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PRAMANA
— journal of
c Indian Academy of Sciences
°
physics
Vol. 70, No. 6
June 2008
pp. 1047–1053
Assessing the quality of stochastic oscillations
GUILLERMO ABRAMSON∗ and SEBASTIÁN RISAU-GUSMAN
Statistical and Interdisciplinary Physics Group, Centro Atómico Bariloche, CONICET
and Instituto Balseiro, R8402AGP Bariloche, Argentina
∗
Corresponding author. E-mail: [email protected]
Abstract. We analyze the relationship between the macroscopic and microscopic descriptions of two-state systems, in particular the regime in which the microscopic one
shows sustained ‘stochastic oscillations’ while the macroscopic tends to a fixed point. We
propose a quantification of the oscillatory appearance of the fluctuating populations, and
show that good stochastic oscillations are present if a parameter of the macroscopic model
is small, and that no microscopic model will show oscillations if that parameter is large.
The transition between these two regimes is smooth. In other words, given a macroscopic
deterministic model, one can know whether any microscopic stochastic model that has it
as a limit, will display good sustained stochastic oscillations.
Keywords. Population dynamics; stochastic oscillations.
PACS Nos 87.23.Cc; 02.50.Ey; 05.40.-a
1. Bridging the gap between the mean field and individual-based models
The subject of this contribution is a spin-off of an interdisciplinary research effort
on the epidemiology of Hantavirus in wild rodent populations. This project, devoted to providing a theoretical framework for ecological and epidemiological field
findings, has explained and predicted a number of phenomena in relation with this
widespread epizootic (see, e.g. [1–3]). Most of that work is based on the analysis
of mean field-like models, continuous models of the kind generally used for large
populations, chemical kinetics, etc. They constitute a macroscopic, population level
description of the system under study. While powerful and useful, they are unsatisfactory in many regards. For fundamentally discrete and not very large populations,
a complementary microscopic, individual level description is called for. Moreover,
the stochastic nature of the processes and interactions that take place at the individual level are responsible for a fluctuating dynamics that can only be captured
by a stochastic model. One expects that the continuous macroscopic models, in
certain sense and regimes in which one-individual changes are small, constitute a
sensible limit of the microscopic ones. The determination of those regimes becomes
a pre-eminent question [4,5]. In addition, there exists the possibility of some phenomena that, due to fluctuations, are exclusive to the microscopic models. The
1047
Guillermo Abramson and Sebastián Risau-Gusman
bA=bBA=0.1
m/Ω, n/Ω, φA, φΒ
0.15
dA=0.13
susceptible
infected
dB=0.49
p=3.02
0.10
0.05
0.00
0
200
400
600
t
Figure 1. Persistence of oscillations in a microscopic SI model. The fluctuating populations of susceptibles and infected in a single realization are shown,
together with the smooth lines corresponding to both the deterministic model
and the average of 10,000 stochastic realizations, which are indistinguishable.
stabilization of stochastic oscillations, which is the subject of this work, is one of
these.
This interesting phenomenon is displayed by certain stochastic microscopic models, whose macroscopic counterparts do not oscillate. The oscillations arise purely
from fluctuations, by the collective synchronization of individual transitions through
their interactions. They may explain the non-seasonal oscillations observed in many
epidemiological and ecological systems (such as in measles or chicken pox [6,7]). Microscopic fluctuations are responsible for other interesting phenomena in chemical
or chemical-like systems, such as the transition between a chaotic attractor and a
fixed point suppression of chaotic oscillations [8–10].
Figure 1 shows the persistence of oscillations exemplified in a susceptible-infected
(SI) model. The smooth analytic curves show just damped oscillations. So does an
average of 10,000 realizations of a microscopic model. A single realization, however,
clearly shows persistent oscillations. In this work we characterize the regime in
which the stochastic oscillations are clearly visible against the random fluctuations,
as exemplified in figure 2.
2. Formulation of the stochastic model
The formulation of a stochastic model proceeds by identifying the ‘microscopic’
processes first, and writing down a master equation next, i.e. an evolution equation
for the joint probability distribution of the system.
Let us consider a population, composed of m individuals of one kind, and n of
another kind (for example, susceptible and infected animals in an epidemic system), that dwell in a region of volume (area) Ω. (The generalization to systems
with more than two species is in principle possible, while much of the analysis still
needs to be carried out in detail.) The state of the system is defined by the joint
probability distribution P (m, n, t). The animals are born, reproduce, and die according to certain rules, which are a simplification of the corresponding processes
in a natural population. They are also converted from one class into the other,
such as in the contagion of a susceptible individual after contact with an infected
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Pramana – J. Phys., Vol. 70, No. 6, June 2008
Assessing the quality of stochastic oscillations
0.095
0.02
10
8
6
)
f4
(
S
2
0
0.00
Q = 0.79
0.03
n/Ω
Q = 6.6
4
0.06
0.09
3
)
f( 2
S1
0
0.00
0.090
0.03
0.06
0.09
f
n/Ω
f
0.085
0.01
1500
1600
1700
t
1800
1900
2000
500
600
700
t
800
900
1000
Figure 2. Infected population of a microscopic SI model, showing two
regimes in which the power spectrum has a peak, but with very different
oscillating characteristics. Insets: Power spectra of the time series.
one. These processes correspond to transitions in the configuration space of the
system, and we suppose that they occur stochastically, according to certain transition probabilities W (m0 , n0 |m, n) (the rate at which the system goes from state
(m, n) to state (m0 , n0 )).
The master equation is the gain–lose equation that describes the evolution of P :
X
dP (m, n)
=
W (m, n|m0 , n0 )P (m0 , n0 )
dt
m0 ,n0
X
−
W (m0 , n0 |m, n)P (m, n),
(1)
m0 ,n0
where the sums are taken over all the states that have allowed transitions to, or
from, the state (m, n). A common situation is that of one-step processes, for which
we write the transition probabilities as Wij , with i, j ∈ {−1, 0, 1}.
3. Expansion of the master equation
Even though it is always a linear equation, the master equation can rarely be
solved exactly, and either an approximate method, or a numerical scheme, or a
combination of both, needs to be used. As shown by McKane and Newman [11],
the systematic expansion of the master equation devised by van Kampen [12] is
particularly adequate in this case. The expansion relies on the following ansatz,
that assumes the existence of a size parameter Ω, and decomposes each stochastic
variable into a macroscopic, analytical one, plus a noise of smaller amplitude:
√
(2)
m(t) = Ω φ1 (t) + Ω ξ1 (t),
and correspondingly for n(t). The use of the ansatz (2) into eq. (1) leads to an
expansion in powers of Ω−1/2 . Equating successive orders of the series provides
Pramana – J. Phys., Vol. 70, No. 6, June 2008
1049
Guillermo Abramson and Sebastián Risau-Gusman
corresponding orders of dynamical description: the leading order consists of equations for φ1(2) only, that one identifies as the macroscopic (mean field) law; the
following order consists of a Fokker–Planck equation for the probability distribution function of the fluctuations ξ1(2) , which is the main component of the dynamics
of the fluctuations; higher orders provide additional levels of detail in the fluctuating
dynamics.
√
The leading order in the expansion is that of Ω. This needs to be made null, for
the expansion to behave regularly when Ω → ∞. One arrives, at this order, at the
following set of two ordinary differential equations for the macroscopic variables:
X
dφ1
=
iWij (φ1 , φ2 ) ≡ C1 (φ1 , φ2 ),
dt
ij
(3a)
X
dφ2
=
jWij (φ1 , φ2 ) ≡ C2 (φ1 , φ2 ).
dt
ij
(3b)
These are the equations of the mean field, macroscopic level of description. They
give the dynamics of the smooth population variables that can be observed, for
example, averaging over realizations in a numerical simulation of the system (see
figure 1). Observe, also, that the summation that appears in eqs (3) implies that
many microscopic systems (corresponding to different choices of the transition probabilities) map onto the same macroscopic model.
A linear stability analysis shows that a stable fixed point exists in eqs (3) if
½
T
< 2, stable spiral,
²= √
(4)
> 2, stable node,
∆
where T and ∆ are, respectively, the trace and the determinant of the matrix
Cij = ∂Ci /∂φj . The case ² = 2 corresponds to a center which, being structurally
unstable, is of lesser interest in applications to real problems. The case ² < 2 is of
particular interest, since it corresponds to damped oscillations of the macroscopic
model.
The following order in the expansion, found by equating the terms of order Ω0 ,
represents the main behavior of the deviations from the macroscopic variables. It
takes the form of a linear Fokker–Planck equation for the probability distribution
function of the fluctuations ξ1 and ξ2 . It has the form
X
∂Π(ξ1 , ξ2 , t) X
∂
∂2
=
aij
ξj Π(ξ1 , ξ2 , t) +
bij
Π(ξ1 , ξ2 , t),
∂t
∂ξi
∂ξi ∂ξj
ij
ij
(5)
with coefficients aij and bij that depend on the transition probabilities and, through
φ1 (t) and φ2 (t), on time. Further orders in the expansion modify this Gaussian
nature of the fluctuations, but one need not consider them in general. The next
order, for example, is of order Ω−1 relative to the macroscopic values, and so of the
order of a single animal.
For the purpose of analyzing the oscillations in the fluctuating solutions of the
master equation, it is more convenient to pose eq. (5) in the form of an equivalent
Langevin equation:
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Pramana – J. Phys., Vol. 70, No. 6, June 2008
Assessing the quality of stochastic oscillations
QA, QB
2
0.6
0.4
0.2
0.0
0
2
1
2
^ ,ω
^
ω
B
Α
0.8
-1
10
-2
πε
10
-3
10
-4
0.0
0.2
0.4
ε
0.6
0.8
1.0
f(ε)
0.1
susceptible
infected
1
10
δQ
susceptible
infected
1.0
2
ε
3
4
0
1
ε
2
3
4
Figure 3. Bounds in the frequency (left) and in the quality (right) of stochastic oscillations. Many microscopic systems are shown in each plot, with
circles and crosses corresponding to each population.
ξ˙1 = C11 ξ1 + C12 ξ2 + η1 (t),
ξ˙2 = C21 ξ1 + C22 ξ2 + η2 (t),
(6a)
(6b)
where the noises η1(2) are Gaussian (since eq. (5) is linear): hηi (t)ηj (t0 )i = Dij δ(t −
P
P
P
t0 ), with D11 = ij i2 Wij , D12 = ij ijWij , and D22 = ij j 2 Wij .
Equations (6) can be Fourier-transformed and an analytic expression for the
power spectra can be found. For the first population, for example, it reads
·
¸
F1 + ω̂ 2
D11
S1 (ω) =
,
(7)
(1 − ω̂ 2 )2 + ω̂ 2 ²2 ∆
2
D22 + C22 D11 −
where the normalized frequency ω̂ 2 = ω 2 /∆ and F1 = (C12
2C12 C22 D12 )/(∆D11 ), are parameters defined by the system. It can be shown
easily that S1 (ω) either has a single maximum or is monotonically decreasing (in
which case it has a single maximum at ω = 0). The maximum lies at
p
ω̂12 = −F1 + (F1 + 1)2 − ²2 F1 ,
(8)
which can be naturally associated with the ‘frequency’ of the stochastic oscillation.
There are corresponding expressions for the second population, and one sees that
the two frequencies do not coincide in general. Indeed, it is possible to show that,
for any given microscopic model, both frequencies are bounded by
2
1 − ²2 /2 < ω̂1(2)
< 1.
(9)
Figure 3 (left) shows these bounds as a function of ² as continuous lines. Between
them lie a multitude of points, which correspond to many microscopic models.
The frequency of the damped oscillations of the single deterministic model that
constitute their macroscopic limit is shown as a dashed line. It is seen that, in the
microscopic description, the two populations oscillate with different frequencies,
which collapse to a single one as ² → 0.
Pramana – J. Phys., Vol. 70, No. 6, June 2008
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Guillermo Abramson and Sebastián Risau-Gusman
Now, the following legitimate question arises: does a peak in the power spectrum
identifies, in any sensible way, a ‘good stochastic oscillation’ ? Figure 2 shows
that, naturally, this is not the case. Both panels show the fluctuating infected
population of a simulated SI model. The insets show the power spectra calculated
from them, and the form of the peaks confirm what could be said directly from the
field observations: that the one on the left ‘oscillates’, while the one on the right
has apparently random fluctuations. Both spectra, however, have a peak at a finite
frequency. The difference between them lies on the form of this peak, and to this
feature we turn now our attention.
We can show that the following definition of the quality of the oscillation, based on
a property of the power spectrum, enables a good characterization of the oscillating
regime:
Q1(2) =
ω1(2) hS1(2) (ω1(2) )i
R
,
hS1(2) (ω)idω
(10)
and a corresponding expression for population 2. This definition of the quality is
arbitrary, in the sense that many other definitions of the same concept are possible
based on the form of the power spectrum. It is justified a posteriori because very
good bounds can be derived for it, showing its general behavior for a whole range
of microscopic models. It can be shown that both qualities (10) obey
f (²) < Q1(2) <
2
,
π
(11)
where f (²) has a cumbersome analytical expression that can nevertheless be found
without difficulty from Q1(2) considered as a function of F1(2) for fixed ². These
bounds are shown in figure 3, together with the qualities calculated for many microscopic models which tend to the same deterministic model in the macroscopic limit.
As we see, the existence of the lower bound tells that the quality of the stochastic
oscillation can be large only when ² is small. And, because of the existence of an
upper bound that so tightly follows the lower one, when it is large, it is so for all
microscopic models.
In summary, the definition of an adequate measure of the quality of stochastic
oscillations, and the analytical bounds found for it, show that the parameters that
define a deterministic model are enough to characterize how oscillatory the time
series of any of its stochastic counterparts looks like. We have found that good
stochastic oscillations are clearly present only when ² is small (² . 0.5, from our
survey of known cases). Our bounds provide a quantitative and accurate measure
of this effect. For increasing values of ², there is a smooth transition to a behavior
that is indistinguishable from noise.
Acknowledgements
This work has been partially supported by ANPCyT (PICT-R 02-87/2 and 04-943)
and from CONICET (PIP 5114). GA presented the talk at the PNLD conference
and his participation in the PNLD conference was possible, thanks to the hospitality
of the Abdus Salam ICTP.
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Pramana – J. Phys., Vol. 70, No. 6, June 2008
Assessing the quality of stochastic oscillations
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