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Spreading of sexually transmitted diseases in
heterosexual populations
Article in Proceedings of the National Academy of Sciences · March 2008
DOI: 10.1073/pnas.0707332105 · Source: PubMed
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Spreading of sexually transmitted diseases
in heterosexual populations
Jesús Gómez-Gardeñes†‡, Vito Latora§, Yamir Moreno‡¶, and Elio Profumo†
†Scuola Superiore di Catania, Via San Paolo 73, 95123 Catania, Italy; ‡Instituto de Biocomputación y Fı́sica de Sistemas Complejos (BIFI), Universidad de
Zaragoza, 50009 Zaragoza, Spain; and §Dipartimento di Fisica e Astronomia, Università di Catania, and Istituto Nazionale di Fisica Nucleare (INFN),
Via San Sofia 64, 95123 Catania, Italy
The spread of sexually transmitted diseases (e.g., chlamydia, syphilis, gonorrhea, HIV, etc.) across populations is a major concern for
scientists and health agencies. In this context, both the data
collection on sexual contact networks and the modeling of disease
spreading are intensive contributions to the search for effective
immunization policies. Here, the spreading of sexually transmitted
diseases on bipartite scale-free graphs, representing heterosexual
contact networks, is considered. We analytically derive the expression for the epidemic threshold and its dependence with the
system size in finite populations. We show that the epidemic
outbreak in bipartite populations, with number of sexual partners
distributed as in empirical observations from national sex surveys,
takes place for larger spreading rates than for the case in which the
bipartite nature of the network is not taken into account. Numerical simulations confirm the validity of the theoretical results. Our
findings indicate that the restriction to crossed infections between
the two classes of individuals (males and females) has to be taken
into account in the design of efficient immunization strategies for
sexually transmitted diseases.
bipartite graphs 兩 epidemic threshold 兩 sexual contact networks
D
isease spreading has been the subject of intense research
for a long time (1–3). On the one hand, epidemiologists
have developed mathematical models that can be used as
guides to understanding how an epidemic spreads and to
design immunization and vaccination policies (1–3). On the
other hand, data collections have provided information on the
local patterns of relationships in a population. In particular,
persons who may have come into contact with an infectious
individual are identified and diagnosed, making it possible to
contact-trace the way the epidemic spreads and to validate the
mathematical models. However, up to a few years ago, some of
the assumptions at the basis of the theoretical models were
difficult to test. This is the case, for instance, for the complete
network of contacts, the backbone through which the diseases
are transmitted. With the advent of modern society, fast
transportation systems have changed human habits, and some
diseases that just a few years ago would have produced local
outbreaks now are a global threat for public health systems. A
recent example is severe acute respiratory syndrome (SARS),
which spread very fast from Asia to North America a few years
ago (4 – 6). Therefore, it is of utmost importance to carefully
take into account as many details as possible of the structural
properties of the network in which the infection dynamics
occurs.
Strikingly, a large number of statistical properties have been
found to be common in the topology of real-world social,
biological, and technological networks (7–9). Of particular relevance, because of its ubiquity in nature, is the class of complex
networks referred to as scale-free (SF) networks. In SF networks, the number of contacts or connections of a node with
other nodes in the system, the degree (or connectivity) k, follows
a power-law distribution, Pk ⬃ k⫺␥. Recent studies have shown
the importance of the SF topology on the dynamics and function
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0707332105
of the system under study (7–9). For instance, SF networks are
very robust to random failures but at the same time extremely
fragile to targeted attacks of the highly connected nodes (10, 11).
In the context of disease spreading, SF contact networks lead to
a vanishing epidemic threshold in the limit of infinite population
when ␥ ⱕ 3 (12–15), which is because the exponent ␥ is directly
related to the first and second moment of the degree distribution, 具k典 and 具k2典, and the ratio 具k典/具k2典 determines the epidemic
threshold above which the outbreak occurs. When 2 ⬍ ␥ ⱕ 3, 具k典
is finite whereas 具k2典 goes to infinity; that is, the transmission
probability required for the infection to spread goes to zero.
Conversely, when ␥ ⬎ 3, there is a finite threshold and the
epidemic survives only when the spreading rate is above a certain
critical value. The concept of a critical epidemic threshold is
central in epidemiology. Its absence in SF networks with 2 ⬍ ␥ ⱕ
3 has a number of important implications in terms of prevention
policies: If diseases can spread and persist even in the case of
vanishingly small transmission probabilities, then prevention
campaigns in which individuals are randomly chosen for vaccination are not very effective (12–15).
Our knowledge of the mechanisms involved in disease spreading and the relation between the network structure and the
dynamical patterns of the spreading process has improved in the
last several years (16–19). Current approaches are either individual-based simulations (18) or metapopulation models in
which network simulations are carried out through a detailed
stratification of the population and infection dynamics (20). In
the particular case of sexually transmitted diseases (STDs),
infections occur within the unique context of sexual encounters,
and the network of contacts (19, 21–26) is a critical ingredient
of any theoretical framework. Unfortunately, ascertaining complete sexual contact networks in significantly large populations
is extremely difficult. However, here we show that it is indeed
possible to make use of known global statistical features to
generate more accurate predictions of the critical epidemic
threshold for STDs.
Networks of Sexual Contacts. Data from national sex surveys
(21–25) provide quantitative information on the number of
sexual partners, the degree k, of an individual. Usually, surveys
involve a random sample of the population stratified by age,
economical and cultural level, occupation, marital status, etc.
The respondents are asked to provide information on sexual
attitudes such as the number of sex partners they have had in the
last 12 months or in their entire life. Although in most cases the
response rate is relatively small, the information gathered is
statistically significant, and global features of sexual contact
Author contributions: J.G.-G., V.L., and Y.M. designed research; J.G.-G., V.L., Y.M., and E.P.
performed research; J.G.-G., V.L., Y.M., and E.P. contributed new reagents/analytic tools;
J.G.-G., V.L., Y.M., and E.P. analyzed data; and J.G.-G., V.L., and Y.M. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
¶To
whom correspondence should be addressed. E-mail: yamir@unizar.es.
© 2008 by The National Academy of Sciences of the USA
PNAS 兩 February 5, 2008 兩 vol. 105 兩 no. 5 兩 1399 –1404
APPLIED PHYSICAL
SCIENCES
Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved December 4, 2007 (received for review August 3, 2007)
Table 1. Statistical properties of sexual contact networks from national sex surveys conducted in four different countries: Sweden,
the U.K., Zimbabwe, and Burkina Faso
12 months
Survey
Sweden
U.K.
Zimbabwe
Burkina Faso
Lifetime
Respondents
Ref.
␥F
␥M
␥F
␥M
Total
Female
Male
21
22, 23
23
24
3.54 ⫾ 0.20
3.10 ⫾ 0.08
2.51 ⫾ 0.40
3.9 ⫾ 0.2
3.31 ⫾ 0.20
2.48 ⫾ 0.05
3.07 ⫾ 0.20
2.9 ⫾ 0.1
3.1 ⫾ 0.30
3.09 ⫾ 0.20
2.48 ⫾ 0.15
—
2.6 ⫾ 0.30
2.46 ⫾ 0.10
2.67 ⫾ 0.18
—
2,810
11,161
9,843
466
—
6,399
5,424
179
—
4,762
4,419
287
The exponents ␥F and ␥M are referred to the distribution of number of sexual partners cumulated in 12 months and over the respondent’s lifetime. The number
of respondents also is reported.
patterns can be extracted. In particular, it turns out that the
number of heterosexual partners reported from different populations is well described by power-law SF distributions. Table 1
summarizes the main results of surveys conducted in Sweden, the
U.K., Zimbabwe, and Burkina Faso (21–24).
The first thing to notice is the gender-specific difference in the
number of sexual acquaintances (21–24). This difference is
manifested by the existence of two different exponents in the SF
degree distributions, one for males (␥M) and one for females
(␥F). Interestingly enough, the predominant case in Table 1 (no
matter whether data refer to time frames of 12 months or to
entire life span) consists of one exponent being smaller and the
other ⬎3. This is certainly a borderline case that requires further
investigation on the value of the epidemic threshold.
The differences found in the two exponents ␥F and ␥M have a
further implication for real data and mathematical modeling. In
an exhaustive survey, able to reproduce the whole network of
sexual contacts, the total number of female partners reported by
men should equal the total number of male sexual partners
reported by women. Mathematically, this means that the number
of links ending at population M (of size NM) equals the number
of links ending at population F (of size NF), which translates into
the following closure relation:
NF具k典F ⫽ NM具k典M.
We thus prefer to use lifecycle data collections that integrate all
these patterns and can consequently be regarded as better
statistical indicators. After all, the values reported in Table 1
indicate that both 1-year and cumulative data produce exponents
in the same range.
Theoretical Modeling. The problem of how a disease spreads in a
population consisting of two classes of individuals can be tackled
by invoking the so-called criss-cross epidemiological model (3).
As illustrated in the bipartite network of Fig. 1a, in the criss-cross
model, the two populations of individuals (NM males and NF
females) interact so that the infection can only pass from one
population to the other by crossed encounters between the
individuals of the two populations, incorporating in this way one
of the basic elements of the heterosexual spreading of STDs. We
adopted here the indexes M and F to denote quantities relative
a
[1]
Assuming that the degree distributions for the two sets are truly
␥G
SF, then PkG ⫽ (␥G ⫺ 1) ⫻ k⫺␥G/k1⫺
, with the symbol G standing
0
for the gender (G ⫽ F, M), and k0 being the minimum degree.
Moreover, if NG ⬎⬎ 1 and ␥G ⬎ 2 for any G, Eq. 1 gives the
relation between the two population sizes as
NM ⫽ NF
冉
具k典F
␥M ⫺ 2
⯝ NF
具k典M
␥F ⫺ 2
冊冉
冊
␥F ⫺ 1
,
␥M ⫺ 1
[2]
which implies that the less heterogeneous (in degree) population
must be larger than the other one.
In conclusion, the empirical observation of two different
exponents demands for a more accurate description of the
network of heterosexual contacts as bipartite SF graphs, i.e.,
graphs with two set of nodes and links connecting nodes from
different sets only. In the following section, we will consider a
graph with NM nodes, representing males and characterized by
the exponent ␥M, and NM nodes, representing females and
characterized by ␥F. Concerning the choice of the couple of
exponents from those reported in Table 1, one must be careful
that different STDs have different associated (recovery) time
scales, and that the spreading is based on the assumption that the
links are concurrent on the time scale of the disease. In this
sense, the exponents extracted from 1-year data seem better
suited to most of the STDs, with HIV being an important
exception. However, during the lifetime of sexually active individuals, sexual behavior is likely to change because of changes in
residence, marital status, age-linked sexual attitudes, etc. (27).
1400 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0707332105
b
Fig. 1. Bipartite and unipartite networks. (a) Representation of a bipartite
network accounting for heterosexual contact networks. In such a network, we
have NM (NF) nodes representing males (females), and only male–female links
are allowed. (b) A rewired version of the network in a in which the bipartite
nature is lost and the degree of the nodes is preserved. The two graphs have
the same couple of degree distributions (one for males and one for females),
although only the graph in a reflects the bipartite character of heterosexual
contact networks.
Gómez-Gardeñes et al.
vF
SF ⫹ IM O
¡ I F ⫹ I M,
threshold as a function of the first and second moments of the
male and female degree distributions as
c ⫽
c ⫽
F
I O
¡ S F,
M
¡ S M,
IM O
being M, F, M, and F the infection and recovery probabilities
for males and females. In the case of heterogeneous contact
networks, there is a further compartmentalization of the population into classes of individuals with the same degree k, i.e., the
same number of sexual partners. Denoting the fraction of males
(females) with degree k in the susceptible or infectious state by
skM and ikM (skF and ikF), respectively, and adopting a mean-field
approach (3, 12, 14), the differential equations describing the
time evolution of the densities of susceptible and infected
individuals in each population are
1
F
dt
⫽
⫺i kF共t兲
⫹ Fk关1 ⫺
i kF共t兲兴⌰ kM共t兲
[3]
and
1 dikM共t兲
⫽ ⫺i kM共t兲 ⫹ Mk关1 ⫺ i kM共t兲兴⌰ kF共t兲,
M dt
冑FM ⬎ *c ⫽
冑
具k典F具k典M
,
具k2典F具k2典M
2共␥F ⫺ 3兲共␥M ⫺ 3兲
.
k0关共␥F ⫺ 2兲共␥M ⫺ 3兲 ⫹ 共␥M ⫺ 2兲共␥F ⫺ 3兲兴
[8]
These two thresholds are equal only in the case ␥M ⫽ ␥F (studied
in ref. 15). More importantly, when ␥F ⫽ ␥M, we have *c ⱖ c,
namely the epidemic state occurs in bipartite networks for larger
transmission probabilities than in unipartite networks. This
result is good news and highlights the importance of incorporating the crossed infections scheme in the propagation of STDs.
However, as shown in Table 1, most of the real networks have
at least one exponent ␥G (G ⫽ F, M) ⬍ 3, which means that, in
most of the practical cases, the two epidemic thresholds vanish
as the system size goes to infinity, no matter the formulation used
to model the disease propagation. However, real populations are
finite, and thus the degree distributions have a finite variance
regardless of the exponents. Consequently, an epidemic threshold does always exist and, to compare unipartite with bipartite
networks, one must then pay attention to the scaling of the
threshold with the size of the population.
Finite Populations. We now analyze in more details the differences
between c and *c when one of the two exponents (say, ␥M
without loss of generality) is in the range 2 ⬍ ␥M ⬍ 3, whereas
␥F ⬎ 2. First, we derive the size scaling of the critical threshold
in unipartite graphs, c. Eq. 6 yields
c ⫽
NF具k典F ⫹ NM具k典M
.
NF具k2典F ⫹ NF具k2典M
Manipulating this expression by considering again the limit of
large (but finite) population sizes, NG ⬎⬎ 1 (G ⫽ F, M), we obtain
c ⯝
2共3 ⫺ ␥M兲/共␥M ⫺ 2兲
冋
3⫺␥M
k0 NM␥M⫺1
冉
3⫺␥F
3 ⫺ ␥M ␥F ⫺ 2
⫺1⫹
N␥FF⫺1 ⫺ 1
3 ⫺ ␥F ␥M ⫺ 2
冊册
.
The final expression for c now can be obtained by using the
closure relation of Eq. 2 for the M and F population sizes,
yielding
[5]
yielding that a necessary condition for the absence of the
epidemic threshold is the divergence of at least one of the second
moments of the degree distributions, 具k2典M and 具k2典F. Eq. 5 can
be compared with the condition obtained without taking into
account that, in heterosexual networks, the infection can occur
only between male–female couples (12, 13). In fact, working with
a unipartite representation of a sexual network, such as that
shown in Fig. 1b, with NM ⫹ NF nodes and a degree distribution
Pk ⫽ (NM PkM ⫹ NF PkF)/(NM ⫹ NF), one can express the epidemic
Gómez-Gardeñes et al.
[7]
[4]
where, G ⫽ G/G (G ⫽ F, M) are the effective transmission
probabilities. The quantities ⍜kM(t), ⍜kF(t) stand for the probabilities that a susceptible node of degree k of one population
encounters an infectious individual of the other set. Eqs. 3 and
4 have the same functional form of the equation derived in ref.
12 for unipartite networks. Neglecting degree–degree correlations, the critical condition for the occurrence of an endemic
state reduces to
冑
1 共␥F ⫺ 3兲共␥M ⫺ 3兲
,
k0 共␥F ⫺ 2兲共␥M ⫺ 2兲
*c ⫽
F
[6]
Eqs. 5 and 6 are clearly different. For real SF networks of sexual
contacts, the two thresholds are finite (in the infinity size limit)
only when the two exponents ␥M and ␥F are both ⬎3, for
example, for the 1-year number of partners in Sweden (see Table
1). In such a case, the two expressions read as follows:
vM
¡ I M ⫹ I F,
SM ⫹ IF O
dikF共t兲
2具k典M具k典F
具k典
⫽
.
具k2典 具k2典M具k典F ⫹ 具k2典F具k典M
c ⯝
2共3 ⫺ ␥M兲/共␥M ⫺ 2兲
冋
3⫺␥M
k0 N␥MM⫺1
⫹
冉 冊 冉
3 ⫺ ␥M ␥F ⫺ 2
3 ⫺ ␥F ␥M ⫺ 2
2
␥F⫺1
␥M ⫺ 1
N
␥F ⫺ 1 M
冊 册
3⫺␥F
␥F⫺1
.
In this formula, only one population size NM appears. Finally, if,
for example, ␥F ⬎ ␥M, the above equation reduces to
c ⯝
2共3 ⫺ ␥M兲 ␥M⫺3
N␥M⫺1 ,
k0共␥M ⫺ 2兲 M
[9]
PNAS 兩 February 5, 2008 兩 vol. 105 兩 no. 5 兩 1401
APPLIED PHYSICAL
SCIENCES
to male and female populations in networks of heterosexual
contacts. However, the present approach is more general and
applies to any spreading of diseases in which crossed infections
between two populations occur. In particular, we consider a
susceptible–infected–susceptible (SIS) dynamics, in which individuals can be in one of two different states, namely, susceptible
(S) and infectious (I). If S M and I M (S F, I F) stand for a male
(female) in the susceptible and infectious states, respectively, the
epidemic in the SIS criss-cross model propagates by the following
mechanisms:
case that for a given transmission probability, in the unipartite
representation shown in Fig. 1b, the epidemic would have
survived, infecting a fraction of the population, whereas when
only crossed infections are allowed, as in Fig. 1a, the same
disease would not have produced an endemic state.
Moreover, the difference between the epidemic thresholds
predicted by the two approaches increases with the system size.
This dependency is shown in Fig. 2, where we have reported, as
a function of the system size, the critical thresholds obtained by
numerically solving Eqs. 5 and 6 with the values of ␥M and ␥F
found for the lifetime distribution of sexual partners in Sweden
(21) and the U.K. (22, 23).
Table 2. Scaling exponents of the epidemic thresholds
Network
2 ⬍ ␥F ⬍ 3
␥F ⬎ 3
␣*
1
2
␣
[(3 ⫺ ␥F)/(␥F ⫺ 1) ⫹ (3 ⫺ ␥M)/(␥M ⫺ 1)]
1
(3 ⫺ ␥M)/(␥M ⫺ 1)
(3 ⫺ ␥M)/(␥M ⫺ 1)
(3 ⫺ ␥M)/(␥M ⫺ 1)
2
␣
Scaling exponents, ␣ and ␣*, of the epidemic thresholds, c ⬃ NM
and c ⬃
␣*
NM
, obtained for the SIS model on unipartite networks and when a bipartite
network is considered, respectively. The two situations considered (2 ⬍ ␥F ⬍ 3
and ␥F ⬎ 3) correspond to 2 ⬍ ␥M ⬍ 3.
which contains simultaneously the cases when 2 ⬍ ␥F ⬍ 3 and
␥F ⬎ 3.
Now we calculate the scaling of the epidemic threshold in
bipartite (heterosexual) networks, *c. Manipulating Eq. 5, *c can
also be expressed as a function of the two exponents ␥M and ␥F
and one population size:
*c ⯝
冑冉
Numerical Simulations. To check the validity of the analytical
arguments and also to explore the dynamics of the disease above
the epidemic threshold, we have conducted extensive numerical
simulations of the SIS model in bipartite and unipartite computer-generated networks. Bipartite and unipartite graphs of a
given size are built up (see Methods) having the same degree
distributions, PkM and PkF, and thus they only differ in the way the
nodes are linked. A fraction of infected individuals initially is
randomly placed on the network, and the SIS dynamics is
evolved: At each time step, susceptible individuals get infected
with probability if they are connected to an infectious one and
get recovered with probability ⫽ 1 (hence, the effective
transmission probability is ⫽ ). After a transient time, the
system reaches a stationary state in which the total prevalence of
the disease, 具I(t)典, is measured (see Methods). Finally, the results
are averaged over different initial conditions and network
realizations. Fig. 3 shows the fraction of infected individuals as
a function of /*c for several system sizes and for the bipartite
(Fig. 3 a and b) and unipartite (Fig. 3 c and d) graphs. Here, the
infection probability has been rescaled by the theoretical value
*c given by Eq. 5. The purpose of the rescaling is twofold. First,
it allows us to check the validity of the theoretical predictions
and, at the same time, provides a clear comparison of the results
obtained for bipartite networks with those obtained for the
unipartite case. Again we have used the values of ␥M and ␥F
extracted from the lifetime number of sexual partners reported
for Sweden and the U.K. (21–23). Fig. 3 indicates that the
analytical solution, Eq. 5, is in good agreement with the simulation results for the two-gender model formulation. Conversely,
when the bipartite nature of the underlying graph is not taken
into account, the epidemic threshold is underestimated, being
c/*c ⬍ 1. In addition, the error in the estimation grows as the
population size increases, in agreement with our theoretical
predictions.
B
3⫺␥M
N␥MM⫺1
⫺1
冊冋冉
␥F ⫺ 2 ␥M ⫺ 1
N
␥M ⫺ 2 ␥F ⫺ 1 M
冊
3⫺␥F
␥F⫺1
⫺1
册
,
with B ⫽ (3 ⫺ ␥M)(3 ⫺ ␥F)/[k20(2 ⫺ ␥M)(2 ⫺ ␥F)]. The above
expression, when evaluated for 2 ⬍ ␥G ⬍ 3 (G ⫽ F, M) and, for
example, ␥F ⬎ ␥M, yields
*c ⯝ B
1/2
冉
␥F ⫺ 2 ␥M ⫺ 1
␥M ⫺ 2 ␥F ⫺ 1
冊
␥F⫺3
2共␥F⫺1兲
冉
1 ␥M⫺3 ␥F⫺3
⫹
␥M⫺1 ␥F⫺1
N2M
冊.
[10]
However, when, for example, ␥F ⬎ 3, the expression reduces to
*c ⯝
冑
␥M⫺3
共3 ⫺ ␥M兲共␥F ⫺ 3兲
2共␥M⫺1兲 .
2 NM
共2 ⫺ ␥M兲共2 ⫺ ␥F兲k0
[11]
Comparing the Scalings. Although both epidemic thresholds, c
and *c, tend to zero as the population goes to infinity, the scaling
⫺␣
⫺␣*
relations, c(NM, ␥M, ␥F) ⬃ NM
and *c(NM, ␥M, ␥F) ⬃ NM
, are
characterized by two different exponents, ␣ and ␣*. Table 2
reports the expression of these two exponents as a function of ␥F
and ␥M, showing that ␣* is always smaller than ␣. In particular,
for the most common case (see Table 1), i.e., when one degree
distribution exponent is in the range ]2,3] and the other one is
⬎3, the value of ␣* found for bipartite networks is two times
smaller than ␣. As a consequence, the results show that in finite
bipartite populations the onset of the epidemic takes place at
larger values of the spreading rate. In other words, it could be the
b 1010
a 1010
λ*c
λc
λc , λ c
λ*c
λc
*
10-1
1.6
2.2
λc* 1.4
λc 1.2
10-1
λc* 1.8
λc 1.4
1
1
2
3
4
5
6
10 10 10 10 10 10
1
10
2
10
3
4
10
10
N
10
5
6
10
10-2
101
1
1
2
3
4
5
6
10 10 10 10 10 10
102
103
104
105
106
N
Fig. 2. Epidemic thresholds as a function of the total population size. The thresholds *c (Eq. 5) and c (Eq. 6) obtained for bipartite and unipartite networks
are plotted as a function of the population size N for two networks with degree distributions as those found for the sexual networks of Sweden (a) (21) and the
U.K. (b) (22). Insets show that the ratio *c /c grows as N increases, so that for a typical population size of N ⫽ 106, the thresholds for heterosexual networks are
53% (Sweden) and 130% (U.K.), respectively, larger than the values expected for the unipartite networks.
1402 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0707332105
Gómez-Gardeñes et al.
6
5
4
3
2
1
0
0.06
0.7 0.8 0.9
0.03
1
1.1 1.2
CC
0.04
·
·
·
2
1
0
0.4 0.5 0.6 0.7 0.8 0.9
1
1.1
0.02
a
·
0.04
0.03
0.04
0.7 0.8 0.9
0.02
1
1.1 1.2
0
0.5
1
1.5
2
2.5
·
2
1
0
0.4 0.5 0.6 0.7 0.8 0.9
0.02
c
0
3
0.03
CC
0.01
b
0.01
0
0.05
6
5
4
3
2
1
0
<I(t)>
<I(t)> 10
3
0.01
<I(t)>
·
0.03
0.02
0
0.05
3
3
<I(t)> 10
0.05
<I(t)> 103
0.04
·
<I(t)>
<I(t)>
0.05
<I(t)> 10 3
0.06
1
1.1
d
0.01
0
3
0
CC
λ/λ*c
0.5
1
1.5
2
2.5
3
CC
λ/λ*c
Conclusions
The inclusion of the bipartite nature of contact networks to
describe crossed infections in the spread of STDs in heterosexual
populations is seen to strongly affect the epidemic outbreak and
leads to an increase of the epidemic threshold. Our results show
that, even in the cases when the epidemic threshold vanishes in
the infinite network size limit, the epidemic incidence in finite
populations is less dramatic than actually expected for unipartite
SF networks. The results also point out that the larger the
population, the greater the gap between the epidemic thresholds
predicted by the two models, therefore highlighting the need to
accurately take into account all of the available information on
what heterosexual contact networks look like. Our results also
have important consequences for the design and refinement of
efficient degree-based immunization strategies aimed at reducing the spread of STDs. In particular, they pose new questions
on how such strategies have to be modified when the interactions
are further compartmentalized by gender and only crossed
infections are allowed. We finally stress that the present approach is generalizable to other models for disease spreading
(e.g., the ‘‘susceptible–infected–removed’’ model) and other
processes in which crossed infection in bipartite networks is the
mechanism at work.
Methods
Bipartite Network Construction. Synthetic bipartite networks construction
starts by fixing the number of males, NM, and the two exponents ␥M and ␥F of
the power-law degree distributions corresponding to males and females,
respectively. The first stage consists of assigning the connectivity kiM (i ⫽ 1, . . .,
NM) to each member of the male population by generating NM random
numbers with probability distribution PkM ⫽ AM k⫺␥M (兺k⬁0 AM k⫺␥M ⫽ 1, with
k0 ⫽ 3). The sum of these NM random numbers fixes the number of links Nl of
the network. The next step is to construct the female population by means of
an iterative process. For this purpose, we progressively add female individuals
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the Ramón y Cajal program. This work was partially supported by Spanish
DGICYT Projects FIS2006-12781-C02-01 and FIS2005-00337 and by the Italian
TO61 Istituto Nazionale di Fisica Nucleare project.
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PNAS 兩 February 5, 2008 兩 vol. 105 兩 no. 5 兩 1403
APPLIED PHYSICAL
SCIENCES
Fig. 3. Monte Carlo simulations for the U.K. and Sweden lifetime networks. The SIS phase diagram, 具I(t)典 versus /*c, is reported for synthetic networks of
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