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Virulence
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Discrete dynamics of contagious social diseases:
Example of obesity
J Demongeot, O Hansen & C Taramasco
To cite this article: J Demongeot, O Hansen & C Taramasco (2015): Discrete
dynamics of contagious social diseases: Example of obesity, Virulence, DOI:
10.1080/21505594.2015.1082708
To link to this article: http://dx.doi.org/10.1080/21505594.2015.1082708
Accepted author version posted online: 16
Sep 2015.
Published online: 16 Sep 2015.
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Discrete dynamics of contagious social diseases:
Example of obesity
J Demongeot
1,2,
*, O Hansen
1
, and C Taramasco
2
1
Team AGIM; Laboratory Jean-Raoul Scherrer; UniGe and University J Fourier of Grenoble; Faculty of Medicine; La Tronche, France;
2
Escuela de Ingeniería Civil en Inform
atica;
Universidad de Valparaíso; Valparaíso, Chile
Keywords: contact dynamics, demography, mathematical modelling, obesity, social networks
Modeling contagious diseases needs to incorporate information about social networks through which the disease
spreads as well as data about demographic and genetic changes in the susceptible population. In this paper, we
propose a theoretical framework (conceptualization and formalization) which seeks to model obesity as a process of
transformation of one’s own body determined by individual (physical and psychological), inter-individual (relational, i.e.,
relative to the relationship between the individual and others) and socio-cultural (environmental, i.e., relative to the
relationship between the individual and his milieu) factors. Individual and inter-individual factors are tied to each other
in a socio-cultural context whose impact is notably related to the visibility of anybody being exposed on the public
stage in a non-contingent way. The question we are dealing with in this article is whether such kind of social diseases,
i.e., depending upon socio-environmental exposure, can be considered as “contagious”. In other words, can obesity be
propagated from individual to individual or from environmental sources throughout an entire population?
Introduction
Social diseases are numerous and obesity can be considered as
one of the most characteristic of what could be identified as a
social “contagious” disease. Both stigmatization and mimicking
1
constitute the method of dissemination of obesity into a family
or a social network. Obesity is defined as an abnormal or exces-
sive accumulation of fat in adipose tissue, more or less leading to
important health problems at the individual level. Currently,
obesity would be reaching an epidemic development worldwide:
according to the latest world estimates of WHO (World Health
Organization), obesity rate has tripled between 1980 and
2005.
2,3
This rate of development suggests that this pathology
involves a socio-cultural problem grafted into a predisposition at
the individual level.
All specialists agree that, for decades, we have been witnessing
an increase in worldwide obesity prevalence. This is true in devel-
oped as well as in developing countries. No society seems to be
immunized against this epidemic. Classic data from MONICA
WHO project
2
shows that obesity prevalence in the majority of
European countries increased in 10 y (1992–2002), going from
10% to 20% in men and from 10% to 25% among women. In
France, between 1980 and 2006, obesity prevalence went from
6.4% to 14.3% in men and from 6.3% to 15.7% among women
(International Association for the Study of Obesity 2000; Mail-
lard et al. 1999
3,4
). Based on these facts, several studies have
been performed to identify risk factors associated with this condi-
tion as well as to contain the epidemic, because obesity became a
real public health problem.
5
It is well known that obesity has a
genetic component as a familiar predisposition toward this con-
dition. However, the genetic component does not explain the
increasing (spectacular) progression of the disease prevalence.
Additional behavioral, social, and economic factors must be con-
sidered.
6-8
In this context, Christakis and Fowler recently showed
the possibility of person to person obesity contagion in a social
network.
9
Moreover, Cohen-Cole and Fletcher suggested that
obesity diffusion could occur via a common exogenous source
applied to a set of individuals.
10
Realistic models of contagious diseases now incorporate infor-
mation about the social networks through which the disease
spreads, as well as data about demographic and genetic changes
in the susceptible population. They also include all the possible
knowledge about the contacts between susceptible and sick indi-
viduals. In “Section Methods”, we will present the mathematical
framework necessary to take into account, at a microscopic level,
the dynamics of contacts between susceptible and obese individu-
als. Then we will introduce the description of the dynamics of
obesity in the Results section, taking into account collective
behaviors mimicking some dominant habits of nutrition trans-
mitted through social networks. Obesity spread modeling will
use the notion of homophilic graphs.
To investigate obesity in a multi-factorial manner, we have
taken into account the impact through time that obese individual
transformation may have on the social structure, by developing a
network model in which individual interactions are in part due
to homophilic selection/de-selection, i.e., a process of preferential
*Correspondence to: J Demongeot; Email: Jacques.Demongeot@yahoo.fr
Submitted: 04/02/2015; Revised: 08/05/2015; Accepted: 08/09/2015
http://dx.doi.org/10.1080/21505594.2015.1082708
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RESEARCH PAPER
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attachment and detachment of inter-individual links according to
characteristics of the individuals involved. Homophily is here
defined as the tendency of an individual to create links with other
individuals sharing similar attributes and to sever links with other
dissimilar individuals. Homophily suggests that individuals tend
to interact with those who resemble them. Second, and recipro-
cally, we study if obesity can be considered as a “contagious”
social disease. So we can study the role which could be played by
the structure of the social fabric in the increase and current devel-
opment of obesity.
We evaluate the impact of relations between individuals
(micro-level) as well as the impact of relations between districts
(meso-level) and between countries (macro-level). This approach
highlights the need to integrate the dynamics of each scale to bet-
ter understand the evolution of the pathology. Two stochastic
models are proposed: i) an epidemiological compartmental
model and ii) an individual centered
network model, considering 3 influen-
ces: exogenous heterogeneous (individ-
ual-cultural), exogenous homogeneous
(individual-social) and endogenous
(individual-individual). All together,
this study of obesity will allow investi-
gating the social and cultural dimension
involved in being and transforming
one’s body.
In the Discussion section, we present
elements of demographic dynamics to
add to the social contagion dynamics as
an obesity preventive policy, and even-
tually some perspectives about a new
more realistic modeling of the contact
dynamics.
Methods
General graph framework
Given that each individual is
immersed in a social system, linked
together with other individuals through
diverse and complex interactions, each
individual “i” can then be characterized,
in a first approach, by its number of
neighbors k
i
, whereas the overall system
is characterized by the connection struc-
ture between individuals. To study the
role played by social interactions in
obesity spreading, 5 simple network
topologies are considered to describe
inter-individual connections: random
(Erd€os-Renyi), scale-free, small-world
and 2 empirical networks.
The empirical networks are built
from degree distributions found by
Christakis and Fowler
8
in real networks.
In Figure 1, we can find examples of
simulated architecture following the
above topologies. We will use these
architectures to start from initial config-
urations of the initial network, before
applying the homophilic rule and con-
verging to an “attractor” of its dynamics,
i.e., a stable configuration of links and
Figure 1. Simulation of various initial architectures: random, scale-free, small world, empirical (1 and
2). (A) Radom; (B) scale-free; (C) small-world; (D) empirical-1; (E) empirical-2.
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node states of the interaction graph related to the social network
involved in the social contagion of the obesity.
Social contagion
In Figure 2, each individual is represented in their social
neighborhood: he/she can influence the narrow contexts to which
he belongs, e.g, by transmitting a good
opinion about his/her obese state (red
arrow) or about his normal state (blue
or brown arrows). Hence, each individ-
ual in a given social sub-network will
receive indirect influence linked to his/
her context. Under theses influences,
some individual (in blue in Fig. 2) can
become obese and others may not (in
green in Fig. 2).
We are interested in modeling the
social contagion mechanisms through
which the disease can propagate from
individual to individual or from envi-
ronmental sources through populations,
individuals changing states like in bio-
logical regulatory networks for which
many theoretical and numerical tools
have been recently developed.
11-14
In Figure 3, we have fixed the corporal state (obese, over-
weight and normal) following the distribution of the BMI (Body
Mass Index) in Chilean children (population between the ages of
5 and 17)
15
in 2011: obese (9.6%), overweight (23.2%) and nor-
mal (67.2%). The tolerance has been taken at the 0.25 level and
the connection probability has been chosen following the Version
Figure 2. Inter-individual relationships between obese and non-obese individuals in a social context.
Figure 3. Dynamics with a progressive clusterization (from left to right) inside a small-world directed
network with initial proportion of obese individuals in red (14,5%), overweight in pink (31.9%) and nor-
mal in white (53,6%), 0.25 tolerance and connection probability of the Version 1.
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1. Directed (not directed) networks with 1000 nodes each have
been simulated, with a probability of having forward directional
(resp. bidirectional) links equal to aD0.6 (resp bD0.2). The
node positioning has been done following the attraction-repul-
sion by the Fruchterman and Reingold algorithm.
16
Homophilic Hebb graphs
The function homophily will be defined as the tendency of an
individual to create links with other individuals sharing similar
attributes and to sever links with other dissimilar individuals, by
playing with the probability of having an infectious contact
between agents having the same given state (e.g., normoweight
susceptible S, overweight Wand obese O). The tendency an agent
or node ihas to create or cut a link with another agent jin a
social interaction graph Ghaving Nagents, depends on similarity
distance d(i,j) in the graph, like the Hebbian rule of pruning/
strengthening in neural networks, which is based on state correla-
tions, destroying (resp. reinforcing) links between nodes weakly
(resp highly) correlated.
For example, if the state x(t,i) of the ith node at time tequals 1
(S), 2(W)or3(O), are considered as well as some biological
characteristics like age A(t,i) and Body Adiposity Index B(t,i),
social variables like sizes of the family F(t,i) and of the circle of
friends C(t,i), environmental parameters like the numbers of
accessible green areas G(t,i) and supermarkets M(t,i), behavioral
variables like sedentary lifestyle index L(t,i) (resp. sport index
S(t,i)), that is the number of hours at home (resp on a sports
area) during the last 24h, they are all the components of a large
state vector V(t,i), we can correlate with the vector V(t,j), e.g., by
calculating the average cross-correlation between the Vcompo-
nents during a defined timelapse.
Then, the Hebbian rule eliminates links between uncorrelated
nodes and builds a positive (resp. negative) link between posi-
tively (resp negatively) sufficiently correlated nodes. The dynam-
ics of creation/cancellation of links can be separated from the
state dynamics, if it is slower : then we can first study, with a fixed
architecture, the fast state dynamics, considered as autonomous
and then study the bifurcations (in number and nature) of its
attractors due to the slow link dynamics. Let us suppose that
there are states xand yin the social graph and denote at time tby
L
x,y
(t) (resp. L
x,x
(t), L
x
(t) and L(t)) the number of heterophilic
links (resp homophilic links of type x, links coming from type x
nodes and total links) and by tthe relaxation time. We suppose
in each time lapse of duration t, a certain proportion of nodes
(agents) create (resp. cancel) links toward nodes being in same
(resp different) state, with a certain tolerance threshold, supposed
to be the same in each state group. The simulation follows the
successive steps:
1) At tDt
0
, generate the random value tfrom an exponential
distribution of parameter 1/b.
2) At tDt
0
Ct, do the following operations:
choose a fraction Fof nodes in G. Let MDFN.
for each node iof these Mnodes (i D1,..,M), define
its state x(k) (known initial conditions), its out-degree
k
i
2IN (equal to the number of links exiting from i),
generate its tolerance to the difference, a real 0h
i
1, from a probability distribution g(h) and do the fol-
lowing operations:
for k
i
D0, connection from ito j:
choose a node jby chance among N-1 other nodes.
create a link from ito jwith probability h
id(i,j)
, where d(i,j)
is the direct distance between iand j, with 3 levels: 0, 1
and 2, as follows:
di;jðÞD0;if x iðÞDxjðÞ
D1;if x iðÞDS;xjðÞDW
and vice versa
D1;if x iðÞDW;xjðÞ DO
and vice versa
D2;if x iðÞ DS;xjðÞ DO
and vice versa
for k
i
1, connection or disconnection from ito j:
if V(i) denotes the set of neighbors of i, let choose a node j
among the jV(i)jneighbors of i with the probability 1/k
i
.
We will denote by V(j)nithe set of the neighbors of j,
minus i.
let r(i,j) be the total similarity distance between nodes iand
j. The link between iand jwill be cut with the probability
1-h
ir(i,j)
where the total distance r is defined by:
ri;jðÞDdi;jðÞ;if ci;jðÞD0;
Dadi;jðÞC1¡aðÞci;jðÞ;if ci;jðÞ6¼ 0
where the indirect distance c is given by:
ci;jðÞDSs2VjðÞnidi;sðÞ/kj¡1
D0;if kjD1:
if the link between iand jhas been cut, we choose by
chance a new node kin GnV(i) n(V(j)ni) and we cre-
ate a link from ito kwith the probability:
Prob i !kðÞDfdi;kðÞðÞ
£nk£hdi;kðÞ
i
ns£hdi;kðÞ
iCnw£hdi;kðÞ
iCno£hdi;kðÞ
i
where nkis the number of nodes in GnV (i)n(V(j)ni)
having the same state as k, i.e., n
k
is the number of
nodes with state S(resp. Wand O)ifkis susceptible
(resp overweight and obese). We will consider in the
simulations 3 versions for the function f:
Version 1: f(d(i,k)) D1,
if d(i,k)D0;
D0 elsewhere.
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Version 2: f(d(i,k)) D1,
if d(i,k)D0or1;
D0 elsewhere
Version 3: f(d(i,k)) D1,
if d(i,k)D0, 1or 2,
these versions being used in the individual centered network for
representing 3 types of progressively increasing influence: exoge-
nous heterogeneous (individual-cultural, Version 1), exogenous
homogeneous (individual-social, Version 2), endogenous (indi-
vidual-individual, Version 3).
1) Change the states x(j), for all jat the end of links created, by
increasing their obesity weight of one level (Sto W, W to O,
Oto O).
2) Generate a new tand go to 2.
3) Stop when graph is no longer changing.
Results
Equilibrium configurations
Under the homophilic rule, the network converges until an
attractor configuration of both links of the undirected graph
architecture and node states, regardless of the initial architecture
and initial state distribution (Fig. 3). By using a simulation
engine of the social network, we can study the speed of conver-
gence to this equilibrium for all the topologies proposed in the
Methods section.
Figure 4 shows that the relaxation time to steady state (in con-
nection with the speed of convergence to homophilic equilib-
rium) depends on the network topology. The shape of the initial
and final “in-degree” (numbering of edges entering a node) distri-
butions are about the same after applying the homophilic dynam-
ics (Fig. 5). However, we can show that, paradoxically (with
respect to the random topology) in the small-word initial topol-
ogy, the mean clustering coefficient of a node i is the number of
edges in V(i) divided by the maximum possible number of edges
|V(i)|(1-|V(i)|)/2) calculated by state decreases (this phenomenon
due to the modification of the state distribution).
In order to improve this study, a theoretical estimation of the
speed of convergence to the equilibrium configuration could be
made, as well as the consideration of the robustness of the pro-
cess: Do more than one equilibrium state exist, and if so, are
there other attractors, e.g., only fixed states or possibly periodic
configurations? Which network parameters are critical or sensi-
tive to the dynamics, i.e., which parameter perturbation provokes
a change in number or nature of attractors? Which perturbation
of the initial configuration of the social network leads to a change
of attraction (or stability) basin? All these questions will be
addressed in a future work.
Examples of dynamics of obesity
Homophily defined as above suggests that individuals tend to
interact with those who resemble them in terms of alimentary
behavior. The structure of social fabric is involved in the increase
and current development of obesity.
17-29
By using the proposed
simulation rules, we compare the simulated graphs with real data
in case of obesity. Four situations have been tested: the pure ran-
dom graph (links chosen by chance), the free scale graph (the dis-
tribution of out-degrees follows a power law), the small world
graph (links around hub nodes are reinforced) and homophilic
graphs, with different versions of probability of linking. The
approach described above has highlighted the need to integrate a
random dynamics at each scale to better understand the evolution
of the obesity pathology, e.g., in Figure 5, the connectivity of the
real social network representing obesity spread is better taken
into account in the homophilic network Version 1 (the qualita-
tive differences between homophilic versions being small) than in
the other versions: random, scale-free or small world. Figure 6
shows the variation of the average homophily (left), based on the
average level of tolerance for each studied network topology and
the relaxation time (right) depending on the average tolerance to
reach the steady-state (Fig. 7).
Demographic dynamics
For evaluating the number of susceptibles and defining them
by age and sex (which are important factors in the occurrence of
obesity), we need to develop a dynamic projection model by
using key socio-demographic indicators of the studied popula-
tion. The lack of comprehensive and documented data often not
allow the use of international performance tools like Individual
Based Model (IBM) demographic simulation such as FELICIE,
DESTINIE, OMPHALE, MOGDEN, LIFEPATH,....
17-19
So
we proposed the DOPAMID model, which requires less raw data
for its dynamic projection method.
DOPAMID model overview
The objective of the model is to make a population evolve
according to statistics based on its composition of age classes.
This evolution allows expressing patterns in the composition of
the population. Statistics used are: the distribution of the popula-
tion according to the age and sex of the individuals, mortality,
fertility, and composition of families as well as the dependency of
individuals. Based on a population and respecting these statistics,
the model advances in time over a period of up to 90 y. The
members of this population will therefore age, reproduce,
become dependent, die... Every year, the values of the popula-
tion statistics are recalculated, they are saved on an Excel or
Open Office-readable text format file and given in the classical
pyramid format (Fig. 8).
Model
A study of an important pathology associated to obesity, like
type 2 diabetes,
21
shows that the proportion of diabetics is equal
to 3.5% in the normal weight Iranian population, and 6.4% and
14.3% respectively in overweight and obese population, repre-
senting an odd ratio of respectively 1.7 (the 95%-confidence
interval being equal to [1.1–2.5]) and 4 (the 95%-confidence
interval being equal to [2.7–5.8]). The demographic modeling
allows calculating, for each age class, the proportion of obese and
the risk of type 2 diabetes: in
21
for example, the odd ratio per 10
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y is equal to 1.2 (the 95%-confidence interval being equal to
[1.1–1.4]). A precise distribution with respect to gender and age
class can be found in.
22
Discussion
Toward the proposal of an obesity preventive policy
The BMI was defined about 2 centuries ago by a Belgian
physician (A. Quetelet) and it represents the basic tool for
diagnosing obesity and for therapeutic surveillance. New poli-
cies are now needed to contain the world pandemic and we
suggest the following methods in order to watch and cure
obesity:
1) Defining new optimal threshold for defining obesity states
and associated risks from the classical BMI.
23
2) Using a new index called the Body Adiposity Index (BAI)
allowing differentiating muscular, skeletal and adipose
masses.
24
3) Elucidating all genetic factors involved in the obesity genesis
(endogenous individual factors).
25
4) Searching for all metabolic factors implied in the develop-
ment of the disease: nutrition, as well as predisposition to
use glycolytic pathway more than oxidative phosphorylation
in order to produce energy, like in the Warburg effect.
26,27
5) Identifying all social factors favoring the present epidemic,
particularly exogenous environmental factors, in social
Figure 4. Evolution of the mean clustering coefficient for each state and for the architectures and initial distribution of states (normal in blue, overweight
in green and obese in red) of Section 2, with tolerance equal to 0.25 and connection probability of the Version 3. (A) Radom; (B) scale-free; (C) small-
world; (D) empirical-1; (E) empirical-2.
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Figure 5. Initial (top) and final (bottom) distributions of the “in-degree”for the architectures and initial distributions of states of Section Methods, with
tolerance equal to 0.25 and connection probability of Version 3. (A) Radom; (B) scale-free; (C) small-world; (D) empirical-1; (E) empirical-2.
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networks involving young individuals (educative, sportive,
familial, social,...) in order to actively prevent the disease
before adult age (for example, cf. www.repop.fr) at school or
during hospital stays.
28
6) Studying all psycho-social factors leading to obesity stigmati-
zation in relation to mental body image and self-esteem.
29
7) Modeling carefully the connection between the demographic
dynamics and the social networks. In the future, it needs
deep knowledge (presently absent) regarding the structure
by age class into the social networks, as well as on the rules
of transmission and intergenerational inheritance of the ali-
mentation and adapted physical activity habits. Nevertheless,
the evolution of the size of the whole population has to be
already introduced in order to fix the number of nodes and
interaction links for calibrating our social networks models.
The Dynamics of Contacts
Influence of the contact duration
Let us now introduce contact duration tand a contagion coef-
ficient bpossibly depending on t.
30
It is possible to retrieve the
quadratic term of interaction already present in all the classical
models of contagion
30-40
by using a stochastic approach coming
from the random chemistry of contacts
41-57
for interpreting the
proposed rules, more developed in.
58
We have the demographic dynamics voiding the overweight
transition:
PfStCdtðÞk;Ot CdtðÞDN¡kgðÞ
¡PfStðÞDk;OtðÞDN¡kðÞ
D¡bN¡kðÞdtZT
0
P.fS.t¡t/Dk;O
.t¡t/DN¡kg/dtCbkC1ðÞN¡k¡1ðÞdtZT
0
P.fS.t¡t//
DkC1;O.t¡t/DN¡k¡1g/dt;
where S(t) (resp. O(t)) is the size of the susceptible (resp obese)
population at time t. The microscopic equation above leads to
the mean differential equation ruling the expectations of the ran-
dom variable S:
dE S tðÞðÞ
dt D¡bZ
T
0
ESt¡t
ðÞðÞ
EIt¡t
ðÞðÞ
dt
and to the macroscopic equation:
dS.t/
dt D¡
Z
T
0
St¡tðÞIt¡tðÞdt,
in which we found the quadratic term of the classical models of
contagion.
This quadratic term is also present in the interaction potential
of Hopfield like networks in which the study of the robustness
with respect to the contagion parameter changes has been per-
formed
59-69
as well as in recent studies taking into account the
spatial character of the disease spread.
70-78
Confinement and Saturation
The localization of contamination has been treated by differ-
ent authors.
79,80
When contagion occurs in confined locations
(like professional, educational or residence buildings), we can use
saturation dynamics terms coming from the enzymatic kinetics
(cf. for example
81,82
) for expressing all the possibilities of meeting
together kindividuals from the Ssusceptible population and i
from the Oobese population in ncontagion sites located in B
buildings.
We call this quantity the partition function P(S,O) and
B
@2LogP
@LogS@LogO
nis the total mean number of occupied sites, consid-
ered as proportional to the infection rate. Then, we have:
dS tðÞ=dt D¡bB
@2LogP
@LogS@LogO
n
CfS ¡mSCrO
dO tðÞ=dt DbB
@2LogP
@LogS@LogO
n
Cf’O¡m’O¡rO;
where the demographic parameters f
(fecundity) and m(mortality) are taken
into account for the susceptible as well
as for the obese population (f’ and m’)
and where rdenotes the recovering
rate at which an obese recovers to
healthy weight.
An example of such a dynamics is
the saturation Micha€elian one, if there
Figure 6. Left: with connection probability of the Version 3, evolution of the homophily coefficient at
equilibrium as function of the mean tolerance. Right: Evolution of the relaxation time to equilibrium
as function of the mean tolerance. (A) Connection version 3; (B) connection version 3.
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is only one contagion site:
PS;OðÞD1CvC;SS
1CvC;OO
where vC;S(resp. vC;O) is the probability for a susceptible (resp
obese) to access a contagion site. If vC;SD1 and vC;O<< 1, then
the infection rate equals about bSO/1CSðÞand the equations of
the dynamics are:
dS tðÞD¡bStðÞOtðÞ
1CStðÞ Cf¡mðÞStðÞCrO.t/
dO tðÞDbStðÞOtðÞ
1CStðÞðÞ
Cf’¡m’¡rðÞOtðÞ
Complex threshold dynamics used in classical Hopfield like
models
59,69
are non-linear, but take into account only pair con-
tacts, neglecting possible additional effects due to the presence
and mutual interaction of more than 2 individuals in the conta-
gion process.
It is now possible to introduce a synergy model, i.e., a formal-
ism to be able of define non-linear n-uples interactions
83
and
simulate the model in a spatial Markovian context like in this
study or in certain cases of remote spatial influence (due to new
social networking on the web) in a renewal context,
84
as well
with different time scales modeling complex dynamics, for sepa-
rating the local dynamics from the global trend of the obesity
epidemic.
85
Confrontation with data
Results shown in this paper about social networks involved in
obesity have been obtained by modeling and simulating networks
with various initial architectures (random, scale-free, small-
world, empirical) evolving under the so-called social homophilic
constraint. The computed evolution of these networks seems to
be similar to the real one observed in developed countries for a
socially “contagious disease:” obesity.
Complementary studies are now required based on large sam-
ples estimating the unobservable parameters linked both to initial
network architecture (taking into account the specificity of the
sub-populations of susceptibles, e.g., the differences between the
schoolchildren, professional and elderly people networks) and to
its interaction weights evolution, as well as incorporating the
demographic dynamics and a more accurate model of social con-
tacts through which the disease can spread. The choice of the
homophilic rule is in agreement with the final proportions of
each state group in Chilean population. Moreover, if we want to
confront the residual number of the intergroup interactions (e.g.,
the number of links between overweight and obese), we have to
look at inquiries in the field like those performed in Chile in
2011.
86
In addition, data from such surveys should, in the future,
document a differential tolerance, specific for each group in order
Figure 7. Homophilic dynamics representing obesity network for the
architectures and initial distribution of states of Figure 5, with tolerance
equal to 0.25 and connection probability of Version 3. Initial conditions
(A), (D) and (G); middle configurations (B), (E) and (H) and final configura-
tions (C), (F) and (I).
www.tandfonline.com 9Virulence
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to relax the constraint of the conservation of the total number of
links. Eventually, we could introduce a new state, the obese with
type II diabetes: indeed one-third of the population of obese suf-
fer from type II diabetes,
86,87
hence we would have the possibility
of monitoring the effectiveness of prevention policies (e.g., using
therapeutic education serious games
88
) for diabetes in the elderly.
Additionally, 66% of cases of type II diabetes, 52% of cases of
cholelithiasis, 29% of cases of hypertension and 22% of cases
associated with cardiovascular disease can be attributed to obe-
sity,
89
while about 20% of all cancers in Chile could be avoided
with strategies for the control and prevention of obesity, espe-
cially in children.
90
Disclosure of Potential Conflicts of Interest
No potential conflicts of interest were disclosed.
Funding
We acknowledge the Projects “Investissements d’Avenir”
VHP (Visual Home Presence inter@ctive), the PHC Maghreb
SCIM (Systemes Complexes et Ingenierie Medicale) and Conicyt
- FONDEF “Plataforma de Integracion Tecnologica para el
Registro, Vigilancia y Alerta de Enfermedades de Notificacion
Obligatoria” IT13I10059.
References
1. Rosen J, Myers A. Obesity stigmatization and coping: relation
to mental health symptoms, body image, and self-esteem. Int J
Obesity Relat Metab Disord 1999; 23:221-230; PMID:
10193866; http://dx.doi.org/10.1038/sj. ijo.0800765
2. Tunstall-Pedoe H, Project M. World’s largest study of
heart disease, stroke, risk factors, and population trends
1979-2002. Int J Epidemiol, WHO, Geneva, 2003.
3. ObEpi-Roche. Enqu^ete epidemiologique nationale sur
le surpoids et l’obesite. International Obesity Task
Force Prevalence Data, Roche, Paris, 2009 and 2012.
4. MaillardG,CharlesM,ThibaultN,ForhanA,SermetC,
Basdevant A, Eschwege E. Trendsin the prevalence of obesity
in the French adult population between 1980 and 1991. Int J
Obesity Relat Metab Disord 1999; 23:389-394;
PMID:10340817; http://dx.doi.org/10.1038/sj.ijo.0800831
5. Barth J. What should we do about the obesity epi-
demic?. Pract Diab Inter 2002; 19:119-122; http://dx.
doi.org/10.1002/pdi.341
6. de Saint-Pol, T. Obesite et milieux sociaux en France :
les inegalites augmentent, Bulletin Epidemiologique
Hebdomadaire 2008; 20:175-179
7. Laitinen J, Power C, Jarvelin M. Family social class,
maternal body mass index, childhood body mass index,
and age at menarche as predictors of adult obesity. Am
J Clin Nutrit 2001; 74:287-294; PMID:11522550
8. Scharoun-Lee M, Adair L, Kaufman J, Gordon-Larsen P. Obe-
sity, race/ethnicity and the multiple dimensions of socio-eco-
nomic status during the transition to adulthood: A factor
analysis approach. Social Sci Med 2009; 6:708-716;
PMID:19136186; http://dx.doi.org/10.1016/j.socscimed
.2008.12.009
9. Christakis N, Fowler J. The spread of obesity in a large social
network over 32 years. New Eng J Med 2006; 17:77-82.
10. Cohen-Cole,AM, Fletcher, J. Is obesity contagious? Social net-
works vs. environmental factors in the obesity epidemic. J
Health Economics 2008; 27:1382-1387; PMID:18571258;
http://dx.doi.org/10.1016/j.jhealeco.2008.04.005
11. Demongeot J, Amor HB, Elena A, Gillois P, Noual M,
Sene S. Robustness of regulatory networks. a generic
approach with applications at different levels: physio-
logic, metabolic and genetic. Inter J Mol Sci 2009;
10:4437-4473; PMID:20057955; http://dx.doi.org/
10.3390/ijms10104437
12. Demongeot J, Elena A, Noual M, Sene S, Thuderoz F.
Immunetworks, intersecting circuits and dynamics. J
Theoretical Biol 2011; 280:19-33; PMID:21439971;
http://dx.doi.org/10.1016/j.jtbi.2011.03.023
13. Demongeot J, Goles E, Morvan M, Noual M, SeneS.
Attraction basins as gauges of environmental robustness
in biological complex systems. PloS One 2010; 5,
e11793; PMID:20700525; http://dx.doi.org/10.1371/
journal.pone.0011793
14. Demongeot J, Noual M, Sene S. Combinatorics of Boolean
automata circuits dynamics. Dis Appl Mathe 2012; 160:398-
415; http://dx.doi.org/10.1016/j.dam.2011.11.005
15. MINSAL. Encuesta nacional de salud. 2010 http://web.
minsal.cl/portal/url/item/bcb03d7bc28b64dfe04001016
5012d23.pdf
16. Fruchterman T, Reingold E. Graph drawing by force-
directed placement. Software: Practice and Exp 1991;
21:1129-1164.
17. Gaymu J, Festy P, Poulain M, Beets G. From elderly
population projections to policy implications, in:
Future elderly living conditions in Europe (Felicie). Les
cahiers de l’INED 2008; 162:267-290.
Figure 8. Aging algorithm applied each year for each human being, with DOPAMID simulations (bottom left) and World Bank data (top left from
20
).
10 Volume 7 Issue 2Virulence
Downloaded by [82.122.10.40] at 23:48 08 January 2016
18. Duee M, Rebillard C. La dependance des personnes
^agees: une projection en 2040. Donnees sociales - La
societe fran¸caise, INSEE, Paris, 2006.
19. LegareJ,Decarie Y. Using statistics Canada life paths
micro-simulation model to project the health status of
Canadian elderly. SEDAP Research Paper 227,
SEDAP, Hamilton, 2008.
20. World Bank. http://web.worldbank.org/WBSITE/EXTER
NAL/DATASTATISTICS/, Washington, 2009
21. Harati H, Hadaegh F, Saadat N, Azizi F. Population-
based incidence of type 2 diabetes and its associated
risk factors: results from a six-year cohort study in Iran.
BMC Public Health 2009; 9, 186; PMID:19531260;
http://dx.doi.org/10.1186/1471-2458-9-186
22. Hosseinpanah F, Barzin M, Eskandary P, Mirmiran P,
Azizi F. Trends of obesity and abdominal obesity in
Tehranian adults: a cohort study. BMC Public Health
2009; 9, 429; PMID:19930668; http://dx.doi.org/
10.1186/1471-2458-9-429
23. Dauphinot V, Wolff H, Naudin F, Gueguen R, Sermet
C, Gaspoz J, Kossovsky M. New obesity body mass
index threshold for self-reported data. J Epidemiol
Com Health 2009; 63:128-132; PMID:18801799;
http://dx.doi.org/10.1136/jech.2008.077800
24. Bergman R, Stefanovski D, Buchanan T, Sumner A, Rey-
nolds J, Sebring N, Xiang A, Watanabe R. A better index of
body adiposity. Obesity 2011; 19:1083-1089;
PMID:21372804; http://dx.doi.org/10.1038/oby.2011.38
25. Baranova A, Collantes R, Gowder S, Elariny H,
Schlauch K, Younoszai SKA, Randhawa M, Pusulury
S, Alsheddi T, Ong J, Martin L, Chandhoke V, You-
nossi ZM. Obesity-related differential gene expression
in the visceral adipose tissue. Obesity Surgery 2005;
15:58-65; PMID:15768482; http://dx.doi.org/
10.1381/0960892054222876
26. Demetrius L, Coy J, Tuszynski J. Cancer proliferation
and therapy: the Warburg effect and quantum metabo-
lism. Theoret Biol Med Mod 2010; 7, 2;
PMID:20085650; http://dx.doi.org/10.1186/1742-
4682-7-2
27. Demetrius L, Tuszynski J. Quantum metabolism explains
the allometric scaling of metabolic rates. J. Royal Soc. Inter-
face 2010; 7:507-514; PMID:19734187; http://dx.doi.org/
10.1098/rsif.2009.0310
28. Delhumeau C, Demongeot J, Langlois F, Taramasco
C. Modelling the hospital length of stay in diagnosis
related groups data bases. in: IEEE ARES-CISIS’09
IEEE Press, Piscataway:955-960, 2009.
29. Myers A, Rosen J. Obesity stigmatization and coping:
relation to mental health symptoms, body image, and
self-esteem. Int J Obes Relat Metab Disord 1999;
23:221-230; PMID:10193866; http://dx.doi.org/
10.1038/sj.ijo.0800765
30. Demongeot J, Hansen O, Hessami H, Jannot A, Min-
tsa J, Rachdi M, Taramasco C. Random modelling of
contagious diseases. Acta Biotheoretica 2013; 61:141-
172; PMID:23525763; http://dx.doi.org/10.1007/
s10441-013-9176-6
31. Bernoulli D. Essai d’une nouvelle analyse de la mortal-
ite causee par la petite verole, et des avantages de
l’inoculation pour la prevenir. Mem. Acad. Roy. Sci.,
Paris, 1760.
32. d’Alembert J. Onzieme memoire : sur lapplication du
calcul des probabilites alinoculation de la petite verole
; notes sur le memoire precedent ; theorie mathema-
tique de linoculation, in: Opuscules mathematiques,
David, Paris. 26-95, 1761.
33. Hamer W. The Milroy lectures on epidemic disease in
England: the evidence of variability and persistency of
type. Bedford Press, London. 733-739, 1906.
34. Ross R. Prevention of malaria. John Murray, London,
1910.
35. Brownlee J. On the curve of the epidemic. Br Med J,
1915; 1:799-800; PMID:20767626; http://dx.doi.org/
10.1136/bmj.1.2836.799
36. Ross R. Some a priori pathometric equations. Br Med J
1915; 1:546-547; PMID:20767555; http://dx.doi.org/
10.1136/bmj.1.2830.546
37. Ross R. An application of the theory of probabilities to
the study of a priori pathometry. part I. Proc Royal Soc
London Series A 1916; 92:204-230; http://dx.doi.org/
10.1098/rspa.1916.0007
38. Gibson M. Sir Ronald Ross and his contemporaries. J
Royal Society of Medicine 1978; 71:611-618;
PMID:20894263
39. Kermack W, McKendrick A. A contribution to the
mathematical theory of epidemics. Proc Royal Soc Lon-
don, Series A 1927; 115:700-712; PMID:23797790;
http://dx.doi.org/10.1098/rspa.1927.0118
40. Kermack W, McKendrick A. Contributions to the
mathematical theory of epidemics. iii. further studies of
the problem of endemicity. Proc Royal Soc London
Series A 1933; 121:141-194.
41. Delbr€uck M. Statistical fluctuations in autocatalytic
reactions. J Chem Phys 1940; 8:120-124; http://dx.
doi.org/10.1063/1.1750549
42. Bartholomay A. On the linear birth and death processes
of biology as Markov chains. Bull Math Biophys 1958;
20:97-118; http://dx.doi.org/10.1007/BF02477571
43. Bartholomay A. Stochastic models for chemical reac-
tions: I. theory of the uni-molecular reaction process.
Bull Math Biophys 1958; 20:175-190; http://dx.doi.
org/10.1007/BF02478297
44. Bartholomay A. Stochastic models for chemical reac-
tions: II. the unimolecular rate constant Bull Math Bio-
phys 1959; 21:363-373; http://dx.doi.org/10.1007/
BF02477895
45. McQuarrie D. Kinetics of small systems I. J Chem Phys
1963; 38:433-436; http://dx.doi.org/10.1063/
1.1733676
46. McQuarrie D, Jachimowski C, Russell M. Kinetics of
small systems. II. J Chem Phys 1964; 40:2914-2921;
http://dx.doi.org/10.1063/1.1724926
47. Jachimowski C, McQuarrie D, Russell M. A stochastic
approach to enzyme-substrate reactions. Biochem
1964; 3:1732-1736; PMID:14235340; http://dx.doi.
org/10.1021/bi00899a025
48. McQuarrie D. Stochastic approach to chemical kinet-
ics. J Appl Probab 1967; 4:413-478; http://dx.doi.org/
10.2307/3212214
49. Hoare M. Molecular Markov processes. Nature 1970;
226:599-603; PMID:16057421; http://dx.doi.org/
10.1038/226599a0
50. Bailey N. The simple stochastic epidemic: A complete
solution in terms of known functions. Biometrika
1963; 50:235-240; http://dx.doi.org/10.1093/biomet/
50.3-4.235
51. Dietz K. Epidemics and rumors: A survey. J Royal Stat
Soc. Series A (General) 1967; 130:505-528; http://dx.
doi.org/10.2307/2982521
52. Demongeot J. A stochastic model for the cellular
metabolism. in: Recent Developments in statistics,
North Holland, Amsterdam:655-662, 1977.
53. Allen L. An introduction to stochastic epidemic mod-
els. Math Epidemiol 2008; 1945:81-130; http://dx.doi.
org/10.1007/978-3-540-78911-6_3
54. Rhodes C, Demetrius L. Evolutionary entropy deter-
mines invasion success in emergent epidemics. PLoS
One 2010; 5, e12951; PMID:20886082; http://dx.doi.
org/10.1371/journal.pone.0012951
55. Britton T. Stochastic epidemic models: a survey. Math
Biosci 2010; 225:24-35; PMID:20102724; http://dx.
doi.org/10.1016/j.mbs.2010.01.006
56. Gillespie D. A general method for numerically simulat-
ing the stochastic time evolution of coupled chemical
reactions. J Comput Phys 1970; 22:403-434; http://dx.
doi.org/10.1016/0021-9991(76)90041-3
57. Magal P, Ruan S. Susceptible-infectious- recovered
models revisited: From the individual level to the popu-
lation level. Math Biosci 2014; 250:26-40;
PMID:24530806; http://dx.doi.org/10.1016/j.mbs.
2014.02.001
58. Taramasco C, Impact de l’obesite sur les structures
sociales et impact des structures sociales sur l’obesite?
Facteurs individuels et environnementaux, PhD thesis,
Ecole Polytechnique, Paris 2011.
59. Aldana M, Cluzel P. A natural class of robust networks.
Proc Natl Acad Sci USA 2003; 100:8710-8714;
PMID:12853565; http://dx.doi.org/10.1073/pnas.
1536783100
60. Demongeot J, Elena A, Sene S. Robustness in neural
and genetic networks. Acta Biotheoretica 2008; 56:27-
49; PMID:18379883; http://dx.doi.org/10.1007/
s10441-008-9029-x
61. Amor HB, Demongeot J, Elena A, Sene S. Structural
sensitivity of neural and genetic networks. Lecture
Notes Comp Sci 2008; 5317:973-986; http://dx.doi.
org/10.1007/978-3-540-88636-5_92
62. Elena A, Demongeot J. Interaction motifs in regulatory
networks and structural robustness. in: IEEE ARES-
CISIS’08, IEEE Press, Piscataway:682-686, 2008.
63. Elena A, Ben-Amor H, Glade N, Demongeot J. Motifs
in regulatory networks and their structural robustness.
in: IEEE BIBE’08, IEEE Press, Piscataway:234-242,
2008.
64. ElenaA, Robustesse des reseaux dautomatesa seuil, PhD thesis,
Universite Joseph-Fourier, Grenoble, 2009.
65. Demongeot J, Elena A, Noual M, Sene S. Random
Boolean networks and attractors of their intersecting
circuits. in: Proceedings IEEE AINA’11, IEEE Pro-
ceedings, Piscataway:483-487, 2011.
66. Glade N, Elena A, Fanchon N, Demongeot J, Amor
HB. Determination, optimization and taxonomy of
regulatory networks. the example of Arabidopsis thali-
ana flower morphogenesis. in: IEEE AINA’11, IEEE
Press, Piscataway:488-494, 2011.
67. Demongeot J, Waku J. Robustness in genetic regula-
tory networks II, application to genetic threshold Bool-
ean random regulatory networks (getbren). Comptes
Rendus Mathematique, 2012; 350:225-228; http://dx.
doi.org/10.1016/j.crma.2012.01.019
68. Demongeot J, Waku J. Robustness in genetic regula-
tory networks IV, application to genetic networks con-
trolling the cell cycle. Comptes Rendus Mathematique
2012; 350:293-298; http://dx.doi.org/10.1016/j.
crma.2012.02.005
69. Demongeot J, Drouet E, Moreira A, Rechoum Y, Sene
S. Micro-RNAs: viral genome and robustness of the
genes expression in host. Phil Trans Royal Soc A 2009;
367:4941-4965; PMID:19884188; http://dx.doi.org/
10.1098/rsta.2009.0176
70. Koopman J, Longini I. The ecological effects of individual
exposures and nonlinear disease dynamics in populations.
Am J Public Health 1994; 84:836-842; PMID:8179058;
http://dx.doi.org/10.2105/AJPH.84.5.836
71 Kretzschmar M. Measures of concurrency in networks
and the spread of infectious disease. Math Biosci 1996;
133:165-195; PMID:8718707; http://dx.doi.org/
10.1016/0025-5564(95)00093-3
72. Sathik M, Rasheed A. Social network analysis in an
online blogosphere. Inter J Eng Sci Technol 2011;
3:117-121,
73. Gaudart J, Giorgi R, Poudiougou B, Ranque S,
Doumbo O, Demongeot J. Spatial cluster detection:
principle and application of different general methods.
Rev Epid Sante Pub 2007; 55:297-306; http://dx.doi.
org/10.1016/j.respe.2007.04.003
74. Gaudart J, Toure O, Dessay N, Dicko A, Ranque S,
Forest L, Demongeot J, Doumbo O. Modelling
malaria incidence with environmental dependency in a
locality of Sudanese savannah area, Mali. Malaria J
2009; 8, 61; PMID:19361335; http://dx.doi.org/
10.1186/1475-2875-8-61
75. Gaudart J, Ghassani M, Mintsa J, Rachdi M, Waku J,
Demongeot J. Demography and diffusion in epidemics:
Malaria and black death spread. Acta Biotheoretica
2010; 58:277-305; PMID:20706773; http://dx.doi.
org/10.1007/s10441-010-9103-z
76. Demongeot J, Gaudart J, Mintsa J, Rachdi M. Demog-
raphy in epidemics modelling. Commun Pure Appl
Analysis 2012; 11:61-82; http://dx.doi.org/10.3934/
cpaa.2012.11.61
77. Demongeot J, Gaudart J, Lontos A, Promayon E, Mintsa J,
Rachdi M. Least diffusion zones in morphogenesis and
www.tandfonline.com 11Virulence
Downloaded by [82.122.10.40] at 23:48 08 January 2016
epidemiology. Int J Bifurcat Chaos 2012; 22: 50028; http://
dx.doi.org/10.1142/S0218127412500289
78. Demongeot J, Ghassani M, Rachdi M, Ouassou I, Tara-
masco C. Archimedean copula and contagion modelling in
epidemiology. Networks and Heterogeneous Media 2013;
8:149-170; http://dx.doi.org/10.3934/nhm.2013.8.149
79. Amarasekare P. Interactions between local dynamics and
dispersal: insights from single species models. Theoret Pop-
ulat Biol 1998; 53:44-59; PMID:9500910; http://dx.doi.
org/10.1006/tpbi.1997.1340
80. Filipe J, Gibson G. Comparing approximations to spa-
tio-temporal models for epidemics with local spread.
Bull Math Biol 2001; 63:603-624; PMID:11497160;
http://dx.doi.org/10.1006/bulm.2001.0234
81. Demongeot J, Laurent M. Sigmoidicity in allosteric
models. Mathematical Biosciences 1983; 67:1-17;
http://dx.doi.org/10.1016/0025-5564(83)90015-9
82. Demongeot J, Kellershohn N. Glycolytic oscillations : an
attempt to an “in vitro” reconstitution of the higher part of
glycolysis. Lectures Notes in Biomaths 1983; 49:17-31;
http://dx.doi.org/10.1007/978-3-642-46475-1_2
83. Demongeot J, Sene S. The singular power of the envi-
ronment on nonlinear Hopfield networks. in:
CMSB11, ACM Proceedings, New York:55-64, 2011.
84. Demongeot J, Fricot J. Random fields and renewal
potentials. NATO ASI Serie F 1986; 20:71-84.
85. Barthelemy M, Nadal J, Berestycki H. Disentangling
collective trends from local dynamics. Proc Natl Acad
Sci USA 2010; 107:7629-7634; PMID:Can’t; http://
dx.doi.org/10.1073/pnas.0910259107
86. Garmendia M, Ruiz P, Uauy R. Obesidad y cancer en
Chile: estimacion de las fracciones atribuibles poblacio-
nales. Revista Medica de Chile 2013; 141:987-994;
PMID:Can’t; http://dx.doi.org/10.4067/S0034-
98872013000800004
87. Fried M, Hainer V, Basdevant A, Buchwald H, Deitel
M, Finer N, Willem J, Greve JW, Horber F, Mathus
Vliegen E, Scopinaro N, Steffen R, Tsigos C, Weiner
R, Widhalm K. Interdisciplinary European guidelines
for surgery for severe (morbid) obesity. Obesity Surgery
2007; 17:260-270; PMID:17476884; http://dx.doi.
org/10.1007/s11695-007-9025-2
88. Demongeot J, Elena A, Taramasco C, Vuillerme N.
Serious games and personalization of the therapeutic
education. Lecture Notes in Comp Sci 2015;
9102:270-281; http://dx.doi.org/10.1007/978-3-319-
19312-0_22
89. Finkelstein E, Khavjou O, Thompson H, Trogdon J,
Pan L, Sherry B, Dietz W. Obesity and severe obesity
forecasts through 2030. Am J Prevent Med 2012;
42:563-570; PMID:22608371; http://dx.doi.org/
10.1016/j.amepre.2011.10.026
90. Lipnowski S, LeBlanc C. Canadian Paediatric Society,
healthy active living and sports medicine committee.
healthy active living: Physical activity guidelines for
children and adolescents. Paediatrics and Child Health
2012; 17:209-210; PMID:23543633
12 Volume 7 Issue 2Virulence
Downloaded by [82.122.10.40] at 23:48 08 January 2016
