050003 2013 G. Mindlin 050003

Since its first 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 finally provide a brief
description of the most relevant results in the field.

# Invited review: Epidemics on social networks

M. N. Kuperman[inst1, inst2] E-mail<EMAIL_ADDRESS>
(6 April 2013; 3 June 2013)

††volume: 5

99 inst1 Consejo Nacional de Investigaciones Científicas y Técnicas,
Argentina. inst2 Centro Atómico Bariloche and Instituto Balseiro, 8400 S. C.
de Bariloche, Argentina

##  1 Introduction

With the development of more precise and powerful tools, the mathematical
modeling of infectious diseases has become a crucial tool for making decisions
associated to policies on public health. The scenario was completely different
at the beginning of the last century, when the first mathematical models
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 flu. The lack of a mathematical
understanding of the evolution of epidemics gave place to an inaccurate
analysis of the epidemiological situation and subsequent failed assertion of
the success of the immunization strategy. During the influenza pandemic of
1892, a viral disease, Richard Pfeiffer isolated bacteria from the lungs and
sputum of patients. He installed, among the medical community, the idea that
these bacteria were the cause of influenza. At that moment, the bacteria was
called Pfeiffer’s bacillus or Bacillus influenzae, while its present name
keeps a reminiscence of Pfeiffer’s wrong hypothesis: Haemophilus influenzae.
Though there were some dissenters, the hypothesis of linking influenza with
this pathogen was widely accepted from then on. Among the supporters of
Pfeiffer hypothesis was William Park, at the New York City Health Department,
who in view of the fast progression of the flu in USA, developed a vaccine and
antiserum against Haemophilus influenzae on October 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. influenzae bacteria. Several other attempts to develop similar vaccines
followed this initiative. However, none of these vaccines prevented viral
influenza infection. The present consensus is that they were even not
protective against the secondary bacterial infections associated to influenza
because the 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 epidemiological
knowledge led to the conclusion that the vaccine was effective. If we look at
Fig. 1 corresponding to the weekly influenza death rates in a couple of U.S.
cities taken from Ref. [1], we observe a remarkable decay after vaccination,
in week 43. This decay was inaccurately attributed to the effect of
vaccination as it corresponds actually to a normal and expected development of
an epidemics without immunization.

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

The inaccurate association between H. influenzae and influenza persisted until
1933, when the viral etiology of the flu was established. But Pfeiffer’s
influenza bacillus, finally named Haemophilus influenzae, accounts in its
denomination for this persistent mistake.

The formulation of mathematical models in epidemiology has a tradition of more
than one century. One of the first successful examples of the mathematical
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 discovered and described the life-cycle of
the malaria parasite in mosquitoes and developed a mathematical model to
analyze the dynamics of the transmission of the disease [2, 3, 4]. His model
linked the density of mosquitoes and the incidence of malaria among the human
population. Once he had identified 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 differential
equations. He divided the human population into two groups, the susceptible,
with proportion $S_{h}$ and the infected, with proportion $I_{h}$. After
recovery, any formerly infected individual returned to the susceptible class.
This is called a SIS model. The mosquito population was also divided into two
groups (with proportions $S_{m}$ and $I_{m}$), with no recovery from
infection. Considering 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

$\displaystyle\frac{dI_{h}}{dt}$ $\displaystyle=$ $\displaystyle
abfI_{m}(1-I_{h})-rI_{h}$ (1) $\displaystyle\frac{dI_{m}}{dt}$
$\displaystyle=$ $\displaystyle acI_{h}(1-I_{m})-\mu_{m}I_{m},$

where $a$ is the man biting rate, $b$ is the proportion 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 $\mu_{m}$
is the rate of mosquito mortality.

One of the parameters to quantify the intensity of the epidemics propagation
is the basic reproductive rate $R_{0}$, that measures the average number of
cases produced by an initial case throughout its infectious period. $R_{0}$
depends on several factors. Among them, we can mention the survival time of an
infected individual, the necessary dose for infection, the duration of
infectiousness in the host, etc. $R_{0}$ allows to determine whether or not an
infectious disease can spread through a population: an infection can spread in
a population only if $R_{0}>1$ and can be maintained in an endemic state when
$R_{0}=1$ [5]. In the case of malaria, $R_{0}$ is defined as the number of
secondary cases of malaria arising from a single case in an susceptible
population. For the model described by Eq. (1)

$R_{0}=\frac{ma^{2}bc}{r\mu_{m}}.$ (2)

It is clear that the choice of the parameters affects $R_{0}$. The main result
is that it is possible to reduce $R_{0}$ 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 predictability power of the original epidemiological model [5, 6,
7, 8, 9, 10, 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: susceptible (S), infected (I) and recovered (R).

On the one hand, Kermack and McKendrick [12] proposed a SIR model that
expanded Ross’ set of differential 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] developed 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
epidemic spread that was intended mainly for teaching purposes, but that is
the starting point of many modern epidemiological studies. The model can be
mapped into a recurrence equation that defines what will happen at a given
moment depending on what has happened in the previous one,

$I_{t+1}=S_{t}(1-(1-\rho)^{I_{t}}),$ (3)

where $I_{t}$ is the number of cases at time $t$, $S_{t}$ is the number of
susceptible individuals at time $t$ and $\rho$ 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
become infectious. After a defined period of time, the infected individuals
heal and remain permanently immune. The interaction between any two
individuals of the population is considered as a stochastic process with a
defined probability of occurrence that most of the deterministic model
translates into a contact rate.

Given a closed population and the number of individuals in each state, the
calculation of the evolution of the epidemics is straightforward. The epidemic
event is over when no infective individuals remain.

While many classic deterministic epidemiological 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 developed [14,
15, 16, 17, 18, 19]. Sometimes, deterministic models introduce some colateral
mistakes due to the continuous character of the involved quantities.An example
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 prevalence of the rabies up to
negligible levels, followed by an unexpected new outbreak of infected foxes.
The spontaneous outbreak after the apparent disappearing of the rabies is due
to a fictitious very low endemic level of infected foxes, as explained in Ref.
[20]. The former one is one among several examples of how stochastic models
contributed to a better understanding and explanation of some observed
phenomena but, as their predecessors, they considered a mean field scheme in
the set of differential equations.

Traditional epidemiological models have successfully describe the generalities
of the time evolution of epidemics, the differential effect 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
population. The real underlying pattern of social contacts shows that each
individual has a finite set of acquaintances that serve as channels to promote
the contagion. While the fully mixed approximation allows for writing down a
set of differential equations 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,
reflects some emergent behavior that cannot be reproduced with a system based
on a set of differential equation 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 significant
portion of the population is enough to block the advance of the infection on
other non vaccinated individuals. Additionally, some network models allow also
for an analytic study of the described process. It is not surprising then that
during the last decade, a new tendency in epidemiological modeling emerged
together with the inclusion of complex networks as the underlying social
topology in any epidemic event. This new approach proves to contribute with a
further understanding of the dynamics of an epidemics and unveils the crucial
effect of the social architecture in the propagation of any infectious
disease.

In the following section, we will introduce some generalities about
traditional epidemiological models. In section III, we will present the most
commonly used complex networks when formulating an epidemiological model. In
section IV, we will describe the most relevant results obtained by modeling
epidemiological processes using complex networks to describe the social
topology. Next, we will introduce the concept of herd protection or immunity
and a discussion of some of the works that treat this phenomenon.

##  2 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
different groups considered as components of the population or epidemiological
compartments: S corresponds to susceptible, I to infected and R to removed.
The S group represents the portion of the population that has not been
affected by the disease but may be infected in case of contact with a sick
person. The I group corresponds to those individuals already infected and who
are also responsible for the transmission of the disease to the susceptible
group. The removed group R includes those individuals recovered from the
disease who have temporary or permanent immunity or, eventually, those who
have died from the illness and not from other causes. These models may or may
not include the vital dynamics, associated with birth and death processes. Its
inclusion depends on the length of time over which the spread of the disease
is studied.

###  2.1 The SIR Model

As mentioned before, in 1927, Kermack and McKendrick [12] developed a
mathematical model in which they considered a constant population divided into
three epidemiological groups : susceptible, infected and recovered. The
equations of a SIR model are

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI$
$\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma I$
(4) $\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma I,$

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 individuals after recovery or, in extreme cases, it kills them.
After the contagious period, the infected individual recovers and is included
in the R group. These models are suitable to describe the behavior of
epidemics produced by virus agent diseases (measles, chickenpox, mumps, HIV,
poliomyelitis) [22].

The model formulated through Eq. (2.1) assumes that all the individuals in the
population have the same probability of contracting the disease with a rate of
$\beta$, the contact rate. The number of infected increases proportionally to
both the number of infected and susceptible. The rate of recovery or removal
is proportional to the number of infected only. $\gamma$ represents the mean
recovery rate, ( $1/\gamma$ 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

$S(0)>0,\,\,\,I(0)>0\,\,\mbox{and }R(0)=0.$ (6)

It is straightforward to show that

$\left.\frac{dI}{dt}\right|_{t=0}=I(0)(\beta S(0)-\gamma),$ (7)

and that the sign of the derivative depends on the value of
$S_{c}=\frac{\gamma}{\beta}$. When $S(t)>S_{c}$, 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. 2.1. We can write

$\displaystyle\frac{dS}{dR}$ $\displaystyle=$ $\displaystyle-\frac{S}{\rho}$
(8) $\displaystyle\Rightarrow$ $\displaystyle S=S_{0}\exp[-R/\rho]\geq
S_{0}\exp[-N/\rho]>0$ $\displaystyle\Rightarrow$ $\displaystyle
0<S(\infty)\leq N.$

The epidemics stops when $I(t)=0$, so we can set $I(\infty)=0$, so
$R(\infty)=N-S(\infty)$. From (8),

$\displaystyle S(\infty)$
$\displaystyle=S_{0}\exp\left[-\frac{R(\infty)}{\rho}\right]$
$\displaystyle=S_{0}\exp\left[-\frac{N-S(\infty)}{\rho}\right].$ (9)

The last equation is a transcendent expression with a positive root
$S(\infty)$.

Taking (9), we can calculate the total number of susceptible individuals
throughout the whole epidemic process

$I_{\mbox{\small total}}=I_{0}+S_{0}-S(\infty).$ (10)

As $I(t)\to 0$ and $S(t)\to S(\infty)>0$, we conclude that when the epidemics
end, there is a portion of the population that has not been affected

The previous model can be extended to include vital dynamics [23], delays
equations [24], age structured population, migration [25], and diffusion. In
any case, all these generalizations only introduce some slight changes on the
steady states of the system, or in the case of spatially extended models,
travelling waves [26].

Figure 2 displays the typical behavior of the density of individuals in each
of the epidemiological compartments described by Eq. (2.1). Compare this with
the pattern shown in Fig. 1.

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

###  2.2 The SIS Model

The SIS model assumes that the disease does not confer immunity to infected
individuals after recovery. Thus, after the infective period, the infected
individual recovers and is again included in the S group. Therefore, the model
presents only two epidemiological 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,

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI+\gamma
I$ $\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma
I.$ (11)

As the relation $S+I=1$ holds, Eq. (2.2) can be reduced to a single equation,

$\frac{dI}{dt}=(\beta-\gamma)I-\beta I^{2}.$ (12)

The solution of this equation is

$I(t)=(1-\frac{\gamma}{\beta})\frac{C\exp[(\gamma-\beta)t]}{1+C\exp[(\gamma-\beta)t]},$
(13)

where $C$ is defined by the initial conditions as

$C=\frac{\beta i_{0}}{\beta(1-i_{0})-\gamma}.$ (14)

If $I_{0}$ is small and $\beta>\gamma$, the solution is a logistic growth that
saturates before the whole population is infected, the stationary value is
$I_{s}=\frac{\beta-\gamma}{\beta}$. It can be shown that $R_{0}=\beta/\gamma$.
This sets the condition for the epidemic to persist.

###  2.3 Other models

The literature on epidemiological models includes several generalizations
about the previous ones to adapt the description to the particularities of a
specific infectious disease [27]. One possibility is to increase the number of
compartments to describe different 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 recovered
individuals and after some time they rejoin the susceptible group,

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\beta SI++\lambda
R$ $\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle\beta SI-\gamma
I$ (15) $\displaystyle\frac{dR}{dt}$ $\displaystyle=$ $\displaystyle\gamma
I-\lambda R.$

Other models include more epidemiological groups or compartments, such as the
SEIS and SEIR model, that take into consideration the exposed or latent period
of the disease, by defining 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 reflected in the
title of Ref. [27]: “A thousand and one epidemiological models”. There are a
lot of possibilities to define the compartment structure. Usually, this
structure is represented as a transfer chart indicating the flow between the
compartments and the external contributions. Figure 3 shows an example of a
diagram for a SEIRS model, taken from Ref. [27].

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

Horizontal incidence refers to a contagion due to a contact between a
susceptible and infectious individual, vertical incidence account for the
possibility for the offspring of infected parents to be born infected, such as
with AIDS, hepatitis B, Chlamydia, etc.

Many of the previous models have been expanded, including stochastic terms.
One of the most relevant differences between the deterministic and stochastic
models is their asymptotic behavior. A stochastic model can show a solution
converging to the disease-free state when the deterministic counterpart
predicts an endemic equilibrium. The results obtained from the stochastic
models are generally expressed in terms of the probability of an outbreak and
of its size and duration distribution [14, 15, 16, 17, 18, 19].

##  3 Complex Networks

A graph or network is a mathematical representation 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 characterized by
several topological properties, some of which will be introduced later. Social
links are preponderantly 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 pioneering works of Erd s
and Renyi in the 1950s [28], their gradual colonization of the modern
epidemiological models has only started a decade ago. The attention of
modelers was drawn to graph theory 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
individuals are distributed in social and geographical space and how the
contacts between them are promoted, reinforced or inhibited, according to the
rules of social dynamics. When the population is fully mixed, each individual
has the same probability of coming into contact with any other individual.
This assumption makes it possible to calculate the effective contact rates
$\beta$ as the product of the transmission rate of the disease, the effective
number of contacts per unit time and the proportion of these contacts that
propagate the infection. The formulation of a mean field model is then
straightforward. However, in real systems, the acquaintances of each
individual are reduced to a portion of the whole population. 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 difference between a structured population
and a fully mixed one fades out. The differences 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 necessary condition for all the
networks used to model disease propagation. In the following paragraphs, we
will introduce the most common families of networks 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. Individuals are regularly located and
connected with adjacent neighbors; therefore, contacts are localized in space.
Figure 4 shows, among others, an example of a two dimensional square lattice

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

Small-world networks. The concept of Small World was introduced by Milgram in
1967 in order to describe the topological properties of social communities and
relationships [29]. Some years ago, Watts and Strogatz introduced a model for
constructing networks displaying topological features that mimic the social
architecture revealed by Milgram. In this model of Small World (SW) networks 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 random
graph [30]. The construction of these networks starts from a regular, one-
dimensional, periodic lattice 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 modified 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 presented in Ref. [30]. As the disorder degree increases, there
number of shortcuts grow replacing some of the original (ordered network)
links.

To characterize the topological properties of the SW networks, two magnitudes
are calculated. The first one, $L(p)$, measures the mean topological distance
between any pair of elements in the network, that is, the shortest path
between two vertices, averaged over all pairs of vertices. Thus, an ordered
lattice has $L(0)\sim N/K$, while, for a random network, $L(1)\sim
ln(N)/ln(K)$. The second one, $C(p)$, measures the mean clustering of an
element’s neighborhood. $C(p)$ is defined in the following way: Let us
consider the element $i$, having $k_{i}$ neighbors connected to it. We denote
by $c_{i}(p)$ the number of neighbors of element $i$ that are neighbors among
themselves, normalized to the value that this would have if all of them were
connected to one another; namely, $k_{i}(k_{i}-1)/2$. Now, $C(p)$ is the
average, over the system, of the local clusterization $c_{i}(p)$. Ordered
lattices are highly clustered, with $C(0)\sim 3/4$, and random lattices are
characterized by $C(1)\sim K/N$. Between these extremes, small worlds are
characterized by a short length between elements, like random networks, and
high clusterization, like ordered ones.

Figure 6: In this figure, we show the mean values of the clustering
coefficient $C$ 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 corresponding to total
disorder, while the value adopted by $C$ is close to the one corresponding to
the ordered case.

Other procedures for developing similar social networks have been proposed in
Ref. [31] where instead 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 presented in Ref. [31]. As the disorder degree increases, three
number of shortcuts as well as the number of total links grow.

Random networks. There are different families of networks with random genesis
but displaying a wide spectra of complex topologies. In random networks, the
spatial position of individuals is irrelevant 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 existing 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 approximated by a Poisson distribution [32].

Figure 8: This figure shows examples of (a) ER and (b) BA networks. The figure
also displays the connectivity distribution $P(k)$, that follows a binomial
distribution for the ER networks and a power law for BA networks.

In Ref. [33], the authors propose a formalism based on the generating function
that permits to construct random networks with arbitrary degree distribution.
The mechanism of construction also allows for further analytic studies on
these networks. In particular, 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 networks 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 mechanism that imitates the underlying process that rules 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 algorithm 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
$p_{i}$ is proportional to the number of links that the existing node already
has

$p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},$

where $k_{i}$ is the degree of node $i$. That means that the new nodes have a
preference to attach themselves to the most “popular” nodes. One salient
feature of these networks is that their degree distribution is scale-free,
following a power law of the form

$P\left(k\right)\sim k^{-3}.$

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 figure 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 coefficient, higher moments of the degree distribution [35, 36, 37,
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 implicit assumption:
the underlying social topology is frozen. However, this situation does not
reflect the observed fact that in real populations, social and migratory
phenomena, sanitary isolation or other processes can lead to a dynamic
configuration 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 reflected if the topology remains
fixed. 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 infectious disease. For this reason, some models consider a
coevolving network, with dynamic links that change the aspect of the networks
while the epidemics occur.

##  4 Epidemiological Models on Networks

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

Lattices. Lattices were the first attempt to represent the underlying topology
of the social contacts and thus to analyze the possible effect of interactions
at the individual level. These models took distance from the paradigmatic
fully mixed assumption and focused on looking for those phenomena that a mean
field model could not explain. Still, the lattices cannot fully capture the
role of inhomogeneities. As the individuals are located on a regular grid,
mostly two dimensional, the neighborhood 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. The dynamic of the epidemics evolves through a contact process
[39] and the evolutive rules do not differ too much from traditional cellular
automata models [40]. Disease transmission is modeled as a stochastic process.
Each infected node has a probability $p_{i}$ of infecting a neighboring
susceptible node. Once infected, the individuals may recover from infection
with a probability $p_{r}$; i.e., the infective stage lasts typically
$1/p_{r}$. From the infective 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 initial stage involves the growing of a cluster, with the
infection propagating at its boundary, like a traveling wave. After the
initial transient, SIS, SIR and SIRS models behave in different 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 figure reflects the results found in Ref.
[42]. The parameter $f$ is associated to the infectivity of infectious
individuals, closely related to the contact rate. We observe the inset
displaying the scaling of the data with a power-like curve
$A|f-f_{c}|^{\alpha}$, with $\alpha\approx 0.5$ [42].

Figure 9:  SIS model. Asymptotic value of infected individuals as a function
of the infectivity of infectious individuals. The inset displays the scaling
of the data with a power-like curve $A|f-f_{c}|^{\alpha}$, with $\alpha\approx
0.5$. Adapted from Ref. [42].

As mentioned before, Kermack and Mckendrick [12] proved the existence of a
propagation threshold for the disease invading a susceptible population. The
lattice based SIR models introduce a different threshold. The simulations show
that epidemics can just remain localized around the initial focus or turn into
a pandemic, affecting 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.

Figure 10:  SIR model. Asymptotic value of susceptible individuals as a
function of the infectivity of infectious individuals. The inset displays the
scaling of the data with a power-like curve $A|f-f_{c}|^{\alpha}$, with
$\alpha\approx 0.5$. Adapted from Ref. [42].

Some works about epidemic propagation on lattices are analogous to forest fire
models [43], with the characteristic feature that the frequency distributions
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.

Random networks. Most of the models based on random graphs were previous to
the renewed interest on complex networks. A simple but effective 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 transient growth is smaller than the corresponding to
similar models in fully-mixed populations. This effect 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 number 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 different analytical approach to random networks 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 generating function method and appealing
to the formalism of percolation models. The author analyzes the propagation of
a disease in networks with arbitrary degree distributions and heterogeneous
infectiveness times and transmission probabilities. The results include the
particular case of scale-free networks, 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 distant individuals, but a negligible clustering.
Both features affect the propagative behavior on any modeled disease. The
spread of infectious diseases 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 unexpected macroscopic effects. By
sharing topological properties from random and ordered networks, SW networks
can display complex propagative patterns. On the one hand, the high level of
clustering 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
authors find an important analytic result. It was a study of a SIRS models
that showed for the first time the evidence of a dramatic change in the
behavior of an epidemic due to changes in the underlying social topology [52].
By specifically analyzing the effect of clustering on the dynamics of an
epidemics, the authors show that a SIRS model on a SW network presents two
distinct types of behavior. As the rewiring parameter $p$ increases, the
system transits from an endemic state, with a low level of infection to
periodic oscillations in the number of infected individuals, reflecting an
underlying synchronization phenomena. The transition from one regime to the
other is sharp and occurs at a finite 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].

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

It would not be responsible to affirm that SW networks reflect all the real
social structures. However, they capture essential aspects of such
organization that play central roles in the propagation of a diseases, namely,
the clustering coefficient and the short social distance between individuals.
Understanding that there are certain limitations, SW networks help to mimic
different social organizations 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 distribution 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 captured 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 number of contacts implies a greater
risk of infection and correspondingly, a higher “success” as an infectious
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 monogamous relationships [53]. Models of
disease spread through scale-free networks showed that the infection is
concentrated among the individuals with highest degree [54, 48]. One of the
most surprising 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, independently of $R_{0}$. Remember that when
analyzed under the fully mixed assumption, the studied SIS model has a
threshold. The authors perform analytic 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 distribution will show no epidemic threshold. The B A network fulfills
this condition. In Ref. [56, 57], the authors analyze the structure of
different networks of sexual encounters, to find 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 policies promote the isolation of infectious individuals
[58]. The behavior of models based on adaptive network is determined by the
interplay of two different dynamics that sometimes have competitive effects.
On the one hand, we have the dynamics of the disease 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
affects the fraction of susceptible individuals connected to infective ones.
The most obvious choice is to eliminate the infectious contacts of a
susceptible individual by deleting or replacing them with noninfectious ones.
The net effect is an effective 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 bistability appears for small rewiring rates [58,
59, 60, 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 neighbor, 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
effective control strategy. Figure 12 displays the results found in Ref. [61],
where two completely different 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.

$\lambda$

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

##  5 Immunization in networks

Any epidemiological model can reproduce the fact that the number of
individuals in a population who are effectively immune to a given infection
depends on the proportion of previously infected individuals and the
proportion who have been efficiently vaccinated.

For some time, the epidemiologists knew about an emerging effect called herd
protection (or herd immunity). They discovered the occurrence of a global
immunizing effect verified when the vaccination of a significant portion of a
population provides protection for individuals who have not or cannot
developed immunity. Herd protection is particularly important for diseases
transmitted from person 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 effect 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 vaccinated individuals will not
contract neither transmit the disease, thus establishing a firewall between
infected and susceptible individuals.

While taking profit 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 effect is equivalent to reduce the $R_{0}$ 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 efficacy of the vaccine but also on the virulence of the disease and the
contact rate. If the herd effect reduces the risk of infection among the
uninfected enough, then the infection may no longer be sustainable within the
population and the infection may be eliminated. In a real population, the
emergence of herd immunity is closely related to the social architecture.
While many fully mixed models can describe the phenomenon, the real effect is
much more accurately reproduced by models based on Social Networks. One of the
most expected result is to quantify how the shape of a social network can
affect the level of vaccination required for herd immunity. There is a related
phenomenon, not discussed here, that consists in the propagation of real
immunity from a vaccinated individual to a non vaccinated one. This is called
contact immunity and has been verified for several vaccines, such as the OPV
[62].

The models to quantify the success of immunization of the population propose a
targeted immunization of the populations.

It is well established that immunization of randomly 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 effects of immunization on an SIR
epidemiological model evolving on a SW network. In the absence of
immunization, 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 effect of immunization reveals through two different phenomena. First,
there is an overall decrease in the fraction of the population affected 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 effective average
number of susceptible neighbors per individual decreases. Targeted
immunization that is applied by vaccinating those individuals with the highest
degree, produces a substantial improvement in disease control. It is
interesting to point out that this improvement occurs even when the degree
distribution over small-world networks is relatively uniform, so that the best
connected sites do not monopolize 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
different levels of vaccination, $\rho$, and different degrees of disorder of
the SW network $p$, as defined in Ref. [30].

Figure 13:  Fraction $r$ of the non-vaccinated population that becomes
infected during the disease propagation, 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 individuals 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 discussed
susceptibility of these networks to epidemic hinders the implementation of a
prevention strategy different from the trivial immunization of all the
population [54, 55, 66].

Taking into account the inhomogeneous connectivity 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 efficient, as targeting some of the super-
spreaders can be sufficient to prevent an epidemic [67, 55].

The vaccination of a small fraction of these individuals increases quite
dramatically the global tolerance 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 prevalence, the targeted immunization
inhibits the propagation of the infection even at very low immunization
levels. These conclusions are particularly relevant when dealing with sexually
transmitted diseases, as the number of sexual partners of the individuals
follows a distribution pattern close to a power law.

Targeted immunization of the most highly connected individuals [64, 65, 67]
proves to be effective, but requires global information about the architecture
of network that could be unavailable in many cases. In Ref. [68], the authors
proposed a different immunization strategy that does not use information 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 individuals are the acquaintances of randomly
selected nodes. The procedure consists of choosing a random fraction $p_{i}$
of the nodes, selecting a random acquaintance per node with whom they are in
contact and vaccinating them. The strategy operates at the local level. The
fraction $p_{i}$ 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 vaccination 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.

##  6 Final remarks

The mathematical modeling of the propagation of infectious diseases transcends
the academic interest. Any action pointing to prevent a possible pandemic
situation or to optimize the vaccination strategies to achieve critical
coverage are the core of any public health policy. The understanding of the
behavior of epidemics showed a sharp improvement during the last century,
boosted by the formulation 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
advance of an infectious disease. The new types of models developed during the
last decade made an important contribution to the field by incorporating a
mean of describing the effect of the social pattern. While a quantitative
analysis of a real situation still demands huge computational resources, the
mathematical foundations to develop it are already 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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