Download Timing and severity of immunizing diseases in rabbits is controlled

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Research
Cite this article: Wells K et al. 2015 Timing
and severity of immunizing diseases in rabbits
is controlled by seasonal matching of host
and pathogen dynamics. J. R. Soc. Interface 12:
20141184.
http://dx.doi.org/10.1098/rsif.2014.1184
Received: 29 October 2014
Accepted: 10 December 2014
Subject Areas:
biomathematics
Keywords:
biological control, individual-based model,
infectious diseases, invasive species,
myxomatosis, rabbit haemorrhagic disease
Author for correspondence:
Konstans Wells
e-mail: [email protected]
Timing and severity of immunizing
diseases in rabbits is controlled
by seasonal matching of host
and pathogen dynamics
Konstans Wells1, Barry W. Brook1, Robert C. Lacy2, Greg J. Mutze3,
David E. Peacock3, Ron G. Sinclair3, Nina Schwensow1, Phillip Cassey1,
Robert B. O’Hara4 and Damien A. Fordham1
1
The Environment Institute and School of Earth and Environmental Sciences, The University of Adelaide,
Adelaide, South Australia 5005, Australia
2
Chicago Zoological Society, Brookfield, IL 60513, USA
3
Department of Primary Industries and Regions, Biosecurity SA, Adelaide, South Australia 5001, Australia
4
Biodiversity and Climate Research Centre (BIK-F), Senckenberganlage 25, 60325 Frankfurt am Main, Germany
Infectious diseases can exert a strong influence on the dynamics of host populations, but it remains unclear why such disease-mediated control only occurs
under particular environmental conditions. We used 16 years of detailed field
data on invasive European rabbits (Oryctolagus cuniculus) in Australia, linked
to individual-based stochastic models and Bayesian approximations, to test
whether (i) mortality associated with rabbit haemorrhagic disease (RHD) is
driven primarily by seasonal matches/mismatches between demographic
rates and epidemiological dynamics and (ii) delayed infection (arising from
insusceptibility and maternal antibodies in juveniles) are important factors
in determining disease severity and local population persistence of rabbits.
We found that both the timing of reproduction and exposure to viruses
drove recurrent seasonal epidemics of RHD. Protection conferred by insusceptibility and maternal antibodies controlled seasonal disease outbreaks by
delaying infection; this could have also allowed escape from disease. The persistence of local populations was a stochastic outcome of recovery rates from
both RHD and myxomatosis. If susceptibility to RHD is delayed, myxomatosis
will have a pronounced effect on population extirpation when the two viruses
coexist. This has important implications for wildlife management, because it is
likely that such seasonal interplay and disease dynamics has a strong effect on
long-term population viability for many species.
1. Introduction
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2014.1184 or
via http://rsif.royalsocietypublishing.org.
The perpetuation and spread of pathogens in host populations is influenced by a
range of epidemiological processes and host-demographic factors. These are not
independent, because birth and mortality determine the pool of individuals
accessible to infections and infectious disease can significantly reduce host populations or the reproductive behaviour of individuals. Consequently, changes
in either demographic or epidemiological traits in space and time can strongly
influence coupled host–pathogen dynamics [1–4].
The population density and behaviour of species vary over space and time,
often owing to periodic (seasonal) forcing and stochastic environmental conditions [5]. Seasonality can therefore be a prominent feature in infectious
disease dynamics [6,7]. Previous studies have established the importance of seasonal variation in disease transmission rates [8] and the importance of seasonality
in birth rates on disease outbreaks [9–12]. For immunizing infections (i.e. infections in which host individuals that survive infection become resistant against
re-infection), recurrent patterns are primarily driven by the availability of susceptible juveniles [13–16]. Whereby, the time between birth and susceptibility can be
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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infection delay
(a)
early
disease escape
(c)
early
early
(d)
late
late
late
10 20 30 40 50 10 20 30 40 50 10 20
calendar week
Figure 1. Illustration of two possible mechanisms by which recurrent seasonal epizootics for an immunizing disease, with maternal antibody protection of juveniles, can
be explained: via the annual matching of the timing of host reproduction (dashed grey lines) and via virus introduction periods (shaded areas). Each panel represents 2
years of recurrent dynamics. Grey rabbits represent immune individuals and red ones susceptible individuals. Adults (large individuals) are left permanently immune if they
survive infection, whereas juveniles (small individuals) may lose immunity because of the waning efficacy of maternal antibodies. The left panels (a,b) illustrate ‘infection
delay’ scenarios: if juveniles do not become infected when they are insusceptible and/or protected by maternal antibodies, they cannot develop lifelong immunity. In the
absence of prolonged insusceptibility and/or maternal antibody protection, only juveniles born relatively late are likely to escape the risk of dying from disease early in life
(i.e. the encircled red juvenile); this increases their chance to reproduce prior to being diseased in subsequent annual cycles. The right panels (c,d) illustrate ‘disease escape’
scenarios: if juveniles with maternal antibodies do become infected, juveniles with maternal antibodies born and infected during the annual disease period may develop
life-long immunity without being diseased (i.e. the encircled grey juveniles). Whether these two suggested mechanisms impact long-term population dynamics will
depend on seasonal processes, demographic rates such as survival (black lines) or reproduction (green lines) and their interaction with virus dynamics, because infection
and disease escape dynamics determine birth and death processes over multiple cycles.
delayed by natural immunity or maternally acquired antibodies
[17–19]. The re-introduction rate of pathogens is assumed to be
crucial for perpetuating immunizing diseases [20]. However,
the role of the timing of pathogen introduction—when and
for how long during an annual (seasonal) cycle—and its phenological matching with birth pulses have been rarely explored for
immunizing diseases in wildlife species.
Long-term population growth of hosts affected by immunizing diseases is dependent on individuals reproducing
efficiently. However, little is known about whether seasonal
reproduction interacts with mechanisms of host–virus interactions such as natural immunity of juveniles or maternal
antibodies (immunoactive compounds transferred to offspring)
to prevent infection or to attenuate the risk of disease in juveniles [21]. In this context, it is important to distinguish between
infection (invasion of host tissue by virus) and disease (pathogenic change in host body condition due to the virus). High
concentrations of maternal antibodies or natural immunity
of newly born individuals can, for example, delay the infection of juveniles under seasonal scenarios of virus activity if
disease outbreaks are of short duration. These delays can
increase the chance that juveniles survive to develop into reproducing adults before being exposed to the disease. This strategy
is reliant on disease outbreaks of short duration. Alternatively, if
juveniles with maternal antibodies become infected with lower
risk from the disease (early in life), then they may gain permanent resistance and escape disease [22]. Which of these two
mechanisms is most prevalent is not clear for many organisms,
but both inevitably impact fitness and long-term recruitment
and may also impact disease dynamics [23]. We expect both
mechanisms (infection delay or disease delay) to cause seasonally recurrent epizootics brought about by a seasonal matching
of host reproduction and virus activity (figure 1).
If delay of infection is the prevalent mechanism at work, we
expect the recovery rates of adults from disease to have a strong
impact on long-term host population persistence, because susceptible individuals growing into adults will likely reproduce if
they do not die from disease. If escape from disease is the
prevalent mechanism—maternal antibodies do not prevent
infection, but slow the course of disease and increase juvenile
survival, allowing for permanent resistance—then individuals
exposed to the disease should be predominantly juveniles with
waning maternal antibodies. We might expect, in this latter
situation, juvenile recovery rates would exert a strong influence
on population persistence.
European rabbits, Oryctolagus cuniculus, are herbivorous
and iteroparous mammals with a native distribution in temperate grasslands in the Iberian Peninsula (southwestern Europe),
where pronounced seasonal environmental changes influence
resource supply. Reproduction in this species is adapted to
optimize resource use in variable environments, and such plasticity is likely to have enhanced the invasiveness of rabbits,
which are now found in many temperate and Mediterranean
ecosystems around the world [24]. Rabbit haemorrhagic disease virus (RHDV, an RNA calicivirus) and myxoma virus
(MYXV, a DNA poxvirus) have caused significant reductions
in rabbit abundances in their native range [25], and they have
been used successfully as biocontrol agents in the rabbits’
exotic range [26]. Epizootics of RHD (the disease caused by
RHDV) are highly seasonal in some rabbit populations, and
in cooler regions of Australia, RHDV is less effective as a biocontrol agent than in drier regions [26].
Both RHDV and MYXV are associated with various arthropod vectors and induce lifelong immunity in rabbits that
survive infection. Survival rates of juveniles infected with
RHDV tend to be higher than those of infected adults [18],
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10 20 30 40 50 10 20 30 40 50 10 20
calendar week
late
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(b)
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early
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population size (MNA)
250
200
150
100
50
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
year
Figure 2. Patterns of host (introduced European rabbit) demography and epidemics at a long-term monitoring site (Turretfield, South Australia) 1998 – 2013. The
minimum number of host individuals known to be alive (MNA) were recorded during 79 captures sessions (black points). Multiple iterative regression with calendar
week, year and a continuous-time vector modelled as covariates was used to generate a continuous time series of population fluctuations based on MNA (black thin
line). This was also used to reveal the seasonal trend in population size using a sinusoidal model (blue line). The number of rabbits that were observed to die from
rabbit haemorrhagic disease is shown as red triangles, with almost all cases recorded between calendar weeks 30 to 50. Vertical dashed lines indicate the first
calendar week of each year.
possibly because histo-blood group antigen expression on epithelial cells for virus binding is largely absent in very young
rabbits [17]. Therefore, juveniles newly born are resistant to
RHD (i.e. they do not die when exposed to the virus) and
may or may not support virus replication—this period of insusceptibility lasts for approximately three weeks [21]. Disease
survival results in antibodies that are passed by the mothers
as maternal antibodies to the embryo, further attenuating infections and disease early in the life of juveniles [17–19]. In
contrast to RHD, juveniles infected with MYXV have a lower
chance of survival than infected adults [27]. This fundamental difference in age-dependent host–virus interactions for
these two virus species could influence seasonality of disease
outbreaks. Earlier disease susceptibility of juveniles to myxomatosis, in turn, is likely to diminish delays in infection and
disease and mitigate the effect of maternal antibodies against
myxomatosis. Therefore, we expect strong indirect interactions
between the disease dynamics of the two viruses, because
high mortality early in life induced by MYXV can reduce the
pool of susceptible juveniles available to RHD during annual
birth pulses.
Here, we use long-term field data, stochastic individualbased models and approximate Bayesian computation (ABC)
to model the synergistic effect of seasonal demographic and
disease processes (two co-circulating viruses with different epidemiological traits) on invasive rabbit population dynamics in
Australia. We used sensitivity analysis to establish which conditions determine rabbit population persistence and establish
the most important drivers of disease severity under complex
seasonal dynamics in Australia. We also test the role of escape
from infection or disease as two important yet little understood mechanisms that might explain disease severity under
recurrent seasonal matching of reproduction and virus activity.
2. Material and methods
2.1. Study system: rabbits in Australia
European rabbits were deliberately introduced to Australia in the
mid-to-late nineteenth century for hunting, but subsequently
caused major adverse effects on native biodiversity and agriculture
[28]. In Australia, rabbits typically reproduce with the onset of
rainfall-driven increases in food supply (native and invasive
grass and herb species). Rainfall varies enormously across the
Australian continent, causing wide-scale spatial heterogeneity in
the duration of rabbit reproduction and seasonal peaks in recruitment [29,30]. MYXV and later RHDV were released into Australia
as biocontrol agents against rabbits, and both led to large
reductions in rabbit populations shortly after their release (in the
1950s and 1990s, respectively). Both viruses spread quickly over
large geographical areas, most likely through arthropod vectors
[27,31]. While RHDV can be classified at a global scale into distinct
genotypes, all genotypes identified in Australia originated from
the Czech (genogroup 2) genotype [32], upon which our model
is based.
Rabbits and their diseases have been intensively studied at a
long-term field site in South Australia; Turretfield (348330 S,
1388500 E) [33,34] where they have been live-trapped for at least
4–5 consecutive days at 8–12 week intervals since 1996. All live
captured animals were marked and surveyed serologically for
RHDV and MYXV antibodies [32]. Additionally, regular searches
were made for dead rabbits and samples were collected to determine the cause of death. Almost all cases of RHD at Turretfield
were found between calendar weeks 30 and 50 [34] (figure 2).
2.2. Coupling of individual-based epidemiological
and demographic models
We linked a stochastic individual-based demographic simulation to
an epidemiological model [35,36] to simulate the interaction of
rabbit demography and virus epidemiology. We derived demographic and disease parameters from published values and
expert knowledge (full details provided in the electronic supplementary material). In brief, we modelled a single (aspatial)
rabbit population with density-dependent reproduction [37] and
a carrying capacity of 500 individuals, which is the assumed maximum size at the Turretfield site. We modelled rabbit population
dynamics simultaneously as weekly time steps with the dynamics
of both RHDV and MYXV operating on daily time steps. The two
viruses (parameters indexed with V ) were assumed to spread
through modelled rabbit populations with transmission probabilities bV sampled across a range of possible (unknown)
values. When host individuals encountered each other, they
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We used the model to explore how seasonal population and disease dynamics (and their interactions) can impact emerging
patterns of epidemics, disease severity and population viability.
To do this, we used Latin hypercube sampling [38] to sample
across an exhaustive range of demography/disease scenarios
likely to be found under variable natural conditions. A total of
17 key parameters ux were sampled (i) three parameters describing
the extent and timing of seasonal rabbit reproduction; (ii) 12 parameters (six for each disease) modelling the annual onset, time
duration and extent of virus introduction, virus transmission rate
and age-structured disease recovery rates ( juveniles/adults); and
(iii) a parameter approximating the time window of maternal
2.4. Statistical analysis
We examined seasonal patterns in population counts from Turretfield, using capture–mark–recapture data [33]. We used data
imputation to generate a continuous time series (weekly steps) of
the minimum number of individuals known to be alive (MNA,
based on all capture and carcass-recovery records) in different
weeks between 1998 and 2013. To do this, we used a multiple iterative regression approach [40] with three covariates (calendar
weeks, years and a continuous time vector). The imputed MNA
can be expected to provide a good surrogate for population fluctuations over time, although MNA might underestimate true
population size if capture probability is small [41]. We fitted
to these estimates a sinusoidal model to identify key summary
statistics of variation in rabbit population abundance: phase
(Fdata ¼ 43; based on calendar week) and amplitude divided by
the average population size (Adata ¼ 0.22). We also calculated
the mean (m.RHDdata ¼ 40.52 weeks) and standard deviation
(s.RHDdata ¼ 3.26 weeks) of RHD outbreaks according to the
calendar weeks of field-based RHD fatality records (n ¼ 470).
We were not able to derive validation data for the seasonality in
myxomatosis outbreaks, because too few carcasses with signs
of myxomatosis have been recovered at Turretfield.
Similarly, we calculated these summary statistics for each
simulated scenario by fitting sinusoidal models to the simulated
population sizes (i.e. output from the demographic/disease
model) and estimated phase (Fsimulation) and amplitude divided
by the average population size (Asimulation). For each simulation,
we further calculated m.RHDsimulation and s.RHDsimulation.
To compare summary statistics from field data using the
outputs of the simulation model and therefore resolve the most
realistic model structures and assumptions from a wide range
of possibilities, we used the ABC methods. This approach
allows the most likely parameter values to be approximated
based on the distance between observed and simulated summary
statistics [42– 44]. We used the neuralnet regression method in the
R package abc [45] to weight the parameter values for each simulation scenario according to its match with observed summary
statistics. Prediction error was minimized by determining the
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2.3. Simulation scenarios
antibody protection from RHDV. Furthermore, we sampled
environmental stochasticity in birth and survival rates, because
temporal fluctuations in demography driven by environmental
variation might influence disease dynamics (see table 1 for a summary with details in the electronic supplementary material). For
derived parameters describing the seasonal matching of reproduction and virus activity (two for each disease), we calculated
the proportion of the annual reproductive effort (% offspring of
the total annual number) prior to any annual virus activity and
after the end of the viral time window (table 1). We sampled
Nsim ¼ 10 000 different combinations of key parameters in order
to cover a large range of potential parameter combinations. This
approach allowed us to consider the sampled parameters as
sufficiently large random samples (i.e. flat prior distributions) for
investigating which simulations best fitted to field data. Each
hypercube sample was treated as a different scenario and run for
10 independent iterations to account for stochastic effects [39].
We initiated all simulations with 200 rabbit individuals, of which
10 were infected randomly with RHDV and/or MYXV, respectively. We then simulated demographic and epidemiological
dynamics for a total period of 10 years after burn-in periods of
5 years (to minimize the impact of initial values and transitory
dynamics). This first 5-year time period of simulations were run,
but not considered in the interpretation and analysis of modeloutput. Simulations commenced in calendar week 36 when
RHDV outbreaks were most frequently observed in the field at
Turretfield. Preliminary inspection of results showed that 5-year
initialization periods were sufficient to reach equilibria states.
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caused disease state transitions (susceptible to exposed, leading
subsequently from exposed to infected, then infected to resistant
or dead). We assumed homogeneous mixing of host individuals
within this population, because field observations have shown
that rabbits frequently moved between adjacent warrens. For
both viruses, we simulated virus introduction periods (i.e. periods
during which individuals were exposed to the viruses from sources
other than infected individuals from the same population) during
certain time windows (intrperiodV), for which the annual onset
was defined by the first annual week of virus introduction
(wkintroV). Infected rabbits were assumed to recover from disease
with recovery rates ( gV) of adults (more than 90 days old) sampled
across a range of possible values. We modelled recovery rates of
juveniles (less than or equal to 90 days old) with logistic logit-link
regression models to account for increasing susceptibility of juveniles to clinical RHD [18] and increasing recovery rates for
myxomatosis [27]. For this, we assumed the decrease in juvenile
recovery rates from RHD to follow a continuous decrease from
100% after the insusceptibility period towards a rate of recovery
typical for adults, which we determined by sampling values of
the regression slope aJuv,RHD of a logistic model.
For myxomatosis, we implemented a similar (inverse relationship) model, but sampled the intercept mJuvMyxo of the logistic
model, which determines maximum susceptibility of juveniles
after the insusceptibility period; details are provided in the
electronic supplementary material, appendix A.
We modelled infection and disease resistance explicitly for
an initial time period (tMV) owing to maternal antibody protection. After this period of full protection (tMV), we assumed that
maternal antibody concentrations would decline linearly (1% per
day), making infection of individuals with waning antibodies
possible, causing juvenile recovery rates to decrease; note that
the effects of aV/mV, tMV and the linear decline in maternal antibodies are additive. Owing to sparse data availability, maternal
antibodies and insusceptibility are assumed to have a similar
effect on survival of challenged juveniles and the effects are not
fully discernible. This model does not, however, account for individual heterogeneity in infection and virus transmission rates owing
to maternal antibodies; to keep our model simple (despite a large
number of unknown parameters), we varied time periods of full
protection (tMV) and assumed that this captured variation in infection and virus transmission rates (i.e. we assume prolonged tMV to
also represent less infection probability of juveniles with waning
antibodies).
Models were built using META MODEL MANAGER (http://www.
vortex10.org), which allows individual age- and sex-structured
demographic models (built in VORTEX v. 10.0 population viability
analysis software) to be coupled to disease transition models
(built in OUTBREAK v. 2.1) [36]. All demographic events and
state transition dynamics are probabilistic. The introduced
stochasticity represents demographic and environmental variation. Full details of the model’s structure and assumptions are
given in the electronic supplementary material.
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parameter
range/unit
description
annual reproductive
effort
RepEff
50 – 200 (% female
reproducing)
total reproduction quantified as proportion of reproducing females
(individuals can reproduce multiple times each year)
annual peak in
reproduction
RepPeak
20 – 50 (calendar week)
seasonal reproduction assumed to peak during optimal conditions,
quantified as the calendar week with most reproduction (Gaussian
variation in reproduction
RepVar
2 – 10 (s.d. of assumed
rabbit demography
normal distribution)
environmental
EV
stochasticity
disease epidemiology
recovery rate
recovery adjustment
rates, weekly)
gV
aJuv,V, mJuv,V
factors ( juveniles)
transmission rate
0.01 –0.5 (s.d. of vital
bV
0.2 – 0.9
aJuv,RHD: 20.1 to 21
aJuv,Myxo: 20.13
mJuvRHD: 6
mJuvMyxo: 26 to 6
0.3 – 0.9
distribution)
reproductive efforts may be allocated over different time windows
owing to seasonal conditions (Gaussian distribution based on
RepPeak and RepVar)
variation in birth and survival rates reflects the stochasticity
commonly encountered in natural populations
recovery rate of infected adult individuals, which are at risk of dying
from the disease
age-dependent changes in recovery rate of juveniles towards gV
(with opposite effects for RHD and myxomatosis); the dynamic
process can be described by a logistic model with slope aJuv,V and
intercept mJuvV
transmission rate of virus from infection to susceptible individual
during encounters
maternal antibody
protection period
tMV
1 – 50 (days)
virus introduction rate
pIntrV
0 – 0.1 (%)
progeny from females resistant to disease are protected from
infection and disease, before waning maternal antibodies result in
increasing disease and infection risks
introduction of viruses from external sources such as arthropod
vectors, carcasses or any other agents
first calendar week of
virus introduction
wkIntroV
1 – 52 (calendar week)
first calendar week each year in which viruses are introduced/host are
exposed to the virus
virus introduction period
IntrPeriodV
1 – 52 (weeks)
virus exposure may be limited to certain time windows, e.g. owing
to seasonal vector activity
phenological matching (derived parameters)
reproduction prior to
virus introduction
RepPrecV
0 – 100 (%)
the proportion of annual reproduction prior to the time period of
virus introduction
reproduction after virus
introduction
RepLateV
0 – 100 (%)
the proportion of annual reproduction after the time period of virus
introduction
most accurate tolerance rate through a subsampling crossvalidation procedure. Leave-one-out cross validation was used
to evaluate the out-of-sample accuracy of parameter estimates
(using a subset of 100 randomly selected simulated scenarios),
with a prediction error estimated for each parameter [45]. Parameter values were calculated using a tolerance rate of 0.002,
resulting in Npost ¼ 200 samples for posterior estimates. Based
on the importance weight w(i) of each accepted scenario i
(with i ¼ 1, . . . , Npost), we then calculated the mean posterior
estimates u x0 of all key parameters such that
PN post
[ux (i)w(i)]
0
:
(2:1)
ux ¼ i¼1
PN post
w(i)
i¼1
All parameter estimates from ABC are reported as means and
95% credible intervals (CIs) of weighted posterior values.
Following model fitting and verification against the field
data, we explored which simulations most impacted disease
severity and led to rabbit extirpation (local extinction of a simulated population owing to demographic or diseases-induced
dynamics). For this, we classified ‘disease severity’ as the average
number of fatal cases for each disease per year, and we classified
‘local extinction risk’ as a binary variable of whether rabbit
populations went extinct or not for each simulated scenario.
We used boosted regression trees using the gbm.step() routine in
the R package dismo [46] to estimate the relative importance of key
parameters on disease severity (Poisson error structure, learning
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Table 1. Parameter definitions and their sampled ranges for the individual-based model of seasonal demographic and epidemiological dynamics in rabbits.
These are specified independently for the two studied viruses (indexed by V ). The components of phenological matching are derived parameters for quantifying
the demographic and epidemiological interplay. Full details of the model specification and software implementation are provided in the electronic
supplementary material, appendix A.
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6
10
20
30
calendar week
40
50
Figure 3. Illustration of the seasonal matching of annual rabbit reproduction and rabbit haemorrhagic disease virus (RHDV) introduction based on posterior estimates from approximate Bayesian computation (200 selected posterior values out of 100 000 simulations). Black lines show the distribution of relative reproductive
efforts between calendar weeks 1 – 52 for each posterior estimate (based on the parameters RepPeak and RepVar according to table 1). Grey lines show the posterior
estimates for the onset and time duration of annual RHDV introduction (based on the parameters wkIntroRHD and IntrPeriodRHD).
rate 0.001, tree complexity of 5, k-fold cross-validation procedure)
and extinction risk (binomial error structure). Similar approaches
were recently used to identify the importance of disease effects
on prey availability for black-footed ferrets [47], and life-history
and spatial traits on the extinction risk of vertebrates [48].
antibody protection from disease is also likely to play a role in
RHD dynamics. The prediction errors were high for MYXV
parameters and parameters relating to recovery rates from
disease (table 2).
3.2. Drivers of disease severity and rabbit extinction
3. Results
3.1. Inference from field data
The ABC approximation results showed that only models
with seasonal dynamics, which accounted for the matching
between rabbit reproduction and virus introduction, were
able to reproduce the patterns of abundance fluctuations and
RHD epizootics observed in the field (figure 3 and table 2).
The annual reproductive peak of rabbits at Turretfield occurred
around calendar week 37 (CI: 29–42; RepPeak) and the first
week of RHDV introduction usually occurred before this
date, around calendar week 30 (CI: 8–42; wkIntroRHD).
RHDV introduction periods typically lasted for approximately
23 weeks of the year (CI: 10–45; IntrPeriodRHD), indicating that
continuous virus introduction and/or perpetuation throughout a year is unlikely (at least at Turretfield; figure 3). The
proportion of the annual reproductive effort after the introduction of RHDV was small, with a mean estimate of only
1% (CI: 0–16; RepLateRHD, table 2), indicating that at least at
Turretfield, juveniles are unlikely to escape RHD infection
and disease because they are born late; rather, the cause of
resistance is likely to occur through insusceptibility and/or
acquiring maternal antibodies to the virus. Maternal antibodies
against RHD were estimated to provide full protection from
both infection and disease for approximately 30 days (CI:
7–48; tMRHD). This suggests that infection delay is likely to be
an important mechanism affecting rabbit population dynamics.
No individuals were susceptible prior to the first week of
RHDV introduction in 47% of the posterior samples. Therefore,
disease escape through infection during periods of maternal
In our simulations, recovery rate from myxomatosis had the
largest influence on fatalities from RHD (27% relative importance weights; gMyxo), whereas annual reproductive effort
had the largest influence on fatalities from myxomatosis
(24% relative importance weights; RepEff; figure 4). Recovery
rates from RHD also had a considerable impact on the proportion of fatalities from both viruses, indicating further
evidence of potentially strong synergistic effects between
the disease dynamics of the coexisting viruses. In addition,
juvenile disease susceptibility to myxomatosis (mJuv,Myxo)
had an impact on fatalities from both diseases (figure 4),
suggesting that susceptibility to myxomatosis reduces the
severity of RHD by reducing the number of RHD-susceptible
juveniles (figure 4).
Recovery rates from both diseases exerted the largest influence on whether rabbit populations went extinct in our
simulations (43% and 16% relative importance weight; gMyxo,
gRHD), i.e. local extinction risk increased with decreasing
recovery rates with a pronounced sensitivity to changes in
gMyxo (figure 5). Juvenile disease susceptibility to myxomatosis also had a considerable influence on extirpation probability
(14% relative importance weight; mJuv,Myxo); earlier disease
susceptibility of juveniles to myxomatosis led to an increased
likelihood of population extinction (figure 5). In contrast, the
transmission rates of both diseases and time period of
maternal antibody protection had negligible influences on
population persistence (less than 1% relative importance
weights; bRHD, bMyxo, tMRHD and tMMyxo), whereas annual
reproductive effort had a moderate impact (9% relative importance weights; RepEff). Overall, local extinction risk of rabbit
J. R. Soc. Interface 12: 20141184
posterior estimates
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95%
credible
interval
prediction
error
RepEff
RepPeak
103
37
53 – 175
29 – 42
0.96
0.27
RepVar
EV
5
0.21
2 – 10
0.01 – 0.45
0.78
0.92
parameter
rabbit demography
disease epidemiology
gRHD
0.47
0.08 – 0.86
aJuv,RHD
0.68
0.28 – 1
bRHD
0.68
0.43 – 0.93
30
7 – 48
tMRHD
0.06
0 – 0.11
pIntrRHD
30
8 – 42
wkIntroRHD
IntrPeriodRHD
23
10 – 45
gMyxo
0.45
0.08 – 0.72
mJuv,Myxo
1.32
25.19 to 6.03
bMyxo
0.62
0.34 – 0.92
27
2 – 50
tMMyxo
0.05
0 – 0.11
pIntrMyxo
34
7 – 66
wkIntroMyxo
10
1 – 35
IntrPeriodMyxo
phenological matching (derived parameters)
33
15 – 92
RepPrecRHD
RepLateRHD
1
0 – 16
32
0 – 100
RepPrecMyxo
35
0 – 100
RepLateMyxo
0.82
1.0
0.9
0.98
0.89
0.19
0.4
0.66
0.85
1
0.94
1
0.9
0.87
—
—
—
—
populations from disease was moderate over the simulation,
with 26% of the sampled scenarios leading to extirpation
within 10 years.
4. Discussion
The seasonal interplay of host recruitment and pathogen
exposure dynamics has been given little attention in previous
rabbit-disease dynamics models [49– 51]. This is primarily
because of a paucity of long-term and site-specific field
data on RHD and myxomatosis epizootics and the absence
(until recently) of modelling frameworks that are able to
couple multi-disease epidemiological processes dynamically
with stochastic –demographic models. However, understanding the timing of interactions between hosts and viruses, and
how these interactions can vary in time and space owing to
environmental conditions and global change [52] is crucial
for both accurate prediction of the timing of epizootics and,
subsequently, for understanding their impact on long-term
population growth of hosts [4,53].
7
J. R. Soc. Interface 12: 20141184
posterior
mean
By accounting for seasonal interplay between demographic
and epidemiological dynamics, our simulation study provides
strong evidence that the population fluctuations of European
rabbits and outbreaks of RHD in Australia are influenced by
complex seasonal interactions between host reproduction and
virus activity. In particular, seasonality and environmental
variability appear to cause shifts in rabbit abundance, altering
the intensity of biotic interactions and leading to changes in
host–pathogen dynamics over multi-year cycles.
Seasonal outbreaks of RHD observed in rabbit populations
in South Australia could be approximated realistically only
when the virus introduction period was limited to no more
than 10–45 weeks per year and rabbit reproduction took place
prior or during the period of virus introduction. Delayed juvenile susceptibility to the disease along with maternal antibodies
against RHD are likely to provide important protection for
juveniles from infection and this could be the mechanism that
leads to recurrent RHD outbreaks. However, we cannot exclude
completely the competing hypothesis of escape from disease
through early infection during periods of maternal antibody
protection as an additional (though probably not principal)
mechanism in controlling the timing and severity of recurrent
outbreaks. This is because some infection of juveniles with
maternal antibodies is likely. Recently, it was suggested that
rabbit recruitment was primarily facilitated by increased
RHDV-infection of juveniles (with maternal antibodies and
age-related insusceptibility) during RHD outbreaks based on
the age and time when carcasses were recovered [34]. The apparent contradiction between this conclusion and support for
infection delay (opposed to a dominant role of escape from
disease) based on our ABC approach warrants further investigation. In particular, more detailed data on the timing of
juvenile infection during times of early resistance (as a basis
for escape from disease) are needed to better understand
which mechanism is most likely at work. For example, if
maternal immunity prevents individuals from seroconverting
(‘blocking effect’, [5]), early infected individuals become fully
susceptible after their maternally acquired immunity has
waned, and are likely to die if they are reinfected during the
same outbreak, which is feasible if the outbreak lasts several
weeks. Under this scenario, at least those individuals borne
early are unlikely to escape disease. Among the factors that
remained unexplored is the interannual variability in host–
virus interactions, including the years of field survey without
any observed outbreaks (figure 2). The force of infection
may also vary within years, being high during short epizootics,
allowing infection and seroconversion, but greatly reduced
at other times of the year, allowing susceptible rabbits to
accumulate and support recurrent RHD outbreaks. Although
it is not clear what could drive a variable force of infection—
seasonal effects on virus persistence outside of the host, insect
behaviour and the presence of susceptible adult rabbits are all
among the possibilities—it is unlikely that infection rates are
temporally static.
Our sensitivity analysis revealed that the factors determining disease severity did not necessarily represent the same
processes that had the largest effect on population persistence.
For example, annual reproductive effort strongly influenced
disease severity (typically, most individuals dying from disease are juveniles or subadults), but this did not translate to
large effects on long-term population persistence, which was
determined predominantly by the recovery rates from the
two diseases. We suggest that disease severity and population
rsif.royalsocietypublishing.org
Table 2. Posterior estimates and prediction error for key parameters
estimated via approximate Bayesian computation (see Material and
methods for details). Cross validation was used to evaluate the outof-sample accuracy of parameter estimates, with a prediction error
estimated for each parameter.
Downloaded from http://rsif.royalsocietypublishing.org/ on June 15, 2017
g Myxo
8
rsif.royalsocietypublishing.org
RepEff
g RHD
IntrPeriodRHD
mJuv,Myxo
IntrPeriodMyxo
wkIntroRHD
RepPeak
aJuv,RHD
J. R. Soc. Interface 12: 20141184
RepSD
tMRHD
wkIntroMyxo
pIntrRHD
pIntrMyxo
tMMyxo
bRHD
EV
bMyxo
0
10
20
30
relative importance (%)
Figure 4. Relative importance of sampled parameters on the average annual number of fatal cases due to RHD (black bars) and myxomatosis (grey bars) as
evaluated by boosted regression trees. For abbreviations, see table 1.
gMyxo
0.1
0.9
gRHD
0.1
0.9
mJuv,Myxo
–6
6
RepEff
IntrPeriodMyxo
50
200
1
52
RepSD
2
10
RepPeak
IntrPeriodRHD
20
50
1
51
pIntrMyxo
aJuv,RHD
0.0
0.1
0.1
1.0
wkIntroRHD
1
52
wkIntroMyxo
1
52
pIntrRHD
0.0
0.1
bMyxo
0.3
0.9
tMRHD
1
50
bRHD
0.3
0.9
1
50
tMMyxo
EV
0.01
0.50
min
max
parameter matches
0
45
relative importance (%)
Figure 5. Sensitivity analysis: density of parameter matches to simulated rabbit extinction events over the full sampled ranges (left panel; darker colours indicate
more matches in the respective range of values) and relative importance on sampled key parameters on the extinction risk of rabbit populations based on boosted
regression trees (right panel). Minima and maxima of sampled ranges are shown at the tip of bars. A high influence of a parameter in the regression tree can be
interpreted as stronger evidence that certain values from the simulated parameter space are more likely to result in population extinction, when also taking the
interactions among parameters into account. For abbreviations, see table 1.
persistence can only be understood when accounting for the
interaction between the two diseases. For example, recovery
rates from myxomatosis had the largest influence on modelled
fatalities from RHD by reducing the pool of juveniles that could
become susceptible to RHD during or after an outbreak of
myxomatosis. Therefore, the seasonal interplay of host reproduction and delay in infection may have comprised a
mechanism important for the coexistence of the two viruses,
i.e. earlier susceptibility of juveniles to myxomatosis could
have allowed MYXV to efficiently coexist with a lower virulence compared with RHDV [54]. However, we emphasize
that a prominent role of myxomatosis under current conditions
in Australia is unlikely because of only moderate virulence
of common circulating field genotypes of MYXV and an
Downloaded from http://rsif.royalsocietypublishing.org/ on June 15, 2017
5. Conclusion
Our models indicate that for immunizing diseases, the timing of
individual infection can have fundamental consequences for
disease severity and the persistence of the host population. In
particular, recovery rates from myxomatosis were found to be
a major influence on the severity of RHD owing to prolonged
latency in disease exposure to RHD, arising from delayed juvenile susceptibility and persisting post-birth maternal antibodies.
More generally, by revealing the importance of seasonal
matching between rabbit recruitment and timing of disease
introduction, we provide important insights into how environmental variation can influence disease-affected population
dynamics. We expect these observations to be generalizable
across a wide range of infectious disease systems. Advancing
epidemiological models to account for the possible matching
of host and pathogen dynamics under seasonal conditions
might be particularly useful to anticipate possible changes in
disease severity in wildlife under global change.
Acknowledgements. We are grateful to everyone involved in the Turretfield rabbit surveys over almost two decades. Phil Miller and John
Kovaliski provided advice on model parameterization. We thank
Brian Cooke, Emilie Roy-Dufresne, Alice Jones and Thomas Prowse
for constructive comments and discussions.
Funding statement. The study was supported by an Australian Research
Council (ARC) Linkage Project Grant (LP-1202002A). B.W.B, P.C. and
D.A.F were supported by ARC grants (FT100100200, FT0991420
and FS110200051). R.C.L. was supported by US NSF grant DEB1146198. Financial support for R.B.O’H. was provided by the Landesoffensive zur Entwicklung wissenschaftlich-ökonomischer Exzellenz’
(LOEWE) of the state Hesse in Germany through the Biodiversity
and Climate Research Centre (BiK-F). eResearch provided high
performance computation.
9
J. R. Soc. Interface 12: 20141184
potential interactions of RHDV with non-pathogenic Australian
rabbit caliciviruses. These benign caliciviruses confer partial
cross-protection to RHD [62] and reduce the pool of susceptible
individuals for RHDV, affecting disease dynamics particularly
in wetter regions in Australia [63]. Future studies should also
explore whether the strong seasonal dynamics revealed in our
study hold true in geographical areas outside of Australia
with different RHDV genotypes [21], or any future scenario of
new genotype appearance and rapid coevolutionary changes
in host–virus interactions. Lastly, a more rigorous understanding of host–virus coevolutionary dynamics and resistance
over time would provide crucial information for model
improvement. In our model, we assume constant host–virus
interaction over time, whereas viruses such as RHDV are
likely coevolving in response to adaptation and variation in
host defence mechanisms [64]. While approaches that incorporate such dynamics into population-level demographic
and epidemiological models could improve understanding
of coevolutionary dynamics, they require detailed information
for translating the temporal scaling of molecular mechanisms
of host defence and virus invasion strategies into epidemiological and demographic parameters [65,66]. Currently, we lack
a sufficiently detailed understanding to translate relevant
mechanisms of transmission and coevolutionary dynamics
for both myxomatosis and RHD into a model. Further work
is needed to better understand the potential importance of
plasticity in rabbit reproduction and the coevolutionary
dynamics of virulence and immune responses on host defence
against infection [65].
rsif.royalsocietypublishing.org
overriding effect of RHD [30,55]. Importantly, the actual
impact of the two diseases under natural conditions will
depend on the seasonal timing of disease introduction, as a
later introduction of MYXV into a host population will
dampen the positive effect on disease severity and population
extirpation in relation to delayed susceptibility to RHDV. Some
of the impact of myxomatosis on RHD severity, however, is
likely underestimated in our modelling, because we did not
take immunosuppressive characteristics of MYXV [56] into
account.
With a relatively low impact of myxomatosis on rabbit
populations, RHDV infection and disease delay may become
an important driver of disease severity and population extirpation. Because introduction of RHDV is unlikely to occur
year-round (based on data from Turretfield), juvenile insusceptibility and maternal antibody protection early in life, which
causes delayed infection, results in relatively more susceptible
adults surviving through to subsequent years, and they then
become part of the reproductive pool. If adult recovery rates
during subsequent epizootics are low, the pool of reproducing
individuals declines, causing population reduction.
The simulation outputs predict that populations did not
persist in 26% of all sampled scenarios. This result is not consistent with evidence from field observations, which suggest
that the introduction of RHDV has not caused local extinctions
of Australian rabbit populations. However, the simulation
model was purposefully parametrized using a large, but plausible, range of possible parameter values [57]. This included
best estimates for disease parameters where empirical data
were lacking.
It has been suggested that population growth may be buffered against mismatches in predator –prey interactions owing
to density-dependent vital rates [58–60]. Our modelling
reveals that complex host –pathogen dynamics are not only
influenced by density-dependent vital rates, but are also the
result of other processes such as the timing of susceptibility
of juveniles to diseases (in this case, myxomatosis, and to a
lesser extent RHD). If recovery rates are low and the diseases
have the potential to kill a large number of individuals, seasonal mismatches are particularly likely to limit the impact of
diseases on population growth. With high recovery rates, in
turn, we may expect reproductive effort and disease dynamics
to have equally important effects on rabbit extinction. This is
because in rabbits, density-dependent population regulation is
expected to disproportionately affect juvenile survival [33], and
seasonally coinciding density-dependent and disease-induced
mortality may counterbalance each other.
Caution should be exercised before using our results to
assess the overall feasibility and cost effectiveness of using
RHDV or MYXV to manage rabbit numbers in their exotic
range. This is because we know that rabbit recruitment
varies spatio-temporally in response to climatic variation
[24], most probably conditioning host –pathogen dynamics.
Although we have provided robust simulation evidence that
seasonal interactions can explain RHD epidemics in Australia,
future studies should target factors such as the occurrence of
possible arthropod vector species, the timing of epidemics
and their subsequent relationship to recruitment to further
improve our predictions [26]. We therefore emphasize that parameter estimates from our ABC approach should be interpreted
cautiously [61]. Furthermore, to reduce possible bias in predictions of RHD epidemics, future simulation studies should also
consider variation in host–virus interaction over time and the
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