plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USA08-PLCB-RA-0980R310.1371/journal.pcbi.1000346Research ArticleComputational Biology/Ecosystem ModelingInfectious Diseases/Viral InfectionsThe Role of Environmental Transmission in Recurrent Avian Influenza
EpidemicsTransmission Dynamics of Avian Influenza
VirusesBrebanRomulus^{1}^{*}^{¤}DrakeJohn M.^{1}StallknechtDavid E.^{2}RohaniPejman^{1}^{3}^{4}Odum School of Ecology, University of Georgia, Athens, Georgia, United
States of AmericaSoutheastern Cooperative Wildlife Disease Study, University of Georgia,
Athens, Georgia, United States of AmericaCenter for Tropical and Emerging Global Diseases, University of Georgia,
Athens, Georgia, United States of AmericaFogarty International Center, National Institutes of Health, Bethesda,
Maryland, United States of AmericaFraserChristopheEditorImperial College London, United Kingdom* E-mail: breban@gmail.com
Current address: Unité d'Epidémiologie des
Maladies Emergentes, Institut Pasteur, Paris, France
Conceived and designed the experiments: RB JMD PR. Performed the experiments:
RB. Wrote the paper: RB JMD DES PR.
The authors have declared that no competing interests exist.
42009104200954e100034631120082320092009Breban et alThis is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited.
Avian influenza virus (AIV) persists in North American wild waterfowl, exhibiting
major outbreaks every 2–4 years. Attempts to explain the patterns of
periodicity and persistence using simple direct transmission models are
unsuccessful. Motivated by empirical evidence, we examine the contribution of an
overlooked AIV transmission mode: environmental transmission. It is known that
infectious birds shed large concentrations of virions in the environment, where
virions may persist for a long time. We thus propose that, in addition to direct
fecal/oral transmission, birds may become infected by ingesting virions that
have long persisted in the environment. We design a new host–pathogen
model that combines within-season transmission dynamics, between-season
migration and reproduction, and environmental variation. Analysis of the model
yields three major results. First, environmental transmission provides a
persistence mechanism within small communities where epidemics cannot be
sustained by direct transmission only (i.e., communities smaller than the
critical community size). Second, environmental
transmission offers a parsimonious explanation of the 2–4 year
periodicity of avian influenza epidemics. Third, very low levels of
environmental transmission (i.e., few cases per year) are sufficient for avian
influenza to persist in populations where it would otherwise vanish.
Author Summary
Avian influenza viruses (AIVs) in wild waterfowl constitute the historic source
of human influenza viruses, having a rich pool of genetic and antigenic
diversity that often leads to cross-species transmission. Although the emergence
of H5N1 avian influenza virus onto the international scene has captured the most
attention, we do not as yet understand the mechanisms that underpin AIV
persistence and dynamics in the wild. We developed a novel
host–pathogen model intended to describe the epidemiology of low
pathogenic AIV in temperate environments. Our model takes into account
seasonality in migration and breeding together with multiple modes of
transmission. AIVs have been detected in unconcentrated lake water, soil swabs,
and mud samples. Laboratory experiments show that AIVs persist and remain
infectious in water for extended periods. However, so far, the possibility of
environmental transmission of AIV has been largely overlooked. Our work shows
that environmental transmission provides a parsimonious explanation for the
patterns of persistence and outbreaks of AIV documented in the literature. In
addition to their scientific importance, our conclusions impact the design of
control policies for avian influenza by emphasizing the dramatic and long-term
role that environmental persistence of pathogens may play at the epidemic
level.
RB, DES, and PR were supported by a grant from the Centers for Disease Control
and Prevention (5U19Cl000401). JMD was supported by the National Science
Foundation (EF-0723601) and the James S. McDonnell Foundation. The authors
declare that they do not have any financial interest. The funders had no role in
study design, data collection and analysis, decision to publish, or preparation
of the manuscript.Introduction
Many important infectious diseases persist on a knife-edge: rapid rates of
transmission coupled with brief infectious periods generate boom-and-bust epidemics
that court extinction. Such violent epidemic behavior has been observed in measles
[1]–[4], plague [5],
cholera [6],
meningitis [7],[8], and pertussis [9], among
others. Several distinct mechanisms have been proposed to explain the long-term
dynamics and persistence of these pathogens. For example, measles persistence is
primarily determined by the rate at which the susceptible pool is replenished,
leading to Bartlett's concept of critical community size,
the minimum population size above which an infectious disease remains endemic [4]. In
contrast, plague is enzootic in rodents and their fleas and thus its persistence in
human populations is explained by intermittent reintroduction from the animal
reservoir [10]. King et. al [6] argue that rapid loss of
immunity to cholera may replenish the human susceptible pool so quickly that large
amplitude cholera outbreaks can be observed semiannually. Finally, rich strain
polymorphism allows echoviruses –responsible for aseptic
meningitis– to circumvent host immunity and thus reinvade the population
[7],[8]. These examples
illustrate the need for understanding alternate persistence/re-invasion mechanisms
of infectious diseases for effective management and control.
In this paper, we investigate the persistence and dynamics of low pathogenic avian
influenza virus (AIV) in North American bird populations. Avian influenza viruses in
wild waterfowl constitute the historic source of human influenza viruses [11], with
a rich pool of genetic and antigenic diversity [11],[12] that often leads to
cross-species transmission. Perhaps the best-known and most topical example is the
transmission of H5N1 avian influenza virus to humans [13]. Human infection with
H5N1 is associated with a significant risk of mortality; to date, approximately
50% of infected individuals have died from the infection (see [13] and
references therein). Developing a better understanding of the ecology of avian
influenza viruses is, therefore, very timely.
AIVs infect more than 90 species of birds from 13 orders, mostly Anseriformes (ducks)
and Charadriiformes (shorebirds). Long-term studies of AIV prevalence in North
America [14],[15] have gathered time series of annual estimates
that extend over 26 years for Anseriformes and 20 years for Charadriiformes. The
data is stratified over influenza subtype: H3, H4, and H6 were the most prevalent
subtypes isolated from Anseriformes. Most interestingly, the prevalence of infection
with these subtypes as well as the aggregate prevalence exhibit recurrent outbreaks
in duck populations at 2–4 year intervals.
It is well established that birds infected with avian influenza are infectious for
approximately a week (range 6–10 days), during which they continuously
shed vast concentrations of viral particles in their feces [11],[16],[17]. These
virions are then ingested by susceptible birds, completing the fecal/oral
transmission route [18],[19]. However, attempts to recover the patterns of
periodicity and persistence in avian influenza epidemics in waterfowl from simple
modeling principles using only this essentially direct transmission mechanism are
unsuccessful (see, for example, Text S1 and Discussion). We propose that the
missing ingredient in direct transmission models is the additional indirect
contribution made to transmission by the ingestion of infectious virions that
persist in the environment. It has been demonstrated, for example, that the avian
influenza strain H2N4 (A/Blue-winged teal/TX/421717/01) can persist for extended
periods in the environment, with an estimated one log decay time of 490 days in
water at temperature 4°C, pH 7.2, salinity 0 ppt [16],[20].
Additionally, these persistent virions are known to be infectious [16],[20],[21], arguing
for a potentially significant epidemiological contribution by environmental
transmission.
Here we examine whether environmental transmission provides a more parsimomious
explanation for the observed patterns of avian influenza epidemics. The phenomenon
of environmental transmission is known to be significant for viral infections in
humans (e.g., gastroenteritis [22]) and animals (e.g., rabbit haemorrhagic disease
[23]), water-borne pathogens (e.g., cholera [6],[24] and
avian cholera [25]), some bacterial infections (e.g., tetanus [26],
salmonella [27] and epizootics of plague [28]), prion diseases (e.g.,
chronic wasting disease [29] and bovine spongiform encephalopathy [30]) and
zoonoses (e.g., Nipah and Hendra viral diseases [31]). Despite these
examples, the epidemiological consequences of environmental transmission remain
poorly understood [32]–[38].
Here we propose a new host-pathogen model that combines within-season transmission
dynamics, with a between-season component that describes seasonal migration,
reproduction and environmental variation. Analysis of deterministic and stochastic
versions of this model shows that environmental transmission plays a critical role
for the persistence of avian influenza and its inter-annual epidemics. We conclude
that environmental transmission may provide a parsimonious explanation of the
observed epidemic patterns of avian influenza in wild waterfowl.
Model
Our model is designed to represent a typical population
(∼5,000–10,000 individuals) of ducks (Anseriformes) that migrates
twice a year between a northern breeding ground and a southern wintering ground. As
shown in Figure 1, the model
assumes two geographically distinct sites linked by rapid migration (thick black
arrows). The duration of the breeding and the wintering seasons are assumed to be
the same. At the beginning of each breeding season, new susceptible chicks are added
to the flock (Figure 1, open
thick arrow); i.e., we assume pulsed reproduction.
10.1371/journal.pcbi.1000346.g001Illustration of the model.
The decay curves of the virus during winter and summer are sketched in blue
and red, respectively. The corresponding symbols of the viral persistence
rates within each ground are also illustrated. The persistence rates of
avian influenza strains in the breeding and wintering grounds are quite
different because they increase strongly with the temperature of the
environment. Since water temperatures where the ducks are present (i.e.,
breeding grounds in the summer and wintering grounds in the winter) may be
similar, we chose the corresponding persistence rates to be similar, as
well. The persistence rate is much reduced (i.e., the persistence time of
the virus increases) in the breeding grounds during the winter as the
temperature drops. Also, the persistence rate is significantly increased
(i.e., the persistence time of the virus decreases) in the wintering grounds
during the summer as the temperature increases.
Given the uncertain and possibly complex patterns of cross-immunity in wild ducks, we
focus on the dynamics of a single subtype. Hence, we assume that after recovery from
infection, ducks acquire life-long immunity. Thus, within each season, the
epidemiological dynamics are of the familiar type with two transmission routes: direct and environmental. To
derive the environmental transmission functional form, we denote the probability
that a duck escapes infection when exposed to virions by ; note that must decrease with and . Next, we consider a bird that is exposed to virions in two steps: first virions and then virions . Therefore, where is the conditional probability that the duck will escape infection
when exposed to virions after escaping infection when exposed to virions. It is assumed that there is no immunological consequence
of unsuccessful exposure; that is, the probability of escaping infection is
independent of past AIV challenges that did not result in infection (). Thus, we obtain the exponential Cauchy equation [39]. Since is a decreasing probability function defined on all non-negative
real numbers, the only acceptable solution is where is a constant with unit of . Therefore, environmental infection is modeled using a continuous
Markov chain with a constant rate . Note that the parameter is related to the empirically determined (the dose at which there is a 50% probability of
infection) by the following equation , giving . However, a bird is exposed to virus in the environment via
continuous ingestion of lake water. To model this, we introduce a constant rate that expresses the per capita fraction of the virions ingested per unit time. Thus , which we call exposure rate, is given by the
per capita consumption rate scaled by the lake volume. The
transmission rate per susceptible due to environmental contamination is given by .
Infected ducks shed virus in the environment where the virus persists. We assume that
the viral population is large enough so that these two processes can be captured by the
following differential equationwhere is the number of infecteds, is the shedding rate and is the decay rate of the virus in the environment. If we divide
the above equation by and use the variable instead of then the equation no longer contains the parameter . Using instead of amounts to measuring the number of virions in units of per shedding rate (i.e., ) which is the unit that we adopt for the rest of the paper. The
environmental transmission rate now becomes , where is the number of susceptibles. Thus, the dynamics of the model
depends on and through their product , which is a re-scaled environmental infectiousness.
Model variables and parameters are presented in Tables 1 and 2, respectively. We use capital subscripts to
denote the season (i.e., for the breeding season and for the wintering season) and lower case superscripts for
geographical location (i.e., for the breeding grounds and for the wintering grounds). For a deep understanding of the
system, we develop two versions of the model: (i) a deterministic system, with
continuous state variables, and (ii) a hybrid framework that consists of discrete
population variables, and stochastic demographic and transmission transition
probabilities together with deterministic virus kinetics. The transmission dynamics
within the continuous model are expressed as coupled ordinary differential equations
and are useful in examining the underlying deterministic clockwork of the system.
Not surprisingly, however, this framework often predicts biologically unrealistic
fractional numbers of infecteds (Mollison's so-called
“atto-fox” phenomenon [40]). Since we are
particularly interested in the processes of extinction and persistence of AIV, we
further refined our study by constructing a stochastic model, where the host
population variables are integer-valued.
10.1371/journal.pcbi.1000346.t001The variables of the model.
Symbol
Definition
Unit
number of ducks
susceptible ducks
infected ducks
recovered ducks
viral population
in the breeding grounds during the summer
in the breeding grounds during the winter
in the wintering grounds during the summer
in the wintering grounds during the winter
10.1371/journal.pcbi.1000346.t002The parameters of the model.
Symbol
Definition
Value/Range
Unit
Reference
habitat carrying capacity
3000
–
duck fecundity
2
[56],[57]
direct transmissibility
0–0.05
–
exposure rate
10^{−3}
–
environmental infectiousness
–
virus shedding rate
10^{5}–10^{6}
[58]
re-scaled environmental infectiousness
1–10^{6}
–
natural death rate
0.3
[56]
recovery rate
52
[11]
virus clearance rate in the breeding grounds during the
summer
5
[21]
virus clearance rate in the breeding grounds during the
winter
1.3
[21]
virus clearance rate in the wintering grounds during
the winter
5
[21]
virus clearance rate in the wintering grounds during
the summer
50
[21]
For further explanation of the parameter values see the Text
S1.
Model with continuous variables
The model proceeds as follows.
The start of the Breeding Season. We start with the
initial conditions , , , and at the beginning of the breeding season. Then, we add
new chicks to the flock. As with many natural reservoirs, the
pathogenicity of AIV to birds is neglible, thus we assume that ducks
have a fixed realized fecundity, , irrespective of infection history. We further assume
that chick survival is density-dependent and is determined by , where is the total number of ducks and is the carrying capacity of the habitat. Therefore,
the number of chicks that join the flock every breeding season is ; i.e., .
Breeding Grounds Dynamics. We now integrate the
variables , , , and for the duration of the breeding season (i.e., half a
year) according to the following set of differential equations:The first three equations describe the well-known model [5], with the
addition of an environmental transmission term. The last two equations
describe the dynamics of the virus at the breeding and wintering
grounds, respectively. They reflect the fact that during the summer at
the breeding grounds, virus is shed by infected birds and decays in the
environment. On the wintering grounds, however, there are no ducks
during the summer, hence virion kinetics are only affected by viral
degradation.
Wintering Grounds Dynamics. At the end of the breeding
season, we introduce viral population variables for the wintering season and and continue the integration for another half of a
year using the following set of differential equations that implicitly
accounts for the migrationAt the end of the wintering season we set and and resume with step 1. with the next breeding season
in a similar fashion.
Hybrid model
In this model, the bird population variables are discrete, evolving through a
continuous-time Markov chain integrated using Gillespie's direct method
[41]. The processes that take place throughout a season and their
corresponding rates are summarized in Table 3. The algorithm of the model is as
follows.
10.1371/journal.pcbi.1000346.t003The processes that take place within a season.
Process
Definition
Rate
Direct infection
,
Environmental infection
,
Death of susceptible
Death of infected
Death of recovered
Recovery
,
The variables and the parameters are explained in Table 1.
The start of the Breeding Season. Start with the initial
conditions , , , and at the beginning of the breeding season. New chicks
are added as before except that the number of chicks is given by a
binomial stochastic variable .
Breeding Grounds Dynamics. We stochastically integrate
the variables , and according to Gillespie's algorithm for one
half of a year (i.e, one season). The variables and are integrated within a season using Eq. (1). For a
time interval where is constant,For the wintering ground we getwhere , as there are no ducks at the wintering grounds.
Breeding Grounds Dynamics. At the end of the breading
season, we introduce viral population variables for the wintering season and . The variables , and are integrated as before. and are integrated as follows. is given bywhere , as the ducks have left the breeding grounds. is given byfor every time interval where is constant. At the end of the wintering season we set and and continue with step 1. in a similar fashion.
We note that a continuous-time Markov chain where all the variables , and are evolved using point processes can be easily constructed by
adding birth (i.e., with rate for and and rate 0 for and ) and death processes (i.e., with rate ) for to the list presented in Table 3. First, it can be shown that if the
variables of this Markov chain are approximately uncorrelated and normally
distributed, then their expectations satisfy the equations of the continuous
model presented in the previous section [42]; i.e., that the
mean-field approximation of this Markov chain is the continuous model
represented by Eqs. (2)–(11). Second, our hybrid model is a good
approximation of the continuous-time Markov chain when the variables are large and the sum of their rates is much larger than the
sum of all the other rates. Indeed, under these conditions, most processes are
births and deaths of virions and other processes occur only sporadically. In
between these processes, the stochastic dynamics of the viral load provided by
the continuous-time Markov chain can be satisfactorily approximated by the
deterministic equations of the hybrid model. We thus conclude that in the case
where virus is abundant the continuous model represents the mean-field
approximation to our hybrid model described above.
ResultsModel without environmental transmission
As a baseline, we first explored a simplified model that includes fecal/oral
transmission, migration, seasonality and pulsed reproduction, without
environmental transmission. Whether stochastic or deterministic, this model is
unable to reproduce the recurrent pattern of avian influenza epidemics. The
continuous model shows unrealistic infected populations as low as
10^{−8} individuals (see Text S1)
while the stochastic model undergoes rapid extinction when the infected
population drops to zero.
Deterministic orbits of mixed transmission model
Figure 2 shows numerical
results for a typical orbit of our deterministic model with both direct and
environmental transmission mechanisms (for definitions of the technical terms in
this section the reader is referred to [43]–[46]).
The orbit rapidly settles to an attractor with a period of two years. Figure 2A–C show the
number of susceptibles, infected and recovered versus time, respectively. The
Fourier power spectrum density of the infected time series is presented in Figure 2D; a peak at is easily noted. Figure 2E and 2F show bifurcation diagrams versus the direct
transmisibility and the re-scaled environmental infectiousness , respectively. The orbits are sampled annually at the end of
the wintering season when, each year, the number of infected is the lowest.
Panel (e) shows a period doubling and an inverse period doubling bifurcation,
while no bifurcations are present in Panel (f). The position of the orbit
presented on the left is marked with dotted lines.
10.1371/journal.pcbi.1000346.g002Simulation results obtained using our deterministic model.
Panels A, B, and C show , and versus ; note the logarithmic scale for . The initial conditions are , , , and . The parameters are as in Table 1 with and . Panel D shows the Fourier power spectrum density of over a time interval of 25,000 years. Panels E and F
show bifurcations diagrams of the model versus and , respectively. The orbits are sampled yearly, at the
end of the wintering season. The dotted lines mark the positions of the
orbit presented on the left within the corresponding bifurcation
diagrams.
However, the continuous model for the parameters of avian influenza in
populations of 5,000 to 10,000 individuals regularly predicts numbers of
infecteds less than one. Thus, the epidemic would often go extinct as the number
of infected would reach zero. This phenomenon is not captured by a
continuous-state model. Furthermore, note that in Figure 2E model dynamics are predicted to be
rigidly biennial, in contrast to the erratic 2–4 year outbreaks
observed in the wild.
Stochastic orbits of mixed transmission model
To understand the extinction and persistence dynamics of avian influenza we
integrated the stochastic version of the model. Figure 3A–C show the number of
susceptibles, infected and recovered versus time, respectively, in a simulation
of our stochastic model. In this case, the infected population often goes
extinct and the epidemic is then reignited by environmental transmission. In
direct contrast to the predictions of the deterministic model, a major epidemic
does not occur every two years as such an event is sparked probabilistically
(Figure 3B). In general,
the periodicity of stochastic orbits is larger than that of corresponding
deterministic orbits. If an epidemic does not occur then susceptibles continue
to build up and the next epidemic will thus be more severe. Note that the
incidence peaks of the sporadic epidemics of the stochastic model are higher
than those of the biannual epidemics of the continuous model by about a factor
of three. The Fourier power spectrum density of the infected time series clearly
shows a sequence of peaks corresponding to the annual inflow of susceptibles;
Figure 3D. A peak around is still visible; however, the peak is now very flat, covering
a broad frequency range. The Fourier transform does not appear to provide a very
insightful characterization of the epidemic dynamics owing to tall and narrow
prevalence peaks that do not occur at very regular intervals.
10.1371/journal.pcbi.1000346.g003Simulation results obtained using our stochastic model.
Panels A, B, and C show , and versus ; note the logarithmic scale for . The initial conditions are , , , and . The parameters are as in Table 1 with and . The blue line in panel D shows the Fourier power
spectrum density of over a time interval of 3,500 years. The yellow line
represents the moving average of the spectrum density. Panels E and F
show the global spectral decomposition in Difference-of-Gaussians (DoG)
wavelets of stochastic orbits versus and , respectively. Each spectrum is an average over 100
wavelet transforms of individual stochastic realizations of the orbit
over 3,300 years (this time interval gives 95% confidence to
the peaks of each wavelet transform; the fluctuations are due to the
stochasticity of the realizations of the model). The color map
represents the power scale measured in . The dotted lines mark the positions of the stochastic
realization presented on the left within the corresponding panels.
A more useful approach to revealing periodic patterns in the stochastic time
series is a wavelet spectral decomposition. Here we use the
Difference-of-Gaussians (DoG) wavelet since it fits well the tall and narrow
prevalence peaks of the time series (see Text S1). Figure 3E and 3F show the global spectral
decomposition of stochastic orbits in DoG wavelets versus the direct
transmissibility and the re-scaled environmental infectiousness , respectively. Each spectrum is an average over 100 wavelet
transforms of individual stochastic realizations of the orbit. The white solid
lines in Figure 3E and 3F
trace the positions of the local peaks in the spectra versus the corresponding
system parameters. Note that stochastic time series show periodicity larger than
one year (i.e., at ∼2 years and above) for a significantly broader range
of than deterministic time series. Also, note that the dominant
periodicity of the stochastic time series changes very little with , similar to the findings presented in Figure 2F.
The disease-free/endemic transition
It is important to distinguish the parameter sets for which AIV is endemic. While
many model parameters have empirically-established ranges (e.g., host breeding
traits and the duration of infectiousness [21]), the values of
other key parameters, such as the direct transmission rate and the environmental infectiousness are less certain. Therefore, we explore the plane (, ) with all the other parameters of the model given in Table 1. For the continuous
model, the disease-free state is a periodic attractor with period of one year.
This disease-free state loses stability through a transcritical bifurcation
which marks the disease-free/endemic transition. Since the bifurcation is
codimension one, the transition occurs on a line segment in the (, ) plane; see Figure 4. The segment was obtained by numerically solving for the
value of where the transcritical bifurcation of the continuous model
with a given value of occurs.
10.1371/journal.pcbi.1000346.g004Color map of the time-average of the number of infected <inline-formula><inline-graphic mimetype="image" xlink:href="info:doi/10.1371/journal.pcbi.1000346.e239" xlink:type="simple"/></inline-formula> versus the direct transmissibility <inline-formula><inline-graphic mimetype="image" xlink:href="info:doi/10.1371/journal.pcbi.1000346.e240" xlink:type="simple"/></inline-formula> and the environmental infectiousness <inline-formula><inline-graphic mimetype="image" xlink:href="info:doi/10.1371/journal.pcbi.1000346.e241" xlink:type="simple"/></inline-formula>.
Each colored point is calculated by averaging the results of 100
stochastic realizations. For each realization, a transient of 100 years
was discarded and the time average was performed over 200 years. The
white line indicates the epidemic threshold of the mean-field model: for
parameters in the circled area around the origin there are no epidemics,
otherwise epidemics occur. In the Text
S1, we present the results of extensive sensitivity analyses.
For the stochastic model, the disease-free/endemic transition is defined in a
more subtle way. The disease-free region is defined by all the parameter sets
for which, in all of the realizations of the model, the number of infected reaches zero in finite time and stays at zero for all
subsequent time, irrespective of the initial conditions. The epidemic region is
defined by all the parameter sets for which there exist realizations of the
model such that, for any moment of time , is not zero for all time once . In the disease-free region the probability of an epidemic is
zero; however, in the endemic region, the probability of an epidemic increases
from zero (close to the boundary with the non-epidemic region) toward one.
Therefore, in the case of the stochastic model, it is more difficult to
numerically obtain a precise border between the disease-free and the endemic
regions. Here we computed the time-average of the infected over 200 years in 100
realizations of the model for a region in the (, ) plane; see Figure 4. (A transient of 100 years was discarded for each
stochastic realization. Numerical analysis reveals that the results are robust
and accurate at these parameters.) Thus, dark blue region corresponds to an
epidemic probability of less than ∼1% and encloses the
disease-free region. Note that for the probability of a sustained epidemic to be
larger than 1%, the re-scaled environmental infectiousness must exceed 10^{3}. Simulations did not show sustained
epidemics for low or absent environmental transmission.
Because of the way in which the disease-free/endemic transition is defined for
the stochastic model, it is difficult to compare the epidemic threshold of the
stochastic model with that of the deterministic model. In our case, however, we
may expect that they disagree. The mean-field approximation of a stochastic
model is obtained in two steps. First, one derives an infinite set of ordinary
differential equations that describes how the moments of all orders of the
stochastic variables change with time. Second, under the assumption that all
stochastic variables are uncorrelated and normally distributed, the set of
equations is truncated at the first moment (i.e., moment
closure) which is the expectation [42]. Disagreement
between a stochastic model and its mean-field approximation is expected if the
assumptions on normality or correlations are violated. This typically happens
when any of the population compartments is small. Here, the disagreement at low
numbers of infecteds might be particularly enhanced because of the fact that the
continuous model allows for the number of infected birds to be less than one so
that we always have two different transmission routes of avian influenza. When
the epidemic is at its nadir in the continuous model, the direct transmission
rate does not vanish (the number of infected always stays larger than zero even
though it may be substantially smaller than one) and thus the the chain of
transmission is maintained by both direct and environmental transmission
mechanisms. In contrast, in the stochastic model the numbers of infecteds often
reaches zero. Therefore, AIV maintenance is exclusively due to environmental
transmission. We thus expect that the disease-free region of the stochastic
model is larger than that of the deterministic model.
The interplay between direct and environmental transmission
In Figure 5A and 5B we
present the time-averages of the direct and environmental transmission rates,
respectively. Note that the environmental transmission rate is two orders of
magnitude smaller than the direct transmission rate, yet critical in maintaining
the epidemic. The time-average of the direct transmission rate increases with and , following the pattern of the time-average of the number of
infected in Figure 4.
However, the time-average of environmental transmission rate has a very
different pattern, attaining high values at low values of and decreasing at high ; Figure
5B. Another picture of these contrasting patterns is Figure 6. At low , the environmental transmission rate is relatively high and
increases with as more infected individuals shed more virus in the
environment. A turning point in this scenario happens at when direct transmission starts to dominate. As the direct and
environmental mechanisms of transmission compete for susceptibles, a marked
increase in direct transmission results in a decrease of environmental
transmission.
10.1371/journal.pcbi.1000346.g005Direct versus environmental transmission.
Color maps versus the direct transmissibility and the environmental infectiousness of the time average of the A direct transmission rate;
B environmental transmission rate and the average (over stochastic
realizations) fraction of time when the C direct transmission is not
zero; D environmental transmission is not zero. The simulation details
are the same as for Figure
4.
10.1371/journal.pcbi.1000346.g006Re-plot of data from <xref ref-type="fig" rid="pcbi-1000346-g005">Figure 5</xref>.
Direct and environmental transmission rates versus the direct
transmissibility at environmental infectiousness .
A fundamental feature of environmental transmission is the fact that it persists
(i.e., does not vanish) even when the number of infecteds (and hence the rate of
direct transmission) is zero. As a result, environmental transmission may
reignite the epidemic. To contrast the persistence characteristics of direct and
environmental transmission, we calculated the average (over stochastic
realizations) fraction of time when direct transmission does not vanish (Figure 5C) and environmental
transmission does not vanish (Figure 5D). The direct transmission rate vanishes when either or while the environmental transmission rate vanishes when either or, quite unlikely, . (Here we assumed that the environmental transmission is
virtually zero when . Computations with yield very similar results.) From Figure 5C, we obtain that direct transmission
at is non-zero at most 30% of the time with a
relatively prominent peak at . In contrast, the environmental transmission is non-zero at
most 70% of the time and the peak is much more shallow over the
chosen range of . Therefore, even though much smaller than the direct
transmission, environmental transmission is much more persistent and may
re-ignite the epidemic when there are no infected left.
An investigation of the time-averaged environmental transmission rate when the
epidemic is reignited was performed as follows (Figure 7). Given a stochastic realization of
the model, we selected the events where the number of infected increases from
zero to one. Say that these events occurred at times and that the corresponding preceding events occurred at times (i.e., for every , the event at time is immediately followed by the event at time ). For each event , we integrated the environmental transmission rate over the time interval (,). Then, the time-averaged transmission rate when the epidemic
is reignited is given bywhere is the number of susceptibles in the time interval (,) and is a constant. In the analysis presented in Figure 7 we further averaged
over 100 realizations of the stochastic model. The pattern in Figure 7 is comparable to that
in Figure 5B. Note that the
environmental transmission rate that re-ignites the epidemic is less than a
factor of two larger than the average.
10.1371/journal.pcbi.1000346.g007Color map of the time-average of the environmental transmission rate
when the epidemic is re-ignited versus the direct transmissibility <inline-formula><inline-graphic mimetype="image" xlink:href="info:doi/10.1371/journal.pcbi.1000346.e282" xlink:type="simple"/></inline-formula> and the environmental infectiousness <inline-formula><inline-graphic mimetype="image" xlink:href="info:doi/10.1371/journal.pcbi.1000346.e283" xlink:type="simple"/></inline-formula>.
The simulation details are the same as for Figure 4.
Discussion
In this paper, we have explored the epidemiological dynamics and persistence of avian
influenza viruses, with a view to understanding the respective roles of
environmental transmission and demographic stochasticity. We have found that an framework that includes seasonal migration, pulsed reproduction
and fecal/oral, but not environmental transmission is unable to reproduce the
documented recurrent pattern of avian influenza epidemics. The continuous version of
the model predicts unrealistic infected populations, with values as low as
10^{−8} individuals (see Text S1), while
the stochastic analogue predicts rapid extinction (similar to the depletions of
infected in Figure 3B). The
unrealistically low infection prevalence is also observed in the continuous model
with added environmental transmission Figure 2E and 2F. Including the interaction between the deterministic
clockwork of the continuous system and demographic noise is fundamental in obtaining
realistic dynamics (with periodicity of 2–4 years), as it is for other
infectious diseases; e.g., see [47],[48] and references therein.
In our full hybrid model, we observe that even small levels of environmental
transmission (a few cases per year) facilitate AIV persistence. Environmental
transmission rates are –on average– hundreds of times smaller
than direct transmission rates, yet they appear critical in sustaining the virus.
The ability of the pathogen to survive in the environment for a long time before
infecting susceptible hosts may thus have profound epidemiological consequences.
The relative influence of environmental transmission for epidemic persistence depends
on the population size. If the population is substantially larger than the critical
community size, then the number of infecteds does not go to zero in between
recurrent epidemics [4],[43],[49] and direct transmission dominates the course of
the epidemic. If, however, the population is small and the number of infecteds goes
to zero, then environmental transmission is a key factor in sustaining the epidemic.
Thus, environmental transmission provides an epidemic persistence mechanism within
populations smaller than the critical community size.
Our results hold for low pathogenicity AIV. The extension to high pathogenicity AIVs,
as evidenced by outbreaks in tufted ducks and pochards [50], awaits additional
empirical information. Another limitation of our model is that we have restricted
our consideration to a single immunological subtype that confers life-long immunity.
We note that partial cross-immunity in a multi-serotype model would enhance the
effective number of susceptibles and, therefore, should be expected to promote
persistence. In reference [51], we address the conditions under which
environmentally and directly transmitted pathogens may coexist.
The actual mechanism of persistence of avian influenza in wild waterfowl may be
complex, including a number of other factors such as spatial and age structures,
waning immunity and strain polymorphism leading to immune escape. Several studies
address the role of spatial heterogeneity in a general framework. For example, Lloyd
and May [52] show in a metapopulation model that persistence of
epidemics (asynchrony of within-subpopulation dynamics) occurs only if the
immigration in between the subpopulations is small. A more recent and thorough
analysis by Hagernaas et al. [53] discussing both oscillatory and non-oscillatory
population dynamics arrives at the same conclusion. Further modeling work is needed
in order to evaluate the relative contribution of other possible persistence
mechanisms.
Further work is also needed to explore our modeling assumption that host populations
form (nearly) closed systems. Empirical evidence suggests that the interaction
between the Eurasian and American clades of migratory birds is so small (despite
overlap in their Alaskan migratory routes) that their exchange of full genome
influenza viruses has yet to be documented [54]. While this observation
supports our modeling assumption, the data on the smaller scale interaction between
flocks of migratory birds within the American continent is insufficient for
validation. Alternate modeling assumptions could be explored theoretically.
Using mathematical modeling, we have investigated the role of environmental
transmission for the pattern and persistence of avian influenza in wild waterfowl
and demonstrated that indeed environmental transmission is a fundamental ingredient
for the modeling of this epidemic. The persistence mechanism induced by enviromental
transmission raises novel problems of epidemic control since traditional strategies
may prove ineffective in the presence of an environmental viral reservoir [55]. Thus,
environmental transmission remains a topic of increasing interest in theoretical
epidemiology.
Supporting Information
Additional explanations of the parameters, wavelet analysis and further
simulations for uncertainty analyses.
(2.05 MB PDF)
We thank Benjamin Roche and three anonymous reviewers for insightful comments on this
manuscript.
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