decision making for mitigating wildlife diseases: from theory to practice … · 2019. 2. 9. · 1...

Decision making for mitigating wildlife diseases: from

theory to practice for an emerging fungal pathogen of amphibians

Journal: Journal of Applied Ecology

Manuscript ID Draft

Manuscript Type: Research Article

Date Submitted by the Author: n/a

Complete List of Authors: Canessa, Stefano; Universiteit Gent, Department of Pathology, Bacteriology and Avian Disease Bozzuto, Claudio; Wildlife Analysis GmbH Grant, Evan; USGS Patuxent Wildlife Research Center, Cruickshank, Sam; University of Zurich, Department of Environmental Studies and Evolutionary Biology Fisher, Matthew; Imperial College London, Division of Epidemiology, Public Health and Primary Care Koella, Jacob; Université de Neuchâtel, Laboratoire d'écologie et d'épidémiologie parasitaire, Institut de Biologie Lötters, Stefan; Trier University, Biogeography Department Martel, An; Universiteit Gent, Department of Pathology, Bacteriology and Avian Disease Pasmans, Frank; Universiteit Gent, Department of Pathology, Bacteriology and Avian Disease Scheele, Benjamin; Australian National University, Spitzen-van der Sluijs, Annemarieke; Reptielen Amfibieën Vissen Onderzoek Nederland Steinfartz, Sebastian; Technische Universität Braunschweig, Evolutionary Biology Schmidt, Benedikt; University of Zurich, Institut für Evolutionsbiologie und Umweltwissenschaften; karch,

Key-words: Amphibian, basic reproduction number, chytrid, epidemiology, expert elicitation, host-pathogen, integral projection model, salamander, disease spread

Confidential Review copy

Journal of Applied Ecology

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Decision making for mitigating wildlife diseases: from theory to practice for an emerging 1

fungal pathogen of amphibians 2

3

Stefano Canessa1*†, Claudio Bozzuto

2*, Evan H. Campbell Grant

3, Sam S. Cruickshank

4, Matthew 4

C. Fisher5**, Jacob C. Koella

6, Stefan Lötters

7, An Martel

1, Frank Pasmans

1, Ben C. Scheele

8, 5

Annemarieke Spitzen-van der Sluijs9, Sebastian Steinfartz

10, Benedikt R. Schmidt

4,11 6

Author affiliations 7

1 Department of Pathology, Bacteriology and Avian Diseases, Faculty of Veterinary Medicine, 8

Ghent University, Salisburylaan 133, 9820 Merelbeke, Belgium 9

2 Wildlife Analysis GmbH, Oetlisbergstrasse 38, 8053 Zurich, Switzerland 10

3 United States Geological Survey, Patuxent Wildlife Research Center, SO Conte Anadromous Fish 11

Laboratory, 1 Migratory Way, Turners Falls, MA 01376 12

4 Department of Evolutionary Biology and Environmental Studies, University of Zurich, 13

Winterthurerstrasse 190, 8057 Zurich, Switzerland 14

5 Department of Infectious Disease Epidemiology, School of Public Health, Imperial College, 15

London, St Mary's Hospital, London W2 1PG, United Kingdom 16

6 Laboratoire d'écologie et d'épidémiologie parasitaire, Institut de Biologie, Université de 17

Neuchâtel, Neuchâtel, Switzerland 18

7 Trier University, Department of Biogeography, Universitätsring 15, 54296 Trier, Germany 19

8 Fenner School of Environment and Society, Australian National University, 2601, Canberra, 20

Australia 21

9 Reptile, Amphibian and Fish Conservation the Netherlands, PO Box 1413, 6501 BK, Nijmegen, 22

the Netherlands 23

10 Zoological Institute, Technische Universität Braunschweig, Mendelssohnstrasse 4, 38106 24

Braunschweig, Germany 25

11 Info fauna karch, Passage Maximilien-de-Meuron 6, 2000 Neuchâtel, Switzerland. 26

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* Equal contributions 27

** Authors 5-12 in alphabetical order 28

† Corresponding author. Email [email protected] 29

Running title. Exploring mitigation of salamander chytrid 30

Word count. Summary: 271; main text: 4407; acknowledgements: 62; references: 1224; tables: 232; 31

figure legends: 398 32

Number of figures/tables: 4/1 33

Number of references: 40 34

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Summary 35

1. Conservation science can be most effective in its decision-support role when seeking answers to 36

clearly formulated questions of direct management relevance. Emerging wildlife diseases, a 37

driver of global biodiversity loss, illustrate the challenges of performing this role: in spite of 38

considerable research, successful mitigation is uncommon. Decision analysis is increasingly 39

advocated to guide mitigation planning, but its application remains rare. 40

2. Using an integral projection model, we explored potential mitigation actions for avoiding 41

population declines and the ongoing spatial spread of the fungus Batrachochytrium 42

salamandrivorans (Bsal). This fungus has recently caused severe amphibian declines in north-43

western Europe and currently threatens Palearctic salamander diversity. 44

3. Available evidence suggests that a Bsal outbreak in a fire salamander (Salamandra salamandra) 45

population will lead to its rapid extirpation. Treatments such as antifungals or probiotics would 46

need to effectively interrupt transmission (reduce probability of infection by nearly 90%) in 47

order to reduce the risk of host extirpation and successfully eradicate the pathogen. 48

4. Improving the survival of infected hosts is most likely to be detrimental as it increases the 49

potential for pathogen transmission and spread. Active removal of a large proportion of the host 50

population has some potential to locally eradicate Bsal and interrupt its spread, depending on 51

the presence of Bsal reservoirs and on the host’s spatial dynamics, which should therefore 52

represent research priorities. 53

5. Synthesis and applications. Mitigation of Bsal epidemics in susceptible host species is highly 54

challenging, requiring effective interruption of transmission and radical removal of host 55

individuals. More generally, our study illustrates the advantages of framing conservation 56

science directly in the management decision context, rather than adapting to it a posteriori. 57

Keywords. Amphibian; basic reproduction number; chytrid; disease spread; epidemiology; expert 58

elicitation; host-pathogen; integral projection model; salamander. 59

60

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Introduction 61

Conservation biology has been called a “crisis discipline” aimed at solving problems about 62

declining species (Soulé 1985). Such problems typically consist of understanding how to manage 63

natural systems, and conservation science seeks to suggest solutions by providing relevant 64

information (Arlettaz et al. 2010). This decision-support role is best performed by trying to find 65

answers to clearly formulated questions directly relevant to ecosystem management ("strong 66

inference"; Chamberlin 1890; Platt 1964; Burnham & Anderson 2001), rather than simply by 67

collecting data and seeking patterns a posteriori (Nichols & Williams 2006). 68

A particularly challenging problem for conservation science is the development of responses to 69

emerging infectious diseases, which are increasingly driving biodiversity loss worldwide (Fisher et 70

al. 2012). Infectious diseases often demand rapid decision-making in the face of scarce knowledge, 71

limited time for learning, and challenges turning the available scientific knowledge into actions 72

(Grant et al. 2017). For example, the amphibian chytrid fungus Batrachochytrium dendrobatidis 73

was identified about twenty years ago (Longcore, Pessier & Nichols 1999); its role in global 74

amphibian declines was clarified over the following decade (Skerratt et al. 2007); potential 75

mitigation strategies were then considered (Woodhams et al. 2011; Scheele et al. 2014), yet to date 76

implementation has been rare and success in maintaining susceptible populations of amphibians in 77

the pathogen’s presence remains elusive (Bosch et al. 2015; Geiger CC et al. 2017). For this reason, 78

a more critical evaluation of potential mitigation strategies for amphibian chytridiomycoses has 79

been recently advocated (Garner et al. 2016; Grant et al. 2017), with the use of specific decision-80

analytic methods to assess research priorities and plan management actions (e.g. Converse et al. 81

2016; Gerber et al. 2017). The lack of applications of decision analysis in the treatment of emerging 82

wildlife diseases such as amphibian chytridiomycoses contrasts with its common application in 83

human healthcare management (e.g. Smith, Hillner & Desch 1993; Claxton & Sculpher 2006; Nutt, 84

King & Phillips 2010). However, this implementation gap may occur because conservation 85

managers, researchers and decision-makers remain unsure about what a decision-analytic approach 86

to emerging wildlife diseases entails and how to implement it in practice, including which research 87

should be pursued to improve population viability. 88

Management decisions arise when, in order to achieve one or more objectives, a choice must be 89

made among two or more alternative actions. Decision analysis is the ensemble of principles and 90

methods to make such choices rationally in the face of uncertainty (Keeney 1982). All revolve 91

around a basic, iterative process: (1) state the decision problem at hand; (2) identify the key 92

management objectives, with associated measures of success; (3) list the potential actions available; 93

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(4) use a model of the system to predict the expected outcome of respective actions, measured 94

against the objectives; (5) recognise uncertainty and solve trade-offs; (6) make and implement a 95

decision. To summarise, decision analysis requires science to perform its “strong inference” role 96

firmly in the management decision context, with explicit statements about objectives, formalisation 97

of current (prior) knowledge and comparable predictions of the outcomes of management actions. 98

In this study, we put in practice this approach as advocated by recent studies (Garner et al. 2016; 99

Grant et al. 2017; Russell et al. 2017), following the decision-analytic process to explore the 100

potential effect of mitigation actions for the recently discovered salamander chytrid fungus 101

Batrachochytrium salamandrivorans (Bsal; Martel et al. 2013). Strong evidence suggests this 102

fungus is native to Asia and causes lethal chytridiomycosis in several species of Palearctic 103

salamanders, threatening to cause loss and disturbance of biodiversity in Europe and North America 104

(Martel et al. 2014). The recent arrival of Bsal in Europe has been implicated in the collapse of 105

several populations of fire salamanders (Salamandra salamandra) in the north-western range of this 106

species (Spitzen-van der Sluijs et al. 2016). A recent model of the dynamics of host populations 107

suggests a potential rapid spread of Bsal across the range of fire salamanders in Middle Europe 108

(Schmidt et al. 2017), requiring urgent management decisions. In the United States, which host 109

amphibian populations at high risk for Bsal (Richgels et al. 2016), pre-arrival actions to restrict the 110

potential importation of infected salamanders are being implemented while proactive management 111

is considered (Grant et al. 2016). Here, we focused on the possible management context at the 112

invasion front and epidemic stage (sensu Langwig et al. 2015; note that for consistency with their 113

classification, we use "epidemic" rather than "epizootic" throughout). We modelled the host-114

pathogen dynamics in a population of a vulnerable salamander species to seek answers to a key 115

management question: how, if at all, can a Bsal epidemic be locally managed? 116

117

Materials and methods 118

Problem definition 119

For our study, we evaluated a range of possible management actions to respond to an infectious 120

disease outbreak. We used the European fire salamander Salamandra salamandra as the target 121

species, since it is a widespread species in Europe at great extinction risk from Bsal (Stegen et al. 122

2017). We considered the management of a single salamander population over a short time frame of 123

three months, chosen on the basis of the rapid Bsal-driven population crashes in fire salamanders 124

(Schmidt et al. 2017; Stegen et al. 2017). We assumed optimal climatic conditions for Bsal growth 125

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and transmission (Stegen et al. 2017). We envisaged management would have two general 126

objectives: minimising the risk of population extinction and minimising the potential for further 127

geographic disease spread. 128

Predicting disease impacts: integral projection model 129

Predictions of the outcomes of mitigation actions are generally based on a simplified representation 130

of the system (a model). Most conservation actions rely on implicitly drawing predictions from 131

previous successes and failures ("experience-based model"; Dicks, Walsh & Sutherland 2014); 132

however, an explicit mathematical description of the system provides a stronger basis to compare 133

and update current knowledge, identify key uncertainties and address trade-offs (Russell et al. 134

2017). We adapted the dynamic population model by Schmidt et al. (2017) into an integral 135

projection model (IPM; Ellner & Rees 2006; all equations in the Supporting Information). An IPM 136

is a generalization of a classic stage-structured population model, where demographic processes are 137

described by one or more continuous covariate(s) rather than discrete stages. This makes IPMs 138

particularly suitable for fungal infections, where rates are influenced both by infection status and 139

infection load, the latter typically measured continuously using quantitative PCR (Wilber et al. 140

2016). As an initial template we used the IPM developed by Wilber et al. (2016) for 141

Batrachochytrium dendrobatidis. We parameterised our model using formal expert elicitation and 142

available data from the literature. Using these sources of information, we then modified model 143

components to reflect on how management could seek to influence key processes to mitigate Bsal 144

impacts. 145

The model considers two disease-related compartments, S(t) and I(z, t), respectively, the number of 146

(uninfected) susceptible, and infected individuals at each time step t (Fig 1). Infected individuals are 147

classified according to their infection load z (eq. S2). The transitions over time and between states 148

are defined by several functions (Fig 1 and eq. S2); below, we provide a brief summary of all 149

functions used in our IPM. Given the short time frame of our management scenario, reflecting the 150

speed of Bsal-driven local population declines in fire salamanders, we focused on a period of the 151

year where recruitment would be negligible, and we also excluded density-dependent mortality 152

effects. We also did not include a “recovered” state or a loss-of-infection function (the probability 153

that an individual transitions from infected to uninfected), since available evidence suggests fire 154

salamanders are unable to clear Bsal infection (Stegen et al. 2017). 155

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Survival (eq. S4). s0 and the function s(z) describe, respectively, the survival of uninfected and 156

infected individuals from time t to t+1. For infected individuals, survival is a function of the 157

infection load z at time t. 158

Transmission (eqs. S3, S5). The function describes the probability that uninfected individuals at 159

time t transition into the infected group at time t+1, and is a function of the load-related distribution 160

of infected individuals in the population at time t and the load-dependent effect of infected 161

individuals on the transmission probability. We assumed a mass-action, density-dependent 162

transmission function for our population. 163

Initial load (eq. S7). G0(z’) describes the probability density of individuals having an infection load 164

z’ upon infection. 165

Load growth (eqs. S8-S10). G(z’, z) describes the probability density of infected individuals having 166

an infection load z’ at time t+1, given load z at time t. 167

Host movement. In addition to the dynamics of a closed population, we calculated – independently 168

from these dynamics – how far Bsal would spread via infected dispersing individuals, given our 169

current knowledge captured in the IPM functions. Starting with an initial load distribution G0(z’), 170

we calculated the life span of a pool of infected individuals given the load growth function and 171

load-dependent survival (i.e. using the IPM kernel). We defined life span as the time taken for the 172

cumulative survival probability to fall below 1%, so that, because of the initial load distribution, 173

infected individuals would travel different distances before dying. Using this formulation, rather 174

than the mean time to death for an infected individual, allows us to better express the uncertainty in 175

the initial load function (see next section). We calculated the distance travelled by dispersing 176

infected individuals based on their life spans and the mean distance travelled per day by fire 177

salamanders in a typical middle European habitat (Tab. S1). 178

To evaluate the ability to meet our objectives of minimising the reduction in the density of 179

susceptible individuals and pathogen spread, we calculated (i) the Bsal basic reproduction number 180

R0 (R0<1 indicates a disease dying out, and R0>1 implies a disease outbreak; see Wilber et al. 181

(2016) for the derivation of R0 as used in our study), (ii) the ratio between the predicted final 182

density of susceptible individuals under each action and the predicted density with neither infection 183

nor mitigation, and (iii) the movement distance of infected individuals. 184

185

186

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Parameterising the model 187

Given the recent emergence of Bsal in Europe, its infection dynamics are not yet entirely 188

understood, although information is rapidly being accrued. Therefore, we used a formal expert 189

elicitation process to estimate the model parameters; this ensured all evidence available was 190

incorporated, while making explicit any additional uncertainty expressed by experts. We carried out 191

the elicitation in a group of ten experts (all listed co-authors except SC, CB and EG, who acted as 192

facilitators) during a four-day workshop, using a best-practice approach (Martin et al. 2012). For 193

each parameter, each expert was initially asked to provide estimates individually (minimum, most 194

likely and maximum values), then allowed to revise those after group discussions based on critically 195

evaluated published and unpublished evidence. For the survival and transmission functions, experts 196

were asked to estimate the respective probabilities at infection loads of 0, 10, 100, 1000 and 10000 197

genomic equivalents (a measure of infection load); for the initial load distribution, estimates of 198

mean, dispersion, and skewness; for the growth function, the maximum observable load and the 199

load growth rate. Figure 1 shows a graphical depiction of the elicited values and fitted functions; 200

further details can be found in the SI, including parameter values in Table S1. The elicitation and 201

discussion also highlighted limited knowledge of the contact rate between host individuals. We 202

therefore chose to derive this parameter from the only published source describing a Bsal outbreak 203

in the wild using individual host mark-recapture (Stegen et al. 2017). To this end, we used a final 204

size relation for a simplified model (see below, Exploring the management-related parameter 205

space). Details of the derivation are provided in the SI. 206

Predicting the outcomes of mitigation actions 207

To understand how management could seek to influence key disease processes, we considered a 208

small set of potential management actions (Table 1), devised during our workshop and from 209

literature (Scheele et al. 2014; Garner et al. 2016; Grant et al. 2017). Fundamentally, actions could 210

be considered different combinations of modifications to pathogen transmission (reducing the 211

probability that an uninfected individual becomes infected upon contact with an infected one), 212

pathogen growth (slowing the growth of Bsal on infected individuals) and host density (reducing 213

density prior to, or right after the arrival of Bsal in the population). Repeating the expert elicitation 214

process, we thought critically about which model parameters each action would seek to modify, and 215

by how much. Since no mitigation of Bsal has been attempted to date, the initial parameterisation 216

described above reflected a “no action” scenario, which we used as a baseline reference after group 217

discussions among experts confirmed that the elicited functions approximated the dynamics 218

observed in the field (Stegen et al. 2017). A further reference level was obtained by modelling the 219

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outcome metrics for an uninfected population. For all scenarios, we expressed uncertainty using 220

probability distributions. Following a best-practice approach, we used beta-PERT distributions for 221

all parameters elicited from experts (Vose 1996). Further, for the initial density derived from Stegen 222

et al. (2017) we used a negative binomial distribution, and for the mean survival probability of 223

infected individuals, also from Stegen et al. (2017), we used a beta distribution. We simulated 1,000 224

datasets by drawing random combinations of parameters from the respective distributions, and then 225

fit the IPM functions using those values to assess model outcomes across the range of parametric 226

uncertainty. 227

Exploring the management-related parameter space 228

The set of actions presented in Table 1 was not a comprehensive evaluation of all potential actions. 229

Rather, it reflected initial creative thinking about mitigation, and may be interpreted as snapshot of 230

a general sensitivity analysis, easier to discuss and link to management than a multi-dimensional but 231

abstract exploration of the parameter space. This approach is particularly useful considering that 232

decision analysis is an iterative process, where different components can be revisited as needed. For 233

example, new actions could be devised using the initial results as an indication of the effects of 234

manipulating different model parameters (N’Guyen et al. 2017). 235

Therefore, after observing the initial results we carried out a further exploration of possible model 236

outcomes, with the aim of identifying the requirements of any new action, or combination thereof, 237

which sought to influence specific model parameters. We used a simplified dynamical model, 238

collapsing the continuous classification of infected individuals as a function of load into a single 239

“Infected” state, to obtain a discrete time Susceptible-Infected model (eq. S11). First, we studied the 240

effects of management by modifying the equation that calculates R0 for this model (eq. S12). Here, 241

on the basis of our initial results, we assumed actions would seek to reduce transmission (β in eq. 242

S12), initial host density (S0 in eq. S12) and/or host survival (s0 and � in eq. S12; see Supporting 243

Information). We multiplied each of these parameters by a term indicating the proportional 244

reduction caused by management (1-m), and we calculated the proportional reductions that would 245

result in R0 = 1. Further, as for the IPM model, we also calculated the density ratio of susceptible 246

hosts between a given reduction and no reduction (i.e., with and without management). For ease of 247

comparison with the initial IPM results, we obtained the density ratios of susceptible hosts by 248

iterating the simplified model (S11). Further details, including a final size relation for the simplified 249

model, can be found in the Supporting Information. 250

251

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Results 252

The results of our IPM simulations reflect the high virulence of Bsal in fire salamander populations, 253

observed in field and laboratory studies to date and confirmed by the expert judgment used to 254

parameterise the model (Fig 1, Table S2). In a “no action” scenario, susceptible hosts would almost 255

inevitably become infected upon contact with infected individuals at high burdens of infection 256

(>100 genomic equivalent) that occur soon after exposure, leading to a rather high Bsal basic 257

reproduction number (median R0 ≈ 9.6). Host survival would decrease to almost certain death at 258

intermediate to high infection loads. Therefore, an unmitigated Bsal-breakout is expected to lead to 259

the effective extirpation of the host population within our three-month time frame (Fig 2). The rapid 260

growth of Bsal and consequential reduced host survival meant that, in the absence of mitigation, an 261

infected dispersing individual would move on average less than 100 m during the study period (Fig 262

3a). However, rare extreme movements may be more relevant for disease spread, where one or a 263

few individuals may reach longer distances. In our case, without mitigation, the mean distance for 264

those rare individuals could be approximately double the mean movement, as illustrated by the 265

uncertainty shown in Fig. 3a. Note that our proportional formulation implies that a larger population 266

will produce a greater absolute number of such long-range dispersers. 267

The potential actions described in Table 1, analysed using the IPM, were mostly unable to prevent 268

population extirpation, even under optimistic parameterisation (Fig 2). An “individual 269

manipulation” that reduced transmission probabilities for low infection loads by 50% did not 270

prevent the collapse of the population or reduce Bsal R0 (Fig. 2); a treatment such as our generalised 271

“probiotic”, that reduced transmission while slowing pathogen growth, actually worsened 272

outcomes, failing to prevent the collapse of the host population while increasing R0 and the 273

distances across which infected hosts moved (Fig 2; Fig 3b,c). A highly effective “antifungal” 274

treatment (treatment “d” in Fig 2) that reduced transmission by 98% was likely to avoid a 275

population collapse within our time frame and eradicate the pathogen (R0<1), although this 276

treatment also increased host movement distances (Fig 2, Fig 3d). However, decreasing coverage 277

(the proportion of individuals treated with the “antifungal”) from 100% to 80% (“e” in Fig 2) was 278

already sufficient to negate these results: although the collapse did not occur during our time frame, 279

it was only delayed beyond our three-month simulation period, since eradication could not be 280

achieved (R0>1), while still increasing host movements (Fig 2, Fig 3e). “Pre-emptive thinning” that 281

manipulated host density by removing 50% of the individuals before Bsal entered the population 282

was not effective under the conditions we simulated; when the proportion removed was increased to 283

90%, the chance of eradicating Bsal was approximately 50% (Fig 2; Fig 3f,g). Rapid and efficient 284

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“post-detection removal” of both susceptible and infected individuals starting immediately after 285

Bsal entry obviously led to almost complete extirpation of the population, but was the only action 286

likely to both eradicate Bsal and minimise host movements, and thus the risk of disease spread to 287

other populations, which was otherwise largely unaffected by other actions. (Fig 2: Fig 3h). We 288

found only one case in which parametric uncertainty resulted in significant uncertainty between 289

success and failure of the same action, i.e., for which the predicted R0 included values lower and 290

greater than 1: as mentioned above, this was pre-emptive removal of 90% of the population, which 291

was equally likely to succeed or fail in eradicating Bsal. The application of the highly effective 292

antifungal with 80% coverage had only a marginal chance of being successful (less than 2.5% of the 293

simulation runs resulted in R0<1; Fig. 2). 294

In addition to the IPM model, we used a simplified model to further assess the efficacy of single 295

and/or combined management actions. The simplified model without management produced R0 ≈ 296

7.9, in good agreement with R0 ≈ 9.6 (5.8 – 14.3 the 95% confidence interval, Fig. 2) from the IPM 297

model. Exploration of this simplified model confirmed that any potential action, or combination of 298

actions, targeting host density, transmission and/or survival would require (very) high effectiveness 299

to achieve R0<1 and prevent a disease outbreak (Fig 4a-c). As suggested by our initial exploratory 300

actions, management strategies targeting single parameters would need to reduce survival by at least 301

75% or initial density or transmission by at least 85%. Acting on several parameters at once would 302

slightly reduce these requirements. For example, R0 = 1 could be achieved by reducing 303

transmission, initial host density and host survival by approximately 42% each (Fig 4c), or by 304

reducing transmission and host survival by 87% each (Fig 4a). Avoiding the collapse of the host 305

population (i.e., obtaining a high density ratio between susceptible individuals with and without 306

intervention) would require a similarly high effectiveness (Fig 4d). As expected, reducing the initial 307

density has approximately a linear effect on the final density; however, this reduction would still 308

need to be paired with a highly effective reduction in transmission to result in a final viable 309

population (Fig 4d). 310

Discussion 311

Our modelling approach revealed that mitigation actions at the Bsal invasion front and during a 312

Bsal epidemic event are highly unlikely to be effective, at least for very susceptible species such as 313

those in the genus Salamandra, particularly S. salamandra (see Sabino-Pinto et al. 2015). 314

Treatments that seek to reduce transmission would require almost complete effectiveness (i.e., 315

perfect interruption of transmission) and very high coverage, with >80% of the population treated 316

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within three days. Moreover, since current evidence suggests fire salamanders do not clear Bsal 317

infection and do not acquire immunity (Stegen et al. 2017), unless the pathogen can be eradicated, 318

even highly effective treatments would likely only slow the decline of host populations. Treatments 319

that only slow the growth of the pathogen, although they may appear intuitively appealing, 320

effectively create a larger pool of active infected individuals and increase the potential for spread of 321

the pathogen to other populations. This inherent risk in treatments that prolong survival but do not 322

interrupt transmission is a recognized concern in the management of virulent pathogens (Read et al. 323

2015). 324

If the persistence of affected populations is unlikely to be achievable after Bsal entry, this objective 325

may become irrelevant and the focus may shift to interrupting the further spread of the pathogen to 326

other populations, by interrupting its transmission and minimising host movements. Our results 327

suggest these objectives might only be achieved with reasonable certainty by applying radical 328

interventions, such as the almost complete removal of infected populations. Moreover, recent 329

evidence suggests that Bsal reservoirs may exist, either in other amphibian species or as free-living 330

encysted spores (Stegen et al. 2017). Once their role in the wild is clarified, such reservoirs can be 331

incorporated into the model (Wilber et al. 2016). From a management perspective, since reservoirs 332

increase the probability of infection, their presence may negate even the only potential benefit of 333

removal-based management (minimising spread by host movement). Removal actions could be 334

revised to include amphibian reservoirs, but this may be less feasible for other taxa or 335

environmental reservoirs. Additional knowledge about the site-specific density and encounter rates 336

of host individuals may help refine predictions and understand the potential for pathogen spread and 337

identify promising management options (e.g. spacing of quarantine fences and radius of capture 338

searches for host removal). 339

Our initial comparison of actions was obviously not exhaustive and more creative thinking is to be 340

encouraged (Grant et al. 2017). However, irrespective of the management strategy, our conclusions 341

are likely to remain broadly applicable; ultimately any action that seeks to address an epidemic 342

would rely on the manipulation of one of the processes described in the model, such as pathogen 343

growth, host survival or rates of transition between infection states. Our exploration of the 344

management-related parameter space using the simplified model suggests that management 345

strategies that target multiple processes are likely to be necessary, and even if they can be devised, 346

all those components would still need to be highly effective to achieve management objectives. 347

Given the limited ability to mitigate epidemics, alternative actions aimed at preventing the entry of 348

the pathogen remain a priority. At the local level, besides mandatory biocontrol precautions, more 349

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radical options could involve restriction of access by humans or isolation of host populations in-350

situ, for example through quarantine fences. Such tactics may however only reduce the risk of Bsal 351

entry, not the ultimate outcome of an epidemic, and their effectiveness may be limited given the 352

multiple possible passive and active vectors of Bsal (Stegen et al. 2017). Ex-situ rescue would 353

likely still face the same challenges highlighted by our model at the time of reintroduction, unless 354

the pathogen could be entirely removed or mechanisms of augmenting resistance were developed 355

(e.g., vaccination; a possibility contrary to current evidence, at least for fire salamanders which do 356

not appear to acquire immunity; Stegen et al. 2017). 357

Recognising the current uncertainty surrounding those processes provides an ideal basis for rational 358

planning of future research and implementation of adaptive management, by focusing on the 359

uncertain parameters that directly influence the choice of management actions (Russell et al. 2017). 360

For example, Wasserberg et al. (2009) describe an adaptive management model for white-tailed 361

deer affected by chronic wasting disease, where they determined a relationship between disease 362

transmission and the effectiveness of culling. In the Bsal case, our results suggest, for example, that 363

understanding whether a candidate antifungal can reduce Bsal transmission by 85% or 95% would 364

be of immediate relevance for deciding whether to use it or not; conversely, understanding whether 365

a probiotic-based treatment reduces transmission by 20% or 50% may reveal important biological 366

processes, but this improved knowledge is still unlikely to make application of such a treatment an 367

optimal management strategy. In terms of more general improvements to our model, better 368

knowledge of the effect of reservoirs, host density and non-amphibian vectors in the transmission 369

function may provide the greatest benefit for decision-making. 370

Perhaps the most urgent need is to clearly define the real, rather than theoretical, decision context 371

for Bsal management. We were able to obtain useful information and to provide an initial 372

assessment of potential actions in a hypothetical management situation. However, we also recognise 373

that natural resource managers (the actual decision makers) face additional objectives and potential 374

constraints, such as the social acceptability of host removal and the allocation of limited funding. 375

These additional complexities will vary with the spatial and temporal scale of possible disease 376

management, and will require the direct involvement of those decision makers. 377

Our study provides a first practical demonstration of the advantages of embedding scientific 378

analysis of emerging diseases in a realistic decision context, as recently advocated for amphibian 379

chytridiomycoses (Garner et al. 2016; Grant et al. 2017; Russell et al. 2017). Essentially, decision 380

analysis required us to approach the problem from a manager’s perspective, stating clear objectives, 381

thinking creatively about actions and making comparable predictions about specific management 382

Page 13 of 33



14

actions, thus performing that ideal role of scientific “strong inference”. Setting our model in this 383

context allowed us to clarify potential actions and formalise current knowledge for use in a model 384

that can be easily updated as new information becomes available. This approach thus creates the 385

foundations for a transparent discussion about the potential of any proposed action to mitigate the 386

effects of disease. 387

Acknowledgments 388

We thank the Schweizerischer Nationalfonds for funding the workshop (1Z32Z0_168399 to BRS). 389

SC is supported by the Research Foundation Flanders (FWO16/PDO/019). This is contribution 390

number 584 of the Amphibian Research and Monitoring and Initiative (ARMI) of the US 391

Geological Survey. Any use of trade, product, or firm names is for descriptive purposes only and 392

does not imply endorsement by the U.S. Government. 393

Author contributions 394

SC, CB and BRS conceived the study; SC and CB performed all analyses (respectively, elicitation- 395

and IPM-related); SC and CB wrote the manuscript with input from all authors. 396

Data accessibility 397

All data supporting the results in this paper will be archived in an appropriate public archive, if the 398

paper is accepted for publication. 399

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517

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Table 1. Summary of exploratory actions for Bsal mitigation and their implementation in the 518

integral projection model. The column “Parameters modified” refers to the parameters in the IPM 519

equations (Supporting Information) and how they were modified to simulate the prospective 520

management actions. “Data” indicates the modification was applied directly to the values elicited 521

from experts (e.g., the estimated transmission rates at different infection loads). 522

Action Description Parameters modified

(a) No action Unmitigated course of Bsal outbreak in

a population

–

(b) Improve body

condition

Improve body condition of individuals,

for example by food supplementation at

larval stage, with the aim of increasing

their resistance to infection at low

infection burdens by 50%

0.5 ∙ �� for ≤ 100 GE

(c) Probiotic treatment (a) Pre-emptive treatment of susceptible

individuals increasing their resistance to

infection at low infection burdens by

50%, and (b) slowing Bsal growth once

infected by 80%

(a) 0.5 ∙ �� for ≤ 100 GE

(b) = �√0.2 ∙ ��

(d) Antifungal

treatment, perfect

coverage

(a) Treatment of both susceptible and

infected individuals, increasing the

resistance of susceptible individuals to

infection by 98%, and (b) slowing Bsal

growth once infected by 80%

(a) 1 − 0.98� ∙ ��

(b) = �√0.2 ∙ ��

(e) Antifungal

treatment, incomplete

coverage

(a) Treatment of both susceptible and

infected individuals, increasing the

resistance of susceptible individuals to

infection by 98% (only 80% of

individuals in the population treated at

each time step), and (b) slowing Bsal

growth once infected by 80%.

(a) 1 − 0.80 ∙ 0.98� ∙ ��

(b) = �√0.2 ∙ ��

(f) Pre-emptive

removal – low thinning

Removal of 50% of individuals prior to

entry of Bsal

0.5∙S0

(g) Pre-emptive Removal of 90% of individuals prior to 0.1∙S0

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removal – high

thinning

entry of Bsal

(h) Post-detection

removal

Removal of 90% of all individuals (per

time step) starting immediately after

entry of Bsal, i.e., imposing an

additional mortality probability of 90%.

0.1∙s(z)

523

524

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525

Figure 1. Summary flow-chart for the fire salamander-Bsal integral projection model. The arrows 526

indicate how hosts can transition within and between the susceptible and infected states from time t 527

to time t+1 (three days apart); z and z’ indicate infection loads at t and t+1 respectively. Inserts 528

represent the elicited parameter values; for Survival and Transmission (probability of transmitting 529

infection from an infected to a susceptible host), curves indicate the most likely elicited values, with 530

shaded areas indicating minimum-maximum ranges. For Initial load, the insert represents the 531

elicited probability distribution; for Load Growth, the insert represents the probability density 532

function of Bsal load at time t+1, given the load at time t. 533

534

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22

535

Figure 2. Comparison of predicted outcomes for potential Bsal mitigation actions in a fire 536

salamander population over a three-month period, obtained from the IPM. The x-axis indicates the 537

basic reproduction number of Bsal (R0), and the y-axis the host population decline, expressed as the 538

ratio between the final number of susceptible individuals for a given action and that simulated for a 539

scenario without infection. Values shown are median (markers) and 95% confidence intervals (CI) 540

for each action (error bars). Note that, given the strong within-action correlation between R0 and 541

final density ratio (not shown), the error bars for the latter are the associated results of the 95% CIs 542

in R0: within a management action, the highest final density ratio is associated to the lower 95% CI 543

bound of R0, et vice versa. 544

545

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23

546

Figure 3. Distance covered by infected individuals during their life-span, under different mitigation 547

actions, obtained from the IPM. Action labels correspond to those indicated in Fig 2 and Table 1. 548

The y-axis indicates the proportion of individuals that moved at least the distance given by the 549

respective value on the x-axis. Dashed lines indicate minimum/maximum values. 550

551

Page 23 of 33



24

552

Figure 4. Results of management-related parameter space exploration, obtained from the simplified 553

model. Panels (a), (b), (c) depict the combinations of management effects (reduction of transmission 554

β, initial host density S0, or host survival) that are required to obtain R0 = 1. In panel (b), reducing 555

transmission or initial density leads to the same graph (see eq. S12). In panel (c), parameter 556

combinations under the plotted surface lead to R0 > 1. Panel (d) indicates the ratios of final host 557

densities (with and without management) as a function of two management parameters (with a 558

quasi-extinction threshold of 0.01). 559

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1

Decision making for mitigating wildlife diseases: from theory to practice for an 1

emerging pathogen of salamanders 2

3

Online Supporting Information 4

5

Bsal Integral projection model (IPM): derivation 6

To set up the Bsal IPM used in this study, we started with the continuous-time Bsal model 7

presented in Schmidt et al. (2017). The latter considered the states susceptible, latent, and 8

infectious. Because an IPM offers the possibility of structuring the infected compartment according 9

to fungal load and defining load-dependent functions, we first eliminated the state latent. Further, 10

because Bsal-caused population crashes in fire salamanders are expected to happen within a few 11

months or even weeks (see main text), we focused on a post-reproductive period of the year, 12

neglecting recruitment and density-dependent effects on population growth for this period. 13

Compared to the original model, the simplified one is given by the system of equations eq. S1, 14

known in the epidemiological literature as SI-model: 15

�� = −�� − �� = �� − �� + ��

(S1)

16

Next, we translated this simplified model into a discrete-time IPM. The resulting model used in the 17

present study is given by the system of equations eq. S2: 18

�� + 1� = �� 1 − ��, ��

��, � + 1� = �� , �� + � ��, �� , ��

(S2)

19

In the following we give a brief description of the model and its components (see also Fig. 1); for 20

more details please refer to Wilber et al. (2016). The points in time t and t+1 are three days apart. 21

Since we focused on a timespan of three months, we got 30 times steps. Parameter � is a constant 22

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2

reflecting survival probability of susceptible individuals over three days; parameterization details of 23

eq. S2 will be presented in the section Bsal Integral projection model (IPM): parameterization. The 24

function ��, �� gives the infection probability, and thus �1 − ��, �� is the probability of 25

not getting infected. The infection probability is a function of the load-dependent transmission 26

effect, ��, of the load-related distribution of infected individuals, ��, ��, and is given by eq. S3: 27

��, �� = 1 − exp �−� ��, ��

� (S3)

28

This function would also allow including a reservoir effect (Wilber et al. 2016). However, lacking 29

any reliable information with regard to Bsal, we omitted this effect. 30

Newly infected individuals enter the infected group following an initial load distribution ��. 31

Already infected individuals, on the other hand, either die or survive according to a load-dependent 32

survival probability s(z). Those individuals surviving will experience a growth in fungal load, given 33

by the function ��, ��; see the next section for details on these functions. 34

35

Bsal Integral projection model (IPM): expert elicitation and parameterization 36

As introduced in the main text, we carried out the elicitation in a group of ten experts during a four-37

day workshop. Each expert was initially asked to provide estimates individually (minimum, most 38

likely and maximum values), then allowed to revise those after group discussions based on critically 39

evaluated published and unpublished evidence. For the survival and transmission functions, experts 40

were asked to estimate the respective probabilities at infection loads of 0, 10, 100, 1000 and 10000 41

genomic equivalents (a measure of infection load); for the initial load distribution, estimates of 42

mean, dispersion, and skewness; for the growth function, the maximum observable load and the 43

load growth rate. Because our IPM is based on a log-transformed load scale, for all subsequent 44

analyses we used the following infection loads for the elicited values: -1, log(10), log(100), 45

log(1000), log(10000) , where log refers to the natural logarithm. 46

For the load-(in)dependent survival probabilities, s0 and ��, we used expert elicitation; for all 47

load-dependent functions, we used a logarithmic scale for the fungal load. Because survival is a 48

probability 0 ≤ ≤ 1, we used logistic regression to fit the elicited load-dependent survival 49

probabilities (eq. S4): 50

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3

�� = exp�"� + "#��1 + exp�"� + "#�� (S4)

51

This function and all following functions are shown in Fig 1 in the main text. The survival 52

probability for susceptible individuals, s0, using these elicited values is very similar to one based on 53

the mean lifespan of an adult salamander: the latter is approximately eight years (see Schmidt et al. 54

2017 and references therein). These and all following parameter values are presented in Table S1. 55

Similar to survival, for the transmission probability we elicited load-dependent probabilities 56

of infection, given an encounter between two salamanders. Here, too, we used logistic regression to 57

fit these values (eq. S5). Further, we scaled this function by an encounter rate ��. 58

�� = �� exp�"$ + "%��1 + exp�"$ + "%�� (S5)

59

Note that we did not directly elicit the exact values needed for the model; rather, since it would be 60

easier for experts to interpret and discuss probabilities, we used these values as an approximation in 61

the transmission probability function eq. S3, where �� is present in an exponentiated form. As 62

mentioned in the main text, identifying an exact value for �� was accompanied by high uncertainty 63

during the workshop. We thus based our estimated value of �� on data published in Stegen et al. 64

(2017) and on a final size relation, as explained in the following. For a variety of epidemiological 65

models a so-called final size relation has been established, giving the expected long-term density of 66

susceptible individuals escaping infection. One approximate final size relation for our simplified 67

model (presented below, see Eq. S11) is given by eq. S6 (Brauer et al. 2010): 68

log )��*+ = ,� )1 − �*�� + (S6)

69

Here, ,� = ��1 − �.#, and log() refers to the natural logarithm. We used this equation to solve 70

for �, based on the following data from Stegen et al. (2017): these authors followed a Bsal outbreak 71

in a Belgian fire salamander population, using mark-recapture and multistate modelling to describe 72

the demography of the host population. Given the reported 475-m-long transect and assuming a 10-73

m search width, we derived �� = 503 adults / ha (95% CI : 236, 966) from Stegen et al. (2017), i.e. 74

a population at carrying capacity, matching published records for this species in core habitat (Table 75

7.3 in Thiesmeier 2004). From the same article, we used the Poisson regression fit to data in Fig. 1a 76

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4

of Stegen et al. (2017), corrected for their average reencounter probability of approximately 10% 77

(B. Schmidt, pers. comm.) to calculate �* = 7 adults / ha after nine months. This time span was 78

chosen because in the present work we considered a post-reproductive period, lasting approximately 79

from May to February in the following year. Finally, we also set the survival of infected individuals 80

to = 0.542 (95% CI 0.191, 0.761), again based on Stegen et al. (2017). Note that Eq. S6 does not 81

include s0: because we assumed a population at carrying capacity and emergence of new adults 82

happens approximately between June and November (Sparreboom 2014), we realistically assumed 83

that over nine months, recruitment and (natural) mortality balance out. In summary, given Eq. S6 84

and the above-mentioned parameter uncertainties, we drew 1000 random values for both parameters 85

and used Eq. S6 to derive an empirical distribution for ��, resulting in a median �� = 0.007 (95% 86

CI 0.0037, 0.0125). We then used one of these 1000 pairs of �� and the corresponding ��in each of 87

the IPM simulations. 88

For the initial load function, G0(z’), we elicited information about mean, dispersion and 89

skewness of such a distribution, based on expert knowledge. For an untransformed load scale, a 90

lognormal distribution was a good approximation to this expert information. On a log-transformed 91

load scale, we got a normal distribution with mean 567 and variance 867$ (eq. S7): 92

��~;�567 , 867$ � (S7)

93

We also approached the load growth function with expert elicitation. This function describes 94

the probability density of infected individuals reaching an infection load z’ at time t+1, given load z 95

at time t. It is easiest to visualize such a function as an auto-correlated linear regression model; the 96

latter is a simple but not at all the only possible function. The adjective auto-correlated refers to the 97

fact that we would use the same data time series (fungal load as a function of time) as both predictor 98

and response. In terms of a linear regression, given a predictor value – in this case load z at time t – 99

the aim is to know the expected value – in this case load z’ at time t+1 – and dispersion around this 100

expected value. Without any load-related data at hand, it was difficult for experts to tie this function 101

to their related knowledge. Thus, we instead chose to elicit simpler information with regard to load 102

growth on a single individual. Here, we were interested in knowing the number of zoospores 103

produced by a single zoospore in a limited amount of time, and the maximum load, <=>?, that can 104

be expected. The latter was motivated by the fact that the fungus spreads mostly in two dimensions 105

on an animal’s body, i.e., skin. Thus, we translated this information into a discrete-time fungal load 106

growth model, based on the Beverton-Holt model often used in population ecology (eq. S8): 107

Page 28 of 33



5

<@A# = B<@1 + �B − 1� <@<=>?

(S8)

108

Parameter B is the multiplicative density-independent load growth rate. Experts gave the number of 109

zoospores produced by a single zoospore in five days, thus the daily multiplicative growth rate, 110

B#C>D, is the 5th

root of this number, and B = B#C>D% . We started the iteration with a low load L0 = 10 111

to get a load time series. Finally, we log-transformed this time series and auto-regressed it with time 112

lag one, i.e., log�<@A#� = E�log�<@�� or �′ = E��. This approach allowed us to derive the expected 113

value of ��, ��, and a 2nd

degree polynomial best described this relationship (eq. S9). 114

�′ = "G + "H� + "I�$ (S9)

115

Because we worked with the hypothesis of a maximum possible load per animal, Lmax, we 116

introduced a cap for the load growth function: for all � ≥ log�<=>?�, 117

��~;�log�<=>?� , 86$�log�<=>?��. An integral component of the load growth function is the 118

probability density associated with every load. We were not able to gather necessary information to 119

this end, and thus we used the variance reported in Wilber et al. (2016) for Batrachochytrium 120

dendrobatidis (eq. S10). 121

86$�� = 86$ exp�"K�� (S10)

122

In summary, a load z at time t will generate a load z’ at t+1 following a normal distribution with 123

mean given by eq. S9 and variance given by eq. S10. 124

To numerically implement our IPM, we chose a rather broad range for the log-transformed 125

load scale, ranging from l = –10 to u = 15 (see eq. S2). Further, to numerically approximate the 126

integrals, we used 300 mesh-points. We did not explicitly calculate an eviction measure: following 127

some representative simulations it seemed that eviction was not a complication in our IPM. All 128

modeling related work was carried out in Matlab R2015a (The MathWorks Inc., Natick, MA, 129

2015). 130

Exploring management-related parameter space: implementation details 131

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6

To more broadly understand the effectiveness of mitigation actions, alone or combined, we 132

simplified our IPM as follows. As suggested by Wilber et al. (2016), our IPM (eq. S2) can be 133

“collapsed” by removing the explicit load-dependency of infected individuals, resulting in a discrete 134

time SI-model given by eq. S11: 135

�� + 1� = �� 1 − ��

�� + 1� = �� + ��

(S11)

136

The transmission probability is now defined as �� = 1 − exp�−��. The survival 137

probability of infected individuals, , reflects a mean survival related to the original load-dependent 138

distribution (see below). The basic reproduction number for this model is ,� = �� 1 − �.# 139

(Brauer et al. 2010, Wilber et al. 2016). For our exploration we added three terms for the effect of 140

management to this equation, resulting in eq. S12: 141

,� = ��1 −LM��1 − LN7� ��1 − LO�PQ�1 − �1 − LO�PQ� (S12)

142

This formulation would allow, in a subsequent stage of the decision-analysis process, to relate such 143

reduction to the effort needed to achieve it. For example, the reduction in initial density can be 144

formulated using a Poisson catchability model, so that L = 1 − exp�−RS�, where c is the 145

catchability coefficient and E is effort data; both can be estimated from data (Seber 1982). 146

As is evident from eq. S12, reduction in transmission and initial density have the same effect on R0, 147

as opposed to survival reduction. We introduced the latter for both parameters, � and , on the 148

grounds that such actions will be realistically targeted at all individuals present in a population. The 149

derivation of parameter values is described above, in section Bsal Integral projection model (IPM): 150

expert elicitation and parameterization), and all parameter values are presented in Table S1. Note 151

that reduction in transmission in eq. S12 only affects transmission, whereas for our actions (Tab. 1) 152

we assumed these interventions to induce at the same time a slowed pathogen growth rate (actions 153

c-e); the latter increases . For the results presented in Fig. 4, we assumed LM to only affect 154

transmission, because slowing down pathogen growth has adverse effects on management outcomes 155

(see main text and eq. S12). Finally, to produce Fig 4a-c, we set ,� = 1 in eq. S12 and solved for 156

one management parameter that depended on the remaining management parameter(s). 157

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7

Additional References 158

Brauer, F., Feng, Z. & Castillo-Chavez, C. (2010) Discrete epidemic models. Mathematical 159

Biosciences and Engineering, 7, 1-15. 160

Seber, G.A. (1982) Estimation of animal abundance. Oxford University Press. 161

Sparrebooom, M. (2014) Salamanders of the Old World. KNNV Publishing, Zeist, the Netherlands. 162

Stegen, G., Pasmans, F., Schmidt, B.R., Rouffaer, L.O., Van Praet, S., Schaub, M., Canessa, S., 163

Laudelout, A., Kinet, T., Adriaensen, C., Haesebrouck, F., Bert, W., Bossuyt, F. & Martel, A. 164

(2017) Drivers of salamander extirpation mediated by Batrachochytrium salamandrivorans. Nature, 165

544, 353-356. 166

Thiesmeier, B. (2004) Der Feuersalamander. Laurenti, Bielefield, Germany. 167

168

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8

Table S1. Parameter values used in this study for the integral projection model. z and z’ refer to the 169

log-transformed load scale. Elicited parameter values shown in this table correspond to the best 170

estimates of experts. For the full distribution of each parameter value see archived data. 171

172

Parameter Description Value Source

Survival (eq. S4)

s0 Survival probability of

susceptible individuals

0.9906 s(z) fit to elicited

survival values;

s0 = s(z=0)

b0 Intercept logit(s(z)) 4.3712 s(z) fit to elicited

survival values

b1 Slope logit(s(z)) –0.8413 s(z) fit to elicited

survival values

Transmission (eq. S5)

�� Encounter rate

0.007 Derived from

Stegen et al. (2017)

b2 Intercept logit( �� ) –4.1651 �� fit to elicited

transmission values

b3 Slope logit( �� ) 1.1612 �� fit to elicited

transmission values

Initial load (eq. S7)

567 Mean of normal distribution 5.2983 Elicitation

867$ Variance of normal distribution 0.5 Elicitation

Load growth (eqs. S8-S10)

b4 Intercept (expected value z’) 1.0194 Elicitation (see text)

b5 Coefficient z (expected value z’) 1.5288 Elicitation (see text)

b6 Coefficient z2 (expected value

z’)

–0.0692 Elicitation (see text)

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9

86$ Variance 86$�� 5.92 Wilber et al. (2016)

b7 Coefficient z (86$��) –0.049 Wilber et al. (2016)

Host movement

vdisp Mean velocity of moving fire

salamanders in a typical middle

European habitat

0.5 km year-1

Schulte et al. 2007

Initial densities for simulations

S0 Initial density of susceptible

individuals at carrying capacity

503 adults ha-1

Stegen et al. (2017)

I0 Initial density of infected

individuals entering a healthy

population

0 or 1 adults

km-2

Schmidt et al.

(2017)

Exploring management-related parameter space

Average survival probability of

infected individuals

0.542 Stegen et al. (2017)

173

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