Bioinformatics: Network Analysis
Networks as a Guiding Tool
COMP 572 (BIOS 572 / BIOE 564) - Fall 2013
Luay Nakhleh, Rice University
1
✤
Networks have been used to
✤
guide GWAS
✤
predict protein function
✤
model epidemics
✤
...
2
Vol. 29 ISMB/ECCB 2013, pages i171–i179
doi:10.1093/bioinformatics/btt238
BIOINFORMATICS
Efficient network-guided multi-locus association mapping
with graph cuts
Chloé-Agathe Azencott1,*, Dominik Grimm1, Mahito Sugiyama1, Yoshinobu Kawahara2 and
Karsten M. Borgwardt1,3
1
Machine Learning and Computational Biology
Researchet
Group,
C.-A.Azencott
al. Max Planck Institute for Developmental Biology & Max
Planck Institute for Intelligent Systems Spemannstr. 38, 72076 Tübingen, Germany, 2The Institute of Scientific and
Industrial Research (ISIR) Osaka University 8-1 Mihogaoka, Ibaraki-shi, Osaka 567-0047, Japan and 3Zentrum für
Bioinformatik, Eberhard Karls Universität Tübingen, 72076 Tübingen, Germany
(a)
(b)
Downloaded from http://bioinformatics.oxfordjournals.org/ at Rice University on November 21, 2013
sets of genetic variants to the phenotype has the potential to help
explain part of this missing heritability (Marchini et al., 2005).
Motivation: As an increasing number of genome-wide association
Although efficient multiple linear regression approaches (Cho
studies reveal the limitations of the attempt to explain phenotypic
et al., 2010; Rakitsch et al., 2012; Wang et al., 2011) make the
heritability by single genetic loci, there is a recent focus on
detection of such multivariate associations possible, they often
associating complex phenotypes with sets of genetic loci. Although
remain limited in power and hard to interpret. Incorporating
several methods for multi-locus mapping have been proposed, it is
biological knowledge into these approaches could help boosting
often unclear how to relate the detected loci to the growing knowledge
their power and interpretability. However, current methods are
about gene pathways and networks. The few methods that take
limited to predefining a reasonable number of candidate sets to
biological pathways or networks into account are either restricted to
investigating a limited number of predetermined sets of loci or do not
investigate (Cantor et al., 2010; Fridley and Biernacka, 2011; Wu
scale to genome-wide settings.
et al., 2011), for instance by relying on gene pathways. They
(c)
Results: We present SConES, a new efficient method to discover sets
consequently run the risk of missing biologically relevant loci
of genetic loci that are maximally associated with a phenotype while
that have not been included in the candidate sets. This risk is
being connected in an underlying network. Our approach is based on
made even likelier by the incomplete state of our current bioloa minimum cut reformulation of the problem of selecting features
gical knowledge.
under sparsity and connectivity constraints, which can be solved
For this reason, our goal here is to use prior knowledge in a
exactly and rapidly.
more flexible way. We propose to use a biological network,
SConES outperforms state-of-the-art competitors in terms of runtime,
defined between SNPs, to guide a multi-locus mapping approach
scales to hundreds of thousands of genetic loci and exhibits higher
that is both efficient to compute and biologically meaningful: We
power in detecting causal SNPs in simulation studies than other methaim to find a set of SNPs that (i) are maximally associated with a
ods. On flowering time phenotypes and genotypes from Arabidopsis
given phenotype and (ii) tend to be connected in a given biological
thaliana, SConES detects loci that enable accurate phenotype predicnetwork. In addition, this set must be computed efficiently on
tion and that are supported by the literature.
genome-wide data. In this article, we assume an additive model
Availability: Code is available at http://webdav.tuebingen.mpg.de/u/
to characterize multi-locus association. The network constraint
karsten/Forschung/scones/.
stems from the assumption that SNPs influencing the same
Fig. 1. Small examples of the three types of networks considered
Contact: chloe-agathe.azencott@tuebingen.mpg.de
phenotype are biologically linked. However, the diversity of the
Supplementary information: Supplementary data are available at
type of relationships that this can encompass, together with the
Bioinformatics online.
current incompleteness of biological knowledge, makes providing a networknetwork:
in which all
the relevant
connections as
are in
present
! GM (gene membership)
SNPs
are connected
unlikely. For this reason, although we want to encourage the
1 INTRODUCTION
the sequence network
described earlier in the text; in addSNPs to form a subnetwork of the network, we also do not
Twin and family/pedigree studies make it possible to
estimate
the near
ition,
SNPs
same that
genethey
aremust
linked
as well.
wantthe
to enforce
form atogether
single connected
compoABSTRACT
pairs of features connected by an e
a given size yields the so-called gra
taken by Huang et al. (2009): th
encourages selecting a small nu
blocks are sets of features define
of the problem. In the case of a g
are defined as small connected
shown in Mairal and Yu (2011)
aforementioned is a relaxation o
number of linear subgraphs or co
size grows exponentially with the
which can reach millions in the ca
only the edge-based version of the
our problem. It is however unclea
ture long-range connections betw
Li and Li (2008) propose a netw
Lasso that imposes the type of g
sirable. However, their approach
works of genes (rather than of SN
easily to the datasets we envisio
they propose relies on a singula
Laplacian of the network, whic
cannot be stored in memory.
Chuang et al. (2007) also sear
protein interaction networks that
a phenotype; however, their greed
3
Network-Assisted Investigation of Combined Causal
Signals from Genome-Wide Association Studies in
Schizophrenia
Peilin Jia1,2, Lily Wang3, Ayman H. Fanous4,5,6,7, Carlos N. Pato7, Todd L. Edwards8,9, The International
Schizophrenia Consortium", Zhongming Zhao1,2,10*
1 Department of Biomedical Informatics, Vanderbilt University School of Medicine, Nashville, Tennessee, United States of America, 2 Department of Psychiatry, Vanderbilt
University School of Medicine, Nashville, Tennessee, United States of America, 3 Department of Biostatistics, Vanderbilt University School of Medicine, Nashville,
Tennessee, United States of America, 4 Department of Psychiatry and Virginia Institute for Psychiatric and Behavior Genetics, Virginia Commonwealth University,
Richmond, Virginia, United States of America, 5 Washington VA Medical Center, Washington, D.C., United States of America, 6 Department of Psychiatry, Georgetown
University School of Medicine, Washington, D.C., United States of America, 7 Department of Psychiatry, Keck School of Medicine of the University of Southern California,
Los Angeles, California, United States of America, 8 Center for Human Genetics Research, Vanderbilt University School of Medicine, Nashville, Tennessee, United States of
America, 9 Division of Epidemiolgy, Department of Medicine, Vanderbilt University School of Medicine, Nashville, Tennessee, United States of America, 10 Department of
Cancer Biology, Vanderbilt University School of Medicine, Nashville, Tennessee, United States of America
Abstract
With the recent success of genome-wide association studies (GWAS), a wealth of association data has been accomplished
for more than 200 complex diseases/traits, proposing a strong demand for data integration and interpretation. A
combinatory analysis of multiple GWAS datasets, or an integrative analysis of GWAS data and other high-throughput data,
has been particularly promising. In this study, we proposed an integrative analysis framework of multiple GWAS datasets by
overlaying association signals onto the protein-protein interaction network, and demonstrated it using schizophrenia
datasets. Building on a dense module search algorithm, we first searched for significantly enriched subnetworks for
schizophrenia in each single GWAS dataset and then implemented a discovery-evaluation strategy to identify module genes
with consistent association signals. We validated the module genes in an independent dataset, and also examined them
through meta-analysis of the related SNPs using multiple GWAS datasets. As a result, we identified 205 module genes with a
joint effect significantly associated with schizophrenia; these module genes included a number of well-studied candidate
genes such as DISC1, GNA12, GNA13, GNAI1, GPR17, and GRIN2B. Further functional analysis suggested these genes are
involved in neuronal related processes. Additionally, meta-analysis found that 18 SNPs in 9 module genes had
Pmeta,161024, including the gene HLA-DQA1 located in the MHC region on chromosome 6, which was reported in previous
studies using the largest cohort of schizophrenia patients to date. These results demonstrated our bi-directional networkbased strategy is efficient for identifying disease-associated genes with modest signals in GWAS datasets. This approach can
be applied to any other complex diseases/traits where multiple GWAS datasets are available.
Citation: Jia P, Wang L, Fanous AH, Pato CN, Edwards TL, et al. (2012) Network-Assisted Investigation of Combined Causal Signals from Genome-Wide Association
Studies in Schizophrenia. PLoS Comput Biol 8(7): e1002587. doi:10.1371/journal.pcbi.1002587
Editor: Frederick P. Roth, Harvard Medical School, United States of America
Received October 31, 2011; Accepted May 15, 2012; Published July 5, 2012
Copyright: ! 2012 Jia et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted
4
Network-based prediction of protein func
REVIEW
Network-based prediction of protein function
Network-based function prediction
R Sharan et al
1
Roded Sharan , Igor Ulitsky1 and Ron Shamir*
networks, with nodes
networks, with nodes representing
representing proteins
and ed
Roded Sharan1, Igor Ulitsky1 and Ron Shamir*
the detecte
representing the detected PPIs.
In this review, we su
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
In this review, we survey the growing body of works
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Corresponding
School of
Computer
Science, Tel Aviv
University,
** Corresponding
author. author.
School of Computer
Science,
Tel Aviv University,
functional
annotation offunctional
proteins via annotation
their network
Tel
69978,
Israel. Tel.:
þ 972
3 6405383;
Fax:
972 3 6405384;
TelAvivAviv
69978,
Israel.
Tel.:
þ 972
3þ
6405383;
Fax: þ 972interactions
3 6405384;(summarizedinteractions
in Table I). We(summariz
distinguish t
E-mail: rshamir@tau.ac.il
types of approaches (Figure 2): direct annotation schem
1 E-mail: rshamir@tau.ac.il
These authors contributed equally to this work.
types
ofbased
approaches
(F
1
which infer the function of
a protein
on its connecti
These authors contributed equally to this work.
in the network, and module-assisted
which fi
which inferschemes,
the function
Received 20.9.06; accepted 9.1.07
identify modules of related proteins and then annotate e
in the network, and m
Received 20.9.06; accepted 9.1.07
module based on the known functions of its memb
of rel
Naturally, the presentedidentify
methods modules
and the emphasis
particular ones reflect themodule
opinions of based
the authors.
on the
Functional annotation of proteins is a fundamental problem
in the post-genomic era. The recent availability of protein
Naturally, the present
interaction networks for many model species has spurred
particular ones reflect th
on
the development
of computational
methods
interFunctional
annotation
of proteins
is for
a fundamental
problem
Direct
methods
preting such data in order to elucidate protein function. In
in the post-genomic era. The recent availability
of protein
this review, we describe the current computational
common principle underlying all direct methods
interaction
networks
for many
modelwhich
speciesThe
has
spurred
approaches
for the
task, including
direct methods,
functional annotation is that proteins that lie closer to
propagate
information
the network, and
on the functional
development
of through
computational
methods
forin interanother
the PPI network are more likely to have sim
module-assisted methods, which infer functional modules
As can
preting such data in order to elucidate proteinfunction.
function.
Inbe seen in Figure 3, there is an evid
within the network and use those for the annotation task.
correlation between network distance and functional distan
this review,
we describe
current
Although
a broad variety
of interestingthe
approaches
has computational
that is, the closer the two proteins
are in the network
the m
The common
principle
been
developed,
further
progress
in
the
field
will
depend
on
approaches for the task, including direct methods,
similar arewhich
their functional annotations. The methods
functional
annotation
i
systematic evaluation of the methods and their disseminascribed
below
differ
in
the
way
they
capture
and
exploit
propagate
information
the network,
andthe functionsanother
Figure
2 biological
Directfunctional
versus module-assisted
approaches for functionalthrough
annotation. The scheme
shows a network in which
of some proteins are known (top),
tion
in the
community.
in isthe
netw
correlation.
In
the
following,
we denote
PPIPPI
network
a
where each function is indicated by a different color. Unannotated proteins are in white. In the direct methods (left), these proteins are assigned
a color thatthe
unusually
Molecular
Systems
Biology
13
March
2007;
module-assisted
methods,
functional
modules
prevalent among their neighbors.
The direction of thewhich
edges indicatesinfer
the influence
of the annotated
proteins
on the unannotated
In the
module-assisted methods
graph
G¼(V,E)
(see Boxones.
1function.
for
graph-theoretic
definitions).
As
can
be se
(right), modules are first identified based on their density. Then, within each module, unannotated proteins are assigned a function that is unusually prevalent in the
doi:10.1038/msb4100129
Direct methods
within the network and use those for the annotation task.
approaches has
module.
In both methods,
proteins may bemethods;
assigned withproteins
several functions.
Subject
Categories:
computational
Although
a broadfunction
variety
of protein
interesting
Keywords:
data integration;
prediction;
interaction
correlation between net
5
that is, the closer the tw
Epidemics and Networks
6
Diseases and the Networks that
Transmit Them
✤
The patterns by which epidemics spread through a population is
determined not just by the properties of the pathogen carrying it
(contagiousness, the length of the infection period, severity, etc.), but
also by network structures within the population it is affecting.
✤
The opportunities for a disease to spread are given by a contact
network: there is a node for each person, and an edge if two people
come into contact with each other in a way that makes it possible for
the disease to spread from one to the other.
7
Diseases and the Networks that
Transmit Them
✤
Accurately modeling the underlying network is crucial to
understanding the spread of an epidemic.
8
pling, a, of
e observed
ille equation
mT. We then
n, the change
ic field in
e also
T in static
orientation
MF to equal
xperiments.
...how travel patterns within a city affect the spread of disease
..............................................................
ARTICLE IN PRESS
Modelling disease outbreaks in
realistic urban social networks
1
2
1
1
Stephen Eubank
,
Hasan
Guclu
,
V.
S.
Anil
Kumar
,
Madhav
V.
Marathe
,
Journal of Theoretical Biology 232 (2005) 71–81
Aravind Srinivasan3, Zoltán Toroczkai4 & Nan Wang5
www.elsevier.com/locate/y
1
Basic and Applied Simulation Science Group, Los Alamos National Laboratory,
MS M997,
Los Alamos,
New
Mexico 87545,
USA
Network
theory
and
SARS:
predicting
outbreak diversity
2
Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic
a,b,!Troy,
,1
c,1,2
b,d
Institute
110
8th Street,
New York
12180-3590, USA
Lauren
Ancel
Meyers
,
Babak
Pourbohloul
,
M.E.J.
Newman
,
–64 (1972).
3
c,2 Studies,
Department
of Computer
Science andc,2
Institute
for Advanced
Computer
5).
Danuta
M. Skowronski
, Robert
C. Brunham
University of Maryland, College Park, Maryland 20742, USA
n animals.
4
ion of Integrative Biology
and for
Institute
for Cellular
and Molecular
Biology, University
Texas at
Austin,
1 University
Station C0930, Austin,
Centre
Nonlinear
Studies
and Complex
Systems of
Group,
Los
Alamos
National
78712, USA
b
c. Lond. B 263,
Laboratory, Santa
MS B258,
Los Alamos,
Mexico
87545,
USA
Fe Institute,
1399 HydeNew
Park Road,
Santa
Fe, NM
87501, USA
5
c
University of British
Columbia Centre
for DiseaseScience,
Control, 655
West 12th of
Avenue,
Vancouver,
British Park,
Columbia, Canada V5Z 4R4
Department
of Computer
University
Maryland,
College
ter for the Study of Complex Systems, University of Michigan, Randall Laboratory, 500 E. University Ave., Ann Arbor, MI 48109-1120, U
agnetite-based
Maryland 20742, USA
9
...how travel patterns via the worldwide airline network affect the spread of
disease
The role of the airline transportation network in the
prediction and predictability of global epidemics
Vittoria Colizza*, Alain Barrat†, Marc Barthélemy*‡, and Alessandro Vespignani*§
*School of Informatics and Center for Biocomplexity, Indiana University, Bloomington, IN 47401; and †Centre National de la Recherche Scientifique,
Unité Mixte de Recherche 8627, Université Paris-Sud, Ba! timent 210, F-91405 Orsay, France
Communicated by Giorgio Parisi, University of Rome, Rome, Italy, December 8, 2005 (received for review June 18, 2005)
The systematic study of large-scale networks has unveiled the
ubiquitous presence of connectivity patterns characterized by
large-scale heterogeneities and unbounded statistical fluctuations.
These features affect dramatically the behavior of the diffusion
processes occurring on networks, determining the ensuing statistical properties of their evolution pattern and dynamics. In this
article, we present a stochastic computational framework for the
forecast of global epidemics that considers the complete worldwide air travel infrastructure complemented with census population data. We address two basic issues in global epidemic modeling: (i) we study the role of the large scale properties of the airline
transportation network in determining the global diffusion pattern of emerging diseases; and (ii) we evaluate the reliability of
forecasts and outbreak scenarios with respect to the intrinsic
model including the full International Air Transport Association
(www.iata.org) database, aiming at a detailed study of the
interplay among the network structure and the stochastic features of the infection dynamics in defining the global spreading
of epidemics. In particular, whereas previous studies have generally been focused in the a posteriori analysis of real case studies
of global epidemics, the large-scale modeling presented here
allows us to address more basic theoretical issues such as the
statistical properties of the epidemic pattern and the effect on it
of the complex architecture of the underlying transportation
network. Finally, such a detailed level of description allows for
the quantitative assessment of the reliability of the obtained
forecast with respect to the stochastic nature of the disease
10
Diseases and the Networks that
Transmit Them
✤
Contact networks are also important in understanding how diseases
spread through animal populations (e.g., the 2001 foot-and-mouth
outbreak in the UK) and plant populations.
✤
Similar models have been employed for studying the spread of
computer viruses...
11
Diseases and the Networks that
Transmit Them
✤
The pathogen and the network are closely intertwined: even within
the same population, the contact networks for two different diseases
can have very different structures, depending on the diseases’
respective modes of transmission.
✤
(Think of airborne transmission based on coughs and sneezes,
compared to a sexually transmitted disease, and think of the density
of the contact networks!)
12
Branching Processes
✤
The simplest model of contagion: every person is in contact with k
people
✤
First wave: a person carrying a new diseases enters a population
and transmits it to each of his contacts independently with
probability p.
✤
Second wave: each person in the first wave transmits to each of his
contacts independently with probability p (the contacts of people
are mutually exclusive)
✤
and so on..
13
Branching Processes
648
CHAPTER 21. EPIDEMICS
(a) The contact network for a branching process
14
Branching Processes
(a) The contact network for a branching process
(b) With high contagion probability, the infection spreads widely
15
Branching Processes
(b) With high contagion probability, the infection spreads widely
(c) With low contagion probability, the infection is likely to die out quickly
Figure 21.1: The branching process model is a simple framework for reasoning about the
spread of an epidemic as one varies both the amount of contact among individuals and the
level of contagion.
16
The Basic Reproductive Number R0
✤
The basic reproductive number, denote R0, is the expected number of
new cases of the disease caused by a single individual.
✤
For the simple branching process we saw, we have R0=kp.
17
n the tree, no one in any future wave will be infected either.
So there are really only two possibilities for a disease in the branching process model:
ches a wave where it infects no one, thus dying out after a finite number of steps; or
tinues to infect people in every wave, proceeding infinitely through the contact netwo
0
d it turns out that there is a simple condition to tell these two possibilities apart, bas
a quantity called the basic reproductive number of the disease.
The basic reproductive number, denoted R0 , is the expected number of new cases of t
ase caused by a single individual. Since in our model everyone meets k new people a
cts each with probability p, the basic reproductive number here is given by R0 = p
outcome of the disease in a branching process model is determined by whether the ba
roductive number is smaller or larger than 1.
The Basic Reproductive Number R
Claim: If R0 < 1, then with probability 1, the disease dies out after a finite
number of waves. If R0 > 1, then with probability greater than 0 the disease
persists by infecting at least one person in each wave.
give a proof of this claim in Section 21.8. Even without the details of the proof, howev
can see that the basic condition expressed in the claim — comparing R0 to 1 — ha
ural intuitive basis. When R0 < 1, the disease isn’t able to replenish itself: each infect
son produces less than one new case in expectation, and so — even if it grows brie
to the outcome of random fluctuations — the size of the outbreak is constantly trendi
18
The Basic Reproductive Number R0
✤
Implication: It’s always good to reduce the value of R0!
✤
Quarantining people reduces k and encouraging behavioral measures
such as sanitary practices reduces p.
19
✤
Clearly, the branching process is too simplistic!
20
The SIR Epidemic Model
✤
An individual node goes three potential stages during the course of
the epidemic:
✤
Susceptible: Before the node has caught the disease, it is susceptible
to infection from its neighbors.
✤
Infectious: Once the node has caught the disease, it is infectious
and has some probability of infecting each of its susceptible
neighbors.
✤
Removed: After a particular node has experienced the full
infectious period, this node is removed from consideration.
21
The SIR Epidemic Model
✤
Given this three-stage “life cycle” for the disease at each node, a
model for epidemics on networks can be defined.
✤
The network structure: a directed graph representing the contact
network (edge u to v means that if u becomes infected, the disease has
the potential to spread to v).
✤
Two other quantities: p (the probability of contagion) and tI (the
length of infection)
22
between people, where either has the potential to directly infect the other, we can put in
directed edges pointing each way: both from v to w and also from w to v. Since contacts
between people are often symmetric, it is fine to use networks where most edges appear in
each direction, but it is sometimes convenient to be able to express asymmetric contacts as
well.
Now, each node has the potential to go through the Susceptible-Infectious-Removed
cycle, where we abbreviate these three states as S, I, and R. The progress of the epidemic
is controlled by the contact network structure and by two additional quantities: p (the
probability of contagion) and tI (the length of the infection).
The SIR Epidemic Model
• Initially, some nodes are in the I state and all others are in the S state.
• Each node v that enters the I state remains infectious for a fixed number of steps tI .
• During each of these tI steps, v has a probability p of passing the disease to each of its
susceptible neighbors.
• After tI steps, node v is no longer infectious or susceptible to further bouts of the
disease; we describe it as removed (R), since it is now an inert node in the contact
network that can no longer either catch or transmit the disease.
This describes the full model; we refer to it as the SIR model, after the three disease states
that nodes experience. Figure 21.2 shows an example of the SIR model unfolding on a
particular contact network through successive steps; in each step, shaded nodes with dark
borders are in the I state and shaded nodes with thin borders are in the R state.
23
The SIR Epidemic Model
652
CHAPTER 21. EPIDEMICS
t
t
s
u
s
u
y
y
z
x
r
z
x
v
r
v
w
w
(a)
(b)
t
t
s
u
s
u
y
y
r
z
x
z
x
v
r
v
w
w
(c)
(d)
24
The SIS Epidemic Model
✤
The SIR epidemic model is appropriate for epidemics in which each
individual contracts the disease at most once.
✤
To allow for nodes that can be reinfected multiple times, a model can
have only the S and I, but not R, states.
25
ividual contracts the disease at most once. However, a simple variation on these models
llows us to reason about epidemics where nodes can be reinfected multiple times.
To represent such epidemics, we have nodes that simply alternate between two possible
tates: Susceptible (S) and Infectious (I). There is no Removed state here; rather, after a
ode is done with the Infectious state, it cycles back to the Susceptible state and is ready to
atch the disease again. Because of this alternation between the S and I states, we refer to
he model as the SIS model.
Aside from the lack of an R state, the mechanics of the model follow the SIR process
ery closely.
The SIS Epidemic Model
• Initially, some nodes are in the I state and all others are in the S state.
• Each node v that enters the I state remains infectious for a fixed number of steps tI .
• During each of these tI steps, v has a probability p of passing the disease to each of its
susceptible neighbors.
• After tI steps, node v is no longer infectious, and it returns to the S state.
igure 21.5 shows an example of the SIS model unfolding on a three-node contact network
with tI = 1. Notice how node v starts out infected, recovers, and later becomes infected
gain — we can imagine this as the contact network within a three-person apartment, or a
hree-person family, where people pass a disease on to others they’re living with, and then
26
The SIS Epidemic Model
56
CHAPTER 21. EPIDEMIC
u
v
u
v
v
w
(a)
u
v
w
(b)
u
v
w
(c)
u
w
(d)
w
(e)
igure 21.5: In an SIS epidemic, nodes can be infected, recover, and then be infected aga
n each step, the nodes in the Infectious state are shaded.
27
A Connection Between SIR and SIS
658
CHAPTER 21. EPIDEMICS
u
v
u
v
u
v
u
v
u
v
w
w
w
w
w
step 0
step 1
step 2
step 3
step 4
(a) To represent the SIS epidemic using the SIR model, we use a “‘time-expanded” contact network
u
v
u
v
u
v
u
v
u
v
w
w
w
w
w
step 0
step 1
step 2
step 3
step 4
(b) The SIS epidemic can then be represented as an SIR epidemic on this time-expanded network.
28
The SIRS Epidemic Model
✤
Combines SIR and SIS:
✤
After an infected node recovers, it passes briefly through the R
state on its way back to the S state.
29
ti !t 1 1" ! 0
if ti !t" ! t0 .
NUMBER
130, until
PHYSICAL REVIEW LETTERS
26 MARCH 2001
VOLUME
usceptible element
stays86,
as such,
at t !
initial state is random with ninf !0" ! 0.1. The 400 time
infected. Once infected, it goes (deterministia cycle that lasts t0 time steps. During the first
steps shown are representative of the stationary state. We
canWorld
see clearly
a transition
from
an endemic situation toModel
an
ps, it is infected and can potentially transmit the
Small
Effect
in an
Epidemiological
oscillatory one. At p ! 0.01 (top), where the network is
susceptible neighbor. During the last tR time
nearly a regular lattice,
e cycle, it remains in state R, immune but not
1, the stationary state is a fixed point,
1,2, †
Marcelo
Kuperman
*
and
Guillermo
Abramson
with fluctuations. The situation corresponds to that of an
After the cycle is complete, it returns to the
1
Centro
Atómico
Bariloche
and Instituto
Balseiro,
8400 S.fraction
C. de Bariloche,
Argentina
endemic
infection,
with a low
and persistent
of instate.
2
Consejofected
Nacional
de Investigaciones
Científicas
y Técnicas,
Argentina
individuals.
At high values
of p — like
the case with
agion of a susceptible element by an infected
(Received
)
5 October
2000
p ! 0.9 shown
in the figure
(bottom)
— large
amplitude,
e subsequent excitation of the disease cycle in
oscillations
develop.
The situation
is almost
ected, occur stochastically atAa model
local level.
Say
for the
spreadself-sustained
of an infection
is analyzed
for different
population
structures. The interactions
with
a
very
well
defined
period
and
small
flucment i is susceptible, andwithin
that it the
has population
ki neigh- areperiodic,
described by small world networks, ranging from ordered lattices to random
tuations
in amplitude.
period is endemic
slightly longer
than
ich kinf are infected. Then,
i
will
become
ingraphs. For the more ordered systems,
there isThe
a fluctuating
state of
low infection. At a finite
t
probability kinf $ki . Observe
that
i
will
become
,
since
it
includes
the
average
time
that
a
susceptible
in0
value of the disorder of the network, we find a transition to self-sustained oscillations in the size of the
th probability 1 if all its neighbors are infected.
dividual remains at state S, before being infected. Epideinfected subpopulation.
s parameter-free mechanism, there may be other
miologically, the situation resembles the periodic epidemic
choices. For example, ifDOI:
the susceptible
had
patterns typical of large populations
[3]. A
mean field
10.1103/PhysRevLett.86.2909
PACS numbers:
89.75.Hc,
05.65.+b, 87.19.Xx, 87.23.Ge
y q of contagion with each infected neighbor,
model of the system, expected to resemble the behavior at
ave a probability of infection %1 2 !1 2 q"kinf &.
p ! 1, can easily be shown to exhibit these oscillations.
transition
apparent
— in
ted that both these
criteria give qualitatively
the the The
show thatisa sharp
transition
in the behavior of an infection
I. Introduction.—
How does
dynamics
of between
an infec-both behaviors
this
relatively
small
system—
at
the
intermediate
value
of
s for valuestious
of q &
0.2.
For
other
values
of
q,
dynamics exists at a finite value of p.
disease depend on the structure of a population? A
rs are outlined
at
the
end
of
Section
III.
II. Epidemic model.—We analyze a simple model of the
great amount of work has been done on the phenomenoerical results.—We have performed extensive
spread of an infectious
disease. We want, mainly, to point
description
of particular
p = 0.01
imulations logical
of the described
model.
Networks epidemic0.4situations [1–4].
to the role played by the network structure on the temporal
classical
to these problems deals
103 to 106 A
vertices
havemathematical
been explored,approach
with
0.2
dynamics of the epidemic. The disease has three stages:
well mixed
populations,
0. A typicalwith
realization
starts with
the genera- where the subpopulations
susceptible (S), infected (I), and refractory (R). An eleinvolved
susceptible,
and removed) inrandom network
and(typically
the initialization
of the infected, 0.0
0.4
p = 0.2
elements. teract
An initial
fraction of 0.1
infected,
ment of the population
is described by a single dynamical
in proportion
to their
sizes. With these zero dimensusceptible,
was used
in allitthe
0.2
variable adopting one of these three values. Susceptible
sional
models
hasresults
beenshown
possible to study,
among other
initial conditions
havefeatures,
been explored
as well, of threshold values for
elements can pass to the infected state through contagion
epidemic
the existence
0.0
nges have been observed in the behavior. After
0.4
= 0.9 Infected elements pass to the refracby an infected pone.
the spread of an infection [5], the asymptotic
solution for
a stationary state is achieved, and the computatory state after an infection time tI . Refractory elements
density
of time
infected
the effect of
0.2
llowed for the
several
thousand
steps people
to per- [6–8], and
stochastic fluctuations on the modulation of an epidemic
return to the susceptible state after a recovery time tR . This
ical averages.
0.0
situation
[9].time
A series
second
classical approach
describes
spa- 5400
kind of system
5200
5300
5500 is usually
5600 called SIRS, for the cycle that a
in Fig. 1 part
of three
displaying
t
of infected tially
elements
in the system,
ninf !t". The such as elements in a
extended
subpopulations,
single
element goes over. The contagion is possible only
s correspond
to systems
with different
values spread of an epidemic
lattice.
In this,
the geographic
duringasthe
S phase,
and only by an I element. During the
FIG. 1. Fraction of infected elements
a function
of time.
rder parameter:
p !analyzed
0.01 (top),
(middle),
can be
as 0.2
a reaction-diffusion
process
R phase, to
thedifferent
elements
are immune and do not infect. SIRS
Three time
series[10–13],
are shown, corresponding
values
4
ottom). Thebearing
three systems
have N ! 10
and
of the disorder
parameter
p,
in theare
legends.
Other systems,
paclose similarity
to paradigmatic
reactions
such
as as shown
models
excitable
known to display relaxation
4
infection cycles with tI ! 4 and tR ! 9. The
rameters are N ! 10 , K ! 3, tI ! 4, tR ! 9, Ninf !0" ! 0.1.
Recall: the small-world model (p: probability of rewiring an edge)
ninf (t)
The number of infected people (ninf(t)) by SIRS epidemic:
Belousov-Zhabotinskii’s.
oscillations in mean field or well-mixed approaches. In
30
Acknowledgments
✤
Slides on epidemics and networks are based on the book “Networks,
Crowds, and Markets” by Easley and Kleinberg.
31