1
Supporting Information
2 3
Timing and severity of immunizing diseases in rabbits is
4
controlled by seasonal matching of host and pathogen dynamics
5
Konstans Wells1,*, Barry W. Brook1, Robert C. Lacy2, Greg J. Mutze3, David E. Peacock3, Ron G.
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Sinclair3, Nina Schwensow1, Phillip Cassey1, Robert B. O’Hara4, Damien A. Fordham1
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1 The Environment Institute and School of Earth and Environmental Sciences, The University of
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Adelaide, SA 5005, Australia
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2 Chicago Zoological Society, Brookfield, Illinois, United States of America
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3 Department of Primary Industries and Regions, Biosecurity SA, Adelaide, SA 5001, Australia
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4 Biodiversity and Climate Research Centre (BIK-F), Senckenberganlage 25, 60325 Frankfurt am
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Main, Germany
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*
Author for correspondence:
[email protected]
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Demographic and epidemiological model
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An individual-based model of seasonal rabbit demography and two co-circulating diseases
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in a meta-model framework – overview and specification
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Overview
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To build our individual-based model of rabbit demography and coupled epidemiology of rabbit
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haemorrhagic disease (RHD) and myxomatosis we used the freely available software packages Vortex
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10.0 and Outbreak 2.1, which were linked through the software Meta-Model Manager 1.0 [1, 2]. The
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software and user manuals can be downloaded at http://www.Vortex10.org. In brief, a demographic
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model in Vortex simulates the fate and reproduction of individuals over discrete time steps with
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various deterministic and stochastic forces [3, 4]. The software was initially conceptualized to model
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population viability over years as discrete time steps [3]. However, ‘days per year’ can be specified
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for modelling shorter time steps, allowing parameters such as demographic rates to be varied over
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shorter time steps. Linking epidemiological models in Outbreak through Meta Model Manager,
Supporting Information 30
allows the disease state and individual fate (i.e. death through disease) of organisms to be modified
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for discrete time steps encapsulated within the time steps defined in Vortex (typically, disease
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transmission dynamics are modelled with daily intervals). The different models are linked through
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defined state variables describing age, disease state, and other characteristics of individuals, and
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Meta-Model Manager facilitates the matching of events in time and space.
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We developed a seasonal model in which we varied reproductive efforts over weeks. We
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defined ‘years’ in Vortex as 7-day time steps and allowed reproductive efforts to vary in different time
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steps. We achieved this by modelling seasonal parameters as functions. For example, if reproduction
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is assumed to vary among calendar weeks in Vortex, the parameter ‘PercentBreed’ can be specified as “= ((Y%52=0)*X1 *(K-N)/K) + ((Y%52=1)*X2 *(K-N)/K) +((Y%52=2)*X3, …, +
40 41
((Y%52=51)*X52 *(K-N)/K)”,
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with Xw being the weekly parameter value in week w and (K-N)/K) accounts for density-dependent
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regulation of reproduction.
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We modelled the increasing susceptibility of infants (1-90 days old) to rabbit haemorrhagic
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disease virus (RHDV)[5] and the decreasing susceptibility to myxoma viruses (MYXV)[6] with a
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logistic model as a function of age. For this, we assumed that the recovery rate from RHD after the
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insusceptibility period declined from a maximum of 100 % (i.e. a fixed intercept μJuv,RHD = 6 in the
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logit-link logistic regression model) to the recovery rates to those of adults (V). The regression
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coefficient (Juv-RHD, ‘recovery adjustment factor’ for juveniles) for infant age, which measure the rate
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of change in infant recovery with increasing age was sampled during simulations (Figure A.1) and
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translates into the dynamic change in infant recovery rate with Juv,RHD: logit(Juv,RHD) = μRHD + Juv,RHD Ageadjusted .
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Here, the variable ‘Ageadjusted’ accounting for the insusceptibility period, i.e. the onset for an
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increasing RHD susceptibility is at an assumed age of 22 days for RHD (insusceptibility period for 21
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days, [7]).
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In Outbreak, the dynamic model for recovery rates can be expressed as a function for ‘probRecovery’
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as
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“= ((Age90) *RHD” whereby Juv,RHD = exp(μJuv,RHD + Juv,RHD *A - 21)/(1+exp(μ Juv,RHD + Juv,RHD *A - 21).
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An inverse relationship was assumed for myxomatosis (the negative value of the logistic
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model) to account for decreasing susceptibility of infants with increasing age. Here, we sampled the
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regression intercept (μJuv,Myxo) rather than the slope as an unknown parameter; the model assumes
2
Supporting Information 66
maximum susceptibility to myxomatosis for juveniles after the insusceptibility period, which is
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determined by μJuv,Myxo and thus assume to be the critical factor for disease dynamics (see Figure A.1).
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69 70
Figure A.1. Illustration of the dynamic modelling of decreasing recovery rates of infants for RHD and
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increasing recovery rates for myxomatosis. The rate of change in infant recovery towards the values of adult
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recovery rats is models with a logistic function after the assumed time of insusceptibility (grey bars) to the age
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of 90 days, when recovery rates are assumed to approach those of adults (V). For this, regression slopes
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(RHD,Juv) are sampled for RHD and regression intercepts (μJuv,Myxo) are sampled for myxomatosis disease
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dynamics. Dashed lines indicate of possible behaviour of the dynamic model across the sampled parameter
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values (Juv,RHD, μJuv,Myxo).
77 78 79
We modelled waning maternal antibodies against RHDV and MYXV and the resulting
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decrease in recovery rates from disease of infants (1-90 days of age) as an additional component for
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‘probRecovery’, assuming a linear and constant decline.
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For the implementation in Outbreak, we assumed a state variable ImAB of value ImAB = 100 if infants
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received antibodies from resistant does and ImAB = 0 otherwise. A is a function variable of individual
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age (see Vortex/Outbreak software manual). Since the effect of maternal antibodies is additive to the
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changes in infant recovery rates due to increasing/decreasing susceptibility the model for
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‘probRecovery’ changes to
87
“= ((Age90) *RHD)”,
89
whereby
90
RHD,Juv =RHD + (1 - RHDt)* Juv,RHD.
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Supporting Information 91 92
Note that the actual effect of maternal antibodies in each simulation are also determined by
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the simulated time period of full protection (tMRHD), during which no infections are possible. For the
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sake of model parsimony and lack of detailed information, we assumed transmission probability βRHD
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to be constant and not reduced by maternal antibodies after the time period of full protection.
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Age of individuals in Vortex are counted as time steps (i.e. weeks in our model), while the age
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of individuals in Outbreak are defined in ‘days’, so that the time periods of disease states such as
98
maternal antibodies can be flexibly accounted for.
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Meta-Model Manager communicated all relevant information between the different software
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
components to fully take the changes in individual state variables over time into account.
4
Supporting Information
117 118
Figure A.2. Flow diagram for an individual-based epidemiological model, used for examining the effect of
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seasonal timing of host reproduction and virus exposure (RHD and myxomatosis) on disease dynamics and
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population persistence. Panel (a) shows the possible individual progression through different disease states (P:
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Pre-/Insusceptible, S: Susceptible, E: Exposed, I: Infected, R: Resistant) for an individual without maternally
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acquired antibodies, (b) the respective dynamics for individuals with maternally acquired antibodies (M:
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Maternal antibodies full protection from infection and disease in juveniles) is illustrated in panel (b); The
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subscript juv indicates that an individual is juvenile (< 12 weeks old), for which changes in susceptibility to
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diseases is a function of age and the parameters and (i.e. increase in RHD, decrease in Myxo). Disease
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exposure of susceptible individuals depends on the transmission rate β(I) and/or the probability [t] that a virus
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is introduced at time t, capturing the seasonality of virus activity. For infected juveniles, the recovery rate is a
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function of the dynamical state of juvenile susceptibility and adult recovery rate . For individuals with maternal
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antibodies, changes in juvenile susceptibility and recovery rates are also dependent on the parameter , which
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describes the linear decline of maternal antibodies of aging juveniles. The parameters P and I describe the time
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length of transition steps. Note that individual fate in our model depends also on demographic dynamics and the
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disease-induced mortality from the co-circulating virus. In particular, the interaction with seasonal birth rates
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determines the phenological matching between demographic and epidemiological dynamics.
134 135 136 137
Using R for sampling and simulation in Meta Model Manager
5
6
Supporting Information 138
The utilised software can be both operated from ready-to-use user interfaces or, alternatively, via
139
command-line operation. We ran Meta-Model Manager from the freely available software R [8]. In
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brief, we defined relevant parameters as either single numerical constants (i.e. fixed parameters) or
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vectors with variable values for each simulation sampled from a latin hypercube with the R package
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LHS [9]. We established template files for each submodel (1 Vortex, 2 Outbreak, and 1 Meta-Model
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Manager for the study). From the R environment, we then loaded these template files, replaced the
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values for key parameters for each simulation and saved the edited files with the simulation number as
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an index. The full set of simulations was then operated with system commands, i.e. a command that
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runs Meta-Model Manager and calls for each simulation. By scanning the output files with R, we
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assembled output values such as the number of individuals in different disease states each day of the
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different simulations into arrays that allowed a large range of subsequent analysis and graphical
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display. The code for repeating our study is available in the Vortex library at
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www.vortex10.org/Downloads/OzRabbitDzFiles.zip.
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Model specification
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Table A.1. First-order independent model parameters and their ranges (minimum/ maximum) used for
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simulations. For each simulation, key parameters of most interest were drawn from a hypercube
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(‘Sample’; abbreviation given as used in the main article). Some parameters are further varied over
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time steps by sampling from uniform distributions over the defined ranges (indicated with ‘uniform’).
Parameter name
Range / Unit
Sampling/
Description
uncertainty
Justification/ Reference
Rabbit demography Survival probability
juv: 0.42 subad: 0.55 ad: 0.75
Sample
Survival probability (annual rate) depending on age with juveniles < 12 weeks, subadults 1224 weeks, adults > 24 weeks.
[10]
7
Supporting Information Annual peak in reproduction (RepPeak)
20 – 50
Sample
[calendar week]
Calendar week of
Field data, [11, 12]
maximum reproduction (used as mean in normal distribution to fit relative reproductive efforts over weeks.
Variation in reproduction (RepVar)
2 – 10
Sample
[calendar week]
Variation in relative
Field data, [11, 12]
weekly reproductive efforts in different simulations (used as SD in normal distribution).
Annual reproductive effort (RepEff)
Litter size
50 – 200 [%
Sample
Total reproductive
female
effort [%] in relation
reproducing]
to populations size N.
1-5
uniform
Number of offspring
[12, 13]
Expert guess
per litter. Reproductive age
26 [weeks]
Age of first
Field data
reproduction. Maximum age
416 [weeks]
Maximum lifetime of
[14]
rabbits in calendar weeks. Carrying capacity
500
Number of
[individuals]
Field data, [10]
individuals defining carrying capacity; we assumed density dependent reproduction, scaled by (K-Nt)/K.
Environmental stochasticity (EV)
0.01 – 0.5 [SD
Sample
Explorative
of vital rates, weekly]
Disease epidemiology Transmission probability (βV)
0.3 – 0.9
Sample
Probability that a
Explorative
virus is transmitted from an infected individual. Maternal antibodies (tMV)
1 – 50
RHD: Sample
Time period of
Explorative
8
Supporting Information [ days]
Myxo: none
maternal antibody protection against infection and disease.
Insusceptibility (newborns)
RHD: 21
Time period before
Myxo: 1
newborns may
[days] Exposure period
RHD: 1
[6, 7]
become susceptible. uniform
[6]
uniform
[6]
Myxo: 1 – 4 [days] Infection period
RHD: 1-2 Myxo: 8 – 12 [days]
Recovery rate (V)
0.2 – 0.9
Sample
Probability of
Explorative
survival of infected individuals > 90 days old. Juvenile susceptibility factor (Juv,RHD, μJuv,Myxo)
Juv,RHD: - 0.1 –
Sample
Decreasing (RHD) /
Explorative,
-1
increasing (Myxo)
similar dynamics
μJuv,Myxo: -6 - 6
recovery of infants
described in [5]
aged 1-90 days, modelled as a coefficient in logitlink logistic model. Virus introduction rate (pIntrV)
0 – 0.1 [%]
Sample
Introduction of
Explorative
viruses from external sources such as arthropod vectors or carcasses. First calendar week of virus
RHD: 1 – 52
introduction (wkIntroV)
Myxo: 1 – 52
Sample
First calendar week
Explorative
each year in which
[calendar week]
viruses are introduced/ host are exposed to the virus.
Last calendar week of virus
RHD: 1 – 52
introduction
Myxo: 1 – 52
Sample
Last calendar week a
Explorative
virus may be (value
[calendar week]
first week of introduction).
Exposure time
RHD: 1 Myxo: 1 - 4
Infection time
RHD: 1 - 2
uniform
[15]
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Supporting Information Myxo: 8 12 [days]
Model initialisation Initial N
200
Initial population size
arbitrary (first 5
for all simulations.
years of simulation not considered in analysis)
Initial infection rate
5%
Proportion of
arbitrary (first 5
individuals of N0
years of simulation
infected.
not considered in analysis)
161 162 163 164 165 166 167 168
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