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arXiv:0810.4683v3 [q-bio.PE] 2 Jun 2009

Fluctuations and oscillations in a simple epidemic model

G. Rozhnova1and A. Nunes1

1Centro de F´ ısica Te´ orica e Computacional and Departamento de F´ ısica,

Faculdade de Ciˆ encias da Universidade de Lisboa, P-1649-003 Lisboa Codex, Portugal

We show that the simplest stochastic epidemiological models with spatial correlations exhibit two

types of oscillatory behaviour in the endemic phase. In a large parameter range, the oscillations

are due to resonant amplification of stochastic fluctuations, a general mechanism first reported for

predator-prey dynamics. In a narrow range of parameters that includes many infectious diseases

which confer long lasting immunity the oscillations persist for infinite populations. This effect is

apparent in simulations of the stochastic process in systems of variable size, and can be understood

from the phase diagram of the deterministic pair approximation equations. The two mechanisms

combined play a central role in explaining the ubiquity of oscillatory behaviour in real data and in

simulation results of epidemic and other related models.

PACS numbers: 87.10.Mn; 87.19.ln; 05.10.Gg

Cycles are a very striking behaviour of prey-predator

systems also seen in a variety of other host-enemy sys-

tems — a case in point is the pattern of recurrent epi-

demics of many endemic infectious diseases [1]. The con-

troversy in the literature over the driving mechanisms

of the pervasive noisy oscillations observed in these sys-

tems has been going on for long [2], because the sim-

plest deterministic models predict damped, instead of

sustained, oscillations. One of the aspects of this contro-

versy is whether these mechanisms are mainly external or

intrinsic, and the effects of seasonal forcing terms [3], [4]

and of higher order non-linear interaction terms [5] have

been explored in the framework of a purely determinis-

tic description of well-mixed, infinite populations. These

more elaborate models exhibit oscillatory steady states

in certain parameter ranges, and have led to successful

modelling when external periodic forcing is of paramount

importance [6], but they fail to explain the widespread

non-seasonal recurrent outbreaks found, for instance, in

childhood infectious diseases [7].

During the last decade, important contributions have

come from studies that highlight the inherently stochastic

nature of population dynamics and the interaction pat-

terns of the population as important endogenous factors

of recurrence or periodicity [8]. A general mechanism of

resonant amplification of demographic stochasticity has

been proposed to describe the cycling behaviour of prey-

predator systems [9] and applied recently to recurrent

epidemics of childhood infectious diseases [10]. The role

of demographic stochasticity modelled as additive Gaus-

sian white noise of arbitrary amplitude in sustaining in-

cidence oscillations had long been acknowledged in the

literature [11]. The novelty in [9] and [10] was that of pro-

viding an analytical description of demographic stochas-

ticity as an internal noise term whose amplitude is de-

termined by the parameters and the size of the system

using a method originally proposed by van Kampen [12].

Our goal is to extend this approach by relaxing the

homogeneous mixing assumption to include an implicit

representation of spatial dependence. We show that the

inclusion of correlations at the level of pairs leads to dif-

ferent quantitative and qualitative behaviours in a re-

gion of parameters that corresponds to infectious diseases

which confer long lasting immunity. Our motivation was

twofold. On one hand, the homogeneous mixing assump-

tion is known to give poor results for lattice or network

structured population [13], [14]. On the other hand, sys-

tematic simulations of infection on small-world networks

have shown that the resonant amplification of stochastic

fluctuations is significantly enhanced in the presence of

spatial correlations [15]. Therefore, apart from stochas-

ticity, the correlations due to the contact structure are

another key ingredient to understand the patterns of re-

current epidemics. One of the main difficulties in in-

cluding this ingredient is that the relevant network of

contacts for disease propagation is not well known [14].

In this paper we shall consider a stochastic Susceptible-

Infective-Recovered-Susceptible (SIRS) epidemic model

that leads to the ordinary pair approximation (PA) equa-

tions of [13] in the thermodynamic limit as the simplest

representation of the spatial correlations on an arbitrary

network of fixed coordination number k. The power spec-

trum of the fluctuations around the steady state can be

computed following the approach of [9] and [12]. The

combined effect of stochasticity and spatial correlations

has been much studied through simulations, but this is

an analytical treatment of a model that includes both

these ingredients.

Consider then a closed population of size N at a given

time t, consisting of n1 individuals of type S, n2 indi-

viduals of type I, and (N − n1− n2) individuals of type

R, modelled as network of fixed coordination number k.

Recovered individuals lose immunity at rate γ, infected

individuals recover at rate δ, and infection of the suscep-

tible node in a susceptible-infected link occurs at rate λ.

Let n3 (respectively, n4 and n5) denote the number of

links between nodes of type S and I (respectively, S and

R and R and I). In the infinite population limit, with

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the assumptions of spatial homogeneity and uncorrelated

pairs, the system is described by the deterministic equa-

tions of the standard or uncorrelated PA as follows [13]:

˙ p1 = γ(1 − p2− p1) − kλp3,

˙ p2 = kλp3− δp2,

˙ p3 = γp5− (λ + δ)p3+(k − 1)λp3

˙ p4 = δp3+ γ(1 − p1− p2− p5− 2p4) −(k − 1)λp3p4

˙ p5 = δ(p2− p3) − (γ + 2δ)p5+(k − 1)λp3p4

(1)

p1

(p1− p4− 2p3) ,

p1

,

p1

.

In the above equations the variables stand for the limit

values of the node and pair densities p1 = n1/N, p2 =

n2/N, p3= n3/(kN), p4= n4/(kN), and p5= n5/(kN)

as N → ∞. As expected, neglecting the pair correlations

and setting p3= p1p2in the first two equations leads to

the classic equations of the randomly mixed or mean-field

approximation (MFA) SIRS model,

˙ p1 = γ(1 − p2− p1) − kλp1p2,

˙ p2 = kλp1p2− δp2.

(2)

The phase diagrams of the two models are plotted in Fig.

1 [solid lines for Eq. (1) and dashed line for Eq. (2), both

with k = 4]. We have set the time scale so that δ = 1.

The critical line separating a susceptible-absorbing phase

from an active phase where a stable steady state exists

with nonzero infective density is given by λMFA

(dashed black line) for the MFA, and by λPA

[solid orange (gray) line] for the PA. In addition, in the

active phase of the PA we find for small values of γ a new

phase boundary [solid blue (black) line] that corresponds

to a Hopf bifurcation and seems to have been missed in

previous studies of this model [13]. This boundary sep-

arates the active phase with constant densities (region

I) from an active phase with oscillatory behavior (region

II). The maximum of the curve is situated at λ ≈ 2.5,

γ ≈ 0.03, which means that the PA model predicts sus-

tained oscillations in the thermodynamic limit when loss

of immunity is much slower than recovery from infection.

According to published data for childhood infections in

the pre-vaccination period [4], taking the average immu-

nity waning time to be of the order of the length of the

elementary education cycles at that time (10 years for

the data points in Fig. 1) many of the estimated param-

eter values for these diseases fall into oscillatory region

II, and the others are in region I close to the boundary.

Different data points for the same disease correspond to

estimates for λ based on different data records. Although

small enough to be missed in a coarse grained numeri-

cal study, the oscillatory phase is large in the admissible

parameter region of an important class of diseases. A

systematic study of the dependence of this oscillatory

c

=

γ+1

3γ+2

1

4

c(γ) =

057.59 10 15 20

0

0.01

0.02

0.03

λ

γ

ΙΙ

Ι

FIG. 1: (Color online) Phase diagram in the (λ,γ) plane for

the MFA and the PA deterministic models and parameter val-

ues for measles (△), chicken pox (◦), rubella (?) and pertussis

(⋄) from data sources for the pre-vaccination period. The blue

stars are the parameter values used in Fig. 3.

phase on the parameter k and of its relevance to under-

stand the behaviour of simulations on networks will be

reported elsewhere [16]. Preliminary results indicate that

the oscillatory phase persists in the range 2 < k ? 6, and

that it gets thinner as k increases. There are indications

that this oscillatory phase is robust also with respect to

variations of the underlying model [17].

Let us now study the combined effect of correlations

and demographic stochasticity in region I by taking N

large but finite. In the stochastic version of the MFA

SIRS model the state of the system is defined by n1and

n2which change according to the transition rates as

Tn1+1,n2= γ (N − n1− n2) ,

Tn1,n2−1= δ n2,

Tn1−1,n2+1=

(3)

kλn1

N

n2,

associated to the processes of immunity waning, recov-

ery and infection. Here Tn1+k1,n2+k2denotes the transi-

tion rate from state (n1,n2) to state (n1+ k1,n2+ k2),

ki ∈ {−1,0,1}, where i = 1,2. As in [9], the power

spectrum of the normalized fluctuations (PSNF) around

the active steady state of system (2) can be computed

approximately from the next-to-leading-order terms of

van Kampen’s system size expansion of the correspond-

ing master equation. Setting n1(t) = Np1(t) +√Nx1(t)

and n2(t) = Np2(t) +√Nx2(t), the equations of motion

for the average densities (2) are given by the leading-

order terms of the expansion. The next-to-leading-order

terms yield a linear Fokker-Planck equation for the prob-

ability distribution function Π(x1(t),x2(t),t). The equiv-

alent Langevin equation for the normalized fluctuations

is ˙ xi(t) =?2

of Eq. (2) at the endemic equilibrium and Li(t) are Gaus-

sian white noise terms whose amplitudes are given by the

expansion. Taking the Fourier transform we obtain for

j=1Jijxj(t)+Li(t), where J is the Jacobian

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the PSNF,

Pi(ω) ≡ ?|˜ xi(ω)|2? =

?

j,k

M−1

ik(ω)BkjM−1

ji(−ω) ,(4)

where Mik(ω) = iωδik − Jik and ?˜Li(ω)˜Lj(ω′)? =

Bijδ(ω + ω′). For k = 4 and δ = 1 this expression be-

comes

PMFA

S

=

B11(J2

(D − ω2)2+ T2ω2,

B11(J2

(D − ω2)2+ T2ω2

12+ ω2)

(5)

PMFA

I

=

11+ J11J21+ J2

21+ ω2)

, (6)

where D and T are the determinant and the trace of J

and B11 = B22 = −2B12 = −2B21 =

susceptible and the infected PSNFs, respectively.

In a stochastic version of the PA SIRS model the

state of the system is defined by the integers ni, where

i = 1,...,5, and recovery, loss of immunity, and infec-

tion induce different transitions according to the pairs

or triplets involved in the process. The simplest set of

transitions and transition rates compatible with Eq. (1)

is

γ(4λ−1)

2λ(γ+1), for the

Tn1+1,n2,n3+k,n4,n5−k=γ

kn5,(7)

Tn1+1,n2,n3,n4−k,n5=γ

kn4,

Tn1+1,n2,n3,n4+k,n5=γ

k(k(N − n1− n2) − n4− n5) ,

Tn1,n2−1,n3−k,n4+k,n5=δ

kn3,

Tn1,n2−1,n3,n4,n5+k=δ

k(kn2− n3− n5) ,

Tn1,n2−1,n3,n4,n5−k=δ

kn5,

Tn1−1,n2+1,n3−k,n4,n5=λ

k

n3

n1n3,

Tn1−1,n2+1,n3−1,n4−(k−1),n5+(k−1)=λ

k

n3

n1n4,

Tn1−1,n2+1,n3+(k−2),n4,n5=λ

k

n3

n1(kn1− n3− n4) .

This is a coarse grained description where the effect of

the change in state of a given node on the k pairs that it

forms is averaged over each pair type. For instance, the

event of loss of immunity occurs at a rate γnR, where

nRis the number of recovered nodes, and changes the k

pairs formed by the node that switches from recovered

to susceptible. On average, each pair type will change

by k units at a rate proportional to its density, accord-

ing to the equation γnR = γnR(n4

where nRR is the number of pairs of recovered nearest

neighbours. Taking this level of description and using

knR= n4+n5+2nRRand n1+n2+nR= N, we obtain

the first three equations of Eq. (7) for the rates of the

knR+

n5

knR+2nRR

knR),

three different pair events associated with loss of immu-

nity. A full microscopic description would require con-

sidering separately all possible five-node configurations

for the central node that switches from R to S and its

four nearest neighbours. We have checked that the de-

tailed stochastic model involving 40 different transitions

for k = 4 gives essentially the same results [16] as the

coarse grained model (7) that we consider here.

For the fluctuations of the pair densities we set n3(t) =

Nkp3(t)+√Nkx3(t), n4(t) = Nkp4(t)+√Nkx4(t), and

n5(t) = Nkp5(t) +√Nkx5(t). The leading order terms

of van Kampen’s system size expansion of the master

equation associated to (7) yield the deterministic PA Eqs.

(1). An approximate analytical expression for the PSNF

can be obtained as before from the next-to-leading-order

terms. Formula (4) is still valid taking now J as the

Jacobian of Eq. (1) at the endemic equilibrium and the

noise cross correlation matrix B computed directly from

the expansion.

00.5

ω

1

0

10

20

30

40

PI(ω)

00.51

ω

1.52

0

0.1

0.2

0.3

0.4

PI(ω)

0 0.51

ω

1.52

10

−2

10

0

10

2

10

4

10

6

PI(ω)

012

λ

34

0

0.05

0.1

0.15

γ

10

5

10

N

6

10

7

0

30

60

A

10

5

10

N

6

10

7

10

3

10

4

10

5

A

a)

c)

e)

b)

d)

f)

I

II

FIG. 2: (Color online) (a) Analytical and numerical PSNFs

of the infectives for model (3) with γ = 0.1 and λ = 2.5; (b)

the same for model (7); (c) a similar plot for model (7) with

γ = 0.034 and λ = 2.5, notice the lin-log scale; (d) location of

the parameter values chosen for (a) and (b) (circle) and for (c)

(square); and (e) and (f) plots of the peak amplitude of the

PSNF of the PA model as a function of N for the parameter

values chosen for (b) and (c).

In Fig. 2 the approximate PSNFs given by Eq. (4)

are plotted (black lines) and compared with the numer-

ical power spectra of stochastic simulations for N = 106

[green (gray) lines] for models (3) and (7) [Figs. 2(a)

and 2(b)].For this system size, there is almost per-

fect agreement between the analytical approximate ex-

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pressions and the results of the simulations across the

whole region I. The plots show that for the same param-

eter values the fluctuations are larger and more coherent

for model (7), in agreement with the results of simula-

tions reported in [15] for small-world networks on a lat-

tice and variable small-world parameter. This effect be-

comes much more pronounced as the boundary between

regions I and II, where the analytical PSNF of model

(7) diverges, is approached. Close to this boundary [see

Fig. 2(c)], there is a significant discrepancy between the

analytical (black line) and the numerical [green (gray)

line] PSNFs associated with the appearance of secondary

peaks at multiples of the main peak frequency. This is

a precursor of the oscillatory phase, and the breakdown

of van Kampen’s approximation for this system size may

be understood as an effect of the loss of stability of the

endemic equilibrium close to the boundary. Relaxation

towards equilibrium becomes slow compared with the pe-

riod of the damped oscillations, and a significant part of

the power spectrum energy shows up in the secondary

harmonics. For these parameter values, van Kampen’s

expansion becomes a good approximation only for larger

system sizes. Also shown in Fig. 2(e) [respectively, Fig.

2(f)] is the scaling with system size of the peak ampli-

tude of the infectives PSNF of the PA model [pink (black)

dots] for the parameter values considered in (b) [respec-

tively, (c)] and the peak amplitude (dashed blue line) of

the approximated PSNF given by Eq. (4). Away from

the phase boundary of the oscillatory phase we find that

the simulations exhibit the amplitude and scaling pre-

dicted by Eq. (4) down to system sizes of ∼ 105. By con-

trast, close to the phase boundary the match is reached

only for system sizes larger than 5 × 106.

0.03

a)

11001200

t

1300

0

0.01

0.02

pi

11501250

t

1350

0

0.01

0.02

pi

b)

FIG. 3:

given by the PA deterministic model (dashed blue lines), and

by simulations of the PA and the MFA stochastic models [solid

black and green (gray) lines, respectively] in regions II and I

for N = 107. Parameters are (a) λ = 7.5, γ = 0.01 and (b)

λ = 9, γ = 0.01.

(Color online) The steady-state infective density

Examples of typical time series predicted by the PA

model in the parameter region of childhood infectious

diseases are shown in Fig. 3 [dashed blue lines for the de-

terministic model (1) and solid black lines for simulations

of the stochastic model (7)]. The results of simulations

of the MFA stochastic model (3) for the same parameter

values are also shown for comparison [solid green (gray)

lines]. Fig. 3(a) illustrates the regular high-amplitude

oscillations of region II. All over this region, simulations

of the stochastic model (7) reproduce the behaviour of

the solutions of Eq. (1) with added amplitude fluctua-

tions. The only limitation to observe these oscillations

in finite systems is that N has to be taken large enough

for the deep interepidemic troughs to be spanned with-

out stochastic extinction of the disease. In region I [Fig.

3(b)] there are no oscillations in the thermodynamic limit

but, in contrast to the stochastic MFA model, the reso-

nant fluctuations in the PA model are large and coherent

enough to provide a distinct cycling pattern, which is

partially described by van Kampen’s expansion (4).

In conclusion, we have considered a stochastic version

of the basic model of infection dynamics including a rep-

resentation of the spatial correlations of an interaction

network through the standard PA. We have shown that

in general the resonant amplification and the coherence

of stochastic fluctuations are much enhanced with re-

spect to the MFA model. This quantitative difference

becomes qualitative in a region of parameter space that

corresponds to diseases for which immunity waning is

much slower than recovery. In this region the nonlineari-

ties of the model and demographic stochasticity give rise

either to oscillations that persist in the thermodynamic

limit or to high amplitude, coherent resonant fluctua-

tions, providing realistic patterns of recurrent epidemics.

These results are relevant for other population dynam-

ics models in the slow driving regime that corresponds to

small γ in our model, suggesting that in systems of mod-

erate size intrinsic stochasticity together with the sim-

plest representation of spatial correlationsmay be enough

to produce distinct oscillatory patterns. This favours the

view that, for a large class of systems, noisy oscillations in

population dynamics data may be intrinsic, rather than

externally driven.

Financial support from the Foundation of the Uni-

versity of Lisbon and the Portuguese Foundation for

Science and Technology (FCT) under Contracts No.

POCI/FIS/55592/2004 and No.

is gratefully acknowledged.

wasalsosupportedby

SFRH/BD/32164/2006 and by Calouste Gulbenkian

Foundation under its Program ”Stimulus for Research”.

POCTI/ISFL/2/618

The first author (G.R.)

FCTunderGrantNo.

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