment across all mesocosms.81

We gave the zoospore death rate d∗ day−1 a normal prior distribution with mean 0.3 day−1,82

standard deviation 0.03, and a lower bound of 0. This tight prior was based on laboratory83

estimates of the zoospore death rate (Woodhams et al. 2008). To aid in the identifiability of our84

model, we set σ = 1 and fT = fA = f and gave f a vague half Cauchy prior distribution with a85

3

scale parameter equal to 1.86

We fit this model for each of the φ functions given in Table 2 in the main text using Hamil-87

tonian Monte Carlo with the RStan package (2.12.1). Three chains were run for each model and88

we assessed convergence of the model parameters by determining if the Gelman-Rubin statis-89

tic R was less than 1.05 (Gelman et al. 2014). We also confirmed that this statistical model90

could recover known transmission functions by simulating our mesocosm experiment in silico91

and testing whether the above model could both recover known parameters in φ and also cor-92

rectly distinguish between different forms of φ using information criteria (see accompanying code93

code/simulate lab data.py at https://github.com/mqwilber/host extinction/).94

We also tested how robust our conclusions were to the assumption that σ = 1 using two95

different approaches. First, we allowed this parameter to have a minimally informative half96

Cauchy prior with a scale parameter equal to three. This allowed for enormous variability in97

the dynamics of the zoospore pool. Incorporating this vague prior led to larger mean estimates98

of transmission from the zoospore pool (i.e. larger β0), but also larger uncertainty around this99

estimate. However, the relative ranking of the different transmission models given in Table 2 in100

the main text did not change. The models with a dynamic zoospore pool were always better101

than corresponding models without the zoospore pool and the density-dependent model with a102

dynamic zoospore model was the best model followed by the frequency-dependent model with a103

zoospore pool.104

Second, we explored how fixed values of σ ranging from 0.25 to 4 affected the relative rank105

of the models as well as the estimates of the coefficients (Figure 7, 8). Across a range of values106

of σ and two different measures of information criteria (WAIC and DIC), the relative rankings107

of the three transmission models with a dynamic zoospore pool (Table 2 in main text) stayed108

largely consistent, with some notable deviations between σ = 1.5 and σ = 2 (Fig. 7). We used109

two different information criteria as there is some question over WAIC’s accuracy for time series110

models (Gelman et al. 2014). For values of σ ≤ 1 the estimates of the transmission coefficients111

from the zoospore pool (β0) remained relatively constant, but began increasing for σ > 1 due112

to the large variability in the zoospore pool dynamics (Fig. 8A). In contrast, the coefficient113

estimates for the density/frequency dependent transmission stayed relatively constant with a114

slight decreasing trend with increasing σ (Fig. 8B). Considering these sensitivity analyses, we115

felt that σ = 1 was a reasonable choice because 1) it did not lead to unrealistically large variability116

in the zoospore pool as did the uninformative prior on σ 2) it provided a conservative estimate117

for effect of the zoospore pool on transmission 3) the density/frequency dependent transmission118

4

coefficient β1 was largely unaffected by the choice and 4) it resulted in model ranks that were119

consistent between two different information criteria and largely consistent across most values120

of σ that we explored.121

2 R0 for host-parasite IPMs with an environmental reser-122

voir123

2.1 R0 for a discrete-time SIS model with an environmental reservoir124

Wilber et al. (2016) showed how to calculate R0 for basic host-parasite IPMs using the general125

methodology developed in Klepac & Caswell (2011). This method can be extended to calculate126

R0 for an IPM with an environmental reservoir. To begin, we illustrate the procedure with a127

simple discrete time Susceptible-Infected-Susceptible (SIS) model with a dynamic zoospore pool128

(Z). Take the following set of discrete dynamical equations129

S(t+ 1) = S(t)s0[1 − φ(I(t), Z(t))] + I(t)sI lI (6)

I(t+ 1) = I(t)sI(1 − lI) + S(t)s0φ(I(t), Z(t)) (7)

Z(t+ 1) = Z(t)ν + I(t)f (8)

where s0 is the survival probability for an uninfected host in a time step, sI is the survival130

probability for an infected host in a time step, lI is the probability of losing an infection in a131

time step, ν is the survival probability of a zoospore in a time step, f is the average number of132

zoospores produced by an infected individual in a time step, and φ(I(t), Z(t)) is the probability133

of gaining an infection in a time step. Based on the results from the transmission experiment134

given in Table 2 in the main text, we assume that transmission depends on density-dependent135

host to host contact as well as transmission from the zoospore pool. Given this, we can write136

φ(I(t), Z(t)) = 1 − exp[−(β1I(t) + β0Z(t))] (9)

We can calculate R0 for the above system of equations by rewriting them as the following137

matrix model138

5

SP

(t+ 1) =

0 0

M(P(t)) U

SP

(t) (10)

where the zeros simply indicate that these values do not contribute to the calculation of R0, not139

that they are actually zero in the model. P(t) is the vector [I(t) Z(t)]T and 0 is a row vector140

[0 0]. Just focusing on the vector P, we can write141

P(t+ 1) = M(P(t))S(t) + UP(t) (11)

M(P(t)) is the vector [s0φ(P(t)) 0]T . U is the 2 x 2 matrix142

U =

sI(1 − lI) 0

f ν

(12)

To calculate R0, we linearize P(t + 1) about a vector n∗. We set this vector to be a host143

population with only susceptibles n∗ = [S∗ 0] (Rohani et al. 2009; Klepac & Caswell 2011),144

where 0 is a vector of zeros of length n = 2. We then compute the Jacobian matrix evaluated145

at n∗146

J =dP(t+ 1)

dP(t)

∣∣∣∣n∗

(13)

which allows us to compute R0 (Klepac & Caswell 2011). Computing this Jacobian requires147

computingdM(P(t))S(t)

dP(t)and

dUP(t)

dP(t).148

We immediately see thatdUP(t)

dP(t)

∣∣∣∣n∗

= U.dM(P(t))S(t)

dP(t)

∣∣∣∣n∗

requires application of the149

chain rule for differentiation and we see that150

dM(P(t))S(t)

dP(t)

∣∣∣∣n∗

= M∗ =

s0S∗β1 s0S∗β0

0 0

(14)

R0 is given by (Klepac & Caswell 2011)151

R0 = max eig(M∗(1 −U)−1) (15)

and plugging into M∗ and U given above we get152

R0 =s0S∗β1

1 − sI(1 − lI)+

s0S∗β0f

(1 − ν)[1 − sI(1 − lI)](16)

Similar to the continuous time result given in Rohani et al. (2009), we see that R0 is a153

6

combination of both transmission from the hosts (first term) and the environment (second term).154

Setting β0 = 0, we recover the simple SIS R0 given in Wilber et al. (2016) and Oli et al. (2006).155

2.2 R0 for the host-parasite IPM with an environmental reservoir156

To generalize this to a host-parasite IPM model with an environmental reservoir where the I class157

is now potentially infinitely many classes, we can take the following steps. First, we recognize158

that in practice IPMs are analyzed using the mid-point rule (Easterling et al. 2000) such that159

there are a finite number of I classes, namely n classes. Therefore, we can think of our IPM as160

a generalization of the SISZ model presented above such that P(t) is a vector of length n + 1,161

M(P(t)) is now a vector of length (n+ 1) and U is a (n+ 1) x (n+ 1) matrix. The plus one is162

because the environmental reservoir Z is part of the P vector.163

For the first n x n elements in U, the element in the ith row and the jth column is given by164

uij = s(xj)(1 − l(xj))G(xi, xj)∆ (17)

which gives the probability of an individual in the jth load class with parasite load xj , surviving165

(s(xj)), not losing its infection (1 − l(xj)), and transitioning to the load class of xi in a time166

step (G(xi, xj)). ∆ is a result of using the midpoint rule to discretize the continuous IPM and167

is used to convert the probability density G(xi, xj) into a probability (e.g. see Easterling et al.168

2000; Wilber et al. 2016). The n+ 1th row of U is the vector [fxx ν] of length n+ 1, where we169

assume that an infected host in class j produces an average of fxxj parasites in a time step into170

the zoospore pool. x is a vector of length n that contains the corresponding parasite loads for171

the n infected classes. The n+ 1th column of U is the vector [0 ν]T , where 0 is of length n.172

M(P(t)) is given by a vector of length n + 1 where the first i = 1, . . . , i, . . . n elements are173

given by174

mi = s0φ(βT I(t))G0(xi)∆ (18)

where βT is a vector of length n+ 1 with the first n elements being β1 and the n+ 1th element175

being β0.dM(P(t))S(t)

dP(t)

∣∣∣∣n∗

= M∗ is given by a (n+1) x (n+1) matrix where the first n columns176

are given by [s0S∗β1G0(x)∆ 0]T and the n + 1th column is given by [s0S

∗β0G0(x)∆ 0]T .177

G0(x) indicates that the function G0(x) is evaluated at each element in x, resulting in a vector178

of length n.179

Given M∗ and U we can again calculate R0 as (Klepac & Caswell 2011)180

7

R0 = max eig(M∗(1 −U)−1) (19)

In the main text, we use this formulation to numerically calculate R0 for the R. muscosa-Bd181

IPM model with and without an environmental reservoir.182

3 The hybrid model183

3.1 Within-year component of the hybrid model184

The model described by equations 6-8 in the main text may be sufficient to describe the dynamics185

of an initial epizootic, but in order to examine Bd-induced extinction dynamics in R. muscosa186

populations a number of additions need to be made. First, the tadpole stage of R. muscosa has187

been shown to play an important role in generating enzootic dynamics in R. muscosa populations188

(Briggs et al. 2005, 2010). R. muscosa can spend three years as tadpoles and thus we include189

three additional tadpole stages into the model T1, T2, and T3 (Briggs et al. 2005). Considering190

equations 6-8, we can add this tadpole class and update our zoospore pool equation as follows191

Ti(t+ 1) = Ti(t)sTi (20)

Z(t+ 1) = Z(t)ν +

3∑i=1

µTiTi(t) + µA

∫ Ux

Lx

exp(x)IA(x, t)dx− ψ(SA(t), Z(t)) (21)

The equation Ti(t + 1) describes how the number of tadpoles in class i changes from time192

t to time t + 1. We do not explicitly model the fungal load on tadpoles. Instead, we assume193

all tadpoles are immediately infected with Bd and have a constant contribution to the zoospore194

pool. This is justified by the observation that most tadpoles in R. muscosa populations carry195

high fungal loads, even in enzootic populations (Briggs et al. 2010). R. muscosa tadpole survival196

is not affected by Bd infection. Therefore, the within-season dynamics of Ti are simply given by197

the probability of a tadpole surviving from time t to t + 1, which is sTi. Notice that infected198

tadpoles are now also contributing µTi zoospores to the zoospore pool at each time step. Previous199

models of this system have included a subadult stage after metamorphosis that can last 1-2 years200

(Briggs et al. 2005, 2010). For simplicity, we are ignoring it here.201

R. muscosa populations also experience seasonal temperature fluctuations in which lake202

temperatures drop to approximately 4 ◦C in the winter (in the unfrozen portion of a lake where203

8

the frogs overwinter) and reach approximately 20 ◦C in the summer (Knapp et al. 2011). We204

account for this seasonal variability by imposing a deterministically fluctuating environment on205

the R. muscosa-Bd IPM. We assumed that temperature follows a sinusoidal curve with a period206

of 1 year and a minimum temperature of 4 ◦C and a maximum temperature of 20 ◦C (Fig.207

1A). At each discrete time point within a season, a new temperature is calculated based on the208

sinusoidal curve and the temperature-dependent vital rate functions are updated accordingly (see209

Table 1 in the main text and Appendix Fig. 1 for temperature-dependent vital-rate functions).210

3.2 Between-year component of the hybrid model211

In the within-year component of the hybrid model a time step is 3 days. This time step is212

on the same scale as Bd dynamics. However, R. muscosa demography occurs on a slower213

scale. We assume that R. muscosa demographic dynamics occur on a yearly time scale (Briggs214

et al. 2005), such that once a year tadpoles either age a year or metamorphose and adults215

reproduce. The mortality for each stage as well as all disease dynamics are accounted for in the216

within-year component of the IPM, thus the between-year component only includes the following217

demographic events (Fig. 1C in the main text)218

T1(t+ 1) = pAλ0SA(t) + pA

∫ Ux

Lx

λ(x)I(x, t)dx (22)

T2(t+ 1) = pT1T1(t) (23)

T3(t+ 1) = pT2T2(t) (24)

SA(t+ 1) = [(1 − pT1)mT1

T1(t) + (1 − pT2)mT2

T2(t) +mT3T3(t)] exp(−A(t)/K) (25)

IA(x′, t+ 1) = IA(x′, t) (26)

Z(t+ 1) = Z(t) (27)

where these events only occur once per year (Fig. 1 in the main text). When these demographic219

events occur each year, they occur after the disease dynamics defined above, but in the same time220

step from t to t+1. Equation 22 gives the contribution of adult frogs to the T1 tadpole class. pA221

defines the probability of an adult reproducing, λ0 is the mean reproductive output of uninfected222

adults and λ(x) is the mean reproductive output of an adult with a Bd load of x. Equation223

23 gives the probability of a T1 tadpole not metamorphosing (pT1) and transitioning to a T2224

tadpole. Equation 24 describes the changes in the T3 class via the probability of a T2 tadpole225

9

not metamorphosing (pT2) and transitioning to a T3 tadpole. Equation 25 describes T1, T2226

and T3 tadpoles metamorphosing, surviving metamorphosis, and recruiting as uninfected adults.227

Recruitment of tadpoles to the adult stage is a density-dependent process following a Ricker228

function with adult density (A(t)) and a parameter that is proportional to the carrying capacity229

(K) (Briggs et al. 2005). Because we have no empirical evidence for Bd-induced fertility reduction230

in R. muscosa, we assumed that reproduction in uninfected adults was the same as reproduction231

in infected adults. We also assume that tadpoles lose their infection during Bd metamorphosis232

(Briggs et al. 2010). Finally, equations 26 and 27 indicate that infected individuals (IA(x′, t+1))233

and the zoospore pool (Z(t+ 1)) do not change during the demographic update.234

4 Converting the hybrid model into an individual-based235

model with demographic stochasticity236

4.1 Details of the individual-based model237

The simplest way to include demographic stochasticity into the hybrid model is to assume238

that demographic stochasticity is given by sampling error and allow all demographic transitions239

to occur following some probability distribution (Caswell 2001; Schreiber & Ross 2016). To240

illustrate this, recall that to analyze the hybrid Integral Projection Model we discretized the241

continuous class∫ baI(x, t)dx (which gives the number of frogs with a ln Bd load between a242

and b at time t) into 30 load classes using the midpoint rule (Easterling et al. 2000). We can243

now loosely think about our hybrid model as a matrix model with 3 tadpole + 1 susceptible244

adult + 30 infected adult + 1 zoospore pool = 35 stages. Let’s call the transition matrix A.245

Following Caswell (2001) and Schreiber & Ross (2016), we can decompose our hybrid model into246

components for infection dynamics (i.e. growth, loss, initial infection gain), host survival, disease247

transmission, and demographic transitions (T) and reproduction (F) such that A = T+F. The248

matrix T gives the probabilities of an individual in stage j transitioning to stage i in a 3 day249

time step. However, T is not a stochastic matrix (i.e. the columns do not sum to one) because250

individuals in each stage also have a survival probability and one potential transition during each251

time step is to a “dead” class. We can augment this matrix T with an extra row d specifying the252

“dead” class where each entry in this row can be defined as d36,j = 1−sum(T,j). sum(T,j) gives253

the sum of the jth column of the T matrix. The new augmented matrix T∗ is fully stochastic254

(Caswell 2001; Schreiber & Ross 2016). Assuming each individual transitions independently of255

each other, for each time step t the transition of a single individual in class j to another class256

10

i (including the “dead” class) follows a multinomial distribution with a probability vector pj257

given by the jth column of T∗: pj = T∗,j. At any time step we could simulate what happens258

to n individuals in class j by drawing from a multinomial distribution with the total number of259

trials equal to n and the probability vector equal to pj. The resulting random vector would be260

of length 36 specifying the stages that the n individuals of stage j at time t now occupy at time261

t+ 1 (including death).262

In addition to individual transitions (or “births”), we also need to account for the births263

defined in the F matrix. In the hybrid model, there are two types of births we account for264

in the F matrix: the production of T1 tadpoles from adults (infected and uninfected) and the265

production of zoospores from tadpoles and infected adults. As is, the fecundity matrix F only266

specifies the mean production of T1 tadpoles and zoospores. To make this probabilistic, we need267

to specify a distribution around this mean. We assumed that the production of tadpoles followed268

a Poisson distribution with mean A(t)pAλ, where A(t) = SA(t) +∫ Ux

LxIA(x, t)dx is the number269

of adults in the population at time t, pA is the probability of an adult reproducing and λ is270

the mean number of tadpoles produced. We used a Poisson distribution for reproduction for271

consistency with previous modeling studies of R. muscosa (Briggs et al. 2005, 2010). Moreover,272

as our goal was to understand the relative effects of resistance, tolerance, and transmission on273

extinction risk, the exact form of the reproduction distribution has little influence of the effects of274

these three process relative to each other. In contrast, the shape of the reproductive distribution275

will have enormous implications if one were to consider the evolution of resistance and tolerance276

in a host population. While this was beyond the scope of this study, an important next step is277

to understand how standing variation in resistance and tolerance can rescue populations from278

extinction (Orr & Unckless 2014). For this, particular attention will need to be paid to both the279

reproductive distribution and any trade-offs between increased resistance/tolerance and mean280

host reproduction (Boots et al. 2009).281

We also assumed that the number of zoospores shed into the zoospore pool from tad-282

poles and adults at each time step followed a Poisson distribution with mean∑3i=1 µTi

Ti(t) +283

µA∫ Ux

Lxexp(x)IA(x, t)dx. The Poisson distribution is justified by first noting that the best esti-284

mate for the rate of zoospore shedding from an adult or tadpole at any point in a time step is285

the mean shedding rate over the entire time step. Given no other information, the best guess286

is to assume a constant rate of zoospore shedding over a time step. Taking this constant rate287

of zoospore shedding over a time step, probability theory then tells that we should expect a288

Poisson distribution of zoospores produced per time step (Grimmett & Stirzaker 2001). If the289

11

rate of zoospore shedding actually varies over a time step, we would no longer expect a Poisson290

distribution of zoospores released per individual over a time step. However, because we have291

no information on how zoospore shedding rate varies on a per individual basis over a three-day292

time step, the Poisson distribution provides a reasonable null hypothesis.293

With these distributional assumptions, we can then calculate the contribution of T1 tadpoles294

and zoospores over a time step t as a draw from a Poisson distribution with a mean given by the295

appropriate entry in the F matrix. Combining these draws from a Poisson distributions with296

the draws from the multinomial distribution described above, we can simulate the individual-297

based representation of our hybrid model with any density-dependent assumptions about recruit-298

ment or disease transmission in addition to any assumptions about temperature-dependent vital299

rates. This can be done by updating the various transition probabilities using the appropriate300

densities or temperatures at each time step and performing the previously described stochastic301

draws with the updated transition probabilities. See the function multiseason simulation in302

IPM functions for R.R for the R code necessary to implement this stochastic simulation (all303

code available at https://github.com/mqwilber/host extinction).304

4.2 Testing the assumptions regarding how tadpoles contribute zoospores305

to the zoospore pool306

In the above stochastic model, we assumed that tadpoles were always infected and shed a Poisson-307

distributed random variable Zshed of zoospores into the zoospore pool at each time step, where308

Zshed has a constant mean µT . This was a simplifying assumption of our model based on309

field data showing that tadpoles have high Bd prevalence, even in endemic populations (Briggs310

et al. 2010). However, this prevalence tends to be less than 1 (e.g. ≈ 0.84 in the mescosm311

experiment and between 0.3 and 1 at certain sites the field, Briggs et al. 2010), which is not312

exactly consistent with what we assumed in the model. To test the influence of our assumption313

that tadpole prevalence is one, it would be ideal to explicitly model tadpole load dynamics as we314

do with adult load dynamics. However, this was not possible as we could not uniquely identify315

tadpoles in the mesocosm experiment as they were too small to PIT tag.316

Instead, we modified our assumption that an individual tadpole contributed Zshed ∼ Poisson(µT )317

zoospores to the zoospore pool per time step. We allowed an individual tadpole to contribute318

Zshed ∼ Zero-inflated Poisson(µT , p) zoospores to the zoospore pool per time step, where p gives319

probability that a tadpole is infected. The justification for the Poisson distribution is the same320

as given in the previous section, but now we allow some tadpoles to be uninfected in a time step321

12

with probability p, such that they do not contribute to the zoospore pool in that time step. This322

modification allows us to relax our assumption that all tadpoles are infected in any given time323

step, without having to explicitly model tadpole infection dynamics. Based on the data from324

the mesocosm experiment used in this study, we set p = 0.84 and µT = 1487.036.325

Rerunning our model with the above changes, we found that allowing for tadpole prevalence326

to be the empirically-derived value of p = 0.84 did not qualitatively affect the extinction dynamics327

in this system (Fig. 5). The reason for this is because infected tadpoles were still able to shed a328

sufficient number of zoospores to keep the force of infection in the system high. To understand329

when reduced tadpole Bd prevalence did begin to affect host extinction dynamics, we reduced330

tadpole prevalence to 0.1, which is far lower than the observed levels of tadpole Bd prevalence331

in this system (Briggs et al. 2010). This low prevalence did reduce extinction risk relative to332

the baseline model and the model with prevalence set to 0.84, but had less of an effect on333

extinction risk than increasing the rate of zoospore decay in the environment (Fig. 5). Given334

these additional analyses showing that large changes in tadpole prevalence had small effects on335

the extinction dynamics, we felt confident that our simplifying assumption that tadpoles were336

always infected did not influence our main conclusions.337

References338

Boots, M., Best, A., Miller, M. R. & White, A. (2009). The role of ecological feedbacks in the339

evolution of host defence: what does theory tell us? Philosophical Transactions of the Royal340

Society of London. Series B, Biological Sciences, 364, 27–36.341

Boyle, D. G., Boyle, D. B., Olsen, V., Morgan, J. A. T. & Hyatt, A. D. (2004). Rapid quantitative342

detection of chytridiomycosis (Batrachochytrium dendrobatidis) in amphibian samples using343

real-time Taqman PCR assay. Diseases of Aquatic Organisms, 60, 141–8.344

Briggs, C. J., Knapp, R. A. & Vredenburg, V. T. (2010). Enzootic and epizootic dynamics of345

the chytrid fungal pathogen of amphibians. Proceedings of the National Academy of Sciences346

of the United States of America, 107, 9695–9700.347

Briggs, C. J., Vredenburg, V. T., Knapp, R. A. & Rachowicz, L. J. (2005). Investigating the348

population-level effects of chytridiomycosis: an emerging infectious disease of amphibians.349

Ecology, 86, 3149–3159.350

13

Caswell, H. (2001). Matrix Population Models: Construction, Analysis, and Interpretation. 2nd351

edn. Sinauer, Sunderland, Massachusetts.352

Easterling, M. R., Ellner, S. P. & Dixon, P. M. (2000). Size-specific sensitivity: applying a new353

structured population model. Ecology, 81, 694–708.354

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. & Rubin, D. B. (2014).355

Bayesian Data Analysis. 3rd edn. Taylor & Francis Group, LLC, Boca Raton.356

Grimmett, G. & Stirzaker, D. (2001). Probability and Random Processes. Oxford University357

Press, Oxford, United Kingdom.358

Klepac, P. & Caswell, H. (2011). The stage-structured epidemic: Linking disease and demogra-359

phy with a multi-state matrix approach model. Theoretical Ecology, 4, 301–319.360

Knapp, R. A., Briggs, C. J., Smith, T. C. & Maurer, J. R. (2011). Nowhere to hide: impact361

of a temperature-sensitive amphibian pathogen along an elevation gradient in the temperate362

zone. Ecosphere, 2, art93.363

Oli, M. K., Venkataraman, M., Klein, P. A., Wendland, L. D. & Brown, M. B. (2006). Population364

dynamics of infectious diseases: A discrete time model. Ecological Modelling, 198, 183–194.365

Orr, H. A. & Unckless, R. L. (2014). The population genetics of evolutionary rescue. PLoS366

Genetics, 10, e1004551.367

Rohani, P., Breban, R., Stallknecht, D. E. & Drake, J. M. (2009). Environmental transmis-368

sion of low pathogenicity avian influenza viruses and its implications for pathogen invasion.369

Proceedings of the National Academy of Sciences, 106, 10365–10369.370

Schreiber, S. J. & Ross, N. (2016). Individual-based integral projection models: the role of size-371

structure on extinction risk and establishment success. Methods in Ecology and Evolution, 7,372

867–874.373

Vredenburg, V. T., Knapp, R. A., Tunstall, T. S. & Briggs, C. J. (2010). Dynamics of an emerg-374

ing disease drive large-scale amphibian population extinctions. Proceedings of the National375

Academy of Sciences, 107, 9689–94.376

Wilber, M. Q., Langwig, K. E., Kilpatrick, A. M., Mccallum, H. I. & Briggs, C. J. (2016). Integral377

Projection Models for host-parasite systems with an application to amphibian chytrid fungus.378

Methods in Ecology and Evolution, 7, 1182–1194.379

14

Woodhams, D. C., Alford, R. A., Briggs, C. J., Johnson, M. & Rollins-Smith, L. A. (2008). Life-380

history trade-offs influence disease in changing climates: Strategies of an amphibian pathogen.381

Ecology, 89, 1627–1639.382

15

2 0 2 4 6 8 10 12 14ln Bd load at time t

2

0

2

4

6

8

10

12

14

ln B

d lo

ad a

t tim

e t+

1

4 C12 C20 C

A. Growth function, G(x′, x)

2 0 2 4 6 8 10 12 14ln Bd load at time t

0.0

0.2

0.4

0.6

0.8

1.0

R. muscosa s

urvi

val p

roba

bilit

y

B. Survival function, s(x)

2 0 2 4 6 8 10 12 14ln Bd load at time t

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

of in

fect

ion

loss

Temperature = 4 C

2 0 2 4 6 8 10 12 14ln Bd load at time t

Temperature = 12 C

2 0 2 4 6 8 10 12 14ln Bd load at time t

Temperature = 20 C

C. Loss of infection function, l(x)

5 10 15 20Temperature, C

1

0

1

2

3

4

5

ln B

d lo

ad a

cqui

red

upon

infe

ctio

n

D. Initial infection function, G0(x′)

0 5 10 15 20 25 30Temperature, C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pro

babi

lity

of z

oosp

ore

surv

ival

E. Zoospore survival function, ν

Figure 1: The various vital rate functions estimated from laboratory data given in Wilber et al.(2016) and Woodhams et al. (2008). In all plots, points give the laboratory data, lines givethe model fit, and gray regions given the 95% credible interval about the predictions. A. Thetemperature-dependent growth function G(x′, x) where temperature is included as a continuouscovariate in the growth function. 95% CIs where not included for visual clarity. B. The survivalfunction s(x) which dictates the probability of an adult R. muscosa surviving with a given loadover a three day time step. C. The loss of infection function l(x) which gives the load- andtemperature-dependent probability of an adult R. muscosa losing a Bd infection over a threeday time step. D. The initial infection function G0(x′) that specifies the temperature-dependentprobability density of gaining an initial infection of size x′. E. The zoospore survival functionν that gives the temperature-dependent probability of a zoospore surviving over three days asgiven in Woodhams et al. (2008). No uncertainty was included around this prediction.

16

02468

101214

ln(B

d lo

ad +

1) Tank number 2 Tank number 8 Tank number 9

Init.

den

sity

= 1

Tank number 14

02468

101214

ln(B

d lo

ad +

1) Tank number 4 Tank number 6 Tank number 11

Init.

den

sity

= 4

Tank number 15

02468

101214

ln(B

d lo

ad +

1) Tank number 3 Tank number 7 Tank number 12

Init.

den

sity

= 8

Tank number 13

0 10 20 30 40 50 60 70 80

Days

02468

101214

ln(B

d lo

ad +

1) Tank number 1

0 10 20 30 40 50 60 70 80

Days

Tank number 5

0 10 20 30 40 50 60 70 80

Days

Tank number 10

0 10 20 30 40 50 60 70 80

DaysIn

it. d

ensi

ty =

16Tank number 16

Figure 2: The observed Bd load trajectories for individual frogs in the density-dependent meso-cosm experiment described in the main text. Tanks/mesocosms had an initial density of either 1,4, 8, or 16 frogs and were assigned numbers 1 - 16 as indicated on a panel. The different coloredlines are the different Bd load trajectories for the frogs in given tank. The black vertical lineindicates day 32 of the experiment, after which there was an unexplained decrease in zoosporeload in the infected frogs in all mesocosms in which frogs were infected by this time point. Theconsistency of the decline between treatments, between mesocosms, and between frogs suggeststhe involvement of an external environmental driver or a frog immune response. Data after thistime point was not used when fitting the transmission models described in the main text.

17

432101234

0.26 0.44 16.0 0.22 0.05 0.19 0.4 0.53 0.38 0.44 0.33 0.32 0.25 0.22 0.16 nan

Dynamic zoospore pool

Logistic regression

432101234

Logi

stic

regr

essi

onpa

ram

eter

wei

ght

0.26 0.44 16.0 0.22 0.05 0.19 0.4 0.53 0.38 0.44 0.33 0.32 0.25 0.22 0.18 0.21

Frequency-dependent w/ dynamic zoospore pool

Logistic regression

s(x)

inte

rcep

t, b 0,0

s(x)

size

, b1,

0

G(x

′ ,x)

inte

rcep

t, b 0,1

G(x

′ ,x)

tem

pera

ture

, b2,

1

G(x

′ ,x)

size

, b1,

1

G(x

′ ,x)

varia

nce,

ν0,

1

G(x

′ ,x)

var.

effe

ct, c

0,1

G0(x

′ ) in

terc

ept, b 0,3

G0(x

′ ) te

mp,

b1,

3

G0(x

′ ) v

aria

nce,

ν0,

3

G0(x

′ ) v

ar. e

ffect

, c0,

3

l(x)

inte

rcep

t, b 0,2

l(x)

tem

p., b

2,2

l(x)

size

, b1,

2

φ z

oosp

ore

pool

, β0

φ d

ensi

ty/fr

eque

ncy,

β1

432101234

0.26 0.44 16.0 0.22 0.05 0.19 0.4 0.53 0.38 0.44 0.33 0.32 0.25 0.22 0.19 0.2

Density-dependent w/ dynamic zoospore pool

Logistic regression

Toleranceparameters

Resistanceparameters

Transmissionparameters

A.

B.

C.

;

Figure 3: The results of the global sensitivity analysis similar to Figure 4 in the main text.However, instead of perturbing parameters by drawing from a lognormal distribution with me-dian 1 and σ = 0.3, we drew the parameters from their corresponding posterior distributionfrom the fitted Bayesian models of the temperature and transmission experiments described inthe main text. This allowed for correlation between the parameters as well as allowing someparameters to have larger variability than others, strictly based on the variability in the pos-terior distribution. The above plot gives the standardized coefficients (i.e. all predictors werez-transformed before the analysis) from a regularized logistic regression. The height of a barindicates how sensitive extinction risk was to this parameter. The direction indicates whetherincreasing the parameter increased or decreased the probability of extinction. The values undereach bar give the coefficient of variance (CV) for each parameter, calculated from that parame-ter’s posterior distribution (“nan” indicates this parameter was not in the model). The resultsfrom this sensitivity analysis were qualitatively consistent with the approach used in the maintext: disease-induced extinction was far more sensitive to the parameters in the growth function(G(x′, x), resistance) and the survival function (s(x), tolerance) than to parameters in the trans-mission function. Pruned regression trees also confirmed the important interaction between thegrowth function and the survival function.

18

0 5 10 15Density of infected adults

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

of in

fect

ion

over

3 d

ays

0.0 0.2 0.4 0.6 0.8 1.0Proportion of infected adults

A. B.

Figure 4: Predictions from the best fit frequency-dependent and density-dependent trans-mission functions with a dynamic zoospore pool, as estimated from the mesocosm experi-ment described in the main text. Because the data used to fit these transmission modelsare Bernoulli (0 or 1) and there are three dimensions of predictor variables (time, host den-sity/frequency, and zoospore density) it is difficult to give a visual representation of modelfit. Here we show the marginal predictions (black lines) from the frequency-dependent anddensity-dependent transmission models, fixing the time step at 3 days and the zoospore poolat 1096 zoospores. The gray region gives the 95% credible interval around these predictions.The open circles give the observed proportion of frogs that transitioned from uninfected to in-fected with the proportion/density of infected individuals given on the x-axis (pooled acrossdifferent time steps and zoospore pool sizes). The size of the points indicates the sample size ofeach point, with larger points indicating larger sample size (n = 333 total samples distributedamong the points). A. The predictions from the best-fit density-dependent transmission func-tion (φ = 1 − exp (−(β0 ln(Z + 1) + β1I)∆t)) B. The predictions from the best-fit frequency-dependent transmission function (φ = 1 − exp

(−(β0 ln(Z + 1) + β1

IA )∆t

))

19

0 5 10 15 20 25Time (years)

0.0

0.2

0.4

0.6

0.8

1.0

Ext

inct

ion

prob

abili

ty

Dynamic zoospore poolFrequency-dependent w/dynamic zoospore poolDensity-dependent w/dynamic zoospore pool

Density-dependent transmissionwith tadpole Bd prevalence = 0.84Density-dependent transmissionwith tadpole Bd prevalence = 0.1Density-dependent tranmissionwith high rate of zoospore decay

Figure 5: The black lines show the time-dependent extinction curves of the hybrid model withthree different transmission functions described in Table 1: dynamic zoospore pool, frequency-dependent transmission with a dynamic zoospore pool, and density-dependent transmission witha dynamic zoospore pool. The curves were generated from 500 stochastic simulations of eachmodel parameterized with the parameters given in Table 1 and 2. The red line gives the ex-tinction curve when transmission with density-dependent with a dynamic zoospore pool and thezoospore survival probability was set to a constant 0.05 per three day time step, compared tothe laboratory-estimated temperature-dependent survival probability of 0.8 at 15 ◦C (Wood-hams et al. 2008). The blue and green lines given the extinction curves when transmission wasdensity-dependent with a dynamic zoospore pool and tadpole Bd prevalence was 0.84 and 0.1,respectively.

20

G(x

', x)

, siz

e <

0.8

s(x)

, int

erce

pt >

= 4

G(x

', x)

, tem

pera

ture

< 0

.17

G(x

', x)

, tem

pera

ture

< 0

.11

G(x

', x)

, siz

e <

0.72

s(x)

, int

erce

pt >

= 5.

5G(x

', x)

, tem

pera

ture

< 0

.12

G(x

', x)

, siz

e <

0.63

G(x

', x)

, var

ianc

e <

6.5

G(x

', x)

, siz

e <

0.84

G(x

', x)

, tem

pera

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< 0

.099

s(x)

, int

erce

pt >

= 4.

6

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', x)

, siz

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21

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< 0

.12

s(x)

, int

erce

pt >

= 3.

4

G(x

', x)

, tem

pera

ture

< 0

.093

G(x

', x)

, siz

e <

0.72

s(x)

, siz

e >=

−3

G(x

', x)

, siz

e <

0.88

G(x

', x)

, tem

pera

ture

< 0

.078

s(x)

, int

erce

pt >

= 4.

7pe

rsis

t pers

ist

extin

ct

pers

ist

extin

ct

extin

ct

pers

ist

extin

ct

extin

ctextin

ct

yes

no

Fig

ure

6B:T

he

pru

ned

regr

essi

ontr

eefo

rth

em

od

elw

ith

freq

uen

cy-d

epen

den

ttr

an

smis

sion

wit

ha

dyn

am

iczo

osp

ore

pool.

Sh

ows

avis

ualre

pre

senta

tion

ofh

owth

eva

riou

str

ansm

issi

on,

resi

stan

ce,

and

tole

ran

cep

ara

met

ers

inte

ract

edto

aff

ectBd

-in

du

ced

host

exti

nct

ion

.In

part

icu

lar,

para

met

ers

of

the

grow

thfu

nct

ion

det

erm

inin

gh

ost

resi

stan

ce(G

(x′ ,x

))an

dth

esu

rviv

al

fun

ctio

nd

eter

min

ing

host

tole

ran

ce(s

(x))

inte

ract

top

red

ict

exti

nct

ion

or

per

sist

ence

.

22

G(x

', x)

, siz

e <

0.78

G(x

', x)

, tem

pera

ture

< 0

.12

s(x)

, int

erce

pt >

= 4.

3

G(x

', x)

, siz

e <

0.68

s(x)

, int

erce

pt >

= 6.

8

G(x

', x)

, siz

e <

0.82

G(x

', x)

, tem

pera

ture

< 0

.088

pers

ist

pers

ist

pers

ist

extin

ct

extin

ctpe

rsis

tex

tinct

extin

ct

yes

no

Fig

ure

6C:

Th

ep

run

edre

gres

sion

tree

for

the

mod

elw

ith

tran

smis

sion

from

ad

yn

am

iczo

osp

ore

pool.

Sh

ows

avis

ual

rep

rese

nta

tion

of

how

the

vari

ou

str

ansm

issi

on,

resi

stan

ce,

and

tole

ran

cep

aram

eter

sin

tera

cted

toaff

ectBd

-in

du

ced

host

exti

nct

ion

.In

part

icu

lar,

para

met

ers

of

the

gro

wth

fun

ctio

nd

eter

min

ing

hos

tre

sist

ance

(G(x′ ,x

))an

dth

esu

rviv

alfu

nct

ion

det

erm

inin

gh

ost

tole

ran

ce(s

(x))

inte

ract

top

red

ict

exti

nct

ion

or

per

sist

ence

.

23

● ●●

●

●

●

●●

●

● ●●

●

●

●

●

●●

●●

●

●

●

●

●●

●

A.

250

275

300

325

350

0 1 2 3 4σ

WA

IC

model●

●

●

density dependent

frequency dependent

zoospore pool

● ● ●●

● ●

●

●●

● ● ●●

●

●

●

●●

● ●●

●

●

●

●●

●

B.

300

600

900

1200

1500

0 1 2 3 4σ

DIC

model●

●

●

density dependent

frequency dependent

zoospore pool

Figure 7: Comparing two information criteria when fitting the latent zoospore model to themesocosm transmission experiment with different levels of process error in the latent zoosporepool (given by σ). The above plots show WAIC (A.) and DIC (B.) values for three differenttransmission models fit to the experimental data: density-dependent transmission with a dy-namic zoospore pool (pink), frequency-dependent transmission with a dynamic zoospore pool(green), and only dynamic zoospore pool (blue). When σ ≤ 1 and σ ≥ 2.5, the relative modelrankings are consistent for both WAIC and DIC. However, between 1 and 2.5 the informationcriteria for the dynamic zoospore pool transmission function either drastically decreases (WAIC)or drastically increases (DIC).

24

● ● ●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

● ●●

●● ●

●

●

●

A.

−5

−4

−3

−2

0 1 2 3 4σ

ln(β

0: z

oosp

ore

pool

coe

ffici

ent)

model●

●

●

density dependent

frequency dependent

zoospore pool

● ● ● ● ● ● ● ● ●

● ● ●●

●

●● ●

●

B.

0.00

0.25

0.50

0.75

0 1 2 3 4σ

β 1: d

ensi

ty/fr

eque

ncy

coef

ficie

nt

model●

●

●

density dependent

frequency dependent

zoospore pool

Figure 8: Comparing the estimates of the transmission coefficients (β0: the zoospore pool coef-ficient, β1: the density/frequency dependent coefficient) when fitting the latent zoospore modelto the mesocosm transmission experiment with different levels of process error (given by σ).A. The zoospore pool coefficient shows a marked increase after σ = 1. This is driven by thelarge variability in the trajectory of the zoospore pool, such the transmission coefficient needsto increase in order to contribute to transmission when, by chance, the zoospore pool may crashto very low levels under the high σ/process error scenarios. B. In contrast, the transmissioncoefficient β1 determining density or frequency-dependent host contacts is relatively consistentacross σ, with a slight decreasing trend. These is no blue line in B. because the transmissionmodel with only a dynamic zoospore pool does not have a coefficient β1. The error bars are 95%credible intervals around the coefficient estimates.

25