Stochastic oscillations in models of epidemics on a network of cities

Abstract

We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of disease spread on a network of $n$ cities. In the model a fraction $f_{jk}$ of individuals from city $k$ commute to city $j$, where they may infect, or be infected by, others. Starting from a continuous time Markov description of the model the deterministic equations, which are valid in the limit when the population of each city is infinite, are recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple: a unique non-trivial fixed point always exists and has the feature that the fraction of susceptible, infected and recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have an analogously simple dynamics: all oscillations have a single frequency, equal to that found in the one city case. We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation of the deterministic equations at the fixed point.

Full text

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PHYSICAL REVIEW E 84, 051919 (2011)
Stochastic oscillations in models of epidemics on a network of cities
G. Rozhnova,1A. Nunes,1and A. J. McKane1,2
1Centro de F´ ısica da Mat´ eria Condensada and Departamento de F´ ısica, Faculdade de Ciˆ encias da Universidade de Lisboa,
PT-1649-003 Lisboa Codex, Portugal
2Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
(Received 12 September 2011; published 28 November 2011)
We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of
disease spread on a network of n cities. In the model a fraction fjkof individuals from city k commute to city
j, where they may infect, or be infected by, others. Starting from a continuous-time Markov description of the
model the deterministic equations, which are valid in the limit when the population of each city is infinite, are
recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van
Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple:
A unique nontrivial fixed point always exists and has the feature that the fraction of susceptible, infected, and
recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have
an analogously simpledynamics: Alloscillations have asingle frequency, equal to thatfound intheone-city case.
We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation
of the deterministic equations at the fixed point.
DOI: 10.1103/PhysRevE.84.051919PACS number(s): 87.10.Mn, 05.40.−a, 02.50.Ey
I. INTRODUCTION
Two of the ideas that are currently dominating the discus-
sion of modeling epidemic spread are those of stochasticity
and network structure [1–6]. Deterministic models of the
susceptible-infected-recovered (SIR) type have a long history
[7,8] and have been thoroughly investigated [9,10] along with
many extensions of the models, such as age classes or higher-
order nonlinear interaction terms. Although stochasticity, due
to random processes at the level of individuals, and networks,
eitherbetweenindividualsortownsandcities,wererecognized
early on as significant and important factors, the tendency
was to model them through computer simulations. This is not
surprising: It is rather straightforward to deal with stochastic
behavior in simulations, and similarly the analytic methods
available to investigate complex networks, especially adaptive
networks, are limited. There has also been a tendency toward
developing extremely detailed agent-based models to study
disease spread [11–14], which are the antithesis of the simple
analytic approach based on the original SIR deterministic
model.
In parallel with these developments, however, there have
been several efforts to extend analytic studies into the realm
of stochastic and network dynamics. The SIR model can be
formulated as an individual-based model (IBM) which can
form a starting point for both an analytical treatment, based on
the master equation (continuous-time Markov chain) [15,16],
and numerical simulations, based on the Gillespie algorithm
[17]. The analytical studies use the system-size expansion
of van Kampen to reduce the master equation to the set
of deterministic equations previously studied, together with
a set of stochastic differential equations for the deviations
from the deterministic result. As long as one is not too close
to the fade-out boundary, there is no need to go beyond
next-to-leading order in the expansion parameter, 1/√N,
where N is the number of individuals in the system. This
already gives results which are, in general, in almost perfect
agreement with the results of simulations [18,19].
This approach has been used to study the stochastic version
of the standard SIR model [19], the susceptible-exposed-
infected-recovered (SEIR)model[20],boththesemodelswith
annual forcing [20,21], staged models [22,23], and the pair
approximation in networked models [24,25], among others.
In this paper we extend the treatment to a metapopulation
model for disease spread, which consists of n cities (labeled
j = 1,...,n), each of which contains Nj individuals. A
fraction of the population of city k, fjk, commutes to city
j and this defines the strength of the link from node k to node
j in the network of cities. We show that the methods used in
the case of one city carry over to the case where the system
comprises a network of cities and that a surprisingly simple
set of results can be derived which allow us to make quite
general predictions for a class of stochastic networked models
of epidemics.
The starting point for our analysis is a specification of
how commuters move between cities in the network. As will
become clear, the model we arrive at does not depend on the
details of how and when these exchanges take place. We then
write transition rates for the usual SIR process, now taking
into account the city of origin of the infector and infected
individuals. From the resulting equation we can immediately
findthedeterministicequationscorrespondingtothestochastic
model when Nj→ ∞ for all j. Deterministic models of this
typebegantobeconsideredlongago[26]andtheexistenceand
stabilitypropertiesoftheendemicequilibriumwerestudiedfor
different formulations of the coupling between the cities and
of the disease dynamics [27–29]. Stochastic effects in these
systems have also been analyzed from the point of view of the
relation between spatial heterogeneity, disease extinction, and
the threshold for disease onset [27,30–32].
Some rather strong and general results on the uniqueness
and global stability of the fixed points of the deterministic
modelareknown[33].Weusetheseresultsandthengobeyond
this leading-order analysis to determine the linear stochastic
corrections that characterize the quasistationary state of the
finite system. As expected, the qualitative predictions of the
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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)
deterministic model are shown to be incorrect; instead, large
stochastic cycles are found, although their form is much
simpler than might naively have been expected. We show that
this is, in part, a reflection of the special nature of the fixed
points of the deterministic model.
Theoutlineofthepaper isasfollows.InSec. IIwedescribe
the basic model and apply it to the case of two cities. The
generalizationtothen-citycaseingiveninSec.III.Theresults
for the form of the sustained oscillations are given in Sec. IV
and we conclude in Sec. V. Two appendixes contain technical
details which are too cumbersome to include in the main text.
II. TWO-CITY MODEL
In this section we formulate the model when there are only
twocities;thegeneraln-citycasedescribedinSec.IIIdoesnot
introduce any new points of principle and is easily explained
once the two-city case has been understood.
The SIR model consists of three classes of individuals: sus-
ceptibles, infected, and recovered. The number of individuals
in the three classes belonging to city j are denoted by Sj, Ij,
and Rj, respectively. We assume that births and deaths are
coupled at the individual level, so that when an individual dies
another (susceptible) individual is born. This means that the
number of individuals belonging to any one city, Nj, does not
change with time, and so the number of recovered individuals
is not an independent variable: Rj= Nj− Sj− Ij, where
j = 1,2 [19].
TherearefourprocessesintheSIRmodelwhichcausetran-
sitions to a new state: infection, recovery, death of an infected
individual, and death of a recovered individual. The death of
a susceptible individual does not cause a transition, since it is
immediately replaced by another, newborn, individual which
is, by definition, susceptible. Of the four listed processes, the
final three only involve one individual and so only involve one
city. The transition rates are [19] as follows.
(a) Recovery of an infective individual (and creation of a
recovered individual):
T1≡ T(S1,I1− 1,S2,I2|S1,I1,S2,I2) = γI1,
T2≡ T(S1,I1,S2,I2− 1|S1,I1,S2,I2) = γI2.
(b) Death of an infected individual (and birth of a suscepti-
ble individual):
(1)
T3≡ T(S1+ 1,I1− 1,S2,I2|S1,I1,S2,I2) = μI1,
T4≡ T(S1,I1,S2+ 1,I2− 1|S1,I1,S2,I2) = μI2.
(c) Death of a recovered individual (and birth of a suscep-
tible individual):
(2)
T5≡ T(S1+ 1,I1,S2,I2|S1,I1,S2,I2) = μ(N1− S1− I1),
T6≡ T(S1,I1,S2+ 1,I2|S1,I1,S2,I2) = μ(N2− S2− I2).
(3)
Here γ and μ are parameters which respectively character-
ize the rate of recovery and of birth/death.
The infection processes introduce the role of the com-
muters. We let f21be the fraction of the population from city
1 which commutes to city 2, leaving a fraction (1 − f21) of
the population as residents of city 1. Similarly, for commuters
FIG. 1. A fraction fjkof residents of city k commute to city j,
where j,k = 1,2.
from city 2, as illustrated in Fig. 1. We note that the number
of individuals in city j is Mj= (1 − fkj)Nj+ fjkNk, where
j ?= k. We do not specify the nature of the commute in
more detail and assume that the fjk are a property of
the corresponding pair of cities that defines the overall
average fraction of time that an individual from one city
spends in the other city. These coefficients are taken as a
coarse-grained measure of the demographic coupling between
the cities that will be applied to all individuals indepen-
dently of disease status and do not discriminate between
different types of stays with their typical frequencies and
durations.
To see the nature of the infective interactions that occur, we
firstfixourattentiononthoseinvolvingsusceptibleindividuals
from city 1. There are four types of term:
(i) infective residents in city 1 infect susceptible residents
in city 1;
(ii) infective commuters from city 2 infect susceptible
residents in city 1;
(iii) infective residents in city 2 infect susceptible com-
muters from city 1;
(iv) infective commuters from city 1 infect susceptible
commuters from city 1 in city 2.
The rates for these to occur according to the usual
prescription for the SIR model [19] are
(i) β (1 − f21)S1(1 − f21)I1/M1,
(ii) β (1 − f21)S1f12I2/M1,
(iii) β f21S1(1 − f12)I2/M2,
(iv) β f21S1f21I1/M2,
where β is the parameter which sets the overall rate of
infection. Adding these rates together we obtain the total
transition rate for infection of S1individuals as
?
where
c11=(1 − f21)2N1
M1
c12=(1 − f21)f12N2
M1
A similar analysis can be made for the transitions in-
volving susceptible individuals from city 2. Putting these
results together we obtain the transition rates for infection as
follows.
(d) Infection of a susceptible individual:
βc11S1I1
N1
+ c12S1I2
N2
?
,
+f2
+f21(1 − f12)N2
21N1
M2
,
M2
.
T7≡ T(S1− 1,I1+ 1,S2,I2|S1,I1,S2,I2)
= β
N1
?
c11S1I1
+ c12S1I2
N2
?
,
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PHYSICAL REVIEW E 84, 051919 (2011)
T8≡ T(S1,I1,S2− 1,I2+ 1|S1,I1,S2,I2)
= β
N1
?
c21S2I1
+ c22S2I2
N2
?
,
(4)
where
c11=
(1 − f21)2
1 − f21+ f12q+
(1 − f21)f12q
1 − f21+ f12q+
(1 − f12)f21
f21+ (1 − f12)q+
(1 − f12)2q
f21+ (1 − f12)q+
f2
21
f21+ (1 − f12)q,
f21(1 − f12)q
f21+ (1 − f12)q,
f12(1 − f21)
1 − f21+ f12q,
f2
12q
1 − f21+ f12q,
c12=
(5)
c21=
c22=
and q = N2/N1. We assume that N1 and N2 are not too
different, so that q is neither very small nor very large.
ThemodelisdefinedbythetransitionsratesinEqs.(1)–(4).
It is interesting that the transitions due to infection depend on
the fractions fjk only through the constants cjk defined in
Eq. (5). Other ways of accounting for commuting individuals
would typically still give rise to the form given in Eq. (4), but
with the constants cjkdefined in a different way.
Since our counting of the ways that infection takes place
was exhaustive, we expect that the constants cjk are not
independent. It is straightforward to check that they obey the
following relations:
c11+ c12= 1,c21+ c22= 1,c12= c21q.
(6)
So there are only two independent parameters in addition to
the usual SIR parameters β, γ, and μ found in the single-
city case, and we choose these to be c12 and q = N2/N1.
Our results are given in terms of these two parameters. It is
easy to see that, for each q, the range of c12is the interval
[0,q/(q +1)] where the maximum is attained for f21+ f12=
1. While exploring the general behavior of the system we
consider the cjkindependently of the underlying microscopic
model as positive parameters that take values in the wider
admissible range defined by the constraints (6).
Having specified the model it may be investigated in
two ways as indicated in Sec. I. First, it can simulated
with Gillespie’s algorithm [17], or some equivalent method.
Second, it can be studied analytically by constructing the
master equation and performing van Kampen’s system-size
expansiononthisequation.Thisisthemainfocusofthispaper.
For notational convenience we denote the states of the system
by σ ≡ {S1,I1,S2,I2}, recalling that the number of recovered
individuals from each city may be written in terms of these
variables. The master equation gives the time evolution of
P(σ,t), the probability distribution for finding the system in
state σ at time t. It takes the form [15,16]
dP(σ,t)
dt
=
?
σ??=σ
8
?
a=1
[Ta(σ|σ?)P(σ?,t) − Ta(σ?|σ)P(σ,t)],
(7)
where Ta(σ|σ?), a = 1,...,8 are the transition rates from the
state σ?to the state σ given explicitly in Eqs. (1)–(4).
The full master equation (7) cannot be solved, but the van
Kampen system-size expansion when taken to next-to-leading
order usually gives results which are in excellent agreement
with simulations. We will see that this is also the case in the
extensions of the method which we are exploring in this paper.
The system-size expansion starts by making the following
ansatz [15]:
Sj= Njsj+ N1/2
j = 1,2.
limNj→∞Ij/Nj are the fraction of individuals from city
j which are respectively susceptible and infected in the
deterministic limit. The quantities xjand yjare the stochastic
deviations from these deterministic results, suitably scaled
so that they also become continuous in the limit of large
population sizes. The ansatz (8) is substituted into Eq. (7)
and powers of?Njon the left- and right-hand sides matched
equations of the model and the next-to-leading order linear
stochastic differential equations for xj and yj. We do not
describethemethodingreatdetail,sinceitisdescribedclearly
in van Kampen’s book [15] and in many papers, including
severalontheSIRmodel[6,19,22].Insteadweoutlinethemain
results of the approximation in the remainder of this section
and give some explicit intermediate formulas in Appendix A.
Thedeterministicequationswhicharefoundtofirstorderin
the system-size expansion can also be obtained by multiplying
Eq. (7) by S1, I1, S2, and I2in turn and then summing over all
states σ. This yields
j
xj,Ij= Njij+ N1/2
sj= limNj→∞Sj/Nj
j
yj,
(8)
where Here
and
ij=
up. The leading order contribution gives the deterministic
ds1
dt
ds2
dt
di1
dt
di2
dt
= −βs1(c11i1+ c12i2) + μ(1 − s1),
= −βs2(c21i1+ c22i2) + μ(1 − s2),
(9)
= βs1(c11i1+ c12i2) − (γ + μ)i1,
= βs2(c21i1+ c22i2) − (γ + μ)i2.
For the case of cities with equal population sizes, these have
beenpreviouslyfoundandanalyzedinRef.[28].Inthecontext
ofthepresentworkwearemainlyinterestedinthefixedpoints
of these equations. We do not discuss these here, instead we
wait until Sec. III, where the case of n cities is discussed when
we can give a more general treatment.
Of more interest to us in this paper are the variables xjand
yj which describe the linear fluctuations around trajectories
of the deterministic set of equations (9). For convenience we
introduce the vector of these fluctuations z = (x1,x2,y1,y2).
Our focus is on fluctuations in the stationary state, that is,
about the nontrivial fixed point of the deterministic equations
(whichweshowinthenextsectionisunique).Thefluctuations
obtained through the system-size expansion obey a linear
Fokker-Planck equation, which is equivalent to a set of
stochastic differential equations of the form [16]
dzJ
dt
=
4
?
K=1
AJKzK+ ηJ(t),J = 1,...,4,
(10)
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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)
where ηJ(t) are Gaussian noise terms with zero mean which
satisfy ?ηJ(t)ηK(t?)? = BJKδ(t − t?). Since the fluctuations
are about the fixed point, the 4 × 4 matrices A and B
are independent of time, and completely characterize the
fluctuations. They are given explicitly in Appendix A.
The fluctuations are analyzed in detail in Sec. IV, when
they are also compared to the results of numerical simulations.
Before discussing this, we generalize the discussion of this
section to an arbitrary network of n cities.
III. ARBITRARY NETWORK STRUCTURE
InthissectionwegeneralizethecontentofSec.IItoncities
and also find the fixed points of the deterministic dynamics in
this case.
A. n-city model
We use the same notation as in Sec. II, labeling the cities
by j and k which now run from 1 to n. It is convenient to
introduce the quantity
?
so that the number of individuals in city j may be written as
?
k?=j
= (1 − fj)Nj+
fj=
k?=j
fkj,
(11)
Mj=
1 −
?
fkj
?
Nj+
?
?
fjkNk.
k?=j
fjkNk
k?=j
(12)
There are, once again, four types of term in the process of
infection (see Fig. 2) and we again fix our attention on those
involving susceptible individuals from city 1.
(i) Infective residents in city 1 infect susceptible residents
in city 1. This gives a rate of β(1 − f1)S1(1 − f1)I1/M1.
(ii) Infective commuters from city j, j = 2,...,n, infect
susceptible residents in city 1. This gives a rate, summing over
all j, of β(1 − f1)S1
(iii) Infective residents in city j, j = 2,...,n, infect sus-
ceptible commuters from city 1. This gives a rate, summing
over all j, of β?
susceptible commuters from city 1 in city j. This gives a total
rate of β?
?
j?=1f1jIj/M1.
j?=1(1 − fj)Ijfj1S1/Mj.
(iv) Infectivecommutersfromcityk (includingcity1)infect
j?=1fj1S1
?
k?=jfjkIk/Mj.




FIG. 2. Individuals commute between n cities, illustrated for a
particular network when n = 4.
Since the transition rates for recovery and birth/death are
simple extensions of those for two cities we can now write
the transition rates for the n-city model as:
(a) Recovery of an infective individual (and creation of a
recovered individual):
Tj≡ T(S1,I1,...,Sj,Ij− 1,...,Sn,In|σ) = γIj,
(b) Death of an infected individual (and birth of a suscepti-
ble individual):
(13)
Tn+j≡ T(S1,I1,...,Sj+ 1,Ij− 1,...,Sn,In|σ) = μIj,
(14)
(c) Death of a recovered individual (and birth of a suscep-
tible individual):
T2n+j≡ T(S1,I1,...,Sj+ 1,Ij...,Sn,In|σ)
= μ(Nj− Sj− Ij),
(d) Infection of a susceptible individual:
(15)
T3n+j≡ T(S1,I1,...,Sj− 1,Ij+ 1...,Sn,In|σ)
n
?
where σ ≡ {S1,I1,...,Sj,Ij...,Sn,In} and where j =
1,...,n. The coefficients cjkin Eq. (16) may be read off from
theterms(i)–(iv),buttheyaresufficientlycomplicatedtowrite
in full that we only list them in Appendix B. In that appendix
wealsoshowthatrelationsbetweenthecjk,analogoustothose
given in Eq. (6) for the two-city case hold, and are given by
?Nj
= β
k=1
cjkSjIk
Nk
,
(16)
cjj+
?
k?=j
cjk= 1;
ckj=
Nk
?
cjk;
j,k = 1,...,n.
(17)
So in the n-city model, there are n(n − 1)/2 independent
coupling parameters cjkand (n − 1) parameters for city sizes
in additional to the usual epidemiological parameters. Note
that if all city sizes are equal the second relation in Eq. (17)
reduces to ckj= cjk. This symmetry is used in the subsequent
analysis.
Following the same path as in Sec. II, having specified
the model by giving the transition rates, we move on to the
dynamics.TheprocessisMarkovianandsosatisfiesthemaster
equation (7) except now the sum on a goes from 1 to 4n. As
detailed in Appendix A, invoking the van Kampen ansatz (8)
gives the following deterministic equations to leading order:
dsj
dt
= −βsj
n
?
n
?
k=1
cjkik+ μ(1 − sj),
(18)
dij
dt
= βsj
k=1
cjkik− (γ + μ)ij,
where j = 1,...,n. At next-to-leading order the fluctua-
tions are found to satisfy the linear stochastic differential
equation (10), but with J,K = 1,...,2n. The two matrices
A and B are given explicitly in Appendix A. They are
independentoftime,sincetheyareevaluatedatthefixedpoints
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PHYSICAL REVIEW E 84, 051919 (2011)
of the dynamics (18). For the restof this section we investigate
the fixed point structure of these equations.
B. The fixed points of the deterministic equations
The fixed points of the deterministic equations (18) are
denoted by asterisks. Adding the two sets of equations we
immediately see that
(γ + μ)i∗
j= μ(1 − s∗
j),j = 1,...,n.
j,andalsousingEq.(17),
(19)
Usingthisequationtoeliminatethei∗
one finds that
?
s∗
j
(β + γ + μ) − β
n
?
k=1
cjks∗
k
?
=(γ + μ),j =1,...,n.
(20)
Two fixed points can be found by inspection. First, suppose
one of the i∗
s∗
0.Sincethecoefficientscjkarenon-negative(seeAppendixB),
then i∗
zeroasinputintoEq.(18),inthesamewayaswedidoriginally
fori∗
cjk, then they will have no infected individuals present. From
Eq.(19)itfollowsthats∗
solution where no infection is present anywhere in this cluster
ofconnected cities.We assumethatallthecitiesareconnected
either directly or indirectly, so that i∗
Ofmoreinterestiswhatwecall“thesymmetricfixedpoint.”
This has s∗
that the i∗
i∗. Using Eq. (17), s∗and i∗are found to satisfy the equations
jis zero, for instance, i∗
?= 1. From Eq. (18) we see immediately that?n
k= 0 for all k as long as c?k?= 0. Using the i∗
?= 0. Then from Eq. (19)
k=1c?ki∗
k=
kwhich are
?,weseethataslongasthecitiesareconnectedbynonzero
k= 1forthesecities.Thisisthetrivial
k= 0,s∗
k= 1 for all k.
k= s∗, a constant, for all k. From Eq. (19) one sees
kare also independent of k, and we denote them by
s∗[(β + γ + μ) − βs∗] = (γ + μ),
(γ + μ)i∗= μ(1 − s∗),
which are the fixed point equations for the ordinary “one-city”
SIR model [7,8]. As is well known, these may be solved to
give for the nontrivial fixed point
i∗=μ[β − (γ + μ)]
Due to a remarkable theorem, we can assert that the
symmetric solution given by Eq. (22) is the only nontrivial
fixed point of the deterministic equations (18) [33]. This is
proved by finding a Liapunov function for the n-city SIR
model. In fact the result is more general than we require and
was proved for the SEIR model; in Appendix B we give the
explicit form of the Liapunov function for the SIR model
and a brief outline of the proof following the argument in
Ref. [33] for this simpler case. The theorem also tells us that
the nontrivial fixed point (22) is globally stable. Therefore, we
can now go on to study stochastic fluctuations about this well
characterized attractor.
(21)
s∗=γ + μ
β
,
β(γ + μ)
.
(22)
IV. SPECTRUM OF THE STOCHASTIC FLUCTUATIONS
Based on previous studies of stochastic fluctuations in the
SIR model in different contexts, we would expect that the
fixed point behavior predicted in the deterministic limit is
replaced by large stochastic oscillations [18,19]. In effect, the
noise due to the randomness of the processes in the IBM
sustains the natural tendency for cycles to occur and amplifies
them through a resonance effect. Since the oscillations are
stochastic, straightforward averaging will simply wipe out the
cyclicstructure;tounderstandthenatureofthefluctuationswe
needtoFouriertransformthemandthenpickoutthedominant
frequencies.
So we begin by taking the Fourier transform of the linear
stochastic differential equation Eq. (10) (generalized to the
case of n cities) to find
2n
?
where the ˜ f denotes the Fourier transform of the function
f. Defining the matrix −iωδJK− AJK to be ?JK(ω), the
solution to Eq. (23) is
K=1
(−iωδJK−AJK)˜ zK(ω)= ˜ ηJ(ω),J =1,...,2n,
(23)
˜ zJ(ω) =
2n
?
K=1
?−1
JK(ω)˜ ηK(ω).
(24)
The power spectrum for fluctuations carrying the index J
is defined by
PJ(ω)≡?|˜ zJ(ω)|2?=
2n
?
K=1
2n
?
L=1
?−1
JK(ω)BKL(?†)−1
LJ(ω).
(25)
Since? = −iωI − A,whereI isthe2n × 2nunitmatrix,and
since A and B are independent of ω, the structure of PJ(ω)
is that of a polynomial of degree 4n − 2 divided by another
polynomialofdegree4n.Theexplicitformofthedenominator
is |det?(ω)|2.
Oscillations with well-defined frequencies should show up
as peaks in the power spectrum. The structure of the power
spectrum described above—with the ratio of polynomials
of high order potentially giving rise to many maxima—
might lead us to suppose that the spectrum of fluctuations
would have a rather complex structure. In fact, numerical
simulations indicate that only a single peak is present for
a large range of parameter values. An example is shown in
Fig. 3, where typical values for measles [7,10,34] were chosen
0.2 0.4 0.6 0.8 1.0ν
0.00
0.04
0.08
0.12
P4
FIG. 3. (Color online) Power spectrum for the fluctuations of
infectivesfromsimulationofathree-citymodelwithequalpopulation
sizes plotted as a function of the frequency ν = ω/(2π) 1/y. The
spectrum shown corresponds to city 1; the spectra for the other
cities are very similar. Metapopulation model parameters: N1=
N2= N3= 106, c12= 0.06, and c13= c23= 0.02. Epidemiological
parameters: γ = 365/13 1/y, μ = 1/50 1/y, and β = 17(γ + μ).
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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)
fortheepidemiologicalparameters(wekeepthesevaluesfixed
throughout this section).
To understand how this comes about, we first note that the
number of peaks in the power spectrum is given by the form
of the denominator, |det?(ω)|2; the numerator essentially just
shiftsthe positionof these peaks somewhat. Therefore, we can
understand the number and nature of the peaks by studying
the eigenvalues of ?JK, which are those of the matrix AJK
shifted by −iω.
Each pair of complex conjugate eigenvalues of AJK, λc,λ∗
gives rise to a factor in |det?(ω)|2of the form (|λc|2− ω2)2+
[2Re(λc)ω]2, and each real eigenvalue of AJK, λr, yields a
factor of the form (λ2
associatedwithcomplexeigenvaluesλcofAJKwithsmallreal
parts,andtheirpositionisapproximatelygivenbyω ≈ Im(λc).
Inthetrivialcaseofonecity,n = 1,AJKhasapairofcomplex
conjugate eigenvalues λ±
and|λ±
for a pronounced peak for ω close to Im(λ±
fulfilled because μ, the death-birth rate, is small. This carries
over to the general n-city case since, as shown in Appendix
B, λ±
parametervaluesofFig.3thenumericalvaluesofthecommon
eigenvalue pair are λ±
to be located close to ν = Im(λ±
For large demographic coupling, the n-city system behaves
as a well mixed system comprising all the cities and we expect
to find in that limit a power spectrum similar to the case
n = 1, where each city contributes proportionally to its size
to the overall spectral density. In the opposite limit, the n
cities uncouple and we find for each city the power spectrum
of the one-city case. In order to understand why additional
peaks do not show up in simulations for intermediate coupling
strengths, it is useful to consider the case n = 2, for which
the eigenvalues of AJK can be determined analytically and
depend on a single coupling parameter c12and the ratio of the
population sizes q = N2/N1[see Eq. (6) and Appendix B].
An Argand diagram of the two pairs of eigenvalues, λ±
λ±
seen that as the coupling increases, λ±
c,
r+ ω2)2. Peaks in the power spectrum are
1with Re(λ±
1) = −βμ/[2(γ + μ)]
1| =√μ(β − γ − μ)(seeAppendixB).Theconditions
1) ≈ |λ±
1| are
1always belong to the set of eigenvalues of AJK. For the
1= −0.17 ± i 2.99, so we expect a peak
1)/(2π) ≈ 0.48 1/y.
1and
2, for the two-city model is shown in Fig. 4. It can be
2follow the circle C
642 2Re λ
4
2
2
4
Im λ
FIG. 4. (Color online) An Argand diagram of the eigenvalues
for the two-city model with q = N2/N1= 3/2 and c12∈ [0,1]. The
large black dots are the common eigenvalue pair λ±
smaller dark gray (blue) and light gray (green) dots are the remaining
eigenvalues λ±
interval. The eigenvalues with Reλ−
they are found for c12> 0.15. The asterisks show the eigenvalues for
the parameter values used in Fig. 5.
1. The sets of
2computed on a uniform grid of values of c12in the
2< −6 are not shown in the plot;
centered at zero that goes through λ±
the imaginary axis. Real and imaginary parts become of the
same order for very small values of the coupling, and so we
expectthepowerspectrumtobealwaysdominatedbythepeak
associated with λ±
uncoupled case. This behavior carries over to the n-city case
with symmetric coupling, for which a complete analysis of
the eigenvalues of AJKcan also be given (see Appendix B).
In particular, it can be shown that, apart from the common
eigenvalue pair λ±
additional eigenvalue pair that behaves as a function of the
coupling parameter as described above for n = 2.
For the coupling parameter that corresponds to the values
of λ±
choice of population sizes, the infective fluctuations power
spectra for the two-city model obtained from simulations and
from Eq. (25) are shown in Fig. 5. We find a nearly perfect
match between the results of numerical simulations and the
analytical calculations. In agreement with the above argument
the power spectra of city 1 and city 2 are very similar to the
power spectrum of the one-city case, which in turn is very
similar to the spectrum shown in Fig. 3 for three cities with
small coupling. In all cases the functional form of the spec-
tral density is dominated by the peak associated with the
common eigenvalue pair λ±
power spectra PJ(ν), their ratio with respect to the one-city
case, rJ(ν), decreases as the coupling increases. For two
cities and q = 1, the power spectra P3and P4of city 1 and
city 2 are equal and the relative peak amplitudes r3,4(νmax)
decrease with the coupling strength c12down to 0.5. For other
values of q, as in Fig. 5, the different peak amplitudes in two
cities reflect the symmetry P3(ν;c12,q) = P4(ν;c12/q,1/q).
Depending on q, the ratio r3,4(ν) may become even smaller
than 0.5, but due to the symmetry that relates P3and P4, the
amplitude of at least one of these peaks is always comparable
to that of the uncoupled case. More precisely, it is easy
to check that 1 ? r3(ν;c12,q) + r4(ν;c12,q) ? 2, where the
second inequality is satisfied strictly for c12= 0 and the lower
bound corresponds to the large coupling limit c12= 1 and to
ν = νmax.
1, moving away from
1that characterizes the spectrum in the
1, AJKhas a single (n − 1)-fold degenerate
2marked with asterisks in Fig. 4 and for a certain
1. As for the amplitudes of the
0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
P3
0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
0.15
0.20
P4
FIG. 5. (Color online) Power spectra for the fluctuations of
infectives from simulation of the two-city model [(red) dots] and
analytic calculation (black solid curve) plotted as a function of the
frequency ν = ω/(2π) 1/y. The population sizes were chosen to be
N1= 106and N2= 1.5 × 106so that their ratio is 3/2. The coupling
coefficient c12= 0.1. The location of the eigenvalues for this choice
of parameters is indicated in Fig. 4 by asterisks and large dots.
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PHYSICAL REVIEW E 84, 051919 (2011)
The general case of three cities with no symmetry can
also, in principle, be treated analytically because finding the
eigenvalues of AJK reduces to finding the roots of a fourth
order polynomial. However, the problem now depends on
three independent coupling parameters and two parameters
for city sizes and closed form expressions are too lengthy
to be useful. An approximate, concise description of the
behavior of the eigenvalues of AJK can be given in terms
of only two parameters that measure coupling strength and
coupling asymmetry (see Appendix B). In this approximation,
we assume that all the cjk, j ?= k, are of order√μ and treat
μ as the small parameter of the system. Simple expressions
for the real parts and the absolute values of the additional
eigenvalue pairs λ±
derived [see Eqs. (B22) and (B24)]. These show that, in this
approximation, both eigenvalue pairs behave as described for
thesymmetriccase.Asthecouplingincreases,botheigenvalue
pairs follow the circle C centered at zero that goes through
λ±
imaginary parts become of the same order within the scope of
the approximation. Equation (B22) also shows how the asym-
metry lifts the degeneracy of the two pairs λ±
coupling increases, the two eigenvalue pairs move along the
circle C at different speeds. We have checked that Eqs. (B22)
and (B24) give a good approximation to the exact results in
the regime when the eigenvalues are complex.
The same behavior is illustrated in Fig. 6, where a plot of
the exact solutions for λ±
correspond to taking those of Fig. 3 and allowing one of the
coupling coefficients to span the whole admissible range. One
of the eigenvalues is shown only up to c12= 0.12, where its
real part becomes smaller than −6.
In Fig. 7 we show numerical results for the behavior of the
eigenvaluesofAJKinthecaseoffourcitieswithdifferentpop-
ulation sizes and a certain choice of the coupling coefficients
cjk, j,k = 1,2,3,4. We make use of the following notation
for the diagonal and off-diagonal coupling coefficients (see
Appendix B): cjj= 1 − ˆ cjj x√μ and cjk= ˆ cjk x√μ,
respectively. We then calculate the set of three nontrivial
eigenvalue pairs as the coupling strength x varies in a suitable
2, λ±
3of AJKup to terms of order μ can be
1, moving away from the imaginary axis. The real and
2, λ±
3. As the
2,3is shown for parameter values that
642 2Re λ
4
2
2
4
Im λ
642 2Re λ
4
2
2
4
Im λ
FIG. 6. (Color online) An Argand diagram of the eigenvalues for
a three-city model with equal population sizes, c12∈ [0,0.98] and the
other parameters as in Fig. 3. The large black dots are the common
eigenvalue pair λ±
gray (green) dots are the remaining eigenvalues λ±
λ±
interval. The eigenvalues with Reλ−
they are found for c12> 0.12.
1. The sets of smaller dark gray (blue) and light
2(left panel) and
3(right panel) computed on a uniform grid of values of c12in the
2< −6 are not shown in the plot,
6422Re λ
4
2
2
4
Im λ
FIG. 7. An Argand diagram of the eigenvalues for a four-city
model with the coupling strength x ∈ [0,0.52]. The large black dots
are the common eigenvalue pair λ±
λ±
are shown as sets of smaller gray dots. As in the previous
figures we only show eigenvalues whose real part is larger than
−6. Metapopulation model parameters: N2/N1:N3/N1:N4/N1=
2:3:4, ˆ c12= 1/√μ = 2ˆ c13= 5ˆ c14/2 = 5ˆ c23/2 = 3ˆ c24= 4ˆ c34.
1. The remaining eigenvalues
2,3,4computed on a uniform grid of values of x in the interval
interval, keeping the ˆ cjkfixed. These results suggest that the
behavior of the eigenvalues of AJK is essentially given by
the description of the symmetric case and that more general
couplings break the degeneracy as in the case n = 3, with no
effects in the contributions to the peaks in the power spectrum.
V. DISCUSSION AND CONCLUSIONS
In this paper we have extended the analysis of a metapop-
ulation model of epidemics into the stochastic domain.
Frequently, epidemic models involving a spatial component,
such as the interaction between several cities, are studied
purely deterministically [28,29] or through computer sim-
ulations [6,13,27]. We have demonstrated how a stochastic
metapopulation model can be studied analytically by using a
relativelystraightforwardextensionofthemethodologywhich
was used to study a well-mixed population in a single city.
We adopted a simple specification of residents and com-
muters in order to set up the model. However, the coefficients
which appear in the dynamical equations are generic and
would appear in the same form if residents and commuters
were included in a different way. It is evident that there are
many ways of characterizing the interchange of individuals
between cities which will result in the same model; only the
identification of the coefficients with the underlying structure
will be different.
Thedeterministicformofthemodelpredictsthatthesystem
willreachastablefixedpointwheretheproportionofinfected,
susceptible,andrecoveredindividualsisthesameineverycity.
Thestochasticversionofthemodelalsopredictsacleansimple
result: that the large sustained oscillations which replace the
deterministic predictions of constant behavior have a single
frequency which is the same for every city. Moreover, for
small, large, and intermediate coupling between the cities,
the form of the power spectrum of these fluctuations is closely
approximatedbythepowerspectrumofthesingle-citysystem.
It is remarkable that such a simple result occurs in what is a
quite complicated stochastic nonlinear metapopulation model.
We hope to explore the range of validity of this result and
its robustness to the addition of new features to the model in
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G. ROZHNOVA, A. NUNES, AND A. J. MCKANE PHYSICAL REVIEW E 84, 051919 (2011)
the future. In any case, we believe that the work presented
here gives a firm foundation to possible future work, including
comparisons with the data available on childhood diseases.
ACKNOWLEDGMENTS
Financial
for Science and Technology (FCT) under Contract No.
POCTI/ISFL/2/261isgratefullyacknowledged.G.R.wasalso
supported by FCT under Grant No. SFRH/BPD/69137/2010.
support fromthePortugueseFoundation
APPENDIX A: SYSTEM-SIZE EXPANSION
Here we give some of the key steps in the application of
the system-size expansion to the model explored in this paper.
The method has been extensively discussed in the literature
[6,15,19–25], and so we confine ourselves to a brief outline
andtodisplayingthemostimportantintermediateresultsinthe
derivation. We assume that we are carrying out the calculation
for the n-city case discussed in Sec. III; the corresponding
results for Sec. II can be obtained simply by setting n = 2.
The first point to mention is that there are apparently n
expansion parameters: {N1,...,Nn}. The method is valid if
they are all large and of the same order. More formally we
can take, for instance, N1≡ N as the expansion parameter
and express all the other Njin terms of it: Nj= Nqj, where
the qj= Nj/N, j = 2,...,n are of order one. In practice,
the method seems to work well when the qjare significantly
different from one, but this has to be checked a posteriori,
for instance, by comparing the analytic results with those
obtained using computer simulations. In what follows we do
not introduce the qjexplicitly; we simply take all the Nj’s to
be of the same order in the expansion.
The van Kampen ansatz Eq. (8) replaces the discrete
stochastic variables σ by the continuous stochastic variables
z and so we write the transformed probability distribution
P(σ,t) as ?(z,t). Since this transformation is time-dependent,
substituting the ansatz into dP/dt on the left-hand side of
Eq. (7) gives [15]
dP(σ,t)
dt
=∂?(z,t)
∂t
−
n
?
∂?(z,t)
j=1
?Nj
∂?(z,t)
∂xj
dsj
dt
−
n
?
j=1
?Nj
∂yj
dij
dt.
(A1)
The right-hand side of the master equation (7) can be put
into a form from which it is simple to apply the expansion
procedure. To do this one introduces step operators [15]
defined by
?±1
Sjf(S1,...,Sj,...,Sn,I1,...,In)
= f(S1,...,Sj± 1,...,Sn,I1,...,In),
?±1
= f(S1,...,Sn,I1,...,Ij± 1,...,In),
for a general function f and where j = 1,...,n. Using
these operators the master equation (7) may be written
(A2)
Ijf(S1,...,Sn,I1,...,Ij,...,In)
as
dP(σ,t)
dt
=
n
?
j=1
???Ij− 1?Tj+
?1
??Ij
?Sj
??Sj
− 1
?
Tn+j
?
+
?Sj
− 1
?
T2n+j+
?Ij
− 1
T3n+j
?
P(σ,t).
(A3)
Within the system-size expansion these operators have a
simple structure,
?Sj=
∞
?
p=0
N−p/2
j
p!
∂p
∂xp
j
,?Ij=
∞
?
p=0
N−p/2
j
p!
∂p
∂yp
j
,
(A4)
and so all the terms of the right-hand side of Eq. (A3) may
be straightforwardly expanded. Comparing these with the left-
hand side in Eq. (A1) the leading order (∼?Nj) yields the
order (which is of order one) gives a Fokker-Planck equation:
deterministic equations given by Eq. (18). The next-to-leading
∂?
∂t
= −
2n
?
J,K=1
∂
∂zJ
[AJKzK?] +1
2
2n
?
J,K=1
BJK
∂2?
∂zJ∂zK.
(A5)
The 2n × 2n matrices A and B which appear in this equation
have the following form. Writing A in blocks of four n × n
submatrices,
?A(1)A(2)
the elements of these submatrices are
A =
A(3)A(4)
?
,
(A6)
A(1)
jk= −μδjk− βδjk
n
?
?=1
cj?i?,
A(2)
jk= −β
?Nj
?
Nk
?1/2
sjcjk,
(A7)
A(3)
jk= βδjk
n
?=1
cj?i?,
A(4)
jk= −(μ + γ)δjk+ β
?Nj
Nk
?1/2
sjcjk.
Writing B in a similar way to A in Eq. (A6), the elements of
the submatrices are
B(1)
jk= μδjk(1 − sj) + βδjk
n
?
?=1
sjcj?i?,
B(2)
jk= B(3)
jk= −μδjkij− βδjk
n
?
?=1
sjcj?i?,
(A8)
B(4)
jk= (γ + μ)δjkij+ βδjk
n
?
?=1
sjcj?i?.
From Eqs. (A7) and (A8) it is clear that the matrices Ajkand
Bjk depend on the solutions of the deterministic equations
given in Eq. (18). However, since we are interested only
in fluctuations about the stationary state, these matrices are
evaluated at the fixed point. Since the unique stable fixed point
is the symmetric one, the same for all cities, the entries (A7)
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PHYSICAL REVIEW E 84, 051919 (2011)
and (A8) are given by
A∗(1)
jk= −[μ + βi∗]δjk,A∗(2)
jk= −β
?Nj
?Nj
Nk
?1/2
s∗cjk,
A∗(3)
jk= βi∗δjk,A∗(4)
jk= β
Nk
?1/2
s∗cjk− (μ + γ)δjk,
(A9)
and
B∗(1)
jk
B∗(2)
jk
= 2μ(1 − s∗)δjk,
= B∗(3)
B∗(4)
jk
= 2(γ + μ)i∗δjk,
(A10)
jk
= −i∗[μ + βs∗]δjk,
where we have used the fixed-point equation (21) to simplify
some of the entries in Eq. (A10).
Finally, the Fokker-Planck equation (A5) is equivalent to
the stochastic differential equation (10). We work with the
latter, since we wish to use Fourier analysis to analyze the
nature of the fluctuations, and since Eq. (10) is linear, it can
easilybeFouriertransformed,asdiscussedindetailinSec.IV.
APPENDIX B: SOME RESULTS FOR THE n-CITY CASE
In this Appendix we give some of the derivations for the
n-city case discussed in Secs. III and IV which are too long
and cumbersome to be given in the main text.
1. The coefficients cjk
The coefficients cjkappearing in Eq. (16) may be read off
from the four types of term (i)–(iv) given in Sec. III:
(1 − fj)2Nj
?(1 − fj)Nj+?
+
??=j
for j = 1,...,n and
(1 − fj)fjkNk
?(1 − fj)Nj+?
+
?
for j,k = 1,...,n and j ?= k.
To prove the first relation given in Eq. (17), consider the
sum cjj+?
Eq. (B1) combines with the last term in Eq. (B2) to give
?f?jNj+?
??
??
cjj=
m?=jfjmNm
f2
?jNj
?
?
?(1 − f?)N?+?
m?=?f?mNm
?
(B1)
cjk=
m?=jfjmNm
?
fkj(1 − fk)Nk
?(1 − fk)Nk+?
?(1 − f?)N?+?
m?=kfkmNm
f?jf?kNk
?
+
??=j,k
m?=?f?mNm
?
(B2)
k?=jcjk. The first term in Eq. (B1) combines with
the first term in Eq. (B2) to give (1 − fj). The last term in
?
??=j
f?j
?(1 − f?)N?+?
=
??=j
?
k?=j,?f?kNk
m?=?f?mNm
?
?
?
f?j
k?=?f?kNk
m?=?f?mNm
?
?(1 − f?)N?+?
fkj
?(1 − fk)Nk+?
?
?,
=
k?=j
m?=kfkmNm
m?=kfkmNm
?
(B3)
where in the last line we have performed a relabeling.
Combining the middle term of Eq. (B2) with the result in
Eq. (B3) gives
?(1 − fk)Nk+?
?
k?=j
fkj
m?=kfkmNm
m?=kfkmNm
?
?(1 − fk)Nk+?
?
= fj,
(B4)
using Eq. (11). Adding this to the result (1 − fj) found earlier
proves the result cjj+?
the interchange of j and k. Therefore,
k?=jcjk= 1.
We also note from Eq. (B2) that cjk/Nkis symmetric under
cjk
Nk
=ckj
Nj,
(B5)
which is the second relation in Eq. (17).
2. Uniqueness and stability of the fixed point
In Sec. III we asserted that the deterministic equations
(18) have a unique nontrivial fixed point, which was glob-
ally stable. Here we prove this by giving a Liapunov
function forthe dynamical
ant region R = {(s1,...,sn,i1,...,in) : 0 ? sj? 1,0 ? ij?
1,sj+ ij? 1,j = 1,...,n},wherethesystemisdefined.This
isamodificationofthefunctiongiveninRef.[33]fortheSEIR
model. The proof assumes that the matrix of the coupling
coefficients cjk is irreducible, which means that any two
cities have a direct or indirect interaction. Otherwise, the
proof breaks down because the n cities may be split into
noninteracting subsets, and several equilibria may be found by
combining disease extinction in some subsets with nontrivial
equilibrium in other subsets.
Let βjk≡ βcjks∗
point of Eq. (18), and denote by M the matrix defined by
Mkj= βjk,j ?= k, and?n
equation Mv = 0 is spanned by a single vector (v1,...,vn),
vj> 0,j = 1,...,n. Let L(s1,...,sn,i1,...,in) be defined as
?
L has a global minimum in R at the fixed point. Functions of
thisformhavebeenusedintheliteratureasLiapunovfunctions
for fixed points of ecological and epidemiological models,
whose variables take only positive values [33]. Differentiating
L along the solutions of Eq. (18), and following the proof of
Theorem 1.1 in Ref. [33], we obtain
system in theinvari-
ji∗
k, where (s∗
1,...,s∗
n,i∗
1,...,i∗
n) is a fixed
k=1Mkj= 0,j = 1,...,n. It can be
shown ( [33], Lemma 2.1) that the solution space of the linear
L =
n
j=1
vj(sj− s∗
jlnsj+ ij− i∗
jlnij).
˙L ?
n
?
j,k=1
vjMkj
?
2 −
s∗
j
sj
−sj
s∗
j
ik
i∗
k
i∗
j
ij
?
.
(B6)
The properties of the coefficients vj in the definition of L
play a crucial role in the derivation of the second term in this
inequality. Use has been made of the identity
n
?
j=1
vj
n
?
k=1
βcjks∗
jik=
n
?
j=1
vj(γ + μ)ij,
(B7)
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which, in turn, uses the fact that Mv = 0 can be written as
n
?
Following Ref. [33], it can then be shown that the right-
hand side of Eq. (B6) is strictly negative except at
(s∗
for this fixed point in R, and the fixed point is unique and
globallystable.Notethattheresultalsoholdswhenthedisease
transmissibility β, the recovery rate γ, and the birth-death rate
μ are different in different cities, in which case the nontrivial
equilibrium is, in general, not symmetric.
j=1
βckjs∗
ki∗
jvk=
n
?
j=1
βcjks∗
ji∗
kvj,k = 1,...,n.
(B8)
1,...,s∗
n,i∗
1,...,i∗
n). Therefore, L is a Liapunov function
3. Nature of the eigenvalues of the matrix A
In this section we give some results on the eigenvalues of
A which are required for the discussion in Sec. IV.
We first recall that A is closely related to the stability
matrix of the deterministic equations (18). In fact, in most
applications of the system-size expansion they are equal. In
our case because we have n expansion parameters√Nj, they
are not equal, but closely related. A simple calculation of the
Jacobian, J, from Eq. (18), shows that
S = diag(√N1,...,√Nn).
The effect of the transformation is simply that one can obtain
J from A by omitting the terms (Nj/Nk)1/2in A(2)
in Eq. (A7) or in A∗(2)
since it follows from the similarity transformation (B9) that
the eigenvalues of A are also the eigenvalues of J. So we may
study the simpler problem of finding the eigenvalues of the
Jacobian at the symmetric fixed point (22).
For orientation, let us explicitly calculate the characteristic
polynomial of the Jacobian for the cases of one city and two
cities. These are
R1(λ) = Q−1(d2λ2+ d1λ + d0),
where
J = S−1AS,
where (B9)
jkand A(4)
jk
jkand A∗(4)
jkin Eq. (A9). This is useful,
n = 1 :
(B10)
Q = γ + μ,
d0= μ(γ + μ)[β − (γ + μ)],
d2= γ + μ,d1= βμ,
(B11)
and
n = 2 :
Thus,
R2(λ) = Q−2(d2λ2+ d1λ + d0)(g2λ2+ g1λ + g0).
R2(λ) = R1(λ)Q−1(g2λ2+ g1λ + g0),
(B12)
where
g2= γ + μ,
g0= μ(γ + μ)[β − (1 − c12− c21)(γ + μ)].
WeseethatthefactorR1(λ)iscommon,whichsuggeststhatthe
pair of eigenvalues found in the one-city case might always be
present in the n-city case. This is easily proved by considering
the vector v = (v1,...,vn,vn+1,...,v2n)Twith components
satisfying vi= v and vi+n= v?for i = 1,...,n. Then the
eigenvector equation J∗v = λv reduces to that for one city
as required.
g1= βμ + (c12+ c21)(γ + μ)2,
(B13)
A similar method can be used to find the characteristic
polynomialforn ? 3citieswithequalpopulationsizes,where
the couplings are equal, that is,
cjk=
?1 − (n − 1)c,j = k,
j ?= k, c,
(B14)
where j,k = 1,...,n. We now take the components of the
vector to be v1= −v2= v, vn+1= −vn+2= v?, and vi=
vi+n= 0fori = 3,...,n.TheeigenvectorequationJ∗v = λv
now reduces to
?
?
Therefore, both solutions of
−
βμ
γ + μ+ λ
βμ
γ + μ− μ
?
?
v − (1 − nc)(γ + μ)v?= 0,
(B15)
v − [nc(γ + μ) + λ]v?= 0.
Q−1(h2λ2+ h1nλ + h0n) = 0,
(B16)
where
h2= γ + μ, h1n= βμ + nc(γ + μ)2,
h0n= μ(γ + μ)[β − (1 − nc)(γ + μ)],
are eigenvalues of J∗. This procedure can be repeated for
n − 1independentvectorswithonlyfournonzerocomponents
and the same symmetry as v. Therefore, the characteristic
polynomial of J∗, Rn(λ), factorizes as
(B17)
Rn(λ) = R1(λ)[Q−1(h2λ2+ h1nλ + h0n)]n−1.
(B18)
Finally, let us consider three cities with arbitrary coupling
and study the eigenvalues of J∗in the limit when the off-
diagonal coefficients cjkare small and of the same order. It
becomes clear that the coupling range to explore corresponds
to cjkof the order of√μ and it is convenient to introduce the
notation
?1 − ˆ cjjx√μ,
cjk(x) =
j = k,
j ?= k,
positive
ˆ cjkx√μ,
(B19)
where
Equation (B19) represents, for each choice of ˆ cjk, a family
of systems with all the off-diagonal coefficients cjk of the
same order, that reaches the zero coupling limit for x = 0.
The quantity x measures the distance to zero coupling along
each particular family, scaled by√μ. Taking into account
the properties of the matrix cjk, given by Eq. (17), the
characteristic polynomial of J∗is a polynomial of degree six
that can be expressed in terms of this distance x√μ and
of three other independent parameters. We choose these to
be ˆ cjj,j = 1,2,3. We know that this characteristic polynomial
factorizesasR1(λ)(λ4+ p3λ3+ p2λ2+ p1λ + p0),wherep3,
p2,p1,andp0aresomecoefficients.TherootsofR1(λ)arethe
pairofeigenvaluesλ±
of this family of systems. The polynomial of degree four can
j,k = 1,2,3 and
x
isa parameter.
1sharedbyallthecharacteristicequations
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STOCHASTIC OSCILLATIONS IN MODELS OF ...
PHYSICAL REVIEW E 84, 051919 (2011)
be easily found by direct computation. For equal city sizes we
obtain for the coefficients
p3= γσ x√μ + O(μ),
p2= 2(β − γ) μ − 3/4 γ2ˆ p2x2μ + O(μ3/2),
p1= γ(β − γ)σ x μ3/2+ O(μ2),
p0= (β − γ)2μ2+ O(μ5/2),
where
(B20)
σ = ˆ c11+ ˆ c22+ ˆ c33,
ˆ p2= ˆ c2
Keepingonlytheleadingordertermsineachofthecoefficients
given by Eq. (B20) we find a simple approximate expressions
for the two additional eigenvalue pairs λ±
we find
2) = −γ
Re(λ±
(B21)
11+ ˆ c2
22+ ˆ c2
33− 2(ˆ c11ˆ c22+ ˆ c11ˆ c33+ ˆ c22ˆ c33).
2, λ±
3. In particular,
Re(λ±
4(σ + k) x√μ + O(μ),
3) = −γ
(B22)
4(σ − k) x√μ + O(μ),
where
k2= 4?ˆ c2
Assuming without loss of generality that ˆ c33? ˆ c22? ˆ c11, k2
is positive and so k is real. Note that k = 0 in the symmetric
case, and in that case (B22) coincides in the same order
of approximation with the roots of Eq. (B16) for n = 3.
The quantities σ x√μ and k x√μ that determine, in this
approximation, the real parts of the two nontrivial eigenvalue
11+ ˆ c2
22+ ˆ c2
33− ˆ c11ˆ c22− ˆ c11ˆ c33− ˆ c22ˆ c33
?.
(B23)
pairs can be interpreted as the overall coupling strength and
the coupling asymmetry for a system of family (B19). We also
find for the absolute value of the eigenvalues
?β − γ√μ + O(μ),
which shows that, for all families of the form Eq. (B19), the
eigenvaluesλ±
plane centered at zero that goes through λ±
city sizes, the same calculation can be carried out to find that
Eqs. (B22) and (B24) still hold, with Eq. (B23) replaced by
k2= σ2+1 + q21+ q31
where qjk= Nj/Nkand
˜k2= ˆ c2
|λ±
2,3| =
(B24)
2,3ofJ∗moveclosetothecircleC inthecomplex
1. For arbitrary
q21q31
˜k2,
(B25)
11+ (ˆ c22q21− ˆ c33q31)2− 2ˆ c11(ˆ c22q21+ ˆ c33q31).
(B26)
The behavior of the two nontrivial eigenvalue pairs along a
family (B19) can be described, in this approximation, in terms
of the two parameters σ and k that characterize the family
and of the scaled distance x. As x increases away from zero,
both eigenvalue pairs move along C with speeds whose ratio is
given by (σ + k)/(σ − k). The parameter k that measures the
asymmetry of the coupling causes the splittingof the two pairs
with respect to the degenerate, symmetric case. The first pair
to reach the real axis does so for x = 4√β − γ/[γ(σ + k)],
which lies within the scope of the approximation. From then
on the two real eigenvalues keep changing with x in such a
way that the square root of their product verifies the constraint
Eq. (B24) until for large x the approximation breaks down.
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