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OPEN 3 ACCESS Freely available online
A Simple Stochastic Model with Environmental
Transmission Explains Multi-Year Periodicity in
Outbreaks o f Avian Flu
Rong-Hua W ang1'2, Zhen Jin1, Quan-Xing Liu2, Johan van de Koppei2, David Alonso3'4*
1 Department of Mathematics, North University of China, Taiyuan, Shan'xi, People's Republic of China, 2 Spatial Ecology Department, the Netherlands Institute of Ecology,
Yerseke, The Netherlands, 3 Community and Conservation Ecology Group, University of Groningen, Groningen, The Netherlands, 4Center for Advanced Studies, Spanish
Council for Scientific Research, Blanes, Catalunya, Spain
Abstract
Avian influenza virus reveals persistent and recurrent outbreaks in North American w ild w aterfow l, and exhibits major
outbreaks at 2 -8 years intervals in duck populations. The standard susceptible-infected- recovered (SIR) fram ework, which
includes seasonal m igration and reproduction, but lacks environm ental transmission, is unable to reproduce the m ultiperiodic patterns o f avian influenza epidemics. In this paper, we argue tha t a fully stochastic theory based on environm ental
transmission provides a simple, plausible explanation fo r the phenom enon o f m ulti-year periodic outbreaks o f avian flu. Our
theory predicts com plex fluctuations w ith a dom in ant period o f 2 to 8 years w hich essentially depends on the intensity o f
environm ental transmission. A wavelet analysis o f the observed data supports this prediction. Furthermore, using master
equations and van Kampen system-size expansion techniques, we provide an analytical expression fo r the spectrum o f
stochastic fluctuations, revealing how the outbreak period varies w ith the environm ental transmission.
C i t a t i o n : Wang R-H, Jin Z, Liu Q-X, van de Koppel J, Alonso D (2012) A Simple Stochastic Model with Environmental Transmission Explains Multi-Year Periodicity
in Outbreaks of Avian Flu. PLoS ONE 7(2): e28873. doi:10.1371/journal.pone.0028873
E d i t o r : Carmen Molina-Paris, Leeds University, United Kingdom
R e c e i v e d April 12, 2011; A c c e p t e d November 16, 2011; P u b l i s h e d February 17, 2012
C o p y r i g h t : © 2012 Wang et al. This is an open-access article distributed under the terms of the Creative Commons AttributionLicense,
whichpermits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
F u n d i n g : Support was provided by The Netherlands Institution for Scientific Research (NWO), The Netherlands Institute of Ecology (NIOO-KNAW), and National
Natural Science Foundation of China (Grant No. 10901145). The funders had no role in study design, data collection and analysis, decision to publish, or
preparation of the manuscript.
C o m p e t i n g I n t e r e s t s : The authors have declared that no competing interests exist.
* E-mail: dalonso@ceab.csic.es
field stochastic models. However, when one considers finite
populations, stochastic interactions even within a well-mixed
system introduce new phenom ena. For example, disease persis­
tence is determ ined by chance events when the num ber of
individuals carrying the disease is small, during the early phases of
disease invasion, or when total susceptible population size is
reduced due to vaccination a n d /o r immunity. In this case, even if
invasion is predicted to be successful in deterministic models, i.e.,
the basic reproductive num ber (1Zq) is larger than one, it may
totally fail in the corresponding stochastic system, which means
that observing a failed invasion in nature does not necessarily
imply a population below the deterministic invasion threshold. In
general, stochastic effects are quite prom inent in finite populations,
and rem ain im portant both in ecological [16-18] and epidemic
dynamics [19-21]. Usually, individual-based a n d /o r integer-based
event-driven simulations [22] are conducted. However, simula­
tions are inferior in several respects to careful m athematical
analysis. For instance, a single simulation may not be represen­
tative of system average behavior but merely produce an outlier
due to a rare combination of events [23]. Usually a huge ensemble
o f replicates are needed to obtain a good representation of the
average behavior of the system. In fact, it is generally accepted that
deeper insights are obtained from the m athematical analysis of
stochastic systems.
Recently, a method, the so-called van K am pen’s system-size
expansion, which is based on a simple individual-based m athe­
matical formulation of stochastic dynamics, has been applied to
Introduction
U nderstanding the dynamics of infectious diseases in humans
has become a increasing focus in public health science [1-3].
Despite a massive body of research on the epidemiology of
seasonal influenza, overall patterns of outbreak and infection have
not been fully understood, in particularly with regard to its multi­
year periodicity. Disease outbreak, persistence, fadeout and
transmission am ong species rem ain difficult to assess, because
they not only depend on a huge variety of biological factors, e.g.
virulence, immunity [4], but also on some abiotic processes, such
as the characteristics of natural environments [5,6], transport and
immigration [7,8], In spite of all these inherent complexities,
simple mathem atical models can provide some very useful
information for many infectious diseases including measles,
mumps and rubella. From early deterministic com partm ental
models to more recent spatially structured stochastic simulation
models [9,10], dynamic models have im pacted both our
understanding of epidemic spread and public health planning.
M ulti-year periodicity in epidemics is widely observed in time
series from many cities with greatly varying climatic and
demographic conditions [5,11-13]. As reported previously
[14,15], multi-year periodicity and irregular fluctuations were
related to both seasonal forcing and entrainm ent in nonlinear
oscillatory and chaotic models. Deterministic models are typically
assumed to be reasonable approximations for infinitely large,
homogeneous populations, and arise from the analysis of mean
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M ulti-year P eriodicity in Avian Flu
investigate stochastic population dynamics [21,24-28]. This
general m athematical framework provides an exact description
of individual-based (integer-based) event-driven stochastic dynam ­
ics [22]. M ore recendy, these methods have been applied to
epidemiology, which has helped to understand the effects of
stochastic amplification [21,29] and seasonal forcing [30-32] on
disease outbreaks. However, most of these studies are based on
single species models, and mainly considered demographic
stochasticity and seasonality. Roche et al [33], however, have
shown that epidemic outbreaks and migrations are not synchro­
nous, which points to the fact, that, in wild birds, virus persistence
in the water should play a major role in the epidemiological cycles.
This approach to characterize disease fluctuations provides a
unique opportunity to investigate the effects o f stochasticity
imposed by finite population numbers on disease persistence and
outbreaks both in single- and multi-species systems.
Here, we estimate the outbreaks period of avian influenza in
N orth Americas with a wavelet method, which reveals 4 -8 year
periodicity from empirical data. To explain this, we first develop a
fully stochastic two species’ avian influenza model (host and virus)
with two routes of transmission: environmental indirect transmis­
sion and direct transmission through contact between individuals
within the wild bird population. Then, we provide a prediction for
the dom inant period of disease oscillations by analytically
calculating the power spectral density from a stochastic FokkerPlanck equation. From a geographically (environmentally) and
temporally restricted data, the model gives general insights into
long-term patterns of disease dynamics in wild bird populations.
Some conclusions may also apply to other infectious diseases
characterized by two transmission routes. O u r analysis sheds new
light on the im portance o f environmental transmission for avian
influenza outbreaks and persistence. O ur results show that, in
principle, it is possible to reduce the frequency and intensity o f the
outbreaks of avian influenza by controlling the environmental
route of transmission.
pathogen model that assumes global mixing, i.e, random contact
between individuals. H ence the epidemiological dynamics of the
host falls within the susceptible-infected-recovered (SIR) frame­
work [37], in which each individual is either susceptible (S),
infectious (I) or recovered (R), b u t infection occurs through two
different routes: direct contact between susceptible (S) and infected
(ƒ) individuals, and external infection from the environm ent (as
found the case in AIV in ducks or other waterfowl). For the
former, the rate of infection can be expressed by ß S I / N , where N
is the size of the host population and ß is the transmission rate. For
the latter, according to the Poison distribution, the transmission
rate per susceptible individual should be given by pa( l — e ~ aVt)
(see section A3 of Methods S I, and also the discussion in Ref. [34]
for details), which is an increasing function of the virus num ber V
per unit volume. In fact, it can be simplified into its leading-order
term using a Taylor expansion, which leads to p V / N y (here
p = p a logg(2), see Eq. (A29) in section A3 of Methods SI), where
N y is a typical reference virion concentration in the water (see
section A3, Methods SI), and p is the environmental transmission
rate (see Methods S I, for details). For most host-parasite systems,
environmental transmission is often represented as a frequencydependent process, which means that the transmission rate
depends on the frequency o f infected vectors in the environm ent
rather than on its absolute num ber or concentration as is the case
for density-dependent transmission [38]. Similar transmission rates
have been considered in m alaria [39], dengue fever [35], West
Nile Virus [36], and avian influenza [34],
Although births and deaths are intrinsically distinct events, we
assume, for simplicity, that host birth and death rates have the
same value p, which means that the total population size N is kept
constant. In sum, the dynamics of the disease in the host
population can be expressed by the following elemental events:
y
In fection :
S I -w I I and S -> I ,
Methods
The stochastic SIR model w ith environmental
transmission
In avian influenza, susceptible hosts are not only infected by
direct contact with infected individuals with avian influenza viruses
(AIV), but also by virus particles that persist in the aquatic
environment. AIV are transm itted via the fecaloid route of the
host and subsequent drinking or filtering of water while feeding
[3-36]. As a consequence, epidemic outbreaks are not necessarily
induced by the arrival of infected hosts in the population, b u t can
also result from virus particles that persist in the environment. The
persistence and subsequent outbreak of viral particles in the
aquatic environm ent is determ ined by several deterministic causes
[34], However, stochasticity should also play an im portant role
[21,34] because the processes controling the densities of viral
particles, such as ingestion and shedding by hosts, and virus decay
in the environment, are essentially probability processes. Accord­
ingly, we consider the density of virus particles in the environm ent
as a separate stochastic variable, which we couple to the dynamics
of host infection through the environm ental transmission rate. We
provide a detailed description about the way in which, as well as
the assumptions under which this is done in section A3 o f M ethods
SI.
To better understand the effects of demographic stochasticity
and virus persistence in the environm ent on epidemic outbreaks
and extinction, we describe virus population dynamics and
environmental transmission using an explicit stochastic host-
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I
D e a th /B irth :
S and
S,
y
R ecovery : / -> R.
where y is the recovery rate. Since host population (N = S + I + R)
is kept constant, after any individuals dies, at the same time, a new
susceptible host will be born in order to keep the total population
N constant. Therefore, since N can be ju st seen as a model
param eter, we eliminate the variable recovered individual R by
using R = N — S —I from our equations.
O ur individual-based stochastic model fully integrates the
abundance o f virus particles in the environm ent into the SIR
framework. Virus particles V are shed by infected ducks (shedding
rate is t ) , then virion concentration decays in the environm ent at
rate r¡. To keep the model general and applicable to other types of
pathogens, and to consider, effectively, the possibility of replication
o f the virus in alternative hosts whose concentrations are not
explicitly modeled, we introduce a production rate, 5. In this
context, this param eter takes into account the ability of the virus to
replicate outside of the specific host that is consider in the model. It
is im portant to rem ark that in the limit of S vanishing small, all our
conclusions still hold (see section A3 of Methods SI). Therefore, its
dynamics can be captured by:
2
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M ulti-year P eriodicity in Avian Flu
£ 0.9
Sto ch a st ic mod el
Dete rm ini st ic mod el
£ 0-8
v y
j y
\T \^ J X r x A
5 0.5
(a)
F ig u re 1. S to c h a s tic SIR-V m o d e l, (a) S chem atic diagram o f th e baseline SIR host-parasite m odel with direct and environm ental transm ission. The
sym bol S rep re sen ts th e susceptibles, I an d R rep re sen t th e infected and recovered individuals, respectively, and V is th e virus concentratio n in th e
env iro n m en t. For host, th ere is equal birth rate and d e a th rate /i. (b) A realization of a stochastic SIR-V m odel and its determ inistic co u n terp art. The
p aram eters used in th e sim ulations are N — IO3, N y — IO5, /? = 0.05, p — 0.4, y = 5.5, /i = 0.3, ¿ = 0.1, x — IO4 and r¡ — 3. Disease param eters co rresp o n d
to avian influenza ep idem ics derived for typical w ater-borne transm ission from [33] and [2], Detailed descriptions of m odel p aram eters and sources
for th eir num erical values are p resen ted in tab:2. The determ inistic curve w as g e n e ra te d by integrating th e m ean field eq u atio n s (5), and stochastic
sim ulation w as im p lem ented with Gillespie algorithm [22] w ith rates listed in Table 1.
doi:10.1371 /journal, pone.0028873.g001
B irth :
D e a th :
v d+4 r v+\,
V
»r
T ( s + l , i — l,v |i,/,v ) = pi,
T (s ,i,v + 11s,i,v) = xi + ôv,
V — I.
All the transitions o f the host and the virus associated with their
corresponding rates are illustrated graphically in Fig. 1(a).
T he basic ingredients of our new framework are the susceptible
S, infected I and the virus V whose actual numbers are
respectively denoted as s,i and v, which are all of them integer.
T he general state of the system is then denoted as c = (s,i,v). All of
the processes taking place in this model and their corresponding
rates are summarized in Table 1.
T he transition probability per unit time from state ff to the state
a' will be denoted as T(o'\o), in which a' is obtained by shifting
each state variable in a by +1 or —1. According to the information
of Table 1, the events occurring in the system can be divided into
three groups:
3.
(2 )
T ( s , i — l,v |i,/,v ) = y/.
(3)
Recovery
Table 1. List o f events associated w ith transition rates.
P r o b a b i l i t y in
1. Infection
T ( s — 1 ,/+ l,vU,/,v) = ß — i + p s
.
t
' at
F Nr
T (s ,i,v — 11s,i,v) = f/v.
( 1)
Event
T ran sitio n
Direct infection
s->s— \,i^>i+ 1
Environment infection3
s->s—
Death of recovered
s->s+ 1
Death of infectedb
2. D eath/B irth
Rate
\t,t+dt\
psW -
S i• dti
ßn—
N
v
PS-rr- dt
Ny
fi(N —s —i)
fi(N —s —i) dt
+1
¡Ü
fii dt
Recovery
i->i— 1
yi
y i dt
Birth of virus
V—>v—
|—1
ÔV+ Ti
(íSv+
Death of virus
V—>V—1
rji
tji dt
tz)
dt
aNote that here we consider it as a frequency dependent,
h'here is no empty site, and the population size N is constant, thus a new
susceptible individual will be born once an infective individual dies.
doi:10.1371 /journal.pone.0028873.t001
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M ulti-year P eriodicity in Avian Flu
where k =
H aving defined the transition rates between different states by Eq.
(1)—(3), now we can construct a master equation describing the
tem poral evolution o f the system. It takes the general form
[16,17,21,25-28]
= £
a'
7 V |tr ') P ( t/; i) - £
a'
7 V |rr)P (rr; t),
lim
jV,AV->go
■£-.
v
It is simple to verify that these equations have a trivial fixed point
E°:
^ = 1,
(4)
02=0,
t/f° = 0;
and a unique non-trivial fixed point E *:
where ft =(s,i,v) represents the state o f the system, P(o,t) is the
probability of the system in the state ft at time t, and the change of
this quantity with time is given by a balance between the sum of
transitions into the state ft from all the other states ft', and minus
the sum of transitions out o f the state ft into all the other states ft'.
So far we have formulated a fully stochastic host-parasite model
with both direct and indirect environmental transmission,
assuming well-mixed conditions. Given the specified analytical
formulations for transition probabilities P (ft'|ft), the master
equation (4) accurately describes the tem poral evolution of the
probability P (ft,i). This model can now be investigated by a
combination of simulation, by using Gillespie algorithm [22], and,
analytically, by perform ing the van K am pen’s system-size
expansion [21,25] of the master equation. Both methods allow
quantitative prediction o f the power spectrum o f the time
fluctuations of each of the system variables, and, therefore, of
the dom inant period of recurrent epidemic outbreaks.
Using van K am pen’s system-size expansion o f the stochastic
dynamics, as discussed in section A Í of Methods S I, we can derive
the deterministic equations. T he stability of the steady states of this
system is tractable, and can be obtained by deriving the
deterministic limit (see subsection A l.l o f Methods SI). T he nextto-leading order gives the linear stochastic differential equationsFokker-Planck equation, which can be analyzed using the Fourier
method. Now we start by introducing the new variables:
s=
where TZo = —
ß + y
F -,
rr?
r is the basic reproductive
( t / - ö ) ( ^ + y) _
num ber. From the stability’s analysis in section A2 of M ethods
S I, we know when TZo < 1, the trivial fixed point E° is stable; when
TZo > 1, the non-trivial fixed point E* exists and is stable.
The periodicity o f the stochastic model
It is im portant to investigate w hether the existence of a stable
fixed point in the deterministic system generates oscillations and
multi-year periodicity in the corresponding stochastic system. In
order to investigate this and describe the stochastic fluctuations of
the system by an analytical method, the higher-order terms should
be included in the van K am pen system-size expansion [25]. As
shown in the section A Í of Methods S I, the fluctuations obey a
linear Fokker-Planck equation, which is equivalent to a set of
Langevin equations having the form
dx
3
- j - = ' ^ 2 A ki x i + £ k { t ) ,
at
l=l
( k , l = 1,2,3),
(6)
where £¿(í) ( k = 1,2,3) are Gaussian white noises with zero mean.
In the same way, the cross-correlation structure is determ ined by
the expansion, which satisfies < (£ i(i)£ /(0 ) = B k iô (t— tr). As
m entioned above, we are interested in evaluating these fluctua­
tions at the non-trivial fixed point of the deterministic system. For
that reason, we evaluated the entries of the Jacobian m atrix Aki
and B k i of the noise covariance m atrix at this stable fixed point.
Explicit expressions for these two matrices are given in the
supporting information in subsection A 1.2 of Methods SI.
T he Langevin equations (6) describe the tem poral evolution of
the norm alized fluctuations o f variables around the equilibrium
state. By Fourier transformation o f these equations, we are able to
analytically calculate the power spectral densities (PSD) that
correspond to the norm alized fluctuations, independent of
comm unity size N . By taking the Fourier transform of Eqs. (6),
we transform them into a linear system o f algebraic equations,
which can be solved, after taking averages, into the three expected
power spectra of the fluctuations o f the susceptible, infectious and
viral densities around the deterministic stationary values:
N $ y + \ / Ñ x \,
i = N(/>2 + \/NX2,
V= Nylj/ + \ J NyXo,
where <(q, (/>2, t/r are the fractions o f the susceptible hosts, the
infected hosts and viruses in the environment, respectively, with x¡
(I = 1,2,3) describing the stochastic corrections to the variables s, i
and V . Full technical details of model analysis are given in the
section A Í of M ethods S I. To leading order, the deterministic
equations for the fractions are
=
= ( t l - S ) ( n + y)
^1 = 7
+
n < \
/i~ ¡ \|2 \
a s + -ßiim 4 + T iü j2
P s (üj) = < |x i(ü j)| > = --------,
\D(w)\
n !
\
/\~
!
u2\
O C 1 + B 2 2 0 / + T 2O J2
B¡(w) = <1*2(ra)I > = -------
—— = ô\j/ + Knj>2 — t]\ß,
2
,
(7)
|X>(ffl)|
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M ulti-year P eriodicity in Avian Flu
V ! \
/\~ ! \ l 2\
P F(w )= < |x 3(w)| > = -
Oly+BiiW + T 3W
T he complete derivation of these PSDs and detailed descriptions
about the way the functions ccj, B 22, T 2, and T>(co) depend on
model param eters are discussed in section A 1.3 of Methods SI.
Wavelet power spectrum
Unlike Fourier analysis, wavelet analysis is well suited for the
study of signals whose spectra change with time. This time—
frequency analysis provides information on the different frequen­
cies (i.e. the periodic components) as time progresses [40,41], The
wavelet power spectrum estimates the distribution o f variance
between frequency, cu, and different times, 1.
If we denote the time-series as x(t), then the wavelet transform
of a signal x (t) is defined as:
W x (co,r)=
In the definition, param eters cu and t denote the dilation
(periodicity) and translation (time shift position). <F*(i) denotes the
wavelet functions. T here are three wavelet basis functions (Morlet,
Paul and D OG ) commonly used in the wavelet analysis. T he M orlet
wavelet is the one used in our analysis. Cazelles et al [40] presents a
detailed description o f the wavelet power spectrum m ethod and a
summary of its applications to disease and ecological data.
Results
Prevalence o f influenza A viruses in w ild ducks over tim e
Previous studies over 15 years from 1976 to 1990 established a
cyclic pattern of occurrence of influenza A viruses in wild ducks
[42], with high prevalence in some years followed by reduced
prevalence in subsequent years. Avian influenza data o f a yearly
time series is described in Ref. [13], for wild aquatic birds from 1976
to 2001 in N orth America. Those records contain samples collected
on wild ducks and shorebirds. To determine w hether these patterns
show multi-year periodicity, we examined avian influenza preva­
lence over the period from 1976 to 2001, as is shown in Fig. 2.
T he data on aquatic wild birds revealed a clear periodicity in
the outbreaks of avian influenza in agreem ent with literature [13].
These periodic patterns are confirmed from the case records
through wavelet analysis (see Fig. 2(b)), as well as through its
wavelet power spectrum analysis versus the frequency with the
largest long-term detectable power (Fig. 2(c) and (d)). Wavelet
analysis performs a time-scale decomposition o f a time signal,
which involves the estimation of the spectral characteristics of the
signal as a function o f time. It reveals how the different periodic
components of the time series change over time. T he oscillations of
avian influenza A virus in ducks species have a considerable
variation as periodicity during these years. However, the wavelet
analysis based on these data reveals significant multi-annual cycles
from 2 to 8 years. By using our model predictions with reasonable
param eter values (presented in Table 2), we can estimate the
environmental transmission rate, p, that yields fluctuating periods
ranging from 2 to 8 years (see curves in Fig. 3(c and d) for different
values of p). For instance, it should be lower than 0.42 y ear-1 for a
reproductive num ber equal to 2.4, y = 5.5, and the rest of
param eters chosen according to Table 2.
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Effect o f stochasticity and environm ental transmission on
disease outbreaks
D irect comparison of the deterministic and stochastic simula­
tions reveals that demographic noise and environmental trans­
mission can induce rich multi-period patterns, corresponding to
deterministically dam ped oscillations (see Fig. 1(b)). O ur analysis
can help us to understand the effect of indirect transmission on the
type of expected fluctuations o f disease incidence. We com pared
the analytical predictions for the PSDs to simulated results in
Fig. 3(a), using biologically reasonable param eter values (see
Table 2). O ur results reveal very good agreem ent between
predictions and stochastic simulations.
The original PSD formula (7) further allows us to examine how the
period o f the epidemic outbreak varies with changes of the
environmental transmission rate p. W e show in Fig. 3(b) that, for
typical parameters of avian influenza, as listed in Table 2, increased
environmental transmission rate p can enhance the frequency of
disease outbreaks. W e can see from Fig. 3(c,d) that, within the
deterministic model, the effects of the basic reproductive num ber on
outbreak periodicity o f the disease are most pronounced when the
pathogen invasion is close to the critical value (fZo « 1). Furthermore,
the PSD surface becomes flatter as the basic reproductive number TZo
increases, indicating that more frequencies are involved in the
stochastic fluctuations, and that the overall variance of infected time
series is more evenly distributed among these frequencies. Simulta­
neously, as the basic reproductive number increases, the dominant
period decreases (the dominant frequency increases), as is elucidated
in Fig. 3(c an d). Finally, coherence disappears and the PSD becomes
totally flat at larger values o f the basic reproductive number, TZo. In
that regime, time fluctuations around average stationary values do
not show a dominant frequency and become white noise.
W e characterize the region of the param eter space that allows
for disease persistence both in the deterministic model (TZo > 1)
and, through simulations, in the corresponding stochastic system.
W e also m ap the dom inant period, which is calculated with the
inverse of the frequency at which the PSD peaks (the dom inant
frequency) in year units (see Fig. 3(d)). From Fig. 3(d), one can see
that the larger the basic reproductive num ber o f the deterministic
model is, the higher outbreak frequencies in the stochastic model
tend to be. This can also be seen by looking at the analytical
prediction of the PSD from Eq. (7) (see Fig. 3(c)). Furtherm ore, we
notice that disease stochastic extinction occurs even if the basic
reproductive num ber is slighdy above its critical deterministic
threshold. This is a comm on difference between deterministic
models and its finite-size stochastic counterparts which is usually
difficult to quantify. T hrough simulation, we have have approx­
im ated the boundary separating disease persistence from stochastic
extinction by the curve o f TZo « 1 .3 (see Fig. 3(d)).
These reported results are robust to changes in model
param eters within the ranges given in Table 1. For instance,
when we take S to zero, the param eter representing pathogen self­
m aintenance in the environment, very m inor changes are seen in
the predicted power spectra. For details on model sensitivity to
param eter changes, see section A4 of Methods SI.
Discussion
In this paper, we have developed a general, fully stochastic hostpathogen model with two routes of transmission: individual-toindividual and environmental transmission. O ur theory provides a
simple, plausible explanation for the phenom enon of multi-year
periodic outbreaks of avian flu. Even in the absence of external
seasonal forcing, our theory predicts complex fluctuations with a
dom inant period o f 2 to 8 years for reasonable param eters values,
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M ulti-year P eriodicity in Avian Flu
50
60
40
40
30
>
iii
1975
1980
1985
1990
1995
2000
1975
1980
1985
1990
Time (year)
1995
2000 0
20
10
(a) 1975
1980
1985
Year
1990
1995
2000
1000 2000
Power
F ig u re 2 . T e m p o ra l p e r io d ic ity a n a ly s is o f a v ia n in flu e n z a in N orth A m erica u s in g t h e w a v e le t m e th o d , (a) Yearly prevalence of
influenza A virus for wild ducks from 1976 to 2001 an d for shorebirds from 1985 to 2000, w here th e d a ta with green sq u are and red circle sym bols
rep re sen t wild d u ck and shorebird, respectively. Annual prevalence w as calculated as a p ercen tag e o f th e total n u m b er of sam ples teste d for a given
year th a t co n tain ed influenza A virus. We have redraw n this figure here with d a ta kindly provided by Dr. W ebster [13]. Panel (b) show s th e tim e series
w ith yearly prevalence of influenza A virus in wild ducks from 1976 to 2001. (c) The w avelet spectrum analysis co rre sp o n d s to tim e series of panel (b),
w h ere tim e runs along th e v-axis and th e co n to u rs limit areas o f pow er a t th e periods indicated in th e v-axis. High pow er values are colored in dark
red; yellow and g reen d e n o te interm ediate pow er; cyan and blue, low. Note th e bold co n tin u o u s black line is know n as th e eo n e of influence and
delim its th e region n o t influenced by e d g e effects. Only p attern s w ithin th ese lines are therefo re considered reliable. Finally, th e right panel (d)
co rresp o n d s to th e average w avelet spectrum (black line; see section: w avelet pow er spectrum ) w ith its significant threshold value of 5% (d o tted
line). W avelet softw are provided by C. T orrence and G. C om po, is available a t h ttp ://w w w .p ao s.co lo rad o .ed u /research /w av elets/.
doi:10.1371 /journal, pone.0028873.g002
which essentially depends on the intensity of environmental
transmission. Since our model does not consider the specificities of
bird migration or seasonal reproduction in any way, in fact, it
applies to any infectious disease with two routes of transmission,
such as cholera. This further justifies the analysis we have done
which assures that infectious agents in the environm ent can not
only persist in the environm ent but also reproduce.
Practically all infectious diseases exhibit fluctuations. Childhood
diseases [9,20], dengue fever [43], cholera [44,45], m alaria [39,46],
and avian influenza [34] are but a few examples where disease
incidence strongly fluctuates. Emerging largely from a deterministic
framework, the standard paradigm is that seasonal a n d /o r climatic
extrinsic forcing and intrinsic host-pathogen dynamics are both
required to understand the character of different types of disease
Table 2. T h e
oscillations from regular to rather erratic patterns [15]. However,
more recently, it has become clear that the interaction between the
deterministic dynamics and demographic stochasticity is funda­
mental to understand realistic patterns of disease [20] including
vaccine-induced regime shifts [21].
Breban et al [34] developed a host-pathogen model for avian
influenza combining within-season transmission dynamics with a
between-season com ponent that describes seasonal bird migration,
and pulse reproduction. In their model, virus dynamics in the
environm ent is modeled as a deterministic process. T heir model is
designed to apply specifically to avian flu. By contrast, our model
applies m ore generally, and considers a m uch simpler dynamics
(without either seasonal pulse reproduction or seasonal bird
migration). In spite of these simplifications, we are still able to
d e f in itio n s o f t h e p a r a m e te r s In th is m o d e l a n d th e i r v a lu e s fo r t h e sp e c ia l c a s e (AIV).
D efinition
V alu e /ran g e
N
host population size
IO3
duck
Ny
viral reference concentration
IO5
virion ml-1
ß
direct transmissibility
0 -0 .0 5
duck-1year-1
Pb
environmental transmissibility
0 ~ 0.425
year-1
[0,3] (years h
P
host birth and death rate
0.3
year-1
[55]
S
virus replication rate
0.1
year-1
[0,1] (years-1)
X
virus shedding rate
IO4
virion/duck/d ay
[56,57]
n
virus clearance rate
3
year-1
[58]
y
recovery rate
0 -5 2
year-1
[34]
Symbol
U n it
Source a
[33]
[2]
Param eter values are based on empirical studies in literature. Since no data are available for p and
their values influence the patterns of interest (see Fig. 3).
bSee Methods S1, section A3 for its biological significance.
doi:10.1371 /journal.pone.0028873.t002
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b,
we let them vary within a reasonable range. We have studied how
February 2012 | V olum e 7 | Issue 2 | e28873
M ulti-year P eriodicity in Avian Flu
Susceptible
Q
Infectious
Virus
0.4
4x10
0.3
3x10
0.2
2x10
10
t/3
1x10
1
(a)
Frequency (y-1)
3
0.08:
< 1years
p=0.35/ V
p=0.55
2
O.Ofr
3
P
0.04
v l - 5years
CL
0.02
5 -10 years
0
0
(b)
1
0.5
1.5
10000
CO
20000
30000
40000
(d)
F ig u re 3 . P o w e r S p ectra l D e n s ity (PSD ), (a) C om parisons b etw een th e theoretical prediction of th e PSD (Eq. (7)) and th e average PSD calculated
from full sto chastic sim ulation as th e o n e show n in Fig. 1 (b), for th e fluctuations of th e total n u m b er of th e susceptible, th e infected an d th e virus. The
black lines rep re sen t th e po w er spectra of tim e series o b tain ed from stochastic sim ulations, and red lines rep re sen t th e analytical prediction. The
p aram eters are listed in Table 2 and y = 5.5, w here 7?o is equal to 2.387, an d a m ain oscillatory period a b o u t 7 years, (b) C hanges in th e PSD as a
function of an increasing environm ental transm ission rate w ith y — 10.0. (c) Three-dim ensional rep resen tatio n o f th e PSD for th e variable I, (Eq. (7)),
for a co n tin u u m of values o f 7?. on th e y axis, with th e restriction 7?o>l. (d) D om inant period and persistence o f th e disease as a function of
p aram eters p and r. Flere w e divide th e dom ain of th e param eter space w here sthochastic fluctuations occur in th re e different regions characterized
by p eriods less th an 1, from 1 to 5, from 5 to 10 years, respectively. We also rep re sen t th e hyperbolic-shaped instability boundary, sep aratin g th e
d om ain o f disease persistence (7?o > 1) from th e region o f disease extinction (7?o < 1), w hich is determ in ed by th e basic reproductive nu m b er 7?o = 1
in th e determ inistic system (5). The sam e boun d ary can be calculated th ro u g h sim ulation for th e full stochastic m odel. It co rre sp o n d s approxim ately
to 7?o = 1.3. Sym bols ( ) rep re sen t 100-year-long sim ulations, w here th e tran sien t dynam ics have b een discarded (the first 50 years).
doi:10.1371/jo u rn al.pone.0028873.g003
O ur general framework can be seen essentially as a stochastic
SIR model with two types of disease transmission: individual-toindividual and environmental transmission, which takes into
account the fact that disease agents are released to the
environm ent by infected individuals and, once there, they follow
a simple dynamics of decay and self-maintenance. O f course, virus
particles cannot self-reproduce independently from the host. In
our application to avian influenza, this term would take into
account virus reproduction in other host species different from the
focal host. In our model, the influence of the “reproduction”
param eter on our predicted power spectrum is very small (see
Methods SI). In addition, our framework readily apply to the large
num ber o f infectious diseases where reproduction of the infectious
agents in the environm ent is not negligible, and the interplay
between these two routes o f transmission is known to be im portant
[47-53],
O ur work points to the fact that seasonal forcing, taking into
account pulse reproduction and seasonal bird migration, is not
essential to understand avian flu fluctuating patterns of disease
incidence. W e argue that this is basically a consequence of the
reproduce with similar accuracy realistic patterns of disease
fluctuations for avian flu. Although our explanation is simpler,
both models show that the interplay between the stochastic
com ponent of disease dynamics and environmental transmission is
essential to understand the erratic outbreak patterns o f avian
influenza, characterized by dom inant periods from 2 to 8 years
(see Fig.2c). H ere we confirm this previous conclusion [34], and
show that it does not critically depend on bird migration and pulse
reproduction. In addition, in this paper, we are able to predict
analytically how the whole spectrum of such fluctuations depends
on model param eters.
In particular, in order to derive the power spectrum, we have
applied van K am pen expansion [25] to the full stochastic model.
This m ethod allows to study the correct interaction between the
deterministic and the stochastic components of the system in a
formal way in the case of finite populations. W e have shown that
the predicted power spectrum is in excellent agreem ent with
model simulations for realistic param eter values. In particular, our
study reveals that higher values of environmental transmission
increase the frequency of epidemic outbreaks.
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M ulti-year P eriodicity in Avian Flu
inherent stochasticity of the system. This type of ‘endogenous’
stochastic resonance [28] has been also described in childhood
diseases [21]. This result does not m ean th at seasonal dynamics is
not im portant in realistic situations. M igration and seasonal
reproduction are the most reasonable minimal ingredients o f any
disease model with applications to migratory birds, and they surely
control other im portant processes in these systems.
T he extend to which the seasonal cycle controls disease
fluctuating patterns has been recently studied in a fully stochastic
framework (both SIR [31] and SEIR [32]) with applications to
childhood diseases. These powerful analytic methods apply also to
infectious diseases with two transmission routes, such as avian
influnza, and further work on this area should be done. However,
these prelim inary studies have already revealed that a complex
interaction of seasonal forcing and the inherently stochastic, non­
linear dynamics of the disease occurs only in very restricted areas
of the param eter space, in particular, close to bifurcation points
[31]. For the most p art of the param eter space, apart from a rather
thin seasonal peak, the predicted, non-forced power spectral
density (PSD) agrees reasonably well with the PSD averaged over
seasonally forced, stochastic model simulations [21,31,32].
Simple non-linear systems have the potential to predict the
complex spatio-temporal patterns observed in nature. T he role of
stochasticity and the way it interacts with nonlinearity are central
issues in our attem pt to understand such complex population
patterns. As new tools and approaches become available
[21,23,31,32,54], here we have argued that the interaction of
external forcing with nonlinearity should be addressed within a
fully stochastic framework. G oing back to avian influenza, we may
well be in a situation w here seasonal migration and reproduction
are rather punctual events that would probably lock the phase of
disease fluctuations w ithout strongly influencing the way the
overall spectral power is distributed am ong the different
frequencies at play, which is basically determ ined by the intrinsic
non-linear stochastic dynamics o f the system. This hypothesis
applies to other infectious diseases as well as to, quite generally,
fluctuating populations in ecological systems. It deserves, on itself,
further investigation.
Supporting Inform ation
Methods SI In the supporting information file, we
provide, essentially, detailed mathematical derivations
o f the different theoretical results presented in the main
text. Supporting information is divided in four sections. T he first
one is devoted to the link between the deterministic and stochastic
description of the system and the system-size expansion used to
calculate power spectral densities. T he second one analyzes the
dynamical stability of fixed points o f the deterministic system. The
third one justifies the functional form used to represent
environmental transmission, and finally, the last one includes a
sensitivity analysis of the m ain fluctuation periodicity with respect
to two model param eters.
(PDF)
Acknowledgments
W e thank Prof. R obert G. W ebster for providing an d perm itting us to use
the data o f yearly prevalence o f influenza A virus for wild ducks and
shorebirds. T he authors would like to thank the two reviewers for their
helpful com m ents and valuable suggestions.
Author Contributions
Conceived an d designed the experiments: R H W ZJ DA. Perform ed the
experiments: R H W Q X L ZJ DA. Analyzed the data: R H W Q X L DA.
W rote the paper: R H W ZJ D A Jv d K . C onducted num erical simulations
and analytical calculations: R H W Q X L ZJ DA.
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February 2012 | V olum e 7 | Issue 2 | e28873