The effect of loss of immunity on noise-induced
sustained oscillations in epidemics
J. Chaffee∗and R. Kuske†
January 21, 2011
Abstract
The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The
analysis captures the key elements supporting the nearly regular oscillations of the
infected and susceptible populations, namely, the interaction of the deterministic
and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic
amplitude equation describing the envelope of the oscillations yields two criteria
providing explicit parameter ranges where they can be observed. These conditions
are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent
without noise. In this context the criteria indicate how loss of immunity and other
factors can lead to a significant increase in the parameter range for prevalence of
the sustained oscillations, without any external driving forces. Comparison of the
power spectral densities of the full model and the approximation confirms that the
multiple scales analysis captures nonlinear features of the oscillations.
Keywords: SIRS, sustained oscillations, coherence resonance, multiple scales, stochastic averaging
∗
previously Department of Mathematics, University of British Columbia, presently Applied Physics
Laboratory, Johns Hopkins University
†
Department of Mathematics,University of British Columbia, rachel@math.ubc.ca
1
1
Introduction
Understanding sustained oscillations in the levels of infected populations has attracted increased attention in recent years. A number of different mechanisms
can support these sustained oscillations in the infected and susceptible populations, ranging from population diffusion [1] to nonlinear interactions [2] to spatial
heterogeneity [3]. The potential for resonance with other drivers of these regular
fluctuations, via seasonality [4, 5, 6] or spatial interactions [7, 8], has been recognized through modeling and computation. The fluctuations in infected levels can
propagate via spatial networks, for example created by waterways [9] or travel [8].
More recently computations and analysis demonstrate how the interaction of noise,
nonlinearity, transients, and multiple time scales can support sustained oscillations
[10] [11].
In this study we focus on understanding those oscillations that are sustained via
a coherence resonance-type mechanism. Coherence resonance (CR) in its broadest
sense is observed when noise excites coherent oscillations in a system that exhibits
stable quiescence without noise. Typically there is a maximum coherence at an optimal noise level. This phenomenon is most commonly cited for transitions between
steady equilibria with large excursions, such as relaxation oscillations [12, 13], but it
is also observed for noise-induced quasi-regular oscillations near a Hopf bifurcation,
also exhibiting CR-type behavior [14, 15]. CR was originally studied in the context of neural models. More recently purely noise-driven oscillations, without other
temporal variations, have been observed widely in other systems that are quiescent
without noise, including gene regulatory circuits [16], machine tool dynamics[17],
and predator-prey dynamics [18].
In the context of infectious disease, the phenomenon of sustained oscillations
driven by CR has been demonstrated analytically as a fundamental feature that
can occur even in the simplest of models, in particular in oscillations about the endemic equilibrium in a stochastic SIR model [4, 6, 11]. The sustained oscillations are
observed without additional drivers such as seasonality or other regular temporal
fluctuations in the rates of the interacting processes. These oscillations have critical
features in common with CR-driven oscillations in systems near a Hopf bifurcation:
2
namely, the corresponding deterministic system exhibits weakly damped oscillations
with a single dominant frequency, and the noise interacts with these damped oscillations to amplify and sustain them. It is these features that are critical in the
analysis of [6], where an SIR model is linearized about the endemic equilibrium, with
additive noise modeling random variation in infection rates. Analytical expressions
for the power spectrum of the linearized system are used to consider amplification
factors and sensitivities to parameter values. The effect of seasonal variation is considered in [6] as well. In [11], the full stochastic SIR model for interacting individuals
is approximated with a diffusion approximation, similar to the type considered in
[19]. The equations for the stochastically varying amplitude of the oscillations are
derived, yielding a multiple scales approximation to the dynamics and two parametric criteria for the CR phenomenon. Similar criteria for CR-type oscillations are
commonly found in other applications [15, 17, 18].
The first criterion in [11] is prevalence of a slow time scale for the weak damping
of the regular oscillations, which are unstable in the deterministic model. Earlier
studies of the basic SIR model [20, 21] already suggested that sustained oscillations
occur for parameter values where multiple time scales play an important role. As
shown in the analysis of [11] the presence of the multiple time scales is observed
over a significant parameter range. In this range the method of multiple scales
can be used to derive a reduced model, approximating the sustained oscillations as
a stochastically modulated sinusoidal oscillation with amplitudes described by an
Ornstein-Uhlenbeck (OU) process.
For the SIR model, the first condition in [11] is given by the critical parameter
ǫ2SIR , capturing the decay rate of the oscillations. The second criterion in [11] for
CR type oscillations is the balance of stochastic fluctations relative to this weak
damping, given by the ratio D2 /ǫ2SIR . These conditions are expressed directly in
terms of all parameters of the model: birth/death rate, infection rate, population
size, recovery rate, and basic reproduction number. It was shown that in parameter
regions where ǫ2SIR and D 2 /ǫ2SIR were both small, CR can drive regular oscillations
close to a single frequency. If the ratio D 2 /ǫ2SIR is O(1) or larger, erratic fluctuations
completely driven by noise lead to large excursions or extinctions, and CR does not
play a role.
3
For the SIR model, the explicit analytical conditions indicated that for larger
populations, reproduction numbers R0 well above unity, and moderate infection
rates γ, CR can drive these types of oscillations. Comparison with data showed
that a number of childhood diseases fall into the parameter regions where these
oscillations could be sustained, at least for larger total populations. Then the
oscillations in the infected population are an order of magnitude larger than the
stochastic fluctuations of the full model, with a clear peak in the power spectrum
indicating oscillations with a dominant period. In contrast for larger values of γ, or
for smaller total populations, the random fluctuations can dominate the dynamics
and drive even larger fluctuations, but no longer with regularity.
The previous study [11] demonstrated the principle of CR-type oscillations in
the context of the simplest possible model, SIR, and the use of a multiple time scales
analysis to approximate the dynamics of the amplitude of the oscillations as an OU
process in this context. The present study is based on that method and considers
two new aspects of the noise-driven sustained oscillations. The first aspect considers
the effect of the loss of immunity, captured via the SIRS model, on the parameter
range for sustained oscillations and consequently the robustness of the oscillations
in this context. Applying the stochastic multiple scales approach of [11] on the SIRS
model yields conditions for the sustained oscillations. These conditions again appear
via the derivation of the stochastic amplitude equations that describe the envelope
for these oscillations. The analysis demonstrates how the loss of immunity can lead
to a dramatic change in the parametric conditions for the oscillations, without any
external driving forces such as seasonal changes or other types of external temporal
fluctuations. The CR-type oscillations are observed for populations an order of
magnitude smaller than those required for CR-type oscillations in SIR models, and
they appear for a larger range of R0 and γ. Since the analysis provides explicit
formulae via the parametric conditions for the CR behavior, we can also compare
with other models in which loss of immunity is introduced in other ways, for example,
through reduction of birth-death rates, as in [4].
A second novel aspect of this analysis is the inclusion of nonlinear effects. In
[11] the stochastic amplitude of the oscillations are approximated by an OrnsteinUhlenbeck process, a linear mean-reverting diffusion, on the slow time scale. While
4
that approximation of the amplitude equation is linear for the sake of simplicity, as
discussed below and demonstrated in this paper, the approach of [11] is not limited
to linear approximations. Using the approach of [11] we derive reduced models
that allow us to identify the parameter regions where we expect to observe different
types of oscillations, specifically larger nonlinear oscillations, and to describe the
character of those oscillations. Including nonlinear effects in the analysis illustrates
how noise-driven oscillations are not necessarily sinusoidal, but can include larger
nonlinear oscillations. As we see from the parametric criteria for CR, the nonlinear
oscillations occur in regions for smaller values of R0 and larger values of γ, on
the edge of parameter regions for CR-driven oscillations. The nonlinear behavior,
captured by the multiple scales approximation, is indicated by additional peaks in
the power spectral density (PSD) with multiple peaks at different frequencies, in
contrast to the single peak of a sinusoidal oscillation of the linearized system. The
contribution of higher order harmonics is also observed in [10] where the effects of
spatial correlation are considered in a small world network model for the epidemic.
Our analysis indicates that the higher order harmonics can appear without any
spatial effects, directly following from the nonlinear interactions.
The paper is organized as follows: in Section 2 we provide the discrete SIRS
model, and indicate how the continuous stochastic diffusion model approximates
the discrete model. A dimensionless form of the model is derived to support the
identification of multiple time scales. In Section 3 a preliminary analysis of the
linearized non-dimensionalized deterministic model illustrates the separation of time
scales. This provides the basis for the stochastic multiple scales analysis, resulting
in equations for the stochastic amplitude equations. In Section 4, we use the results
from the multiple scales analysis to define parametric conditions for observations of
the noise-driven quasi-regular oscillations. From these comparisons we see clearly
that the loss of immunity extends the parameter range over which CR-type behavior
is observed, and we compare these ranges for different parameter combinations. We
also illustrate the nonlinear character of the oscillations via the PSD, and show that
the multiple scales approximation captures these nonlinear effects.
5
2
The Model
In a deterministic SIRS model with demography, there is a susceptible population
of size S, an infectious population of size I, and a recovered immune population of
size R. The populations evolve according to the rates,
transition
rate
transition
rate
S →S+1
µN + δR
S →S−1
βSI/N + µS
I →I +1
βSI/N
I →I −1
(γ + µ)I
R → R + 1 γI
(1)
R → R − 1 (µ + δ)R ,
where N is fixed equal to the total population size. The annual rate of birth or death
per individual is µ for all classes. It is assumed that the disease does not affect the
mortality rate of the population. The recovery rate is γ, inversely proportional to the
mean infectious period, and β is the annual contact rate. The recovered population
reverts to being susceptible at a rate δ. The basic reproduction number, R0 , is the
number of new infections produced by one infectious person who is introduced to
an entirely susceptible population [22], defined by R0 =
β
µ+γ .
Since N = S + I + R
is a constant, the deterministic model can be written as the system,
SI
dS
= µ(N − S) − β
+ δ(N − S − I)
dt
N
dI
SI
=β
− (γ + µ)I,
dt
N
(2)
with a disease-free equilibrium (I = 0, S = N ) which is stable for R0 < 1 and
unstable for R0 > 1. If R0 > 1, there is a stable endemic equilibrium (I 6= 0).
The corresponding stochastic model is a continuous time Markov process {(S(t), I(t), R(t)) :
t ∈ [0, ∞)}. S(t), I(t), and R(t) are random variables such that S(t), I(t), R(t) ∈
{0, 1, . . . , N }. Denoting the change in these random variables over the time period
from t to t+∆t by ∆S(t), ∆I(t), and ∆R(t), the conditional transition probabilities
of the stochastic (Poisson) process are expressed in terms


(µN + δR)∆t + o(∆t)





 µS∆t + o(∆t)
P {(∆S, ∆I) = (i, j)|(S, I)} =


(γ + µ)I∆t + o(∆t)




 β SI ∆t + o(∆t)
N
P {(∆S, ∆I) = (0, 0)|(S, I)} = 1 − [β
of Eq. (1),
(i, j) = (1, 0)
(i, j) = (−1, 0)
,
(i, j) = (0, −1)
(i, j) = (−1, 1)
SI
+ (γ + µ)I + µS + (µN + δR)]∆t + o(∆t) .
N
6
The process starts with S + I + R = N and the expected sum remains N .
The resulting equations for the Poisson increments are then expressed in a form
that is easily compared with the equations (2) of the deterministic continuous model,
µ(N − S) + δR − β SI
N ∆t + ∆Z1 − ∆Z2
SI
∆I =
β N − (µ + γ)I ∆t + ∆Z2 − ∆Z3 ,
∆S =
(3)
where ∆Z1 is the combined centered increment obtained from births and deaths
of susceptibles and recovereds becoming susceptible again, ∆Z2 corresponds to the
increment due to infections of susceptibles, and ∆Z3 is combined from deaths and
recoveries of infectives, all with mean zero and variances as derived in Appendix A.
In order to study parametrically the prevalence and behavior of sustained oscillations in the populations, we look for an analytical representation of them. Ideally
we would like to use a diffusion approximation for the process (3), but a good
approximation of that type is not always possible. In this paper we obtain analytical expressions (30) for the parameter ranges within (6) that support sustained
nearly regular oscillations. Under the conditions of (30) the standard deviations
of the Poisson increments ∆Zi , i = 1 − 3, are large enough so that ∆Zi can be
well-approximated by a normal random variable for small but non-zero ∆t. For
example, in the range for sustained oscillations and for ∆t ∼ .01 corresponding to a
time interval of a few days (parameters are in terms of annual rates), the standard
deviations of ∆Zi are well-above the value 10, typically quoted as the minimal value
for a reasonable approximation of a Poisson random variable by a normal random
variable. Then we replace the conditionally centered Poisson variables ∆Zi in Eq.
(3) with normal random variables with mean zero and the same standard deviations
as the Poisson increments ∆Zi . It is straightforward to verify computationally that
the behavior of the sustained oscillations in (3) is well-approximated by the behavior
of the sustained oscillations generated by
µ(N − S) + δR − β SI
N ∆t + ∆w1 − ∆w2
SI
∆I =
β N − (µ + γ)I ∆t + ∆w2 − ∆w3 ,
∆S =
(4)
where ∆wj are normally distributed with zero mean and variance Var∆Zi . In
addition, we have verified computationally (not shown) that the PSDs for (3) and (4)
7
are virtually indistinguishable for the range of parameters used to produce Figures
1 and 3 and similar examples of sustained oscillations.
In the limit ∆t → 0 of (4) we then have a continuous process described by
stochastic differential equations (SDE’s),


 dw1





µ(N − S) + δR − β SI
G1 −G2
0
dS

N 

 =
dt + 
dw2 ,

0
G2 −G3 
dI
−(µ + γ)I + β SI
N
dw3


(5)
with wj , j = 1 . . . 3 independent standard Brownian motions and
r
p
p
SI
, G3 = (µ + γ)I.
G1 = µ(N + S) + δR, G2 = β
N
The behavior of the stochastically driven sustained oscillations and the PSDs obtained by simulating (5) for smaller values of ∆t agree well with those computed for
(3) and (4) with larger ∆t corresponding to intervals of a few days. As would be
expected, for small values of N , S, or I, away from the range of sustained oscillations, the normal approximation to the Poisson is no longer appropriate, and thus
the diffusion approximation no longer holds. This is consistent with the analysis we
provide for the sustained nearly periodic oscillations: we determine the parameter
ranges analytically where this behavior is prevalent, and it is in this same range
of parameters that the normal approximation to the Poisson distribution is valid
and there is clear agreement in the behavior between (3), (4), and (5). As shown
in Section 3, the asymptotic approximation is obtained by an expansion about the
endemic steady state solution of the deterministic system. This expansion yields
constant noise coefficients to leading order, resulting in additive noise terms.
Similar modeling approaches, approximating discrete interacting systems with
stochastic differential equations are used, for example, in random molecular reactions [23] and approximations of shot noise [24]. Note that Eq. (5) has drift terms
corresponding to Eq. (2). As demonstrated in [19] there is more than one representation for the diffusion terms, as long as the correlations are correct, and our choice
here is for the sake of transparency. Note that Eq. (3) does not converge to a diffusion as N → ∞, but rather when normalized it converges to a deterministic system.
In this large N limit, the random fluctuations ∆Zi appear formally as higher order
corrections to the deterministic dynamics, yet we show below that via the influence
8
of CR these random fluctuations play an important role in significant fluctuations in
the disease dynamics about the endemic state. As we show in the next section, there
is another important small parameter, ǫ, related to the slow decay of the oscillations
about the endemic state. Using the asymptotic expansion in ǫ and a comparison of
ǫ and the noise coefficents in the amplitude equation for the sustained oscillations
we provide analytically the parameter ranges for the sustained oscillations. The
relative sizes of ǫ, 1/N , and the noise coefficients under the conditions for sustained
oscillations are discussed further in Section 4.
The interpretation associated with the stochastic integration represented by dwi
in (5) should also be specified, as either Stratonovich or Ito [25]. In this paper we
take the Ito interpretation for convenience, since the Stratonovich interpretation
yields no appreciable difference in the dynamics for the sustained oscillations. To
show this, one can write down a modified version of (5) that includes a correction
term to account for the Stratonovich interpretation of dwi [25]. This correction term
can be shown to be a higher order correction in the asymptotic expansions below,
using the asymptotic behavior of the small parameters as discussed in Section 4.
We have verified this computationally by comparing the amplitude dynamics and
the PSDs of the sustained oscillations for both the Ito and Stratonovich interpretations of (5). Furthermore, in the asymptotic analysis of the dimensionless system
below, we approximate the diffusion (noise) coefficients with their leading order
contributions, which are constants. In the case of constant diffusion coefficients, the
Stratonovich and Ito interpretations are identical.
2.1
The dimensionless problem
In the following we consider the ranges of parameters:
10 < γ < 40
0 < δ < .25
1 < R0 < 20
1/10 < µ < 1/55
50000 < N < 2000000 .
(6)
Note that specifying R0 yields β = R0 (γ + µ) ∼ R0 γ for the parameters considered here. As the coefficients in the stochastic model involve products of small
and large quantities, it is often not possible to determine whether or not the noise
is playing a dominant role in the disease dynamics. This is particularly true in
9
looking for oscillations around the endemic equilibrium, where multiple time scales
can be hidden among the combinations of small and large parameters. An important step in determining the significance of the random fluctuations is therefore
non-dimensionalization, which reveals the conditions for these multiple time scales.
Once identified, multiple scales analysis can be used to identify the critical balance
between the size of the noise and the slow time scale, a key element in sustained
oscillations.
For R0 > 1 in Eq. (2), the deterministic system has a unique nontrivial stable
endemic equilibrium point (Seq , Ieq ) at
Seq =
N
,
R0
Ieq =
(µ + δ)N
(R0 − 1).
β + δR0
(7)
We introduce dimensionless variables
u = (S − Seq )/Seq , v = (I − Ieq )/Ieq ,
p
τ = Ωt, Ω = β(µ + δ)(R0 − 1)/R0 .
(8)
The choice of dimensionless time appears unmotivated at this point, but below we
see that this transformation yields a system with oscillations of unit frequency for
Eq. (2) linearized about the endemic equilibirum in Eq. (7) for µ ≪ 1 and δ ≪ 1.
The change of variables allows a consistent separation of time scales in the analysis
below. The resulting non-dimensionalized equations for the stochastic system in
Eq. (5) are

du

 = 1
Ω
dv
where


β
β
0
−(µ + δ) − N
Ieq ) u − β+δR
I
v
−
I
uv
eq
eq
N
N
dτ + GdW(τ ) , (9)

β
β
β
N Seq u + N Seq − (µ + γ) v + N Seq uv




W1 (τ )


G1 −G2
0



W(τ ) = W2 (τ ) G = 


0
G3 −G4
W3 (τ )
s
δ
µ
(N + Seq (u + 1)) + 2 (N − Seq (u + 1) − Ieq (v + 1)) ,
G1 =
2
Seq Ω
Seq Ω
s
βIeq
(u + 1)(v + 1), G3 = Seq G2 /Ieq = G4 .
G2 =
ΩN Seq
10
Note that the Brownian motions Wi are given in terms of the time τ .
Figure 1 shows the time series for I − Ieq for different parameter combinations.
Differences between smaller nearly sinusoidal-type oscillations (dash-dotted lines)
and larger nonlinear oscillations (solid lines) are apparent. The parameter regions
for these different types of oscillations are given by the criteria presented in Section
4.
1500
I−Ieq
1000
500
0
−500
350
360
370
380
390
400
410
420
430
440
450
410
420
430
440
450
time
800
600
I−Ieq
400
200
0
−200
−400
350
360
370
380
390
400
time
Figure 1: Time series using (3) for variation of I from its equilibrium value for different parameter values. In the top graph, γ = 20, R0 = 15 and µ = 1/55. The solid and dash-dotted
line both correspond to N = 500000, with δ = 0 and Ieq ≈ 423 and δ = .1 and Ieq ≈ 2741
respectively. In the bottom graph N = 50000, δ = .125, and µ = 1/55. For the solid line γ = 30,
R0 = 6, with Ieq ≈ 197, and γ = 30, R0 = 10, Ieq ≈ 213 for the dash-dotted line. The time axis
is scaled by Ω in both graphs.
3
Multiple scales analysis of the noisy nonlin-
ear problem
We illustrate the basis for the multiple scales approach by first considering the nondimensionalized deterministic system. Setting G = 0 in Eq. (9) and then linearizing
11
about u = v = 0 yields
 
u
du
  = M dτ
v
dv

where


0
− (µ+δ)(δ+β)R
(β+δR0 )Ω

M=
(10)
0 −1)
− (µ+δ)(R
Ω
β
ΩR0
0

.
We find that solutions of Eq. (10) are given in terms of the eigenvalues λM of M,
which are
λM
= −ǫ2 ±
p
ǫ2 =
ǫ4 − 1 ,
(µ + δ)(δ + β)R0
.
2Ω(β + δR0 )
(11)
For the parameter ranges given above, ǫ2 takes values in the range .02 < ǫ2 < .2,
p
with ǫ2 increasing roughly as δ/γ for 1 > δ ≫ µ. Then for ǫ2 ≪ 1 the leading order
solution of the non-dimensionalized deterministic problem near the equilibrium is
given by slowly damped oscillations with frequency near unity, that is,
 




u
b
cos
τ
b
sin
τ
  ∼ C1 exp−ǫ2 τ 
 + C2 exp−ǫ2 τ 
,
v
sin τ
− cos τ
(12)
if we drop O(ǫ2 ) corrections to the frequency. This result shows that the time scale
of the decay of the amplitude of the oscillations is an order of magnitude smaller
than the time scale of the frequency, thus indicating dynamics on separate time
scales. It is this observation that motivates the analysis that follows. Later we see
that it is also a critical feature for sustained oscillations via the CR mechanism. In
the following we derive equations for the amplitude of the oscillations driven by CR,
and obtain expressions for the effective stochastic contribution to these amplitude
equations, parametrized by λ, given in (27).
Here we also introduce additional non-dimensional parameters,
α1 =
γ+µ
,
µ
α2 =
δ+µ
.
µ
(13)
For the parameter ranges (6) considered in this paper, α1 is at least an order of
p
magnitude larger than α2 , yielding α2 /α1 ≪ 1. From the expression for ǫ in
terms of α1 /α2 it then follows how the relative magnitude of the key parameters
contribute to ǫ2 ≪ 1,
ǫ
2
=
1
2
r
α1 R0 + α2 − 1
α2
.
α1 (R0 − 1) α1 + α2 − 1
12
(14)
Since the introduction of these non-dimensional parameters does not simplify
the derivation of the stochastic amplitude equations, those calculations are given
below in terms of the original parameters. In Section 3, where the noise coefficient
λ2 /ǫ2 and ǫ2 parameters are used to find parameter regions supporting the sustained oscillations, we give the results in terms of α1 and α2 . This allows simplified
asymptotic expressions that highlight the influence of the key parameters.
3.1
Derivation of the stochastic amplitude equations
The behavior of the solution to the linearized deterministic problem in Eq. (12)
suggests a form of approximation to the solution to Eq. (10) with oscillations
on the τ scale and with modulations described by a slowly varying amplitude or
envelope on the slow time ǫ2 t . We might expect that the slow variation of the
envelope would be destroyed by the stochastic fluctuations, but we use the method
of multiple scales to show that for relatively small noise, this slow variation survives.
For the stochastic model Eq. (9) we show that the noise is indeed small in this sense
over a significant parameter range, which expands with increasing loss of immunity
δ.
Writing the linear operator M = M0 + ǫ2 M1 as

 

o −1)
0
−b
0
− (µ+δ)(R
Ω
,
=
M0 = 
β
1/b 0
0
ΩRo


−2 0
,
M1 = 
0 0
where b = R0 Ω/β, the non-dimensionalized system Eq. (9) is then
 
 
 
 
P
u
u
du
  = M0  dτ + ǫ2 M1  dτ +  uvdτ + GdW
Q
v
v
dv
β
0 −1)
where P = − ΩN
=
Ieq = − β(µ+δ)(R
Ω(β+δR0 )
−Ieq
bSeq
and Q =
β
ΩN Seq
=
β
ΩR0
(15)
(16)
= 1/b.
We propose an ansatz for approximating the solution of the stochastic model
given by Eq. (16), in particular in the neighborhood of the endemic equilibrium. In
order to capture the influence of noise over the slow time scale T = ǫ2 τ , we look for
a solution that is periodic on the fast time scale τ with a slowly varying envelope
13
of the oscillations,
 





u
b cos τ
b sin τ
  ∼ǫ A(T )
 + B(T )

v
sin τ
− cos τ

 
 

a
(T
)
s
(T
)
sin
2τ
s
(T
)
cos
2τ
1
 .
+ 1
+ 3
+ ǫ2  
a2 (T )
s2 (T ) cos 2τ
s4 (T ) sin 2τ
(17)
Here A(T ) and B(T ) are stochastic amplitudes varying on the slow time scale T ,
and our goal is to determine them in order to approximate u and v. Through A
and B we have a direct measurement of the magnitude of the oscillations. The
coefficients si (T ), i = 1, . . . , 4, and aj , j = 1, 2, are functions of A(T ) and B(T ), as
shown below.
Substituting Eq. (17) into Eq. (16) and keeping only the leading order contribution in ǫ for the noise terms, we find



 
 
n
b sin τ
b cos τ
du

 + B
  ∼ M0 + ǫ2 M1 ǫ A 
− cos τ
sin τ
dv

 
  
o
s1 cos 2τ
s sin 2τ
a
+ 3
 +  1  dτ
+ǫ2 
s4 sin 2τ
s2 cos 2τ
a2
 
P +   ǫ[Ab cos τ + Bb sin τ ] + ǫ2 [s1 cos 2τ + s3 sin 2τ + a1 ] ·
Q
ǫ[A sin τ − B cos τ ] + ǫ2 [s4 sin 2τ + s2 cos 2τ + a2 ] dτ + gdW.
(18)
Here g gives the leading order contributions to the coefficients G for u ≪ 1, v ≪ 1,
with

g = 
g1 −k2 g2
0
g2
0
−g2


g1 =
s
δ
µ
(N + Seq ) + 2 (N − Seq − Ieq )
2 Ω
Seq
Seq Ω
g2 =
s
βSeq
,
ΩN Ieq
(19)
k2 = Ieq /Seq .
Note that with this approximation the leading order contribution to the noise is
additive.
To complete the approximation for u, v in Eq. (17) we derive equations for the
slowly varying coefficients A(T ), B(T ). For ǫ ≪ 1 and relatively small noise it is
14
reasonable to look for a form of the approximate amplitude equations to be SDE’s
with constant diffusion coefficients,
  



dA
f1 (A, B)
dξ1 (T )
 =
dT + Σ
,
dB
f2 (A, B)
dξ2 (T )

Σ=
σ1 σ2
σ3 σ4

,
(20)
where ξ1 , ξ2 ’s are independent standard Brownian motions on the slow time-scale,
T and the coefficients σk , k = 1, . . . , 4 are constant.
The drift coefficients, fj , j = 1, 2 and the diffusion coefficients σk , k = 1 . . . 4
are unknown and must be determined by relating Eq. (18) to Eq. (20). Using Eq.
(17) and Eq. (20) together with Ito’s formula [25], yields equations for du and dv
(33) given in Appendix B. The leading order contributions to these equations are
given in Eq. (34). We determine the fj ’s and σk ’s by equating Eq. (18) and Eq.
(34) and collecting terms of the same order in powers of ǫ.
We first consider the drift terms. Setting the coefficients a1 = a2 = 0 and sj
j = 1 . . . 4 to be slaved to A and B as,
s1 =
s3 =
b
1
k2
k2 2
(A − B 2 ) − AB , s2 = − (A2 − B 2 ) − AB ,
3
3
3
3b
2
2
2k
k
2
b 2
(A − B 2 ) +
AB , s4 = (A2 − B 2 ) − AB ,
6
3
6b
3
(21)
yields that the O(ǫ) and O(ǫ2 ) terms cancel. This choice is consistent with deriving
stochastic amplitude equations for A and B that balance the leading order noise
terms with the O(ǫ3 ) contributions to the drift from the cubic nonlinearities and the
correction in the linear terms involving M1 . The correction in the linear operator
ǫ2 M1 is directly related to the slow decay of the amplitude on the T time scale for
the modes cos τ, sin τ , and the choice of sj removes any higher modes like sin 2τ in
the corrections in the amplitude equations at O(ǫ2 ).
At O(ǫ3 ) the drift terms from Eq. (18) are equated to the drift terms in Eq.
15
(34) using aj = 0, j = 1, 2, Eq. (21), and dT = ǫ2 dτ , giving



 
−2b sin τ
−2b cos τ
 dT
 + B
ǫ A
0
0
 
P
+ ǫ  [ (Ab cos τ + Bb sin τ )(s4 sin 2τ + s2 cos 2τ ) + (A sin τ − B cos τ )(s1 cos 2τ + s3 sin 2τ ) ] dT
Q




b cos τ
b sin τ
f1 dT + ǫ
f2 dT .
= ǫ
(22)
sin τ
− cos τ
From Eq. (22) we wish to determine equations for the slowly varying functions of
T only. To accomplish this, we use a projection of Eq.(22) onto the solution to
the adjoint problem of the linearized problem given in Eq.(10). This projection is
then averaged over one period of the fast oscillations, averaging only the terms that
depend on the fast time scale τ , and treating the functions of the slow variable as
constant. Then this projection is simply
Z 2π
Γ(22)dτ ,
(23)
0
with the matrix Γ given in Eq. (36), and functions of T in Eq. (22) treated
as constants. This projection and averaging is a standard step in a deterministic
multiple scales analysis. There a solvability condition, equivalent to an orthogonality
condition on higher order terms in the asymptotic expansion, yields equations for the
slowly varying coefficients in a multiple scales expansion [26]. A similar averaging
approach was used in the context of stochastic averaging of a Duffing-van der Pol
system with multiplicative noise (see [27] and references therein). Then Eq. (23)
yields equations on the slow time scale T for fj (T ) in terms of A(T ), B(T ),
1
(P 2 + 1)B A2 + B 2 − A
24
1
f2 = (P 2 + 1)A A2 + B 2 − B.
24
f1 = −
(24)
Some of the details for this calculation are given in Appendix B.
Similarly, we compare the diffusion terms from Eqs. (18) and (34) and project
as in the case of the drift terms. Then, following [27], to determine σj we look
for consistency in the diffusion terms in the generator for the process in Eq. (20)
with those of the averaged generator for the process in Eq. (9). In the present
16
setting, this consistency reduces to a straightforward statement: the coefficients for
the diffusion terms in the equation for the probability density of A and B should be
the same as the coefficients of the averaged diffusion operator corresponding to the
right hand side of Eq. (38), where the noise terms are projected onto the adjoint
solution space using Γ as above. We then obtain
p
g12 + k4 g22 + 2b2 g22
√
,
σ1 = σ4 =
2bǫ2
σ2 = σ3 = 0
(25)
The details of the calculation are given in Appendix B.
3.2
Parameter ranges for coherence resonance
The stochastic amplitude equations (20) for A and B are approximated by


 
 

  
 
1
0
0
−1
dξ
A
A
−1 0
A
 1  ,
 (A2 + B 2 ) dT + λ 
  + 1 
d  = 
24 1 0
ǫ2 0 1
dξ2
B
B
0 −1
B
(26)
where
√
2bλ =
q
g12 + (k4 + 2b2 )g22 ,
(27)
and ξj (T ) are independent Brownian motions on the slow time scale.
From Eq. (26) we identify the parametric conditions under which one expects
to observe the phenomenon of CR, focusing on the effect of noise in driving oscillations in this way. We demonstrate that under these conditions the multiple
scales construction of the previous section captures the character of the SIRS process. We note that the nonlinear terms in the drift contribute to slow variation in
the phase, so the primary contributor to the magnitude of the oscillations is the
diffusion coefficient.
To identify the range of parameter values that result in significant oscillations
for u and v around the endemic equilibrium, we write


 
b cos(τ − φ)
u

  = R
sin(τ − φ)
v
(28)
for ǫA = R cos φ and ǫB = R sin φ. Then the equation for the amplitude R of the
oscillations is
dR = −RdT +
λ
λ
cos φdξ1 + sin φdξ2
ǫ
ǫ
17
(29)
The variability of the amplitude R is driven by the magnitude of λ2 /ǫ2 , the critical
parameter in variations of the oscillation envelope. This ratio is a measure of the
magnitude of the noise in the process. For small values of λ2 /ǫ2 , we expect to see
relatively small oscillations. For larger values of λ2 /ǫ2 but still well below unity,
the stochastic fluctuations balance with the deterministic decay on the slow time
scale T . In this case, both stochastic and deterministic features are apparent in
the dynamics, as evidenced by the CR-type phenomenon. For λ2 /ǫ2 = O(1), the
noise dominates the behavior so that the PSD is no longer concentrated around a
single frequency. Then the magnitude of λ2 /ǫ2 indicates parameter ranges where
regular fluctuations are driven by CR, as well as parameter ranges where stochastic
fluctuations dominate the variability in the population levels. Furthermore, we
observe that the multiple scales approximation is inappropriate for values of λ2 /ǫ2
approaching unity. Then the stochastic variations govern the dynamics, and an
approximation based on a slowly varying modulation is no longer appropriate.
The multiple scales approach is valid for parameter ranges in which there is
a separation of time scales and there is a balance of the deterministic dynamics
and stochastic variation. The validity region indicates conditions for observing
the sustained quasi-regular oscillations, since the approximation is based on the
characteristics of CR. The criteria for the CR-driven oscillations are then quantified
as,
ǫ2 ≪ 1,
λ2
≪ 1.
2ǫ2
(30)
The first of these criteria is the standard assumption in a multiple scales expansion,
which states that the critial time scales are well-separated. In this case it is the time
scale of the period of the oscillations that is well-separated from the slow decay rate
of their amplitude. The second criterion is the condition that the noise does not
dominate the dynamics, but rather balances with the deterministic dynamics. We
write ǫ2 and λ2 in terms of non-dimensionalized parameters R0 , α1 and α2 :
ǫ
2
=
k2 =
g12 =
r
α1 R0 + α2 − 1
1
g2 + (k4 + 2b2 )g22
α2
,
λ2 = 1
,
(31)
2 α1 (R0 − 1) α1 + α2 − 1
2b2
α2 (R0 − 1)R0
α2 (R0 − 1)
α1 (α1 + α2 − 1)R0
p
, b2 =
, g22 =
α1 + α2 − 1
α1
α2 N (R0 − 1) α1 α2 (R0 − 1)
R0
p
R0 + 1 + (α2 − 1)(R0 − 1 − k2 ) .
N α1 α2 (R0 − 1)
18
As discussed in Section 3, α2 ≪ α1 . Then ǫ2 and λ2 /ǫ2 can be written in terms
of their asymptotic behavior for N −1 ≪ α2 /α1 ≪ 1.
1
ǫ ∼
2
2
r
R
α2
√ 0
,
α1 R0 − 1
λ2
2α21
∼
.
ǫ2
N (R0 − 1)α22
(32)
In Figure 2 we show the ranges where the criteria (30) are satisfied, and thus
where regular oscillations driven by noise via coherence resonance is predicted to
be prevalent. These are shown for the full expressions given in (31), and it is
straightforward to show that the asymptotic expressions (32) give virtually the same
boundaries for parameters satisfying the criteria (30). Note that these criteria also
indicate how the small parameter ǫ follows from the fact that the rate δ is smaller
than the rate γ, reflected in α2 ≪ α1 . In addition, the asymptotic behaviors indicate
√
the relative magnitudes of ǫ and 1/ N that support the CR-type phenomenon for
√
R0 = O(1): combining λ2 ≪ ǫ2 and (32) yields N ≫ ǫ−4 . The boundaries for the
regions (30) are shown in Figure 2 for relatively large values (0.3) of ǫ2 and λ2 /ǫ2 ,
since the asymptotic expansion gives a reasonable approximation up to this value
for these expressions in (30), as evidenced by comparisons with the PSD’s below.
In the top row of Figure 2 we show these regions for δ = 0 (results for SIR
model), and how the region is reduced for smaller population size N . In the middle
row figures, we show the parameter regions for sustained oscillations in the SIRS
model, indicating that loss of immunity opens the possibility for sustained oscillations over a larger range of parameters, both for larger values of α1 (γ) and for
R0 above but closer to unity. Furthermore, in the middle row right figure, the
boundaries indicate that these oscillations are more widely observed in populations
an order of magnitude smaller with loss of immunity than without loss of immunity.
In the last figure, we compare the SIRS model in Eq. (9) with a different approach
for modeling loss of immunity, as in [4], where a larger value of µ was used in an
SIR model, rather than including the extra term with coefficient δ as in Eq. (2).
The simplified expressions for ǫ2 and λ2 given in (32) are valuable for identifying
the key influence of the main parameters. For increased loss of immunity δ the
region in the R0 − α1 parameter space expands for sustained oscillations, and shifts
the multiple scales boundary towards larger α1 (γ) and smaller R0 parameter com-
19
binations. For increased µ, the boundaries for the criteria in (30) shift to smaller
values of α1 corresponding to the reduced values of α1 ∼ γ/µ. In the R0 − γ plane
(not shown) the boundaries for (30) for the two different combinations of δ and µ
shown in the lower right of Figure 2 are very close, consistent with the fact that the
ratio α1 /α2 takes on similar values for these two cases.
For parameter values chosen near the boundary of the second criterion in Eq.
(30), the oscillations are larger and are characterised by peaks in the PSD at more
than one frequency. The additional peak is due to the nonlinear nature of these
oscillations, as shown in Figure 3. The approximation in Eq. (17) captures this
behavior up to parameter values where the oscillations are not sinusoidal, but are
fluctuating about the endemic equilibrium, by including the correction terms with
coefficients sj . For larger values of noise, where the approximation is no longer
valid, the system regularly escapes to either extinction or to a dramatic increase in
I corresponding to an epidemic onset.
In Figure 3 we compare the PSD of I obtained from the the full model in Eq.
(9) and that obtained from the multiple scales approximation in Eq. (17) with
the amplitudes A, B given by Eq. (26). As noted in Section 2, we can reproduce
these figures using (3), (4), and (5). We focus mainly on parameter values near
the boundary for the criterion on λ2 /ǫ2 . In the first two figures, we compare the
PSD for larger populations levels N = 500000, and show how the nonlinear effects
contribute to the behavior of the oscillations on the edge of the criteria for CR-type
oscillations.
The extra contribution is evident in the smaller peak for the higher
modes, as captured by the O(ǫ2 ) terms in the asymptotic approximation (17). Note
that the edge of the second criterion shifts to signficantly smaller values of R0 for
non-zero loss of immunity than for no loss of immunity. In the top right figure we
compare the PSD for smaller values of N = 50000, smaller R0 and larger γ than in
the first two figures. We again see the contribution of higher order harmonics, and
in general there is a longer tail in the PSD as compared with larger population sizes
N . For the full model, the PSD shows additional contributions for a frequency below
that of the main peak, corresponding to occasional escapes to larger fluctuations,
not captured by the multiple scales approximation. In the bottom row, we show
the PSD for larger values of µ = 1/10 and N = 50000. Given a similar shift in the
20
2000
1500
1500
α1
α1
2000
1000
500
500
0
5
10
R0
0
15
2000
2000
1500
1500
α1
α1
0
1000
1000
5
0
5
10
15
10
15
R0
1000
500
500
0
0
0
0
5
0
5
10
15
10
15
R0
R0
2000
α1
1500
1000
500
0
R0
Figure 2: Parameter regions bounded by ǫ2 = .3 and λ2 /ǫ2 = .3. Regions for ǫ2 < .3 are above
and to the right of ǫ2 = .3, and regions for λ2 /ǫ2 < .3 are below and to the right of λ2 /ǫ2 = .3.
(Top row, left)Parameters are δ = 0, µ = 1/55, with ǫ2 = .3 given by the dotted line. For
λ2 /ǫ2 = .3, solid line is for N = 500000 and o’s for N = 1000000. (Top row, right)Parameters
are δ = 0, µ = 1/55, with ǫ2 = .3 given by the dotted line. For λ2 /ǫ2 = .3, solid line is for
N = 500000 and +’s for N = 50000. (Middle row, left) N = 500000 for all curves, with ǫ2 = .3
the dotted line and λ2 /ǫ2 = .3 the solid line for δ = 0. For δ = .25, the dash-dotted line is
ǫ2 = .3 and +’s for λ2 /ǫ2 = .3 (Middle row, right) δ = .125 for all curves, and dash-dotted is
ǫ2 = .3 for both values of N . λ2 /ǫ2 = .3 given by solid line N = 500000 and +’s for N = 50000.
d) (Bottom row, left) δ = .125, N = 50000 for all curves. For µ = 1/55 and δ = .125, ǫ2 = .3
is the dotted line, λ2 /ǫ2 = .3 is the solid line; For µ = 1/10, δ = .05, ǫ2 = .3 is the dash dotted
line, λ2 /ǫ2 = .3 is +’s .
21
region for sustained oscillations for larger µ as seen for larger δ, we show the PSD for
larger µ with contributions from higher harmonics, corresponding to the edge of the
parameter region. For larger population values N = 500000 and larger µ = 1/10,
we observe a single peak for the PSD, even for smaller values of δ, indicating the
4
4
3
3
3
2
1
0
PSD
4
PSD
PSD
parameter values are well within the region for sustained CR-type oscillations.
2
1
0
5
10
0
15
1
0
4
3
3
2
1
0
5
10
15
frequency
PSD
PSD
frequency
4
2
0
0
5
10
15
frequency
2
1
0
5
10
15
frequency
0
0
5
10
15
frequency
Figure 3: Comparison of averaged PSD of I, obtained computationally from 100 realizations of
the full model in Eq. (9) (solid blue line) and the multiple scales approximation using Eq. (26)
(dash-dotted red line) on a time interval equivalent to t = 200. Top Left: N = 500000, µ = 1/55,
γ = 20, R0 = 15, δ = 0 ; Top Center: N = 500000, µ = 1/55, γ = 45, R0 = 2 δ = .125; Top
Right: N = 50000, µ = 1/55, γ = 32, R0 = 6, δ = .125 ; Bottom Left: N = 50000, µ = 1/10,
γ = 32, R0 = 3, δ = .125; Bottom Center N = 500000, µ = 1/10, γ = 50, R0 = 6, δ = .05
4
Conclusion
A multiple scales analysis is used to study the effect of loss of immunity on sustained
oscillations about an endemic equilibrium. The oscillations are studied in the context where nearly regular oscillations in the infected and susceptible populations
are sustained by the random fluctuations in the model, rather than by external
temporal variation such as seasonality or by spatial variations. Even though these
22
oscillations are driven by noise, the interaction of the noise with the deterministic
dynamics results in nearly regular behavior.
For large enough populations, a diffusion approximation to the discrete SIRS
model of interacting individuals provides a starting point for this analysis. The
nondimensionalized model indicates that multiple time scales are possible under a
range of parameter combinations. In particular, the weak damping of the oscillations of the system linearized about the endemic equilibrium motivates an appropriate approximation to the dynamics. The form is then a regular oscillation on the
original time scale with a stochastically varying amplitude on the slow time scale.
The derivation of a nonlinear stochastic amplitude equation not only completes the
asymptotic approximation to the solution of the model, but also yields two criteria
in Eq. (30) for the prevalence of the nearly regular oscillations. The conditions
are consist with the key elements for the phenomenon of coherence resonance (CR),
where noise drives nearly regular oscillatios in a system that is quiescent without
noise. Both a balance in the interaction of deterministic and stochastic dynamics
and a separation of time scales are critical features in the support of CR-type oscillations. The parameter combinations are derived directly from expressions for the
weak damping of the oscillations in the linearized non-dimensionalized system, and
for the noisy variation of the oscillation envelope given by the diffusion coefficients
in the stochastic amplitude equation.
From the criteria directly in terms of the model parameters for CR-type sustained oscillations, we see that the loss of immunity can lead to a significant difference in parameter regions where sustained oscillations can be observed, without
any external temporal driving forces. Compared with an SIR model, with no loss of
immunity, the CR-type oscillations in the SIRS model are robust for smaller population levels, and are observed over a wider range of basic reproduction number R0
and infection rates γ. We also show that modeling loss of immunity by varying the
birth and death rates as in [4] has a similar effect on the range of parameters for
sustained oscillations.
Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of
the oscillations. These nonlinear features are observed near the boundaries of the
23
parameter regions for the CR-type oscillations, consistent with the predictions of
these validity criteria that the oscillations are larger near the criteria boundaries
and are not purely sinusoidal. The additional peaks in the power spectral density
(PSD) near these boundaries are observed consistently between the full SIRS model
in Eq. (9) and the multiple scales approximation Eqs. (17) with (26).
The analysis applied here to capture nonlinear features of the oscillation and
parametric predictions for sustained oscillations is adapted from more general methods of stochastic multiple scales analysis or stochastic averaging. Then this approach
is not limited to this particular system, but rather is readily generalizable to other
models. For example, preliminary results indicate its direct applicability to larger
systems, such as for cholera, where bacterial and host populations interact in the occurence and recurrence of epidemics [28]. Here we have considered only the leading
order contributions to the noise in the approximations, but additional contributions
from parametric or multiplicative noise can be treated in similar ways, as demonstrated in [27, 29].
Acknowlegdments
This research was supported in part by an NSERC Discovery Grant.
5
Appendix A
Here we show how the equations for ∆S and ∆I can be written in terms of centered
Poisson increments and conveniently compared to the deterministic model. To each
increment we add and subtract the conditional expectation, conditioned on the value
of the process at the beginning of the time increment of length ∆t.
∆S = E(∆S) + ∆S − E(∆S),
∆I = E(∆I) + ∆S − E(∆I)
From the conditional transition rates, it is obvious that
E(∆S) = (µN + δR)∆t − µS∆t − β
SI
∆t,
N
E(∆I) = −(µ + γ)I∆t + β
SI
∆t
N
Next, write ∆S − E(∆S) and ∆I − E(∆I) as the sums of centered increments [30]:
∆S − E(∆S) = ∆Z1 − ∆Z2 ,
24
∆I − E(∆I) = ∆Z2 − ∆Z3
As centered Poisson increments ∆Z1 , ∆Z2 , and ∆Z3 all have mean 0 and the
following variances:
Var(∆Z1 ) = (µN + δR + µS)∆t,
6
Var(∆Z2 ) = β
SI
∆t,
N
Var(∆Z3 ) = (µI + γI)∆t
Appendix B
Here we give the details for the application of Ito’s formula [25] and the projection
used in the multiple scales analysis. Using Ito’s formula and Eq. (17) we obtain the
equations for du, dv,

  
2
ǫ[−Ab sin τ + Bb cos τ ] + ǫ [−2s1 sin 2τ + 2s3 cos 2τ ]
du
dτ
  =
2
ǫ[A cos τ + B sin τ ] + ǫ [2s4 cos 2τ − 2s2 sin 2τ ]
dv


∂s3
1
ǫb cos τ + ǫ2 [ ∂s
cos
2τ
+
sin
2τ
]
∂A
∂A
(f1 dT + σ1 dξ1 + σ2 dξ2 )
+
∂s2
∂s4
2
ǫ sin τ + ǫ [ ∂A sin 2τ + ∂A cos 2τ ]


∂s3
1
cos
2τ
+
sin
2τ
]
ǫb sin τ + ǫ2 [ ∂s
∂B
∂B
(f2 dT + σ3 dξ1 + σ4 dξ2 )
+
∂s4
2
2
cos
2τ
]
−ǫ cos τ + ǫ [ ∂B sin 2τ + ∂s
∂B


2
2
∂ s3
∂ s1
2
2
cos
2τ
+
sin
2τ
σ + σ2  ∂A2
∂A2
dT
+ ǫ2 1
∂ 2 s2
∂ 2 s4
2
sin
2τ
+
cos
2τ
∂A2
∂A2


∂ 2 s3
∂ 2 a1
∂ 2 s1
2
2
cos
2τ
+
sin
2τ
+
σ + σ4  ∂B 2
∂B 2
∂B 2 
+ ǫ2 3
dT
∂ 2 s2
∂ 2 a2
∂ 2 s4
2
sin 2τ + ∂B 2 cos 2τ + ∂B 2
∂B 2


∂ 2 s3
∂ 2 a1
∂ 2 s1
cos 2τ + ∂A∂B sin 2τ + ∂A∂B
dT .
+ ǫ2 (σ1 σ2 + σ3 σ4 ) ∂A∂B
(33)
∂ 2 s4
∂ 2 s2
∂ 2 a2
sin
2τ
+
cos
2τ
+
∂A∂B
∂A∂B
∂A∂B
In the analysis of this paper, we find below that the multiple scales approxima-
tion is valid for σj2 ≪ ǫ. Therefore the terms contributing to the approximation to
O(ǫ3 ) are
 


du
ǫ[−Ab sin τ + Bb cos τ ] + ǫ2 [−2s1 sin 2τ + 2s3 cos 2τ ]
 = 
dτ
2
dv
ǫ[A cos τ + B sin τ ] + ǫ [2s4 cos 2τ − 2s2 sin 2τ ]


ǫb cos τ
(f1 dT + σ1 dξ1 + σ2 dξ2 )
+
ǫ sin τ


ǫb sin τ
(f2 dT + σ3 dξ1 + σ4 dξ2 ) .
+
−ǫ cos τ
25
(34)
Using the method of multiple scales, we look for a projection on the fast time
scale τ of the equations for (du, dv), (18) and (34), in order to express the unknowns
from Eq. (20) in terms of the contributions in Eqs. (18) for u, v.
The appropriate projection is identified by considering the noise-free linear system
 
u
∂
L  = 0, L =
− M0 with solution
∂τ
v
 
 
  
c
b cos τ b sin τ
c1
u
  = V 1  ,
 =
c2
c2
sin τ − cos τ
v
for c1 , c2 constant. For L∗ the adjoint operator of L,
 
 
∗
u∗
u
∂
− M0 T   = 0,
L∗   = −
∂τ
v∗
v∗
 
 
  
d
d1
b−1 cos τ b−1 sin τ
u∗
   = ΓT  1 
 =
d2
d2
sin τ − cos τ
v∗
(35)
(36)
for d1 , d2 constant. Note that ΓV = 1.
Then to determine the drift and diffusion terms in Eq. (20) we consider the
projection of Γ on Eqs. (18) and (34), together with the appropriate average over
one period of the solution on the fast time scale. For the drift terms given in Eq.
(22), the projection of the drift terms together with the averaging is given by
Z
0
2π
Γ((−2Ab cos τ − 2Bb sin τ
+ P [(Ab cos τ + Bb sin τ )(s4 sin 2τ + s2 cos 2τ ) + (A sin τ − B cos τ )(s1 cos 2τ + s3 sin 2τ )]),
(Q[(Ab cos τ + Bb sin τ )(s4 sin 2τ + s2 cos 2τ ) + (A sin τ − B cos τ )(s1 cos 2τ + s3 sin 2τ )]))T dτ
 
Z 2π
f1
ΓV  dτ
=
0
f2
Averaging over one period of the oscillations of the fast variable τ , with the multiple
scales treatment of the slow time variables as constant with respect to this fast
variable, yields f1 and f2 in terms of A, B, and si , i = 1, . . . 4. Substituting Eq.
(21) and simplifying we get Eq. (24).
26
Next we consider the projection of the diffusion terms in Eqs. (18) and (34).
The projection applied to the diffusion terms from Eqs. (18) and (34) and yields





dξ1 (T )
dξ1 (T )
σ1 σ2

 from Eq. (34) ,
 = ǫΣ
ǫΓV
dξ2 (T )
dξ2 (T )
σ3 σ4
g
(37)
ΓgdW(τ ) = Γ dW(T) from Eq. (18) ,
ǫ
where we use properties of Brownian motion to write both sides on the slow time
scale T . Then the averaged diffusion operator corresponding to these noise terms
must be equal. Therefore the averaged coefficients in the diffusion operator must
be equal,
ǫ2
Z
2π
1
ǫ2
ΣΣT dτ =
0
Z
2π
ΓggT ΓT dτ
(38)
0
where the average is taken entry by entry for the matrices. Since Σ is a constant
matrix, then we need only compute the averages on the right hand side, yielding
ΣΣT =
λ2
I
ǫ4
(39)
and σj as in Eq. (25) with λ is given in Eq. (27).
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