Papers in Physics, vol. 5, art. 050003 (2013)

Received: 6 April 2013, Accepted: 3 June 2013
Edited by: G. Mindlin
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.050003

www.papersinphysics.org

ISSN 1852-4249

Invited review: Epidemics on social networks

M. N. Kuperman1,2∗

Since its �rst formulations almost a century ago, mathematical models for disease spreading
contributed to understand, evaluate and control the epidemic processes. They promoted a
dramatic change in how epidemiologists thought of the propagation of infectious diseases.
In the last decade, when the traditional epidemiological models seemed to be exhausted,
new types of models were developed. These new models incorporated concepts from graph
theory to describe and model the underlying social structure. Many of these works merely
produced a more detailed extension of the previous results, but some others triggered a
completely new paradigm in the mathematical study of epidemic processes. In this review,
we will introduce the basic concepts of epidemiology, epidemic modeling and networks, to
�nally provide a brief description of the most relevant results in the �eld.

I. Introduction

With the development of more precise and powerful
tools, the mathematical modeling of infectious dis-
eases has become a crucial tool for making decisions
associated to policies on public health. The sce-
nario was completely di�erent at the beginning of
the last century, when the �rst mathematical mod-
els started to be formulated. The rather myopic
comprehension of the epidemiological processes was
evidenced during the most dramatic epidemiologic
events of the last century, the pandemic 1918 �u.
The lack of a mathematical understanding of the
evolution of epidemics gave place to an inaccurate
analysis of the epidemiological situation and subse-
quent failed assertion of the success of the immu-
nization strategy. During the in�uenza pandemic
of 1892, a viral disease, Richard Pfei�er isolated

∗E-mail: kuperman@cab.cnea.gov.ar

1 Consejo Nacional de Investigaciones Cientí�cas y Técni-

cas, Argentina.

2 Centro Atómico Bariloche and Instituto Balseiro, 8400

S. C. de Bariloche, Argentina

bacteria from the lungs and sputum of patients.
He installed, among the medical community, the
idea that these bacteria were the cause of in�uenza.
At that moment, the bacteria was called Pfei�er's
bacillus or Bacillus in�uenzae, while its present
name keeps a reminiscence of Pfei�er's wrong hy-
pothesis: Haemophilus in�uenzae. Though there
were some dissenters, the hypothesis of linking in-
�uenza with this pathogen was widely accepted
from then on. Among the supporters of Pfei�er hy-
pothesis was William Park, at the New York City
Health Department, who in view of the fast pro-
gression of the �u in USA, developed a vaccine and
antiserum against Haemophilus in�uenzae on Oc-
tober 1918. Shortly afterwards the Philadelphia
municipal laboratory released thousands of doses
of the vaccine that was constituted by a mix of
killed streptococcal, pneumococcal, and H. in�uen-
zae bacteria. Several other attempts to develop
similar vaccines followed this initiative. However,
none of these vaccines prevented viral in�uenza in-
fection. The present consensus is that they were
even not protective against the secondary bacte-
rial infections associated to in�uenza because the

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

37 39 41 43 45 47 49 51 1 3 5
0

50

100

150

200

 

 

D
ea

th
 R

at
e 

(p
er

 1
05

 )

WEEK

Figure 1: Weekly �Spanish in�uenza� death rates
in Baltimore (circles) and San Francisco (squares)
from 1918 to 1919. Data taken from Ref. [1].

vaccine developers at that time could not identify,
isolate, and produce all the disease-causing strains
of bacteria. Nevertheless, a wrong evaluation of the
evolution of the disease and a lack of epidemiologi-
cal knowledge led to the conclusion that the vaccine
was e�ective. If we look at Fig. 1 corresponding
to the weekly in�uenza death rates in a couple of
U.S. cities taken from Ref. [1], we observe a re-
markable decay after vaccination, in week 43. This
decay was inaccurately attributed to the e�ect of
vaccination as it corresponds actually to a normal
and expected development of an epidemics without
immunization.

The inaccurate association between H. in�uenzae
and in�uenza persisted until 1933, when the viral
etiology of the �u was established. But Pfei�er's
in�uenza bacillus, �nally named Haemophilus in-
�uenzae, accounts in its denomination for this per-
sistent mistake.

The formulation of mathematical models in epi-
demiology has a tradition of more than one century.
One of the �rst successful examples of the mathe-
matical explanation of epidemiological situations is
associated with the study of Malaria. Ronald Ross
was working at the Indian Medical Service during
the last years of the 19th century when he discov-
ered and described the life-cycle of the malaria par-
asite in mosquitoes and developed a mathematical
model to analyze the dynamics of the transmission

of the disease [2�4]. His model linked the density
of mosquitoes and the incidence of malaria among
the human population. Once he had identi�ed
the anopheles mosquitoes as the vector for malaria
transmission, Ross conjectured that malaria could
be eradicated if the ratio between the number of
mosquitoes and the size of the human population
was carried below a threshold value. He based his
analysis on a simple mathematical model.
Ross' model was based on a set of deterministic

coupled di�erential equations. He divided the hu-
man population into two groups, the susceptible,
with proportion Sh and the infected, with propor-
tion Ih. After recovery, any formerly infected in-
dividual returned to the susceptible class. This is
called a SIS model. The mosquito population was
also divided into two groups (with proportions Sm
and Im), with no recovery from infection. Consid-
ering equations for the fraction of the population
in each state, we have S + I = 1 for both humans
and mosquitoes and the model is reduced to a set
of two coupled equations

dIh
dt

= abfIm(1− Ih)−rIh (1)

dIm
dt

= acIh(1− Im)−µmIm,

where a is the man biting rate, b is the propor-
tion of bites that produce infection in humans, c
is the proportion of bites by which one susceptible
mosquito becomes infected, f is the ratio between
the number of female mosquitoes and humans, r is
the average recovery rate of human and µm is the
rate of mosquito mortality.
One of the parameters to quantify the intensity

of the epidemics propagation is the basic reproduc-
tive rate R0, that measures the average number of
cases produced by an initial case throughout its
infectious period. R0 depends on several factors.
Among them, we can mention the survival time of
an infected individual, the necessary dose for infec-
tion, the duration of infectiousness in the host, etc.
R0 allows to determine whether or not an infec-
tious disease can spread through a population: an
infection can spread in a population only if R0 > 1
and can be maintained in an endemic state when
R0 = 1 [5]. In the case of malaria, R0 is de�ned
as the number of secondary cases of malaria arising
from a single case in an susceptible population. For

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

the model described by Eq. (1)

R0 =
ma2bc

rµm
. (2)

It is clear that the choice of the parameters a�ects
R0. The main result is that it is possible to re-
duce R0 by increasing the mosquito mortality and
reducing the biting rate. For his work on malaria,
Ross was awarded the Nobel Prize in 1902.
Ross' pioneering work was later extended to

include other ingredients and enhance the pre-
dictability power of the original epidemiological
model [5�11].
Some years after Ross had proposed his model, a

couple of seminal works established the basis of the
current trends in mathematical epidemiology. Both
models consider the population divided into three
epidemiological groups or compartments: suscepti-
ble (S), infected (I) and recovered (R).
On the one hand, Kermack and McKendrick [12]

proposed a SIR model that expanded Ross' set of
di�erential equations. The model did not consider
the existence of a vector, but a direct transmission
from an infected individual to a susceptible one.
A particular case of the original model, in which
there is no age dependency of the transmission and
recovery rate, is the classical SIR model that will
be explained later.
On the other hand, Reed and Frost [13] devel-

oped a SIR discrete and stochastic epidemic model
to describe the relationship between susceptible,
infected and recovered immune individuals in a
population. It is a chain binomial model of epi-
demic spread that was intended mainly for teach-
ing purposes, but that is the starting point of many
modern epidemiological studies. The model can
be mapped into a recurrence equation that de�nes
what will happen at a given moment depending on
what has happened in the previous one,

It+1 = St(1− (1−ρ)It), (3)

where It is the number of cases at time t, St is the
number of susceptible individuals at time t and ρ
is the probability of contagion.
The basic assumption of these SIR models, which

is present in almost any epidemiological work, is
that the infection is spread directly from infectious
individuals to susceptible ones after a certain type
of interaction between them. In turn, these newly

infected individuals will develop the infection to be-
come infectious. After a de�ned period of time, the
infected individuals heal and remain permanently
immune. The interaction between any two individ-
uals of the population is considered as a stochas-
tic process with a de�ned probability of occurrence
that most of the deterministic model translates into
a contact rate.
Given a closed population and the number of in-

dividuals in each state, the calculation of the evo-
lution of the epidemics is straightforward. The epi-
demic event is over when no infective individuals
remain.
While many classic deterministic epidemiologi-

cal models were having success at describing the
dynamics of an infectious disease in a population,
it was noted that many involved processes could be
better described by stochastic considerations and
thus a new family of stochastic models was devel-
oped [14�19]. Sometimes, deterministic models in-
troduce some colateral mistakes due to the contin-
uous character of the involved quantities.An exam-
ple of such a case is discussed in Ref. [20]. In Ref.
[21], the authors proposed a deterministic model
to describe the prevalence of rabies among foxes in
England. They predicted a sharp decaying preva-
lence of the rabies up to negligible levels, followed
by an unexpected new outbreak of infected foxes.
The spontaneous outbreak after the apparent dis-
appearing of the rabies is due to a �ctitious very
low endemic level of infected foxes, as explained in
Ref. [20]. The former one is one among several ex-
amples of how stochastic models contributed to a
better understanding and explanation of some ob-
served phenomena but, as their predecessors, they
considered a mean �eld scheme in the set of di�er-
ential equations.
Traditional epidemiological models have success-

fully describe the generalities of the time evolu-
tion of epidemics, the di�erential e�ect on each
age group, and some other relevant aspects of an
epidemiological event. But all of them are based
on a fully-mixing approximation, proposing that
each individual has the same probability of getting
in touch with any other individual in the popula-
tion. The real underlying pattern of social contacts
shows that each individual has a �nite set of ac-
quaintances that serve as channels to promote the
contagion. While the fully mixed approximation al-
lows for writing down a set of di�erential equations

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

and a further exploitation of a powerful analytic
set of tools, a better description of the structure
of the social network provides the models with the
capacity to compute the epidemic dynamics at the
population scale from the individual-level behavior
of infections, with a more accurate representation
of the actual contact pattern. This, in turn, re�ects
some emergent behavior that cannot be reproduced
with a system based on a set of di�erential equa-
tion under the fully mixing assumption. One of
the most representative examples of this behavior
is the so called herd immunity, a form of immunity
that occurs when the vaccination of a signi�cant
portion of the population is enough to block the
advance of the infection on other non vaccinated
individuals. Additionally, some network models al-
low also for an analytic study of the described pro-
cess. It is not surprising then that during the last
decade, a new tendency in epidemiological model-
ing emerged together with the inclusion of complex
networks as the underlying social topology in any
epidemic event. This new approach proves to con-
tribute with a further understanding of the dynam-
ics of an epidemics and unveils the crucial e�ect of
the social architecture in the propagation of any
infectious disease.

In the following section, we will introduce some
generalities about traditional epidemiological mod-
els. In section III, we will present the most com-
monly used complex networks when formulating an
epidemiological model. In section IV, we will de-
scribe the most relevant results obtained by mod-
eling epidemiological processes using complex net-
works to describe the social topology. Next, we will
introduce the concept of herd protection or immu-
nity and a discussion of some of the works that treat
this phenomenon.

II. Basic Epidemiological Models

Two main groups can be singled out among the
deterministic models for the spread of infectious
diseases which are transmitted through person-to-
person contact: the SIR and the SIS. The names
of these models are related to the di�erent groups
considered as components of the population or epi-
demiological compartments: S corresponds to sus-
ceptible, I to infected and R to removed. The S
group represents the portion of the population that

has not been a�ected by the disease but may be in-
fected in case of contact with a sick person. The
I group corresponds to those individuals already
infected and who are also responsible for the trans-
mission of the disease to the susceptible group. The
removed group R includes those individuals recov-
ered from the disease who have temporary or per-
manent immunity or, eventually, those who have
died from the illness and not from other causes.
These models may or may not include the vital dy-
namics, associated with birth and death processes.
Its inclusion depends on the length of time over
which the spread of the disease is studied.

i. The SIR Model

As mentioned before, in 1927, Kermack and McK-
endrick [12] developed a mathematical model in
which they considered a constant population di-
vided into three epidemiological groups : suscep-
tible, infected and recovered. The equations of a
SIR model are

dS

dt
= −βSI

dI

dt
= βSI −γI (4)

dR

dt
= γI,

where the involved quantities are the proportion
of individuals in each group. As the population is
constant,

S(t) + I(t) + R(t) = 1. (5)

The SIR model is used when the disease under
study confers permanent immunity to infected in-
dividuals after recovery or, in extreme cases, it kills
them. After the contagious period, the infected in-
dividual recovers and is included in the R group.
These models are suitable to describe the behav-
ior of epidemics produced by virus agent diseases
(measles, chickenpox, mumps, HIV, poliomyelitis)
[22].
The model formulated through Eq. (4) assumes

that all the individuals in the population have the
same probability of contracting the disease with
a rate of β, the contact rate. The number of in-
fected increases proportionally to both the number

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

of infected and susceptible. The rate of recovery or
removal is proportional to the number of infected
only. γ represents the mean recovery rate, ( 1/γ
is the mean infective period). It is assumed that
the incubation time is negligible and that the rates
of infection and recovery are much faster than the
characteristic times associated to births and deaths.
Usually, the initial conditions are set as

S(0) > 0, I(0) > 0 and R(0) = 0. (6)

It is straightforward to show that

dI

dt

∣∣∣∣
t=0

= I(0)(βS(0)−γ), (7)

and that the sign of the derivative depends on the
value of Sc =

γ
β
. When S(t) > Sc, the derivative

is positive and the number of infected individuals
increases. When S(t) goes below this threshold, the
epidemic starts to fade out.
A rather non intuitive result can be obtained

from Eq. 4. We can write

dS

dR
= −

S

ρ

⇒ S = S0 exp[−R/ρ] ≥ S0 exp[−N/ρ] > 0
⇒ 0 < S(∞) ≤ N. (8)

The epidemics stops when I(t) = 0, so we can set
I(∞) = 0, so R(∞) = N −S(∞). From (8),

S(∞) = S0 exp
[
−
R(∞)
ρ

]
= S0 exp

[
−
N −S(∞)

ρ

]
. (9)

The last equation is a transcendent expression with
a positive root S(∞).
Taking (9), we can calculate the total number of

susceptible individuals throughout the whole epi-
demic process

Itotal = I0 + S0 −S(∞). (10)
As I(t) → 0 and S(t) → S(∞) > 0, we conclude

that when the epidemics end, there is a portion of
the population that has not been a�ected
The previous model can be extended to include

vital dynamics [23], delays equations [24], age struc-
tured population, migration [25], and di�usion. In

0 10 20 30 40 50
0.0

0.2

0.4

0.6

0.8

1.0

 

 

P
ro

po
rt

io
n

Time (arb. units)

Susceptible

Recovered

Infected

Figure 2: Temporal behavior of the proportion of
individuals in each of the three compartments of
the SIR model.

any case, all these generalizations only introduce
some slight changes on the steady states of the sys-
tem, or in the case of spatially extended models,
travelling waves [26].
Figure 2 displays the typical behavior of the den-

sity of individuals in each of the epidemiological
compartments described by Eq. (4). Compare this
with the pattern shown in Fig. 1.

ii. The SIS Model

The SIS model assumes that the disease does not
confer immunity to infected individuals after recov-
ery. Thus, after the infective period, the infected
individual recovers and is again included in the S
group. Therefore, the model presents only two epi-
demiological compartments, S and I. This model
is suitable to describe the behavior of epidemics
produced by bacterial agent diseases (meningitis,
plague, venereal diseases) and by protozoan agent
diseases (malaria) [22]. We can write the equations
for a general SIS model assuming again that the
population is constant,

dS

dt
= −βSI + γI

dI

dt
= βSI −γI. (11)

As the relation S + I = 1 holds, Eq. (11) can be
reduced to a single equation,

dI

dt
= (β −γ)I −βI2. (12)

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

The solution of this equation is

I(t) = (1−
γ

β
)

C exp[(γ −β)t]
1 + C exp[(γ −β)t]

, (13)

where C is de�ned by the initial conditions as

C =
βi0

β(1− i0)−γ
. (14)

If I0 is small and β > γ, the solution is a logistic
growth that saturates before the whole population
is infected, the stationary value is Is =

β−γ
β

. It can

be shown that R0 = β/γ. This sets the condition
for the epidemic to persist.

iii. Other models

The literature on epidemiological models includes
several generalizations about the previous ones to
adapt the description to the particularities of a spe-
ci�c infectious disease [27]. One possibility is to
increase the number of compartments to describe
di�erent stages of the state of an individual during
the epidemic spread. Among these models, we can
mention the SIRS, a simple extension of the SIR
that does not confer a permanent immunity to re-
covered individuals and after some time they rejoin
the susceptible group,

dS

dt
= −βSI + +λR

dI

dt
= βSI −γI (15)

dR

dt
= γI −λR.

Other models include more epidemiological
groups or compartments, such as the SEIS and
SEIR model, that take into consideration the ex-
posed or latent period of the disease, by de�ning
an additional compartment E.
There are several diseases in which there is a

vertical transient immunity transmission from a
mother to her newborn. Then, each individual is
born with a passive immunity acquired from the
mother. To indicate this, an additional group P is
added.
The range of possibilities is rather extended, and

this is re�ected in the title of Ref. [27]: �A thou-
sand and one epidemiological models�. There are a

Figure 3: Transfer diagram for a SEIRS models.
Taken from Ref. [27].

lot of possibilities to de�ne the compartment struc-
ture. Usually, this structure is represented as a
transfer chart indicating the �ow between the com-
partments and the external contributions. Figure 3
shows an example of a diagram for a SEIRS model,
taken from Ref. [27].
Horizontal incidence refers to a contagion due to

a contact between a susceptible and infectious indi-
vidual, vertical incidence account for the possibility
for the o�spring of infected parents to be born in-
fected, such as with AIDS, hepatitis B, Chlamydia,
etc.
Many of the previous models have been ex-

panded, including stochastic terms. One of the
most relevant di�erences between the deterministic
and stochastic models is their asymptotic behavior.
A stochastic model can show a solution converg-
ing to the disease-free state when the deterministic
counterpart predicts an endemic equilibrium. The
results obtained from the stochastic models are gen-
erally expressed in terms of the probability of an
outbreak and of its size and duration distribution
[14�19].

III. Complex Networks

A graph or network is a mathematical represen-
tation of a set of objects that may be connected
between them through links. The interconnected
objects are represented by the nodes (or vertices) of
the graph while the connecting links are associated
to the edges of the graph. Networks can be char-
acterized by several topological properties, some of
which will be introduced later. Social links are pre-
ponderantly non directional (symmetric), though
there are some cases of social directed networks.
The set of nodes attached to a given node through
these links is called its neighborhood. The size of
the neighborhood is the degree of the node.
While the study of graph theory dates back to the

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

pioneering works of Erdös and Renyi in the 1950s
[28], their gradual colonization of the modern epi-
demiological models has only started a decade ago.
The attention of modelers was drawn to graph the-
ory when some authors started to point out that
the social structure could be mimicked by networks
constructed under very simple premises [30, 34].
Since then, a huge collection of computer-generated
networks have been studied in the context of disease
transmission. The underlying rationale for the use
of networks is that they can represent how individu-
als are distributed in social and geographical space
and how the contacts between them are promoted,
reinforced or inhibited, according to the rules of so-
cial dynamics. When the population is fully mixed,
each individual has the same probability of coming
into contact with any other individual. This as-
sumption makes it possible to calculate the e�ective
contact rates β as the product of the transmission
rate of the disease, the e�ective number of contacts
per unit time and the proportion of these contacts
that propagate the infection. The formulation of a
mean �eld model is then straightforward. However,
in real systems, the acquaintances of each individ-
ual are reduced to a portion of the whole popula-
tion. Each person has a set of contacts that shapes
the local topology of the neighborhood. The whole
social architecture, the network of contacts, can be
represented with a graph.
In the limiting case when the mean degree of the

nodes in a network is close to the total number of
nodes, the di�erence between a structured popula-
tion and a fully mixed one fades out. The di�er-
ences are noticeable when the network is diluted,
i.e., the mean degree of the node is small compared
with the size of the network. This will be a neces-
sary condition for all the networks used to model
disease propagation. In the following paragraphs,
we will introduce the most common families of net-
works used for epidemiological modeling.
Lattices. When incorporating a network to

a model, the simplest case is considering a grid
or a lattice. In a squared d dimensional lattice,
each node is connected to 2d neighbors. Individu-
als are regularly located and connected with adja-
cent neighbors; therefore, contacts are localized in
space. Figure 4 shows, among others, an example
of a two dimensional square lattice
Small-world networks. The concept of Small

World was introduced by Milgram in 1967 in order

Figure 4: Scheme of four kinds of networks: (a)
Lattice, (b)scale free, (c) Exponential, (d) Small
World.

to describe the topological properties of social com-
munities and relationships [29]. Some years ago,
Watts and Strogatz introduced a model for con-
structing networks displaying topological features
that mimic the social architecture revealed by Mil-
gram. In this model of Small World (SW) net-
works a single parameter p, running from 0 to 1,
characterizes the degree of disorder of the network,
ranging from a regular lattice to a completely ran-
dom graph [30]. The construction of these networks
starts from a regular, one-dimensional, periodic lat-
tice of N elements and coordination number 2K.
Each of the sites is visited, rewiring K of its links
with probability p. Values of p within the interval
[0,1] produce a continuous spectrum of small world
networks. Note that p is the fraction of modi�ed
regular links. A schematic representation of this
family of networks is shown in Fig. 5.

Figure 5: Representation of several Small World
Networks constructed according the algorithm pre-
sented in Ref. [30]. As the disorder degree in-
creases, there number of shortcuts grow replacing
some of the original (ordered network) links.

To characterize the topological properties of the

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

SW networks, two magnitudes are calculated. The
�rst one, L(p), measures the mean topological dis-
tance between any pair of elements in the network,
that is, the shortest path between two vertices,
averaged over all pairs of vertices. Thus, an or-
dered lattice has L(0) ∼ N/K, while, for a ran-
dom network, L(1) ∼ ln(N)/ln(K). The second
one, C(p), measures the mean clustering of an ele-
ment's neighborhood. C(p) is de�ned in the follow-
ing way: Let us consider the element i, having ki
neighbors connected to it. We denote by ci(p) the
number of neighbors of element i that are neigh-
bors among themselves, normalized to the value
that this would have if all of them were connected
to one another; namely, ki(ki −1)/2. Now, C(p) is
the average, over the system, of the local clusteri-
zation ci(p). Ordered lattices are highly clustered,
with C(0) ∼ 3/4, and random lattices are charac-
terized by C(1) ∼ K/N. Between these extremes,
small worlds are characterized by a short length
between elements, like random networks, and high
clusterization, like ordered ones.

10-3 10-2 10-1 100

0.0

0.2

0.4

0.6

0.8

1.0

 

 

L & C

p

Figure 6: In this �gure, we show the mean values of
the clustering coe�cient C and the path length L
as a function of the disorder parameter p. Note the
fast decay of L and the presence of a region where
the value adopted by L is similar to the one corre-
sponding to total disorder, while the value adopted
by C is close to the one corresponding to the or-
dered case.

Other procedures for developing similar social
networks have been proposed in Ref. [31] where in-
stead of rewiring existing links to create shortcuts,
the procedure add links connecting two randomly
chosen nodes with probability p. In Fig. 7, we show

an example, analogous to the one shown in Fig. 5.

Figure 7: Representation of several Small World
Networks constructed according the algorithm pre-
sented in Ref. [31]. As the disorder degree in-
creases, three number of shortcuts as well as the
number of total links grow.

Random networks. There are di�erent fami-
lies of networks with random genesis but displaying
a wide spectra of complex topologies. In random
networks, the spatial position of individuals is irrel-
evant and the links are randomly distributed. The
iconic Erdös-Rényi (ER) random graphs are built
from a set of nodes that are randomly connected
with probability p, independently of any other ex-
isting connection. The degree distribution, i.e., the
number of links associated to each node, is binomial
and when the number of nodes is large, it can be ap-
proximated by a Poisson distribution [32]. In Ref.
[33], the authors propose a formalism based on the
generating function that permits to construct ran-
dom networks with arbitrary degree distribution.
The mechanism of construction also allows for fur-
ther analytic studies on these networks. In partic-
ular, networks can be chosen to have a power law
degree distribution. This case will be presented in
the next paragraphs.
Scale-free network. As mentioned before, one

of the most revealing measures of a network is its
degree of distribution, i.e., the distribution of the
number of connections of the nodes. In most real
networks, it is far from being homogeneous, with
highly connected individuals on one extreme and
almost isolated nodes on the other. Scale-free net-
works provide a means of achieving such extreme
levels of heterogeneity.
Scale-free networks are constructed by adding

new individuals to a core, with a connection mecha-
nism that imitates the underlying process that rules

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

Figure 8: This �gure shows examples of (a) ER and
(b) BA networks. The �gure also displays the con-
nectivity distribution P(k), that follows a binomial
distribution for the ER networks and a power law
for BA networks.

the choice of social contacts. The Barabási - Albert
(BA) model algorithm, one of the triggers of the
present huge interest on scale-free networks, uses
a preferential attachment mechanism [34]. The al-
gorithm starts from a small nucleus of connected
nodes. At each step, a new node is added to the
network and connected to m existing nodes. The
probability of choosing a node pi is proportional to
the number of links that the existing node already
has

pi =
ki∑
j kj

,

where ki is the degree of node i. That means that
the new nodes have a preference to attach them-
selves to the most �popular� nodes. One salient
feature of these networks is that their degree dis-
tribution is scale-free, following a power law of the
form

P (k) ∼ k−3.

A sketch of the typical topology of the last two
networks is shown in Fig. 8. While the degree
distribution of the ER network has a clear peak
and is close to homogeneous, the topology of the
BA network is dominated by the presence of hub,
highly connected nodes. The �gure also displays
the typical degree distribution P(k) for each case.

Over the last years, many other attachment
mechanisms have been proposed to obtain scale-
free networks with other adjusted properties such
as the clustering coe�cient, higher moments of the
degree distribution [35�38].

Coevolutive or adaptive topology. When
one of the former examples of networks is chosen
as a model for the social woven, there is an im-
plicit assumption: the underlying social topology is
frozen. However, this situation does not re�ect the
observed fact that in real populations, social and
migratory phenomena, sanitary isolation or other
processes can lead to a dynamic con�guration of
contacts, with some links being eliminated, other
being created. If the time span of the epidemics
is long enough, the social network will change and
these changes will not be re�ected if the topology
remains �xed. This is particularly important in
small groups. The social dynamics, including the
epidemic process, can shape the topology of the
network, creating a feedback mechanism that can
favor or attempt against the propagation of an in-
fectious disease. For this reason, some models con-
sider a coevolving network, with dynamic links that
change the aspect of the networks while the epi-
demics occur.

IV. Epidemiological Models on Net-

works

In this section, we will discuss several models based
on the use of complex networks to mimic the so-
cial architecture. The discussion will be organized
according to the topology of these underlying net-
works.

Lattices. Lattices were the �rst attempt to rep-
resent the underlying topology of the social con-
tacts and thus to analyze the possible e�ect of in-
teractions at the individual level. These models
took distance from the paradigmatic fully mixed
assumption and focused on looking for those phe-
nomena that a mean �eld model could not explain.
Still, the lattices cannot fully capture the role of in-
homogeneities. As the individuals are located on a
regular grid, mostly two dimensional, the neighbor-
hood of each node is reduced to the adjacent nodes,
inducing only short range or localized interactions.
A typical model considers that the nodes can be in
any of the epidemiological states or compartments.

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

The dynamic of the epidemics evolves through a
contact process [39] and the evolutive rules do not
di�er too much from traditional cellular automata
models [40]. Disease transmission is modeled as
a stochastic process. Each infected node has a
probability pi of infecting a neighboring suscepti-
ble node. Once infected, the individuals may re-
cover from infection with a probability pr; i.e., the
infective stage lasts typically 1/pr. From the infec-
tive phase, the individuals can move back to the
susceptible compartment or the recovered phase,
depending on whether the models are SIS or SIR.
Usually, a localized infectious focus is introduced
among the population. The transient shows a local
and slow development of the disease that at the ini-
tial stage involves the growing of a cluster, with the
infection propagating at its boundary, like a travel-
ing wave. After the initial transient, SIS, SIR and
SIRS models behave in di�erent ways. The initially
local dynamics that can or cannot propagate to the
whole system is what introduces a completely new
behavior in this spatially extended model. In Ref.
[41], the author argued the infective clusters behave
as the clusters in the directed percolation model.
Figure 9 shows an example of the behavior of the
asymptotic value of infected individuals under SIS
dynamics in a two dimensional square lattice. The
�gure re�ects the results found in Ref. [42]. The
parameter f is associated to the infectivity of in-
fectious individuals, closely related to the contact
rate. We observe the inset displaying the scaling of
the data with a power-like curve A|f − fc|α, with
α ≈ 0.5 [42].
As mentioned before, Kermack and Mckendrick

[12] proved the existence of a propagation thresh-
old for the disease invading a susceptible popula-
tion. The lattice based SIR models introduce a
di�erent threshold. The simulations show that epi-
demics can just remain localized around the initial
focus or turn into a pandemic, a�ecting the entire
population. The most dramatic examples of real
pandemic are the Black Plague between the 1300
and 1500 and the Spanish Flu, in 1917-1918. Both
left a wake of death and terror while crossing the
European continent. The predicted new threshold
established a limit below in which the pandemic
behaviour is not achieved.
Some works about epidemic propagation on lat-

tices are analogous to forest �re models [43], with
the characteristic feature that the frequency dis-

Figure 9: SIS model. Asymptotic value of infected
individuals as a function of the infectivity of infec-
tious individuals. The inset displays the scaling of
the data with a power-like curve A|f − fc|α, with
α ≈ 0.5. Adapted from Ref. [42].

Figure 10: SIR model. Asymptotic value of suscep-
tible individuals as a function of the infectivity of
infectious individuals. The inset displays the scal-
ing of the data with a power-like curve A|f −fc|α,
with α ≈ 0.5. Adapted from Ref. [42].

tributions of the epidemic sizes and duration obey
a power-law. In Ref. [44, 45], the authors exploit
these analogies to explain the observed behavior of
measles, whooping cough and mumps in the Faroe
Islands. The observed data display a power-like
behavior.

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

Random networks. Most of the models based
on random graphs were previous to the renewed in-
terest on complex networks. A simple but e�ective
idea for the study of the dynamics of diseases on
random networks is the contact process proposed
in Ref. [46] that produces a branching phenomena
while the infection propagates. In Ref. [47], the
authors use a E-R network with an approximately
Poisson degree distribution. A common feature to
all these models is that the rate of the initial tran-
sient growth is smaller than the corresponding to
similar models in fully-mixed populations. This ef-
fect can be easily understood noting that, on the
one hand, the degree of a given initially infected
node is typically small, thus having a limited num-
ber of susceptible contacts. On the other hand,
there is a self limiting process due to the fact that
the same infection propagation predates the local
availability of susceptible targets.
A di�erent analytical approach to random net-

works is presented in Ref. [48]. The author shows
that a family of variants of the SIR model can be
solved exactly on random networks built by a gen-
erating function method and appealing to the for-
malism of percolation models. The author analyzes
the propagation of a disease in networks with arbi-
trary degree distributions and heterogeneous infec-
tiveness times and transmission probabilities. The
results include the particular case of scale-free net-
works, that will be discussed later.
Small-world networks. As mentioned above,

regular networks can exhibit high clustering but
long path lengths. On the other extreme, random
networks have a lot of shortcuts between two dis-
tant individuals, but a negligible clustering. Both
features a�ect the propagative behavior on any
modeled disease. The spread of infectious dis-
eases on SW networks has been analyzed in several
works. The interested was triggered by the fact
that even a small number of random connections
added to a regular lattice, following for example
the algorithm described in Ref. [30], produces un-
expected macroscopic e�ects. By sharing topologi-
cal properties from random and ordered networks,
SW networks can display complex propagative pat-
terns. On the one hand, the high level of cluster-
ing means that most infection occurs locally. On
the other hand, shortcuts are vehicles for the fast
spread of the epidemic to the entire population.
In Ref. [51], the authors study a SI model and

show that shortcuts can dramatically increase the
possibility of an epidemic event. The analysis is
based on bond percolation concepts. While the
result could be easily anticipated due to the long
range propagative properties of shortcuts, the au-
thors �nd an important analytic result. It was a
study of a SIRS models that showed for the �rst
time the evidence of a dramatic change in the be-
havior of an epidemic due to changes in the under-
lying social topology [52]. By speci�cally analyzing
the e�ect of clustering on the dynamics of an epi-
demics, the authors show that a SIRS model on a
SW network presents two distinct types of behav-
ior. As the rewiring parameter p increases, the sys-
tem transits from an endemic state, with a low level
of infection to periodic oscillations in the number of
infected individuals, re�ecting an underlying syn-
chronization phenomena. The transition from one
regime to the other is sharp and occurs at a �nite
value of p. The reason behind this phenomenon
is still unknown. Figure 11 shows the temporal
behavior of the number of infected individuals for
three values of the rewiring parameter p, as found
in Ref. [52].

0 250 500 750 1000
0.0

0.2

0.4

 

t

 p = 0.9 

0.0

0.2
 

I(
t)

 p = 0.2 

0.0

0.1

0.2

 

 

 p = 0.01

Figure 11: Asymptotic behavior of the number of
infected individuals in three SW networks with dif-
ferent degrees of disorder p. The emergence of
a synchronized pattern is evident in the bottom
graph.

It would not be responsible to a�rm that SW
networks re�ect all the real social structures. How-
ever, they capture essential aspects of such orga-
nization that play central roles in the propagation

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

of a diseases, namely, the clustering coe�cient and
the short social distance between individuals. Un-
derstanding that there are certain limitations, SW
networks help to mimic di�erent social organiza-
tions that range from rural population to big cities.
There are more sophisticated models of networks
with topologies that are more closely related to real
social organizations at large scale. These networks
are characterized by a truncated power law distri-
bution of the degree of the nodes and by values of
clustering and mean distance corresponding to the
small world regime.
Scale-free networks. Scale-free networks cap-

tured the attention of epidemiologists due to the
close resemblance between their extreme degree
distribution and the pattern of social contacts in
real populations. A power law degree distribution
presents individuals with many contacts and who
play the role of super-spreaders. A higher num-
ber of contacts implies a greater risk of infection
and correspondingly, a higher �success� as an in-
fectious agent. Some scale-free networks present
positive assortativity. That translates into the fact
that highly connected nodes are connected among
them. This local structures can be used to model
the existence of core groups of high-risk individuals,
that help to maintain sexually transmitted diseases
in a population dominated by long-term monoga-
mous relationships [53]. Models of disease spread
through scale-free networks showed that the infec-
tion is concentrated among the individuals with
highest degree [48, 54]. One of the most surpris-
ing results is the one found in Ref. [54]. There, the
authors show that no matter the values taken by
the relevant epidemiological parameter, there is no
epidemic threshold. Once installed in a scale-free
network, the disease will always propagate, inde-
pendently of R0. Remember that when analyzed
under the fully mixed assumption, the studied SIS
model has a threshold. The authors perform ana-
lytic and numerical calculations of the propagation
of the disease, to show the lack of thresholds. Later,
in Ref. [55], it was pointed out that networks with
divergent second moments in the degree distribu-
tion will show no epidemic threshold. The B A
network ful�lls this condition. In Ref. [56,57], the
authors analyze the structure of di�erent networks
of sexual encounters, to �nd that it has a pattern of
contact closely related to a power law. They also
discuss the implications of such structure on the

propagation of venereal diseases

Co-evolutionary networks. Co-evolutionary
or adaptive networks take into account the own
dynamics of the social links. In some occasions,
the characteristic times associated to changes in
social connections are comparable with the time
scales of an epidemic process. Some other times,
the presence of n infectious core induces changes in
social links. Consider for example a case when the
population of susceptible individuals after learning
about the existence of infectious individuals try to
avoid them, or another case when the health poli-
cies promote the isolation of infectious individuals
[58]. The behavior of models based on adaptive net-
work is determined by the interplay of two di�erent
dynamics that sometimes have competitive e�ects.
On the one hand, we have the dynamics of the dis-
ease propagation. On the other hand, the network
dynamics that operates to block the advance of the
infection. The later is dominated by the rewiring
rate of the network, which a�ects the fraction of
susceptible individuals connected to infective ones.
The most obvious choice is to eliminate the infec-
tious contacts of a susceptible individual by delet-
ing or replacing them with noninfectious ones. The
net e�ect is an e�ective reduction of the infection
rate. While static networks typically predict either
a single attracting endemic or disease-free state,
the adaptive networks show a new phenomenon, a
bistable situation shared by both states. The bista-
bility appears for small rewiring rates [58�61]. In
Ref. [61], the authors consider a contact switching
dynamics. All links connecting a susceptible agents
with an infective one is broken with a rate r. The
susceptible node is then connected to a new neigh-
bor, randomly chosen among the entire population.
The authors show that reconnection can completely
prevent an epidemics, eliminating the disease. The
main conclusion is that the mechanism that they
propose, contact switching, is a robust and e�ec-
tive control strategy. Figure 12 displays the re-
sults found in Ref. [61], where two completely dif-
ferent types of behavior can be distinguish as the
rewiring parameter r changes. The crossover from
one regime to the other is a second order phase
transition.

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

λ

Figure 12: These two panels show the equilibrium
fraction number of infected individuals, as a func-
tion of the infectivity of the disease, λ. Lines are
analytic results, symbols are numerical simulations.
Adapted from Ref. [61].

V. Immunization in networks

Any epidemiological model can reproduce the fact
that the number of individuals in a population who
are e�ectively immune to a given infection depends
on the proportion of previously infected individu-
als and the proportion who have been e�ciently
vaccinated.

For some time, the epidemiologists knew about
an emerging e�ect called herd protection (or herd
immunity). They discovered the occurrence of a
global immunizing e�ect veri�ed when the vaccina-
tion of a signi�cant portion of a population provides
protection for individuals who have not or cannot
developed immunity. Herd protection is particu-
larly important for diseases transmitted from per-
son to person. As the infection progresses through
the social links, its advance can be disrupted when
many individuals are immune and their links to non
immune subjects are no longer valid channels of
propagation. The net e�ect is that the greater the
proportion of immune individuals is, the smaller
the probability that a susceptible individual will
come into contact with an infectious one. The vac-
cinated individuals will not contract neither trans-
mit the disease, thus establishing a �rewall between
infected and susceptible individuals.

While taking pro�t from the herd protection is

far from being an optimal public health policy, it
is still taken into consideration when individuals
cannot be vaccinated due, for example, to immune
disorders or allergies. The herd protection e�ect is
equivalent to reduce the R0 of a disease. There is
a threshold value for the proportion of necessary
immune individuals in a population for the disease
not to persist or propagate. Its value depends on
the e�cacy of the vaccine but also on the virulence
of the disease and the contact rate. If the herd
e�ect reduces the risk of infection among the un-
infected enough, then the infection may no longer
be sustainable within the population and the infec-
tion may be eliminated. In a real population, the
emergence of herd immunity is closely related to the
social architecture. While many fully mixed mod-
els can describe the phenomenon, the real e�ect is
much more accurately reproduced by models based
on Social Networks. One of the most expected re-
sult is to quantify how the shape of a social net-
work can a�ect the level of vaccination required for
herd immunity. There is a related phenomenon,
not discussed here, that consists in the propaga-
tion of real immunity from a vaccinated individual
to a non vaccinated one. This is called contact im-
munity and has been veri�ed for several vaccines,
such as the OPV [62].
The models to quantify the success of immuniza-

tion of the population propose a targeted immu-
nization of the populations.
It is well established that immunization of ran-

domly selected individuals requires immunizing a
very large fraction of the population, in order to
arrest epidemics that spread upon contact between
infected individuals.
In Ref. [63], the authors studied the e�ects of im-

munization on an SIR epidemiological model evolv-
ing on a SW network. In the absence of immuniza-
tion, the model exhibits a transition from a regime
where the disease remains localized to a regime
where it spreads over a portion of the system. The
e�ect of immunization reveals through two di�er-
ent phenomena. First, there is an overall decrease
in the fraction of the population a�ected by the
disease. Second, there is a shift of the transition
point towards higher values of the disorder. This
can be easily understood as the e�ective average
number of susceptible neighbors per individual de-
creases. Targeted immunization that is applied by
vaccinating those individuals with the highest de-

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

gree, produces a substantial improvement in dis-
ease control. It is interesting to point out that this
improvement occurs even when the degree distri-
bution over small-world networks is relatively uni-
form, so that the best connected sites do not mo-
nopolize a disproportionately high number of links.
Figure 13 shows an example of the results found in
Ref. [63], where the author compare the amount of
non vaccinated individuals that are infected for dif-
ferent levels of vaccination, ρ, and di�erent degrees
of disorder of the SW network p, as de�ned in Ref.
[30].

Figure 13: Fraction r of the non-vaccinated popula-
tion that becomes infected during the disease prop-
agation, as a function of the disorder parameter p,
for various levels of random immunization (upper)
and targeted immunization (bottom). Adapted
from Ref. [63].

In a scale-free network, the existence of individu-
als of an arbitrarily large degree implies that there
is no level of uniform random vaccination that can
prevent an epidemic propagation, even extremely
high densities of randomly immunized individuals
can prevent a major epidemic outbreak. The dis-
cussed susceptibility of these networks to epidemic
hinders the implementation of a prevention strat-
egy di�erent from the trivial immunization of all
the population [54,55,66].
Taking into account the inhomogeneous connec-

tivity properties of scale-free networks can help to
develop successful immunization strategies. The
obvious choice is to vaccinate individuals according
to their connectivity. A selective vaccination can

be very e�cient, as targeting some of the super-
spreaders can be su�cient to prevent an epidemic
[55,67].

The vaccination of a small fraction of these indi-
viduals increases quite dramatically the global tol-
erance to infections of the network.

When comparing the uniform and the targeted
immunization procedures [67], the results indicate
that while uniform immunization does not produce
any observable reduction of the infection preva-
lence, the targeted immunization inhibits the prop-
agation of the infection even at very low immuniza-
tion levels. These conclusions are particularly rel-
evant when dealing with sexually transmitted dis-
eases, as the number of sexual partners of the in-
dividuals follows a distribution pattern close to a
power law.

Targeted immunization of the most highly con-
nected individuals [64,65,67] proves to be e�ective,
but requires global information about the architec-
ture of network that could be unavailable in many
cases. In Ref. [68], the authors proposed a di�erent
immunization strategy that does not use informa-
tion about the degree of the nodes or other global
properties of the network but achieves the desired
pattern of immunization. The authors called it
acquaintance immunization as the targeted indi-
viduals are the acquaintances of randomly selected
nodes. The procedure consists of choosing a ran-
dom fraction pi of the nodes, selecting a random
acquaintance per node with whom they are in con-
tact and vaccinating them. The strategy operates
at the local level. The fraction pi may be larger
than 1, for a node might be chosen more than once,
but the fraction of immunized nodes is always less
than 1. This strategy allows for a low vaccina-
tion level to achieve the immunization threshold.
The procedure is able to indirectly detect the most
connected individuals, as they are acquaintances of
many nodes so the probability of being chosen for
vaccination is higher.

VI. Final remarks

The mathematical modeling of the propagation of
infectious diseases transcends the academic inter-
est. Any action pointing to prevent a possible
pandemic situation or to optimize the vaccination
strategies to achieve critical coverage are the core

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Papers in Physics, vol. 5, art. 050003 (2013) / M. N. Kuperman

of any public health policy. The understanding of
the behavior of epidemics showed a sharp improve-
ment during the last century, boosted by the for-
mulation of mathematical models. However, for a
long time, many important aspects regarding the
epidemic processes remained unexplained or out of
the scope of the traditional models. Perhaps, the
most important one is the feedback mechanism that
develops between the social topology and the ad-
vance of an infectious disease. The new types of
models developed during the last decade made an
important contribution to the �eld by incorporat-
ing a mean of describing the e�ect of the social
pattern. While a quantitative analysis of a real sit-
uation still demands huge computational resources,
the mathematical foundations to develop it are al-
ready laid. The is too much to do yet, but the
breakthrough produced by these new models based
on complex networks is already undeniable.

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